improving inferential performance of flexible motion systems · gb gain balancing gm gain margin gs...

Improving Inferential Performance of Flexible Motion Systems

S.L.H. Verhoeven

DCT Report 2010.006

APT536-10-0060

Master of Science Thesis

Committee: Dr. ir. M.M.J. van de Wal§ (main supervisor)Dr. ir. J.J.M. van Helvoort§ (supervisor)Ir. T.A.E. Oomen‡ (supervisor)Prof. ir. O.H. Bosgra‡ (graduate professor)Dr. ir. C.M.M. van Lierop♮

Prof. dr. ir. M. Steinbuch‡

§ Philips Applied Technologies

Mechatronics Program

Drives and Control Group

♮ Eindhoven University of Technology

Department of Electrical Engineering

Control Systems Group

‡ Eindhoven University of Technology

Department of Mechanical Engineering

Control Systems Technology Group

Eindhoven, February 2010

Summary

Philips Applied Technologies has a long history of research on advanced control for high pre-cision mechatronic positioning systems. This research has especially been done for ASML,which is a leading company in the market for chip manufacturing machines, i.e., wafer scan-ners. New trends cause such systems to become more flexible and taking these flexibilitiesinto account will eventually become necessary.

Due to this increased flexibility, the actual performance, which is also referred to as theinferential performance, is not necessarily represented well by the measured variables. Thissituation occurs if internal dynamics between the measured variables and the performancevariables becomes relevant. In addition, interaction is inevitable when systems become moreflexible and MIMO controller design techniques are crucial to achieve high performance. Thegoal of this research is to analyze, by using simulation models, whether flexible behavior isimportant in next generation high performance motion systems whereby the performancevariables are not measured during normal operation and, if so, to investigate what the newlimitations on performance are and how these limitations can be reached by control design.

By performing simulations on a next generation wafer stage FEM model it is shown thatconventional controller design, i.e., controller design whereby the system is assumed to behaveas a rigid body, is not sufficient; the performance is severely limited while this is not observedby the sensors. Moreover, this research shows that the single-degree-of-freedom control struc-ture, which is used in conventional control, is inadequate for the control of next generationhigh performance motion systems. An extra control degree-of-freedom therefore needs to beincluded in the control structure and three alternative structures are introduced. One ofthese structure, which is referred to as the inferential control structure, is used in the samesimulation environment and increases the inferential performance in z-direction by 80%.

Furthermore, important interpolation and integral constraints are derived for the standardplant setup that are able to deal with the inferential nature of the problem. Classical perfor-mance limitations, including the Poisson integral constraints, are also valid in the standardplant setup. However, it is shown that these constraints do not necessarily limit performancein case the performance variables are not measured. The reverse is also true: there may bestrong limitations on the performance variables that do not limit the measured variables.

Finally, it is established in this research that actuator/sensor selection is important for flexiblemotion systems, since it determines the extent to which the flexible dynamics is actuated,sensed and observed in the performance variables. The optimal actuator/sensor configurationstrongly depends on the system (and its disturbances) and there is a need for profound toolsfor actuator/sensor selection in next generation high performance motion systems.

iii

Nomenclature

Abbreviations

ATO Actuator-To-OriginCLHP Closed Left Half PlaneCOG Center of GravityCRHP Closed Right Half PlaneDOF Degree-Of-FreedomFB FeedbackFF FeedforwardFEM Finite Element MethodFRF Frequency Response FunctionGB Gain BalancingGM Gain MarginGS Gain SchedulingIC Integrated CircuitIO Input-OutputI/O Input/OutputLQG Linear Quadratic GaussianLTI Linear Time InvariantMIMO Multi-Input Multi-OutputMP Minimum PhaseMSD Mass-Spring-DamperMM Modulus MarginNMP Nonminimum PhaseOLHP Open Left Half PlaneORHP Open Right Half PlanePM Phase MarginPOC Point-Of-ControlPSD Power Spectral DensityRHS Right Hand SideSISO Single-Input Single-OutputSS State-SpaceSVD Singular Value DecompositionTF Transfer FunctionTFM Transfer Function Matrix

v

vi Nomenclature

WS Wafer Stage

Notation and symbols

R field of real numbersC field of complex numbers

C− and C− open and closed left half plane

C+ and C+ open and closed right half plane

∈ belong to

� end of proof

= equal to≡ identically equal to

, defined as& asymptotically greater than. asymptotically less than≫ much greater than≪ much less than

α complex conjugate of α ∈ C

|α| absolute value of α ∈ C

In n× n identity matrixdiag(a1, . . . , an) an n× n diagonal matrix with aj as its i-th diagonal elementAT transpose of AAH complex conjugate transpose of AA−1 inverse of AA† Moore-Penrose pseudoinverse of Adet(A) determinant of Aλi(A) i-th eigenvalue of Aσi(A) i-th singular value of Aσ(A) maximum singular value of Aσ(A) minimum singular value of Ac(A) condition number of Arank(A) Rank of A‖A‖p p-norm of A, e.g., ‖A‖∞

∠ angle

Fl(G,C) lower linear fractional transformation

Contents

Summary iii

Nomenclature v

Contents vii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Project motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Outline of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Elements of linear system theory 7

2.1 System descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 State-space description . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Transfer function description . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Frequency response function . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.4 Coprime factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Controllability and observability . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Directionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Singular value decomposition . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Condition number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5.1 Poles and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.5.2 Pole direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6.1 Zero direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7 Signal and system norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.7.1 Signal norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7.2 System norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

vii

viii Contents

3 Controller design for flexible motion systems 19

3.1 Controller design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Control objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Control architecture design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.1 I/O selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3.2 Control structure design . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3.3 Controller configuration selection . . . . . . . . . . . . . . . . . . . . . 23

3.4 Norm-based controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4.1 Selecting a norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4.2 Incorporating dynamic weighting filters . . . . . . . . . . . . . . . . . 25

3.4.3 Selecting exogenous outputs and inputs . . . . . . . . . . . . . . . . . 26

3.5 Internal stability and well-posedness . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Control problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.7.1 Time domain performance . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.7.2 Frequency domain performance . . . . . . . . . . . . . . . . . . . . . . 31

3.8 Control structures for inferential servo problems . . . . . . . . . . . . . . . . 313.8.1 Limitations of the single-DOF control structure . . . . . . . . . . . . . 31

3.8.2 Alternative control structures for inferential servo problems . . . . . . 34

3.8.3 Formulating a nine-block control problem . . . . . . . . . . . . . . . . 37

3.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 H∞ loop-shaping 39

4.1 Motivation for loop-shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 The design procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2.1 Shaping filter selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.2 Robust stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Relation with H∞ optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Using γmin as performance parameter . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4.2 Guarantees on the achieved open-loop shape . . . . . . . . . . . . . . 46

4.5 Design examples: H∞ loop-shaping applied to a two MSD system . . . . . . . 48

4.5.1 SISO controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5.2 MIMO shaping filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.6 Guidelines for shaping filter selection . . . . . . . . . . . . . . . . . . . . . . . 56

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Fundamental performance limitations 59

5.1 Fundamental limitations in SISO systems . . . . . . . . . . . . . . . . . . . . 60

5.1.1 S plus T is one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.1.2 Interpolation constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.1.3 Bode integrals for S and T . . . . . . . . . . . . . . . . . . . . . . . . 60

5.1.4 Poisson integrals for S and T . . . . . . . . . . . . . . . . . . . . . . . 625.1.5 Bode gain-phase relation . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1.6 LHP zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1.7 Design examples: limitations due to a NMP zero . . . . . . . . . . . . 65

5.2 Fundamental limitations in MIMO systems . . . . . . . . . . . . . . . . . . . 72

Contents ix

5.2.1 S plus T is identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.2 Interpolation constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.3 Bode integrals for S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.4 Poisson integral for S . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2.5 Design example: Bode sensitivity constraint in a MIMO system . . . . 75

5.3 Performance limitations for non-square plants . . . . . . . . . . . . . . . . . . 78

5.4 Performance limitations in the standard plant setup . . . . . . . . . . . . . . 79

5.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4.2 Systems reducible to feedback loop . . . . . . . . . . . . . . . . . . . . 81

5.4.3 Interpolation constraints due to CRHP zeros in Gzu or Gyw . . . . . . 82

5.4.4 A generalized Bode integral . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4.5 Design example: limitations in the standard plant setup . . . . . . . . 86

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Control of a two MSD system 89

6.1 Conventional control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.1.1 Rigid body assumption . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.1.2 Collocated control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.3 Non-collocated control . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.1.4 Limitations in the standard plant setup . . . . . . . . . . . . . . . . . 99

6.2 Inferential control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2.1 State observer design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2.2 Collocated measurement, non-collocated performance . . . . . . . . . 105

6.2.3 Non-collocated measurement, collocated performance . . . . . . . . . . 107

6.2.4 Motivation for a general two-DOF control structure . . . . . . . . . . 107

6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7 Control of a wafer stage FEM model 111

7.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.2 Collocated and non-collocated control . . . . . . . . . . . . . . . . . . . . . . 113

7.3 I/O selection for flexible stages . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.4 Controller design and performance evaluation for P 3 . . . . . . . . . . . . . . 115

7.4.1 Control objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.4.2 Model reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.4.3 Controller design for I/O set 3 . . . . . . . . . . . . . . . . . . . . . . 120

7.4.4 Performance evaluation for I/O set 3 . . . . . . . . . . . . . . . . . . . 122

7.4.5 Error-based shaping filter selection . . . . . . . . . . . . . . . . . . . . 126

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8 Conclusions and recommendations 131

8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

x Contents

A More on transmission zeros 135A.1 Blocking effect of a zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135A.2 Physical interpretation of transmission zeros in mechanical systems . . . . . . 136

A.2.1 Collocated actuators and sensors . . . . . . . . . . . . . . . . . . . . . 137A.2.2 Non-collocated actuators and sensors . . . . . . . . . . . . . . . . . . . 138A.2.3 Distributed parameter systems . . . . . . . . . . . . . . . . . . . . . . 138

A.3 Interpretation using wave propagation theory . . . . . . . . . . . . . . . . . . 138A.4 Initial undershoot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139A.5 Creating transmission zeros by manipulating the plant inputs and outputs . . 141A.6 Zero assignment problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

B Case study: a flexible cart system 145B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145B.2 Conventional feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . 145B.3 Explicit distinction between y and z (option 1) . . . . . . . . . . . . . . . . . 146B.4 Explicit distinction between y and z (option 2) . . . . . . . . . . . . . . . . . 150B.5 Effect of changing mass and inertia . . . . . . . . . . . . . . . . . . . . . . . . 151B.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

C Simulation parameters 155C.1 Flexible cart system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155C.2 Mass-spring-damper systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

D FEM model 157D.1 FEM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157D.2 Input and output selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161D.3 Constructing the physical plant . . . . . . . . . . . . . . . . . . . . . . . . . . 161D.4 Rigid body decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

D.4.1 Static sensor transformation . . . . . . . . . . . . . . . . . . . . . . . . 163D.4.2 Static actuator transformation . . . . . . . . . . . . . . . . . . . . . . 164D.4.3 Effect of decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

E Frequency domain results for the WS FEM model 169

Bibliography 177

Chapter 1

Introduction

1.1 Background

Philips Applied Technologies has a long history of research on advanced control for highprecision motion systems. This research has especially been done for ASML, which is aleading company in the market for chip manufacturing machines, i.e., wafer scanners. Chips,or Integrated Circuits (ICs), are miniaturized electronic circuits that are produced on siliconwafers by a photolithographic process. This process is schematically depicted in Figure 1.1.

Single wavelet lightbundle

Reticle stagecontaining the reticle

Lens that projects thereticle pattern on the

wafer

Wafer stage containingthe wafer

Figure 1.1: Schematic representation of the photolithographic process in a wafer scanner.

A wafer is first covered with a photo resistant layer and then placed on the so-called waferstage. To etch the IC pattern in the wafer, a monochromatic light bundle is sent through areticle containing an enlarged version of the IC pattern. An advanced lens and mirror system

1

2 Introduction

then focusses the light bundle on a small part of the wafer and only those parts of the waferthat need to be removed are exposed. Because an IC contains multiple layers with differentpatterns, the exposure process needs to be repeated several times with different reticles. Inaddition, only parts of the wafer can be exposed at the same time due to limitations of thelens system. These two conditions require that the reticle and wafer need to be positionedwith high accuracy with respect to each other and the lens system.

High-accuracy stages are nowadays controlled in six Degrees-Of-Freedom (DOFs): three rigidbody translations and rotations. Controller design is mainly based on the plant Input-Output1 (IO) behavior and six independent actuators and sensors are used to control thesix rigid body DOFs, while the flexible dynamics is regarded as parasitic. In addition, tokeep FeedBack (FB) controller design simple, PID-like Single-Input Single-Output (SISO)controllers are used, despite the Multi-Input Multi-Output (MIMO) nature of the prob-lem (Van de Wal et al., 2002).

However, the actual performance of a motion system is not necessarily represented well bythe plant IO behavior. This situation occurs if internal dynamics between the measuredvariables and the performance variables becomes relevant. For a wafer stage, the truly relevantperformance is in terms of positioning that part of the silicon wafer that is subject to lightexposure. The measured variables can only be used to estimate the performance variablesby using geometric relations, since it is based on laser interferometer data measured at theedges of the wafer stage. A similar reasoning applies to the actuator side, where internaldeformations refute the validity of the traditional actuator transformation.

1.2 Project motivation

Because of fierce competition in the IC market, it is desirable to put more and smallerelectronic components on a single IC and to increase machine throughput. ASML is thereforefaced with the industrial challenge to build bigger and lighter stages, while at the same timethe performance requirements become ever demanding, see, e.g., Jansen (2008). Similar trendsemerge for general high precision motion systems. Demanding a higher accuracy motivatescontactless operation and hence weight minimization. In addition, by virtue of Newton’slaw F = m · a, decreasing the mass of the system allows higher accelerations without moreenergy consumption, which saves money. Unfortunately, these trends imply that the nextgeneration motion systems become more flexible, see, e.g., Balas (1990), and hence it willeventually become necessary to explicitly address the internal plant dynamics. Such systemsare referred to as “flexible motion systems” in the remainder of this research.

In 2005, Philips Applied Technologies recognized the importance of taking flexible dynamicsinto account and this resulted in the Ph.D assignment of Tom Oomen, see Oomen (2010).The main focus herein is on model identification for advanced control design, i.e., obtainingaccurate plant and uncertainty models, but more motion control challenges exist. A completeoverview of these challenges can be found in Van de Wal (2009) and Verhoeven (2009). Thechallenges that are elaborated on in this research are introduced in the next section.

1Input as used here refers to the physical plant inputs (actuator forces) and output to the measured plantoutputs (displacements).

1.3 Problem formulation 3

1.3 Problem formulation

Due to the increased flexibility of next generation motion systems, conventional controller de-sign, i.e., controller design whereby the system is assumed to behave as a rigid body, eventuallydoes not lead to the desired performance anymore. To illustrate this, consider the controlstructure depicted in Figure 1.2. Herein zp denotes the unmeasured performance variables,yp are the measured variables, up are the plant inputs, u are the controller outputs, and ris the reference. The structure depicted by the solid lines is common practice in the field ofmotion control and is therefore also referred to as the conventional control structure. Sincethe system is assumed to behave as a rigid body, the assumption that yp = zp can be madewithout loss of generality. Relatively simple PID-like SISO controllers are then designed tokeep r−yp small (reference tracking) under the influence of disturbances du and measurementnoise η. However, due to the presence of internal dynamics, the actual system performanceshould be evaluated at zp and cannot be measured directly. A model-based controller designapproach is therefore essential. Furthermore, notice that zp and yp do not necessarily havethe same dimensions. Hence, the dashed lines in Figure 1.2. The goal of this research projectis then formulated as:

Analyze, by using simulation models, whether flexible behavior is important in nextgeneration high performance motion systems whereby the performance variables are notmeasured during normal operation. And, if so, investigate what the limitations on theachievable performance are and how these limitations can be reached by control design.

C Pypr upu

du

η

ey

Cff

uff

Pz

zp ez

Figure 1.2: Conventional control structure (solid line). The dashed structure points out theactual control goal of making ez small. Notice that ez cannot be determined unless yp and zphave the same dimensions.

It is well-known that a well-designed FeedForward (FF) signal (uff in Figure 1.2) greatlyimproves the performance of conventional servo systems. Using FF control, however, is basi-cally nothing more than using setpoint information and plant knowledge2 to steer the systemwithout evaluating the effect of the plant inputs during normal operation. Since model uncer-tainties and unknown disturbances are always present in real systems, a FB loop is essentialand FF controller design3 is regarded as a tool to further boost the performance after a FBcontroller has been designed. Hence, this research mainly focusses on FB controller design.

2Although not explicitly visible in Figure 1.2, known disturbances can also be accounted for in uff .3FF controller design for flexible motion systems is considered in different research project, see, e.g., Lunen-

burg (2009).

4 Introduction

In the context of this research project, the following three research issues are addressed:

1. Explicit distinction between the measured and performance variables. Theconventional control structure, which is depicted by the solid lines in Figure 1.2, doesnot represent the true control goal of keeping ez small. It is merely an approximationand is only justified when yp ≈ zp.

In this research, shortcomings of the conventional control structure are investigatedand alternative two-DOF control structures are proposed that are better suited for thecontrol of next generation high performance motion systems.

2. Norm-based controller design. FB controllers are often designed using SISO con-troller design techniques, despite the MIMO nature of the problem. When systemsbecome more flexible, interaction is inevitable and MIMO controller design techniquesare essential to achieve high performance.

The standard plant setup, which is depicted in Figure 1.3, offers the possibility toformulate a wide variety of control problems that are able to deal with the MIMOnature of the problem. In addition, it allows for an explicit distinction between sensedoutputs y and (weighted) exogenous outputs z on one side, and (weighted) exogenousinputs w and control signals u on the other side, which makes this setup extremely usefulto distinguish between measured and performance variables. The goal of control is thento minimize a norm of the (closed-loop) mapping between w and z. Hence, the term“norm-based control”. In literature, several examples can be found that demonstratethe usefulness of this setup, see, e.g., McFarlane and Glover (1990) and Steinbuch andNorg (1998). Within Philips Applied Technologies, norm-based controller design hasalready been used to design MIMO controllers for ASML wafer stages, see Van de Walet al. (2002).

C

G

w z

u y

(Weighted)exogenous outputs

Sensed outputs

(Weighted)exogenous inputs

Control signals

Figure 1.3: Standard plant setup

3. Fundamental limitations on achievable performance. Fundamental limitationssuch as the Bode sensitivity integral are well understood for the conventional controlstructure, see, e.g., Freudenberg and Looze (1985, 1986) and Skogestad and Postleth-waite (2005). However, for the standard plant setup these limitations are less wellunderstood. Obtaining insights in these limitations is considered relevant, since limita-tions in the standard plant setup may severely limit the performance (at zp), even whenthere are no limitations on the measured variables (yp). Possible limitations in the stan-

1.4 Outline of the report 5

dard plant setup should thus be taken into account when altering the actuator/sensorconfiguration.

In this research, an overview is given of the fundamental performance limitations in theconventional control structure and the standard plant setup that are considered relevant(at present) for the control of next generation high performance motion systems. Theword “fundamental” is used here to indicate that only limitations are considered thatare caused by poles and zeros in the closed right half plane. Causality and practicallimitations, like, e.g., a finite sampling frequency, are not yet considered.

1.4 Outline of the report

This thesis is organized as follows. Chapter 2 introduces basic elements of linear systemtheory that are considered relevant for the control of flexible motion systems and are neededin the remainder of this thesis.

Chapter 3 discusses (part of) the controller design procedure for flexible motion systems. Thefocus is limited to steps related to FB controller design. Although the procedure to design FBcontrollers seems trivial when a plant model is available, the presence of internal dynamicscreates new aspects that need to be considered. The main focus of Chapter 3, however, is onformulating sensible norm-based control problems for conventional (stiff) and flexible motionsystems (research issues 1 and 2). For the latter, it is shown that an extra control DOFneeds to be included in the control structure and alternative two-DOF control structures areproposed that are better suited for the control of flexible motion systems.

Chapter 4 proposes H∞ loop-shaping as an alternative for conventional norm-based controllerdesign in which the desired closed-loop transfer functions between w and z need to be specified.One of the benefits of H∞ loop-shaping, which was originally introduced by McFarlane andGlover (1990), is that the open-loop singular values are shaped, such that knowledge frommanual loop-shaping can be used (research issue 2). In addition, design examples are givenfor a simple flexible motion system and these examples are used to derive guidelines for tuningcontrollers.

In Chapter 5, an overview is given of fundamental performance limitations in the conven-tional control structure and the standard plant setup that are caused by poles and zeros inthe closed right half plane and are considered relevant (at present) for the control of nextgeneration motion systems. Illustrative examples are included as well to show the effect ofthese limitations.

Subsequently, FB controllers are designed in Chapters 6 and 7 for two flexible motion systems.In Chapter 6, for a relatively simple two Mass-Spring-Damper (MSD) system and in Chapter 7for a Finite Element Method (FEM) model of a Wafer Stage (WS) short stroke device. Forboth systems, the effect of applying current state-of-the-art control and adding an extracontrol DOF is compared in the time and frequency domain.

Finally, in Chapter 8 the main conclusions and recommendations are stated.

Chapter 2

Elements of linear system theory

The goal of this chapter is to summarize important results from linear system theory and tointroduce definitions that are needed in the remainder of this thesis. For a more thoroughtreatment, see, e.g., Skogestad and Postlethwaite (2005) and Zhou et al. (1996).

2.1 System descriptions

2.1.1 State-space description

A linear system with nu inputs (vector u), ny outputs (vector y), and an internal descriptionof nx state variables (vector x), can be represented by a linear State-Space (SS) model of theform

x(t) = Ax(t) +Bu(t), x(0) = x0, (2.1a)

y(t) = Cx(t) +Du(t), (2.1b)

where x(t) ≡ dx/dt. Matrix A ∈ Rnx×nx is called the “state matrix”, B ∈ R

nx×nu the “inputmatrix”, C ∈ R

ny×nx the “output matrix”, and D ∈ Rny×nu the “feedthrough matrix”. If,

just as in (2.1), these matrices are independent of time, the system is called a Linear TimeInvariant (LTI) system.

Remark 2.1. To avoid notational confusion, variables y(t) and u(t) are sometimes replacedby yp(t) and up(t), respectively. At the sensor side, the subscript “p” is added to distinguishbetween the measured variables yp(t) and the sensed variables y(t) in the standard plant setup.At the actuator side, the subscript is used to differentiate between controller outputs u(t) andplant inputs up(t) in the presence of input disturbances. When there is no danger of confusion,the subscript “p” is omitted.

The SS representation is not a unique system description, because multiple realizations(A,B,C,D) can give the same IO behavior. Firstly, because there can be additional un-observable and/or uncontrollable states, which never show up in the IO behavior. Secondly,even if there are no unobservable and uncontrollable states, there are still infinitely many

7

8 Elements of linear system theory

state realizations. To see this, consider a state transformation x(t) = Tx(t), where T is aninvertible constant matrix. The new SS realization has the same IO behavior, but is of theform

A = TAT−1, B = TB, C = CT−1, D = D. (2.2)

2.1.2 Transfer function description

An alternative system representation is the Transfer Function (TF) description. A TF maybe obtained directly from the SS model by assuming a zero initial state, i.e., x0 = 0, andtaking the Laplace transform of (2.1). This leads to

sx(s) − x0 = Ax(s) +Bu(s) → x(s) = (sI −A)−1Bu(s), x0 = 0 (2.3a)

y(s) = Cx(s) +Du(s) → y(s) = (C(sI −A)−1B +D)︸︷︷︸P (s)

u(s), (2.3b)

where P (s) is the TF, or Transfer Function Matrix (TFM) in case of a MIMO system. Becausea TF only describes the IO behavior, unobservable and uncontrollable states are not included.Since only physical systems are considered, all SISO TFs (or elements of TFMs) are rationalfunctions of the form

P (s) =βnzs

nz + · · · + β1s+ β0

αnpsnp + · · · + α1s+ α0

. (2.4)

For SISO systems, np is also defined as the order of the system and δ(P ) = np−nz as the poleexcess or relative degree of the system. Additionally, it is assumed that a TF P (s) is alwaysa minimum realization, meaning that common terms in the denominator and numeratorpolynomial are omitted from (2.4).

Definition 2.2. Consider a TFM P (s) with elements pij. Then:

• System P (s) is strictly proper if np > nz for all its elements pij(s), i.e., |pij(s)| → 0as s→ ∞.

• System P (s) is semi-proper or bi-proper if np = nz for at least one element pij(s),i.e., |pij(s)| → c, c ∈ R, c > 0 as s→ ∞, and the other elements are strictly proper.

• A system P (s) which is strictly proper or semi-proper is proper.

• A system P (s) is improper if np < nz for at least on element pij(s), i.e., |Pij(s)| → ∞as s→ ∞.

Notice that it is always possible to go from a SS representation to a TF description. However,in order to go back from a TF description to a SS representation, the system should at leastbe proper.

2.1.3 Frequency response function

By evaluating s over jω, the Frequency Response Function (FRF) is obtained. This sys-tem representation has the advantage of having a clear physical interpretation. At eachfrequency ω, the complex number P (jω) (or complex matrix for a MIMO system) relates a

2.1 System descriptions 9

sinusoidal input signal to a sinusoidal output signal. If this input signal is persistent, i.e.,applied at t = −∞, and of the form

u(t) = u0 sin(ωt+ φ). (2.5)

Then, the output signal is also a persistent sinusoid of the same frequency, i.e.,

y(t) = y0 sin(ωt+ ψ). (2.6)

The magnitudes u0 and y0 and phase shift ψ − φ can be directly obtained from P (jω) by

y0

u0= |P (jω)| and ψ − φ = ∠P (jω) = arctan

(Im(P (jω))

Re(P (jω))

). (2.7)

2.1.4 Coprime factorization

Another useful way to describe systems is by means of a coprime factorization, see, e.g., Damenand Weiland (2002, p. 72).

Definition 2.3 (Left coprime factorization). Any rational transfer function matrix Pcan be factored as

P = M−1N, (2.8)

in such a way that:

• Both M and N are stable TFMs.

• M is square and N has the same dimensions as P .

• There exist stable transfer function matrices X and Y such that

NX +MY = I, (2.9)

which is called the Bezout identity.

Such a factorization is called a left coprime factorization of P .

A right coprime factorization can be defined is a similar way. Left and right coprime fac-torizations are not unique and therefore often normalized. The left coprime factorization iscalled a normalized left coprime factorization if M and N are chosen such that

NNH +MMH = I, (2.10)

where {.}H denotes the complex conjugate transpose or Hermitian transpose.

Remark 2.4. The terminology comes from number theory where two integers n and m arecalled coprime if ±1 is their greatest common divisor. It follows that n and m are coprime ifand only if there exist integers x and y such that xn+ ym = 1.


2.2 Controllability and observability

Definitions in literature about controllability and observability are often about the states ofa system. In this thesis, the definitions from Skogestad and Postlethwaite (2005) are used.

Definition 2.5 (State controllability). The dynamical system x(t) = Ax(t)+Bu(t) or thepair (A,B) is said to be state controllable if, for any initial state x(0) = x0, any time t1 > 0,and any final state x1, there exists an input u(t), such that x(t1) = x1. Otherwise, the systemis said to be state uncontrollable.

Definition 2.6 (State observability). The dynamical system x(t) = Ax(t)+Bu(t), y(t) =Cx(t) + Du(t) or the pair (A,C) is said to be state observable if, for any time t1 > 0, theinitial state x(0) = x0 can be determined from the time history of the input u(t) and theoutput y(t) in the interval [0, t1]. Otherwise, the system is said to be state unobservable.

A SS realization (A,B,C,D) of a system is said to be a minimal realization of a system P (s)if all states, or equivalently, all modes of the system are controllable and observable. Thestate matrix A then has the smallest possible dimension.

2.3 Directionality

The main difference between SISO systems and MIMO systems is the presence of directionsin the latter. For a SISO system, the gain |y0|/|u0| at a given frequency is given by (2.7).The gain clearly depends on the frequency ω and is independent of the input signal.

For MIMO systems, the input and output signals are both vectors and to get a similarinterpretation of gain, the elements of these vectors need to be “summed up” by use of anorm. In Section 2.7, signal and system norms are further discussed. If the vector 2-norm1

is used for this, the gain of an n×m system P at a given frequency is given by

‖y(ω)‖2

‖u(ω)‖2=

√∑j |uj(ω)|2

√∑j |yj(ω)|2

=

√y210 + · · · + y2

n0√u2

10 + · · · + u2m0

. (2.11)

Clearly, for MIMO systems the gain still does not dependent on the input magnitude, but itdoes depends on the direction of u. The maximum gain of P as the direction of the input isvaried, is given by the maximum singular value of P , i.e.,

maxu 6=0

‖Pu‖2

‖u‖2= max

‖u‖2=1‖Pu‖2 , σ(P ). (2.12)

Similarly, the minimum gain equals the minimum singular value of P , i.e.,

minu 6=0

‖Pu‖2

‖u‖2= max

‖u‖2=1‖Pu‖2 , σ(P ). (2.13)

Besides changing the magnitude of the input signal, P also changes the directionality. Thedirection of y, i.e., the output direction, is therefore generally different than the input direc-tion.

1The vector 2-norm (or Euclidian norm) is a spatial norm defined as: ‖a‖2 ,√∑

i|ai|2.

2.3 Directionality 11

2.3.1 Singular value decomposition

Definition 2.7 (Singular Value Decomposition). Any complex n×m matrix A may befactorized using the Singular Value Decomposition (SVD)

A = UΣV H , (2.14)

where U ∈ Cn×n and V ∈ C

m×m are unitary2 and matrix Σ ∈ Rn×m contains a diagonal

matrix Σ1 of real, non-negative singular values σi, arranged in descending order as in

Σ =

[Σ1

0

]if n ≥ m, (2.15)

orΣ =

[Σ1 0

]if n ≤ m, (2.16)

whereΣ1 = diag {σ1, . . . , σk} , with k = min(n,m) (2.17)

andσ ≡ σ1 ≥ σ2 ≥ · · · ≥ σk ≡ σ. (2.18)

The column vectors of U , denoted by ui, are orthogonal and represent the output directionsof a plant. Similarly, the orthogonal column vectors of V , denoted by vi, represent theinput directions. If there is an input in direction vi, the output is in direction ui with anamplification of a factor σi. The maximum gain of any system is thus given by the maximumsingular value σ and the minimum gain by the minimum singular value σ.

The eigenvalues λ of a matrix also give an indication of the system gain and are related tothe singular values by

σi(P ) =√λi(PHP ). (2.19)

Eigenvalues, however, can be a very misleading measure for the system gain as illustrated inExample 2.8 and can only be calculated for square systems.

Example 2.8. Consider the system y = Pu, with

P =

[0 1000 0

]. (2.20)

Input vector u =[0 1

]Tgives output vector y =

[100 0

]T. Although both eigenvalues are

zero, the maximum system gain is σ = 100.

2.3.2 Rank

Definition 2.9 (Rank). The rank of an n×m matrix A is equal to the number of non-zerosingular values of the matrix. If rank(A) = r, then the matrix A is called rank deficient ifr < k = min(n,m), and singular values σi = 0 for i = r+1, . . . , k. In case of a square matrix,rank-deficiency also implies that the matrix is singular.

2A complex matrix is unitary if UH = U−1.


If a matrix has rank r, it has r independent rows (or columns) and the other k − r rows(or columns) are zero or are a linear combinations of the first r rows (or columns). In caseof a TFM, the term “normal rank” is often used, because the rank depends on the Laplacevariable s.

Definition 2.10 (Normal rank). The normal rank of an n×m TFM P (s) is the maximumrank of the matrix P (s) evaluated over all values of s.

In most situations, the normal rank of an n×m system is equal to k = min(n,m). However,if two or more sensors (or actuators) are located at exactly the same point, two or more rows(or columns) are equal and the normal rank can be less than k.

2.3.3 Condition number

If the gain of a system varies heavily with the input and output direction, the system hasstrong directionality and is called ill-conditioned. The condition number of a matrix is ameasure for this and is defined as the ratio between the maximum and minimum singularvalue (see Definition 2.11).

Definition 2.11 (Condition number). The condition number of a matrix is defined as theratio between the maximum and minimum singular value, i.e.,

c(P ) ,σ(P )

σ(P ). (2.21)

2.4 Stability

Several definitions of stability exist in literature. In this thesis, the notion of asymptoticstability is used.

Definition 2.12 (Stability). An LTI system is (asymptotically) stable if the injection ofa bounded external signal or a non-zero initial condition does not result in an unboundedoutput signal. Moreover, if the system is given a fixed, finite input, i.e., a step, then anyresulting oscillations in the output will decay at an exponential rate and the output will tendasymptotically to a new final, steady-state value.

2.5 Poles

The poles p of a system may be loosely defined as the values s = p, where P (p) equals infinity.In case of a rational TF as (2.4), the poles are equal to the roots of the denominator polyno-mial. Practically, poles are related to the modes of a system and are therefore independentof the inputs and outputs. Hence, they only depend on the state matrix A. Appendix Adiscusses the physical interpretation of poles and zeros in more detail.

Definition 2.13 (Poles). The poles pi of a system with a SS realization (A,B,C,D) are theeigenvalues λi(A), i = 1, . . . , nx.

2.6 Zeros 13

2.5.1 Poles and stability

For linear systems, the pole locations determine stability. Denote the open and closed leftand right halves of the complex plane by OLHP, CLHP, ORHP, and CRHP, respectively. Theimaginary axis, which is exactly in between the two halves, is included in the domain in caseof closed half and not included in case of an open half.

Theorem 2.14 (Stability). A continuous time system x(t) = Ax(t) + Bu(t) is stable ifall poles pi are in the OLHP, i.e., Re(pi) = Re(λi(A)) < 0 ∀ i. A state matrix with such aproperty is called “Hurwitz”. A system is unstable if it has any poles in the CRHP.

Proof. The proof follows straightforwardly by considering the time response of x(t), i.e.,

x(t) = eA(t−t0)x(t0) +

∫ t

t0

eA(t−τ)Bu(τ) dτ, (2.22)

where

eAt =

nx∑

i=1

tieλitqH

i , (2.23)

with ti and qi the left and right eigenvectors of A, respectively.

Remark 2.15. The term “neutrally stable” is sometimes used for systems with non-repeatedpoles (Franklin et al., 2002) on the imaginary axis. For example, a single integrator systemis neutrally stable, since a zero initial condition results in a constant output. However, anon-zero constant input results in an unbounded response. Hence, the system is unstable byDefinition 2.12.

2.5.2 Pole direction

As mentioned earlier, directionality is important in multivariable systems and hence it is notsufficient to define multivariable poles in terms where the denominator polynomials of P (s)have a singularity. Multivariable poles have directions associated with them and to quantifythese directions the input and output normalized pole vectors are defined as

ypi=

Cti‖Cti‖2

and upi=

BHqi‖BHqi‖2

. (2.24)

The matrix gain is infinite in the direction of the pole, which can be written as (Skogestadand Postlethwaite, 2005, p. 137)

P (pi)upi= ∞ · ypi

(2.25)

2.6 Zeros

Zeros of a system arise when competing internal effects are such that the output is zero, evenwhen the inputs (and the states) are not identically zero. Whether or not a system contains


zeros is therefore determined by the placement of the actuators and sensors relative to theunderlying dynamics.

For SISO systems, the zeros qi are simply the solutions to P (qi) = 0, i.e., the roots of thenumerator polynomial in (2.4). However, to capture this “blocking” nature for MIMO systemsis not that straightforward and Definition 2.16 should be used.

Definition 2.16 (Transmission zeros). The (transmission) zeros qi of a system P (s) arethe locations in the complex plane where the rank of P (qi) is less than the normal rank ofP (s).

Definition 2.16 is based on the TFM description, which corresponds to a minimum realizationof the system. If no minimum realization is available, numerical computations may yieldadditional invariant zeros. These invariant zeros plus the transmission zeros then form theso-called system zeros (Skogestad and Postlethwaite, 2005, p. 141). Due to the multivariablenature, transmission zeros are also called “multivariable zeros” to distinguish them from zerosin the elements of the TFM. For simplicity, however, transmission zeros are often simplyreferred to as “zeros”.

2.6.1 Zero direction

Similar to pole directions in the previous section, there are also directions associated with thetransmission zeros of a system. If P (s) loses rank at s = qi, then there exist non-zero vectorsuqi

and yqisuch that

P (qi)uqi= 0 · yqi

, (2.26)

where uqiand yqi

are defined as the zero input and output direction, respectively.

Remark 2.17 (Pinned zeros). A zero is pinned to a subset of outputs if the zero outputdirection yqi

has one or more elements equal to zero. Pinned zeros are common in practiceand their effect cannot be moved freely to any output. For example, the effect of measurementdelay at one output cannot be moved to another output. Example 5.12 shows how the effectof a zero can be moved to a specific output. If the ith element of the zero output direction iszero, the effect of the corresponding zero cannot be moved towards this output. Similarly, azero is pinned to certain inputs if the zero input direction uqi

has one or more elements equalto zero.

Remark 2.18. For square systems the poles and zeros of P (s) are essentially not the sameas the poles and zeros of det(P (s)). Using det(P (s)) to determine the system poles andzeros fails when pole/zero cancellations occur between elements of P (s) when calculating thedeterminant.

2.7 Signal and system norms

The basic idea behind a norm is to have a single number that gives an overall measure of thesize of a vector, matrix, signal, or system. Norms are therefore particulary useful to recastthe control problem as a mathematical optimization problem (see Section 3.4). The material

2.7 Signal and system norms 15

presented here is presented without going into the mathematical details and is mainly basedon Oomen (2004, Section 2.4). A more comprehensive treatment of norms is given by Damenand Weiland (2002, Chapter 5) and Zhou et al. (1996, Chapters 2 and 4).

Regardless whether e represents a vector, matrix, signal, or a system, the norm of e shouldsatisfy the following definition.

Definition 2.19. A norm of e is a real-valued number, denoted by ‖e‖, that satisfies thefollowing properties:

1. Non-negative: ‖e‖ ≥ 0.

2. Positive: ‖e‖ = 0 ⇔ e = 0.

3. Homogeneous: ‖αs‖ = |α| · ‖s‖, α ∈ C.

4. Triangle inequality:

‖e1 + e2‖ ≤ ‖e1‖ + ‖e2‖ (2.27)

For matrices (and systems) the following definition should also hold for the norm to qualifyas a matrix norm.

Definition 2.20. A norm of matrix E, denoted by ‖E‖, is a matrix norm if, in addition tothe properties in Definition 2.19, it also satisfies the multiplicative property, i.e.,

‖AB‖ ≤ ‖A‖ + ‖B‖. (2.28)

Property (2.28) is very important when combining systems and forms the basis of the small-gain theorem (Zhou et al., 1996, Theorem 9.1). Notice that there exist norms on matrices(thus satisfying the properties in Definition 2.19) that do not qualify as a matrix norm.

2.7.1 Signal norms

A signal e is a function that quantifies how a certain variable evolves in time and is given bythe mapping

e : T →W, (2.29)

where T denotes the time set and W is the signal space. In this research, only continuous timesignals are discussed, i.e., T ⊂ R. Furthermore, only real-valued multi-dimensional signalsare considered. At each time instant t ∈ T , e is thus a vector with l entries, representing lreal valued quantities, i.e., W = R

l. To deal with the multi-dimensional nature of signal, thevector norm is defined first.

Definition 2.21 (Vector norm). Consider a vector x ∈ Cl. The p-norm of x, denoted by

‖x‖p, is defined as

‖x‖p ,

(∑li=1 |xi|p

) 1

p,

maxi |xi|,

for 1 ≤ p <∞,

for p = ∞.(2.30)


After the channels of a vector-valued signal are “summed up” at a given time instant usinga vector norm, a second summation is made over the time values using a temporal norm. Itis common practice, see, e.g., Skogestad and Postlethwaite (2005, p. 537), to use the samep-norm to sum up the vector and the time signal.

Definition 2.22 (Signal norm). The temporal norm, denoted as the Lp norm 3, of a signale : T → Rl is given by

‖e‖Lp ,

(∫t∈T (‖e(t)‖p)

p dt) 1

p ,

supt∈T ‖e(t)‖∞,

for 1 ≤ p <∞,

for p = ∞,(2.31)

where t denotes the time instant (either for a finite time set T = [a, b] or an infinite time setT = R).

Signal norms can also be determined in the frequency domain as in done in Section 2.3. Thefirst summation is then done by applying a vector norm at a given frequency and the secondsummation is done over the frequencies. For T = R, several signals norms are of specialinterest. The 1-norm (p = 1) of a signal equals the integrated absolute value, the squared2-norm equals the energy of a signal, and the ∞-norm equals the maximum absolute value ofa signal. By using Parseval’s Theorem, see, e.g., Ambardar (1999), it can be shown that thetime domain 2-norm as defined by (2.31) is equivalent to frequency domain 2-norm as usedin Section 2.3.

Since not all signals have a finite energy, like, e.g., periodic signals, an additional norm isintroduced:

‖e‖pow ,

√

limT→∞

1

2π

∫ T

−T|e(t)|2 dt. (2.32)

The norm ‖e‖pow is called the power-norm and defined for all signals with a finite power,i.e., ‖e‖pow < ∞. Note that the power-norm does not satisfy the second property in Defini-tion 2.19. The power-norm is therefore only a semi-norm.

2.7.2 System norms

The norm of a system is used to quantify how large (in terms of a signal norm) the outputsignals can be if a certain sized (in terms of a norm) input are applied. Two norms arediscussed that are popular in the field of norm-based controller design: the H2 and the H∞norm.

H∞ norm

Consider a stable MIMO system H(s) with inputs w and outputs z.

3L stands for the fact that the signals should be Lebesgue integrable. Sometimes this notations is shortenedto the p-norm. The context should then avoid confusion with the vector 2-norm as defined by (2.30).

2.7 Signal and system norms 17

Definition 2.23 (H∞ norm). Let H(s) be a stable TFM with FRF H(jω). The H∞ normof H(s), denoted by ‖H(s)‖∞, is then defined as

‖H(s)‖∞ , supω∈R

σ (H(jω)) . (2.33)

For SISO systems, the H∞ norm thus equals the peak value in the Bode magnitude diagramand for MIMO systems this peak value is generalized to the maximum singular value. TheH∞ norm is a so-called induced norm, see Damen and Weiland (2002, Section 5.3), because

supω∈R

σ (H(jω)) = supw∈L2,w 6=0

‖Hw‖2

‖w‖2= sup

w 6=0

‖z(t)‖L2

‖w(t)‖L2

= supw 6=0

‖z(t)‖pow

‖w(t)‖pow. (2.34)

The proof involves using Parseval’s Theorem and can be found in Damen and Weiland (2002,Section 5.3). The H∞ norm can thus be interpreted in the time domain as the largestamplification of energy (or power) for an input signal in a particular direction and at aparticular frequency. Hence, the H∞-norm is also called an L2- (or power-) induced norm.In case a system is unstable, i.e., has poles in the CRHP, the output energy is unbounded.Hence, the H∞ norm is not defined for unstable systems.

Remark 2.24. In robust control literature, see, e.g., Zhou et al. (1996), the H∞ norm ofH(s) is often denoted by ‖H‖∞ rather than by ‖H‖H∞

. The context should avoid confusionwith the matrix ∞-norm. A similar reasoning holds for the H2 norm.

H2 norm

Consider the stable SISO system H(s) with input w and output z.

Definition 2.25 (H2 norm for SISO systems). Let H(s) be a stable TF with FRF H(jω).The H2 norm of H(s), denoted by ‖H(s)‖2, is then defined as

‖H(s)‖2 ,

(1

2π

∫ ∞

−∞

(|H(jω)|2

)dω

) 1

2

. (2.35)

Although the H2 norm is not an induced norm, several interpretations exist. By using Parse-val’s theorem, it can be shown that the H2 norm equals the 2-norm of the impulse responseof H. This interpretation is often referred to as the deterministic interpretation and is notdiscussed further.

A stochastic interpretation also exist. Only the general idea is presented here; the completederivation is given in Damen and Weiland (2002, p. 51–53). Suppose input w has a PowerSpectral Density (PSD) Sw(ω), it can then be shown that the PSD of z is given by

Sz(ω) = Sw(ω)|H(jω)|2. (2.36)

Next, it can be shown that (Skogestad and Postlethwaite, 2005, Section 9.3.2)

‖z‖pow =

(1

2

∫ ∞

−∞

(|H(jω)|2

)dω Sw(ω)

) 1

2

, (2.37)


where ‖z‖pow is defined in (2.32). Now consider a white noise input u with Sw(ω) = 1 ∀ω ∈ R.Then, (2.37) equals (2.35) and the H2 norm can be interpreted as the system response to awhite noise input signal. In other words: the H2 norm is the squared area under the Bodemagnitude plot of H.4

To conclude this section, a generalization of the H2 norm is given for MIMO systems.

Definition 2.26 (H2 norm for MIMO systems). Let H(s) be a stable TFM with FRFH(jω). The H2 norm of H(s), denoted by ‖H(s)‖2, is then defined as

‖H(s)‖2 ,

(1

2π

∫ ∞

−∞Trace{HH(−jω)H(jω)} dω

) 1

2

, (2.38)

=

(1

2π

∫ ∞

−∞

k∑

i=1

σi(H(jω))2

) 1

2

, (2.39)

where the “Trace” of a square matrix is the sum of the entries at its diagonal, σi is the ith

singular value, and k is the number of singular values.

It follows from (2.39) that the H2 norm is a measure for the area under all the singular valuesof H(s) in a singular value diagram.

Remark 2.27. The symbol H in H∞ and H2 stands for “Hardy space”. The symbol H∞stands for the set of TFMs with bounded ∞-norm, which is the set of proper and stable TFMs.Similarly, the symbol H2 stands for the set of TFMs with bounded 2-norm, which is the setof strictly proper and stable TFMs.

2.8 Conclusions

In this chapter, basic elements of linear system theory are introduced and definitions areincluded that are needed in the remainder of this research. The topics discussed are varioustypes of system descriptions, directionality in multivariable systems, controllability and ob-servability, system stability, (multivariable) poles and zeros, and signal and system norms.The next chapter discusses (part of) the controller design procedure for (flexible) motionsystems, whereby the main focus is on norm-based controller design.

4Note that the magnitude is taken to the second power and the frequency is evaluated over a linear axis.

Chapter 3

Controller design for flexiblemotion systems

This chapter deals with controller design for flexible motion systems. By going through thecontroller design process, shortcomings of current state-of-the-art controller design for thecontrol of flexible motion systems are discussed. In Section 3.1, the steps of the controllerdesign process that are important for this research are introduced and in Sections 3.2 – 3.7these steps are treated in detail. Subsequently, alternative control structures are proposedin Section 3.8 that are able to deal with the distinction between measured and performancevariables (research issue 1).

In addition, this chapter discusses the formulation of sensible norm-based control problems forflexible motion systems whereby the measured variables are assumed to be the performancevariables, i.e., conventional norm-based controller design, and whereby internal dynamics isnot neglected (research issue 2).

3.1 Controller design procedure

Several step-by-step controller design procedures are available, see, e.g., the procedures pre-sented in Skogestad and Postlethwaite (2005, p. 1) and Van de Wal et al. (2002, p. 18). Onlysteps considered relevant for controller design for flexible motion systems are discussed hereand because no practical setup is available, these procedures can be further reduced to thefollowing four steps:

1. Specify the control objective (Section 3.2).

2. Control architecture design (Section 3.3). Control architecture design as definedhere1 involves three parts:

• Actuator/sensor selection or Input/Output (I/O) selection: the choice of the type,number, and locations of the actuators (inputs) and sensors (outputs).

1Similar as in Van de Wal (2009)

19

20 Controller design for flexible motion systems

• Control structure design (Section 3.8) or control configuration: definition of thesignal routing (disturbance, performance, measured, and manipulated signals) inthe control scheme.

• Structure of the controller or controller configuration selection: usage of diagonal,block-diagonal, or full MIMO controllers.

3. Design a controller (Section 3.4 and Chapter 4) such that the control objectivesare met. As already stated in Chapter 1, the main focus is on FB controller design.Within Philips, it is common practice to design FB controllers by SISO loop-shaping,see, e.g., Angelis et al. (2005). In this research, however, the focus is on norm-basedcontroller design to systematically deal with plant interaction.

4. Evaluate the performance (Section 3.7).

3.2 Control objective

Simply stated, the goal of control is to let the system behave in a desired way. The controlledsystem needs to achieve a certain level of performance, while being robust for model uncer-tainties and stay within the limitations of the system. In Bosgra et al. (2006), a distinctionis made between three types of control problems:

• Regulator problems. The control objective in a regulator problem is to keep a desig-nated output within tolerances of a predetermined value, despite the effects of externaldisturbances.

• Servo problems. In a servo problem, the measured system outputs need to follow aprescribed reference signal, while also rejecting external disturbances.

• Tracking problems. Basically the same as a servo problem, but in a tracking problemthe reference signal is not predetermined.

So, because servo problems aim at keeping the measured error small (r−yp in Figure 1.2), themeasured variables either are the performance variables or are assumed to be the performancevariables, i.e., yp = zp. If the system behaves as a rigid body, the assumption that yp = zp isjustified. For the flexible motion systems considered in this research, however, the rigid bodyassumption is not necessarily justified and the performance variables are also not measuredduring normal operation. In addition, zp may even contain a set of performance variablesor be a spatial norm. Either way, zp 6= yp and to emphasize this difference these type ofservo problems are referred to as the inferential servo problems in Oomen et al. (2009) andthe actual performance is often referred to as inferential (servo) performance. The extent towhich the control objective is achieved can be divided into four categories (Skogestad andPostlethwaite, 2005):

• Nominal stability. The controlled system is stable and model uncertainties are nottaken into account.

3.3 Control architecture design 21

• Nominal performance. The controlled system is stable and achieves the desired levelof performance while model uncertainties are not considered.

• Robust stability. The controlled system is stable in the presence of specified modeluncertainties.

• Robust performance. The controlled system is stable and achieves the desired levelof performance in the presence of specified model uncertainties.

3.3 Control architecture design

3.3.1 I/O selection

The process of I/O selection involves the choice of the type, number, and locations of theactuators (inputs) and sensors (outputs). For most conventional motion systems, however,the process of I/O selection is part of the mechanical system design process and thus notconsidered part of the control problem. I/O selection is then often aimed at creating a stiffrelation between the actuators and sensors. In case of ideal actuators and sensors, the ac-tuators and sensors do not change the system dynamics. For a system in SS representation,this means that the A matrix is fixed and that only the B (actuators), C (sensors), and D(actuators and sensors) matrices are affected by I/O selection. Different I/O sets may there-fore lead to systems with different zero locations, of which some zeros may be in the CRHP.In Appendix A, additional information is given about zeros, how they can be interpretedphysically and how they can be created by manipulating the plant inputs and outputs.

I/O Selection becomes more important for flexible motion systems, since a specific I/O setstrongly determines the extent to which the flexible dynamics can be observed and controlled.Van Wingerden (2004) already showed that for flexible motion systems changing the locationsof the actuators and sensors can have a greater effect on the achievable performance than thecontroller design itself. In addition, a bad I/O set may impose fundamental limitations thatcannot be resolved by advanced linear controller design.

Regarding the actuator placement, a logical choice for disturbance attenuation is to locate theactuators such that the flexible dynamics is maximally excited. In this way, the least amountof energy is needed. Unfortunately, this also means that the flexible dynamics is heavilyexcited by the FF signal. The optimal actuator configuration is therefore system dependent.

A similar reasoning applies to the sensor locations. For collocated (see Definition 3.1) actua-tors and sensors, the dynamics between an actuator/sensor pair shows a repeating sequenceof complex poles and zeros, see, e.g., Preumont (2002, Chapter 4), which is beneficial for sta-bility. The phase of the plant then never drops below -180 [◦]. Although from this viewpointcollocated control is preferred, it ignores the fact that the performance variables are not mea-sured. There may be internal dynamics present that severely deteriorates the performancevariables, while this is not observed by the sensors. It is therefore proposed that it is betterto place the sensors close to the performance variables.

Definition 3.1 (Collocated variables). The concept “collocated” refers to the positionof one variable with respect to another. If two (or more) variables are collocated, they are


located at the same physical position. In case of collocated control, the actuators and sensorsare located at the same physical position and in non-collocated control they are not.

Remark 3.2. In practice, actuators and sensors can never be completely collocated due tovarious reasons, like, e.g., the size of the actuators and sensors.

In the foregoing reasoning about the optimal actuator and sensor locations, the effect ofNonMinimum Phase (NMP) zeros, i.e., zeros in the CRHP, is not taken into account. Theeffect of NMP zeros on the achievable performance in case the measured variables are not theperformance variables is discussed in detail in Chapter 5.

In conventional controller design, the number of independent2 actuators and sensors is oftenequal to the number of rigid body DOFs. Adding extra actuators (over-actuation) and/or sen-sors (over-sensing) offers the possibility to explicitly control the flexible dynamics. Althoughsome promising results have been achieved using this approach, see, e.g., Cloosterman et al.(2003) and Schneiders et al. (2003, 2004), this control objective is beyond the scope of thisresearch.

3.3.2 Control structure design

The control structure that is commonly used in high performance motion control is the struc-ture depicted in Figure 1.2. Since the focus of this research is not on FF design, the FFfilter Cff is omitted. The resulting structure is depicted in Figure 3.1 and is referred to asthe single-DOF control structure. As already stated in Chapter 1, zp denotes the unmea-sured performance variables, yp are the measured variables, up are the plant inputs, u arethe controller outputs, and r is the reference. Input disturbances du, outputs disturbancesdy, and measurement noise η are depicted as well. The single-DOF structure works well forsystems that can be approximated by a rigid body, see, e.g., Steinbuch and Norg (1998), butis inadequate to deal with inferential servo problems, as is discussed in Section 3.8.

C Pypr upu

du

η

ey

dyPz

zp

Figure 3.1: Single-DOF control structure.

For the single-DOF control structure, the open-loop L is defined as the system as seen whenbreaking the loop at the output of the plant. Thus, if the loop consists of a plant P andcontroller C, L is given by

L , PC. (3.1)

2In some systems there are more actuators and/or sensors than rigid body DOFs, but this is often done forpractical rather than system-theoretic reasons. For example, extra actuators can be used to achieve sufficientactuator power into the direction of a rigid body mode. Effectively, however, this can be treated as a singleactuator.

3.3 Control architecture design 23

The sensitivity and complementary sensitivity are then defined as

S , (I + L)−1 and T , L(I + L)−1. (3.2)

In Figure 3.1, T is the mapping T : r → yp and S is the mapping S : dy → yp and S : r → ey,where ey , r− yp − η the servo error. Notice that the servo error is the tracking error r− yp

contaminated with measurement noise. In a similar fashion, LI is defined as the system seenwhen breaking the loop at the input of the plant, i.e.,

LI , CP. (3.3)

The input sensitivity and input complementary sensitivity are then defined as

SI , (I + LI)−1 and TI , LI(I + LI)

−1. (3.4)

In Figure 3.1, −TI is the mapping −TI : du → u and SI is the mapping SI : du → up. Theprocess sensitivity SP and control sensitivity CS are defined next as3

SP , (I + L)−1P = SP = PSI , (3.5)

CS , C(I + L)−1 = CS = SIC. (3.6)

In Figure 3.1, SP is the mapping SP : du → yp and CS is the mapping CS : r → u. Noticethat CS and SP are not dimensionless (unlike SI , T , and TI), so proper scaling is important.

Although a FB controller is usually implemented in a structure as depicted in Figure 3.1, thecontrol structures in Figure 3.2 allow for more design freedom. These structures are referredto as the standard plant setup, whereby G is the so-called standard plant. The standardplant formulation allows for an explicit distinction between sensed outputs y and (weighted)exogenous outputs z on one side and (weighted) exogenous inputs w and control signalsu on the other side. It is therefore regarded a more suitable structure for the control offlexible motion systems. Notice that the conventional control structure can also be writtenin the standard plant formulation. The real plant P (and possible Pz) is then containedin G: G(P ). In addition, it is also possible to explicitly account for model uncertainty inthe problem formulation. An uncertainty block ∆ then needs to be included in the controlstructure as depicted in Figure 3.2(b).

3.3.3 Controller configuration selection

Controller configuration, which is not to be confused with control configuration as discussedin Section 3.3.2, refers to the internal structure of the controller. The structure can eitherbe centralized or decentralized. In centralized control, each controller input possibly affectseach controller output. In decentralized control, each controller input only affects a subsetof outputs. This would be a block-diagonal controller or - in an extreme version - a diagonalcontroller. It often depends on the controller design methodology (see Section 3.4) whichcontrol configuration is used. In case of manual loop-shaping, which is still common practice

3The equalities SP = PSI and CS = SIC follow from the so called push-through rule, see, e.g., Skogestadand Postlethwaite (2005, p. 68): C(I + PC)−1 = (I + CP )−1C. This relation is readily proven by pre-multiplying both sides by (I + CP ) and post-multiplying both sides by (I + PC).


C

G

w z

u y

(a) Without uncertainty

C

Gwperf zperf

u y

∆

wu zu

(b) With uncertainty

Figure 3.2: Standard plant setup.

for most applications, see, e.g., (Steinbuch and Norg, 1998), separate SISO controllers are de-signed for the diagonal plant entries. Such controllers are called “multiloop SISO controllers”.Interaction can either be completely neglected (independent SISO control) or control loopscan by closed one at a time (sequential loop closing). Both methodologies result in diagonalcontrollers. For the standard plant setup, the computation of a norm-based controller oftenresults in a centralized controller. Computing structured controllers is possible, but in gen-eral this gives rise to non-convex optimization problems, which are much harder to solve, see,e.g., Hol (2006).

3.4 Norm-based controller design

Various controller design methodologies exist. The well-known method of SISO loop-shapingis insightful and has proven its value in various applications, even for MIMO systems, see,e.g., Steinbuch and Norg (1998). Although the effect of plant interaction can be taken intoaccount by sequentially closing FB loops, norm-based controller design techniques explicitlyaccount for the interaction among the various motion DOFs. The essence of norm-basedcontroller design is to formulate the control problem (as discussed in Section 3.2) as anoptimization problem, where the norm of a particular closed-loop TF or TFM is minimized.Hence, the term “optimal control” is also used.

As stated in Section 3.3.2, the standard plant setup is very useful for formulating a well-definedcontrol problem. Since, however, no uncertainty models are available at present, uncertaintyis not explicitly taken into account during controller synthesis. The control objective is thenformulated as the minimization of a norm of the TF(M) between the (weighted) exogenousinputs w, e.g., disturbances or the reference, and the (weighted) exogenous outputs z, i.e.,the variables to be kept small, e.g., error signals. Let G(s) in Figure 3.2(a) be partitioned asfollows [

zy

]=

[G11 G12

G21 G22

] [wu

]. (3.7)

3.4 Norm-based controller design 25

The closed-loop TF(M) from w to z is then given by the lower linear fractional transformation

M = Fl(G,C) , G11 +G12K(I −G22C)−1G21. (3.8)

Once the control problem has been specified as an optimization problem in which the normof a TF(M) needs to be minimized, the actual computation of a controller that achieves thisminimum is easy, fast and reliable. The most time consuming part, however, is to formulatea sensible optimization problem that will result in a well performing control system.

3.4.1 Selecting a norm

Both the H2 and H∞ norm are often encountered in the area of robust control, see, e.g., Zhouet al. (1996). The most important reason for using of the H∞ norm is that, in contrast tothe H2 norm, the H∞ norm is an induced norm and thus satisfies the multiplicative property,which is given by (2.28). As a consequence, by determining the H∞ norm of individualsystems, information can be obtained about the behavior of a series interconnection of thosesystems. In robust control, this property is exploited to derive guaranteed stability marginsfor unstructured uncertainty models. From a performance perspective, however, the H2 normwould be a more logical choice, since it takes the singular values at all frequencies into account,while the H∞ norm is only concerned with the largest singular value at a single frequency.Therefore, σ(M(jω)) needs to be manipulated such that the desired performance objectivesare met.

3.4.2 Incorporating dynamic weighting filters

These manipulations are carried out by choosing appropriate weights for the unweightedexogenous inputs (w) and outputs (z), as depicted in Figure 3.3. Herein G represents thestandard plant without weighting filters. The weighted exogenous inputs and outputs arethen obtained by

z = Wz z, w = W−1w w, (3.9)

where Wz and Ww have to be bi-proper and stable4 TFMs (Skogestad and Postlethwaite,2005, Section 9.3).

Incorporating these weights yields the control criterion

J(P,C) = ‖M(P,C)‖∞ = ‖WzM(P,C)Ww‖∞ = ‖WzFl(G(P ), C)Ww‖∞, (3.10)

leading to the following control problem.

Definition 3.3. Given a system P and controller C in the interconnection structure as de-picted in Figure 3.3, determine

Copt = arg minC

J(P,C), (3.11)

where J(P,C) is given by (3.10).

4A TF H(s) that is bi-proper and stable has no poles in the CRHP and has a non-zero D-matrix. SuchTFs are also called bi-stable, since bi-properness and stability mean that both H(s) and H(s)−1 need to bestable


G

C

Ww Wz

w z zw

G

M

yu

M

Figure 3.3: Standard plant setup with performance weights.

This type of controller design is referred to as H∞ optimization and the resulting controller iscalled an H∞ controller. How to guarantee that the resulting controller is internally stabilizingis discussed in Section 3.6.

The most obvious method to select weighting filters is the signal-based approach, as is discussedin Skogestad and Postlethwaite (2005, Section 9.3.6). Weights are then used to describe theknown or expected frequency content of the exogenous input signals and desired frequencycontent of the error signals. For practical applications, however, it is often difficult andtime consuming to estimate power spectra for disturbances and measurement noise. A loop-shaping-based approach, which does not require accurate disturbance models, is thereforepreferred and adopted in this research. The signals included in w and z then need to be chosensuch that relevant TFMs are included in M . In doing so, the interpretation that z contains thesignals to be kept small and that w contains disturbances is lost; only minimizing the normof the mapping between the weighted signals, i.e., the mapping M : w → z, is meaningful.Weighting filters are hereby used to roughly specify up to which frequency the control systemis effective. For general MIMO systems, this is done by specifying the desired shape of thesingular values of the closed-loop TFMs. For instance, by requiring σ(S(jω)) small for lowfrequencies. Notice that selecting w and z in order to weigh particular closed-loop TFMs isessentially part of the control structure design step as discussed in Section 3.3.2.

3.4.3 Selecting exogenous outputs and inputs

Selecting w and z such that M contains meaningful closed-loop TFMs is discussed in thissection. Consider the single-DOF control structure depicted in Figure 3.1. For servo problems(the situation whereby yp 6= zp is discussed in Section 3.8) S is of great importance, because itdescribes the mappings r → ey (servo performance) and dy → yp (disturbance attenuation).In standard plant formulation, this yields

z = Mw, M = S, (3.12)

where either w = r and z = ey or w = dy and z = yp. Weighting filters Ww and Wz thenneed to be specified such that σ(WzSWw) ≫ 1 at low frequencies. Subsequently, standard

3.5 Internal stability and well-posedness 27

controller synthesis algorithms can be used to perform the minimization in (3.11).

However, when only S is included in M , the control synthesis algorithm has the freedom totune C such that small improvement in S are allowed at the cost of undesired behavior, like,e.g., pole/zero cancellations and extremely large control signals. Other TFMs are thereforeoften included in M . Two popular alternatives in the area of robust control are the mixed-sensitivity control structure (Skogestad and Postlethwaite, 2005, Section 2.8.3) and the four-block control structure (Englehart and Smith, 1990). In the former, S, T , and CS are stackedin M , while the latter contains four TFMs to guarantee internal stability. Because the resultingcontroller always needs to be internally stabilizing, the four-block control structure is adoptedin this research. In Section 3.6, the four-block control structure is discussed in detail. Theconcept of internal stability, however, is discussed first in the next section.

3.5 Internal stability and well-posedness

If a FB controller is designed such that a particular closed-loop TFM, e.g., S and/or SP ,has the desired shape, stability of this closed-loop TFM alone does not guarantee stability ofthe overall FB system. Although one closed-loop TFM may have all its poles in the OLHP,unstable pole/zero cancellations can still occur between the system components, leading toan unbounded internal signal. This is referred to as internal instability. To achieve internalstability, all signals in the control structure should be bounded as a result of signals that areinjected at any location in the control structure. The FB loop in Figure 3.1 has two systemcomponents: the plant P and controller C, implying that the four TFMs in

yp = Tr + PSIdu, (3.13)

up = SICr + SIdu. (3.14)

need to be stable. This is formalized in Theorem 3.4.

Theorem 3.4 (Internal stability). Assume that the components P and C do not containunstable modes. The feedback interconnection of Figure 3.1 is internally stable if all fourTFMs in (3.13) and (3.14) are stable. If, however, there are no unstable pole/zero cancella-tions between C and P , stability of one closed-loop TFM implies stability of the others, andis sufficient to guarantee internal stability.

Proof. See Zhou et al. (1996, Section 5.3) or Francis (1987, p. 3)

Remark 3.5. Internal stability requires that the TFMs between all inputs and outputs in theloop have to be stable. This, however, does not mean that C and P have to be stable. Inaddition, it also means that input ry in (3.13) and (3.14) may just as well be replaced by η,and output yp by ey.

As a consequence of internal stability, the following two statements apply:

1. If P (s) has a CRHP zero at q, then also L, T , SP , LI , and TI each have a CRHP zeroat q.


2. If P (s) has a CRHP pole at p, then also L and LI have a CRHP pole at p, while S, CSand SI each have a CRHP zero at p.

Proof. The proof follows straightforwardly from realizing that no CRHP pole/zero cancel-lations between system components, such as P and C, are allowed. Such cancellations are,however, allowed between two TFMs, such as L and S. See Skogestad and Postlethwaite(2005, p. 146) for the complete proof.

Similar to the internal stability requirement, well-posedness is related to properness of theTFMs in (3.13) and (3.14).

Definition 3.6. The feedback interconnection of Figure 3.1 is said to be well-posed if allclosed-loop TFMs in (3.13) and (3.14) exist and are proper.

3.6 Control problem formulation

In this section, the four-block control structure is discussed in detail. To guarantee internalstability, w and z should be chosen such that at least the four TFMs from (3.13) and (3.14)are included in M . Consider the single-DOF control structure in Figure 3.1, one possibilityis to select

w =

[rdu

], z =

[yp

up

], (3.15)

which results in the mapping

M(P,C) =

[PI

]SI

[C I

]=

[T SPCS SI

]. (3.16)

To systematically formulate the control problem such that it can be dealt with using standardalgorithms, the standard plant setup is employed. By extracting the controller from thestructure in Figure 3.1, the unweighted standard plant

[zy

]= G(P )

[wu

](3.17)

is obtained (see Figure 3.4). Including bi-stable weighting filters

Wz =

[Wz,1 0

0 Wz,2

], Ww =

[Ww,1 0

0 Ww,2

], (3.18)

leads to the control criterion J(P,C) in Definition 3.3. This control problem is, withoutregard of the norm, often referred to as the four-block control problem, see, e.g., Englehartand Smith (1990). Clearly, employing an H∞ norm in (3.11) leads to an internally stabilizingcontroller C and a well-posed FB system. Finally, some remarks regarding the control goal(Definition 3.3) and the algorithm to perform the optimization are made:

1. Although specifying the desired singular values of the closed-loop TFMs seemingly givesa lot of design freedom, the only design freedom is the controller C. Dependencies

3.6 Control problem formulation 29

C

{ z

yu

P

G(P )r

du

yp

up

w }

Figure 3.4: Recasting the setup of Figure 3.1 into the standard plant configuration.

between the elements of M , like, e.g., due to the analytical constraint S + T = I,need to be taken into account by the designer. In Chapter 4, an approach is proposedwhereby the open-loop singular values are shaped. In doing so, these dependencies arealready taken into account implicitly.

2. It follows from Zhou et al. (1996, Corollary 18.7) that

∥∥∥∥[PI

](I + CP )−1 [ C I

]∥∥∥∥∞

=

∥∥∥∥[CI

](I + PC)−1 [ P I

]∥∥∥∥∞. (3.19)

An equivalent four-block control problem can thus be formulated in terms of S. In Van deWal (2002), this equivalent problem formulation is used to design controllers for high-performance stages.

3. It is possible to select extra exogenous inputs and outputs compared to (3.15), suchthat M consist of more than four TFMs. For instance, select

w =

rdu

dy

, z =

[yp

up

], (3.20)

which results in the mapping

M(P,C) =

[T SP SCS SI −CS

]. (3.21)

Although this potentially leads to a better formulated control problem, there are somegeneral warnings:

• Because the essence of the loop-shaping-based approach for weighting filter selec-tion is to roughly specify up to which frequency control is effective, the dependen-cies between the extra elements in (3.21) further complicate the selection of goodweighting filters. In addition, notice that CS even occurs twice in (3.21), whichgenerally leads to more conservative stability bounds. A more elaborate motiva-tion for keeping the dimensions of M as small as possible is given in Damen andWeiland (2002, Section 7.2).


• The order of the resulting controller is generally equal to the sum of the order ofthe plant P , and the orders of the weighting filters (Damen and Weiland, 2002,p. 68). High order filters thus lead to high order controllers, which is generallyundesirable.

4. Another advantage of the four-block structure in combination with the H∞ norm is thatfor weakly damped systems pole/zero cancellations in the OLHP are also less likely tooccur, because these cancellations lead to undamped resonances in one or more elementsof M(P,C). Such cancellations are often not exact in practice and may therefore resultin oscillatory behavior.

5. Determining the optimal controller in (3.11) is computationally (and theoretically) dif-ficult and often not necessary in practice. A suboptimal one, i.e., one close to theoptimal one in the sense minimizing the H∞ norm, is usually sufficient. Let γmin bethe minimum value of J(P,C) over all stabilizing controllers C. Then the suboptimalcontrol problem is: given a γ > γmin, find all stabilizing C such that J(P,C) < γ. Ageneral, reliable, and computationally effective method to solve this problem is proposedin Doyle et al. (1989) and is referred to as “DGKF” solution. The minimum value ofJ(P,C) is hereby approximated by using a bisection algorithm, i.e., by starting withhigh and low values for J(P,C) and then iterating towards the minimum value (Balaset al., 2008, p. 6–128).

3.7 Performance evaluation

Although internal stability is a necessity, certain performance specifications also need to beachieved. These can either be specified in the time domain, frequency domain, or both.Without giving details, some common characteristics and definitions are included.

3.7.1 Time domain performance

Performance evaluation in the time domain is often done by considering the response to a(unit) step in reference. It then considers the following characteristics:

• Settling time: the time after which the output remains within ±ǫ% (typically, ǫ = 5)of its final value.

• Overshoot: the peak value divided by the final value.

• Decay ratio: the ratio between the first and second peak. This is related to dampingin the system.

• Steady-state error: the difference between the final value and the desired final value.

3.8 Control structures for inferential servo problems 31

3.7.2 Frequency domain performance

Closed-loop performance may also by analyzed by considering the FRFs of relevant TFs.Which TFs are relevant depends on the system of interest and the type and amount ofdisturbances. For all FB systems, however, the open-loop L is of great importance andvarious definitions related to performance and stability exist that are related to properties ofL(jω).

Definition 3.7 (Bandwidth or cross-over frequency). The BandWidth (BW) or cross-over frequency of a system is the frequency where L(jω) crosses the 0 dB line from above forthe first time.5

Definition 3.8 (Phase margin). The Phase Margin (PM) of a system is the amount bywhich the phase of L(jω) exceeds -180 [◦] at the BW, i.e.,

PM , ∠L(jωbw) + 180. (3.22)

Definition 3.9 (Gain margin). The Gain Margin (GM) of a system is the amount by whichthe controller gain be raised before instability occurs. In a Bode plot it is the distance betweenthe 0 dB line and |L(jω)| at the point where ∠L(jω) = −180 [◦].

Definition 3.10 (Modulus margin). The Modulus Margin (MM) of a system is definedas the maximum value of S|(jω)|, and this is equal to the inverse of the smallest distancebetween the point (-1,0) and L(jω) in the complex plane.

The definitions introduced above only hold for SISO plants, but can also be applied to decou-pled MIMO plants. For coupled MIMO plants, however, no such widely accepted definitionsexist. Doyle and Stein (1981) present generalizations to MIMO systems that are in terms ofthe singular values6 (for BW) and eigenvalues7 (for PM) of L(jω).

3.8 Control structures for inferential servo problems

This section consists of three parts. In the first part, it is shown that the single-DOF controlstructure in Figure 3.1 is inadequate for general inferential servo problems. In the subsequenttwo parts, alternative structures are proposed that are able to deal with the inferential natureof the problem.

3.8.1 Limitations of the single-DOF control structure

When internal dynamics that is not directly sensed becomes relevant, the control objective isin general not to keep ey small. For inferential servo problems, the performance should ratherbe evaluated at zp as is shown in Figure 3.5 and ez, which is defined as inferential servo error,

5In literature, the BW is also often defined in terms of S or T .6The frequencies where the singular values of L cross the 0-[dB] line are referred to as “cross-over frequen-

cies” to emphasize the MIMO nature of the problem.7The eigenvalues of L(jω) are also called characteristic loci


is the variable to be kept small. Notice that measurement noise is omitted from this figure,because it is not needed to formulate the four-block control structure in (3.16). The controlstructure shown in Figure 3.5, however, is in general not able to deal with inferential servoproblems and an extra control DOF needs to be included. The reason for this is twofold.

C Pyprz upu

du

ey

Pz

zp ez

Figure 3.5: Single-DOF control structure applied to an inferential servo problem. Noticethat ey can only be determined if zp and yp have the same dimensions.

First, the reference signal rz is specified in terms of the performance variables zp. Hence,the subscript z for rz. A useful servo error ey can therefore only be determined if dim(yp) =dim(zp) and if vectors yp and zp contain the same type of scalar variables, i.e., subtractingvariables with the same units. Second, even if a useful servo error can be determined, the linearcontroller C in Figure 3.5 is not able to achieve good tracking and disturbance attenuationsimultaneously, as is exemplified next for a SISO controller.

Example 3.11. Consider a similar motion system as depicted in Figure 6.3, but assume

P (0) = 1 and Pz(0) = α. (3.23)

When using the single-DOF control structure depicted in Figure 3.5, two situations can bedistinguished.

1. Suppose that the controller needs to achieve good responses in terms of yp. It is well-known from the internal model principle that to guarantee zero steady-state errors fora unit step in rz and for a unit step in du, an integrator needs to be included in thecontroller. Recall from Section 3.3.2 that

yp = Trz + SPdu. (3.24)

In case C is internally stabilizing and contains an integrator, then application of a unitstep in rz leads to

yp = rz for t→ ∞, (3.25)

because T (0) = 1 − S(0) = 1. In a similar fashion, applying a unit step in du leads to

yp = 0, (3.26)

since S(0)P(0) = 0. Clearly, the integrator in C achieves good responses in terms of yp

with respect to reference tracking and disturbance attenuation. For the performancevariable zp, the response is given by

zp = Tzrz + PzSdu, (3.27)


where

Tz = αT. (3.28)

In case C is internally stabilizing and contains an integrator, applying a unit step in rzyields

zp = α for t→ ∞, (3.29)

since Tz(0) = α. Hence, if α 6= 1, a steady-state error remains after application of astep in reference input. Similarly, application of a unit step in du yields

zp = 0 for t→ ∞, (3.30)

since Pz(0)S(0) = 0. Concluding, including an integrator in the controller in generaldoes not lead to perfect reference tracking if the performance variable is not measured.Therefore, keeping S small does not imply good inferential performance.

2. To resolve the above deficiencies, one can modify C such that C(0) = 1α−1 . So, C does

not contain integral action if α 6= 1, i.e., if Pz(0) 6= P (0). It then follows immediatelythat for a unit step on rz

zp = 1 for t→ ∞, (3.31)

since Tz(0) = 1. Similarly, applying a unit step in du leads to

zp = Pz(0)S(0) = α

(1 − 1

α

)= α− 1 for t→ ∞. (3.32)

Hence, if the performance variable is not measured good tracking and disturbance atten-uation cannot be achieved simultaneously if a single-DOF control structure is used.

Although only steady-state errors are considered in Example 3.11, similar results can bederived for other frequencies. In addition, notice that in the first situation T (0) = I andS(0) = 0, while the mapping relevant for inferential performance, i.e., Tz : rz → zp, is givenby Tz = αT . Standard closed-loop TFMs, like, e.g., S and T , which are suitable for evaluatingperformance in case yp = zp, are thus inadequate for evaluating the inferential performance.

Remark 3.12. In Van de Wal (2009), the need for a two-DOF control structure is alsoestablished but the motivation is given by including a FF controller Cff (see Figure 1.2).For standard FF design, plant knowledge is used to steer the system. For conventional servoproblems this means Cff = P−1 if no knowledge about the external disturbances is included.By the same reasoning, Cff = P−1

z for inferential servo problems. However, this choice leadsto ez 6= 0 if Pz 6= P . Hence, an extra control DOF is needed.

Although a two-DOF control structure is essential to appropriately deal with general inferen-tial servo problems, other controller design methodologies for single-DOF control structuresmay also lead to performance improvements. Basically, all model-based controller design pro-cedures in which the performance is evaluated at zp should be able to improve the inferentialperformance. Besides using a two-DOF control structure, two other possibilities are:

1. Using a model Pz to evaluate the performance during manual loop-shaping.


2. Using norm-based controller design in the single-DOF control structure whereby ez isincluded as an (additional) exogenous output variable.

Remark 3.13. Regarding the second possibility, it follows from inspecting the structure of thecontroller, which is not the same as the control structure, that ez needs to be reconstructed inC. This is, however, only partly possible, because C is not able to appropriately differentiatebetween rz, which is an exogenous input, and yp, which is caused by plant dynamics.

3.8.2 Alternative control structures for inferential servo problems

To deal with inferential servo problems, two natural extensions of the single-DOF controlstructure are proposed:

• The indirect control structure (Figure 3.6(a)), where an intermediate reference signalis constructed (ry = Carz) for yp, such that a conventional FB controller (Cb) can bedesigned to keep ey small (Skogestad and Postlethwaite, 2005, Section 10.4). Controlblock Ca acts as an “input shaper” and needs to be stable to achieve internal instability.

• The inferential control structure (Figure 3.6(b)), where a FB controller Cb acts on anestimate ez of ez. The extra control block then acts as a filter zp = Ca(u, yp) to estimate(infer) zp (Parrish and Brosilow, 1985). Hence, Ca is referred to as a performance filterfrom here on.

Cb Pyprz uey

Pz

zp ez

ryCa

(a) Indirect control structure

Cb Pyprz uez

Pz

zp ez

Ca

(b) Inferential control structure

Figure 3.6: Alternative control structures for inferential servo control.


To reveal that these structures can resolve the deficiencies of the single-DOF control structure,consider the following example, which is a continuation of Example 3.11. Also notice that thestructures in Figure 3.6 can deal with the general multivariable case where dim(yp) 6= dim(zp).

Example 3.14. Two cases need to be considered: indirect control and inferential control.

1. Consider the control structure depicted Figure 3.6(a) and let Cb contain the integratorand let Ca = 1

α . Then, it can easily be verified that Tz : rz → zp satisfies

Tz(0) = αT (0)1

α= 1. (3.33)

Hence, application of a unit step in rz yields

zp = 1 for t→ ∞. (3.34)

Similarly, the response to a unit step in du is given by

zp =Pz

1 + PCbdu. (3.35)

Since Pz(0)1+P (0)Cb(0)

= 0, applying a unit step in du leads to

zp = 0 for t→ ∞. (3.36)

2. Consider the control structure depicted Figure 3.6(b) and let Cb contain the integratorand let Ca =

[0 α

]. Then, it can easily be verified that Tz : rz → zp satisfies

Tz(0) =αP (0)Cb(0)

1 + αP (0)Cb(0)= 1. (3.37)

Hence, application of a unit step in rz yields

zp = 1 for t→ ∞. (3.38)

In addition, the response to a unit step in du is given by

zp =Pz

1 + αPCbdu. (3.39)

Since Pz(0)1+αP (0)Cb(0)

= 0, applying a unit step in du yields

zp = 0 for t→ ∞. (3.40)

Clearly, both control structures in Figure 3.6 can appropriately deal with inferential servoproblems. Similar to conventional controller design, controller blocks Ca and Cb need to bedesigned such that a certain level of robustness is obtained. In case of the indirect controlstructure, input shaper Ca needs to be stable and exhibits no robustness to model uncertain-ties, because it is placed outside the FB loop. Imperfections in the mapping Ca : rz → ry thatare caused by uncertainties in the plant models of P and Pz are therefore not measured and


no correcting control action can be taken. In addition, how to select a stable and dynamicfilter Ca such that the reference for yp leads to good performance at zp is also unclear atpresent. Hence, this structure is not investigated further in this research.

In case of the inferential control structure, classical observers, like, e.g., a Kalman filter (Kalmanand Bucy, 1961), can be used to determine zp under the influence of measurement noise andmodel uncertainties. Using the inferential control structure therefore leads to a certain levelof robustness and allows Ca to be designed using well-established observer design methodolo-gies. Hence, this structure is preferred over the indirect control structure. In case a perfectestimate of zp can be obtained, i.e., zp = zp, controller block Cb could be designed as if zp ismeasured directly.

As simple version of the inferential control structure is the single-DOF control structurewhereby the static sensor transformation matrix Ty (see Appendix D) is not considered to bepart of the plant, but part of the controller. The control structure is depicted in Figure 3.7 andis commonly used at Philips Applied Technologies for the control of high performance stages.For such systems, the measured variables yp are the translations at the sensor locationsand the matrix Ty is used to estimate zp, i.e., the translations and rotations of the pointthat is subject to light exposure, from the sensor data. This estimation, however, is basedon geometric relations (rigid body assumption) and therefore not valid if internal dynamicsbecomes relevant.8 By incorporating a model of the internal dynamics in the estimation, amore accurate estimate of zp can be obtained, which potentially improves performance.

C Pyprz uez

Pz

zp

Ty

zp

C2DOF

ez

Figure 3.7: Two-DOF control structure that is commonly used within Philips Applied Tech-nologies for the control op high performance stages.

Unfortunately, the indirect and inferential control structure lead to structured control prob-lems in the standard plant setup, which cannot be dealt with by using standard optimal con-troller synthesis algorithms. This problem is also observed in Yaesh and Shaked (1991) and Hol(2006). To enable the use of standard controller synthesis algorithms to solve inferential servoproblems, a general two-DOF controller (see Figure 3.8)

u =[Ca Cb

]︸︷︷︸

, C

[rzyp

], (3.41)

is proposed in Oomen et al. (2009), which includes the indirect and inferential control struc-tures as special cases.

8The notation used in Figure 3.7 is slightly different from the notation used in Chapter 7 and Appendix D,because Ty is usually seen as part of the plant. Variables zp and yp as used here are therefore the same as yp

and yp, respectively, in Chapter 7 and Appendix D.


3.8.3 Formulating a nine-block control problem

Next, exogenous inputs and outputs have to be selected to ensure that the resulting controlleris useful. The controller should be internally stabilizing and enforce tracking of a reference rzin the presence of disturbances and measurement noise. A meaningful selection of unweightedexogenous inputs and outputs is given in Oomen et al. (2009) and is depicted in Figure 3.8.Here, r3 represents the reference signal rz, r2 represents measurement noise η, and r1 repre-sents disturbances at plant input. The exogenous output variable z contains the inferentialservo error ez, plant input up to penalize large control efforts, and measured plant outputsyp to ensure internal stability. In Figure 3.8, Tr is a stable reference model and is generallyneeded to ensure a sensible problem formulation (to avoid plant inversion in Ca).

C Pyp

r3upu

r1

r2

Pz

zp ez

Tr

Figure 3.8: General two-DOF control structure for inferential servo problems.

This choice of unweighted exogenous outputs and inputs leads to the mapping M : w → z,where

M =

Pz

PI

(I + CbP )−1

[Ca Cb I

]−

Tr 0 00 0 00 0 0

, (3.42)

w =

r3r2r1

, z =

ezyp

up

.

To formulate the inferential control problem such that it can be dealt with using standardalgorithms, the structure shown in Figure 3.8 is cast into the standard plant formulation

[zy

]= G(P,Pz)

[wu

], with y =

[r2 − yp

r3

]. (3.43)

The final step is to include diagonal weighting matrices Wz (z = Wz z) and Ww (w = Www)to specify performance specifications. Incorporating these weights yields the criterion

J(P,Pz , C) = ‖M(P,Pz , C)‖∞ = ‖WzM(P,Pz , C)Ww‖∞, (3.44)

leading to the following control problem

Copt = arg minCJ(P,Pz , C). (3.45)

In Oomen et al. (2009), this control problem is referred to as the nine-block control problem.Enforcing an H∞ norm in (3.45) clearly enforces internal stability and enables the incorpo-ration of model uncertainty in the design procedure. Notice that the blocks M22, M23, M32,


and M33 in (3.42) are the same as in (3.16). The four-block control problem is thus part ofthe nine-block control problem. In Oomen et al. (2009), additional guidelines are given toweigh the other blocks in M .

Since the nine-block control problem is relatively new, no clear guidelines and tools to tuneWz and Ww are available at present. More research and experimental work is needed before itcan be successfully applied to control complex systems, like, e.g., flexible wafer stages. Recallthat in the inferential control structure both controller blocks can be designed independentlyusing classical controller (for Cb) and observer (for Ca) design methodologies. Hence, theinferential control structure is adopted in Chapters 6 and 7 for the control of two flexiblemotion systems.

Remark 3.15. For regulator problems, rz = 0 and (3.41) reduces to

u = Cb (r2 − yp) , (3.46)

where r2 can be regarded as measurement noise or a reference signal for yp. So, for inferentialregulator problems no extra control DOF is needed.

3.9 Conclusions

The main results from this chapter are summarized below:

• Motion control problems where the performance variables are not measured are referredto as inferential servo problems. The reference is hereby specified for the performancevariables zp, where zp can either be a single point, set of points, or a spatial norm.

• I/O selection is important for flexible motion systems, since it determines the extentto which the flexible dynamics is actuated, sensed, and observed in the performancevariables. Creating a stiff relation between the actuators and sensors therefore does notimply a stiff relation between the actuators and the performance variables. The optimalactuator/sensor configuration strongly depends on the system (and its disturbances).

• The single-DOF control structure and the well-known four-block control problem arenot able to deal with inferential servo problems and an extra control DOF needs to beadded (research issue 1). Standard performance indicators, like, e.g., S and T , are thusalso inadequate to evaluate the inferential performance.

• The indirect and inferential control structures are proposed as alternatives for the single-DOF control structure. In the inferential control structure, the extra controller block canbe designed using well-established observer design methodologies, like, e.g., a Kalmanfilter. In doing so, robustness to model uncertainty and measurement noise is also takeninto account to some extent.

• Both the indirect and inferential control structures, however, lead to structured controlproblems in the standard plant setup and, as a consequence, the resulting controllersynthesis problem cannot be dealt with by standard optimal controller synthesis al-gorithms. To enable the use of these standard algorithms, a more general two-DOFcontroller (see (3.41)) needs to be adopted that leads to the nine-block control problem.

Chapter 4

H∞ loop-shaping

In this chapter, H∞ loop-shaping is discussed and proposed as an alternative for solving thefour-block control problem by H∞ optimization as discussed in Chapter 3 (research issue 2).The main difference is that H∞ optimization is concerned with specifying weighting filters forthe closed-loop TFMs, while H∞ loop-shaping, which was originally introduced by McFarlaneand Glover (1990), is about shaping the open-loop singular values. In doing so, knowledgefrom manual loop-shaping can be used and dependencies between the closed-loop TFMs aretaken into account. H∞ loop-shaping therefore delivers a more straightforward approach fordesigning MIMO controllers for systems whereby the performance variables are assumed to bemeasured. The method also has proven its value in various other applications, see, e.g., Hydeand Glover (1993) for the design of a flight controller for a Harrier jet. In Section 4.1, themotivation for H∞ loop-shaping is discussed in detail and Sections 4.2 – 4.4 explain thecontroller design procedure. Subsequently, design examples are given in Section 4.5 and theseexamples are used in Section 4.6 to derive guidelines for tuning controllers.

4.1 Motivation for loop-shaping

Consider the single-DOF control structure depicted in Figure 3.1. The principle idea be-hind “loop-shaping” is that the singular values of the closed-loop TFMs can be determineddirectly from the singular values of the open-loop. Designing a controller that achieves cer-tain specifications in terms of the closed-loop TFMs, can therefore be done by appropriately“shaping” the open-loop singular values. The relation between the open-loop singular valuesand six well-known closed-loop TFMs are shown below, whereby the following singular valueidentities and inequalities are invoked

σ(A) =1

σ(A−1), (4.1a)

σ(AB) ≤ σ(A)σ(B), (4.1b)

σ(A+B) ≤ σ(A) + σ(B), (4.1c)

σ(A+B) ≥ σ(A) − σ(B). (4.1d)

39

40 H∞ loop-shaping

1. The sensitivity S is related to the open-loop singular values by

σ((I + PC)−1

)=

1

σ(I + PC)

≤ 1

σ(PC) + 1

≈ 1

σ(PC)for frequencies where σ(PC) ≫ 1. (4.2)

2. The input sensitivity SI is related to the open-loop singular values by

σ((I + CP )−1

)=

1

σ(I + CP )

≤ 1

σ(CP ) + 1

≈ 1

σ(CP )for frequencies where σ(CP ) ≫ 1. (4.3)

3. The process sensitivity SP is related to the open-loop by the following inequality

σ((I + PC)−1P

)= σ

{((PC)−1 + I

)−1(PC)−1 P

}

≤ σ(C−1

)

σ ((PC)−1 + I)if C is square

≤ σ(C−1

)

1 − σ ((PC)−1)=

σ(C−1

)

1 − (σ(PC))−1

≈ 1

σ(C)for frequencies where σ(PC) ≫ 1. (4.4)

4. The control sensitivity CS is related to the open-loop singular values by

σ(C(I + PC)−1

)≤ σ(C)

σ (I + PC)≤ σ(C)

1 − σ (PC)

≈ σ(C) for frequencies where σ(PC) ≪ 1. (4.5)

5. The complementary sensitivity T is related to the open-loop by the following inequality

σ((I + PC)−1PC

)≤ σ(PC)

σ (I + PC)≤ σ(PC)

1 − σ (PC)

≈ σ(PC) for frequencies where σ(PC) ≪ 1. (4.6)

6. The input complementary sensitivity TI is related to the open-loop singular values by

σ((I + CP )−1CP

)≤ σ(CP )

σ (I + CP )≤ σ(CP )

1 − σ (CP )

≈ σ(CP ) for frequencies where σ(CP ) ≪ 1. (4.7)

4.2 The design procedure 41

It is well-known from manual loop-shaping that the magnitudes of S, SI and SP should besmall for good servo performance and disturbance attenuation, while the magnitudes of Tand CS should be small to achieve good robustness against uncertainties and measurementnoise. Inspection of (4.2)- (4.7) shows that the requirement of good performance (σ(PC),σ(CP ), σ(C) large) is in conflict with the robustness requirement (σ(PC), σ(CP ), σ(C)small), which demonstrates the well-known design trade-off between performance and robuststability. Clearly, arbitrarily good performance and robustness cannot be achieved over allfrequencies. However, an acceptable compromise can be formulated, since performance is typi-cally important at low frequencies, while model uncertainties and measurement noise typicallyemerge at high frequencies. Suppose the frequency domain is divided into three parts, asillustrated in Figure 4.1. Controller C should then be designed such that for ω ∈ [0, ωl),σ(PC) ≫ 1, and for ω ∈ (ωh,∞), σ(PC) ≪ 1.

Furthermore, recall from SISO loop-shaping that the Bode gain/phase relation of the open-loop is important near the cross-over frequency (|PC| ≈ 1). As was already shown by Bode(1945), the rate of transition from high to low gain is limited by phase requirements. Theinterval (ωl, ωh) in Figure 4.1, can therefore not be arbitrarily small. Doyle and Stein (1981)generalized this limitation to MIMO systems and showed that a similar limitation existsfor the roll-off rate of the magnitude of the open-loop eigenvalues. The control engineer istherefore required to manipulate σ(PC) and σ(PC) to achieve the desired loop shape, whilerestricting the roll-off rate of the eigenvalues of PC around the cross-over frequencies.

log ω

0 dBωl

ωh

σ(PC)

σ(PC)

Figure 4.1: Shaped open-loop singular values.

4.2 The design procedure

This section discusses the controller design procedure as it was originally presented in McFar-lane and Glover (1990). It incorporates the relatively simple performance/robustness trade-offin loop-shaping, with the guaranteed stability properties of H∞ design methods.


The H∞ loop-shaping design procedure

1. Loop-shaping (Figure 4.2(a)): using a post-compensator W1 and/or a pre-compensatorW2 to shape the singular values of the (nominal) plant P . The plant P , and “shapingfilters”1 W1, W2 are combined to form the “shaped plant” Ps, where

Ps = W2PW1. (4.8)

It is important that Ps does not contain hidden unstable modes.

2. Robust stabilization (Figure 4.2(b)): synthesize a FB controller C∞ that robustly stabi-lizes the shaped plant Ps. This process always results in a stabilizing controller.

3. Final controller (Figure 4.2(c)): the final FB controller C can then be constructed bycombining C∞ with the shaping filters W1 and W2 such that

C = W1C∞W2. (4.9)

4.2.1 Shaping filter selection

The main difference with H∞ optimization is that H∞ optimization is concerned with speci-fying weighting filters for the closed-loop TFMs, while H∞ loop-shaping is about shaping theopen-loop singular values. In doing so, knowledge from manual loop-shaping can be used anddependencies between the closed-loop TFMs, like, e.g., S + T = I, are taken into account.Moreover, in contrast with classical SISO loop-shaping, H∞ loop-shaping is done without ex-plicit regard of plant phase information, because the robust stabilization step always results ina stabilizing controller. Closed-loop stability requirements are thus not regarded at this stage.

The first step is to scale the plant inputs and outputs. In general, scaling improves the condi-tioning of the problem and it enables a meaningful comparison between outputs (or inputs).This is especially true if outputs (or inputs) are measured in different units. Decouplingmatrices, which are often used to facilitate controller design (see Appendices A and E), canalso be included here.

Next, a set of filters is specified to shape the open-loop singular values. There is a lot ofdesign freedom in this step, but it is appealing to use diagonal filters. This allows everyinput and output to be weighted separately. For most motion systems the filters are chosento achieve integral action at low frequencies (ω ∈ [0, ωl)), roll-off rates of approximately 20[dB/decade] (corresponding to a -1 slope in a Bode magnitude diagram) around the the cross-over frequencies, and a higher roll-off rate at high frequencies (ω ∈ (ωh,∞)). In section 4.5,examples are presented that illustrate the effect of different shaping filters and the results aresummarized in Section 4.6.

Some procedures, see, e.g., the procedure presented by Hyde and Glover (1993), place theintegral action in W1 and roll-off action in W2. However, the claimed advantages of these

1Filters W1 and W2 are here referred to as shaping filters to distinguish them from the closed-loop weightingfilters Ww and Wz. In literature, however, W1 and W2 are usually also referred to as weighting filters.

4.2 The design procedure 43

P W2W1

(a) The shaped plant

P W2W1

Ps

C∞

(b) Robust stabilization

P

W2W1

C

C∞

(c) Final controller

Figure 4.2: The H∞ loop-shaping procedure.

design choices are only valid when the controller is implemented in a so called “observer-based” form, which is depicted in Figure 4.3. In this structure, the measured outputs yp arefirst filtered by W2 (filtering out measurement noise) and the error signal ey is filtered firstby W1 (smoothing the plant input signals). In general, this leads to a smoother response andless overshoot. However, this structure is only beneficial when no FF signal is used. Sincea FF controller is always present in high performance motion systems, these choices for W1

and W2 are not adopted in this research. In Section 4.5, the effect of placing certain filters ineither W1 or W2 is also illustrated.

W1 Pry

C∞ W2

C∞(0)W2(0)u yp

Figure 4.3: Observer based implementation form.


As a final optional step, the singular values can be aligned at a desired cross-over frequencyby adding an additional constant non-diagonal weighting matrix in W1 and/or W2. In doingso, all singular values of PC obtain the same gain at a certain frequency, which effectivelymeans that the shaped plant is approximately decoupled around the BW. Alignment of theopen-loop singular values is useful if diagonal shaping filters do not lead to the desired open-loop singular values. In literature, see, e.g., Skogestad and Postlethwaite (2005, p. 371), it isrecommended that singular value alignment should not be used if the plant is ill-conditioned,because decoupled ill-conditioned plants are in general not robust to input uncertainty.

Remark 4.1. For some systems, diagonal weighting filters do not lead to the desired open-loop singular values. Therefore, in Papageorgiou and Glover (1997) a systematic procedureis presented for designing non-diagonal weighting filters such that the singular values can beshaped independently. This procedure, however, is beyond the scope of this research.

4.2.2 Robust stabilization

In the second step, the shaped plant is robustly stabilized. Consider the robust stabilizationof a plant P (the subscript s is left out for notational convenience), which has a normalized2

left coprime factorization

P = M−1N. (4.10)

A perturbed plant model, which contains the nominal plant and all possible plants, can thenbe written as

Pp = (M + ∆M)−1 (N + ∆N ) , (4.11)

where ∆M and ∆N are stable norm-bounded unknown TFMs that represent the uncertainty inthe nominal plant model P . This type of uncertainty description is called coprime uncertainty.Although this description seems less intuitive, it is practically more useful than other morecommon descriptions, like, e.g., additive or multiplicative uncertainty, which both can bewritten in terms of coprime uncertainty. Moreover, since uncertainty blocks are includedfor both the plant numerator and denominator, (4.11) can be used to describe a perturbedplant set where poles and zeros are allowed to cross the imaginary axis, which is not possiblewith, e.g., an additive uncertainty description. The objective of robust stabilization is then tostabilize all plants in the set of perturbed plants given by

Pp ={(M + ∆M )−1 (N + ∆N ) :

∥∥[ ∆N ∆M

]∥∥∞ < ǫ

}, (4.12)

where ǫ > 0 is the stability margin (see Figure 4.4). Maximization of ǫ is the problem of robuststabilization of coprime factor plant descriptions as introduced and solved in McFarlane andGlover (1990, Chapter 4).

2Left and right coprime factorizations are not unique and are therefore often normalized as discussed inChapter 2. The left coprime factorization is normalized if NNH + MMH = I .

4.3 Relation with H∞ optimization 45

The FB system in Figure 4.4 is stable, if and only if the nominal FB system is stable and if

∥∥∥∥[PI

](I − CP )−1 [ C I

]∥∥∥∥∞

≤ γ, (4.13)

where γ , ǫ−1 and (I−CP )−1 the input sensitivity function in this positive FB arrangement.In contrast to most H∞ optimization problems, the controller leading to γmin, i.e., the lowestvalue of γ over all stabilizing controllers, can be determined directly (no γ-iteration needed)by solving two algebraic Riccati equations. For a more thorough treatment of the calculationof C, see, e.g., McFarlane and Glover (1990, Chapter 4). For this research, it is sufficient tostate that the stabilizing controller C and γmin can be obtained directly by using the commandncfsyn in the Matlab Robust Control Toolbox (Balas et al., 2008).

∆N

N M−1

∆M

C

Pp

Figure 4.4: Robust stabilization of (normalized) coprime factor plant descriptions.

4.3 Relation with H∞ optimization

The H∞ loop-shaping procedure and the four-block control problem discussed in Chapter 3are strongly related. To see this more clearly, consider the control criterion (3.10) with Mgiven by (3.16). Weighting filters Wz and Ww are hereby used to shape the desired closed-loopfunctions, whereas in H∞ loop-shaping W1 and W2 are used to shape the open-loop. Thefollowing proposition shows how these filters are related.

Proposition 4.2. Open-loop weighting filter design and robust stabilization of the shapedplant matches the minimization of the control criterion J(P,C) in (3.10), with a particularselection of closed-loop weights, i.e.,

J(Ps, C∞) = ‖M(Ps, C∞)‖∞ = ‖WzM(P,C)Ww‖∞, (4.14)

with Wz = diag(W2,W

−11

)and Ww = diag

(W−1

2 ,W1

).

Proof. The proof follows directly from substituting Ps = W2PW1 and C∞ = W−11 CW−1

2 into


M(Ps, C∞). This leads to

M(Ps, C∞) =

[Ps

I

](I + C∞Ps)

−1 [ C∞ I]

=

[W2PW1

I

] (I +W−1

1 CW−12 W2PW1

)−1 [W−1

1 CW−12 I

]

=

[W2PW1

I

]W−1

1 (I + CP )−1W1

[W−1

1 CW−12 I

]

=

[W2P

W−11

](I + CP )−1 [ CW−1

2 W1

]

=

[W2 0

0 W−11

]

︸︷︷︸Wz

[PI

](I + CP )−1 [ C I

] [ W−12 00 W1

]

︸︷︷︸Ww

, (4.15)

which concludes the proof.

Remark 4.3. The weighting filters used in H∞ optimization can only contain bi-stable TFMs.It is therefore not possible to use pure integrators and pure roll-off filters. In H∞ loop-shaping,these restrictions do not exist and the only limitation is that W1 and W2 should be chosensuch that no unstable pole/zero cancellations occur when forming Ps.

4.4 Using γmin as performance parameter

The parameter γmin can be seen as a measure of success of the loop-shaping procedure. Asa rule of thumb, γ ≤ 4 for successful controller design (McFarlane and Glover, 1990) andγmin ≫ 4 always indicates incompatibility between the achieved open-loop and the specifiedopen-loop, and low stability margins as is discussed next.

4.4.1 Stability

Regarding stability, γmin ≥ 1 is a measure of robustness for a level of coprime uncertaintyand should be as small as possible. If, for example, γmin = 4, the controlled system cantolerate 25% of coprime uncertainty. For SISO controller design, γmin = 4 corresponds toapproximately 45 [◦] PM and 6 [dB] GM (Papageorgiou and Glover, 1999). However, toevaluate for robust stability and robust performance, evaluating γmin is not sufficient andrealistic uncertainty models are needed. These uncertainty models can then be included inthe problem formulation as shown in Figure 3.2(b).

4.4.2 Guarantees on the achieved open-loop shape

Besides being a measure of robust stability, γmin also indicates how well the open-loop approx-imates the desired open-loop, i.e., the shaped plant. To show this, bounds on the singularvalues of L and LI are derived and compared with the singular values of the shaped plant.

4.4 Using γmin as performance parameter 47

In the low frequency range (ω ∈ [0, ωl)), the deterioration of L can be obtained by comparingσ(L) with σ(Ps), i.e.,

σ(PC) = σ(PW1C∞W2) ≥ σ(W2PW1)σ(C∞)/c(W2) = σ(Ps)σ(C∞)/c(W2), (4.16)

where c(·) denotes the frequency dependent condition number. In a similar fashion, σ(LI) iscompared with σ(Ps), i.e.,

σ(CP ) = σ(W1C∞W2P ) ≥ σ(W2PW1)σ(C∞)/c(W1) = σ(Ps)σ(C∞)/c(W1). (4.17)

In the high frequency range (ω ∈ (ωl,∞)), the deterioration of L can be obtained by comparingσ(L) with σ(Ps), i.e.,

σ(PC) = σ(PW1C∞W2) ≥ σ(W2PW1)σ(C∞)/c(W2) = σ(Ps)σ(C∞)/c(W2). (4.18)

Similarly, the deterioration of LI can be obtained by comparing σ(LI) with σ(Ps), i.e.,

σ(CP ) = σ(W1C∞W2P ) ≥ σ(W2PW1)σ(C∞)/c(W1) = σ(Ps)σ(C∞)/c(W1). (4.19)

Notice that the condition numbers c(W1) and c(W2) are selected by the designer and are com-monly of order one. Therefore, σ(C∞) (at low frequencies) and σ(C∞) (at high frequencies)are needed to obtain bounds in the deterioration of the loop shape. Expressions for σ(C∞)and σ(C∞) are given as a function of γ by the following lemma.

Lemma 4.4. If σ(Ps(jω)) ≫√γ2 − 1, then

σ(C∞(jω)) &1√γ2 − 1

, (4.20)

where “&” denotes “asymptotically greater than or equal to as σ(Ps(jω)) → ∞”. If σ(Ps(jω)) ≪1√

γ2−1, then

σ(C∞(jω)) .√γ2 − 1, (4.21)

where “.” denotes “asymptotically less than or equal to as σ(Ps(jω)) → 0”.

Proof. See McFarlane and Glover (1990, p. 110–117).

Lemma 4.4 shows that at frequencies where σ(Ps(jω)) ≫ 1 or σ(Ps(jω)) ≪ 1, the deteriora-tion in loop shape due to C∞ is bounded by γmin. A low value of γmin indicates a tighter lowerbound in (4.16) and (4.17) and a tighter upper bound in (4.18) and (4.19). This confirmsthat γmin is also a measure for the compatibility between the specified loop shape and theachieved loop shape.


4.5 Design examples: H∞ loop-shaping applied to a two MSD

system

In this section, the H∞ loop-shaping procedure is used to design controllers for a two MSDsystem. The main objectives are to illustrate the process of shaping filter selection and to showthe effect of the robust stabilization step. Since the same system is discussed in Chapter 6, thecontrollers Ci from that chapter, which are designed by manual loop-shaping, are sometimesused as shaping filters in this section to show the effect of the robust stabilization step. Thecorresponding controllers resulting from robust stabilization are then denoted by CH

i .

Consider the two MSD system shown in Figure 4.5 with control input u1. Either y1 (collocatedcontrol), y2 (non-collocated control), or y1 and y2 can be used for FB control. The systemparameters are given in Appendix C and the Bode diagrams of the TFs between u and y1

(P1) and u and y2 (P2) are given in Figure 4.6.

m1 m2

u

y1 y2

k

d

Figure 4.5: Two MSD system with input u and outputs y1 and y2.

101

102

103

−180

−160

−140

−120

−100

−80

−60

Magnitude

[dB

]

101

102

103

−400

−300

−200

−100

0

Phase

[◦]

Frequency [Hz]

Figure 4.6: Bode diagram of P1 (thin line) and P2 (thick line).

4.5.1 SISO controller design

For SISO controller design PC = CP and therefore there is no difference between shapingthe open-loop with W1 or W2 as long as the controller is implemented as a single block C andnot in separate blocks as, for example, shown in Figure 4.3.

4.5 Design examples: H∞ loop-shaping applied to a two MSD system 49

Collocated control

Consider the two MSD system of Figure 4.5 where y1 is used for FB control. To illustratethe effect of the robust stabilization step, consider static shaping filters

W1 = 1, W2 = 78.9, (4.22)

that lead to a desired cross-over frequency of 1 [Hz]. In Figure 4.7(a), the Bode diagrams of theshaped plant Ps and the resulting open-loop L are depicted. Clearly, the robust stabilizationalgorithm creates sufficient PM to “robustly” stabilize the system. Recall that although theperformance parameter γmin = 2.61 < 4 implies successful controller design, robust stabilityand robust performance can only be evaluated by using uncertainty models. In Figure 4.7(b),the controller resulting from H∞ loop-shaping (CH) is compared to controller C1, which isgiven by (6.1) and designed by manual loop-shaping for a BW of 1 [Hz]. Both controllersare quite similar and the only difference, besides the small mismatch in static gain, is theamount of phase lead. If controller C1 were used as a shaping filter, the open-loop wouldcloser resemble the shaped plant, implying a lower value for γmin. Although not shown here,γmin would decrease to γmin = 1.85 for CH

1 .

10−1

100

101

102

103

−150

−100

−50

0

50

Magnitude

[dB

]

10−1

100

101

102

103

−200

−150

−100

−50

0

Phase

[◦]

Frequency [Hz]

Ps

L

(a) Ps and L

10−2

10−1

100

101

102

25

30

35

40

45

50

Magnitude

[dB

]

10−2

10−1

100

101

102

0

20

40

60

Phase

[◦]

Frequency [Hz]

CH

C1

(b) CH and C1

Figure 4.7: Bode diagrams of the shaped plant and open-loop (left) and controllers CH andC1 (right).

Because the desired BW is well below the resonance frequency, the robust stabilization al-gorithm only weakly addresses the flexible dynamics. To better illustrate the effect of therobust stabilization step on the flexible dynamics, controller C2 (500 [Hz] BW), which is givenby (6.2), is used as shaping filter for W2 and W1 = 1. The controller resulting from robuststabilization (CH

2 ) is compared to C2 in Figure 4.8. Both controllers are alike, but CH2 has

and additional resonance/anti-resonance pair with the poles located in the ORHP. Hence, thephase is shifted 360 [◦] compared to C2.


101

102

103

120

130

140

150

160M

agnitude

[dB

]

101

102

103

0

100

200

300

400

Phase

[◦]

Frequency [Hz]

(a) Controllers

101

102

103

−20

0

20

40

Magnitude

[dB

]

101

102

103

−200

0

200

400

Phase

[◦]

Frequency [Hz]

(b) Open-loop

Figure 4.8: Bode diagrams of the controllers (left) and the corresponding open-loop (right)for the system controlled with C2 (thin line) and CH

2 (thick line).

Without robust stabilization3, γmin = 3.78. Nevertheless, the additional resonance/anti-resonance pair created by the robust stabilization step leads to γmin = 1.83, implying betterrobustness for coprime uncertainty. To see this more clearly, the closed-loop TFs of thefour-block control problem are depicted in Figure 4.9. For CH

2 , the resonance peaks in Sand T are lowered and slightly more damped, which is beneficial for the control criterionJ(Ps, C∞). Whether practical aspects might prevent the implementation of CH

2 is not anissue here. Despite the higher value of γmin, the manually shaped controller can therefore stillbe preferred.

As is well-known by now, H∞ loop-shaping minimizes a cost criterion J(P,C) whereby theperformance variable are assumed to be measured. Whether this approach works well if theperformance variables are not measured remains to be seen. To illustrate that a manuallyshaped controller can outperform the robust stabilization algorithm if the control objective isnot represented well by the cost criterion, consider controller C4, which is given by (6.4) anddesigned to maximize damping of the controlled system. More specifically, the goal of C4 is tomaximize the dimensionless damping ratio (ζ) of the least damped pole pair of the controlledsystem. If C4 is used as shaping filter, the robust stabilization algorithm, in which y1 acts asperformance variable, leads to controller CH

4 with γmin = 1.86, whereas C4 has γmin = 2.47.Although γmin is lowered by the robust stabilization algorithm, C4 leads to a better dampedclosed-loop system. This is confirmed by Figure 4.10, where the Bode diagrams of T and theresponses to a unit step in reference are depicted for both controllers. The settling time ismuch lower for C4 (better damped), but using C4 also leads to more overshoot in y1 (higherpeak value in T ).

3The value for γmin can be obtained by filling in C∞ = 1 into the control criterion J(Ps, C∞), which isgiven by (3.44)


101

102

103

−60

−50

−40

−30

−20

−10

0

10M

agnitude

[dB

]

Frequency [Hz]

(a) Sensitivity

101

102

103

70

80

90

100

110

120

130

140

150

Magnitude

[dB

]

Frequency [Hz]

(b) Control sensitivity

101

102

103

−150

−145

−140

−135

−130

−125

−120

Magnitude

[dB

]

Frequency [Hz]

(c) Process sensitivity

101

102

103

−20

−15

−10

−5

0

5

10

Magnitude

[dB

]

Frequency [Hz]

(d) Complementary sensitivity

Figure 4.9: Bode magnitude diagrams of the TFs included in the four-block control problemfor the system controlled with C2 (thin line) and CH

2 (thick line).

The main difference between these two controllers is that the goal of C4 is maximize dampingdamping of the controlled system and thus basically uses both y1 and y2 as performancevariables. For the four-block control problem solved here (collocated control), only y1 is usedas performance variable, implying that increased performance of y1 is allowed to come at thecost of decreased performance of y2. The lowest value for the cost criterion J(P,C) is achievedby decreasing the peak value in T (and S), rather than maximizing damping of the controlledsystem. In the next section, y2 is included as an additional performance variable and it isshown that the H∞ loop-shaping procedure then also leads to a more damped response. So,a lower γmin only implies improved performance is the control objective is represented well bythe cost criterion.

Non-collocated control

Consider the two MSD system of Figure 4.5 where y2 is used for FB control. The desired BWis placed below the resonance frequency and it is investigated how the robust stabilization


101

102

103

−40

−30

−20

−10

0

10

Magnitude

[dB

]ry → y1

101

102

103

−40

−30

−20

−10

0

10

Magnitude

[dB

]

Frequency [Hz]

ry → y2

(a) Complementary sensitivity

0 0.01 0.02 0.03 0.04 0.05 0.060

0.5

1

1.5

y 1

0 0.01 0.02 0.03 0.04 0.05 0.060

0.5

1

1.5

2

y 2

Time [s]

(b) Step response data

Figure 4.10: Bode magnitude diagrams of the closed-loop TF between reference r and outputsy1, y2 (left) and time responses of y1 and y2 to a unit step change in reference for the systemcontrolled with C4 (thin line) and CH

4 (thick line).

step deals with the finite GM that is caused by the resonance peak. Consider W1 = 1 andW2 a series connection of a constant gain Wy, an integrator, and a lead-filter, i.e.,

W2 = Wy ·s+ 2πfint

s︸︷︷︸Wint

·1

2πfdesc /α

s+ 1

12πfdes

c ·αs+ 1︸︷︷︸

Wlead

, fint =1

5· fdes

c , α = 3. (4.23)

The static gain parameter Wy is tuned such that the cross-over frequency fc is approximatelyequal to the desired cross-over frequency fdes

c , which is set at 20 [Hz]. The effect of robuststabilization is visualized in Figures 4.11 and 4.12. The lightly damped nature of the plantresonance is mainly included in the control criterion via the process sensitivity. For this exam-ple, minimizing this peak in SP can only be accomplished by locally lowering the sensitivity,i.e., creating a dip in S at the resonance frequency. As a result, damping is added: ζ = 0.0105in P (open loop) versus ζ = 0.0482 in SP (closed-loop). Because decreasing S implies movingfurther away from the point (-1,0) in the Nyquist diagram, this effect can also by observedin Figure 4.12(a) as a circle in the ORHP. The robust stabilization step realizes this circle byusing a complex NMP zero pair at 101 [Hz] to create the necessary phase lag.

By the same reasoning, the resonance peak in SP can be decreased further by extending W2

with an inverse notch filter at the resonance frequency. To illustrate this, two new controllersare designed using the shaping filters depicted in Figure 4.13. Inverse notch filter W 2

notch iswider and has a higher peak value than W 1

notch . Therefore, if W 2notch is included in W2, it

leads to a lower peak in SP than when W 1notch is used. Notice, however, that low peak values

in S or SP do not guarantee improved damping in general. In fact, multiple closed-looppole pairs emerge around 100 [Hz] and these pole pairs are more clearly visible in the Bode


diagram of the complementary sensitivity (see Figure 4.12(b)). Nevertheless, all closed-looppole pairs when using W 2

notch have a dimensionless damping ratio greater than ζ = 0.09 andhence overall damping is improved.

102

−130

−120

−110

−100

−90

−80

Magnitude

[dB

]

Frequency [Hz]

(a) Process sensitivity

102

−20

−15

−10

−5

0

5

Magnitude

[dB

]Frequency [Hz]

(b) Sensitivity

Figure 4.11: Bode magnitude diagrams of the process sensitivity (left) and sensitivity (right)resulting from shaping filter W2 (thin black solid line) and W2 extended with notch filterW 1

notch (thick black solid line) and W 2notch (thick grey solid line). The dashed curve in the left

figure represents the uncontrolled plant and is included to visualize the effect of control.

−2 0 2 4 6 8 10−6

−4

−2

0

2

4

Im

Re

(a) Nyquist diagram

102

−10

−5

0

5

Magnitude

[dB

]

Frequency [Hz]

(b) Complementary sensitivity

Figure 4.12: Nyquist diagram and Bode magnitude diagram when using shaping filter W2

(thin black line) and W2 extended with notch filter W 1notch (thick black line) and W 2

notch (thickgrey line).

4.5.2 MIMO shaping filters

Controllers for MIMO plants (possibly non-square) can also be designed using the sameshaping filter components used in SISO loop-shaping. Properties like integral action or roll-


100

101

102

103

40

45

50

55

60

65

70

Magnitude

[dB

]

Frequency [Hz]

Figure 4.13: Bode magnitude diagram of shaping filter W2 (thin black line) and W2 extendedwith an inverse notch filter W 1

notch (thick black line) and W 2notch (thick grey line).

off then need to be specified at the plant input (in W1) or the plant output (in W2), becausefor general MIMO systems PC 6= CP .

Consider the system of Figure 4.5, where both y1 and y2 are used for FB control. To illustratethe effect of locating filters in W1 or W2, first consider integral action. It follows directly fromsimple matrix multiplication that placing an integrator in W1, which is a SISO filter, isequivalent to placing integrators on both diagonal elements of W2 (2 × 2). Since integratorsare needed to suppress constant disturbances, adding integrators on both diagonal elementsof W2 thus means that it is desired to eliminate the steady-state errors in y1 and y2. However,since only one actuator is available, this can never be achieved and a trade-off exists. This isconfirmed by Figure 4.14, where the singular values of SI and S are depicted for a controllerthat is designed using a diagonal shaping filter W2 with the diagonal elements given by (4.23)and fdes

c = 20 [Hz]. It follows from visual inspection that there is integral action at the plantinput (suppressing constant input disturbances), but only limited integral action at the plantoutput, since σ(S) → c, c > 0, for f → 0. The orthogonal input directions of σ(S) and σ(S)at 1 [Hz] are given by

v1 ≈ 1√2

[1 −1

]Tand v2 ≈ 1√

2

[1 1

]T, (4.24)

respectively. From (4.24) it follows that constant disturbances can only be suppressed if theywork on y1 and y2 simultaneously. For example, using the same reference signal for bothy1 and y2. This is not surprising from a practical point of view, since control actions usedto eliminate the steady-state error in output y1 also decrease the steady-state error in y2.Constant disturbances that work on y1 and y2 with equal magnitude but opposite sign arethus impossible to suppress. Therefore, in order to end up with a meaningful optimizationproblem, W2 is only allowed to contain a single integrator (either for y1 or y2).

Remark 4.5. If an extra actuator is placed on m2, it is possible to completely control m1 andm2. Although this situation is rather trivial, it is now possible to suppress constant outputdisturbances with arbitrary directions. A static spring deformation is thus also allowed.

For roll-off filters, a similar reasoning can be applied as for integral action. There is, however,


100

101

102

103

−30

−25

−20

−15

−10

−5

0

5

10M

agnitude

[dB

]

Frequency [Hz]

(a) Input sensitivity

100

101

102

103

−30

−20

−10

0

10

Magnitude

[dB

]

Frequency [Hz]

(b) Output sensitivity

Figure 4.14: Bode magnitude diagrams of the singular values of SI (left) and S (right).

one difference. Unlike with integral action, actuator deficiency (a similar reasoning is valid forsensor deficiency) does not prevent roll-off for all output directions. This is understandablefrom a practical point of view, since roll-off for one output is generally not in conflict withroll-off for another output. Conflicts may occur if, e.g., different amounts of roll-off are neededfor different outputs.

For lead filters, the reasoning is somewhat different than for manual loop-shaping. Recallthat the open-loop singular values are shaped without taking stability issues into account.Specifying sufficient phase lead is therefore not necessary. It, however, often leads to lowervalues of γmin, because the open-loop closer resembles the shaped plant. The effect of usingdifferent lead filters on the main diagonal of W2 or W1 is therefore similar to specifying thestatic gains of the diagonal elements in W2 and W1. The only difference is that the magnitudeof a static gain filter is constant over all frequencies, while the magnitude of a lead filter isnot. Hence, using lead filters to shape the open-loop singular values is not recommended andvarying the static gains of the diagonal elements of W1 and W2 is suggested as more insightfulalternative.

For SISO controller design, the static gains of W1 or W2 (Wy in (4.23)) are used to specifythe desired BW. For MIMO systems, the concept of BW does not exist and a generalizationis often made in terms of cross-over frequencies of the open-loop singular values (Doyle andStein, 1981). The diagonal entries of W2 and W1 can then be used to specify the relativecontribution of particular inputs and outputs to these singular values. Practically, this meansspecifying the relative importance of the outputs (for W2) and actuator preference (for W1).

To illustrate the effect of specifying relative output importance, consider shaping filtersW1 = 1 and

W a2 =

[C4 00 C4

], W b

2 =

[3 · C4 0

0 C4

], W c

2 =

[C4 00 3 · C4

]. (4.25)

Controller C4 (SISO) is given by (6.4) and is designed to maximize damping for a collocatedactuator/sensor pair. In Figure 4.15, the closed-loop time responses of y1 and y2 are compared


for a simultaneous unit step in reference. Because y2 is included as an extra measured output,the controllers resulting from the H∞ loop-shaping procedure lead to a much more dampedresponse than in Figure 4.10(b), where only y1 is taken as performance variable. Hence,adding damping seems an obvious solution if global performance is required. Increasing theimportance of y1, which is done by using W b

2 , leads to a smaller overshoot for y1, but a largerovershoot for y2 and an overall less damped response. Increasing the importance of y2, whichis done by using W c

2 , also leads to less damping, but m2 is accelerated faster and has lessovershoot. To realize larger accelerations for m2, larger spring forces are needed, which canonly be accomplished by larger displacements of m1.

0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

y 1

0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

y 2

Time [s]

(a) y1 more important

0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

y 1

0 0.01 0.02 0.03 0.040

0.5

1

1.5

2

y 2

Time [s]

(b) y2 more important

Figure 4.15: Closed-loop time responses of y1 and y2 for a simultaneous unit step in referencefor shaping filter W a

2 (thin black line), W b2 (thick black line), and W c

2 (thick grey line).

4.6 Guidelines for shaping filter selection

In this section, guidelines are presented that should be kept when applying H∞ loop-shaping.It is hereby assumed that the controller is implemented in the single-DOF control structuredepicted in Figure 3.1.

• For SISO controller design, there is no difference between shaping the open-loop withW1 or W2.

• For general MIMO systems properties need to be specified at the plant input (in W1)and output (in W2) side. First, the plant needs to be scaled properly. Next, a setof filters is specified to shape the open-loop singular values. There is a lot of designfreedom in this step, but it is appealing to use diagonal filters to allow every input andoutput to be weighted separately. Increasing the gain of a specific output (or input)leads to a stronger contribution from all plant entries related to this specific output (or

4.7 Conclusions 57

input) to the singular values of the shaped plant. For integral action, this allows thedesigner to specify integral action for a subset of outputs in case of actuator deficiency.

• The shaping filters should be chosen such that the shaped plant does not contain un-stable hidden modes, since the robust stabilization step is not able to stabilize thesemodes.

• Choosing shaping filters involves some trial-and-error. Recall from Section 3.4.2 thatweighting filters can be selected in two ways: in a signal-based approach and in a loop-shaping-based approach. The former approach is often not adopted for various reasonsand weighting filters (in case of H∞ optimization) and shaping filters (in case of H∞loop-shaping) are chosen to roughly specify the loop-shape. For H∞ loop-shaping, thisnormally means integral action at low frequencies, roll-off rates of approximately 20[dB/decade] near the cross-over frequencies, and a higher roll-off rate at high frequen-cies. If there are more stringent performance requirements, fine-tuning is essential.Adding notches, applying error-based tuning (see Chapter 7), or using other fine-tuningtechniques (see Chapter 8) are then needed to improve performance.

• If controllers designed by manual loop-shaping are used as shaping filters for H∞ loop-shaping, the design parameter γmin is usually lowered, implying more robustness forcoprime uncertainty. However, it should be kept in mind that this type of robustnessdoes not have to be relevant for the system to be controlled. For example, uncertaintyin a certain plant parameter that is not represented well by coprime uncertainty may bemore detrimental. To evaluate for robust stability and robust performance, evaluatingγmin is therefore not sufficient and realistic uncertainty models are needed.

• For general norm-based controller design, minimization of a cost criterion does notnecessarily mean that a controller has been synthesized that is optimal in terms of trulyrelevant performance objectives. The major difficulty is to formulate the optimizationproblem such that it represents the actual control objectives. A lower value of γmin

therefore does not necessarily imply better performance.

4.7 Conclusions

H∞ loop-shaping is an alternative for solving the four-block control problem by H∞ optimiza-tion as discussed in Chapter 3. The main difference is that H∞ optimization is concernedwith specifying weighting filters for the closed-loop TFMs, while H∞ loop-shaping is aboutshaping the open-loop singular values. This has several advantages:

• Dependencies between the four closed-loop TFMs, like, e.g., the analytical constraintS + T = I, are taken into account.

• Knowledge from manual loop-shaping can be used.

• The shaping filters needed to shape the open-loop singular values do not have to bebistable.


Because of these advantages, H∞ loop-shaping is proposed as an alternative for the approachtaken in Van de Wal (2002). The main shortcoming for the control of flexible motion systemsis that, just as with solving the four-block control problem, H∞ loop-shaping also assumesthat the performance variables are measured. For controlling flexible motion systems, it cantherefore only be used to solve part of the nine-block control problem (see Section 3.8.3) orto design Cb in the indirect or inferential control structure (see Figure 3.6).

Chapter 5

Fundamental performancelimitations

This chapter gives an overview of fundamental performance limitations (research issue 3) inthe conventional control structure (Figure 1.2) and the standard plant setup (Figure 1.3).Following Seron et al. (1997), the word “fundamental” is used here to indicate that themain focus is on the so-called interpolation and integral constraints. Limitations caused bydisturbances, uncertainty, causality of the controller, and various practical issues, like, e.g., afinite sampling time, are thus not considered. It is, however, pointed out that mathematicalformulae can often also be derived for these type of limitations, see, e.g., Skogestad andPostlethwaite (2005, Chapters 5 and 6).

Sections 5.1 and 5.2 discuss the fundamental performance limitations in the conventionalcontrol structure for SISO and MIMO systems, respectively. The theory of design limitationsis well-established for SISO systems and the original work is due to Bode (1945). Around 25years ago, starting with Freudenberg and Looze (1985, 1986, 1988), the theory has beenextended to MIMO systems, but the resulting mathematical formulae are much less insightfulthan for SISO systems.

In Section 5.3, some remarks are made about design limitations in MIMO systems arisingfrom the non-square nature of a plant. Although this topic is not treated in detail, it isimportant to be aware of the problem, since in practice plants are often not square.

In Section 5.4, fundamental performance limitations for the standard plant setup are intro-duced. The focus is hereby limited to NMP zeros, because the systems of interest generallydo not have unstable poles, except pure integrators. Obtaining insights in these limitationsis considered especially relevant, since limitations in the standard plant setup may severelylimit the actual system performance (evaluated in zp), even when there are no limitations onthe measured variables (yp). The reverse is also true. Since the standard plant setup is muchnewer than the conventional control structure, the work regarding these limitations is alsomore recent and the first work is due to Hong and Bernstein (1996, 1998).

Examples are included at the end of each section to demonstrate the theory that is presented.Additional information needed to reproduce the examples is included in Appendix C.

59

60 Fundamental performance limitations

5.1 Fundamental limitations in SISO systems

5.1.1 S plus T is one

From the definitions of the sensitivity S and complementary sensitivity T , which are givenby (3.2), it follows that

S + T = 1. (5.1)

It is therefore impossible to have small S and T at the same time and the difference between|S(jω)| and |T (jω)| at any frequency is at most one for SISO systems.

5.1.2 Interpolation constraints

As a consequence of internal stability (Theorem 3.4), S and T must satisfy the followinginterpolation constraints:

• If p is a CRHP pole of P (s), then

S(p) = 0, and T (p) = 1. (5.2)

• If q is a CRHP zero of P (s), then

S(q) = 1, and T (q) = 0. (5.3)

In other words, the plant characteristics of instability and “nonminimum phaseness” aretranslated into properties that S and T have to satisfy. It is also possible to formulateinterpolation constraints for the open-loop L. However, the locations of the poles and zerosof L are partly under the influence of the control engineer (by choosing C) and hence theseconstraints are not considered fundamental.

5.1.3 Bode integrals for S and T

Analytical limitations are often given by integral relations, such as the Bode sensitivity inte-gral of Theorem 5.1.

Theorem 5.1 (Bode integral for S). Let S be the sensitivity function as defined in (3.2)and let {pi : i = 1, . . . , np} be a set of poles in the ORHP of the rational open-loop L. Then,assuming internal stability,

∫ ∞

0ln

∣∣∣∣S(jω)

S(j∞)

∣∣∣∣ dω =π

2lim

s→∞s[S(s) − S(∞)]

S(∞)+ π

np∑

i=1

pi. (5.4)

Proof. See Seron et al. (1997, p. 54).

5.1 Fundamental limitations in SISO systems 61

If L has at least two more poles than zeros, which is always true in practice due to actuatorand measurement dynamics1, Corollary 5.2 follows directly from Theorem 5.1.

Corollary 5.2. Let S be the sensitivity function as defined in (3.2) and let L have at leasttwo more poles than zeros. Then, assuming internal stability,

∫ ∞

0ln |S(jω)| dω = π

np∑

i=1

pi. (5.5)

Proof. Since the relative degree of L is two or more, the first term in the Right Hand Side(RHS) of (5.4) disappears (Seron et al., 1997, p. 54).

Corollary 5.2 shows that reducing S in one frequency range (disturbance attenuation), mustlead to an increase of S at other frequencies (disturbance amplification), as is illustrated inFigure 5.1. This effect is more commonly known as the “waterbed effect”: pushing the beddown on one side results in an increase somewhere else, or as the conservation of “sensitivitydirt”: dirt obtained from “digging” in |S| in one frequency region can only be shifted to otherfrequencies. In case L does not contain poles in the ORHP, the areas of sensitivity decrease(light grey) and sensitivity increase (dark grey) in Figure 5.1 are equally large if ω would beplaced on a linear axis. In theory, (5.4) could be satisfied over a wide frequency range, i.e.,the maximum achievable BW is only limited by practical aspects.

log ω0 dB

|S(jω)|

Figure 5.1: Illustration of the waterbed effect.

If unstable open-loop poles are present, the RHS of (5.5) is positive, implying that the areaof sensitivity increase (|S| > 1) exceeds the area of sensitivity decrease (|S| < 1). Usuallythis increases the peak of S. Unstable poles located further away from the imaginary axisare more detrimental, which seems plausible because part of the control effort is aimed atstabilizing the system.

Similar integral relations exist for the complementary sensitivity T and ORHP zeros of L.The role of the ORHP poles of L is now taken by the ORHP zeros of L, and vice versa, ascan be seen by writing T as

T =L

I + L=

1

1 + 1L

. (5.6)

1In case of force inputs and position outputs, which is common for most motion systems, the dynamicsalready causes P to have two more poles than zeros.


Theorem 5.3 (Bode integral for T). Let T be the complementary sensitivity function asdefined in (3.2). Let L have two more poles than zeros and let {qi : i = 1, . . . , nq} be the setof zeros of L in the ORHP. Then, assuming internal stability,

∫ ∞

0ln

∣∣∣∣T (jω)

T (0)

∣∣∣∣1

ω2dω = π

nq∑

i=1

1

qi. (5.7)

Proof. See Seron et al. (1997, p. 57) or Goodwin et al. (2001, p. 248).

ORHP zeros located closer to the imaginary axis lead to a larger RHS of (5.7), which generallymeans a higher peak value in T (jω).

5.1.4 Poisson integrals for S and T

Theorem 5.1 and Corollary 5.2 show the effect of ORHP poles of L on S in the absenceof ORHP zeros. If L also contains ORHP zeros, the sensitivity function must satisfy anadditional constraint, which has a stronger implication on the peak value of S. Recall fromthe interpolation constraints that ORHP poles and zeros of L become zeros of S and T ,respectively. Let {qi, i = 1, . . . , nq} be the set of ORHP zeros of L and let {pi, i = 1, . . . , np}be the set of ORHP poles of L. The ORHP zeros of S and T are then given by the sets ZS

and ZT , respectively, where

ZT = {qi : i = 1, . . . , nq}, (5.8a)

ZS = {pi : i = 1, . . . , np}. (5.8b)

Next, the Blaschke products of the ORHP zeros of S and T are introduced as

BS(s) =

np∏

i=1

pi − s

pi + s, and BT (s) =

nq∏

i=1

qi − s

qi + s, (5.9)

where pi and qi denote the complex conjugates of pi and qi, respectively. The Blaschkeproducts are all-pass filters, because their magnitude is always equal to one on the imaginaryaxis. The Blaschke products can be used to factorize L, S, and T as

L(s) = L(s)B−1S (s)BT (s), (5.10a)

S(s) = S(s)BS(s), (5.10b)

T (s) = T (s)BT (s), (5.10c)

where L, S, and T have no zeros in the ORHP and, L also has no ORHP poles. In addition,if the closed-loop system is internally stable, S and T also have no poles in the CRHP. Thecombined effect of ORHP poles and zeros on S can then be described by Theorem 5.4.

Theorem 5.4 (Poisson integral for S). Let S be the sensitivity function as defined in (3.2)and let the ORHP poles and zeros of L be given by (5.8). Then, assuming internal stability,

∫ ∞

−∞ln |S(jω)| · w(qi, ω) dω = π ln |B−1

S (qi)| ∀qi ∈ ZT , (5.11)


where for a zero of the form q = x+ jy, w(qi, ω) is given by

w(q, ω) =x

x2 + (y − ω)2. (5.12)


Remark 5.5. An equivalent representation of (5.11) in which the integral is evaluated overpositive frequencies only, is given by (Skogestad and Postlethwaite, 2005, Theorem 5.2)

∫ ∞

0ln |S(jω)| · w(qi, ω)dω = π ln |B−1

S (qi)| ∀qi ∈ ZT , (5.13)

with

w(q, ω) =x

x2 + (y − ω)2+

x

x2 + (y + ω)2. (5.14)

In the absence of ORHP poles, the term BS is defined to be one and the RHS of (5.11) iszero. The only remaining difference with (5.5) is then the weight w(q, ω) within the integrand.This weight effectively cuts off the contribution from ln |S(jω| to the integral at frequencieswhere ω > |q|. Zeros in the ORHP with a lower undamped eigenfrequency are thus moredetrimental. Effectively, the weight w(q, ω) poses an upper bound on the achievable BW as isillustrated in Figure 5.2.

log ω0 dB

|S(jω)|

Figure 5.2: Illustration of the poisson integral constraint.

The waterbed effect is thus still in effect, except that the trade-off between |S| < 1 and |S| > 1must be satisfied over a limited frequency range, i.e., the size of the waterbed is limited. Alarge peak in |S| is therefore unavoidable if |S| needs to be small at low frequencies. Forpositive zeros (real zeros in the ORHP), the upper bound on the BW can be felt intuitivelyby the initial undershoot behavior (see Appendix A). The response first moves in the “wrong”direction and a low BW controller effectively ignores this effect. A high BW controller,however, will try to correct this movement, which moves the system even further into thewrong direction.

Remark 5.6. As shown in Appendix A, increasing the FB gain causes the closed-loop polesto move from their open-loop locations towards the open-loop zero locations. Hence, open-loopzeros in the CRHP pose an upper bound on the maximum FB gain.


Due to the RHS of (5.11), the amount of disturbance attenuation is further impaired if theopen-loop also contains an ORHP pole. If an ORHP pole pi is close to an ORHP zero qi, thenpi+qi

pi−qiis large. Hence, plants with one or several close ORHP pole-zero pairs are difficult to

stabilize. In the extreme situation of unstable pole/zero cancellations, i.e., pi and qi coincide,the plant contains unobservable and/or uncontrollable modes and stabilization is impossible.

Analogous to Theorem 5.4, a Poisson integral can be derived to describe the effects of ORHPpoles in L on the complementary sensitivity. This integral is given by Theorem 5.7.

Theorem 5.7 (Poisson integral for T). Let T be the complementary sensitivity functionas defined in (3.2) and let the ORHP poles and zeros of L be given by (5.8). Then, assuminginternal stability,

∫ ∞

−∞ln |T (jω)| · w(pi, ω)dω = π ln |B−1

T (pi)| ∀pi ∈ ZS , (5.15)

where, for a complex pole of the form p = a+ jb, w(pi, ω) is given by

w(p, ω) =a

a2 + (b− ω)2. (5.16)


As is well-known, S is required to be small at low frequencies, whereas T needs to be smallat high frequencies. It follows from Theorem 5.7 that if excessive peaking of T at low andintermediate frequencies is to be avoided, T (jω) may only be made small at frequencies thatare lower bounded by the open-loop ORHP poles. Practically, this means that the controllershould react sufficiently fast to stabilize the instabilities. In addition, the achievable reductionof T is further impaired if the plant has ORHP zeros. Similar to the Poisson integral for S,this effect is large when one or several ORHP pole-zero pairs are very close.

Remark 5.8. Theorems 5.4 and 5.7 also hold when L has poles and zeros on the imaginaryaxis. For proof, see Seron et al. (1997, p. 66).

5.1.5 Bode gain-phase relation

The Bode gain-phase relation uniquely relates the phase to the slope in magnitude of a stableMinimum Phase (MP) system H for all frequencies ω. This relation is given by

phase(H) = slope(H) · 90 [◦]. (5.17)

The phase and gain of a controller can therefore not be designed independently, leading tofundamental performance limitations. If, for example, roll-off in magnitude is required athigh frequencies, the phase also decreases leading to a lower PM or even instability.

A single OLHP zero leads to a +1 change in slope and thus +90 [◦] change in phase. Similarly,an OLHP pole leads to a −1 change in slope and thus a -90 [◦] change in phase. In case ofNMP zeros and unstable poles, the reverse is true: NMP zeros lead to an increase in slope,but a decrease in phase, and unstable poles lead to a decrease in slope, but an increase inphase.


5.1.6 LHP zeros

Zeros in the OLHP are often not considered as a fundamental performance limitations, becausethey can (at least in theory) be cancelled by the controller without causing internal instability.Recall, however, that the closed-loop poles mitigate towards the open-loop zeros when the FBgain is increased. OLHP zeros therefore limit arbitrary placement of the closed-loop poles.In Chapter 6 and in Van Wingerden (2004), the effect of this non-arbitrary placement onthe inferential performance is illustrated for a simple two MSD system and a flexible beamsystem, respectively. It is shown that high performance in the measured variables does notguarantee that the actual performance is good.

5.1.7 Design examples: limitations due to a NMP zero

Flexible cart system

To illustrate the performance limiting effect of a NMP zero, consider the flexible cart systemdepicted in Figure 5.3. Either output y1 or y2 (both translations) is used for feedback control.Although not shown, dampers are placed in parallel with the springs.

u

k, d k, d

y2

y1

x

φ

l

Figure 5.3: Flexible cart system.

The parameter l denotes the width of the block, m is the mass, I is the moment of inertia,k is the stiffness of the springs, d is the absolute damping coefficient, x is the horizontaltranslation, and φ is the rotation of the block around its Center of Gravity (COG). Forsimplicity it is assumed that the cart is square and the COG is exactly in the middle of thebody. In addition, it is assumed that the actuator, sensors, springs, and dampers are locatedat the corners of the block. The non-linear equations of motion and output equations arethen given by (C.1) in Appendix C and can be linearized by assuming small rotations φ. Thisleads to the following TFs

y1

u= P1(s) =

1

ms2− 1

4

l2

Is2 + 12dl

2s+ 12kl

2, (5.18a)

y2

u= P2(s) =

1

ms2+

1

4

l2

Is2 + 12dl

2s+ 12kl

2. (5.18b)

Because the horizontal input force u also creates a moment around the COG, u also causes theblock to rotate around its COG. If y1 initially move in the reverse direction (initial undershoot)


as a result of this rotation, the system has a single real NMP zero. It depends on the systemparameters whether this initial undershoot occurs. For this example, the parameters aregiven in Table C.1. Plant P1 is NMP with the zeros located on the real axis at s ≈ −81and s ≈ 88. If y2 is used for feedback control, the system is MP and has a complex zeropair s ≈ −0.75 ± 39j. Both plants have two poles at the origin and a complex pole pair ats = −1.9 ± 62j (poles are system properties and thus invariant under actuator and sensortransformations).

In Figure 5.4(a), the FRFs of both plants are depicted. Notice that the FRF for P1 does notsatisfy (5.17) due to the NMP zero. The combined effect of the two real zeros in P1 is a +2slope starting from approximately 13 [Hz], but no change in phase. Hence, the complex polepair causes the phase to drop 180 [◦]. Although P1 and P2 seem quite similar from a mechanicalpoint of view, the maximum achievable performance is very different. The minimum phasesystem P2, which has an anti-resonance/resonance pair, can never be destabilized by increasingthe feedback gain of a standard PD-controller,2 since the closed-loop poles will never moveinto the RHP. Conversely, for P1, increasing the feedback gain of a PD-controller eventuallyleads to instability.

To allow for a fair comparison between P1 and P2, controllers

C1 = 4.5 · 103 ·1

2π·5s+ 11

2π·50s+ 1·

12π·5s+ 1

12π·50s+ 1

·1

612 s2 + 2·0.03

61 s+ 11

612 s2 + 2·0.561 s+ 1

, (5.19)

and

C2 = C1 ·2π · 2 · s+ 1

2π · 2 · s , (5.20)

are used to control both systems. Both controllers are depicted in Figure 5.4(b) and consistof a series connection of a gain, two lead filters, and notch filter. The lead filters are usedto create PM and the notch filter to cancel the resonance at 61 [rad/s]. The notch filter isnecessary for P1, since the combination of an increase in magnitude (due to the resonancepeak) and loss in phase leads to an unstable closed-loop system. Controller C2 includes anadditional weak integrator to deal with steady state errors.

By using the Nyquist stability criterion, see, e.g., Franklin et al. (2002, Section 6.3), closed-loop stability can be analyzed by considering the open-loop. In Figure 5.5, the Nyquistdiagrams are depicted for the four plant and controller combinations. Clearly, P2C1 andP2C2 cannot be destabilized by increasing the feedback gain, because P2C1 and P2C2 nevercross the line between the origin and the point (−1, 0) (infinite GM). For P1, the GM is finiteand increasing the feedback gain eventually leads to a passing of the point (-1,0) at the leftside, which means closed-loop instability. Because the phase of P1 goes to -360 [◦] for ω → ∞,the use of causal controllers will always lead to a finite GM.

To illustrate the effect of the Bode and Poisson integral constraint for S, the sensitivityfunctions are depicted in Figure 5.6. For P2, (5.5) holds, which implies equal areas (if ωwould be placed on a linear axis) between |S| and the 0 dB line for |S| < 1 and |S| > 1.Pushing |S| down at low frequencies leads to an increase in |S| around 4 [Hz] and 100 [Hz].Adding an integrator thus also leads to a higher peak value of |S|. For P1, (5.13) shouldbe satisfied. This implies that, due to the inclusion of an extra term (see Figure 5.7) in the

2Practical effects like, e.g., time delay due to sampling, are not considered in this reasoning.


10−1

100

101

102

−150

−100

−50

0M

agnitude

[dB

]

10−1

100

101

102

−400

−300

−200

−100

0

Frequency [Hz]

Phase

[◦]

P1

P2

(a) Bode diagram of P1 and P2

10−1

100

101

102

60

80

100

120

Magnitude

[dB

]

10−1

100

101

102

−200

−100

0

100

200

Frequency [Hz]P

hase

[◦]

C1

C2

(b) Bode diagram of C1 and C2

Figure 5.4: Bode diagrams of the plants and controllers.

−3 −2 −1 0 1−2

−1.5

−1

−0.5

0

0.5

1

Im

Re

(a) For the open-loop of P1

−2 −1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

Im

Re

(b) For the open-loop of P2

Figure 5.5: Nyquist diagrams of the open-loop for the system controlled with C1 (thin line)and C2 (thick line).

integrand, the Bode sensitivity integral has to be satisfied over a limited frequency range. Asa result, the peaks in |S| are much higher.

Notice that S and L are related by S−1 = (I+L). Therefore, |S(jω)|−1 is equal to the distancebetween L(jω) and the point (-1,0) in a Nyquist diagram. A large peak value of S thusimplies a close approximation of the point (-1,0) in a Nyquist diagram. If |S(jω)| < 6 [dB],L(jω) will never enter a circle with radius 0.5 around the point (-1,0). To some extent,Bode diagrams of S and Nyquist diagrams give the same information. On the other hand,the integral relations also give extra information that cannot be retrieved from the Nyquistdiagrams. For example, consider plant P1, which is subject to the Poisson integral constraint.


100

101

102

103

−40

−20

0

20

40

Frequency [Hz]

Magnitude

[dB

]

(a) Sensitivity for P1

100

101

102

103

−40

−20

0

20

40

Frequency [Hz]

Magnitude

[dB

]

(b) Sensitivity for P2

Figure 5.6: Bode magnitude diagrams of S for the system controlled with C1 (thin blackline) and C2 (thick black line). The grey lines allow for an easy comparison between the twodiagrams.

10−1

100

101

102

103

0

0.01

0.02

0.03

Frequency [Hz]

w(q,ω

)[-]

Cutt-offfrequency

Figure 5.7: Weighting function w(q, ω) for a real RHP zero at s ≈ 88.

An experienced “loop-shaper” might falsely conclude that it is possible to get a very high BWby pushing L away from the point (-1,0). The Poisson integral constraint, however, statesthat the well-known sensitivity integral should be satisfied over a limited frequency range.Hence, for NMP systems there will always be a trade-off between robustness and performancein a limited frequency range and the peak(s) in |S| can only be made lower by decreasingperformance, i.e., increasing |S|, at lower frequencies.

Remark 5.9. A seemingly attractive solution to get rid of the NMP zero in P1 is to cancel itby an unstable pole. Although this approach eliminates the peak in S, it also causes internalinstability.


m1 m2

u

y1 y2

k

d

Figure 5.8: Two MSD system with input u and outputs y1 and y2.

Two mass-spring-damper system

Consider the two MSD system shown in Figure 5.8. For m1 = m2 = m, the TF between uand y1 is given by

y1

u= P1(s) =

ms2 + ds + k

m2s4 + 2dms3 + 2kms2, (5.21)

and the TF between u and y2 by

y2

u= P2(s) =

ds+ k

m2s4 + 2dms3 + 2kms2. (5.22)

If either y1 or y2 is used for FB control, there are no fundamental performance limitations and,at least in theory, the BW can be infinitely high. This can easily be verified by inspection ofthe FRFs that are depicted in Figure 4.6. For P1 (collocated control), the phase never dropsbelow -180 [◦] and for P2 (non-collocated control), a controller can be designed such that thephase of the open-loop is well above -180 [◦] around the cross-over frequency.

Suppose that a new output z is created as an affine combination of outputs, i.e.,

z = y2 − α(y2 − y1) =[α 1 − α

]︸︷︷︸

Ty

[P1

P2

]u = Pzu. (5.23)

For α = 0, Pz reduces to P2 and the system only has a single OLHP zero. The effect of thiszero manifests itself at high frequencies and physically this can be interpreted as the damperdominating the spring in transmitting force between the two masses. For α = 1, Pz reducesto P1 and the SISO system has a complex zero pair that manifests itself as an anti-resonancein the FRF of Figure 4.6.

The zeros of Pz are located at the locations of s where αP1+(1−α)P2 = 0. Using (5.21), (5.22),and equating the expression for Pz to zero leads to

Pz(s) = αP1 + (1 − α)P2 = 0,

= αms2 + ds+ k

m2s4 + 2dms3 + 2kms2+ (1 − α)

ds+ k

m2s4 + 2dms3 + 2kms2= 0,

= αms2 + ds+ ks = 0, (5.24)

which has solutions of the form

s1,2 =−d

2mα±

√D

2mα, with D = d2 − 4kmα. (5.25)


The zeros can then be given as a function of α:

• For α > d2

4mk , the system has a complex zero pair in the OLHP.

• For α = d2

4mk , the system has a real zero in the OLHP with multiplicity 2.

• For 0 < α < d2

4mk , the system has real zeros in the OLHP.

• For α = 0, the system reduces to P2 and the system has a single zero in the OLHP.

• For α < 0, the system has two real zeros that are symmetric with respect to the point−d

2mα . One zero is located in the ORHP and one in the OLHP. So, by combining outputsit is possible to create a NMP system, even when both original systems, i.e., P1 and P2,are MP.

Several controllers are designed using H∞ loop-shaping (see Chapter 4) with W2 = 1 andW1 = c, with c a constant value. If c is high, high gain feedback is requested. Integral action,roll-off, and phase lead are not specified, because that is is not relevant for the exampleconsidered here. The controller resulting from the robust stabilization step (see Section 4.2),however, is always stabilizing and may therefore not lead to the requested high gain feedback,i.e., Ps 6= L. Results are summarized in Table 5.1 for various values of c and α.

100

102

104

−150

−100

−50

0

50

Magnitude

[dB

]

Frequency [Hz]

Sc=102

Sc=103

Sc=104

Sc=105

Sc=106

Sc=107

(a) For α = 0

100

102

104

−150

−100

−50

0

50

Magnitude

[dB

]

Frequency [Hz]

(b) For α = −10

Figure 5.9: Bode magnitude diagrams of the sensitivity for varying values of c.

The controllers designed for P1 (α = 1) have γmin < 4, which confirms that the collocatedsystem can be stabilized easily, since the phase never drops below -180 [◦]. The small variationsin γmin are most likely caused by the different amounts of phase lead in Ps. Due to the anti-resonance/resonance pair, phase lead is present over the entire frequency range, but theamount of phase differs at each frequency. P2 (α = 0) can also always be stabilized. However,placing the BW in the frequency region where the plant phase is -360 [◦] (a high BW controller)is more difficult (and often impossible in practice), because the controller has to create a lot ofextra phase. This leads to lower stability margins and hence γmin is larger. In Figure 5.9(a),the sensitivity is plotted for plant P2 and various values of c. Since increasing c means thata higher FB gain is requested, |S| is lowered accordingly in the low frequency region.


Table 5.1: Zero locations and achieved performance γmin as a function of α

αZero locations Achieved γmin for

OLHP ORHP c = 102 c = 103 c = 104 c = 105 c = 106 c = 107

0 −3.01 · 104 - 2.61 2.62 2.64 3.01 7.26 11.8

−1 −443 449 2.61 2.62 2.69 4.06 17.3 96.1−10 −141 141 2.62 2.66 3.24 13.2 107 991−100 −44.6 44.6 2.62 3.12 11.4 103.7 1010 11577

1 −3.30 ± 446j - 2.61 2.61 2.59 2.52 2.50 2.58

Certain combinations of α and c lead to γmin ≫ 4, implying very low robustness and, atthe same time, a large difference between the desired open-loop (the shaped plant) and theachieved open-loop. The latter is visualized in Figure 5.10, where the open-loop is depictedfor various values of α and constant c = 104. For α = 0, there is no ORHP zero and the(achieved) open-loop closely approximates the shaped plant. Hence, the relatively low valueof γmin. Setting α < 0 introduces an ORHP zero and this, as discussed earlier, createsfundamental performance limitations. Since the amount of disturbance attenuation at lowfrequencies is limited, the open-loop is not able to closely approximate the shaped plant.Decreasing α shifts the real zeros towards the imaginary axis, creating an even more stringentlimitation. Hence, the values of γmin increase with decreasing values of α. The effect of theORHP zero is also visualized in Figure 5.9(b), where the sensitivity is depicted for α = −10and varying values of c. Clearly, the FB gain at low frequencies cannot be increased infinitelydue to the ORHP zero, whereas there are no limitations for α = 0 (see Figure 5.9(a)).

100

101

102

103

104

−80

−60

−40

−20

0

20

40

60

Magnitude

[dB

]

Frequency [Hz]

Figure 5.10: Bode magnitude diagram of the shaped plant (dashed line) and open-loop forα = −1 (thin black solid line), α = −10 (thick black solid line), and α = −100 (thick greysolid line) with shaping filter W1 = 104.


5.2 Fundamental limitations in MIMO systems

In this section, the performance limitations discussed in the previous section are generalizedto MIMO systems. Most of the results also hold for MIMO systems, but with additionalconsideration for directions. Two approaches exist to generalize the integral constraints forS and T to MIMO systems. One is in terms of singular values and one in terms of matrixelements. Analogous to Seron et al. (1997), the Bode integral for S is presented in terms ofsingular values and the Poisson integral for S in terms of elements of S. Integral relations forT are not further discussed.

In the following, it is assumed that the open-loop is given by (3.1), i.e., a cascade of the plantP and controller C. S and T are therefore defined at the plant output, but similar resultscan be obtained for SI , LI and TI .

5.2.1 S plus T is identity

From the definitions for S and T it follows that

S + T = I, (5.26)

which is the MIMO variant of (5.1). After some rearranging, (5.26) can be written as

|σ(S) − σ(T )| ≤ 1, (5.27)

implying that the magnitudes of σ(S) and σ(T ) differ by at most 1 at a given frequency.

5.2.2 Interpolation constraints

Similar as for SISO systems, S and T must satisfy the following constraints as a consequenceof internal stability (Skogestad and Postlethwaite, 2005, p. 223):

• If p is a CRHP pole of P (s) with output direction yp, then

S(p)yp = 0, and T (p)yp = yp. (5.28)

• If q is a CRHP zero of P (s) with output direction yq, then

yTq S(q) = yT

q , and yTq T (q) = 0. (5.29)

Similar constraints exist for LI , SI , and TI , but these are in terms of input pole and zerodirections.

5.2.3 Bode integrals for S

Several extensions of the Bode integrals for S and T exist, see, e.g., Chen (1995, 2000)and Seron et al. (1997). However, these relations are complex and often not very insightfulfrom a practical point of view. A possible extension in terms of singular values is given byTheorem 5.10

5.2 Fundamental limitations in MIMO systems 73

Theorem 5.10 (Bode integral for S). Let S be the (output) sensitivity function and let{pi : i = 1, . . . , np} be the set of poles in the ORHP of the rational open-loop L. If the largestsingular value of L has a roll-off rate of more than one pole-zero excess and if the closed-loopsystem is internally stable, then for all singular values σj , j = 1, . . . , n, of S(jω)

∫ ∞

0lnσj(S(jω)) dω = Fj +Kj , (5.30)

where

Fj =1

2

∫∫

C+

σ∇2 lnσj

(S(σ + jω)

)dσ dω,

Kj = limR→∞

∫ π/2

−π/2R lnσj

( np∏

i=1

B−1i

(Rejθ

))cos θ dθ,

with S the minimum phase part of S and Bi the all-pass factors needed for this factorization.

Proof. See Chen (1995) or Seron et al. (1997, p. 91).

Theorem 5.10 also implies an integral relation, see, e.g., Chen (1995), that uses the determi-nant of S(jω) and is given by

∫ ∞

0ln |det (S(jω)) | dω =

n∑

j=1

∫ ∞

0lnσj(S(jω)) dω =

n∑

j=1

Fj

︸︷︷︸=0

+π

np∑

i=1

pi, (5.31)

so that ∫ ∞

0lnσ(S(jω))dω ≥ 0 and

∫ ∞

0lnσ(S(jω))dω ≤ 0. (5.32)

Since the flexible motion systems of interest generally do not have ORHP poles, the term Kj

is generally zero. Theorem 5.10 and (5.31) show that there are two types of trade-offs inMIMO systems.

First, there exists a frequency wise trade-off that is also present in SISO systems. If the jth

singular value needs to be small (σj(S) < 1) in one frequency range, then it needs to belarge (σj(S) > 1) in another frequency range. In the absence of open-loop unstable poles,the Laplacian of lnσj, which is large if the direction of σj(S) changes rapidly in a smallfrequency region (Chen, 1995), determines how stringent this trade-off is. Since the largestsingular value of S can be regarded as a “worst-case” direction of disturbance attenuationand F1 ≥ 0, it can be concluded that the sensitivity trade-off is in general more rigorous forMIMO systems than for SISO systems.

In addition, it is also possible to exchange frequency wise trade-offs between singular values,because the term Fj can be non-zero. In Goodwin et al. (2001), this is referred to as a “spatialtrade-off”. More disturbance attenuation in the direction of the largest singular value of S(worst-case direction), therefore implies decreased attenuation in the directions of the othersingular values.


5.2.4 Poisson integral for S

For NMP systems, a generalization of Theorem 5.4 to MIMO systems is given by Theo-rem 5.11. This theorem holds for n× n open-loop systems with an ORHP zero at q = x+ jywith multiplicity νq. In case of multiple ORHP zeros, the integral constraint should be sat-isfied for all zeros whereby the zero with the lowest undamped eigenfrequency leads to themost stringent constraint.

Theorem 5.11 (Poisson integral for S). Consider the (output) sensitivity function S fora square n× n rational open-loop L, having an ORHP zero at s = q = x+ jy with associateddirections hT

1 , hT2 , . . . , h

Tνq

. Then, assuming internal stability, the sensitivity function satisfies

(i)

∫ ∞

−∞ln∣∣hT

i [S(jω)]∗c∣∣ ·w(q, ω) dω = π ln |hic[B

−1i (q)]cc|, c ∈ Hi, i = 1, . . . , νq, (5.33)

and

(ii)

∫ ∞

−∞ln∣∣hT

i [S(jω)]∗c∣∣ · w(q, ω) dω ≥ π ln |hic|, c ∈ Hi, i = 1, . . . , νq. (5.34)

wherew(q, ω) =

x

x2 + (y − ω)2,

with Hi the set of integers corresponding to the non-zero elements of output zero vector hi:Hi = {c | hic 6= 0 } and [S(jω)]∗c is the cth column of S.

Proof. See Goodwin et al. (2001, p. 782) or Seron et al. (1997, p. 99).

The term B−1i (q) is a diagonal matrix in which each diagonal entry is a scalar inverse Blaschke

product [B−1i (q)]jj , constructed such that ln (vi(s)), with vi(s) given by

vi(s) = hTi s(s)B

∗i (q) =

[vi1 vi2 · · · vin

], i = 1, 2, . . . , νq, (5.35)

is an analytic function in the ORHP. This means that

[B−1i (s)]jj =

cj∏

k=1

s+ pjk

s− pjk, (5.36)

where {pjk, k = 1, 2, . . . , cj} is the set of unstable poles of the cth element of the row vectorhT

i S(jω).

The MIMO integral relations are, as expected, more complex as for SISO systems due to thepresence of directions. In the special case of a decoupled plant, the Poisson integral and Bodeintegral constraint reduce to n times the scalar integral relations. From a practical point ofview, the Poisson integral for S shows that it is possible to shift the deteriorating effect of anORHP zero between outputs as long as hT

i has more than one non-zero element. If there isonly one non-zero element, the zero is pinned to a single output (pinned zero) and the effectcannot be shifted to another output. The absolute values of the elements in hT

i are a measureof how difficult it is to push the effect of an ORHP zero to a certain output k. If |hik| islarge compared to the other elements of hT

i , it is easy to push the effect of the ORHP zeroto this output. This is illustrated in Example 5.12, which is partly taken from Skogestad andPostlethwaite (2005, Section 6.6.1).


Example 5.12 (Moving zeros). Consider the 2 × 2 system

P (s) =1

(0.2s + 1)(s + 1)

[1 1

2s+ 1 2

], (5.37)

which has a transmission zero at q = 0.5 with output direction

h =1√5

[2−1

]≈[

0.89−0.45

]. (5.38)

It is desired to have perfect reference tracking, i.e., y = Tr, with T = I. Any allowable T mustalso satisfy the interpolation constraint hHT (q) = 0, see (5.29), and this poses the followingrelationships between the components of T

2t11(q) − t21(q) = 0, (5.39a)

2t12(q) − t22(q) = 0. (5.39b)

A dynamic decoupled design requires t21(s) = t12(s) = 0, and to satisfy (5.39), t11(q) =t22(q) = 0, which means that the ORHP zero is contained in both diagonal elements. In

other words, by requiring a dynamically decoupled system, the multiplicity of the

ORHP zero is increased. One possible choice, which also satisfies T (0) = I, is

T0(s) =

[ −s+qs+q 0

0 −s+qs+q

]. (5.40)

Two other possible choices are T1, in which output 1 is perfectly controlled, and T2, in whichoutput 2 is perfectly controlled. In literature, this is also referred to as “triangular decoupling”.With the requirement that T (0) = I, triangular decoupling leads to

T1(s) =

[1 0

β1ss+q

−s+qs+q

]and T2(s) =

[−s+qs+q

β2ss+q

0 1

]. (5.41)

To satisfy (5.39), β1 = 4 and β2 = 1. So, it is indeed possible to completely move the effectto one output. However, this comes at the cost of some interaction. The magnitudeof the interaction, expressed by β, is the largest for T1. This is plausible, because the zerodirection is mainly in the direction of y1. Pushing the effect to output 2 therefore “has ahigher cost”.

5.2.5 Design example: Bode sensitivity constraint in a MIMO system

Consider the two MSD system depicted in Figure 5.11 with control inputs u1 and u2, and

measured outputs y1 and y2. For m1 = m2 = m, the TFM between inputs u =[u1 u2

]T

and outputs y =[y1 y2

]Tis given by

P =

[P1 P2

P2 P1

]. (5.42)


m1 m2

u1

y1 y2

k

d

-u2

Figure 5.11: Two MSD system with control inputs u1 and u2, and measured outputs y1 andy2.

Analytical expressions for P1 and P2 are given by (5.21) and (5.22), respectively, and theFRFs are shown in Figure 4.6. Suppose that P is controlled using either

C1 =

[C 00 C

]or C2 =

[C 00 0.1C

]. (5.43)

Herein C is a series connection of a static gain, a lead filter, and a weak integrator, of whichthe FRF is depicted in Figure 5.12. Using such a multi-loop SISO controller basically meansthat two separate controllers are used to position each mass independently, while no positioninformation of the other mass is used.

100

101

102

103

104

120

130

140

150

Magnitude

[dB

]

100

101

102

103

104

−100

−50

0

50

Phase

[◦]

Frequency [Hz]

Figure 5.12: Bode diagram of controller C.

Suppose the control objective is to attenuate disturbances at the outputs of the plant, i.e.,making S small. Figure 5.13 shows the singular values of S as a function of frequency whenusing C1 (Figure 5.13(a)) and C2 (Figure 5.13(b)) in a single-DOF control structure. Incase the controlled system is stable, integral constraint (5.30) is satisfied and for the systemconsidered here (5.30) reduces to

∫ ∞

0lnσ1(S(jω))dω = F1 ≥ 0, (5.44a)

∫ ∞

0lnσ2(S(jω))dω = F2 ≤ 0, (5.44b)


100

101

102

103

104

−100

−80

−60

−40

−20

0

20M

agnitude

[dB

]

Frequency [Hz]

(a) Plant controlled with C1

100

101

102

103

104

−100

−80

−60

−40

−20

0

20

Magnitude

[dB

]

Frequency [Hz]

(b) Plant controlled with C2

Figure 5.13: Singular values of S.

with F1 + F2 = 0. For both controllers, the values of F1 and F2 are approximated by takinga summation over a limited frequency range and limited set of data points. Results aresummarized in Table 5.2.

Table 5.2: Evaluation of (5.44a) and (5.44b).

For C1 For C2

F1 1070 3793F2 -1077 -3792

Since F1 > 0, there is, in addition to the frequency-wise trade-off, also a spatial trade-off, i.e.,a trade-off between the two singular values. The spatial sensitivity trade-off is clearly visibleby comparing Figure 5.13(a) with Figure 5.13(b). The peak in σ2 around 103 [Hz] almostcompletely disappears when C2 is used, implying that F2 is lowered. As a result, F1 mustincrease with the same amount (a spatial trade-off), and this is accomplished by increasing thevalue of σ1 over the entire lower frequency range with approximately a factor ten. Practically,this can be interpreted as ten times less disturbance attenuation in the worst-case direction.A SVD of S gives more information on how this worst-case direction relates to input andoutput variables. The SVD of S at 1 [Hz] for controller C2 is given by

S(2πj) =1

1000

[1.79 + 9.63j −1.79 − 9.63j−17.9 − 96.3j 17.9 + 96.3j

]= UΣV H , (5.45)

with

U =[u1 u2

]=

[−1.8 · 10−2 − 9.8 · 10−2j 7.6 · 10−2 + 9.9 · 10−1j1.8 · 10−1 + 9.8 · 10−1j 7.6 · 10−3 + 9.9 · 10−2j

],

Σ =

[1.39 · 10−1 0

0 2.77 · 10−6

],

V H =[v1 v2

]H=

[−0.71 −0.71

0.71 + 2.5 · 10−6j −0.71 − 2.6 · 10−6j

]H

.


The second element of output direction u1 is ten times larger in magnitude than the firstelement. The gain σ1 is therefore mainly in the direction of y2. This is of course not surprising,because the loop-gain of the controller fixed to the second mass is lowered by a factor ten

when C2 is used. The largest singular value has an input in direction v1 ≈[−0.707 0.707

]T,

which means that disturbances that work on y1 and y2 with equal magnitude but oppositesign are the most difficult to suppress. This is also logical from a practical point of view,because in this situation both actuators effectively work against each other.

100

101

102

103

104

−50

−40

−30

−20

−10

0

10

Magnitude

[dB

]

S11

100

101

102

103

104

−80

−60

−40

−20

0

S12

100

101

102

103

104

−80

−60

−40

−20

0

S21

Magnitude

[dB

]

Frequency [Hz]10

010

110

210

310

4−50

−40

−30

−20

−10

0

10S22

Frequency [Hz]

Figure 5.14: Sensitivity function for system controlled with C1 (thin line) and C2 (thick line).

In Figure 5.14, the sensitivity function S is depicted for C1 and C2. The elements of S thatare related to output y2, i.e., S21 and S22, confirm that there is a factor ten difference indisturbance attenuation at low frequencies. In addition, at high frequencies the S12 term is afactor ten lower for the system controlled with C2. This may seem unexpected at first, but alower BW of the controller fixed to m2 also influences y1. Since the BW of this controller islower, high frequency disturbances that work on m2 have a smaller effect on y1.

5.3 Performance limitations for non-square plants

The fundamental performance limitations discussed so far are in terms of poles and zerosof the open-loop L(s) (or LI(s)), which is always a square TFM. However, potential designlimitations arising from using non-square controllers to control non-square plants are notexplicitly addressed. In Woodyatt et al. (2001) and Freudenberg and Middleton (1999), thisissue is investigated for a single-input two-output system and integral constraints are proposedthat describe the relation between the elements of T (jω). These integral constraints, however,are beyond the scope of this research. The reason for this is twofold. First, the integral

5.4 Performance limitations in the standard plant setup 79

constraints are practically difficult to interpret and second, analyzing the performance limitingbehavior of ORHP zeros and poles of non-square plants can still be done using the relationspresented in Sections 5.1 and 5.2.

Non-square plants often occur in practice due to a variety of reasons. One reason is thatnon-square plants are less likely to have zeros than square systems. Example 5.13 showsthat adding extra outputs generally leads to new open-loop zeros. For NMP systems, thisimplies that fundamental performance limitations caused by NMP zeros can be dealt with byadding extra sensors and smart controller design. The non-square controller should then bedesigned such that L and LI do not contain the plant NMP zeros. Leventides and Karcanias(2008) deal with similar problems by considering a system with more outputs than inputs. Anon-square static pre-multiplication matrix is then designed such that the new (transmission)zeros of the square system are located at the desired locations. This procedure, however, isbeyond the scope of this research.

Example 5.13. Consider a SISO system p1(s) that has an ORHP zero at s = q. Adding anextra output leads to a 2 × 1 system P (s)

P (s) =

[p1(s)p2(s)

]. (5.46)

The new system only has a transmission zero if both p1(s) and p2(s) are zero simultaneously.In general, however, this does not happen, unless there is some underlying structure in thesystem. For instance, a zero pinned to a particular input or output, or if the extra sensor isplaced at the same location as the first sensor. For now, assume that P (s) has no transmissionzeros. One could then falsely conclude that the fundamental performance limitation caused bythe ORHP zero in p1(s) is gone. It must, however, be kept in mind that during the controllerdesign new zeros can be created by forming L and LI

L = PC =

[p1

p2

] [c1 c2

]=

[p1c1 p1c2p2c1 p2c2

], (5.47)

LI = CP =[c1 c2

] [ p1

p2

]= p1c1 + p2c2. (5.48)

The normal ranks of LI and L, which is a dyadic matrix, are one and the open-loop zeros arein general located at different locations than at s = q.

5.4 Performance limitations in the standard plant setup

5.4.1 Preliminaries

The theory of fundamental design limitations in the single-DOF control structure is well un-derstood for both SISO and MIMO plants. In this section, an extension for the standard plantsetup is presented that is based on the work of Hong and Bernstein (1996, 1998) and Freuden-berg et al. (2002, 2003). The focus is hereby on NMP zeros, because the systems of interestgenerally do not have unstable poles (except pure integrators). For convenience, the standardplant setup is depicted again in Figure 5.15. Here, y denote the sensed outputs, u the control


signals and w, z are the weighted exogenous inputs and outputs, respectively. The goal ofcontrol is then to attenuate the response from w to z.

C

G

w z

u y

Figure 5.15: Standard plant setup.

If the system G is partitioned as

G =

[Gzw Gzu

Gyw Gyu

], (5.49)

the response from w to z is given by

Tzw = Gzw +GzuC(I −GyuC)−1Gyw. (5.50)

Unless Gyu is identically zero (Gyu ≡ 0), i.e., Gyu is zero for all values of s, the systemcontains a feedback loop and thus stability and robustness issues must be addressed. Forease of reference, Tzw is referred to as the closed-loop disturbance response, even if w is not adisturbance or when there is no feedback loop. For the remainder of this section, the followingassumptions are made:

• The signals z and w are scalar valued. This assumption simplifies the derivation ofthe interpolation and integral constraints and can be removed at the cost of additionalcomplexity (Freudenberg et al., 2003).

• Unless stated otherwise, u and y are scalar valued. This assumption is also imposed forsimplicity.

• The system G is internally stabilizable by feedback from y to u. It follows from Zhouet al. (1996, Lemma 12.1) that G is stabilizable if and only if all CRHP poles of G arepoles of Gyu with the same multiplicity. Under this assumption, C internally stabilizesG if and only if C internally stabilizes Gyu.

• Elements Gyw and Gzu are not identically zero. Otherwise, no controller can influencethe closed-loop disturbance response.

Suppose that Gzw is stable and not identically zero. Then, it is meaningful to directly comparethe controlled and the uncontrolled disturbance response by means of the disturbance responseratio as defined in Definition 5.14.


Definition 5.14 (Disturbance response ratio). The disturbance response ratio is definedas

Hzw , G−1zwTzw. (5.51)

At any frequency where |Hzw(jω)| > 1, the closed-loop disturbance response is worse thanfor the uncontrolled system and at frequencies where |Hzw(jω)| < 1, the closed-loop responseis better. Perfect disturbance attenuation, i.e., Hzw ≡ 0, can then be achieved by using afinite gain cancellation controller that can be determined using Theorem 5.15. In Hong andBernstein (1998), this controller is referred to as a “zero spillover controller”, where spilloveris defined in an uncommon3 way as the area where |Hzw| > 1.

Theorem 5.15 (Cancellation controller). Assume that (i) Gyw 6≡ 0, (ii) Gzu 6≡ 0, and(iii) det(G) = GzwGyu −GzuGyw 6≡ 0. Then the controller C = CC , where

CC =Gzw

GzwGyu −GzuGyw, (5.52)

yields Tzw ≡ 0. If Gzw 6≡ 0, then, if any of assumptions (i) - (iii) is violated, it is impossibleto determine a controller that achieves Tzw ≡ 0.

Proof. Setting Tzw = 0 in (5.50) yields

0 = Gzw +GzuC(I −GyuC)−1Gyw. (5.53)

If Gzu and Gyw are non-zero, then rearranging (5.53) leads to

−G−1zuGzwG

−1yw = (I −G−1

zuGzwG−1ywGyu)C. (5.54)

If det(G) 6≡ 0, then (5.54) may be solved for C yielding (5.52).

The goal of this section can now be formulated more clearly as to investigate what kind oflimitations exist in the standard-plant setup and whether usage of the standard plant setuppotentially leads to better performance than with the single-DOF control structure. An attrac-tive possibility is to simply use the cancellation controller, because this leads to Tzw ≡ 0. Itis, however, remarked in advance that CC is in general improper and cannot be implemented.

5.4.2 Systems reducible to feedback loop

Some systems represented in the standard plant are reducible to a single-DOF control config-uration. Hence, the same fundamental performance limitations as presented in Sections 5.1and 5.2 apply. Examples 5.16 and 5.17 describe two important classes of such systems: col-located z and y (performance variables are measured), and collocated u and w.

3Balas (1978), Meirovitch et al. (1983), and Preumont (2002) use the term “spillover” to describe the effectof actuators on unmodelled plant dynamics (control spillover) or the contamination of the measured outputsby unmodelled plant dynamics (observation spillover)


Example 5.16. Suppose the performance output is measured for feedback, i.e., y = z, andallow u to have dimension nu. Then Gyw = Gzw, Gzu = Gyu, and (5.50), (5.51) reduce to

Tzw = SGzw and Hzw = S, (5.55)

respectively, where S is scalar because z and y are scalar. It is known from Section 5.1 thatif Gyu has no CRHP zeros, high gain FB can be used to make S arbitrarily small over anarbitrarily wide frequency interval. In practice, however, practical issues limit the use of suchcontrol laws and require roll-off at higher frequencies. Therefore, the disturbance responsemust satisfy the Bode sensitivity integral and S must exhibit a peak value greater that one.

Example 5.17. Next, suppose that the control and disturbance signal excite the systemidentically (u = w), and allow y to have dimension ny. Then Gzw = Gzu, Gyw = Gyu,and (5.50), (5.51) reduce to

Tzw = GzwSI and Hzw = SI , (5.56)

respectively, where SI is a scalar TF. Similar to Example 5.16, it is possible to achieve arbi-trary high disturbance attenuation by using high gain feedback.

Remark 5.18. In case of scalar u and y in Examples 5.16–5.17, the need for high gainfeedback can also be established by filling in det(G) = 0 into (5.52).

For more general systems with arbitrary dimensions for w, z, u, and y, the elements of Hzw

and Tzw also have to satisfy the constraints presented in Sections 5.1 and 5.2. For example,consider the four-block control problem discussed in Chapter 3, for which Tzw is given by

Tzw =

[S SPCS TI

]. (5.57)

TF(M)s S and T must satisfy the interpolation and integral constraints presented in Sec-tions 5.1 and 5.2. In addition, for SISO systems TI = T and thus S + T = I. The elementsof Tzw can therefore not be identically zero.

5.4.3 Interpolation constraints due to CRHP zeros in Gzu or Gyw

The requirement of internal stability implies that Tzw and Hzw must satisfy constraints atcertain points of the CRHP. Since these constraints are similar to the interpolation constraintsdiscussed in Sections 5.1 and 5.2, these constraints are referred to as interpolation constraintsfor the standard plant setup. The points at which interpolation constraints must be satisfiedare located at a subset of CRHP zeros of Gzu and Gyw and at a subset of the CRHP poles ofG.

To get acquainted with interpolation constraints in the standard plant setup, consider Exam-ples 5.19–5.20 that are taken from Freudenberg et al. (2002). The first example illustratesthat a CRHP zero in Gzu or Gyw constrains the closed-loop response unless Gzw shares thiszero.


Example 5.19. Consider the generalized plant

G(s) =

[ s−αs+1

s−1s+1

1s+1

2s+1

]. (5.58)

Then, using (5.50) the closed-loop disturbance response is

Tzw(s) =s− α

s+ 1+

s− 1

(s+ 1)2C(s)S(s). (5.59)

Internal stability requires that CS must be stable and hence CS cannot cancel the NMP zeroat s = 1. Therefore, Tzw must satisfy the interpolation constraint

Tzw(1) =1 − α

2. (5.60)

Clearly, if Tzw(1) 6= 0 it is not possible to find a stabilizing controller such that Tzw ≡ 0. Todetermine whether a stabilizing controller exists if Tzw(1) = 0, the cancellation controller isdetermined using Theorem 5.15, i.e.,

CC(s) =(s− α)(s + 1)

s+ (1 − 2α). (5.61)

The associated sensitivity function is then given by

SC(s) =(s+ 1 − 2α)

(−s+ 1). (5.62)

For α = 1, Tzw(1) = 0, (5.61) reduces to CC = (s + 1), and (5.62) reduces to SC = −1.Therefore, there is no conflict between perfect disturbance attenuation and internal stability.However, if α 6= 1, SC has an unstable pole at s = 1.

The next example shows that fundamental design limitations may continue to exist, even ifTzw is equal to zero. To illustrate this, a zero at s = 1 is added to Gyw.

Example 5.20. Consider the new generalized plant

G(s) =

[ s−αs+1

s−1s+1

s−1s+1

2s+1

]. (5.63)

Using (5.50) then leads to

Tzw(s) =s− α

s+ 1+

(s− 1

s+ 1

)2

C(s)S(s), (5.64)

and

Tzw(1) =1 − α

2. (5.65)

Using the cancellation controller for this system leads to

SC(s) =s2 − 4s+ 1 − 2α

s2 − 2s + 1, (5.66)

which is unstable even if α = 1. The reason may be seen by factoring into

Tzw(s) = T 1zw(s)(s− 1), and T 1

zw(s) =1

s+ 1+

s− 1

(s+ 1)2C(s)S(s), (5.67)

leading to T 1zw(1) = 0.5. Since CC must result in T 1

zw ≡ 0, CC cannot stabilize the system.


Examples 5.19–5.20 are now formalized in Theorem 5.21, which was originally introduced inFreudenberg et al. (2002). The number of actuators and sensors is allowed to be arbitrary, soGzu is a row vector and Gyw a column vector.

Theorem 5.21. Suppose the system in Figure 5.15 is internally stable and let y and u havedimensions ny and nu, respectively. Let q be a CRHP zero of Gzu or Gyw and assume that qis not a pole of G.

(i) Then under these conditions

Tzw(q) = Gzw(q) (5.68)

and it follows that Tzw(q) = 0 if and only if Gzw(q) = 0.

(ii) In addition, assume that Gzw 6≡ 0 and that the multiplicity of q (denoted by ν) as a zeroof Gzw, Gzu, and Gyw satisfies the bound

νGzw < νGzu + νGyw. (5.69)

Then the following factorizations can be made

Tzw(s) = T 1zw(s)(s − q)νGzw and Gzw(s) = G1

zw(s)(s − q)νGzw , (5.70)

where T 1zw and G1

zw have no poles at q, and

T 1zw(s) = G1

zw(s) 6= 0. (5.71)

Proof. (i) Internal stability requires that the transfer functions

GzuCS = GzuC(I −GyuC)−1, (5.72)

CSGyw = C(I −GyuC)−1Gyw, (5.73)

must be stable. It then follows that if q is a CRHP zero of Gzu or Gyw, q must also be a zeroof GzuCSGyw. Rewriting (5.50) as

Tzw = Gzw +GzuCSGyw (5.74)

yields (5.68).

(ii) The bound (5.69) implies that the terms Gzw and GzuCSGyw must posses the commonfactor (s − q)νGzw . Since GzuCSGyw has a zero at q with multiplicity greater than themultiplicity of Gzw, (5.71) follows straightforwardly from (5.74).

In a similar fashion, interpolation constraints can be derived for CRHP poles ofG, see Freuden-berg et al. (2002, 2003). These constraints, however, are not relevant for the control of stableflexible motion systems and therefore not discussed here.


5.4.4 A generalized Bode integral

Definition 5.22. Denote the set of all NMP zeros of Tzw by {ξi : i = i, . . . ,Nξ} and separatethis set into a set {βi : i = i, . . . ,Nβ} of zeros that are shared with Gzw and a set of additionalzeros {γ : i = i, . . . ,Nγ}. Then, Tzw can be factored as

Tzw = TzwBξ = TzwBβBγ , (5.75)

where Bξ, Bβ, and Bγ are Blaschke products as defined in (5.9). Denote the set of NMP zerosof Gzw that are not shared with Tzw by {α : i = i, . . . ,Nα} and the set of all ORHP poles ofGzw by {ρ : i = i, . . . ,Nρ}. Gzw may then be factored as

Gzw = GzwBαBβB−1ρ . (5.76)

Finally, define

Rzw , G−1zwGzuC(I −GyuC)−1Gyw, (5.77)

such that Hzw = 1 +Rzw.

In Theorem 5.23, a generalization of the Bode sensitivity integral is presented that holdswhenever a condition involving the relative degree of Rzw (denoted by δ(Rzw)) is satisfied.Furthermore, the relation is valid for scalar w and z and arbitrarily sized u and y.

Theorem 5.23 (Generalized Bode integral). Suppose that the controlled system is stable.

(i) If δ(Rzw) > 1, then

∫ ∞

0ln |Hzw(jω)| dω = π

Nγ∑

i=1

Re γi + π

Nρ∑

i=1

Re ρi − π

Nα∑

i=1

Re αi. (5.78)

(ii) If δ(Rzw) = 1, then

∫ ∞

0ln |Hzw(jω)| dω = π

Nγ∑

i=1

Re γi + π

Nρ∑

i=1

Re ρi − π

Nα∑

i=1

Re αi +π

2kH , (5.79)

with

kH = lims→∞

sRzw(s). (5.80)

Proof. See Freudenberg et al. (2002).

If GyuC is strictly proper and the relative degrees of the TFs in (5.77) satisfy

δ(Gzu) + δ(C) + δ(Gyw) ≥ δ(Gzw) + 1, (5.81)

then (5.79) holds. If (5.81) is satisfied with a strict inequality, then kH = 0 and (5.78) issatisfied.


Consider (5.78) and suppose that both Gzw and Tzw have no NMP zeros and that Gzw isstable. Then, as with the usual Bode sensitivity integral, the area of disturbance attenuation(|Hzw| < 1) must be equal (on a linear frequency scale) to the area of disturbance amplification(|Hzw| > 1). In theory, the area of disturbance amplification may consist of a very smallamount of amplification spread over a very wide frequency range. In practice, however,actuator constraints restrict the length of this interval. Next, suppose that Tzw is minimumphase and that Gzw has at least one NMP zero, i.e., Nα > 0. Then, the last term on theRHS of (5.78) is negative and the area of disturbance attenuation exceeds that of disturbanceamplification. If Gzw is minimum phase and Tzw has a NMP zero, the first term on the RHSof (5.78) will be positive and the area of disturbance amplification exceeds that of disturbancerejection.

Remark 5.24. The existence of a generalized Bode integral may seem in conflict with the can-cellation controller. Notice, however, that the cancellation controller often leads to δ(Rzw) < 1and hence the integral constraints presented in Theorem 5.23 do not have to be satisfied. Theexample of Section 5.4.5 further illustrates this.

More integral relations for the standard plant setup can be derived. A straightforward exten-sion of the generalized Bode integral can be obtained by taking the interpolation constraintsdue to ORHP zeros in Gzu and Gyw into account. This results in a generalized Poisson inte-gral constraint. In Appendix B, the effect of the ORHP zero of the flexible cart system (seeFigure 5.3) discussed earlier in this chapter is further investigated. By using the cancellationcontroller, it is shown that if the NMP zero is included in Gyu, i.e., when measuring y1, thereare no fundamental performance limitations on output y2. Arbitrarily high performance4

(for y2) can therefore be achieved, while the FB loop satisfies the Poisson sensitivity integralconstraint (Theorem 5.4). However, when y2 is measured and performance is evaluated aty1, the NMP zero is located in Gzu and the achievable performance is limited due to theinterpolation constraints presented in this section.

5.4.5 Design example: limitations in the standard plant setup

Consider the two MSD system depicted in Figure 5.16 with control input u, disturbance inputw, measured output y, and performance output z.

m1 m2

u

z y

k

d

-w

Figure 5.16: Two MSD system with control input u, disturbance input w, measured outputy, and performance output z.

For m1 = m2 = m the system G is partitioned as

G(s) =

[P2(s) P1(s)P1(s) P2(s)

], (5.82)

4The performance is only limited by various practical aspects


with P1 and P2 given by (5.21) and (5.22), respectively. Using (5.52) leads to the cancellationcontroller

CC(s) = −ds− k, (5.83)

which is a standard PD-controller. Figure 5.17 shows that this controller indeed yieldsTzw ≡ 0. There are no unstable pole/zero cancellations between CC and Gyu and in-spection of the poles of S is therefore sufficient to guarantee internal stability. Since S onlyhas OLHP poles, the system is internally stable.

Unfortunately, CC is improper and cannot be implemented. A (strictly) proper approximationof CC can be obtained by adding an nth order low pass filter

CaC(s) , CC(s)Cn

f (s), Cnf ,

(1

12πfc

s+ 1

)n

, (5.84)

where n is large enough to make CaC proper and fc is large enough such that Ca

C is stillstabilizing. Obviously, using Ca

C results in Tzw 6≡ 0. Hence, if the relative degree conditionof (5.81) is satisfied, Hzw is constrained by the generalized Bode integral. Figure 5.17 alsoshows Hzw for two approximations of CC with fc = 1.0 · 107 and n = 1, 2. If n = 2, (5.81) issatisfied with strict inequality and (5.78) holds. Since neither Gzw nor Tzw has NMP zeros andGzw is stable, the area of disturbance attenuation is balanced by an equal area of disturbanceamplification, as can be seen by the “small” area |Hzw| > 1 in Figure 5.17(b). For n = 1, Ca

C

is proper and (5.81) is satisfied with strict equality. A numerical approximation of kH showsthat kH is very large (order 107), but negative. This explains why Hzw does not cross the0 [dB] line in Figure 5.17(b).

100

105

1010

−300

−250

−200

−150

−100

−50

0

50

Frequency [Hz]

Magnitude

[dB

]

(a) Hzw

106

107

108

109

−20

−15

−10

−5

0

5

Frequency [Hz]

Magnitude

[dB

]

(b) Hzw (zoomed)

Figure 5.17: Bode magnitude diagrams of the closed-loop disturbance response ratio Hzw forthe system controlled with the cancellation controller CC (thick black line), and a bi-proper(thick grey line) and strictly proper (thin black line) approximate cancellation controller withfc = 1.0 · 107.


5.5 Conclusions

The main focus of this chapter is on interpolation and integral constraints of the single-DOFcontrol structure and the standard plant setup. Regarding the single-DOF control structure,the main results are summarized below:

• Assume that σ(L) has a roll-off rate of more than one and that L contains no zeros inthe ORHP. The Bode sensitivity integral for S states that S exhibits a design trade-offbetween performance and robust stability. If S is to be kept small in one frequencyrange, it necessarily becomes large at other frequencies. The open-loop unstable polesdetermine how stringent this trade-off is.

• The presence of ORHP zeros further aggravates this trade-off in the sense that theintegral relation should be satisfied over a limited frequency range (Poisson sensitivityintegral for S). ORHP zeros therefore pose an upper bound on the achievable BW. Inaddition, systems in which the poles and zeros in the CRHP are located close to eachother are the most difficult to control.

• Similar integral constraints exist for T and the ORHP zeros of L. The role of ORHPpoles of L is now taken by ORHP zeros of L, and vice versa. It is shown that the ORHPpoles pose a lower bound on the achievable BW.

For the standard plant setup, the focus is limited to constraints caused by ORHP zeros only.The main conclusions are summarized below:

• Important interpolation constraints and integral constraints are presented for the stan-dard plant setup. If desired, more constraints can be derived.

• Classical performance limitations, including the Poisson integral constraint for S, arestill valid in the standard plant setup. However, these constraints do not necessarilylimit the performance in terms of z, unless z = y or w = u. The reverse is also true;there may be strong limitations on z that do not limit y. It should, however, be keptin mind that in the latter situation high performance in y does not imply good actualperformance.

• With respect to the optimal sensor placement for a given actuator configuration, itfollows that relocating the sensors such that there are no fundamental performancelimitations on measured outputs does not affect the limitations (that are discussed inthis chapter) on the performance variables. Various practical aspects and causality of thecontroller are, however, not included in this reasoning and are left for future research. Awell-chosen sensor configuration may therefore still increase the performance in practice.

Chapter 6

Control of a two MSD system

This chapter is about controller design for the two MSD system introduced in Chapter 4.For convenience, the system is depicted again in Figure 6.1. Herein u represents the forceinput and either translation y1 (collocated control) or y2 (non-collocated control) is measuredfor feedback, while performance is evaluated at the other output. The system parameters aregiven in Appendix C and Bode diagrams are given in Figure 4.6. Both masses have equalweights m and k and d are chosen such that the undamped eigenfrequency fn = 100 [Hz] anddimensionless damping ratio ζ = 0.0105 [-]. Since the system only contains a single flexibleelement, it allows for comprehensible controller design and performance evaluation from amechanical point of view. Nevertheless, it contains sufficient complexity to demonstrate theneed of taking flexibilities into account for next generation motion systems.

m1 m2

u

y1 y2

k

d

Figure 6.1: Two MSD system with input u and measured outputs y1 and y2.

In Section 6.1, conventional controller design (without FF) is applied. Basic loop-shapingrules, like, e.g., creating sufficient PM and MM, are used to design SISO controllers under theassumption that the performance variable (denoted by zp) is the measured variable (denotedby yp). Recall from Chapter 5 that due to the absence of poles and zeros in the ORHP, thereare no fundamental performance limitations for the measured variables.

In Section 6.2, the inferential control structure (see Figure 3.6(b)) is adopted to allow forcontroller design where the performance variables are not measured during normal operation.The main focus is on designing the performance filter Ca. Similar results can be obtained forindirect control (see Figure 3.6(a)) and the general two-DOF control structure (see Figure 3.8).This, however, is beyond the scope of this research.

89

90 Control of a two MSD system

6.1 Conventional control

6.1.1 Rigid body assumption

If the system is assumed to behave as a single mass (rigid body assumption), PD-controllerscan be interpreted physically as fixing the single mass to the world by a spring (P-action)and a damper (D-action). The rigid body mode is thus replaced by a flexible mode. By usingnormalized responses of second order systems, see, e.g., Franklin et al. (2002, p. 141), thespring and damper coefficients can be chosen such that the desired time-domain specificationsare met.

10−1

100

101

102

103

−150

−100

−50

0

50

Magnitude

[dB

]

10−1

100

101

102

103

−400

−300

−200

−100

0

Phase

[◦]

Frequency [Hz]

(a) Bode diagram of Luy1and Luy2

0 0.5 1 1.5 20

0.5

1

1.5

y 1

0 0.5 1 1.5 20

0.5

1

1.5

y 2

Time [s]

(b) Step responses of y1 and y2

Figure 6.2: Open-loop FRFs (left) for the collocated configuration (thin line) and non-collocated configuration (thick line), and the responses of y1 and y2 on a unit step in reference(right). Due to the low BW, the responses of y1 and y2 are approximately equal.

If these time-domain specifications are such that the required PD-controller leads to a BW thatis well below the first resonance frequency, the rigid body assumption is justified and it doesnot matter whether y1 or y2 is used for FB control. For example, consider the approximatePD-controller (lead filter)1

C1(s) = 30 ·1

2π1/3s+ 1

12π1·3s+ 1

, (6.1)

that achieves a BW of approximately 1 [Hz]. The corresponding open-loop FRFs for collocatedand non-collocated control are depicted in Figure 6.2(a). Clearly, the rigid body assumptionis valid, because differences between the two FRFs only occur well above the target BW.This is confirmed by Figure 6.2(b), where the responses of y1 and y2 to a unit step change

1A PD-controller is often approximated by a lead filter, because the pure derivative action causes thecontroller to be improper. In addition, the high gain at high frequencies also leads to a large amplification ofmeasurement noise, which is undesired in practice.

6.1 Conventional control 91

in reference are depicted. Both masses move with only minor internal deformation (‖y1 −y2‖∞ < 15 · 10−4 [m]). Practically, the spring and damper coefficients of the controller arelow compared to the spring and damper coefficients of the system.

6.1.2 Collocated control

For collocated control, i.e., using plant P1 for controller design, a PD-controller is equivalentto fixing the first mass to the world by a spring and a damper as depicted in Figure 6.3. Thecontrolled system thus contains two flexible modes and no rigid body modes.

m1 m2

y1 y2

dD

kP

Figure 6.3: Physical interpretation of a PD-controller. The controlled system thus has twoflexible modes.

Demanding a faster response than in Figure 6.2(b) requires a higher spring stiffness P , ahigher damping coefficient D, and a higher P/D ratio. A controller that achieves a BW of500 [Hz] is given by

C2(s) = 3 · 106 ·1

2π500/3s+ 1

12π500·3s+ 1

. (6.2)

The FRFs of C2 and the corresponding open-loop are shown in Figure 6.4. Clearly, the BW islocated well above the resonance frequency, meaning that the spring and damper coefficientsare large compared to the spring and damper coefficients of the system.

Figure 6.5 shows the FRFs of S and T . The effect of control on output y1 (the measuredvariable) is given by the (1, 1)-terms and the effect on y2 (the performance variable) by the(2, 1)-terms.2 By fixing the controller spring and damper to m1, the rigid body mode isreplaced by a damped flexible mode with a natural frequency of approximately 500 [Hz]. Themode shape of this flexible mode mainly consists of movement of m1, because the internalspring and damper are not strong enough to let m2 follow m1. Hence, this mode is hardlyvisible in the (2, 1)-terms and m2 is said to be “decoupled”.

As is well known from root-locus analysis, see, e.g., Franklin et al. (2002, Chapter 5), increas-ing the FB gain causes the closed-loop poles to migrate from their open-loop locations towardsthe open-loop zero locations. Hence, the relatively high (lightly damped) resonance peaks inS and T are located close to the zero frequency. Practically, the presence of a complex zeropair in P1 means that the actuator force u and the spring and damper forces caused by themovement of m2 can counteract each other. The effect of the OLHP zero pair is more clearlyshown in Figure 6.6(a), where the time responses of y1 and y2 are depicted for a unit step inreference. The large (in magnitude) undamped oscillations of m2 of approximately 70 [Hz]

2To evaluate the inferential performance, S and T are determined using both plant outputs. Hence, Sand T are 2 × 2 matrices, whereby the (1, 1)-terms are thus the same S and T as used in conventional SISOcontroller design.


101

102

103

104

130

140

150

160

170

180M

agnitude

[dB

]

101

102

103

104

−100

−50

0

50

100

Phase

[◦]

Frequency [Hz]

(a) Bode diagram of C

101

102

103

104

−50

0

50

100

Magnitude

[dB

]

101

102

103

104

−300

−200

−100

0

100

Phase

[◦]

Frequency [Hz]

(b) Bode diagram of L

Figure 6.4: FRFs of the controller (left) and corresponding open-loop (right), for controllerC2 (thin line) and C3 (thick line).

101

102

103

104

−60

−40

−20

0

20

Magnitude

[dB

]

S11

101

102

103

104

−50

0

50

Magnitude

[dB

]

Frequency [Hz]

S21

(a) Bode magnitude diagram of the S

101

102

103

104

−15

−10

−5

0

5

Magnitude

[dB

]

T11

101

102

103

104

−50

0

50

Magnitude

[dB

]

Frequency [Hz]

T21

(b) Bode magnitude diagram of the T

Figure 6.5: FRFs of the sensitivity (left) and complementary sensitivity (right), for thesystem controlled with C2 (thin line) and C3 (thick line). The effect of the controller on y1 isgiven by S11 and T11 and the effect on y2 by S21 and T21, respectively.

are only observed as much smaller oscillations of m1. Since high performance is requested atm2, this result establishes that for general flexible motion systems good performance at themeasured output(s) does not imply good actual performance. So, when only looking at thesensor data, one may falsely conclude that the desired performance specification are met.


0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5y 1

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

y 2

Time [s]

(a) For C2

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

y 1

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5

y 2

Time [s]

(b) For C3

Figure 6.6: Responses of y1 and y2 for a unit step in reference.

For this system, the ripple in y1 can be reduced by increasing the feedback gain. This caneither be achieved by increasing the BW or locally, by applying an inverse notch filter tocancel the complex zero pair. The latter is realized by controller C3 given by

C3(s) = C2(s) ·1

(2π70.996)2 s2 + 2·1

2π70.996s+ 1

1(2π70.996)2

s2 + 2·0.00742π70.996s+ 1

. (6.3)

The effect of the inverse notch filter is also included in Figures 6.4–6.6. It follows fromFigure 6.5 that the resonance peak at approximately 70 [Hz] has been removed3 from S11

and T11 as intended. However, the resonance peak is still present in the (2, 1)-terms. Thisis in line with the conclusion of Chapter 3 that standard performance indicators S and T arenot suitable to evaluate the inferential performance. Due to the local increase in controllergain, the closed-loop poles are moved further towards the open-loop zeros. This implies aless damped resonance and a small change in the corresponding resonance frequency. Theactual performance, which for this system is evaluated at y2, can thus deteriorate, while theperformance at the measured variables is improved. Notice that the performance deteriorationis caused here by a complex zero pair in the OLHP. So, although MP zeros do not posefundamental limitations on the measured performance (see Section 5.1), they can severelylimit the actual performance if it is incorrectly assumed that yp = zp. For this system, theperformance deterioration is more clearly visible in Figure 6.6(b). Output y1 follows the stepin reference almost perfectly, while y2 has a lower decay ratio (decreased damping) than withC2. In addition, suppressing the resonance peak in S11 also leads to a small increase in theheight of the resonance peak around 500 [Hz] (waterbed effect). In the time domain, thismanifests itself by a small increase in overshoot for output y1.

3No exact cancellation takes place, because the plant zeros and controller poles are located at slightlydifferent locations in the complex plane


So far, the MP complex zero pair has shown to limit the achievable performance (at m2). Oneway to improve performance by using a single PD-like controller, is to let the PD-controlleralso damp the movement m2. This can be achieved by lowering both the controller springstiffness and damping coefficient, such that the flexible mode introduced by controller hasroughly the same natural frequency as the system flexible mode. Practically, the controllernow “allows” m1 to have a larger position error (lower spring stiffness) such that oscillationsof m2 are also damped by the controller. In loop-shaping terms, the BW should be placedbetween the plant anti-resonance and resonance as discussed in Den Hamer (2005). A leadfilter that achieves this effect is given by

C4(s) = 1.20 · 105 ·1

2π85/3s+ 1

12π85·3s+ 1

. (6.4)

The FRFs of the corresponding open-loop and the achieved sensitivity and complementarysensitivity function are shown in Figure 6.7. Both closed-loop flexible modes are now dampedheavily, which is also visible in the response to a unit step in reference (see Figure 6.8).Although the achieved BW of C4 is much lower than for C2 and C3, the performance atm2 is much better. However, if a faster response of m2 is desired, a PD-like controller isinsufficient and distinguishing between the measured variables (measured performance) andperformance variables (inferential performance) is essential.4

100

101

102

103

104

−100

−50

0

50

100

Magnitude

[dB

]

100

101

102

103

104

−200

−150

−100

−50

0

50

Phase

[◦]

Frequency [Hz]

(a) Bode diagram L

100

101

102

103

104

−80

−60

−40

−20

0

20

Magnitude

[dB

]

S11, T11

100

101

102

103

104

−80

−60

−40

−20

0

20

Phase

[◦]

Frequency [Hz]

S21, T21

(b) Bode magnitude diagrams of S and T

Figure 6.7: FRFs of the open-loop (left), and sensitivity (thin line) and complementarysensitivity (thick line) (right).

Remark 6.1. Recall from Chapter 3 that taking the inferential performance into account doesnot necessarily mean that a two-DOF control structure has to be used. Other possibilities mayalso lead to performance improvement. For example, by taking zp as an additional exogenousoutput variable in the standard plant setup. However, to appropriately deal with the inferentialnature of the problem, a two-DOF control structure is essential.

4This only holds if no adjustments can be made to the I/O configuration.


0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

y 1

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2

2.5y 2

Time [s]

Figure 6.8: Responses of y1 and y2 for a unit step in reference for system controlled with C4

(thick line) and C2 (thin line).

6.1.3 Non-collocated control

By shifting the sensor to m2 the flexible mode is sensed differently by the sensor. This isdiscussed in detail in Verhoeven (2009, Section 4.3.4). The difference between P2 and P1 isthe absence of the complex zero pair in the former. Hence, there is no dynamics present thatis able to counteract control actions u such that m2 does not move. On the one hand, thisis beneficial for conventional controller design, since large undamped oscillations in y1, whichnow acts as the performance variable, are always noticed by the sensor (at m2). On the otherhand, if the performance (at m1) is good at the cost of large oscillations of m2, as is thecase for controller C3 in Section 6.1.2, assuming that yp = zp, which is done in conventionalcontroller design, will lead to undesired control actions.

Another consequence of the absence of the complex zero pair in P2, is that plant phaseeventually drops to -360 [◦], implying a finite GM. The presence of the undamped resonancepeak even further decreases the GM. Demanding a faster response than in Figure 6.2(b)therefore requires more than simply increasing the controller gain and shifting the phase leadof the controller. Extra phase in the open-loop needs to be created to stabilize the system.

Remark 6.2. Unlike with collocated control, the mechanical analogue between a PD-controllerand a spring and damper is not valid for non-collocated control.

A controller that stabilizes the system and achieves a BW of 500 [Hz] is given by

C5 = 15 · 103 ·(

12π15s+ 1

12π10000s+ 1

)3

. (6.5)

The corresponding FRF of the open-loop is depicted in Figure 6.9(a). Although the classicalPM, GM, and MM are satisfactory, practical issues often limit the implementation of such


a controller. In order to move m2 with high accelerations, the spring and damper need totransmit large forces and this can only be achieved by large internal deformations, as shownin Figure 6.9(b). The large translations of m1 are, however, not measured, implying that goodperformance at the measured output(s) does not imply good actual performance.

100

101

102

103

104

−40

−20

0

20

40

60

Magnitude

[dB

]

100

101

102

103

104

−200

−100

0

100

Phase

[◦]

Frequency [Hz]

(a) Bode diagram of L

0 0.02 0.04 0.06 0.08 0.1

0

100

200

300

y 1

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

y 2

Time [s]

0 0.0005

0

100

200


Figure 6.9: Open-loop FRF (left) and the responses of y1 and y2 on a unit step in reference(right).

A logical attempt to deal with the finite GM in conventional controller design is to supplementthe lead compensator with a skewed notch filter with a complex zero pair that exactly cancelsthe resonance mode. A controller that cancels the plant resonance and achieves a BW of 85[Hz] is given by

C6(s) = 190 · 103 ·1

2π85/3s+ 1

12π85·3s+ 1

·1

(2πfz)2s2 + 2·0.01

2πfzs+ 1

1(2π300)2

s2 + 2·0.72π300s+ 1

, fz = 100.4 [Hz]. (6.6)

The poles of this so-called “skewed notch filter”, which are needed to end up with a propercontroller, are placed at a sufficiently high frequency such that enough PM is created. Thecorresponding Nyquist diagram is shown in Figure 6.10(a) and the time responses of y1 andy2 on a unit step in reference are shown in Figure 6.10(b). The responses of y1 and y2 arenicely damped and cancelling the plant resonance seems to work satisfactorily. In addition,because the BW is much lower than for C5 the required movement of m1 is also much smaller.

Physically speaking, applying a notch filter can be interpreted as filtering the error signal suchthat the flexible mode is only weakly excited by the FB controller. To illustrate the filtereffect, the FRF of C6 is depicted in Figure 6.11(a). There is a clear anti-resonance presentaround 100 [Hz]. However, signals that directly influence the plant without going through thecontroller, like, e.g., a FF signal, can still heavily excite the flexible dynamics. The processsensitivity, which is shown in Figure 6.11(b), describes the mapping between the FF signal


−3 −2.5 −2 −1.5 −1 −0.5 0−2

−1.5

−1

−0.5

0

0.5

1

Re

Im

(a) Nyquist diagram of L

0 0.005 0.01 0.015 0.02 0.0250

0.5

1

1.5

2

y 1

0 0.005 0.01 0.015 0.02 0.0250

0.5

1

1.5

2

y 2

Time [s]


Figure 6.10: Nyquist diagram (left) and the responses of y1 and y2 on a unit step in reference(right).

100

101

102

103

80

100

120

140

Magnitude

[dB

]

100

101

102

103

0

50

100

150

200

Frequency [Hz]

Phase

[◦]

(a) Bode diagram of C6

100

101

102

103

−200

−150

−100

Magnitude

[dB

]

100

101

102

103

−200

−100

0

100

200

Frequency [Hz]

Phase

[◦]

(b) Bode diagram of SP

Figure 6.11: FRFs of the controller (left) and process sensitivity (right).

and the plant output and still contains the lightly damped plant resonance. Cancelling flexibleplant dynamics therefore does not mean that the dynamics is eliminated from the system.

In addition, model uncertainties prevent exact pole/zero cancellations in practice and com-pletely eliminating the resonance peak from the open-loop is therefore not feasible. The zerosshould rather be located slightly below the resonance frequency, such that the phase lead ofthe controller starts prior to the phase lag of the resonance frequency. Preumont (2002, p. 80)


gives the same recommendation, but his reasoning is based on the root-locus. A shift of thecomplex zero pair of the notch filter to fz = 99 [Hz] creates a circle at the safe side, while anequally large shift to higher frequencies destabilizes the system, as is shown in Figure 6.12(a).For both adjustments, the shift of the notch filter leads to lightly damped oscillations for aunit step in reference as is shown in Figure 6.12(b). For fz = 102 [Hz] (thick black line), theoscillation is unbounded. Notice that for both adjustments the MM is approximately equallylarge as for C6. A large MM is therefore a necessary but not sufficient condition for dampingand damping of the controlled system can only be evaluated by considering the closed-looppoles.

−2 −1.5 −1 −0.5 0 0.5−1.5

−1

−0.5

0

0.5

1

Re

Im


0 0.01 0.02 0.03 0.04 0.050

0.5

1

1.5

2

y 1

0 0.01 0.02 0.03 0.04 0.050

0.5

1

1.5

2

y 2

Time [s]


Figure 6.12: Nyquist diagram (left) and the responses of y1 and y2 on a unit step in reference(right) for fz = 99 [Hz] (thin black line), fz = 100.4 [Hz] (thick grey line), and fz = 102 [Hz](thick black line). The controlled system for fz = 102 [Hz] is unstable

Another way to alleviate the effect of the deal with the finite GM is to add phase lag to “turnthe resonance peak into the ORHP and away from the point (-1,0) in a Nyquist diagram”.Two types of controllers are proposed here that achieve this effect. First consider controllers

C7(s) = 10 · 103 ·1

2π7s+ 11

2π60s+ 1· 1

1(2π95)2

s2 + 2·0.42π95s+ 1

, (6.7)

C8(s) = 10 · 103 ·1

2π7s+ 11

2π60s+ 1·

1(2π100.4)2

s2 + 2·0.01052π100.4 s+ 1

1(2π200)2

s2 + 2·0.52π200s+ 1

. (6.8)

Controller C7 achieves a BW of 20 [Hz] and consists of a classical lead filter and a secondorder low-pass filter (corner frequency of 95 [Hz]) to create the necessary phase lag. To enablea fair comparison between this approach and controller C6, controller C8 is designed, which issimilar to C6 but tuned for a BW of only 20 [Hz]. Figure 6.13 shows the corresponding Nyquistdiagram and the FRF of SP . Controller C7 “turns” the weakly damped plant resonance away


from the point (-1,0) in the Nyquist diagram, while the height of the resonance peak remainsunchanged. Although this achieves the primary objective of stability, the damped peak inSP also means that the closed-loop system is more damped.

−4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

4

5

Re

Im


100

101

102

103

−180

−160

−140

−120

−100

−80

Magnitude

[dB

]

102

−110

−100

−90

−80

Frequency [Hz]

Phase

[◦]

(zoomed)

(b) Bode magnitude diagram of SP

Figure 6.13: Nyquist diagram (left) and FRF of the process sensitivity (right) for C7 (thinblack line) and C8 (thick black line). To show the destabilizing effect of the flexible mode,the curve for C7 without the second order low pass filter is depicted as well in the Nyquistdiagram (thick grey line).

It is also possible to use a skewed notch filter with the zeros located in the ORHP. Thissolution is also encountered when using H∞ loop-shaping, as is illustrated in Chapter 4.Both the NMP zeros and the poles then create phase lag. In this way, there is more designfreedom compared to the first approach and the achievable performance is expected to be atleast as good as for C7. An example of such a controller is given by

C9(s) = 21.5 · 103 ·1

2π10s+ 11

2π75s+ 1·

1(2π110)2

s2 + 2·−0.042π110 s+ 1

1(2π121)2

s2 + 2·0.142π121s+ 1

· 11

(2π190)2s2 + 2·0.6

2π190s+ 1. (6.9)

The sensitivity and process sensitivity for C7 and C9 are compared in Figure 6.14. ControllerC9 achieves a BW of 30 [Hz], implying a faster response, but this also leads a to higher peakin S (waterbed effect). However, the dimensionless damping ratio of the resonance in SP isroughly the same for both controllers.

6.1.4 Limitations in the standard plant setup

If follows from Section 5.4 that MP zeros do not limit the achievable performance in thestandard plant setup. As long as w 6= u and y 6= z, a stabilizing cancellation controller can bedesigned such that the mapping Tzw : w → z is identically zero. However, the absence of NMPzeros does not guarantee that the approximate cancellation controller is stabilizing. Various


100

101

102

103

−50

−40

−30

−20

−10

0

10

Frequency [Hz]

Magnitude

[dB

]

(a) Bode magnitude diagram of S

100

101

102

103

−160

−140

−120

−100

−80

−60

Frequency [Hz]

Magnitude

[dB

]

(b) Bode magnitude diagram of SP

Figure 6.14: FRF of the sensitivity (left) and process sensitivity (right) for C7 (thin blackline) and C9 (thick black line).

practical aspects and the requirement that controllers should always be causal can thereforelead to limitations on the achievable performance that are not discussed in Section 5.4. Thisis illustrated by the following two examples. In Appendix B, similar and more elaborateresults are presented for the flexible cart system introduced in Section 5.1.7.

Example 6.3. Consider the system of Section 6.1.2, i.e., output y1 is measured and per-formance is evaluated at y2. If the reference signal is taken as the exogenous input w, thegeneralized plant G is given by

G =

[1 −P2

1 −P1

]. (6.10)

For m1 = m2 = m = 1 [kg], the stabilizing cancellation controller is given by

CC =1

−P1 + P2= −s2 − 2ds− 2k. (6.11)

However, when forming the semi-proper approximate cancellation CaC , which is done by mul-

tiplying CC with a second order low-pass filter (with the corner frequency at 104 [Hz]), thenew S has poles in the ORHP and the controlled system is unstable. The poles of S for bothcontrollers are summarized in Table 6.1.

Table 6.1: Pole locations of S

for CC for CaC

−6.600 ± 630.8j −6.600 ± 630.8j−3.015 · 103 −940.0 ± 1571j

−125.7 · 103

1.870 · 103

6.2 Inferential control 101

Example 6.4. Consider the system of Section 6.1.3, i.e., output y2 is measured and perfor-mance is evaluated at y1. The generalized plant G is now given by

G =

[1 −P1

1 −P2

](6.12)

and the stabilizing cancellation controller by

CC = s2 + 2ds + 2k, (6.13)

Unlike in Example 6.3, however, multiplying (6.13) by the same second order low-pass filterresults in an internally stable system.5 The poles of S for both controllers are summarized inTable 6.2. Notice, however, that for Ca

C the generalized Bode integral needs to be satisfied.The mapping between w and z is thus not identically zero.

Table 6.2: Pole locations of S

for CC for CaC

−3.300 ± 446.1j −1.326 ± 446.1j−6.599 ± 630.8j −6.600 ± 630.8j

−63.30 · 103

−62.37 · 103

6.2 Inferential control

If no hardware changes can be made, i.e., no actuators and/or sensors can be added orrelocated, the controller design process is faced with the task of improving the actual systemperformance for the case that the performance variables are not measured during normaloperation (inferential servo problem). It is motivated in Chapter 3 that an extra controlDOF is needed for inferential servo problems. Two possible extensions are: indirect control(see Figure 3.6(a)) and inferential control (see Figure 3.6(b)), whereby the latter is preferredbecause of robustness issues (see Section 3.8 for a motivation).

The performance filter Ca uses the measured variable yp and controller output u to createan estimate zp of the performance variable zp. In this research, a Kalman filter is used toestimate the performance variables as is discussed below. Other options are, however, alsopossible and this is discussed in Section 6.2.4.

The structure of the plant combined with the performance filter Ca is depicted in Figure 6.15.Herein triple (A,B,C) is the SS representation of the plant P , triple (A, B, C) is the SSrepresentation of the model P of P that is used for state estimation, Cz is the output matrixof Pz, and finally Cz is the output matrix of the model Pz of Pz. The role of disturbancesignals w and v, and static innovation matrix Ko is discussed later in this section.

The estimation of zp is basically split up into two parts:

5Since there are no pole/zero cancellations in the ORHP between Cac and P1, it is sufficient to evaluate the

poles of S to guarantee internal stability (see Theorem 3.4).


uB

A

xC

x

w v

yp

Cz

zp

B

A

˙xC

x yp

Cz

zp

Ko

Ca

1

s

1

s

Figure 6.15: Plant with performance filter Ca.

1. An estimate x of the state variable x is created by using a Kalman filter. Bycomparing the estimated output yp by the measured output yp, corrections can bemade to ˙x and hence there is some robustness against model uncertainties and unknowndisturbances. Section 6.2.1 discusses this step in more detail.

2. An estimate zp of the performance variable zp is constructed from the estimatedstate variable by using the static mapping Cz, i.e.,

zp = Czx. (6.14)

Since there is no FB loop in this step, there is no robustness for mismatches betweenCz and Cz.

To evaluate the effect of imperfections in the performance filter, the TFs between controlleroutput u and yp, and between u and zp need to be determined. To derive these relations, firstconsider the following expression

˙x(t) = Ax(t) + Bu(t) +Ko (yp(t) − yp(t)) ,

= Ax(t) + Bu(t) +Koyp(t) −KoCx(t). (6.15)

Taking the Laplace transform of (6.15) (with zero initial conditions for x) and rewriting leads


to(sI − A+KoC

)x(s) = Bu(s) +Koyp(s)

= Bu(s) +KoC (sI −A)−1Bu(s)

x(s) =(sI − A+KoC

)−1 [B +KoC (sI −A)−1B

]

︸︷︷︸, Px

u(s), (6.16)

where Px is defined as the mapping between u and x. Next, using C and Cz leads to themappings Py : u→ yp and Pz : u→ zp, respectively, where

Py , C · Px, (6.17)

Pz , Cz · Px. (6.18)

6.2.1 State observer design

Since the state variable x is not available from measurements (only yp is measured), it needsto be reconstructed by a state observer. For a more detailed treatment of observers, see,e.g., Kwakernaak and Sivan (1972). The basic idea is illustrated below.

Consider the system

x = Ax+Bu, (6.19a)

yp = Cx. (6.19b)

The goal is to determine a reconstruction x of the true state x by using the observed signals yp

and u. Such a system is called an observer. Assume the following observer expression

˙x = Ax + Ko(yp − Cx︸︷︷︸, yp

) + Bu, (6.20)

where yp denotes the estimated plant output and Ko is defined as the innovation gain. Sub-tracting (6.20) from (6.19a) leads to

x− ˙x = Ax− (A−KoC) x−Koyp

= (A−KoC) (x− x) . (6.21)

Next, define the observation error e , x− x such that (6.21) can be rewritten as

e = (A−KoC)︸︷︷︸, Aobs

e. (6.22)

It follows from (6.22) that e → 0 for t → ∞, if all the poles of the matrix (A−KoC) arestable. These poles are referred to as observer poles and can be placed at any location in theOLHP if the system is observable.

In the reasoning above, some practical issues are left out to illustrate the basic idea of anobserver. In practice, however, extra elements need to be taken into account. First, a perfect


model of the system is never available. Second, the control signal u is not the only input signalthat affects the system state. External disturbances are also present. Third, the measuredsignal y is often disturbed by measurement noise. These three elements are the main issuesthat need to be taken into account when designing observers and are the motivation for theso-called Kalman filter.

The practical issues mentioned above can be packed together in the following system descrip-tion

x = Ax+Bu+ w, (6.23a)

yp = Cx+ v, (6.23b)

where w is called system noise and v is called observation noise. Model uncertainties, distur-bances, and measurement errors are thus translated into extra signals in the system descriptionof the observed system.

System noise w acts as an extra input signal and is therefore able to affect the system state.External disturbances can therefore be accounted for in w and it also strengthens the assump-tion that plant input signal u is available for measurement, rather than the controller output,since potential errors in measuring u can be accounted for in w. In addition, since w is ableto affect the state variable x, model uncertainties can also be accounted for.

To reflect the presence of noise inputs, a small change in terminology is required. Since thestate cannot be reconstructed exactly anymore, the term “reconstructed state” is replaced by“estimated state” and the observation error e is now referred to as estimation error. Finally,the term “observer” is replaced by the term “filter”.

For simplicity, consider scalar values in the following reasoning. In case the innovation gainKo is large, the difference between yp and yp, which is called the innovation, is multiplied bya large number to require a fast convergence of the estimation error. However, this correctionis incorrect in the presence of observation noise, because yp− yp = C (x− x)+v. The value ofKo is thus limited by the presence of measurement noise. On the other hand, choosing a smallvalue for Ko is neither desired, since it decreases the level of corrections on x. Since the pathof x is also influenced by w, which is not included in the observer, a large innovation gain isrequired to obtain a good estimate of x in the presence of system noise. Hence, there existan optimal value for Ko that depends on the magnitude of the two noise signals (actually theratio between the magnitudes).

Next, the magnitude of the two noise signals needs to be quantified. If the sources of uncer-tainty are unknown or if there are other reasons to not specify noise properties in full detail,the rather extreme assumption of white noise can be made, which is often done in filter design.A scalar stationary white noise signal w has a zero mean and contains all frequencies from 0to ∞, with the same power, i.e., the PSD is a constant value over all frequencies. For generalvector-valued signals, the definition (see Definition 6.5) for white noise given by Kwakernaakand Sivan (1972) is used.

Definition 6.5 (White noise). Consider a zero-mean vector-valued stochastic process w(t)with covariance matrix

Rww(t2, t1) = Vw(t1)δ(t2 − t1), Vw(t) ≥ 0. (6.24)


The process w(t) is said to be a white noise stochastic process with intensity6 Vw(t). In casethe intensity is constant, the process is called a stationary7 stochastic process.

In case the white noise signals w(t) and v(t) satisfy

Rw = E{w} = 0, Rv = E{v} = 0, (6.25)

Rww = E{wwT } = Vw ≥ 0, Rvv = E{vvT } = Vv > 0, (6.26)

the innovation gain Ko that minimizes the variance of the estimation error e(t) = x(t) −x(t) can be obtained using a so-called minimum variance estimator, which was first derivedby Kalman and Bucy (1961) and is therefore often referred to as a Kalman-Bucy or Kalmanfilter.

Remark 6.6. In this and the next chapter the performance filter Ca is designed using aKalman filter. Model uncertainties and measurement noise are hereby taken into account bytuning the noise intensity parameters Vv and Vw, which is similar as in Linear QuadraticGaussian (LQG) control problems. Notice that specifying the ratio between Vv and Vw issomewhat similar to specifying roll-off for the complementary sensitivity function T . Basi-cally, specifying roll-off for T means that high (compared to the roll-off frequency) frequencycomponents in the measured variables should be regarded as measurement noise and no cor-recting control actions should be taken.

6.2.2 Collocated measurement, non-collocated performance

In the case of a perfect reconstruction of zp, i.e., Pz = Pz, Cb in Figure 3.6(b) can be designedas if the performance variables are measured directly. Hence, the controllers designed inSection 6.1 can be used. To illustrate the effect of parameter uncertainty, the observers inthis and the next section are designed by using an incorrect plant model8 with +10% errorin the spring stiffness k (natural frequency f ≈ 105 [Hz]) and −5% error in the absolutedamping coefficient d, i.e., A 6= A. The Kalman filter is created using the command kalman

in the Matlab Control System Toolbox, with

Vw = α · I4, Vv = 1, (6.27)

where α is a scalar to scale the ratio between system noise and measurement noise. First,consider the situation where y1 (= yp) is measured and good tracking of y2 (= zp) is required.In Figure 6.16, plants P and Pz are compared to Py and Pz, respectively, for α = 10. Dueto the change in stiffness and damping, there is a difference between the locations of theanti-resonance and resonance frequencies in the actual plant and the plant used to design

6The term intensity is used, because the variance is infinitely large, which is due to the infinitely fastfluctuations in the process with infinitely large jumps.

7Strictly speaking, this is called wide-sense stationary, which is a weaker form of stationarity to indicatethat only the first (the mean) and second (variance) moments need to be constant.

8To test for parameter uncertainty in a simulation environment, the plant parameters should be adjustedrather than the plant parameters on which the observer is based. Both controller blocks Ca and Cb inFigure 3.6(b) are then based on models of P and Pz. In this section, however, the focus is on imperfections inthe observer only. Hence, the approach taken here is justified.


the performance filter. However, due to the relatively low value of α these differences areconsidered to be caused by observation noise and basically no adjustments are made to x(low innovation gain). Since this resonance is shared with Pz, the same mismatch emergesbetween Pz and Pz.

102

−160

−140

−120

−100

−80

−60

Magnitude

[dB

]

102

−200

−150

−100

−50

0

Phase

[◦]

Frequency [Hz]

(a) P and Py

102

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−120

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−100

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−80

Magnitude

[dB

]

102

−400

−350

−300

−250

−200

−150

Phase

[◦]

Frequency [Hz]

(b) Pz and Pz

Figure 6.16: Bode diagrams of the actual plants (thin line) and the plant models obtainedby using the performance observer (thick line) with α = 10.

When α is increased, the differences between P and P are considered to be caused by systemnoise w and hence significant corrections are applied to x (high innovation gain). In Fig-ure 6.17, a similar comparison is made as in Figure 6.16, but for α = 1.0 · 105. As a result ofthe high innovation gain, the difference between the resonances in P and Py is much smaller.Since this resonance is shared with Pz , Pz also more closely resembles Pz. Further increas-ing the innovation gain also leads to a better approximation of the anti-resonance. Notice,however, that the presence of measurement noise poses an upper bound on α as discussed inSection 6.2.1. Taking model uncertainty and measurement into account when designing Ca istherefore essential.

Remark 6.7. It should be kept in mind that the difference between P and Py is evaluatedin absolute units rather that on a logarithmic scale. Differences between the anti-resonancesare therefore much smaller than they appear on a logarithmic scale and the reverse holds formismatches between the resonances.

Remark 6.8. Increasing the innovation gain basically means that less model knowledge, whichis given by the triple (A, B, C), is used to adjust ˙x such that yp − yp is small. However, evenif the innovation gain is very large, there is still a small delay in the innovation loop that iscaused by the integrator. There will therefore always be a small mismatch between P and Py.


102

−140

−120

−100

−80M

agnitude

[dB

]

102

−200

−150

−100

−50

0

50

Phase

[◦]

Frequency [Hz]

(a) P and Py

102

−130

−120

−110

−100

−90

−80

Magnitude

[dB

]

102

−400

−350

−300

−250

−200

−150

Phase

[◦]

Frequency [Hz]

(b) Pz and Pz

Figure 6.17: Bode diagrams of the actual plants (thin line) and the plant models obtainedby using the performance observer (thick line) with α = 1.0 · 105.

6.2.3 Non-collocated measurement, collocated performance

For α = 10, there are no differences compared to Figure 6.16, except that y2 (= yp) now actsas the measured variable and that performance is evaluated at y1 (= zp). Increasing α leadsto a better approximation of the resonance in both P and Pz , but does not lead to a betterapproximation of the anti-resonance in Pz. The reason for this mismatch most likely has todo with the fact that the anti-resonance is not present in P . More importantly, however, isthat anti-resonance in Pz is approximated by a NMP complex zero pair in Pz. Even if α isincreased to α = 1.0 · 1015, Pz still contains a NMP complex zero pair and there is still anequally large mismatch between Pz and Pz . Simply designing a controller as if zp is measureddirectly, i.e., designing a controller for Pz and not Pz, can thus lead to instability. Takingmodel uncertainties into account when designing Cb is therefore also essential.

6.2.4 Motivation for a general two-DOF control structure

In case Cb, i.e., the standard FB controller block, is designed using norm-based controllerdesign (which is done in the next chapter), both controller blocks are designed using anoptimization algorithm. One optimization step is performed over controller block Ca (filterdesign) and one over Cb (controller design). This has two consequences:

1. Because controller block Cb is also observer based, see Doyle et al. (1989), using sucha controller in combination with a performance filter implies that two plant models arereconstructed in the FB loop. Clearly, this leads to redundant states in the completetwo-DOF controller, i.e., the combination of Ca and Cb. Whether this redundancy


102

−130

−120

−110

−100

−90

−80M

agnitude

[dB

]

102

0

50

100

150

200

Phase

[◦]

Frequency [Hz]

(a) P and Py

102

−140

−120

−100

−80

Magnitude

[dB

]

102

−200

−100

0

100

200

Phase

[◦]

Frequency [Hz]

(b) Pz and Pz

Figure 6.18: Bode diagrams of the actual plants (thin line) and the plant models obtainedby using the performance observer (thick line) for α = 1.0 · 105.

leads to implementation problems depends on the application. In general, however, itis desired the have as few controller states as possible.

2. Since both steps aim at achieving high inferential performance, it is expected that asingle optimization over Ca and Cb leads to better inferential performance. However,the inferential control structure leads to structured control problems in the standardplant setup, which cannot be dealt with by using standard optimal controller synthesisalgorithms.

Due to these shortcomings, the general two-DOF control structure, which is discussed inSection 3.8.3, is proposed as a more suitable alternative to deal with inferential servo problems.Because standard optimal controller synthesis algorithms can be used, the general two-DOFcontroller C, which is defined in (3.41), can be designed in a single optimization step, suchthat C does not have redundant states in its internal state observer. Although the nine-blockcontrol problem is a relatively new, Oomen (2010) has already successfully implementedcontrollers on a real flexible motion system where the performance variable is not measuredduring normal operation.

6.3 Conclusions

In this chapter, conventional and inferential control are evaluated for the control of a twoMSD system (see Figure 6.1), whereby the performance variable is not measured. The mainconclusions are summarized below:

• Section 6.1 shows that for conventional controller design, good performance of the mea-

6.3 Conclusions 109

sured variables does not imply good inferential performance. Controller modificationsaimed at increasing the performance of the measured variables, may therefore lead todeterioration of the truly relevant performance. Taking the difference between measuredand performance variables into account during controller design is therefore essential.

• In case of inferential control (see Section 6.2), a perfect reconstruction of the perfor-mance variable would allow the FB controller (Cb in Figure 3.6(b)) to be designed asif the performance variable is measured directly. Model uncertainties and measurementnoise, however, degrade the estimate of the performance variable and should thereforebe taken into account when designing the performance filter Ca and FB controller Cb.

• In case controller block Cb in the inferential control structure is designed using norm-based controller design, the general two-DOF control structure (see Figure 3.8) is pro-posed as a more suitable alternative to deal with inferential servo problems.

Chapter 7

Control of a wafer stage FEMmodel

This chapter focusses on controller design for next generation stages with force actuatorsand displacement sensors. A FEM model is created with Ansys and three I/O sets arecompared. For all three I/O sets, the Point-Of-Control (POC) is located in the middle ofthe top surface. More information regarding the FEM model, its mode shapes, the three I/Osets, and the decoupling procedure can be found in Appendix D. Table 7.1 summarizes themain characteristics of the I/O sets.

The effect of non-collocation on the presence of CRHP zeros is discussed in Section 7.2.I/O sets 1 and 2 have the same actuator configuration, but a different sensor configuration.The sensors of I/O set 1 are collocated with the actuators, whereas the sensors of I/O set 2are placed in a more realistic non-collocated setting.

I/O set 3 contains two additional actuators and sensors and the actuators and sensors arelocated differently compared to I/O set 2. Although adding the extra actuators and sensorsoffers the possibility to explicitly control the flexible dynamics, the main difference that isfocussed on in Section 7.3 is that the flexible modes are excited in a different way.

In Section 7.4, FB controllers are designed using H∞ loop-shaping for the plant with I/Oset 3. It is already concluded in Chapter 3 that the single-DOF control structure is not wellsuited to deal with the inferential servo problem. The extent to which it fails for a FEMmodel of a potential next generation wafer stage is shown in Section 7.4 by time domainsimulations. In addition, inferential control is invoked and error-based tuning is applied toimprove the truly relevant performance.

Table 7.1: Main characteristics of the I/O sets used in this chapter.

I/O setNumber of actua-

Remarkstors and sensors

1 6 Sensors are collocated with actuators on bottom surface2 6 Same actuators as in I/O set 1, but sensors on top surface3 8 Actuators (at bottom) and sensors (at top) are relocated

111

112 Control of a wafer stage FEM model

7.1 Notation

In Chapter 6, yp, zp, up, etc., are all scalar variables and the physical meaning of thesevariables is rather trivial. For the WS FEM model, however, the elements contained inthese vectors depend on whether the static decoupling matrices are considered part of theplant or controller. Within Philips Applied Technologies, the decoupling matrices are oftenconsidered part of the plant as is shown in Figure 7.1. The controllers are then designed forthe decoupled plant. In line with this reasoning, the symbol “P” is used here to represent thedecoupled plant and vectors yp and up are used to describe the plant outputs and inputs afterdecoupling. Hence, yp ∈ R

6×1 contains the displacements of the POC in the directions of thesix rigid body modes: three translations in the x-, y-, and z-directions,1 and three rotationsaround the x-, y-, and z-axis. The right-handed coordinate system (see Figure D.2), in whichcounter-clockwise rotations are defined positively, is used to define directions. Notice thatdue to the rigid body assumption in the decoupling procedure, the translations and rotationsin yp are in fact estimates of the actual displacements of the POC, which are given by zp.Similarly, vector up ∈ R

6×1 contains forces and torques to independently control the POC inthe six rigid body DOFs and the reference rz also needs to be specified in these directions.Vectors yp, zp, up, and rz are now defined as

yp ,[yx yy yRz yz yRx yRy

]T, (7.1)

zp ,[zx zy zRz zz zRx zRy

]T, (7.2)

up ,[ux uy uRz uz uRx uRy

]T, (7.3)

rz ,[rz,x rz,y rz,Rz rz,z rz,Rx rz,Ry

]T, (7.4)

where Rz, Rx, and Ry represent the rotations around the z-, x-, and y-axis. In Figure 7.1, the

physical plant is represented by P . Plant P has inputs up ∈ Rnu×1 and outputs yp ∈ R

ny×1

that are in terms of nu actuator forces (no torques) and ny translations (no rotations).

TyTu˜P

up ypypup

P

Cu

du

Pz

zp

rz

ez

Figure 7.1: Usage of static matrices Ty and Tu to decouple the physical plant P .

1The x, and y-directions are the in-plane directions and the z-direction is the out-of-plane direction.

7.2 Collocated and non-collocated control 113

7.2 Collocated and non-collocated control

It is shown in Appendix A that collocation of actuators and sensors causes a stable linear sec-ond order system to be minimum phase and that non-collocation is a reason for the existenceof NMP zeros. Figure 7.2, in which the pole-zero maps for I/O sets 1 and 2 are depicted,confirms that this also holds for a flexible wafer stage.

−400 −200 0 2000

1

2

3

4

5x 10

4

Re

Im

(a) For I/O set 1

−400 −200 0 2000

1

2

3

4

5x 10

4

Re

Im

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Figure 7.2: Pole-zero maps of the full state plant (100 modes). Because complex zeros inphysical systems solely exist in pairs, only the positive imaginary part is depicted. The polesare represented by crosses (“×”) and the zeros by circles (“◦”).

It follows from the conclusions of Chapter 5 that due to the absence of NMP zeros for I/O set 1the BW can (at least in theory) be set infinitely high. However, as is already shown inChapter 6, simply applying high gain FB in the single-DOF control structure for collocatedactuators and sensors does not necessarily imply high performance if the performance variablesare not measured. So, although a collocated I/O configuration is beneficial for stability, it doesnot necessarily lead to good inferential performance if no extra control DOF is added.

Adding a pre-filter (indirect control) or performance filter (inferential control) are possibilitiesto improve the inferential performance. With indirect control the filter is placed outside theloop, so the zeros of the plant as seen by the FB controller (Cb in Figure 3.6) are unchanged.With inferential control, however, the filter is placed inside the FB loop and possible NMPzeros can be created in the plant that “is seen” by the FB controller.

For I/O set 2, NMP zeros are present and thus there exists an upper bound on the achievablecross-over frequencies. The positive zero at s = 3.76 · 103 (not visible in Figure 7.2(b)) posesthe strongest limitation with a cut-off frequency of 599 [Hz]. Figure 5.7 shows an example ofa filter used to include such a limitation into the sensitivity integral constraint. To establishthat NMP zeros also limit performance for complex systems like, e.g., wafer stages, H∞ loop-shaping is used in Example 7.1 to design controllers for I/O sets 1 and 2 whereby the desiredcross-over frequencies of the singular values of the shaped plant Ps are chosen sufficientlyhigh, i.e., singular value alignment well above 599 [Hz]. If follows that for I/O set 2 similarlarge values for γmin occur as in Table 5.1.


Example 7.1 (H∞ loop-shaping). If the desired cross-over frequencies, i.e., the singularvalues of Ps, are aligned at 4000 [Hz] and no model reduction is applied prior to controllerdesign, γmin = 4.29 for I/O set 1 and γmin = 45.5 for I/O set 2. Since the singular values of Sare constrained by the Poisson integral constraint (Theorem 5.11) for I/O set 2, the singularvalues of L do not resemble the singular values of Ps and γmin is large.

7.3 I/O selection for flexible stages

Before discussing I/O selection for the WS FEM model, first consider three conclusions fromChapters 3 and 5. If the sensors are located such that the flexible dynamics is not or onlyminimally observed, i.e., by placing the sensors at or near the nodes of a mode shapes, thesystem seems to be rigid. However, the flexible dynamics can then still greatly affect theperformance variables. A similar reasoning applies to the actuators. For the FF signal, itis desired to minimally excite the flexible dynamics. However, if the flexible dynamics isuncontrollable, (unknown) external disturbances that excite the flexible dynamics cannot beattenuated. Hence, it is proposed in Chapter 3 that the sensors should be placed close to thePOC and that the optimal actuator configuration is system dependent. In case of inferentialcontrol, the sensors should thus at least be located such that the performance variables can beinferred from the sensor data and controller outputs.

In addition, Chapter 5 shows that NMP zeros in the TFM between the plant inputs up andthe performance variables zp still limit the achievable performance, even if there are no NMPzeros in the TFM between the plant inputs and the measured variables yp. Relocating thesensors such that there are no fundamental performance limitations on measured outputstherefore does not eliminate the possible limitations (that are discussed in Chapter 5) onthe performance variables. So, the only design freedom in I/O selection that is consideredhere is the actuator configuration. Notice, however, that various practical limitations andthe required causality of the controller are not included in this reasoning. The number andplacement of sensors is thus still a potential design freedom to improve inferential performance.

In this section, the effect of two different actuator configurations on the performance variablesis discussed. In analogy with Appendix D, the plant with I/O set 2 is defined by P 2 andthe plant with I/O set 3 by P 3. The corresponding performance plants P 2

z and P 3z , i.e., the

plants with inputs up and performance variables zp as output (see Figure 7.1), are defined ina similar fashion. To allow for a fair comparison, the actuator transformation Tu is includedin both performance plants, such that the inputs up are in terms of the six rigid body DOFsof the POC.2 The Bode magnitude diagram for inputs ux and uy is shown in Figure 7.3. Theother inputs, i.e., the inputs for the Rz-, z-, Rx-, and Ry-directions, are omitted, becausethe largest actuator forces, which are created by the FF controller, occur in the x- and y-directions. Since the performance variables are the same for both I/O sets, the only differencein Figure 7.3 is due to the way in which the flexible dynamics is excited. The main differencesbetween the two I/O sets in Figure 7.3 are:

1. The actuators used for the x-direction excite the flexible mode with natural frequency

2Since the actuator transformation Tu is included in the performance plant, inputs in the in-plane directions,like, e.g., in the ux-direction, also lead to physical actuator forces (up) in the z-direction.

7.4 Controller design and performance evaluation for P 3 115

fn = 818 [Hz] (see Figure D.2(e)) much heavier for P 2z than for P 3

z . This is visualizedbest by the constant difference in magnitude in the mapping from ux to zRy . However,the effect also manifests itself in the output elements. In the (1, 1)-term, the strongerexcitation of this flexible mode for I/O set 2 leads to a more dominant resonance/anti-resonance pair, which is in general not beneficial for stability (lower GM).

2. Similar to the actuators working in the x-direction, the actuators used for the y-directionexcite the flexible mode with natural frequency fn = 821 [Hz] (see Figure D.2(f)) muchharder for P 2

z than for P 3z .

3. The actuators working in the uy-direction for I/O set 3 do not excite the flexible modewith fn = 553 [Hz] (see Figure D.2(b)), while this mode is excited heavily if I/O set 2is used.

4. In general, the flexible dynamics is excited the least when using I/O set 3. The onlyexception occurs in the mapping from ux to zz for the flexible mode with fn = 667 [Hz].

5. Although not visible in Figure 7.3, the most detrimental NMP zero, i.e., the zero withthe lowest undamped eigenfrequency, is located at s = 3.90 · 103 (undamped eigen-frequency at f = 621 [Hz]) for P 3

z and at s = 1.02 · 104 ± 1.22 · 103i (undampedeigenfrequency at f = 1.63 · 103 [Hz]).

When only looking at the location of the NMP zeros, I/O set 2 is preferred. In Section 7.4,however, it is shown that for this specific application other effects are more detrimental forperformance. The effect of NMP zeros is therefore not considered relevant for I/O selectionat present. Following the conclusions from Chapter 3, the actuator configuration of I/O set 3is thus preferred if there are no unknown external disturbances present, because the flexibledynamics is not excited as heavily as for I/O set 2.

Remark 7.2. I/O set 3 has two more actuators than rigid body modes (over-actuation).Although this freedom is not utilized with the current actuator transformation Tu, the actuatorexcess can be used to explicitly control (part of) the flexible dynamics, which potentially leadsto better performance. More information on over-actuation and over-sensing can be foundin Schneiders et al. (2003, 2004) and Van Wingerden (2004).

7.4 Controller design and performance evaluation for P 3

7.4.1 Control objective

The fist step of the controller design process (see Section 3.2) is to specify the control objec-tive. For general wafer stage control, the objective is to let the part of the silicon wafer thatis subject to light exposure follow a prescribed reference trajectory. In this chapter, node 24,which is located in the middle of the top surface, is defined as the center of this area and isreferred to as the POC. Notice that for a real WS this point travels around during normaloperation. The performance variable zp thus contains the three translations and three rota-tions of the POC, as defined by (7.2). The translations of the POC can be obtained directlyfrom the FEM model. The rotations, however, need to be approximated from the translations


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200 400 600 800 1000−240

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Frequency [Hz]

Mag

nit

ude

[dB

]

Figure 7.3: Bode magnitude diagram of Pz for I/O set 2 (thin line) and I/O set 3 (thick line)on input signals ux and uy.


of the surrounding nodes. The procedure to obtain zp from the FEM model is discussed indetail in Appendix D.

Remark 7.3. For wafer stage control, defining the center of the area subject to light exposureas the POC is a straightforward choice. It is, however, also allowed to include multiple pointsor a spatial norm in zp. Adding more points in zp will most likely lead to a controller thatimproves damping of the system. Although this is not investigated for the WS FEM model,it is shown in Chapter 4 that something similar occurs for a two MSD system. Moreover,mode shapes that correspond to the low frequency flexible dynamics are usually such that thereare no large differences between points located close to each other. Hence, it is expected thatincluding points in zp that are located close to the POC will not significantly improve theperformance, but only complicate the controller design process.

Trajectory generation

For next generation stages, the required large acceleration forces in the x- and y-directionsare expected to cause the most significant excitation of the flexible dynamics. Hence, usinga realistic reference signal and FF controller to supply these large forces is essential in asimulation experiment. For the simulations performed in this section, the reference in x-direction is chosen to be the third order point-to-point move, i.e., a single scan movement,that is depicted in Figure 7.4. The references for the other directions are kept zero.

Table 7.2: Reference parameters.

Parameter Value Unit

xb 100 mmvb 1.00 m s−1

ab 100 m s−2

jb 5.00 · 104 m s−3

The values for the x-translation3 xb, the maximum velocity4 vb, the maximum acceleration ab,and the maximum jerk5 jb are given in Table 7.2 and are chosen such that the referencetrajectory is suitable for next generation wafer stages.

The exposure process would ideally take place during the entire point-to-point movement. Inthat way, the highest wafer throughput is achieved. At present, however, the errors duringthe acceleration and deceleration phase are simply too high for accurate exposure. Hence,exposure is only performed during the constant velocity phase tc (see Figure 7.4). The timeinstant at which the constant velocity phase begins is defined here as t∗. Since t∗ may slightlyvary with varying settings of the simulation parameters, like, e.g., the sampling frequency,t∗ , 12 [ms] without loss of generality.

3Under normal operation this value is typically a factor three smaller. For performance evaluation purposes,however, it is desired to have a large time span where the stage moves with a constant velocity.

4This velocity is also often referred to as the scanning velocity, because the original pattern on the reticlestage is also moving.

5The jerk of a signal x(t) is defined as the time derivative of the acceleration a(t) of x(t): j(t) , a(t)


0 20 40 60 80 100 120

0

Pos.

0 20 40 60 80 100 120

0

Vel

.

0 20 40 60 80 100 120

0

Acc

.

0 20 40 60 80 100 120

0

Jer

k

Time [ms]

xb

vb

ab

−ab

jb

−jbtc

t∗

Figure 7.4: Third order reference trajectory for the x-direction.

Due to the large errors in position immediately after acceleration, i.e., at t = t∗, the expo-sure process needs to be further postponed until all performance variables are within theirspecifications. The time instant t at which all the performance variables are within theirspecifications is defined here as the settling time ts ≥ 0, because it is strongly related tosettling time as defined in Chapter 3. Notice that this definition is slightly different fromwhat is common within Philips Applied Technologies.

Performance quantification

For the simulations performed in this section, the maximum allowable tracking error for thetranslations in the x-,y-, and z-direction is set to be 1 [nm]. No error specifications are givenfor the rotations, since these specifications are in general less stringent, see, e.g., Van de Wal(2002), and therefore usually do not limit the achievable performance. Hence, if time domainresponses are discussed, rotations are not included.


Besides creating very fine IC patterns, wafer scanners also need to have a high wafer through-put. Hence, the large maximum jerk and acceleration parameters in Table 7.2. However,if there is a large period of time between t∗ and ts, i.e., if it takes a long time before theperformance variables are within their required specifications, there is no use in demandingsuch aggressive setpoints. It is therefore important to minimize ts. In case ts < t∗, the per-formance variables are also within their specification during the acceleration phase and theexposure process can thus begin before t∗.

Next, settling times similar to ts are defined for each translational direction. Variables ts,x,ts,y, and ts,z are used to represent settling times for the x-, y-, and z-direction, respectively,and are used later in this section for performance evaluation.

7.4.2 Model reduction

The full state plant model contains 200 states (100 modes). Although the full state plantmodel is used to perform the simulations, the order of the plant needs to be reduced to anumber that is manageable for controller design. Only open-loop plant reduction is consideredhere. A balanced system representation is obtained first by using the Matlab commandbalreal. In this way, a state transformation is performed such that the controllability andobservability Gramians are equal and diagonal and their diagonal entries contain the Hankelsingular values in a descending order (from large to small). Small Hankel singular valuesindicate that the corresponding states have a small contribution to the plant IO behaviorand can therefore be removed to simplify the system. Subsequently, the Matlab commandmodred is used to remove these states from the model. Flexible modes with a higher naturalfrequency usually, but not always, have a smaller contribution to the IO behavior and arethus often removed in the model reduction step.

For inferential control, it is crucial that states with a small contribution to the measuredvariables yp, but a large contribution to zp are not removed from the model. Otherwise, it isnot possible to determine a good estimate of zp. To ensure that these states are not removed

from the model, open-loop model reduction is performed on plant Pyz, which is defined as

Pyz ,

[Pz

P

], (7.5)

and where Pz denotes the performance plant without the actuator transformation matrix Tu.The reduced order plant needed for controller design can then be extracted straightforwardlyfrom Pyz . In Figure 7.5, the Hankel singular values of Pyz are depicted for I/O set 3. TheHankel singular values of the first 12 states correspond to the six rigid body modes and arelocated (by definition) at infinity. Hence, the first Hankel singular value that is visible inFigure 7.5 is the thirteenth. The states corresponding to the Hankel singular values that arelocated below the dashed line in Figure 7.5 are removed from the plant model. The resultingreduced order plant model thus contains 26 states: six rigid body modes and seven flexiblemodes.


0 10 20 30 40 50 60 70 80 90 100−180

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Hankel singular value number

Magnitude

[dB

]

Figure 7.5: Hankel singular values of Pyz for I/O set 3. States corresponding to the Hankelsingular values below the dashed line are removed from the model used for controller design.

7.4.3 Controller design for I/O set 3

If a perfect model of the system and its disturbances is available, perfect tracking can beachieved by FF steering. Unfortunately, no such models exist in practice and there will alwaysbe imperfections in the FF signal. FB control is therefore essential. Since there is only littleinformation about the external disturbances in next generation wafer stages, imperfections inthe FF signal are created here by using standard mass FF. So, internal dynamics is not takeninto account and the flexible modes are excited. The goal of this section is now stated moreformally as to compare how conventional control, i.e., using the single-DOF control structureand assume that the performance variables are measured, and inferential control can deal withthese imperfections. For both control approaches, the FB controllers are designed by usingH∞ loop-shaping (see Chapter 4) to deal with plant interaction and the plant translationsand rotations are scaled by 10 [nm] and 100 [nrad], respectively.

Remark 7.4. The nine-block control problem (see Section 3.8.3) does not have the sameshortcomings of inferential control as discussed in Section 6.2.4. However, solving the nine-block problem for a complex system like a wafer stage may lead to other problems that areunknown at present. By adopting the inferential control structure the control problem is splitup into two parts, which both can be solved using knowledge from conventional control, i.e.,Kalman filter design and FB controller design for servo problems.

Conventional control

Conventional control uses the single-DOF control structure depicted in Figure 3.1. H∞ loop-shaping is used to design the controller C for the decoupled plant P 3, i.e., the plant withthe measured variables (yp) as output. In doing so, internal dynamics is neglected to someextent, because it is assumed that good performance at yp also implies good performanceat zp. Plant interaction, however, is taken into account.

The 6 × 6 shaping filters W1 (at plant input) and W2 (at plant output) are chosen to be


W1 = I6 andW2 = Wy ·Wint ·Wlead ·Wroll, (7.6)

where Wy is a constant 6 × 6 diagonal matrix and Wint, Wlead, Wroll are classical SISOloop-shaping filters used to specify integral action, phase lead, and roll-off. These SISO loop-shaping filters are created by using standard loop-shaping rules, see, e.g., Steinbuch and Norg(1998), such that

Wint =s+ 2πfint

s, fint =

1

5· fdes

c (7.7)

Wlead =

12πfdes

c /αs+ 1

12πfdes

c ·αs+ 1, α = 3 (7.8)

Wroll =1

1(2πfroll)

2 s2 + 2β2πfroll

s+ 1, froll = 6 · fdes

c β = 0.7. (7.9)

The only parameters that need to be specified are the desired cross-over frequency fdesc and

the diagonal elements (gains) of Wy. The gains of Wy allow the singular values of the shapedplant Ps to be shifted such that they cross the 0 [dB] line around the desired cross-overfrequency. For this research, fdes

c = 150 [Hz] and

Wy = Wy,y , diag(1.00 · 10−2

[17.5 17.5 8.60 20.0 4.60 4.60

]), (7.10)

for conventional control. The desired cross-over frequency is thus a factor three below thefirst resonance mode. Higher values for fdes

c are possible and also lead to successful controllerdesign. In addition, if the filters in (7.7)–(7.9) were diagonal, it would be possible to specifydifferent desired cross-over frequencies for each singular value. This may be useful whenbetter tracking in a particular output direction is required. For this research, however, thegoal is not to achieve the highest possible cross-over frequency, but to illustrate the effect of theflexible dynamics on the actual performance. Hence, the lower desired cross-over frequency.Since only classical SISO loop-shaping filters are used in W2, W1 and W2 as used here aredefined as “classical shaping filters”.

Inferential control

In the inferential control structure (see Figure 3.6(b)), performance filter Ca creates an es-timate zp of zp and Cb acts as a conventional FB controller. The Kalman filter included inCa uses the reduced order plant model for state estimation. Hence, exact reconstruction ofthe performance variables zp is not possible, i.e., zp 6= zp. The performance filter basicallyacts as a low-pass filter, since the high-frequency dynamics is typically removed in the modelreduction step. Similar as for the two MSD model, the Kalman filter is designed with theMatlab command kalman and the used noise intensity matrices are

Vw = 10 · I26, Vv = I8. (7.11)

Next, H∞ loop-shaping is used to design controller block Cb as if zp is measured directly,i.e., H∞ loop-shaping for the (reduced order) plant P 3

z . Similar shaping filters as for thesingle-DOF control structure are used, except

Wy = Wy,z , diag(1.00 · 10−2

[17.5 17.5 8.60 15.4 4.10 4.10

]), (7.12)


such that the singular values of Ps cross the 0 [dB] line at approximately 150 [Hz] (= fdesc ).

Remark 7.5. In Section 6.2, it is shown that designing Cb as if zp is measured directly canlead to instability. Uncertainty therefore needs to be taken into account. In this chapter,however, the model used in the performance filter closely approximates the model used toperform simulations. The only differences occur at frequencies located well-above (at least afactor 5) the desired cross-over frequency.

7.4.4 Performance evaluation for I/O set 3

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Im

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Figure 7.6: Characteristic loci.

The characteristic loci are computed (see Figure 7.6) to show the achieved stability margins6

and the obtained values of γmin are presented in Table 7.3. Since simulations are performedwith the full state plant model and the controller is based on a reduced order plant model, thecharacteristic loci are determined for the full state plant model to guarantee stability of theimplementation. Furthermore, for inferential control the effect of the performance filter Ca isalso included in Figure 7.6.

From inspection of the characteristic loci, it follows that both implementations are stable.Similar to the controllers designed in Chapters 4 and 6, the resonances of the FEM model areturned away (by the robust stabilization step in H∞ loop-shaping) from the point (-1,0) toachieve closed-loop stability. For some resonances, this is accomplished by using NMP zeros.Bode magnitude diagrams of C (for conventional control) and Cb (for inferential control), thefour closed loop TFMs of the four-block control problem, and the singular value plots of Ps,L, LI are included in Appendix E for completeness.

The performance of both control approaches is evaluated mainly in the time domain byperforming simulations for the reference signal introduced in Section 7.4.1. The performancemeasures used here are the tracking errors during the constant velocity phase tc, i.e., the

6Recall from Chapter 3 that classical PM, GM, and MM are only defined for SISO systems and no trivialextension exist for MIMO systems. Some generalizations to MIMO systems are possible and are included inSection 3.7.


∞-norm of the translation components of the error signal after acceleration, and the settlingtimes ts,x, ts,y, and ts,z. Time domain simulations are performed using Matlab Simulink, witha discrete solver (ODE3) and a sampling frequency (Fs) of 30 [KHz]. The error signals usedfor performance evaluation are

ez = rz − zp, (7.13)

ez = rz − zp, (7.14)

ey = rz − yp, (7.15)

and are depicted in Figures 7.7(a) and 7.7(b). In the single-DOF control structure the per-formance estimator Ca is not present and yp equals zp. Hence, ey is replaced by ez whenconsidering conventional control. Although this may seem confusing at first, recall that theoutput yp of the decoupled plant is in fact also an estimate of zp. This estimate, however,is obtained by assuming rigid body behavior. Practically, for both control structures ez thusdenotes the error for the actual performance variables and ez denotes the estimated (or mea-sured) error, i.e., the error that is available during normal operation to evaluate performance.

The settling times and maximum error values are retrieved from Figures 7.7(a) and 7.7(b)and summarized in Table 7.3. To allow for a better comparison between the error signals andthe control approaches, the PSDs7 of ez(t) and ez(t) are also included in Figure 7.7. Basedon Figure 7.7 and Table 7.3, the following observations and conclusions are made.

Table 7.3: Overview of the settling times and maximum errors for t > t∗ in the x-, y-, andz-directions for the conventional control and inferential control

Conventional Inferentialcontrol control

Performance evaluated at yp (= zp) zp yp zp zp

Settling times [ms]ts,x 47.8 45.5 47.8 45.1 46.3ts,y 31.5 28.1 23.7 12.2 15.3ts,z 15.4 57.5 38.7 56.1 56.1

‖e(t)‖∞ for t > t∗ [nm]x - dir 231 228 247 241 251y - dir 25 24 14 0.97 1.0z - dir 1.6 6.8 6.1 8.6 8.9

γmin 2.365 2.769

For the x-direction

Since the reference is in the x-direction, the largest actuator forces emerge in this direction.During the acceleration phase, the stage temporarily deforms due to the large actuator forcescreated by the FF controller. Since these elastic deformations are neglected in the FF controllerdesign (perfect mass FF is used), the errors are large (> 100 [nm]) as shown in Figure 7.8.

7To obtain more accurate estimates of the PSDs, the time span of the constant velocity phase is increased(by changing the length of the point-to-point move) to approximately 1.4 [s] and only values of the error signalsat t > t∗ are included in the PSD estimation.


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]

0 20 40 60−5

0

5

Err

or

(y)

[nm

]

0 20 40 60−5

0

5

Err

or

(z)

[nm

]

Time [ms]

ts,x

ts,y

ts,z


0 500 1000 150010

−20

10−15

10−10

10−5

PSD

(x)

[nm

2/H

z]

0 500 1000 150010

−20

10−15

10−10

10−5

PSD

(y)

[nm

2/H

z]

0 500 1000 150010

−20

10−15

10−10

10−5

PSD

(z)

[nm

2/H

z]

Frequency [Hz]

666

1276

1276

(c) Conventional control

0 500 1000 150010

−20

10−15

10−10

10−5

PSD

(x)

[nm

2/H

z]

0 500 1000 150010

−20

10−15

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PSD

(y)

[nm

2/H

z]

0 500 1000 150010

−20

10−15

10−10

10−5

PSD

(z)

[nm

2/H

z]

Frequency [Hz]

1280

663

820

1279

(d) Inferential control

Figure 7.7: Time responses (above) and PSDs (below) of ez (black line) and ez (grey line)for conventional (left) and inferential (right) control.


0 2 4 6 8 10 12 14−400

−200

0

200

400E

rror

(x)

[nm

]

Time [ms] (t∗)


0 2 4 6 8 10 12 14−400

−200

0

200

400

Err

or

(x)

[nm

]

Time [ms] (t∗)


Figure 7.8: Error signals ez (black line) and ez (grey line) during the acceleration phase.

Regarding how well ez (which is fed to the FB controller) approximates ez, first consider theconventional control. Since a static matrix Ty is used to approximate ez from the measuredvariables, the elastic deformation is not taken into account and there is a large (> 100 [nm])difference between ez and ez (= ey) (see Figure 7.7(a)). Control actions taken by the FBcontroller during acceleration are thus not aimed at improving the truly relevant performance.At the end of acceleration, i.e., at t = t∗, the elastic deformation caused by the large FFforces is gone8 and hence ez better approximates ez. Small differences, however, remain toexist since the dynamics is observed differently at the sensor locations than at the POC. Theinferential control structure contains a dynamic performance filter Ca that is better suitedto estimate the elastic deformation during the acceleration phase. Hence, ez approximatesez much better during the acceleration phase (see Figure 7.7(b)). Furthermore, since theKalman filter only uses a model that contains the low-frequency dynamics, ez is basically alow-frequency estimate of ez. This is also confirmed by the PSD in Figure 7.7(d). The lowpower value of the flexible mode with fn = 1278 [Hz] (see Figure D.2(g)) implies that it hasa negligible contribution to ez. Notice that for the single-DOF structure the contribution ofthis mode to ez is much higher (see Figure 7.7(c)), because Ty does not act as a low-passfilter.

With respect to the performance achieved by the FB controllers, i.e., C for the single-DOFstructure and Cb for the inferential structure, there are no large differences. The large settlingtimes (±45 [ms]) are mainly due to the imperfect FF controller, which leads to the large errorsimmediately after acceleration (at t∗), and the low BW of the FB controllers. For the systemconsidered here, slightly increasing the BW (not shown here) already leads to much lowersettling times.

For the y-direction

A similar reasoning as for the x-direction applies to the y-direction. The only difference isthe absence of the FF signal, which leads to much smaller error values at t∗ and thus to muchlower settling times.

8Since forces caused by the FB controller are still present, there are still internal deformations. These forces,however, are usually much smaller than the forces created by FF controller and so the elastic deformations arealso much smaller.


For the z-direction

The main difference between the z-direction and the x- and y-directions is the decreasedstiffness of the wafer stage in the z-direction. In conventional control, this decreased stiffnessleads to severe mismatches between the actual performance (ez) and the measured perfor-mance (ey = ez). Because the measured performance (ts,z = 15.4 [ms] for yp) is much betterthan the actual performance (ts,z = 56.1 [ms] for zp), one could falsely conclude that themachine is operating well within the needed specifications for exposure, while this is in factnot true. Evaluating the performance at zp is therefore essential. It follows fromFigure 7.7(c) that the flexible mode with fn = 666 [Hz] severely limits the performance. Al-though the maximum error in zp is only 6.8 [nm], the undamped nature of this flexible modeleads to a very long settling time. Because of the mode shape, this flexible mode is referredto as the “umbrella mode” from here on.

By including a performance estimator (inferential control), a better estimate of zp is availableto the FB controller Cb. However, the controller does not take the opportunity to improvethe performance, since it only designed using classical shaping filters (see Section 7.4.3) witha desired cross-over frequency of only 150 [Hz]. The flexible mode with fn = 666 [Hz], whichlimits the performance, is therefore not explicitly addressed and hence the settling time forzp in Table 7.3 is large. In the next section, an inverse notch filter is added to W2 to betteraddress this flexible mode.

7.4.5 Error-based shaping filter selection

From the previous section it follows that:

• For the x- and y-directions the excitation of flexible dynamics is not the most perfor-mance limiting effect. The combination of an imperfect FF signal and a low gain FBcontroller has a much larger contribution to the large settling times. Model-based FBcontrol for these direction is therefore not a necessity.

• Excitation of the umbrella mode causes the performance in z-direction to be well outof the required specification, while this is not observed in the measured signals. Byadopting the inferential control structure, a better estimate of zp is fed to the FBcontroller.

In this final part, error-based weighting filter selection is used to improve the performance inz-direction when using the inferential control structure. The same performance filter Ca isused as in the previous section. The mapping P 3

z SI : up → zp describes the relation betweenthe FF signal and the performance variable zp. However, because for FB controller designit is assumed that the performance variables are measured directly, P 3

z SI equals the processsensitivity SP for plant P 3

z . Similar as in Section 4.5, SP is reduced locally by applying aninverse notch filter. However, because the system considered here has six outputs, only the(4, 4)-term of W2, which corresponds to output zz, is extended with an inverse notch filter at666 [Hz]. The (4, 4)-term of the error-based shaping filter W2 is compared with the classicalW2, which is given by (7.6), in Figure 7.9(a).


100

101

102

103

104

−30

−20

−10

0

10

20

30M

agnitude

[dB

]

Frequency [Hz]

(a) (4, 4)-Term of W2

−4 −2 0 2 4 6 8−5

−3

−1

0

1

3

5

Re

Im

(b) Characteristic loci

Figure 7.9: Bode magnitude diagram (left) of the (4, 4)-term of the classical shaping filterW2 (thick line) and error-based shaping filter W2 (thin line) and the characteristic loci (right)for controller Ceb

b .

The resulting controller is defined as Cebb and leads to γmin = 3.133. Bode magnitude diagrams

of Cebb , the singular values of Ps, L, LI , and the four closed-loop TFMs of the four-block

control problem are included in Appendix E. Just as in Figure 7.6, the characteristic loci arecomputed (see Figure 7.9(b)) to check for stability of the closed-loop system with the fullstate plant model and the performance filter. From inspection of the characteristic loci itfollows that the controlled system is stable. In Figure 7.10, the (4, 1) component of SP iscompared for Cb and Ceb

b . Clearly, the large resonance for Cb, which corresponds to the largeoscillatory response of zp in Figure 7.7, is gone.

101

102

103

−90

−80

−70

−60

−50

−40

Magnitude

[dB

]

Frequency [Hz]

Figure 7.10: (4, 1)-component of SP for controller Cb (thin line) and Cebb (thick line).

The time domain results and corresponding PSDs are depicted in Figure 7.11. The settlingtimes and maximum error values for t > t∗ are summarized and compared with Cb in Table 7.4.Not surprisingly, there are only minor differences in the x-, and y-directions. As intended,however, there is a large improvement in the z-direction: the settling time for zp reduces from56.1 [ms] to 20.7 [ms]. If the acceleration phase (0 ≤ t < t∗ = 12 [ms]) is subtracted from


0 20 40 60−10

−5

0

5

10E

rror

(x)

[nm

]

0 20 40 60−5

0

5

Err

or

(y)

[nm

]

0 20 40 60−5

0

5

Err

or

(z)

[nm

]

Time [ms]

ts,x

ts,y

ts,z

(a) Time responses

0 500 1000 150010

−20

10−15

10−10

10−5

PSD

(x)

[nm

2/H

z]

0 500 1000 150010

−20

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10−5

PSD

(y)

[nm

2/H

z]

0 500 1000 150010

−20

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10−10

10−5

PSD

(z)

[nm

2/H

z]

Frequency [Hz]

662

820

8191279

1279

(b) PSDs

Figure 7.11: Time responses (left) and PSDs (right) of ez (black line) and ez (grey line) forthe system controlled with Cb and Ceb

b .

these settling times,9 the reduction corresponds to an improvement of 80%. Inferential controlis thus able to deal with the flexible dynamics and there is no need to change the mechanicaldesign of the system. The error in the x-direction now limits the wafer throughput, butexposure can still start approximately 22% earlier: at t = 46.3 [ms] compared to t = 57.6 [ms]for Cb. Further tuning may lead to more performance improvement. It also follows fromTable 7.4 that the measured performance (at yp) is worse for Ceb

b . Practically, the edges ofthe stage oscillate with a large magnitude, while the center oscillates within the requestedtolerance band.

Remark 7.6. Notice that there is still a small (in magnitude) 662 [Hz] oscillation in thez-direction in Figure 7.11(a), while this undamped resonance is not present in Figure 7.10.The reason is that there is still an undamped resonance in the mapping between rz and zp.

To show that the error-based shaping filter W2 only works in the inferential control structure,consider controller Ceb that is synthesized using the same error-based shaping filter W2 asFigure 7.9(a), but for plant P 3 and not P 3

z . The time domain result for the z-direction isshown in Figure 7.12. Although the effect of the umbrella mode on yp is now decreased, i.e.,better measured performance, the actual performance remains unchanged. To some extent,this result is similar to using controller C3 instead of C2 in Chapter 6. The inverse notch in

9This is justified because exposure typically does not start during acceleration.

7.5 Conclusions 129

Table 7.4: Overview of the settling times and maximum errors for t > t∗ in the x-, y-, andz-directions for the inferential control structure with classical shaping filters and error-basedtuning

Classical shaping Error-basedfilter selection filter selection

Performance evaluated at yp zp zp yp zp zp

Settling times [ms]ts,x 47.8 45.1 46.3 48.6 45.4 46.3ts,y 23.7 12.2 15.3 23.2 15.1 15.2ts,z 38.7 56.1 56.1 47.4 19.6 20.7

‖e(t)‖∞ for t > t∗ [nm]x - dir 247 241 251 267 262 269y - dir 14 0.97 1.0 15 1.4 1.3z - dir 247 241 251 8.5 7.5 8.1

γmin 2.769 3.133

C3 significantly improves the measured performance (at m1) at the cost of deterioration ofthe inferential performance (weaker damped oscillations in m2).

0 10 20 30 40 50 60−5

0

5

Err

or

(z)

[nm

]

Time [ms]

(a) Time response

0 500 1000 150010

−20

10−15

10−10

10−5

PSD

(z)

[nm

2/H

z]

Frequency [Hz]

666

(b) PSD

Figure 7.12: Time response (left) and PSD (right) of the z-component of ez (black line) andez (grey line) for controller Ceb in the single-DOF control structure.

7.5 Conclusions

In this chapter, a first step is made towards the control of next generation wafer stages bystudying a wafer stage FEM model with three I/O sets. The main conclusions are summarizedbelow:

• Non-collocation of the actuators and sensors, which can hardly be avoided in practice,leads to NMP zeros in the decoupled FEM model and poses fundamental limitationson the achievable performance when using the single-DOF control structure. However,for the model considered in this chapter these NMP zeros are located at such highfrequencies that other effects are more detrimental for the inferential performance.

• Changing the actuator configuration significantly changes the severity to which the


flexible dynamics is excited. Combined with the conclusions from Chapter 5 regardinglimitations in the standard plant setup, this motivates the importance of profound toolsfor actuator/sensor selection for next generation stages.

• For the high-acceleration single scan movement performed in this chapter, excitationof the umbrella mode severely limits the performance in the z-direction, while thisis not observed in the measured variables when using conventional controller design.Distinguishing between the measured variables and the inferential performance duringcontroller design is therefore essential.

• Adopting the inferential control structure whereby the performance filter contains aKalman filter for state estimation leads to an improved estimate of the performancevariables compared to using the conventional sensor transformation Ty. This allows theperformance limiting effect of the umbrella model to be addressed by error-based shapingfilter selection and there is no need to redo the mechanical design of the wafer stage. Itis possible to decrease the settling time for the z-direction by 80%, which benefits waferthroughput. Fine-tuning may even further increase the inferential performance.

• Due to the higher stiffness of the WS in the x- and y-directions, flexibilities are lessdetrimental for performance in these directions. The combination of a standard massFF signal and a relatively low gain FB controller (BW at 150 [Hz]) has a much largercontribution to the large settling times than the excitation of the weakly damped flexibledynamics.

Chapter 8

Conclusions and recommendations

8.1 Conclusions

New trends in mechatronic positioning systems such as wafer stages cause these systems tobecome more flexible. In case the performance variables of such systems are not measuredduring normal operation, the control problem is called an inferential servo problem and theactual performance is referred to as inferential (servo) performance.

By performing simulations on a FEM model of a potential next generation waferstage, it is established that conventional controller design, i.e., using the single-degree-of-freedom control structure (see Figure 3.1) and assuming rigid body behavior, isnot suitable to deal with inferential servo problems: it is shown that flexibilities limitthe inferential performance, while this is not observed by the sensors. It is therefore essentialto take internal dynamics into account for the control of next generation wafer stages.

Furthermore, it is shown that the single-degree-of-freedom control structure andthe well-known four-block control problem are not able to deal with inferen-tial servo problems. This also implies that standard closed-loop functions, like, e.g., thesensitivity function and complementary sensitivity function, are inadequate to evaluate theinferential performance.

To deal with the inferential nature of the problem by control design, an extracontrol degree-of-freedom needs to be included. Hereto three alternative two-degree-of-freedom control structures are proposed:

1. The indirect control structure (see Figure 3.6(a)).

2. The inferential control structure (see Figure 3.6(b)).

3. The general two-degree-of-freedom control structure (see Figure 3.8).

The following specific conclusions are identified:

1. Adopting the inferential control structure (second alternative listed above)for the control of a wafer stage FEM model leads to improved inferentialperformance. In the inferential control structure, the conventional feedback loop is

131

132 Conclusions and recommendations

extended with a performance filter to estimate the performance variables. In this way, abetter estimate of the performance variables is obtained compared to using the conven-tional sensor transformation. This allows the performance limiting flexibilitiesto be addressed by conventional feedback controller design and there is noneed to redo the mechanical design of the wafer stage. In the same simulationenvironment as used for conventional control, the settling time for the z-direction is de-creased by 80%, which benefits wafer throughput. Fine-tuning of the feedback controllermay even further improve the inferential performance.

Since the extra control degree-of-freedom in inferential control acts as a performancefilter, both controller blocks can be designed independently using classical controllerand observer design methodologies. In addition, because the extra controller block isincluded in the feedback loop, which is not the case in indirect control (first alternativelisted above), robustness to model uncertainty and measurement noise can be taken intoaccount to some extent by using, e.g., a Kalman filter.

2. To enable the use of standard controller synthesis algorithms, the generaltwo-degree-of-freedom controller (third alternative listed above) needs to beadopted. Since the indirect and inferential control structures lead to structured controlproblems in the standard plant setup, the resulting controller synthesis problem cannotbe dealt with by these standard algorithms. Adopting the general two-degree-of-controlstructure leads to a nine-block control problem.

3. Important interpolation and integral constraints are presented for the stan-dard plant setup. If needed, more constraints can be derived. Classical performancelimitations, including the Poisson integral constraints, are also valid in the standardplant setup. However, these constraints do not necessarily limit the inferential perfor-mance in case the performance variables are not measured. The reverse is also true:there may be strong limitations on the performance variables that do not limit themeasured variables. This is also confirmed by simulation examples.

4. Actuator/sensor selection is important for flexible motion systems, because itdetermines the extent to which the flexible dynamics is actuated, sensed, and observed inthe performance variables. Creating a stiff relation between the actuators and sensorstherefore does not imply a stiff relation between the actuators and the performancevariables. The optimal actuator/sensor configuration strongly depends on the system(and its disturbances).

8.2 Recommendations

Regarding the control of flexible motion systems there are numerous control challenges,see Van de Wal (2009), that require further investigation. Related to this research, thefollowing recommendations are made:

1. The general two-degree-of-freedom control structure is proposed as a moresuitable alternative to deal with inferential servo problems, because theindirect and inferential control structure lead to structured control problems

8.2 Recommendations 133

in the standard plant setup. The two-degree-of-freedom controller can then bedesigned in a single optimization step.

However, solving the corresponding nine-block control problem even further complicatesthe selection of good weighting filters. Although some promising results have beenobtained in Oomen (2010) for simple flexible systems, more research on how to selectthese weighting filters is needed to successfully design two-degree-of-freedom controllersfor more complex MIMO systems.

2. Fine-tuning controllers is essential in high performance motion systems. Due to theMIMO nature of the problem, however, fine-tuning the multivariable opti-mal controller, either directly or by adjusting the weighting filters to syn-thesize the controller, is complex and further research is needed. In practiceand in this research, the weighting filters used in norm-based controller design are oftenonly used to roughly specify the loop-shape of particular transfer function matrices.This can either be done by shaping the open-loop singular values (H∞ loop-shaping) orby shaping particular closed-loop transfer functions. It is then up to the optimizationalgorithm to determine a controller, which is referred to as the optimal controller, thatminimizes a norm (often the H∞ norm) of a set of weighted closed-loop transfer func-tion matrices. However, the optimal controller will most likely not achieve the desiredspecifications on the real system and fine-tuning is necessary.

A good stating point for further research is Douma et al. (2003). The basic idea is thatthe controller can also be represented as a Youla-type “perturbation” on the presentcontroller. This results in the so-called Double-Youla parametrization in which smallperturbations on the present controller are allowed. This turns out to be very useful forre-tuning the controller.

3. A mixed H2/H∞ control problem potentially leads to significant performanceimprovements and further research on this topic is recommended. In norm-based controller design, an H∞ norm is often employed to derive guaranteed stabilitymargins for unstructured uncertainty models. Furthermore, employing an H∞ norm infour- and nine-block control problems leads to internally stabilizing controllers. From aperformance perspective, however, minimizing an H2 norm seems to be a more logicalchoice, because it minimizes the singular values over all frequencies. Practically, thisimplies minimizing the output power due to a unit intensity white noise input. Ifaccurate disturbance models can be identified, it is expected that a mixed H2/H∞control problem can lead to significant performance improvements.

Khargonekar and Rotea (1991) is a good starting point for further research on this topic.The problem considered by Khargonekar and Rotea (1991) is the problem of finding aninternally stabilizing controller that minimizes a mixed H2/H∞ performance measuresubject to an inequality constraint on the H∞ norm of another closed-loop transferfunction. Practically, this problem can be interpreted as a problem of optimal nominalperformance subject to a robust stability constraint.

4. The importance of actuator/sensor selection for flexible motion systems andthe existence of integral and interpolation constraints in the standard plantsetup motivate the need for profound tools for actuator/sensor selection innext generation high performance motion systems.

Appendix A

More on transmission zeros

The role of transmission zeros (especially in the ORHP) in limiting the achievable performanceis shown in Chapters 5 and 6. It is therefore desired to understand the mechanisms that giverise to such zeros in flexible structures. Although it is well known that the poles of such asystem correspond to the natural frequencies of the flexible structure, it is more difficult togive a physical interpretation of the transmission zeros.

Consider the class flexible motion systems represented by the following second-order matrixdifferential equation

Mq(t) +Kdq(t) +Ksq(t) = Bou(t), q(0) = q0, q(t) = q0 (A.1a)

y(t) = Coqq(t) + Cov q(t). (A.1b)

In this description, which is commonly used in the area of structural dynamics, see, e.g.,Gawronski (2004), q(t) ∈ R

nq×1 is the displacement vector, u(t) ∈ Rnu×1 the input force

vector, y(t) ∈ Rny×1 the output displacement vector, M ∈ R

nq×nq the positive definite massmatrix, Kd ∈ R

nq×nq the positive semi-definite damping matrix, Ks ∈ Rnq×nq the positive

semi-definite stiffness matrix, Bo ∈ Rnq×nu the input force matrix, and Coq, Cov ∈ R

ny×nq theoutput displacement and velocity matrix. It is possible to switch between this description

and the SS description by taking the state variable x(t) =[q(t) q(t)

]T, which leads to

A =

[0 I

−M−1Ks −M−1Kd

], B =

[0

M−1Bo

], C =

[Coq 0

], (A.2)

where A ∈ Rnx×nx , B ∈ R

nx×nu , and C ∈ Rny×nx . The poles of (A.1) are then the eigen-

values of the A matrix and hence do not depend on the input or output matrix (Hoagg andBernstein, 2007). In contrast to the system poles, transmission zeros are determined by thephysical placement of the actuators and sensors relative to the underlying dynamics. Hence,transmission zeros depend on the matrices A,B, and C.

A.1 Blocking effect of a zero

The zeros of a linear MIMO system can be characterized (Calafiore, 1997) as the values q inthe complex plane for which there exists a vector u0 and an initial state vector x0, such that

135

136 More on transmission zeros

for the inputu(t) = u0e

qt, t ≥ 0 (A.3)

the output is identically zero: y(t) ≡ 0,∀t > 0. In (A.3), q is called a transmission zeroand u0 and x0 are the zero input and zero state direction, respectively. Each transmissionzero therefore blocks a specific input signal multiplied by an arbitrary constant. Practically,there is dynamics present between the input and output that counteracts the effect of theinput signal. In case of a zero in the ORHP, the input signal is unbounded. It can be shown,see, e.g., MacFarlane and Karcanias (1976), that for an input given by (A.3) and initialstate condition x0, the state x(t) evolves as x(t) = x0e

qt. Consequently, to keep the outputidentically zero the following relation must hold

[qI −A −BC 0

] [x0

u0

]= 0, (A.4)

with R(s) =

[sI −A −BC 0

]the Rosenbrock system matrix. Since

[x0 u0

]T 6= 0, rela-

tion (A.4) can only hold if, at least for square systems (nu = ny), R(q) is not of full rank.

Next, suppose that the system is non-square. The Rosenbrock system matrix is then alsonon-square and q is a transmission zero if the rank of R(q) is less than the normal rank ofR(s), which in general equals min(nu, ny). Using the fact that (sI−A) has rank nx, the rankof R(s) can be expressed as

rank R(s) = nx + rank C(sI −A)−1B︸︷︷︸P (s)

, (A.5)

where P (s) is the system TFM. The transmission zeros of a system can therefore also bedefined as the locations of s where P (s) loses rank (Definition 2.16). That is, q is a (trans-mission) zero if the rank of P (q) is lower than the normal rank of P (s).

In general, rank loss for square systems occurs as soon as one column (or row) is a linearcombination of the other columns (rows). For non-square systems, rank loss can only occurif more columns (rows) are a linear combination of other columns. Transmission zeros aretherefore a rare phenomenon in non-square systems.1

Example A.1. Consider the matrices

P1 =[a b

]=

[a1 b1a2 b2

]and P2 =

[a b c d

]=

[a1 b1 c1 d1

a2 b2 c2 d2

]. (A.6)

Both matrices have normal rank of two. However, P1 loses rank if b = γa, while P2 onlylooses rank when three columns are a scalar multiple of the other. Hence, non-square systemsare less likely to have zeros.

A.2 Physical interpretation of transmission zeros in mechan-

ical systems

Although transmission zeros directly result from the mathematical system model, a modeldoes not give a physical interpretation of the zeros. To get a better understanding of trans-

1An exception are the so called pinned zeros that are briefly mentioned in Chapters 2 and 5.

A.2 Physical interpretation of transmission zeros in mechanical systems 137

mission zeros in mechanical systems, consider the lightly damped serially connected lumpedparameter structure in Figure A.1.

m1 m2

x1 x2

k

d

m4 m5

x4 x5

k

d

k

d

m3

x3

k

d

Figure A.1: MSD system with five serially connected masses.

As is well-known, this system has ten poles. Two at the origin, which correspond to the rigidbody mode, and four complex pole pairs corresponding to the natural modes of the system.

A.2.1 Collocated actuators and sensors

In, e.g., Calafiore (1997), it is shown that for MIMO systems of the form (A.1) with D = 0and collocated actuator sensor pairs, the transmission zeros are the natural frequencies of thesubsystems obtained by constraining the movement of all input/output locations to zero, i.e.,by applying infinitely high FB gain. Hence, if the uncontrolled system is stable, it is also MP.

Example A.2. Consider the situation where the actuator and sensor of the five MSD systemare fixed to m1. The transmission zeros then correspond to the natural frequencies of thesystem shown in Figure A.2, where masses m2 −m5 form a so called sub-system. Hence, theoriginal system has four complex zero pairs.

m2

x2

k

d

m4 m5

x4 x5

k

d

k

d

m3

x3

k

d

Figure A.2: Mass-spring-damper system constrained at the first mass.

Example A.3. Suppose that the five MSD system depicted in Figure A.1 has actuators andsensors located on m2 and m4. The three complex pairs of transmission zeros correspond tothe natural modes of the constrained system shown in Figure A.3, where two pairs correspondto the natural mode of m1 and m5, which are located at the same complex frequency s. Thethird pair corresponds to the natural mode of m3.

k

d

m5

x5

k

d

k

d

m3

x3

k

d

m1

x1

Figure A.3: Mass-spring-damper system constrained at m2 and m4.


A.2.2 Non-collocated actuators and sensors

Non-collocation is often mentioned as a source of NMP zeros, see, e.g., Hoagg and Bernstein(2007) and Fleming (1990). If the sensors are displaced from the actuators, the transmissionzeros are relocated. The evaluation of the transmission zeros strongly depends on the type ofsystem. In case of serially connected lumped parameter structures, which have a tridiagonaldamping and stiffness matrix (Ks and Kd in (A.1)), a proposition regarding the zero locationsis given in Calafiore et al. (1997). This proposition is valid for SISO systems, but no proofis given for MIMO systems. The proposition is therefore not included here, but the idea isillustrated by means of an example.

Example A.4. Consider again the five MSD system, but with the actuator located at m2

and the sensor at m4. This system has two negative zeros (real zeros in the OLHP) and twocomplex zero pairs in the OLHP. The complex zeros pairs correspond to the natural modes ofm1 and m5 by constraining the movement of m2 and m4 to zero. These complex zero pairsare thus the same as in Example A.3. However, the third complex zero pair of Example A.3,which is caused by the movement of m3, has been replaced by two negative zeros, of which thelocations depend on the ratio k/d.

A.2.3 Distributed parameter systems

For distributed parameter systems, e.g., beams, a similar reasoning can be applied as forlumped parameter structures. For beams that can be modelled by a second order partialdifferential equation, like, e.g., a free-free beam system under a torsional load, the onlydifference is the infinite size of the A,B, and C matrices. If the actuator is located at one sideof the beam, shifting the sensor away from the actuator means that the sub-system becomessmaller. As a result, the zero frequencies shift towards infinity and then disappear. However,for fourth order beam systems, e.g., transverse bending of an elastic beam, these zeros donot simply disappear, but emerge as real zeros mirrored around the imaginary axis. Hence,non-collocation in such systems leads to nonminimum phase behavior. A physical explanationis given by using wave propagation theory and is discussed in the next section.

A.3 Interpretation using wave propagation theory

Elastic media (assume zero internal damping) under dynamic loading can be characterizedinto two major categories according to their wave propagation characteristics (Miu, 1991).Media exhibiting non-dispersive behavior, such as torsional or axial deformation of elasticbeams, i.e., systems described by (A.1), and those exhibiting dispersive behavior, such asbending of elastic beams, i.e., systems with multiple mechanisms of energy transfer.

Wave propagation in an elastic medium is the result of energy radiating through the variousenergy storage elements. When this energy encounters a boundary, e.g., the end of a beamor an actuator, its reflection is either in-phase or out-of-phase with the incoming wave. Res-onances can then be seen as the frequencies at which all reflected waves are such that theradiation of energy can be sustained indefinitely within the structure.

A.4 Initial undershoot 139

In case of non-dispersive systems, there only exist propagating waves, i.e., all externallyapplied energy must be radiated. The only way that energy exerted by the actuator does notshow up by the sensor is if the energy is confined in a sub-portion of the original structure.For a collocated actuator/sensor pair, the entire structure acts as the sub-structure and thetransmission zeros correspond to the eigenfrequencies of the system constrained at the actuatorand sensor location. When the sensor and actuator are displaced, the length of the sub-system(s) changes to reflect the change in energy absorption characteristics and hence thezeros mitigate along the imaginary axis. If the sensor and actuator are located at the oppositeends of the system, there will be no sub-system and thus there will be no zeros.

In case of dispersive systems, there also exist non-propagation waves, i.e., not all externallyapplied energy has to be radiated. Therefore, there may also exist real zeros of which thelocation depends on the type of system. In case of a series connection of masses, the damperscause the media to be dispersive. The zero locations are then determined by the k/d-ratio and,for serially connected masses, are always in the OLHP. However, for bending of elastic beams,the coupling between multiple energy storage mechanisms, i.e., deformation and bending,causes the media to be dispersive and the real zeros are mirrored with respect to the imaginaryaxis.

Example A.5. Consider the pinned-free beam in Figure A.4. The beam is subject to trans-verse bending by a torque actuator fixed at the left side of the beam (xa = 0) and the verticaldisplacement sensor is located at 0 ≤ xs ≤ L, with L the length of the beam. As long as0 < xs < L, there are two types of sub-systems and thus two types of zeros. The first type arethe complex conjugate zeros corresponding to the resonance frequencies of the right sub-system.Energy exerted by the actuator propagates through the medium and is contained within theright sub-system. The other type of zeros are real zeros and correspond to the left sub-system.In case xs = 0, there are no real zeros and if xs = L, there are no complex conjugate zeros.

yu

sub-structure(real zeros)

sub-structure(complex zeros)

Figure A.4: Pinned-free beam setup.

A.4 Initial undershoot

It is well-known that the effect of a positive zero (real zero in the ORHP) of a SISO TFmanifests itself as an initial undershoot to a step in the input, see, e.g., Leon de la Barra S.(1994) and Theorem A.6. Figure A.5(a) shows initial undershoot for an asymptotically stablesystem. The step response initially moves into the negative direction, but then reverses intothe positive direction. Practically, a real ORHP zero means that at a certain sensor locationthe contribution of all modes is such that the fast and slow dynamics have a different sign.In case of two real zeros, see Figure A.5(b), the step response initially moves into the correct


direction, then reverses into the wrong direction and crosses the zero-line, and finally reversesagain into the correct direction.

Theorem A.6 (Initial undershoot for SISO systems). Consider a strictly proper SISOtransfer function P (s), and suppose the system is stable. This system has initial undershootif and only if its TF has an odd number of real ORHP zeros.

Proof. See Vidyasagar (1986).

0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time [s]

Outp

ut

[-]

(a) P (s) = −(s − 1)/(s + 1)3

0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time [s]

Outp

ut

[-]

(b) P (s) = (s − 1)2/(s + 1)3

Figure A.5: Step responses for systems with a single positive zero (left) and two positivezeros (right).

According to Theorem A.6, a complex zero pair in the ORHP does not lead to initial under-shoot. It can, however, lead to two or zero direction reversals as is shown in Figure A.6.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

Time [s]

Outp

ut

[-]

(a) P (s) = (2s2 − s + 1)/(s + 1)3

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

Time [s]

Outp

ut

[-]

(b) P (s) = 0.25(s2 − s + 4)/(s + 1)3

Figure A.6: Step responses for systems with a complex zero pair in the ORHP.

Theorem A.6 is also valid for every element of a TFM, i.e., the TF between a specific input jand output i. An odd number of positive zeros in element (i, j) of the TFM leads to undershootin the ith output as a result of a step on the jth input.

A.5 Creating transmission zeros by manipulating the plant inputs and outputs 141

A.5 Creating transmission zeros by manipulating the plant

inputs and outputs

As stated earlier in this appendix, the locations of the transmission zeros do not only dependon the dynamics of a system, but also on the locations of the actuators and sensors. Manipu-lating the input and output signals by using static or dynamic pre- and/or post-compensatorsmay therefore create transmission zeros. This is especially true when the “new” system ismade square, because square systems are more likely to have transmission zeros.

The usage of static pre- and/or post-compensators, see Figure A.7, is often to done to decouplethe systems in its rigid body modes. Within Philips Applied Technologies, this process isreferred to as “rigid body decoupling” and it is applied to facilitate controller design. The goalis to make the plant more diagonal by transforming the physical plant inputs (u) and outputs(y) into inputs (u) and outputs (y) of the point subject to exposure. Dynamic compensatorsare also a possibility, but practical issues often prevent satisfactory implementation on a realsystem.

TyTu˜P

P

yu yu

Figure A.7: Usage of pre- and post-compensators to transform the physical plant P , to a newplant P = TyP Tu. As already stated in Chapter 3, Ty can be seen as a performance observerthat does not take the internal dynamics into account.

To investigate whether pre- and post multiplication introduces extra transmission zeros,Sylvester’s Inequality (Petersen and Pedersen, 2008) is used.

Lemma A.7 (Sylvester’s Inequality). If A is an m × n matrix and B an n × r matrix,then

rank (A) + rank (B) − n ≤ rank (AB) ≤ min {rank (A), rank (B)} . (A.7)

Theorem A.8. For square plants and square nonsingular pre- and post multiplication ma-trices, i.e., ny = ny = nu = nu, the zeros of P and P are equal.

Proof. Consider the transformation P Tu (at the actuator side). Since Tu is always of full rank,it follows from (A.7) that the rank of matrix P is fully determined by the rank of P . Hence,if P looses rank, P also looses rank. A similar reasoning can be applied to P = TyP .

Care should be taken when using non-square compensators, because the normal rank of thenew plant is in general different than the rank of the old plant. Example A.9 illustrates this.

Example A.9. Consider a 3× 3 system P that is post multiplied by a constant 3× 1 matrixTu

P (s) = P (s)Tu =

p11 p12 p13

p21 p22 p23

p31 p32 p33

Tu,1

Tu,2

Tu,3

=

p1

p2

p3

.


Then, using (A.7) and rank(Tu) = 1 leads to

rank (A) − 2 ≤ rank (AB) ≤ min {rank (A), 1} . (A.8)

If P (s) is always of full rank, i.e., has no transmission zeros, (A.8) reduces to

1 ≤ rank (P Tu) ≤ 1. (A.9)

Hence, P (s) also has no transmission zeros if Tu 6= 0. If P (s) has at least one transmissionzero, i.e., looses rank for some s = qi, then there exists a non-zero matrix Tu such thatP (s) = 0. The transmission zeros of P (s) are therefore located at the same locations as thetransmission zeros of P (s). The reverse, however, is not true.

A FEM model of the long stroke of a wafer stage is available (Van de Wal, 2008) withinPhilips Applied Technologies. This model has eight inputs, six outputs, and no transmissionzeros. Post-multiplying this model by a constant 8 × 6 matrix Tu creates a square systemwith 88 transmission zeros, of which 21 are in the ORHP. The creation of transmission zerosby forming P (s) is in line with Lemma A.7, which shows that 4 ≤ rank (P (s)Tu) ≤ 6.The process of creating transmission zeros when using non-square compensators is furtherillustrated in Example A.10.

Example A.10. Consider a 1 × 2 system P (s) that is post-multiplied by a 2 × 1 matrix Tu,i.e.,

P (s) = P (s)Tu =[p1(s) p2(s)

] [ Tu,1

Tu,2

]= p1(s)Tu,1 + p2(s)Tu,2.

Plant P (s) has no transmission zeros as long as p1(s) and p2(s) are not zero simultaneously.Using (A.7) yields 0 ≤ rank (P (s)Tu) ≤ 1, which is not surprising since P (s) is a scalar.Unlike in Example A.9, however, a non-zero Tu can be found such that P (s) = 0, evenwhile P (s) has no transmission zeros. For Tu = [ 1 −1 ]T , a zero is created at s = q ifp1(q) = p2(q). If the non-square matrix Tu is replaced by a nonsingular 2 × 2 matrix, then

P (s) = P (s)Tu =[p1(s) p2(s)

] [ Tu,11 Tu,12

Tu,21 Tu,22

]=

[p1(s)Tu,11 + p2(s)Tu,21

p1(s)Tu,12 + p2(s)Tu,22

]T

.

As long as p1 and p2 are not zero simultaneously, plant P (s) always has a rank of one (1 ≤rank (P (s)Tu) ≤ 1). Rank loss only occurs is both elements of P (s) are zero simultaneously.This, however, is not possible because Tu is nonsingular:

p1(s)Tu,11 + p2(s)Tu,21 = 0 → p1(s) = −Tu,21

Tu,11p2(s)

p1(s)Tu,12 + p2(s)Tu,22 = 0

⇓ Use the exppression for p1(s)

p2(s) (Tu,11Tu,22 − Tu,21Tu,12)︸︷︷︸det(T )6=0

= 0

Remark A.11. Lemma A.7 can also be used to analyze the effect of using dynamic compen-sators. The only difference is then that Tu(s) and Ty(s) have a rank dependent on s.

A.6 Zero assignment problem 143

Using dynamics compensators to transform a plant P is, at least mathematically speaking,the same as forming the open-loop L or LI . Lemma A.7 may therefore just as well be used toanalyze the creation of transmission zeros in forming the open-loop. Adding an extra actuatoror sensor to get rid of the plant transmission zeros seems a logical choice at first, but informing the square open-loop new transmission zeros are created that potentially still limit theachievable performance.

A.6 Zero assignment problem

In the previous section, it is shown that using pre- and/or post-compensators potentially leadsto zeros in the new plant. Two questions then arise:

1. Are pre- and/or post-compensators really needed for control design? For example,specifying the desired shape of the TFM between w and z in the standard plant setup,without considering the dimensions of y and u.

2. How can the pre- and/or post-compensators be chosen such that the new zeros are notlimiting the performance? Notice that when extra sensors are added, the controller canalso be regarded as a pre- or post-compensator when forming the open-loop, as shownin Example 5.13.

The second question is central in literature on “zero assignment” and “(structured) squaringdown”. A good example is Leventides and Karcanias (2008), where a plant with more outputsthan inputs needs to be squared down, i.e., made square, by using a pre-compensator to selectand combine various output signals. In doing so, the creation of undesired zeros (such as NMPzeros) needs to be avoided. Another example is Liu et al. (2009), where sensor selection isused to assign structural properties like, e.g., zero locations. Both topics are beyond the scopeof this research.

Appendix B

Case study: a flexible cart system

B.1 Introduction

This case study is a continuation of the example given in Section 5.1.7, which is about theflexible cart system (see Figure 5.3) and where it is shown that the presence of a NMP zero (inP1) poses a fundamental performance limitation on the measured output (y1) in conventionalFB control. To goal of the case study in this appendix is to

use cancellation controllers to investigate whether the limitation imposed by theORHP zero still exists if there is an explicit distinction between the measured andperformance variables in the standard plant setup.

B.2 Conventional feedback control

To get familiar with formulating a control problem in the standard plant setup, consider theclassical FB structure in Figure B.1, where output y1 is used for control.

C P1

w = r u

y = z

y1

ey

Figure B.1: Conventional FB control with collocated measured and performance variables.

The error signal ey resembles the sensed variable y and the exogenous output variable z. Thecorresponding generalized plant G is then given by

[zy

]=

[Gzw Gzu

Gyw Gyu

]

︸︷︷︸G(s)

[wu

]=

[1 −P1

1 −P1

] [wu

]. (B.1)

145

146 Case study: a flexible cart system

The closed-loop disturbance response, i.e., the mapping Tzw : w → z, then reduces toS = (1 + L)−1, with L = P1C. This TF cannot be made identically zero, because:

1. This would require a infinitely high FB gain (practical limitation).

2. The closed-loop poles move towards the open-loop zeros, which are partly in the ORHPplane, when the FB gain is increased (fundamental limitation).

3. The sensitivity has to satisfy the Poisson integral constraint.

∫ ∞

0ln |S(jω)| · w(q, ω) dω, (B.2)

with w(q, ω) depicted in Figure B.2.

10−2

100

102

104

0

0.005

0.01

0.015

0.02

0.025

Frequency [Hz]

w(q,ω

)[-]

Figure B.2: Additional weight in the integrand of the Poisson integral constraint.

B.3 Explicit distinction between y and z (option 1)

Now consider the control problem depicted in Figure B.3, where y1 is used for control andhigh performance is required at y2. So, now there is an explicit distinction between the sensedvariable y and exogenous output variable z. The corresponding generalized plant G is givenby [

zy

]=

[1 −P2

1 −P1

]

︸︷︷︸G(s)

[wu

]. (B.3)

A controller CC such that Tzw ≡ 0, i.e., a cancellation controller, is then given by

CC =Gzw

GzwGyu −GzuGyw=

1

−P1 + P2= 2

I

l2s2 + ds+ k. (B.4)

Remark B.1. For classical FB control (see Section B.2), substituting the components of (B.1)into (B.4) leads to CC = 1

0 . This is of course not unexpected, since it is equivalent to infinitelyhigh gain FB, which is often strived for in classical loop-shaping.

B.3 Explicit distinction between y and z (option 1) 147

C P1

w = r u

yP2

y1

y2

z

Figure B.3: single-DOF control problem with an explicit distinction between measured (y1)and performance variables (y2).

Unfortunately, CC in (B.4) is improper and cannot be implemented. An approximation musttherefore be made. One possibility is to multiply the controller by a third order low-pass filter(to make the controller strictly proper) with the breakpoint fb at 104 [Hz], i.e.,

CaC = CC

(1

τs+ 1

)3

, with τ =1

2π104. (B.5)

The FRFs of CC , CaC , and the corresponding open-loops are depicted in Figure B.4. The

cancelling nature of CC is clearly visible, since the resonance in P1 is not present in the cor-responding open-loop (pole/zero cancellation in the OLHP). In practice, however, exact can-cellation is often not possible due to, e.g., model uncertainties. The closed-loop disturbanceresponse ratio (defined in (5.51)) is depicted for CC (Hzw) and Ca

C (Hazw) in Figure B.5(a).

Clearly Hzw ≡ 0, implying perfect suppression of the effect of w on z. However, for CaC the

generalized Bode integral constraint (given by Theorem 5.23) should be satisfied, because Gyu

is strictly proper and (5.81) is satisfied with strict inequality. Hence, disturbance attenuationcan only be achieved on a limited frequency interval and there is an equally large frequencyrange with disturbance amplification (|Ha

zw > 1|). Although the amount of disturbance at-tenuation is limited for Ha

zw, disturbance attenuation is still much better than for output y1,which is described by the sensitivity S. As shown in Figure B.5(b), S satisfies the Poissonintegral constraint and only achieves disturbance attenuation up to 5 [Hz]. Classical perfor-mance limitations in the FB loop therefore do not necessarily limit the performance, as longas the performance variables are not collocated with the measured variables.

The high peak in S (at 6 [Hz]), which is approximately 30 [dB], implies a low robustnessmargin and this is also visible as a close approximation of the point (-1,0) in the Nyquistdiagram, which is shown in Figure B.6. From inspection of the Nyquist curve, it follows thatthe PM is only 1.73 [◦].

The responses to a unit step change in input (w(t)) at t = 1 [s] for CC and CaC are shown in

Figure B.7(a) and the corresponding error signals are shown in Figure B.7(b). The simulationis performed in Matlab Simulink by using separate blocks for the controller and the plant.These simulations therefore also confirm that the feedback system is internally stable. Thestep responses show that the performance in y2 (z) is much better than in y1 (y) and that themain component of the error signals is a sinusoid with a frequency of approximately 6 [Hz],which is due to the large peak in S at this frequency.


100

102

104

50

100

150

200

250

Magnitude

[dB

]

100

102

104

−100

0

100

200

Phase

[◦]

Frequency [Hz]

(a) CC and CaC

100

102

104

−100

−50

0

50

100

150

Magnitude

[dB

]

100

102

104

−400

−300

−200

Phase

[◦]

Frequency [Hz]

(b) Open-loop

Figure B.4: Bode diagrams of CC (thin line), CaC (thick line) and the accompanying open-loop

FRFs.

100

102

104

−300

−250

−200

−150

−100

−50

0

Magnitude

[dB

]

Frequency [Hz]

(a) Hzw and Hazw

100

102

104

−40

−20

0

20

40

Magnitude

[dB

]

Frequency [Hz]

(b) Sensitivity

Figure B.5: Bode magnitude diagrams of the closed-loop disturbance response ratio (Hzw)and sensitivity for the system controlled with CC (thin line) and Ca

C (thick line).

B.3 Explicit distinction between y and z (option 1) 149

−1 −1.2 −0.8 −0.6 −0.4 −0.2 0 −0.03

−0.02

−0.01

0

0.01

Im

Re

Figure B.6: Nyquist diagram of the open-loop for P1 controlled with CC (thin line) and CaC

(thick line).

0 2 4 6 8 10−1

0

1

2

3

y 1

0 2 4 6 8 100

0.5

1

1.5

y 2

Time [s]

(a) Responses of the measured (y1) and perfor-mance (y2) variable

0 2 4 6 8 10−2

−1

0

1

2

w−y 1

0 2 4 6 8 10−4

−2

0

2

4x 10

−3

w−y 2

Time [s]

(b) “Error” signals w - y1 and w - y2

Figure B.7: Time responses to a unit step change in reference signal w (dashed line) for CaC

at t = 1 [s].


B.4 Explicit distinction between y and z (option 2)

Consider the same system, but with y2 measured for feedback and performance required aty1. The generalized plant is then given by

[zy

]=

[Gzw Gzu

Gyw Gyu

]

︸︷︷︸G(s)

=

[1 −P1

1 −P2

]=

[wu

]. (B.6)

The cancellation controller is determined in a similar fashion, i.e.,

CC =Gzw

GzwGyu −GzuGyw=

1

−P2 + P1= −2

I

l2s2 − ds− k. (B.7)

The only difference with (B.4) is a minus-sign. The new cancellation controller is stable, butthe resulting closed-loop system is unstable. To see this, consider the expression

Tzw = Gzw +Gzu · CS ·Gyw. (B.8)

If the TF from u to z, i.e., Gzu, has a zero located at s = q (Gzu(q) = 0), then Tzw = Gzw = 1.In order to make Tzw ≡ 0, the controller should allow CS to cancel the ORHP zero. So,because CS now has an unstable pole, the FB system in internally unstable.

These type of constraints are interpolation constraints for the standard plant setup and arediscussed in Chapter 5. A straightforward extension of the generalized Bode integral canbe obtained by taking such interpolation constraints into account. This leads to a strongerintegral relation, which is similar to the Poisson integral constraint. Hence, this integral isreferred to as a generalized Poisson integral constraint. For completeness, one (more exist)generalized Poisson integral constraint is included in Theorem B.2.

Theorem B.2 (Generalized Poisson integral). Let q = x + yj be a NMP zero of Gzu

or Gyw that is not a pole of G and Bξ(q), Bα(q), Bγ(q), and Bρ(q) the Blaschke products asdefined in Definition 5.22. Then, if the system is internally stable and if Gzw(q) 6= 0,

∫ ∞

0ln |Tzw(jω)| · w(q, ω) dω = π ln |Gzw(q)B−1

ξ (q)|, (B.9)

and ∫ ∞

0ln |Hzw(jω)| · w(q, ω) dω = π ln |B−1

α (q)B−1γ (q)B−1

ρ (q)|, (B.10)

where

w(q, ω) ,x

x2 + (y − ω)2+

x

x2 + (y + ω)2. (B.11)

Proof. See Freudenberg et al. (2002).

B.5 Effect of changing mass and inertia 151

B.5 Effect of changing mass and inertia

The flexible cart system was originally used because of its initial undershoot characteristic,which can emerge if the actuator and sensor are located at different sides of the COG. Byapplying a positive force u, the COG immediately moves into the positive x-direction. How-ever, if the inertia is low compared to the mass, the rotation of the block causes y1 to initiallymove into the negative x-direction, see Figure B.9(b). Hence, there exist a positive zero (areal zero in the ORHP) and a negative zero. If the inertia is lowered further or if the massis increased (see Figure B.8(a) for the FRF of a ten times heavier system), these zeros movecloser towards the imaginary axis, implying a more detrimental performance limitation wheny1 is used for FB control. Indeed, it follows from comparing Figures B.9(a) and B.9(b) thatthe increase in mass leads to more undershoot. Conversely, if the inertia is increased or if themass is decreased, the zeros move further away from the imaginary axis and are eventually re-placed by a complex zero pair in the OLHP. The resulting FRF is depicted in Figure B.8(b).Physically speaking, the COG accelerates fast enough to prevent movement of y1 into thenegative x-direction (see Figure B.9(d)). Figure B.9(c) also shows the step response at the“tipping point”, i.e., the weight of the system is such that the contribution of the translationand rotation cancel each other. Output y1 then initially remains zero. In this situation thenegative zero is located at s = −1000 and there is no positive zero anymore1.

10−1

100

101

102

103

−200

−150

−100

−50

0

Magnitude

[dB

]

10−1

100

101

102

103

−400

−350

−300

−250

−200

−150

Phase

[◦]

Frequency [Hz]

(a) Mass increased by a factor ten

10−1

100

101

102

103

−200

−150

−100

−50

0

50

Magnitude

[dB

]

10−1

100

101

102

103

−300

−250

−200

−150

Phase

[◦]

Frequency [Hz]

(b) Mass decreased by a factor ten

Figure B.8: Bode diagrams of P1 with an alternative mass parameter.

The unit step simulation performed in Section B.3 is repeated for the system with m = 80 [kg]and m = 0.80 [kg]. The time domain results are shown in Figure B.10 and the correspondingsensitivities in Figure B.11(a) and Figure B.11(b), respectively. For m = 80 [kg], the peak inS is increased, implying even smaller robustness margins. This is not unexpected, becausethe weight in the integrand of the Poisson integral, see Figure B.2, is effectively shifted to the

1It can be shown that for this point m = 4I/l2m. Substituting this value of m into (5.18) shows that thereis only one zero.


0 0.02 0.04 0.06 0.08 0.1−1

0

1

2

3x 10

−4

Displa

cem

ent

[m]

Frequency [Hz]

(a) m = 80 [kg]

0 0.02 0.04 0.06 0.08 0.1−1

0

1

2

3x 10

−4

Displa

cem

ent

[m]

Frequency [Hz]

(b) m = 8 [kg]

0 0.005 0.01 0.015 0.02

0

2

4

6

8

10x 10

−5

Displa

cem

ent

[m]

Frequency [Hz]

(c) m = 5.2 [kg]

0 0.005 0.01 0.015 0.02

0

2

4

6

8

10x 10

−5

Displa

cem

ent

[m]

Frequency [Hz]

(d) m = 0.8 [kg]

Figure B.9: Time response of y1 (thin line) and y2 (thick line) on a unit step: u(t) = 1(t), t ≥ 0for different mass values.

B.6 Discussion 153

left. For m = 0.8 [kg], this weight is not relevant, because the well-known Bode sensitivityintegral (and not the Poisson integral) must be satisfied. Hence, the small peak aroundf = 104 [Hz]. Notice that there is no such peak in S(jω) for the system controlled with theimproper cancellation controller of (B.4), because the relative degree of L is smaller than two.

Remark B.3. The pole and zero locations can also be altered by changing the other systemparameters, like, e.g., the spring stiffness and sensor and actuator locations.

B.6 Discussion

• For the system with y1 measured for feedback and performance required at y2, a stabi-lizing cancellation controller (CC) exists such that Tzw ≡ 0. This controller is improperand a third order low-pass filter is needed to achieve a strictly proper controller thatcan be implemented in a simulation environment. As a consequence, however, the gen-eralized Bode integral, which is presented in Theorem 5.23, must be satisfied. Theresulting sensitivity function S still satisfies the Poisson integral constraint, but be-cause the closed-loop disturbance ratio is not described by S, better performance at zis achieved.

• If there exist a NMP zero in Gzu (or Gyw), the generalized Bode integral needs to bereplaced by a “stronger” version that takes the NMP zero into account. This extendedversion looks a lot like the well-known Poisson integral constraints for the single-DOFcontrol structure. Hence, in such situations there is a fundamental performance limita-tion even when there are no NMP zeros in the TF between the actuators and measuredvariables. It is therefore concluded that if the mapping from u to z is nonminimumphase, there will always exist fundamental performance limitations, regardless of wherethe sensors are located.


0 2 4 6 8 10−2

−1

0

1

2

w−y 1

0 2 4 6 8 10−4

−2

0

2

4x 10

−3

w−y 2

Time [s]

(a) m = 80 [kg]

0 2 4 6 8 10−0.4

−0.2

0

0.2

0.4

w−y 1

0 2 4 6 8 10−2

−1

0

1

2x 10

−3

w−y 2

Time [s]

(b) m = 0.80 [kg]

Figure B.10: Time responses of the error signals w - y1 and w - y2 for the approximatecancellation controller as a result of a unit step change in reference w at t = 1 [s].

100

102

104

−40

−20

0

20

40

Magnitude

[dB

]

Frequency [Hz]

(a) Sensitivity for m = 80 [kg]

100

102

104

−60

−40

−20

0

20

Magnitude

[dB

]

Frequency [Hz]

(b) Sensitivity for m = 0.80 [kg]

Figure B.11: Bode magnitude diagrams of the sensitivity for P1 controlled with CC (thinline) and Ca

C (thick line).

Appendix C

Simulation parameters

In this appendix, additional information regarding the flexible cart system and two MSDsystem are given to reproduce the simulations.

C.1 Flexible cart system

Consider the flexible cart system depicted in Figure 5.3 with input force u and positionoutputs yi, i = 1, 2. Although not shown, dampers are located in parallel with the springs.For simplicity it is assumed that the cart is square, that the COG is exactly in the middle ofthe body, and that the actuator, sensors, springs, and dampers are located at the corners ofthe block. The equations of motion and output equations are then given by

mx = u, (C.1a)

Iφ = −1

2lu− 1

2kl2 sin(φ) − 1

2dl2 sin(φ), (C.1b)

y1 = x+l

2sin(φ), (C.1c)

y2 = x− l

2sin(φ), (C.1d)

where l is the width of the block, m the mass, I the moment of inertia, k the stiffness of thesprings, d the absolute damping coefficients, x the horizontal displacement, and φ the rotationof the block around its COG. The parameters values used in Chapter 5 and Appendix B aregiven in Table C.1. The set of non-linear equations can be linearized by only allowing smallrotations around the COG, i.e., sin(φ) ≈ φ, which leads to

P1(s) =y1(s)

u(s)=

1

ms2− 1

4

l2

Is2 + 12dl

2s+ 12kl

2, (C.2a)

P2(s) =y2(s)

u(s)=

1

ms2+

1

4

l2

Is2 + 12dl

2s+ 12kl

2. (C.2b)

155

156 Simulation parameters

Table C.1: System parameters for theflexible cart system.


m 8 kgI 1.33 · 10−2 kg m2

k 10 · 103 N m−1

d 10 N s m−1

l 0.1 m

C.2 Mass-spring-damper systems

The equations of motion for the more general n MSD systems are given by:

m1x1 = u1 − k(x1 − x2) − d(x1 − x2), (C.3a)

m2x2 = u2 + k(x1 − x2) + d(x1 − x2) − k(x2 − x3) − d(x2 − x3), (C.3b)

...

mixi = ui + k(xi−1 − xi) + d(xi−1 − xi) − k(xi − xi+1) − d(xi − xi+1), (C.3c)

...

mnxn = un + k(xi−1 − xi) + d(xi−1 − xi). (C.3d)

The parameter values for the two MSD system used in Chapters 4 – 6 are given in Table C.2.The values of the parameters are chosen such that the flexible mode has a resonance frequencyof approximately 100 [Hz] and dimensionless damping constant ζ ≈ 0.01.

Table C.2: System parameters for the two MSD system.


m1 1.00 kgm2 1.00 kgk 1.99 · 105 N m−1

d 6.60 · 100 N s m−1

Appendix D

FEM model

The FEM model of a WS short stroke device that is used in Chapter 7 is presented in thisappendix. The operations needed to obtain the three I/O sets and to decouple the systemare discussed as well.

D.1 FEM Model

Although a wafer stage is usually a complex geometry containing ribs, cooling channels, etc.,it is approximated here as a hollow block. The dimensions of this block are: 500 [mm] (l) x500 [mm] (l) x 50.0 [mm] (h) and the other system parameters are given in Table D.1. ThreeI/O sets with point actuators and sensors are compared. The actuator/sensor configurationsfor I/O sets 1 and 2 are depicted in Figure D.1(a). The actuators for both sets are located atthe bottom surface. The sensors for I/O set 1 are collocated with the actuators, but for I/Oset 2 the sensors are located on the top surface. I/O set 3 contains two extra actuators andsensors, and the I/O configuration is depicted in Figure D.1(b). The actuators are locatedon the bottom surface and the sensors on the top surface.

Table D.1: Parameters for the WS FEM model. The exact locations of the actuators andsensors can be retrieved from Table D.2

Parameter Symbol Value Unit

Mass M 60.716 kgMoment of inertia around x-axis Ixx 1.5112 kg · m2

Moment of inertia around y-axis Iyy 1.5112 kg · m2

Moment of inertia around z-axis Izz 2.9837 kg · m2

Sensor offset, see Figure D.1 ds 100 mmActuator radius I/O sets 1 and 2, see Figure D.1 ra 150 mm

Actuator offset I/O set 3, see Figure D.1 da 200 mmRadius performance nodes 25-32, see Figure D.1 rz1 75 mmRadius performance nodes 33-40, See Figure D.1 rz2 150 mm

157

158 FEM model

y

x

z

ra

ds

12

34

5

6

1

2

3

45

6

ds

wafer

l

l1

2ra

(a) I/O sets 1 and 2

y

x

z ds

1

4

3

56

7

ds

ds

ds

da

rz2rz1

2

8

13

2

4

56

7 8

da

da

da

(b) I/O set 3

Figure D.1: Top views of the actuator/sensor configuration for I/O sets 1, 2, and 3. Theactuators are located at the bottom surface and are therefore colored grey. Nodes 24-40,which are used to evaluate performance, are indicated by “×” in the right figure.

The FEM model is created using Ansys and contains 100 modes: six rigid bode modes and94 flexible modes. All flexible modes have a dimensionless damping ratio of ζ = 0.01. Themode shapes belonging to the eight lowest eigenfrequencies are depicted in Figure D.2. Sincethe stiffness in z-direction, which is the out-of-plane direction, is much lower than in the x-,and y-directions, the largest translations are in the vertical direction. In addition, Figure D.2also shows the right handed coordinate framework that is used to define directions. Counter-clockwise rotations are defined positive in this framework.

To apply forces and perform measurements on the FEM model several nodes have to bespecified. Notice that the term “node” is also used to describe the points of a mode shapethat do not move. The context should avoid confusion. In Table D.2, the locations of the40 nodes of the FEM model are listed. Nodes 1 - 23 are used to create the three I/O setsand node 24 acts as the POC. Extra nodes (nodes 25 - 40) are included in two circles aroundthe POC for model validation purposes and to estimate the rotations of node 24, since thesecannot be obtained directly from the FEM data. Table D.2 also shows for which I/O set thenode is used and if it is used as an actuation point (A), measured point (S), or both (AS).The subscripts x, y, and/or z are added to indicate the associated direction. The plant withI/O set 1 is referred to as P 1, P 2 refers to I/O set 2, and P 3 to I/O set 3. Plant P i

z refers tothe plants with the performance output. Since the input depends on which I/O set is used,the superscript i is invoked to distinguish between different actuator locations.

Remark D.1. The FEM software does not allow nodes to be located at the edges of the stage.Therefore, nodes located at the edges or in the corners are shifted 5 [mm] into the block.

D.1 FEM Model 159

(a) 435.9 [Hz] (b) 553.4 [Hz]

(c) 622.2 [Hz] (d) 666.5 [Hz]

(e) 818.2 [Hz] (f) 821.7 [Hz]

(g) 1278 [Hz] (h) 1280 [Hz]

Figure D.2: First eight flexible modes. Blue means no displacement (node) and red meansa large displacement (anti-node). In Figures D.2(e) – D.2(h), the stage is rotated 90 [◦]compared to Figures D.2(a) – D.2(d)

160 FEM model

Table D.2: Nodes of the WS FEM model.

Node x-cor y-cor z-cor I/O set

1 0.5 · 0.5 ·√

3 · ra 0.5 · 0.5 · ra −h P 1(ASxy), P2(Axy)

2 −0.5 · 0.5 ·√

3 · ra 0.5 · 0.5 · ra −h P 1(ASxy), P2(Axy)

3 0 −0.5 · ra −h P 1(ASx), P 2(Ax)

4 0.5 ·√

3 · ra −0.5 · ra −h P 1(ASz), P2(Az)

5 0 ra −h P 1(ASz), P2(Az)

6 −0.5 ·√

3 · ra −0.5 · ra −h P 1(ASz), P2(Az)

7 0.49 · l ds 0 P 2(Sx), P 3(Sx)8 −ds 0.49 · l 0 P 3(Sy)9 −0.49 · l −ds 0 P 2(Sx), P 3(Sx)10 0.49 · l 0.49 · l 0 P 2(Sz), P

3(Sz)11 −0.49 · l 0.49 · l 0 P 2(Sz), P

3(Sz)12 0 −0.49 · l 0 P 2(Syz)

13 da 0 −h P 3(Ay)14 0 da −h P 3(Ax)15 −da 0 −h P 3(Ay)16 0 −da −h P 3(Ax)17 da da −h P 3(Az)18 −da da −h P 3(Az)19 −da −da −h P 3(Az)20 da −da −h P 3(Az)

21 ds −0.49 · l 0 P 3(Sy)22 −0.49 · l −0.49 · l 0 P 3(Sz)23 0.49 · l −0.49 · l 0 P 3(Sz)

24 0 0 0 P iz(Sxyz)

25 rz1 0 0 P iz(Sxyz)

26 0.5 ·√

2 · rz1 0.5 ·√

2 · rz1 0 P iz(Sxyz)

27 0 rz1 0 P iz(Sxyz)

28 −0.5 ·√

2 · rz1 0.5 ·√


29 −rz1 0 0 P iz(Sxyz)

30 −0.5 ·√

2 · rz1 −0.5 ·√


31 0 −rz1 0 P iz(Sxyz)

32 0.5 ·√

2 · rz1 −0.5 ·√


33 rz2 0 0 P iz(Sxyz)

34 0.5 ·√

2 · rz2 0.5 ·√


35 0 rz2 0 P iz(Sxyz)

36 −0.5 ·√

2 · rz2 0.5 ·√


37 −rz2 0 0 P iz(Sxyz)

38 −0.5 ·√

2 · rz2 −0.5 ·√


39 0 −rz2 0 P iz(Sxyz)

40 0.5 ·√

2 · rz2 −0.5 ·√


D.2 Input and output selection 161

D.2 Input and output selection

Next, the FEM model is exported from ANSYS to Matlab, which results in a SS model with200 states (100 modes), 120 inputs (force inputs in the x-, y-, and z-direction for all 40 nodes),and 360 outputs (translation, velocity, and acceleration in the x-, y-, and z-direction for all40 nodes). Inputs numbers 1, 2, and 3 correspond to the forces working at node 1 in x-, y-,and z-direction, respectively. In the same manner, input numbers 4, 5, and 6 correspond tothe second node. A similar reasoning applies to the output numbers. The first 120 outputnumbers hereby correspond to the translational outputs, the second 120 outputs to velocityoutputs, and the last 120 outputs to acceleration outputs.

Plants P 1, P 2, P 3, and P iz need to be created from this SS model by selecting the relevant

inputs and outputs. Table D.3 lists the relevant input and output numbers for these fourplants. The obtained system is referred to as P and the inputs u and outputs y are in termsof forces and translations, respectively, in the x-, y-, or z-direction.

Table D.3: Input and output numbers.

Plant Input numbers Output numbers

P 1 1, 2, 4, 5, 7, 12, 15, 18 1, 2, 4, 5, 7, 12, 15, 18P 2 1, 2, 4, 5, 7, 12, 15, 18 19, 35, 25, 30, 33, 36P 3 38, 40, 44, 46, 51, 54, 57, 60 19, 23, 25, 62, 30, 33, 66, 69P i

z - 70, 71, 72, 74, 79, 86, 91, 75, 81, 87, 93

D.3 Constructing the physical plant

Next, inputs u and outputs y need to be transformed such that actuators and sensors can bepositioned under an angle with respect to the x- and y-axis. These transformations are neededto construct I/O sets 1 and 2. The newly created plant is the physical plant P , which hasphysical plant inputs up = T−1

u u, i.e., actuator forces, and physical plant outputs yp = Ty y,i.e., sensor data. Although often modelled differently, external disturbances work on thesesignals. The process of constructing the physical plant is depicted in Figure D.3.

TyTu P

˜P

u yup yp

Figure D.3: Usage of pre- and post-compensators to transform P into the physical plant P .

162 FEM model

Transformation matrices Ty and Tu are given by

Ty =

cos(60) − cos(30) 0 0 0 0 0 00 0 cos(60) cos(30) 0 0 0 00 0 0 0 −1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

, (D.1)

and

Tu =

cos(60) 0 0 0 0 0− sin(60) 0 0 0 0 0

0 cos(60) 0 0 0 00 sin(60) 0 0 0 00 0 −1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

, (D.2)

respectively. For I/O set 2 and P iz , only Tu is needed, i.e., Ty = I.

D.4 Rigid body decoupling

If it assumed that the system behaves a rigid body, static decoupling matrices can be used todecouple the system in its rigid body DOFs. Figure D.4 shows how the decoupling matricesare located with respect to the physical plant. Decoupling is often done to facilitate controldesign. It makes the MIMO plant more diagonal by transforming the physical inputs andoutputs into inputs up = T−1

u up and outputs yp = Tyyp of the POC in the directions of thesix rigid body DOFs. For the control of stages, the decoupling matrices are often consideredto be part of the system. Hence, the tilde-symbol (∼) is used here for the signals beforedecoupling.

TyGB ˜P

Tu

up ypucyp

GSup

P

Figure D.4: Usage of pre- and post-compensators to decouple P .

Remark D.2. A disadvantage of using pre- and/or post-multiplication matrices is that dis-turbances that work on the physical plant variables u and y are distributed over the manipu-lated inputs and outputs u and y. This is illustrated in Figure D.5 for a disturbance workingat the output of the physical plant. More discussion about this phenomenon can be foundin (Boerlage, 2004).

D.4 Rigid body decoupling 163

TyTu˜P

P

u yu y

Ty

˜dy

dy

Figure D.5: Distribution of physical output disturbances over measured variables.

D.4.1 Static sensor transformation

At the sensor side, geometric relations between the sensor locations and the POC are used to“estimate” the translations and rotation of the POC. Counter-clockwise rotations are herebydefined positive. For P 1, P 2, and P 3 the sensor transformations are given below:

For P 1:

yp =

yx

yy

yRz

yz

yRx

yRy

=

1 0 0 0 0 h0 1 0 0 −h 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

·

13

13 −2

3 0 0 0− 1√

31√3

0 0 0 0

− 23·ra

− 23·ra

− 23·ra

0 0 0

0 0 0 13

13

13

0 0 0 − 0.51.5·ra

11.5·ra

− 0.51.5·ra

0 0 0 − 1√3·ra

0 1√3·ra

y1

y2

y3

y4

y5

y6

︸︷︷︸= yp

. (D.3)

For P 2:

yp =

yx

yy

yRz

yz

yRx

yRy

=

12 0 1

2 0 0 00 1 0 0 0 0

−ds

2 0 ds

2 0 0 00 0 0 1

414

12

0 0 0 12·l

12·l −1

l0 0 0 −1

l1l 0

y1

y2

y3

y4

y5

y6

. (D.4)

164 FEM model

For P 3:

yp =

yx

yy

yRz

yz

yRx

yRy

=

12 0 1

2 0 0 0 0 00 1

2 0 12 0 0 0 0

− 14·ds

− 14·ds

14·ds

14·ds

0 0 0 0

0 0 0 0 14

14

14

14

0 0 0 0 12·l

12·l − 1

2·l − 12·l

0 0 0 0 − 12·l

12·l

12·l − 1

2·l

y1

y2

y3

y4

y5

y6

y7

y8

. (D.5)

For P iz , the translations can be obtained directly from the FEM model and the rotations

need to be estimated from the surrounding nodes. The transformation needed to obtain theperformance variables from the FEM model is given by

zp =

zxzyzRz

zzzRx

zRy

=

1 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 00 0 0 1

4·rz− 1

4·rz− 1

4·rz

14·rz

0 0 0 0

0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 1

2·rz0 − 1

2·rz

0 0 0 0 0 0 0 − 12·rz

0 12·rz

0

y1

y2

y3

y4

y5

y6

y7

y8

y9

y10

y11

.

(D.6)

D.4.2 Static actuator transformation

A similar operation needs to be performed at the actuator side. The derivation of Tu consistsof two steps: Gain Balancing (GB) and Gain Scheduling (GS), as illustrated in Figure D.4.

Gain balancing

First construct the Actuator-To-Origin (ATO) matrix, which describes the relation betweenthe physical plant inputs (up) and the forces working on the COG (uc). It is hereby assumedthat the COG is exactly in the middle of block. This results in:

D.4 Rigid body decoupling 165

For P 1 and P 2:

uc ,

ux

uy

uRz

uz

uRx

uRy

=

12

12 −1 0 0 0

−12

√3 1

2

√3 0 0 0 0

−12 · ra −1

2 · ra −12 · ra 0 0 0

0 0 0 1 1 1

−14

√3 · h 1

4

√3 · h 0 −1

2 · ra ra −12 · ra

−14 · h −1

4 · h 12 · h −1

2

√3 · ra 0 1

2

√3 · ra

u1

u2

u3

u4

u5

u6

︸︷︷︸= up

.

(D.7)

For P 3:

uc =

ux

uy

uRz

uz

uRx

uRy

=

0 1 0 1 0 0 0 01 0 1 0 0 0 0 0da −da −da da 0 0 0 00 0 0 0 1 1 1 1h2 0 h

2 0 da da −da −da

0 −h2 0 −h

2 −da da da −da

u1

u2

u3

u4

u5

u6

u7

u8

. (D.8)

The GB matrix is then given by

GB = (ATO)† , (D.9)

where † represents the Moore-Penrose pseudoinverse. In case the ATO matrix is square, asfor P 1 and P 2, the pseudoinverse reduces to the normal inverse.1

Gain scheduling

Since the POC is in general not the same as the COG, a correction needs to be made totransform uc into forces that work on the POC (up). Because the POC is the same for thethree I/O sets, only one GS matrix has to be derived. For the model used in this research, thePOC is located on the top surface straight above the COG.2 By using the relation betweenthe accelerations of the COG and the POC, the GS matrix can be derived and it follows thatthe relation between uc and up is given by

uc =

ux

uy

uRz

uz

uRx

uRy

=

1 0 0 0 0 − M ·h2·Iyy

0 1 0 0 M ·h2·Ixx

0

0 0 1 0 0 00 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

ux

uy

uRz

uz

uRx

uRy

︸︷︷︸= up

, (D.10)

1Besides using the pseudoinverse, another possibility is to add constraint equations to the ATO matrix suchthat it becomes a nonsingular matrix.

2In practice, the POC continuously moves on the top surface during normal operation.

166 FEM model

Due to the free meshing elements in the FEM model, the COG is not exactly in the middleand a small correction is made on Tu to achieve exact decoupling.

D.4.3 Effect of decoupling

Figure D.6 shows the effect of decoupling for I/O set 3. Similar results can be obtained for theother two I/O sets, but since there are no controllers designed for these I/O sets in Chapter 7,results of decoupling are only presented for I/O set 3. If the flexible modes are eliminatedfrom the FEM model, the system is a rigid body and there is roughly a 300 [dB] differencebetween the diagonal and off-diagonal terms; perfect decoupling is achieved. In case flexibledynamics is present, decoupling can only be achieved up to the frequency where the flexibledynamics starts. Hence, the system is not decoupled at high (above approximately 400 [Hz])frequencies.

D.4

Rigid

body

decou

plin

g167

100

102

104

−600

−400

−200

0

To:y x

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−600

−400

−200

0To:y y

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−600

−400

−200

0

To:y R

z

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−600

−400

−200

0

To:y z

1

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−600

−400

−200

0

To:y R

x

1

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−600

−400

−200

0

To:y R

y

1

From: ux

100

102

104

From: uy

100

102

104

From: uRz

100

102

104

From: uz

100

102

104

From: uRx

100

102

104

From: uRy

Frequency [Hz]

Mag

nit

ude

[dB

]

Fig

ure

D.6

:B

ode

magn

itude

diagram

ofP

3w

ith100

flex

ible

modes

(thin

line)

and

only

rigidbody

modes

(thick

line).

Appendix E

Frequency domain results for theWS FEM model

In this appendix, Bode magnitude diagram and singular value plots are depicted that arenot included in Chapter 7. Three controllers are designed in Chapter 7 using H∞ loop-shaping. Controller C for the single-DOF control structure (see Figure 3.1), controller blockCb for the inferential control structure (see Figure 3.6(b)), and controller block Ceb

b for theinferential control structure with error-based weighting filter selection. For the inferentialcontrol structure, the effect of the performance filter Ca is not taken into account, i.e., it isassumed that an exact estimate of zp can be obtained.

The 6 × 6 Bode magnitude diagrams for C, Cb, and Cebb are depicted in Figures E.1, E.4,

and E.7, respectively. The maximum and minimum singular values of Ps, L, and LI aredepicted for C, Cb, and Ceb

b in Figures E.2, E.5, and E.8, respectively. Finally, the maximumand minimum singular values of the TFMs contained in the four-block control problem, i.e.,SI , SP , CS, and T , are depicted for C, Cb, and Ceb

b in Figures E.3, E.6, and E.9, respectively.

169

170Freq

uen

cydom

ainresu

ltsfor

the

WS

FE

Mm

odel

100

102

104

−100

−50

0

To:y x

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−100

−50

0To:y y

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−100

−50

0

To:y R

z

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−100

−50

0

To:y z

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−100

−50

0

To:y R

x

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

100

102

104

−100

−50

0

To:y R

y

From: ux

100

102

104

From: uy

100

102

104

From: uRz

100

102

104

From: uz

100

102

104

From: uRx

100

102

104

From: uRy

Frequency [Hz]

Mag

nit

ude

[dB

]

Fig

ure

E.1

:B

ode

magn

itude

diagram

ofC

.

Frequency domain results for the WS FEM model 171

101

102

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104

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40

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Magnitude

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Figure E.2: Maximum and minimum singular values of Ps (thin black line), L (thick blackline), and LI (thick grey line) for C.

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Magnitude

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310

4−60

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Frequency [Hz]

Figure E.3: Maximum (thick line) and minimum (thin line) singular values of the TFMscontained in the four-block control problem for C.

172Freq

uen

cydom

ainresu

ltsfor

the

WS

FE

Mm

odel

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From: uRz

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From: uz

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From: uRx

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From: uRy

Frequency [Hz]

Mag

nit

ude

[dB

]

Fig

ure

E.4

:B

ode

magn

itude

diagram

ofC

b .


101

102

103

104

−80

−60

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−20

0

20

40

Frequency [Hz]

Magnitude

[dB

]

Figure E.5: Maximum and minimum singular values of Ps (thin black line), L (thick blackline), and LI (thick grey line) for Cb.

100

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Magnitude

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Magnitude

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Frequency [Hz]10

010

110

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310

4−60

−40

−20

0

20

40SI

Frequency [Hz]

Figure E.6: Maximum (thick line) and minimum (thin line) singular values of the TFMscontained in the four-block control problem for Cb.

174Freq

uen

cydom

ainresu

ltsfor

the

WS

FE

Mm

odel

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To:z x

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From: uRz

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102

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From: uz

100

102

104

From: uRx

100

102

104

From: uRy

Frequency [Hz]

Mag

nit

ude

[dB

]

Fig

ure

E.7

:B

ode

magn

itude

diagram

ofC

ebb

.


101

102

103

104

−80

−60

−40

−20

0

20

40

Frequency [Hz]

Magnitude

[dB

]

Figure E.8: Maximum and minimum singular values of Ps (thin black line), L (thick blackline), and LI (thick grey line) for Ceb

b .

100

101

102

103

104

−80

−60

−40

−20

0

20

Magnitude

[dB

]

PSIC

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20SIC

100

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20

40PSI

Magnitude

[dB

]

Frequency [Hz]10

010

110

210

310

4−60

−40

−20

0

20

40SI

Frequency [Hz]

Figure E.9: Maximum (thick line) and minimum (thin line) singular values of the TFMscontained in the four-block control problem for Ceb

b .

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