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An investigation to the applicability of a nonlinear friction compensator

Hultermans, S.C.

Award date:1995

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Report no: 95.166

An investigation to the applicability of a

nonlinear friction compensator

S.C. Hultermans

November 15, 1995

An investigation to the applicability of a nonlinear

friction compensator

Graduation report

S.C. Hultermans

Supervisor: Prof.Dr.Ir. J.J. Kok

Coaches: Dr.Ir. M.J.G. van de Molengraft Eindhoven University of Technology Faculty of Mechanical Engineering section Control Engineering

Ir. M. J. Vervoordeldonk Philips Centre for Manufacturing Technology Eindhoven

Report no: 95.166

Report no: 95.166

S.C. Hultermans

An investigation to the applicability of a non-linear friction conipensator

ABSTRACT

To improve the system behaviour of friction influenced mechanical servo-systems , several friction compensation methods are described in literature. This research investigates the applicability of a non-linear control concept, proposed in the ‘Journal of Dynamic Systems, Measurement and Control’ by Southward et al. In the article it is claimed that an exact friction model is not required. This non-linear control concept is referred to as the RNL (Robust Non-Linear) control concept. Theoretical analysis and simulations, performed during this research, regarding a RNL controlled one degree of freedom system influenced by friction, showed a very good positioning behaviour.

Experiments on an industrial application, i.e. the LiMMSl axis, which is generally assumed to behave as an one degree of freedom system, showed the occurrence of a limit cycle. Experiments in earlier research showed also the occurrence of limit cycle phenomena when applying the RNL control concept to industrial applications, but no explanation was found. This research revealed that the limit cycle phenomenon is caused by the fact that the LiMMS axis essentially behaves as a two degrees of freedom system when RNL control is applied. The results of the experiments and simulations led to the conclusion that the proposed non-linear control concept is not applicable to this class of positioning devices.

In order to get insight in the limit cycle behaviour, with the purpose to alter the non-linear system’s configuration to prevent limit cycling, the RNL controlled system is analysed using a numerical variant of the sinusoidal describing function (DF). The analysis results described the limit cycle characteristics of the RNL controlled system, in terms of amplitude and frequency, fairly well. These results were used to create a non-linear filter in order te exclude the Emit cycle phenomenon. The design method used, is based on linear superposition of the RNL controlled system’s DF and the DF of the non-linear filter. Subsequently, the RNL controller was extended with this non-linear filter. However, simulations revealed a behaviour other than expected. In other words, limit cycling could not be prevented by applying the non-linear filter to the RNL controller. This is explained by the fact that it was incorrect to assume tbai the Characteristics of the extended non- linear system are only determined by linear superposition of the original RNL controlled system’s DF and the DF of the non-linear adaptation.

Designing an adaptation to the RNL controlled system which could exclude limit cycling might theoretically succeed when an iterative design method, based on the total extended system’s DF, is used. However, due to certain expected difficulties during simulation this possibility is not considered to be a practical one. Therefore, it is recommended to search for or develop other non-linear design methods in order to exclude limit cycling behaviour in non-linear systems.

‘LiMMS: &near Motor ganipulator System.

1

Preface

This report is a result of a graduation project in order to conclude the M.Sc. course at Eindhoven University of Technology. This project has been performed at the Predictive Modelling Group of Philips CFT Mechatronics, Eindhoven.

I want to thank all the people who supported me during this research project. However, a few people have to be mentioned explicitly.

At first, I want to thank Michiel Vervoordeldonk, my mentor at Philips, for all the in- structive discussions we had about the research subject. I also want to thank René van de Molengraft for the constructive meetings we had and for being an encouraging mentor. Furthermore, I address my gratitude to Maarten Steinbuch for his contribution to the realization of this report and for the many inspiring conversations (for the second time). At last I want to thank Adrian Rankers and Jan van Eijk who gave me the opportunity to carry out my graduate project in the pleasant and educational environment of the Pre- dictive Modelling Group as part of the Mechatronics department.

Bas Hultermans Valkenswaard , 1995.

... 111

Contents

Abstract 1

... Preface 111

Contents V

1 Introduction 1 1.1 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Structure of the research project . . . . . . . . . . . . . . . . . . . . . . . . 2

2 RNL-control concept 2.1 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Robust non-linear stick-slip friction compensation . . . . . . . . . . . . . . .

2.2.1 One degree of freedom (1-DOF) system influenced by stick-slip friction 2.2.2 PD-control of a 1-DOF system with friction . . . . . . . . . . . . . . 2.2.3 Robust non-linear control design . . . . . . . . . . . . . . . . . . . .

2.3.1 1-DOF system description . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Controlled system specifications and PD-control design . . . . . . . 2.3.3 Robust non-linear control of 1-DOF system with friction . . . . . . . 2.3.4 Linear control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 RNL-control of 1-DOF systems with friction (simulation) . . . . . . . . . .

3 3 4 5 6 8

10 18 11 12 15

3 Experimental environment 19 3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Process modelling and PD control design . . . . . . . . . . . . . . . . . . . 21 3.3 RNL control design and experiment design . . . . . . . . . . . . . . . . . . . 23 3.4 Experimental results and evaluation . . . . . . . . . . . . . . . . . . . . . . 25 3.5 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

I

4 System analysis 29 4.1 Improvement of the dynamic model . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Simulation of the improved system . . . . . . . . . . . . . . . . . . . . . . . 32

5 Non-linear system analysis 35 5.1 Limit cycle analysis of R.NL controlled system . . . . . . . . . . . . . . . . 36 5.2 39 Generating solutions to the limit cycle problem using the DF . . . . . . . . .

Y

6 Conclusions and recommendations 43 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Bibliography 47

A Bode plot of the PD controller 49

B Application of the RNL compensation in a discrete environment 51 B.l Influence of discrete RNL on system behaviour . . . . . . . . . . . . . . . . 51 B.2 Limit cycling in discrete RNL . . . . . . . . . . . . . . . . . . . . . . . . . . 52 B.3 Maximum compensation force in discrete RNL . . . . . . . . . . . . . . . . 54 B.4 Verification by simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

C Simulation model of 1-DOF system with friction 59 C.l simulation model in blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

C.l.l Leadlag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 C.1.2 Non-linear friction compensation . . . . . . . . . . . . . . . . . . . . 60 (3.1.3 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 (3.1.4 Friction simulation model . . . . . . . . . . . . . . . . . . . . . . . . 61

C.2 Zero Order Hold (ZOH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 C.3 Quantizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 C.4 Numerical integration method and parameters . . . . . . . . . . . . . . . . 63

D Digital control environment 65

E Simulation model of 2-DOF system with friction 67 E.l Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 E.2 simulation model in blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

E.2.1 Lead lag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 E.2.2 Non-linear friction compensation . . . . . . . . . . . . . . . . . . . . 68 E.2.3 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 E.2.4 Friction simulation model . . . . . . . . . . . . . . . . . . . . . . . . 69

b.6 Zero Order Hold (ZOB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 E.4 Quantizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 E.5 Numerical integration method and parameters . . . . . . . . . . . . . . . . 71 E.6 Overview most important system parameters . . . . . . . . . . . . . . . . . 72

T.-.

F Ideniificakion 73 F.l Spectral analysis theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 F.2 Determination of the frequency response of the LiMMS-axis . . . . . . . . . 75

G Determination of the dynamic friction component 77

H Derivation of the theoretical model of the LiMMS-axis 81

I Contrd software 83 1.1 C-sourcedCPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 1.2 Communication pascal source software . . . . . . . . . . . . . . . . . . . . . 90

J Third order setpoint 99

K Sinusoidal-input describing function (DF) theory 101

Using the describing function to determine limit cycle characteristics . . . . 103 K.1 Sinusoidal-input describing function (DF) theory . . . . . . . . . . . . . . . 101 K.2 K.3 Stability of limit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

M Interpolation and filtering 111

N Tuning rules non-linear network NLF 113

O DF calculation NLF 117

i

vii

~! i i

Chapter 1

Introduction n

1.1 Backgrounds

Nowadays the request for, as well as the desired accuracy of, high performance servo systems increases. However, due to the inevitable presence of friction the performance of conventional (for example P(1)D) controlled servo systems is limited. Application of a PD (Proportional Derivative) control concept to a servo system influenced by friction implies a steady state error. Using a PID (Proportional Integral Derivative) control concept eliminates the steady state error but introduces a certain settling time when applied in situations where small positioning errors occur. This i s caused by the time needed to integrate these small errors to create enough power to overcome the static friction level.

To improve the tracking and positioning performance of mechanical servo systems with friction several model based control techniques are developed [13]. The success of these techniques depends on the accuracy of the friction model used. Literature survey [13] shows however that the scientific world shows little consensus on the modelling of the friction phenomenon. This is due to the fact that friction depends on a variety of parameters such as the materials involved, the applied lubricant , temperature, time, relative position and velocity between the two contacting surfaces. Another disadvantage of using these techniques is the increase in complexity of the model. At last, due to the lack of experimental verification, assessment of these model based techniques, is still difficult.

A less complex friction compensation technique for application to servo systems can be found in Southward Southward e t al. [26]. This non-linear control concept is claimed being robust for friction variations. More specific, it is claimed that this technique does not require an exact friction model. This technique is referred to in this report as the RNL1-controZ concept. The claimed robustness for the friction model and some promising experimental results [26], [20] on the one hand and the appearance of limit cycling phenomena when applying the RNL-method to a high performance servo system described in earlier research [ia] on the other hand demands further investigation of the applicability of this technique.

'RNIL: Robust Non Linear

1

Chapter I Introduction

1.2 Problem formulation

The preceding section results in the objective of this research project:

The objective of this research project is to investigate the applicability of a non-linear control concept [2G] for high performance (precision) inechanical servo systems with friction.

1.3 Structure of the research project

In order to get insight in the RNL control concept, this research is started with the description of the RNL theory and the investigation of the behaviour of a RNL controlled one degree of freedom mass by carrying out simulations (chapter 2).

Subsequently, the application of the RNL control concept to an experimental setup is investigated in order to verify the suitability of the RNL control concept in practical applications (chapter 3). The experimental setup involved is a LiMMS2 axis, which is assumed to approximate a one degree of freedom system.

During the experiments an unexpected oscillation with constant amplitude and constant frequency occurred, which was identified to be a limit cycle. A limit cycle is an oscillation in a closed loop system, which is not caused by an oscillation at the reference input. In order to be able to understand the difference (and find the cause for the limit cycling phenomenon) between the one degree of freedom system used in simulation and the one used in experiments, an in depth analysis of the experimental setup is carried out (chapter 4).

Sustained oscillations have to be prevented in mechanical servo systems, because they can cause severe damage to the control system, they consume a lot of energy and they limit the performance of servo systems. Since the RNL control concept however, shows an excellent settling behaviour in simulations, it is tried to eliminate the limit cycling phenomenon in order to be able to apply the non-linear control concept to industrial applications. In this context the non-linear system is analysed using the describing function theory to get insight in the limit cycle behaviour and generate solutions (chapter 5).

The last part of this research describes the general conclusions and the recommendations for further research (chapter 6).

'LiMMS: &near Motor Manipulator System

2

Chapter 2

RNL-control concept

As mentioned in the introduction, most existing friction-model-based control techniques are based on a specific friction model. These models are all simplifications of the real friction characteristic. Because of the unpredictable variability of friction none of these control methods are able to guarantee perfect positioning when applied to servo systems. It is claimed that applying the non-linear friction compensation technique as proposed by Southward et al. [26] to PD-controlled servo systems leads to an errorless positioning behaviour as long as the friction is bounded between certain minimum and maximum limits. It is also claimed that the settling time for the non-linear compensation technique is smaller in comparison with a conventional PID-controller.

For convenience the non-linear control technique is referred to in this report as: "RNL- control concept".

For good understanding of the RNL-control concept firstly the friction phenomenon (section 2.1) is discussed. Subsequently, the proposed non-linear control concept is explained (section 2.2). The last section (2.3) of this chapter describe simulations that visualise the behaviour of a RNL controlled l-DOF1 system affected by friction. In this section also a comparison with the classical control (PID) techniques is made.

2.1 Friction

Whenever there is motion or tendency of motion between two elements, frictional forces exist. The frictional forces encountered in physical systems are usually of a non-linear nature. Figure 2.1 shows the friction as a function of velocity for lubricated systems [Hl. This curve is referred to as the Stribeck curve.

The curve visualises two main components of the friction force, F f :

1. The static friction force also referred to as sticking friction force Fstick. When the relative velocity of two contacting surfaces equals zero (2 = O), the friction force is referred to as Fstick. The sign of the friction force opposes the direction of the resultant force affecting the surface of interest. The value of the static friction force

'DOF: Degree Of Freedom

3

,

Chapter 2 RNL-control concept

Stribeck effect Ff I f

Coulomb friction level - - - - - - - - - - - - _ - - ‘ w o u s friction

l i - relative velocity x I xs

- static friction force level cp = arctan(D)

Fig. 2.1: The Stribeck curve

depends on the external forces affecting the surface of interest and is always between certain minimum and maximum levels. Once the external forces overcome the static friction force levels a relative movement between the surfaces occurs.

2. The dynamic friction force also referred to as slipping friction force Fslip. This force affects both surfaces in case the relative velocity 2 is unequal to zero. This force consists of a viscous friction force D IC where D is the viscous damping coefficient, and a minimum (in absolute sense) constant force referred to as the Coulomb friction level, FCoulomb .

The typical transition from the maximum static friction force to the dynamic friction force when the relative velocity changes from zero to a certain nonzero value, is referred to as the Stribeck effect. The Stribeck effect ends (or begins) when the relative velocity k s is reached (figure 2.1). In this report the distance from zero relative velocity to ks is assumed to be very small which results in the assumption that the Stribeck effect is negligible. When the relative velocity IL. > 9,, the dy;zar;iic frictiori force is represented by

E z i p = FCoulomb + (2.1) The above described friction model is often referred to as stick-slip friction or the stick-slip phenomenon.

For the sake of completeness it is mentioned that the friction force levels also depend on other parameters/variables as mentioned in the introduction. In this investigation the influence of these parameters/variables is neglected, which is generally accepted while investigating lubricated systems [18]

2.2 Robust non-linear stick-slip friction compensation

Robust Non-hear Stick-Slip Compensation (RNL compensation) is a friction compensation technique introduced by Southward, Radcliffe and MacCluer[26] €or the regulation

,

4


of a 1-DOF mass with stick-slip friction. The non-linear control concept is claimed to be robust with respect to the character of the slipping' force which is assumed to lie within a piecewise linear band. Exact knowledge of the static friction force levels is claimed not to be required, only upper bounds for this level. The non-linear stick-slip compensation force is developed to supplement a Proportional + Derivative (PD) control law applied to a 1-DOF mechanical system with friction.

2.2.1 One degree of freedom (1-DOF) system influenced by stick-slip friction

The system under investigation is a mass constrained to move in one dimension with stick- slip friction ( F j ) present between the mass (m) and the supporting surface, see figure 2.2.1. The control force (F,l) (actuator force) necessary to position the mass is assumed to work

x -

mass

Fig. 2.2: Mass influenced by friction

directly upon the mass. The state space model for this system is given by:

o 1 [ ; ] = [ o o ] [ : ] + [ : -i][;]

- - A possible output equation is:

S t i&-slip friction The stick-slip friction force as defined by Southward et al. [26] is:

The sticking force, Fst;,k(Fcl), represents the friction force at zero velocity3 . The positive and negative limits of the sticking force are respectively given by F' and F; representing the positive and negative static friction force as represented in figure 2.2.1. These limits are assumed to be constant, but they are not presumed to be equal in magnitude. The mass cannot move until the driving force FCl is greater in magnitude than the respective static friction force. As long as the velocity equals zero Fstick = Fel.

'The slipping force is a the friction force at nonzero velocity. 3As of now the term velocity refers to the velocity of the mass

5


The slipping force Fslip(X) represents the friction force at nonzero velocity, and is given by:

Fslip(X) = F,S(X)v(X) + Fä(X)v( -X) (2.6) where v(.) is the right continuous unit step function. The function F’(X) defines the slipping force for positive velocity and F T ( X ) defines the the slipping force for negative velocity of the mass. The slipping force can be denoted as:

Fdf = F+ Coulomb +D+X ’d X > O

Fä = F- Coulomb +D-X ’d X < O

where F$oulomb is the constant Coulomb friction level for positive velocity and F;oulomb is the constant Coulomb friction level for negative velocity. D+ is the viscous damping coefficient for positive velocity and D- is the viscous damping coefficient for negative velocity. From now on it is assumed that D+ = -D- = D Furthermore it is assumed that there exist constants bl 2 bo > O,Fs- 5 Fö < O, and FZ 2 F: > O which define the following piecewise linear bounds for the slipping force, see also figure 2.3:

Based on physical observation it is stated that FS and F; are natural bounds for F$ and Fo-. This definition is used in a later stage to determine the non-linear friction compensation.

Fig. 2.3: Slipping force function

2.2.2

In order to describe the functioning of the RNL method orderly first the basic behaviour of a theoretical Proportional + Derivative (PD) controlled 1-DOF mass with stick-slip friction (figure 2.4) is discussed. In figure 2.4 C(s) is used to refer to the PD-controller and P ( s ) is used to refer to the transfer function of the by friction F’ disturbed mass. For the time being friction is modelled as an external disturbance force. The setpoint is referred to as T and e represents the position error.

PD-control of a 1-DOF system with friction

6


Fig. 2.4: Block diagram of a PD controlled system disturbed by friction

The transfer function of the PD-controIler4 is given by [26]:

where K p is the proportional gain and II'~ is the derivative gain. Phase plane analysis of this PD-controlled system yields state trajectories which end up or near the origin with a bounded steady state error [3], [24]. See also figure 2.5.

Fig. 2.5: Phase portrait of a 1-DOF mass system affected by friction

This bounded area of equilibrium points ( k = x = O) is defined as EPD. The EPI) limits are derived as follows: As assumed in subsection 2.2.1 the control force (&) affects the mass (m) directly. This leads in combination with the friction force ( F f ) to the system equation:

mx = FCl - Ff (2.1Q)

where 2 is the acceleration of the mass. Since the actuator force equals the control force, FCl can be represented by:

FCl = KPe f I ide (2.11)

where e = T - x conform figure 2.4. (2.11) and (2.4) in (2.10) yields for the equilibrium

41n a later stage of this report PD-control will be described more extensively

7

Chapter 2 RNE-control concept

points ( 5 = 2 = O) and the assumption T = 05:

- K p X - Fstick(Fcl) = 0 (2.12)

(2.13)

Solving (2.13) using the definition for FStick(F) from subsection 2.2.1 yields the set of eC@briUm puint's Epa:

EPD = { ( ~ , k ) I 5 = O , X L 5 x 5 XH} where the steady state error bounds XL and XH are defined as:

(2.14)

..=-(E) (2.15)

As long as the mass is stagnated between the steady state error bounds (sticking limits), the proportional control force is unable to move the mass. The steady state error is limited by XL and XH.

2.2.3 Robust non-linear control design

In order to alter the set EPD, defined in (2.14), to one unique equilibrium point viz. x = O, the proposed non-linear compensation force F,(z) is added to the PD-control force ( T = O):

Fnl = -K,x - KdZ - F,(x) (2.16)

where F,(x) is chosen to be: Fc = Kpxc (2.17)

The non-linear function x, [26] which insures x = O to be the unique equilibrium point of the closed loop system is described by:

where

(2.18)

(2.19)

The compensation force is only active within the limits: 53, 5 x 5 53, and x # O. Within these limits the position dependent part of the non-linear control force Fni, called F ~ D , is:

Fsf Y o < X < z H FPD, = KPx + F,(x) = o v x = o (2.20)

Fs- v 5h < x < o where F$ = F: + K ~ E and FF = F; - K ~ E .

5 e can be substituted by -x without loss of generality

8


In figure 2.6 a graphical representation is given of FpgZ. The dashed line represents the

I - Z H 2 , position error

Fig. 2.6: RNL adjusted position dependent part of the control action

proportional control force without the non-linear compensation force. When the mass is between the original sticking limits ZL 5 z 5 ZH and the non-linear compensation force is active, any positive value of E is enough to guarantee that the feedback force will always exceed the static friction levels. This insures that the mass continues moving toward the origin. Notice that when the mass’ position reaches zero, the controller force will be essentially a bang-bang force. Southward et al proved that the equilibrium point is global asymptotically stable. The combination of the PD-controller with application of the non- linear adaptation Kpxc is referred to as the RNL-controller.

The first remark regarding the RNL-control concept which has to be made is: by application of RNL-control to a system influenced by friction, an extra discontinuous non-linearity besides the stick-slip friction phenomenon is introduced. In order to get insight in the consequences of this addition with regards to the system’s behaviour simulation of the non-linear system is necessary. Section 2.3 will describe simulation results regarding the application of the RNL-control technique to a 1-DOF mechanical servo system influenced by friction.

The second remark concerns the claimed robustness for variation of the friction levels. Since the maximum (minimum) static friction force levels in practical systems are hard to determine because of the dependence of time, position, etc, a certain minimum overcompensation is necessary. The overcompensation level ( KPc) during simulation is therefore chosen to be at least 10 76 of the maximum determined static friction level in order to have a safe margin.

9


2.3 RNL-control of 1-DOF systems with friction (simulation)

In order to perform realistic simulations, the system data necessary to describe the 1-DOF mass system is taken from an industrial application which is assumed to approximate 1- DOF system behaviour, the LiMMS axis. Identification results with respect to this LiMMS axis are discussed in chapter 3 since the LiMMS axis is siubject of experimental research in a later stage of this project.

2.3.1 1-DOF system description

The 1-DOF RNL-controlled system consists of a mass influenced by friction (G), a RNL- controller (CRNL) and an idealized gain actuator ( A ) (control signal is transformed to control force). Figure 2.7 shows the closed loop block diagram of this system where a double Pined block refers to a non linear element..

I I------------- G

Fig. 2.7: Block diagram of the closed loop 1-DOF system

G is described by (2.2) and (2.3). The friction is modelled as discussed in section 2.2.1. The system’s parameters used in simulation are given in table 2.1. These parameters

parameter designation value unit m Mass 10.3 [ k l FS = -Fs- Static friction level (symmetric) 27 [NI F$(O) = -Fy (0 ) Coulomb friction level (symmetric) 23 [NI

A Amplification 66 [VI inc Encoder resolution le-6 [ml

NS D Viscous damping coefficient 20 [,I

TS Sample time 0.25e-3 [s]

Table 2.1: parameters 1-DOF system

are based on Identification results regarding an experimental setup as mentioned before. In order to be able to control (feedback) this system a stabilizing controller has to be developed which is described in the next subsection.

10

2.3' ENL-control of l -DOF systems with friction (simulation)

2.3.2

Before being able to analyse the behaviour of the R.NL-controfled second order system, a non-linear RNL controller has to be designed. This controller is designed in two steps. At first, a stabilizing PD-controller C(s) is designed. Secondly, the non-linear part F,(z) proposed by Southward et al. is added to the PD-control action which eventually results in the RNL-controller CRNL (section 2.3.3).

Controlled system specifications and PD-control design

The design of the PD-controller C(s) requires some time or frequency domain specifications. In this stage of theoretical investigation only frequency domain specifications are considered. These specifications relate however to linear systems which implies the necessity of linearizing the non-linear system in the case at hand. This is done by neglecting the friction phenomenon during the design of a linear controller. The open loop gain of the linearized system is referred to as H ( s ) .

The frequency domain performance (relative stability) of the closed loop system can be characterized by three quantities described by the following definitions [18]: (i) The bandwidth frequency ( f b [Hz]) is defined6 as the smallest frequency where the open loop gain H ( s ) (figure 2.7) equals 1. (ii) The phase margin ( S , [degrees]) is defined as the angle in degrees through which the Nyquist plot of H ( s ) must be rotated about the origin in order that the gain-crossover point7 on the locus passes through the (-1,O) point . (iii) The gain margin b is defined as the amount of gain that can be allowed to increase in the loop before the closed loop system reaches instability.

In the case at hand (linearized system) the bandwidth, f b , is chosen to be 30 [Hz]. The phase margin, +, is selected to equal 45' and the gain margin, b, is specified to be greater or equal then 0.5 (= -6 dB). The reason of restricting the bandwidth of the controlled system is to approximate a realistic situation [27].

The control structure is chosen to be a lead lag (CLL) control action combined with a lowpass filter (CLP). The control structure is denoted by (2.21).

(2.21)

The designation of the parameters as well as the accompanying values are given by:

-tr, proportional gain 2100 [Vlml K d derivative gain 28 [Vslml Y lead ratio 100 [-I P damping ratio 0.7 [-I on undamped natural frequency 754 b d l s l

'This is the definition used within Philips CFT 7The gain crossover point is the point of intersection between the H ( s ) locus and the unit circle in the

Nyquist plot

11


This choice of the control structure (2.21) is based on the following considerations:

1. A theoretical unlimited derivative action is not realizable, therefore a lead lag filter is chosen. In this application the derivative action is limited and thus realizable

2. A lowpass filter is added to the controller to prevent amplification of high frequent disturbances by the lead lag filter.

In order to meet the specifications the following parameter values are chosen as given in The damping ratio ,8 is chosen to be 0.7 to avoid extra overshoot of the response. A Bode plot of the controller is given in appendix A. The open loop frequency response of the controlled system is given in figure 2.8.

1 O0

50 I

a, D

= C = o

E -50 P

-100 1 oo 1 O’ 1 o2 1 o3

frequency [Hz]

-100

- $ -200 m 0

3 -300 a,

8 P

-400 1 oo 10’ 1 o* 1 o3

frequency [Hz]

Fig. 2.8: Frequency response of H ( s )

The resulting bandwidth of the controlled system is 30 [Hz], the phase margin is 45 [degrees] and the gain margin equals 0.74. The controlled system satisfies the frequency domain specifications. The application of this controller results in a closed loop system behaviour as described in subsection 2.2.2.

As to reduce the steady state error the non-linear F , ( z ) compensation is added to G ( S ) . The next subsection describes the implementation of the RNL-controller as well as the choice of the necessary parameter values.

2.3.3

A numerical simulation of the non-linear stick-slip compensation force applied to the 1- DOF mass system will verify the stability and will contribute to the understanding of the RNL-controlled system behaviour. As mentioned before the measured position is quantized [9] (figure 2.9). Furthermore a zero order hold [7] is added to the controller

Robust non-linear control of 1-DOF system with friction

12

2.3 RNL-control of 1-DOF systems with friction (simulation)

- 1 A l ms r

I

in order to approximate digital control behaviour. The simulations are performed using SIMULINK. Figure 2.9 shows the block diagram of the simulation model.

I I I S

1 RNL controller r - - - - - - - - - i I I

Fig. 2.9: Block diagram of the model used in simulation

A double lined block indicates a non-linear element in the system. The block named LiMMS axis refers to the 1-DOF mass with friction. Block C ( s ) refers to the proportional derivative controller filtered by a lowpass filter. F,(z) refers to the non-linear compensation described by (2.17). The amplifier transforms the controller’s output to a force necessary to move the mass. The parameter values necessary to describe the 1-DOF system (including the friction phenomenon) are given in subsection 2.3.1. The simulation parameters regarding C ( s ) are described in subsection 2.3.2. The only parameter value left to be chosen is the amount of overcompensation E described by (2.19). Theoretically, any small positive value of E is enough to guarantee positioning of the mass at the reference position. In practical systems however, the maximum static friction level varies over the trajectory which leads to the necessity of choosing E with a certain margin. The extra controller force (only present within certain bounds) to overcome the static friction can be described by:

Fe = A.KP~ (2.22)

The margin with respect to Fe is chosen to be at least 10% of the average maximum static friction force over the trajectory. Due to the digital nature of the servo system, the overcompensation force is restricted in order to maintain stability of the controlled system. In this case the margin of 10% is well below this maximum level. See appendix B for further information.

In this case the static friction force is 27 [NI and Fe is chosen 3 [NI which leads to:

E = 2.16e - O5 [m] (2.23)

Appendix C describes the SIbIBTLINK model used for simulations as well as the simulation parameters of concern.

13


The sticking limits (2.14) of the PD-controlled system (without application of RNL) are given by:

(2.24)

(2.25)

Two simulations of interest are carried out. The first one concerns a simulation with an initial position deviation of the mass of 20 [pm] which is within the sticking zone. A second simulation regards an initial position of 500 [pm] which is outside the sticking zone. The initial velocity of the mass concerning both situations equals zero. A constant setpoint with value zero is offered to the controller.The simulated time response of the RNL-controlled mass to the initial condition of 20 [pm] is given in figure 2.10.a. This figure also shows the response of the controlled system without the non-linear adaptation F&).

--- Velocity of the mass

0005 O 0 1 0015 O02 0025 O03 time [SI

Fig. 2.10: RNL-controlled system response to an initial condition of 20 [pm] a: Position of the mass for the RNL b: Simulated velocity and control

force of the RNL-controlled system and PD-controlled system

Figiire 2.10.a shows the reduction of the error to zero by applying the RNL method. In this figure it is also shown that the PD-controller is not capable of moving the mass when the mass is at standstill in the sticking zone, thus a steady state error continuous to exist.

Whenever the velocity of the mass becomes zero there exists a possibility of the mass coming to a standstill. From figure 2.10.b it is concluded that the RNE-control force always exceeds the maximum static friction force in these cases, this means that the mass is continuously moving until the position error becomes zero.

An initial condition concerning the mass position outside the sticking zone (500 [inc]) leads to the responses shown in figure 2.11.a. This figure show three different responses.

14


The response of the RNL-controlled system is given by the solid line. The response of a PD (with lowpass filter) controlled system is indicated by the dash dotted line. A PID- controlled system response is given by the dashed line. The response of the RNL-controlled system reaches the desired position very fast. The PD-controlled mass sticks within the sticking zone as expected. The PID-controlled system response will be discussed in the next paragraph. Figure 2.1i.b velocity and control force of the RNL-controlled mass during the last phase of settling. Also in this case the control force keeps the mass moving uniii the zero error position is reached.

6W 50

40 - -- Velocity of the mass

Upper Friction level .,- - - 30- .- I

I 1 t solid line. RNL controlled system response

- y 20- _ _ *- -- I

- 3

- z m-10- i

I I I I I I

dash401 line PD controlled system response m -*

10- dashed Iihe PID mntroiled system response

B o - - I

o I I

. ._.. ' . . , . . . ,.L..:; . . . I

- -30 - I

1.:- <:-- Control force -40 - .,. . .

-50 0.025 0.03 0.035 0.04 0.045 O 0.W5 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.02

time Is] time [SI

Fig. 2.11: RNL-controlled system response to an initial condition of 500 [pm] a: Position response of the mass for b: Simulated ve-

locity and control force concerning RNL, PD and PID-controlled system the RNL-controlled system

The preceding simulations verify the stability of the non-linear controller applied t o a one degree of freedom mass with friction. The steady state error introduced by applying a PD controller is reduced to zero by using the non-linear controller. Subsection 2.3.4 compares the RNL-control method to an equivalent PID-controller, traditional x e d for reducing a steady state error.

2.3.4 Linear control

In order to compare the performance of the RNL-controller to the performance of 2. PIE (Proportional Integral Derivative) controller simulations have been performed. The PID- controller with lowpass filter used during these simulations has the form:

(2.26)

C(s) is defined in 2.21. To meet the frequency domain specifications mentioned in subsection 2.3.2 the Integrator constant K; is chosen 2n- [rad/s]. The other parameters are chosen as given in (2.21). The rest of the simulation parameters and system parameters are the same as described in subsection 2.3.3.

15


The simulated (SIMULINK) response of the PID-controlled system to a initial position of 20 [inc] and a initial velocity which equals zero, is given in figure 2.10.a. In this case response of the PID-controlled system equals the response of the PD controlled system. It is concluded that the PID-controller does not affect the position of the mass during the plotted time. Eventually the PID-controller moves the mass to the desired position but the settling time is approximately 50 times longer than the settle time needed by the equivalent RNL-controller. Increasing the I-action is not a possibility because of the required system specifications. Furthermore, increasing the I-action without adapting the other parameters of the controller eventually yields a unstable controlled system behaviour.

The response of the PID-controlled system to a initial position of 500 [inc] with the same initial condition regarding the velocity as before, is given in figure 2.11.a. In this case the response of the PID-controlled system equals the response of the PD and RNL-controlled systems during the first phase (till approximately 0.075 [s]). After 0.075 [s] the PID- controlled system shows more overshoot compared to the PD controlled system. This Is explained by the extra control force generated by the integral action. The difference in overshoot causes the PID-controller to have a larger error compared to the P D controlled system during the remaining simulation time showed in the plot. In the end (way past simulation time) the error of the PID-controlled system becomes zero just as the RNL- controlled system but the settle time of the RNL controlled system is very small compared to the time needed by the PID controlled system to settle.

The following two items explain the difference in the time needed to settle the system between RNL and PID-control:

1. If the position error is very small and the system is stuck, the integration action will take a long time to build up enough force to get the mass moving again. The RNL compensation will apply enough force instantly to get the mass moving. The necessary integration time causes the difference in the time needed t o settle the system.

2. If the system moves past the reference position, the position error changes sign. In this situation the I action compensation will need some time to generate the force necessary to change the direction of the movement. The RNL compensation however changes the direction of the compensation immediately to generate a force necessary to change the direction of the system’s movement.

The possibility of sending rapidly varying signals to the ampEfier/actuator due to the non- linear RNL compensation force F, defined in (2.17) is a disadvantage because this implies the need for a high bandwidth amplifier/actuator. If the system is equipped with a low bandwidth amplifier, PID compensation is preferred. The fact that RNL compensation uses friction to settle the mass is another drawback of this application in case the friction is subject t o variation. It is however possible to calculate the minimum allowed friction levels for applying the RNL compensation.

It can be concluded that a 1-DOF system with stick-slip friction controlled by RNL settles much faster compared to an equivalent PID controlled system.


To close this subsection the following remarks concerning application of RNL control in a realistic environment, evaluated by simulation can be made:

1. When a 1-DOF system influenced by friction is controlled by a RNL controller the system can be positioned in a unique equilibrium point (desired position) with a short settling time.

2. oppUcatim Gf the XNL-cGEtrG! car,cept is relatively simple; Gniy a. few adaptatiûns regarding the conventional PD controller are necessary.

3. For setting the RNL-controller only F? and Fs- are needed as extra parameters.

4. RNL-control of a 1-DOF system with stick-slip friction seems to be stable as claimed.

The promising results of the theoretical evaluation of RNL-control described in the previous sections motivates an investigation to the applicability of the RNL control concept with respect to an experimental servo system. Therefore a RNL-controller is implemented in a experimental setup in order to investigate the suitability for industrial applications. The experiments and results are described in chapter 3.

17

Chapter 3

Experimental environment

In order to evaluate the RNL control concept in a practical environment, this chapter describes the application of the non-linear compensation method to an industrial servo positioning device (LiMMS axis) which is designed to behave as a positioning device with one degree of freedom.

This chapter is structured as follows. The first section describes the experimental setup. Section 3.2 treats the modelling of the experimental setup in order to be able t o determine the control parameters concerning the RNL control concept. Subsequently section 3.3 describes the actual design of the RNL controller. At last section 3.4 shows the experimental results and gives the conclusions with regard to the application of the RNL control concept to the LiMMS axis.

3.1 Experimental setup

The industrial application used is the controlled Linear Motor Manipulator System (LiMMS)- axis. The LiMMS axis is part of a modular system. Each module in this system provides one degree of freedom. Figure 3.1 shows a setup with three LiMMS axis.

Fig. 3. A three translatz.mal degrees of freedom system, build of three LiMMS axis

19

Chapter 3 Experimental environment

The evaluation of the RNL control concept is restricted to one LiMMS axis. Figure 3.2 shows a graphical representation of the LiMMS axis.

slider

Fig. 3.2: The one translational degree of freedom system, LiMMS axis

The experimental setup comprises: (i) dSPACE, a hardware/software environment for digital control, (ii) an amplifier and (iii) a LiMMS axis, including the position measurement system.

Since the LiMMS-axis and amplifier are subject of modelling, they are described in the next subsections. A more detailed description of the experimental setup and especially the digital control environment (dSPACE) is given in appendix D.

The LiMMS axis The LiMMS axis consists of a slider mounted on a beam, see figure 3.2. The slider can be translated along two linear ball way guidings with a maximum translation of approximately 0.6 [m]. Friction exists between the slider and the beam.

The slider is actuated by a linear direct drive, brushless AC synchrone motor. The "rotor" component of the linear motor is part of the slider. The stator component (permanent magnets) is placed between the linear ball way guidings. As in all electric motors the coils of the rotor do have preference positions with respect to the slotted stator magnets. This phenomenon results in a position dependent periodic force called cogging force which is caused by the attraction force between the copper coils and iron plates in the rotor on the one hand and the permanent stator magnets on the other.

The input of the linear motor is a current generated by an amplifier which will be discussed in the next subsection. The position of the slider is measured by using an optical encoder system. This system exists of an encoder head and a ruler. The optical incremental encoder is mounted on the slider and the ruler is attached to the beam. The accuracy of the measurement is 1 [pm] in other words 1 increment ([inc]) equals le-6 [m].

Ampliffiei- For proper functioning of the h e a r direct drive brushless A@ synchrone motor a current, generated by the amplifier is used. The force generated by the motor is linear dependent

20

3.2 Process modelling and PD control design

+ Amplifier c t I C

on the input to the amplifier (control signal [VI). The commutation is electronic and based on the measured position signal.

I X L L i M M S axis I

The combination of the amplifier and the AC motor behaves up to approximately 1000 [Hz] as a linear brushless DC motor. This combination is also considered to behave as a linear voltage controlled force source.

3.2 Process modelling and PD control design

In order to be able to design a RNL controller, first a stabilizing PD controller has to be designed. A parametric model of the process including its values is necessary for the PD control design. The process model and its values are derived by using theoretical principles and static measurements. This model is used for the actual PD control design. In order to verify the process model, a dynamic analysis of the controlled system is carried out (’, system identification”).

The experimental setup is schematically represented by the block diagram given in figure 3.3

Fig. 3.3: Block diagram of the experimental setup

In the block diagram the process is modelled in block P, consisting of the amplifier and the LiMMS axis. Block C represents the digital controller (dSPACE). T refers to the setpoint position, c to the controller output voltage, f to the amplifier output current and 2 to the position of the slider mass. The signal e refers to the difference between setpoint and the encoder posi tion.

Theoretical process modelling The slider of the LiMMS axis is able to translate along linear ball guidings on the beam. The beam is considered to be part of the fixed world. The slider is assumed to behave as a single mass with one degree of freedom, in the horizontal plane. Furthermore the control force is assumed to affect the mass directly.

This means that the dynamical behaviour of the 1-DOF slider mass affected by the control force Pc, the friction force Ff and the cogging force Fcogg can be described by the differential equation:

mX = KaKmc - Ff + Fcogg (3.1)


where Ka represents the amplifier gain and Km represents the motor gain. A more detailed derivation of equation (3.1) as well as more information about the determination of the parameter values is given in appendix H. The determination of cogging and friction parameters is described in appendix G. The parameter values given in table 3.1 complete equation (3.1).

paramet er designation value unit

F’ = -Fs- Static friction level (symmetric) 27 [NI Fd+ (O) = - F i (O) Coulomb friction level (symmetric) 23 [NI

m Mass 10.3 [sg]

D Viscous damping coefficient 20 [ml Ka Amplifier gain 1 L A P I Ir, Motor gain 66 [NIAI

N s

Table 3.1: parameters 1-DOF system

PD control design The controlled system specifications, the PD control structure (2.21) and its parameters are given in section 2.3.2.

System identification To determine the frequency response of the system, in order to verify the model, a spectral analysis method is used. For the explanation of the applied identification procedure is referred to appendix F. Figure 3.4 shows the calculated frequency response.

. . . . . . . . . . . . .

. . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . -150 10‘ 1 o2 1 o3

frequency [Hz]

O

- g -100

i? -200 Lu D

a>

8 m a-300

-400 1 o’ 1 o2 1 o3

frequency [Hz]

Fig. 3.4: Bode plot of the measured frequency response with second order model fit

The solid Ene represents the frequency response obtained by spectral analysis and the

22

3.3 RNL control design and experiment design

dashed line represents the frequency response based on the theoretical model (H.1). From 10 [Hz] to approximately 200 [Hz] the theoretical model approximates the measured frequency response. Above 200 [Hz] some resonances can be seen. In the frequency area below 10 [Hz] the measured frequency response is not evaluated because the 1 [Hz] com- muting movement of the slider distorted the measurements.

The theoretical modei is said to be sufficient accurate.

3.3 RNL control design and experiment design

As mentioned in section 2.3.2 the non-linear part F,(x) has to be added to the already chosen PD controller (including lowpass filter). Section 2.3.3 describes the aspects of choosing the overcompensation force Fe. For the system under consideration the overcompensation force is chosen to be 3 [NI (approximately 10 %).

Before experiments can be performed, the controller has to be implemented in' the digital environment of dSPACE. This is realized by discretizing the PD controller (inclusive the lowpass filter) and defining its in and outputs using the software program IMPEX [14]. This software program generates a C-code (appendix I) which is downloaded in the DSP processor. In order to be able to send a setpoint to the controller, communication between the AT386 processor and the DSP processor is realized by using a pascal program which is also given in appendix I. This program also allows the online change of some parameters of the controller.

In order to test the behaviour of the RNL controlled system, two setpoints are presented to the controller. The first setpoint dictates a displacement of 16 [mm] and the second setpoint dictates a displacement of 1 [mm]. The type of setpoint used is a third order setpoint. The used setpoints comply with the safe margins of the LiMMS axis [4]. A force feedforward is also added to the system in order to approximate realistic and practical control behaviour. The block diagram given in figure 3.5 refers to the situation.

I I

I I

LiMMS axis ArripGfier - -j-

Fig. 3.5: Setpoint and feedforward

X -

The feedforward concept is introduced to compensate for nominal inertia and for the

23


modelled viscous friction forces. With r ( t ) as the setpoint signal, the feedforward signal f f ( t ) is given as:

m D f f ( t ) = -q t ) A + -7yt) A ( 3 4

where m is the mass of the slider, D is the viscous friction coefficient and A is the amplifier gain. The setpoint and feedforward in the 16 [mni] case are given in figure 3.6.a and 3.6.b. The setpoint in the 1 [mm] case shows a similar characteristic. Appendix J shows the complete characteristic of the used setpoints in terms of the desired position, velocity and acceleration regarding the slider of the LiMMS axis.

I

0006

O 004

o 002

O (

Fig. 3.6: Setpoint and feedforward a: Setpoint b: Feedforward

Before the measurements are executed the slider is placed at a position where the cogging force is minimal, this is done to ensure that only the static friction force has to be overcome by the control force. Since the period of the cogging force is 16 [mm] the cogging force at the desired endposition in case of the 16 [mm] setpoint is also assumed to be negligible. In case of the I [mmj setpoint the siider is also placed in a minimum cogging position, where the cogging is assumed not to vary over a 1 [mm] distance. The next section describes the experimental results and an evaluation.

24

3.4 Experimental results and evaluation

3.4 Experimental results and evaluation

16 [mm] setpoint

The response of the controlled system to the setpoint of 16 [mm] is represented by the error (setpoint - measured position) as given in figure 3.7.a and 3.7.b. In these figures the posi-

50

O

-50

-100 - = - -150

!5

-200

-250

-300

O O 1 02 03 0 4 0 5 0 6 0 7 0 8 O9 1 -350)

lime [SI

12 I 10

8

6

- I e 4 L e $ 2

O

-2

-4

_f

0.75 0.8 0.85 0.9 time [SI

-" 0.7

Fig. 3.7: System response to the 16 [mm] setpoint a: Complete response b: zoomed response

tion error shows an oscillation after the setpoint has become constant (after approximately 0.19 [s]). The amplitude of the oscillation neither increased nor decreased in the course of time. It is concluded that the origination of a limit cycle is a fact. The large error peak from approximately 0.08 [SI to 0.2 [SI is probably caused by a local high friction level. Once the error reached a certain value the controller was able to overcome this high friction level.

In order to evaluate the limit cycle, plot 3.7.b shows the last 0.2 [SI of plot 3.7.a. The amplitude of the limit cycle is between 7 - 8 [inc] and the accompanying frequency is determined to be 135 [Hz]. In order to exclude this phenomenon to be an incident, the same experiment is performed at different places along the beam (axis) where the initial position of each experiment is chosen to be a place with minimum cogging level. This resulted in the conclusion that every experiment showed limit cycle behaviour with little variation in the amplitude f. 3 [inc] and a variation in the limit cycle frequency of f 3 [Hz]. These differences are probably caused due to a variation in the friction. All experiments are also carried out by placing the slider in succession at the minimum cogging force positions and applying a zero valued reference position to the controlled system. The system response showed a limit cycle in all experiments. The properties of the limit cycle were the same as described before.

B [mm] setpoint

The response of the controlled system to the setpoint of 1 [mm] is represented by the error (setpoint - measured position) as given in figure 3.8.a and 3.8.b. Figure 3.8.b shows the

25


response during the time interval [0.35-0.451 (s). This response also shows a limit cycle

i i I I I .

-15 t O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-I -14' 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44

time [SI time [SI

Fig. 3.8: System response to the 1 [mm] setpoint a: Complete response b: zoomed response

with a frequency of 135 [Hz] and an amplitude of 9 [incl.

3.5 Evaluation

The results are not in accordance with the theoretical behaviour of the RNL controlled 1-DOF system as described in section 2.3.3. In order to exclude the option that the PD controller with lowpass filter causes instability, an experiment is carried out where the PD control parameters are not changed and the non-linear action is excluded (system is now PDflowpass controlled). The response of the system to a setpoint of 16 [mm] showed a steady state error of f 120 [inc],as expected. The PD controlled system with lowpass filter is therefore proved to be stable.

It is concluded that the behaviour of the RNL controlled system is not in accordance with the expected behaviour. A direct cause for the existence of the limit cycle is not (yet) demonstrable. Some ad hoc adaptations have been made to the RNL concept which intuitively could suppress the limit cycling. The influence on the stability is examined:

1. The first adaptation is the variation of the overcompensation force. Application of a non-linear control force F,f which is below the maximum static friction level results in steady state error. This is explained by the fact that the maximum friction level is not totally compensated by the non-linear control force. So the control force is not able to compensate the maximum friction force and thus the mass will not move.

Application of a larger overcompensation force Fe (for example 20 [N])results in a Ernit cycle with a larger amplitude and a lower frequency, see figure 3.9.a and 3.9.b.

26

30 15

- s o - 73

20

10

I - z

?-Ij -15 -20 m-10 -30 -20

-25 O 0.1 O 2 O 3 O4 O 5 O 6 0 7 O8 O9 1 O35

-40

Fig. 3.9: System response to the 1 [mm] setpoint with a overcompensation of 20 [NI a: Complete response b: zoomed response

E l i l l i i, i ~~ i :I i i i ;I 1 11 i: 9 1 I:'! I'I:'!~] 036 0.37 0.38 039 0 4 0.41 0.42 0.43 O44

2. The second adaption concerns the altering of the characteristic of the non-linear control concept. A zone around error zero is introduced where the non-linear force is not active. This means that the non-linear compensation force is not active (zero) for absolute values of the error 2 x,, see figure 3.10.

control force t F:

-Za

non linear

/ /"

Fig. 3.10: Non-linear control concept with deadzone

The area -2, 5 error 5 2, is called the dead zone. Investigation of this non-linear control concept with a dead zone is based on choosing Fe = 3[N] and a setpoint of 1 [mm] (setpoint ends at 0.25 [SI) as described before. When the deadzone is introduced by choosing for example x, = 3[inc] the amplitude of the limit cycle becomes 5 [inc] which is smaller then in the case of no deadzone. The frequency increases to 170 [Hz]. Enlarging the deadzone shows a further decrease of the limit cycle's amplitude and a further increase of the limit cycle's frequency. When xu is chosen to be 5 [inc] the limit cycle no longer appears and a steady state error of 2 [inc] exists, see figure 3.11.a and 3.11.b. However altering the setpoint characteristic (larger velocity when entering the dead zone) showed again the origination of a limit

27


10

5 o p t

- - ..

O

-5

I O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time Is]

-10'

I

J 0.3 0.35 0.4 0.45 0.5

-3' 0.25

time [SI

Fig. 3.11: System response to the 1 [mm] setpoint with a overcompensation of 3 [NI and a deadzone of 5 [inc] a: Complete response b: zoomed response

cycle. i.e. the concept is not suited for general purposes. Another disadvantage of applying a deadzone is the introduction of a certain maximum accuracy which equals the dimension of the dead zone. This is of course only of concern if the deadzone is large enough to prevent limit cycling.

At this point in research the following strategy is chosen to continue the investigation to the applicability of the RNL control concept:

i. Analysis of the non-linear system in order to find the cause for the existence of the limit cycle. This is done by modelling and simulation.

2. In depth analysis of the nature of the limit cycle in order to be able to generate a solution to the problem is possible.

28

Chapter 4

System analysis

A limit cycle occurs when the RNL control concept is applied to an industrial application which is very close to a 1-DOF system. This is in contradiction with the simulation results described in chapter 2.

This leads to the conclusion that the cause of the limit cycling phenomenon must be found in a difference between the actual system and the simulation model. In this chapter the elements of the actual system are re-analysed and compared to the way they are modelled. The elements of concern are:

o Encoder (quantizer);

o Zero Order Hold (ZOH);

o Controller (RNL);

o Dynamics between driving fore and measured displacement of the slider mass;

o Friction.

4.1 Improvement of the dynamic model

It is assumed that the models used for the encoder and ZOH are correctly implemented in the simulation model used in chapter 2. Experiments and simulations showed a similar behaviour between the implemented controller in dSPACE and the controller used in the simulation model.

With respect to the system dynamics it is mentioned that the measured frequency response showed some high-frequent resonances which are not accounted for in the second order simulation model (figure 3.4). These resonances are not modelled because it is generally assumed that they have no significant influence on the system’s behaviour. Therefore, the LiMMS axis under consideration is assumed to be a second order (1-DOF) system. A closer look to the construction of the LiMMS axis is necessary to determine if these resonances are caused by the system dynamics.

The friction model used in the simulations is trivially known to differ from reality. Litera- ture survey [15], [13] and [i] showed a few alternatives but no general accepted simulation

29

Chapter 4 System analysis

model is found. Simulations with different numerical friction models have been carried out and showed that for the LiMMS model the friction model used is not critical. This research is concentrated on the improvement of the modelling of the dynamics. In the previous part of this research as well as in other investigations regarding the LiMMS axis is assumed that the slider of the LiMMS axis approximates a 1-DOF mass system fairly well. Based on the identification results, the LiMMS axis is expected to be a multiple degree of freedom system. The simplest modification of the originally as 1-DOF system modelled slider is an extension to a 2-DOF system. This 2-DOF system must correspond to the measured frequency response up to the first resonance peak (figure 3.4). To determine the two masses, the stiffness and damping of the connection, the slider of the LiMMS axis must be analysed.

The slider is globally constructed of a coil frame attached to a metal block which is connected to the guides by use of ball bearings, see figure 4.1.

direction of movement

A -4 slider I J

A - A’

load

Fig. 4.1: Slider construction

These two elements are also assumed to be the masses of the 2-DOF system mentioned above. For convenience the metal block is referred to as load and the coil frame is referred to as motor. The combination of the motor and the magnets is in essence the h e a r motor. The load experiences a friction force because of the contact between the load and the guides. The connection between thc load and the motor is realized by using four hexagon socket-head screws and four pin joints. The encoder is attached to the load, so the load’s position is measured.

The slider can now be modelled by a 2-DOF system as given in figure 4.2. The following symbols are used:

ml : motor mass F, : motor (driving) force (control force) rn2 : load mass Ff : friction force 21 : motor position 5 : stiffness of connection between load and motor x2 : load position d D damping of Connection between load and motor

30

4.1 Improvement of the dynamic model

//////////////// Fig. 4.2: 2-DOF model of the slider

The stiffness of the connection is estimated to lie within the limits [le7 - 5e7] [N/m] [23]. The mass of the load as well as the mass of the motor are weighted after disassembling the slider. The motor mass is determined to be 3.5 [kg] and the load mass is determined to be 6.8 [kg].

The matrices which define the state space model used to describe the system are given by (4.1) - (4.3).

A =

B = [ O 1 - ml O

O 1 -- ::I m 2

1 0 0 0 O 0 0 1 0 0

0 0 0 1

The states are chosen to be: g = [q 22 2, & I T and the input vector is chosen to be:

F,, is the non-linear control force and Fj is the friction force.

Substitution of the determined mass values in the state space model yield a parametric state space model where the unknown parameters are k and d. These parameters have been estimated by fitting the frequency response of the model's input to the position output 22, upon the measured frequency response given in section 3.2 (figure 3.4). The parameters k and d are determined to be:

5 = 1.2e7 [N/m] d = 3e2 [Ns/m]

The bode plot of the measured frequency response and the fitted model are given in figure 4.3.

31


. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .s - I . . . . . . . . . . . .. . . . : : I , . . . . . . . . . . . . . . . . . : . . . . . . .:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . A . . . . . . . . . . . . .

- I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-200 1 O‘ 1 o2 I o3

frequency [Hz]

UI CCI c a-300

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . , . . . . . . . . . . , .. . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . .

: ; ; I :

I _ : .

.; . . . . . ;.\.i . . . . \ . . . . . . . . . . . . . . . .

-400 ’ 1 O’ 1 o2 I o3

frequency [Hz]

Fig. 4.3: Frequency response extended LiMMS axis model (dashed line) and measured frequency response (solid line)

The parameter value concerning b is within the expected band as mentioned before. It is concluded that the first resonance peak (335 [Hz]) in the measured frequency response, represents the connection between the load and the motor.

The extended model is applied in the previous described simulation environment.

4.2 Simulation of the improved system

Simulations are executed within the same environment as mentioned in chapter 2. The state space model (4.1) - (4.3) is implemented in the simulation model described in section 2.3.3. Appendix E describes the simulation model used as well as the simulation parameters of concern.

All system parameters used in this simulation are given in table E.3 (appendix E section E.G. These parameters are based on identification results described in chapter 3 as well as the fitting results described in this chapter. The control parameters and the overcompensation regarding the BNL controller are the same as described in section 2.3.

32

4.2 Simulation of the improved system

L -

Frequency 135 133 [Hz] Amplitude 7-9 5 [inc]

During the experiments described in chapter 3 several setpoints are used t o excite the system and so cause limit cycling. In order to minimize the simulation time the simulation model is excited by giving the masses certain initial positions. As initial conditions the position $1 = 22 = 2e-5 [m] and zero velocity (51 = x2=0 are chosen. This results in a response as given in figure 4.4. This figure shows the position response of the load (solid line) and the motor (dashed line) position to the initial positions. The simulation showed the origination of a limit cycle after 0.03 [s] with constant amplitude and constant frequency.

Fig. 4.4: Simulation results regarding the load (solid line) and motor position (dashed line)

In order to exclude the possibility that the limit cycle exists due to numerical problems as rounding errors, several simulations with different initial conditions (position and velocity) are perÏormed. AU simulations eventually showed the same limit cycle behaviour, so it is assumed that the limit cycle behaviour is inherent to the controlled system. Table 4.2 shows the characteristics of the limit cycle with regard to the simulations as well as the characteristics of the limit cycle concerning the experiments as described in chapter 3.

I parameter experiment simulation unit I

Table 4.1: Limit cycle characteristics

The results given In table 4.2 confirm the statement that the occurrence of the limit cycle phenomenon by the application of the RNL controller to the LiMMS axis is explained by the presence of higher order dynamics. A qualitative explanation of the existence of limit

33


cycling in this system is as follows:

If the load mass and motor mass are given the same initial position error, a compensation force generated by the non-linear controller will be applied to the motor which starts to move in the direction towards the reference position. The flexibility between the motor and load delays the transfer of the compensation force to the load. When the distance between the motor and load is enough to create a spring force which is able to overcome the static friction force, the load also begins to move towards the reference position. The motor and load are now moving in the same direction. Only when the load passes/reaches the reference position, the RNL force affecting the motor mass will reverse direction. The control force will decelerate the motor and eventually reverse the motor’s direction of movement. The load is however still moving in the opposite direction. Eventually the load’s direction of movement is also changed and the load moves towards the reference position. This explains a time delay of the load’s movement in comparison with the motor movement visible in figure 4.4. The reversed version of the event as described before will occur. Under the right circumstances this could lead to a limit cycling system. It is mentioned that this is not meant to be a proof but this qualitative analysis is only described to make limit cycling in the described system plausible. This analysis is visualized in figure 4.4 by plotting the motor mass (dashed line).

Increasing the stiffness between the two masses will yield a decrease in the limit cycle’s amplitude according to the previous description. This motivates the search for a required stiffness to stabilize the system by simulation. The simulation model used for determining this stabilizing stiffness is the same as described earlier in this section. Several simulations are performed which resulted in the observation that a stiffness of approximately 5e8 [N/m] stabilizes the controlled system and no limit cycling occurred anymore. The response from systems with a stiffness bigger in size then the required value corresponds to the response of the earlier described 1-DOF mass systems.

However, in a practical realization it is very hard if not impossible to increase the stiffness of the connection between motor and load with a factor 25. Moreover, suppose the increase of the stiffness of the connection is a real possibility, than the occurrence of other resonances (for example due to the connection between the encoder and the load) in the system would probably give rise to a limit cycle.

Furthermore, it is mentioned that improving the dynamic behaviour of the LiMMS axis is not part of the goal of this project. Since the LiMMS axis is a good approximation of a 1-DOF system, since the resonance frequency (335 [Hz]) is relative high with respect to the bandwidth (30 [Hz]), it has to be concluded that the RNL control concept is not applicable t o the class of systems represented by the LiMMS axis. This means in practise that the RNL control concept in its present form is not suitable for the existing class of positioning devices since they all contain certain resonances.

The next chapter describes a method to analyse the RNL controlled non-linear system to get more insight in the limit cycling problem. This understanding might lead to an adaptation of the RNL control concept which prevents limit cycling in the discussed type of dynamical systems, but preserves the positive characteristics of the RNL control concept.

34

Chapter 5

Non-linear system analysis

In chapter 3 experiments showed the occurrence of a limit cycle by applying the RNL control concept to an industrial application, the LiMMS axis. This limit cycling behaviour is also detected during simulation as described in chapter 4. Limit cycling wastes control power and the vibration causes wear to the mechanical system. Furthermore, limit cycling leads to poor control accuracy due to the periodic tracking error in the system. So, limit cycling limits the performance of controlled servo systems. From these results the conclusion may be drawn that the proposed RNL concept [26] is not suitable for these kind of industrial applications.

In order to perform a founded investigation to possible solutions of this limit cycle phenomenon, analysis of the non-linear system’s behaviour is a necessity. For linear time- invariant systems several techniques to analyse the stability are developed in the coarse of time. Most of these techniques however are not applicable to non-linear systems. The problem of analysing the non linear system is often avoided by simply replacing each non- linear operation by an approximately linear operation (linearization) and studying the resulting linearized system using linear techniques. It is obvious that this method fails to predict a limit cycle since the non-linear character of the system is neglected.

In literature several methods are proposed to analyse non-linear system stability in relation with limit cycling [8],[25] and [13]. The three most important are mentioned:

The first method is the describing function technique which is a quasi linearization’ technique. The general usage of the describing function is prediction of the limit cycle phenomenon and its stability in non-linear systems.

The second analysis method found in literature is the phase plane method which is a graphical method for finding the transient response of first or second order systems. This method can seldom be applied to non-linear systems greater than second order [8].

The third method is Lyapunov’s direct method (or Lyapunov’s second method). This is a time domain method based on the non-linear state space description of the system. This method is applicable to any non linear system configuration. However, for most systems

‘Quasi linearization is the approximation of a non linear operation by a linear one, which depends on some properties of the input

35

Chapter 5 Non-linear system analpis

it is extremely laborious to find a Lyapunov function which is necessary to determine the stability or instability of the process.

Since the phase plane analysis not suited for third or higher order systems and the Lya- punov method is extremely difficult to apply, the describing function is chosen as analysis method for the non-linear control system described earlier in this report. To be more specific, the sinusoidal-describing function theory is used. For a detailed description is referred to appendix E(. This appendix also describes how the DF-theory is applied to determine limit cycling. Section 5.1 describes a numerical method which is used t o calculate the describing function of the RNL controlled system. At last it is tried to find a solution of the limit cycle problem based on the describing function of the RNL controlled system.

5.1 Limit cycle analysis of RNL controlled system

Because the RNL controlled system exists of various non-linear parts, the (standard) sep- aration between one non-linear and one linear dynamic subsystem is not feasible. Hence, the complete open loop system (figure 5.1) is considered to be one non-linearity. The DF of this non-linearity is calculated numerically to analyse the limit cycle behaviour. Note that the closed loop system is realized by applying a negative unity feedback loop from YNL to u in figure 5.1.

Fig. 5.1: RNL controlled system as non-linear element

The principle of determining the DF by simulation is based on the fact that the fundamental component of the Fourier expansion with respect to a periodical signal2 equals the least-mean-square-error approximation of this periodical signal with a sinusoidal component with the fundamental frequency [3]. By using this method it is also assumed that in case a sinusoidal input is applied to the non -linear element, the output signal is periodic.

The characteristic equation is given by N ( A , w ) = -1. The intersection point (-1 ,O) of the N ( A , w ) curves in the Nyquist plot with the critical point determine the characteristics of the existing limit cycle.

The following steps are executed to derive the DF of the RNL controlled system:

21t is assumed that the open loop response of the non-linear system to a sinusoid input is also periodic.

36

5.1 Limit egde analysis of RNL controlled system

0 1 -

0-

-0 1

E - - 0 2 -

-03-

-04 -

- 0 5 -2

First the response YN,(t) of the open loop RNL controlled system to the input u(t) = Asin(&) is determined by simulation. Appendix L shows the simulation model and all simulation and system parameters used. Since the non-linearity is assumed to be both amplitude and frequency dependent, the quantities A and w are varied. A is varied in a linear way from i [inc] to 10 [inc] (i [inc] = i pum) with an interval of i [incl. w is logarithmic varied from 27r-120 [rad/s] to 2n.142 [rad/s] in 8 steps. The above implies that 64 simulations have been carried out.

005- 142[Hzj

0-

-0 o5 - -amplitude i 0 [incl

- E -01-

-015-

-02- amplriude 3 [inc]

1 -0 25 -

-2 -1 5 -1 - 0 5 O O 5 -16 -15 -14 -13 -12 -11 -1 -09 -08 - 5

Secondly, a sinusoidal approximation of the form p n ( t ) = Msin(wt + p) is determined for each response by using the least-mean-square method. This means that the criterion

is minimized. In (5.1) tl refers to the moment in time at which the response Y,,(t) be- came stationary and T is the reciprocal of the frequency regarding the input signal.

Subsequently, the combination of an approximated response y,(t) and the corresponding input signal u(t) is used to determine one point of the DF curves. This point is characterized by the variables A and w of the input signal and the variables M and cp of the approximated output signal. The point of the DF curve is described by:

( 5 4 M A

N ( A , w ) = -eJq

In this case 64 points are calculated.

At last all calculated points are plotted in a Nyquist plot (figure 5.2.a).

i

7

The lines are curves where the amplitude (A) of the input signal was constant. The frequency increases In the direction indicated by the numbers i to 8. This means that point

37

Chapter 5 Non-linear system analysis

1 corresponds with frequency 120 [Hz] of the input signal u(t) . According to the Nyquist criterion, the system is unstable for input amplitude < 5 [incl. This means that vibrations in the system grow in amplitude to approximately 5 [incl. Oscillations with an amplitude above 5 [inc] will show a decrease in amplitude. This implies the occurrence of a limit cycle with amplitude 5 [incl. Figure 5.2.b shows the frequency of the limit cycle to be approximately 138 [Hz]. It can be concluded that the limit cycle observed in experiments and simulation is predicted fairly well by the describing function analysis.

Before further analysis is described, some remarks about the numerical method used to determine the DF are made.

At first it is mentioned that calculating the response of the open loop RNL controlled system is extremely time consuming. This is caused by the fact that very small integration steps are used in the Runge Kutta routine in order to perform an acceptable friction phenomenon simulation. Furthermore, it is mentioned that tuning the simulation parameters in relation with the system parameters is done in previous closed loop simulations.

Subsequently, it is mentioned that simulations had to be continued until a stationary response for a t least one period existed. This is due to the fact that at least one period of the periodic non-linear output signal is necessary to make a meaningful sinusoidal approximation.

The previous two remarks indicated that much data is generated during simulation. In order to reduce this data to minimize the time needed by the fitting routine, interpolation is applied. The interpolation method used is the interp1.m routine defined in MATLAB [22]. In order to be able to use a digital filter the time series are interpolated in a way so that the individual points are equidistant in time.

At last it is mentioned that the open loop response of the RNL controlled system affected by friction is periodic with subharmonics regarding to the frequency used in the input signal. The response also showed an offset. In order to be able to determine the DF, the response is filtered by using a fourth order digital Butterworth bandpass filter. The frequency band which is passed is chosen to be the frequency of the input signal f 20 [Hz]. It is recalled that the calculated time response was interpolated in order to create a time series with equidistant points in time. The time interval used in the interpolation method is chosen to be 5e-4 [SI. The m-file used in matlab to interpolate and filter the data is given in appendix M.

It is concluded that the numerical determination of the DF requires much effort. The results however predict the existing limit cycle in the non-linear system fairly well. The next section describes if it is possible to generate solutions to the limit cycle problem based on information obtained by evaluating the numerically determined DF.

38

5.2 Generating solutions to the limit cycle problem using the DF.

The previous section showed the prediction of the limit cycle by calculating the describing function of the RNL controlled system affected by friction. In order to generate possible solutions the produced Nyquist plots are evaluated.

The evaluation of the DF deduced the following proposition : If the curves in the Nyquist plot which pass at the left of point ( - l , O ) , could be changed in a way so they would pass at the right side of the critical point, no limit cycle would occur.

In other words, a solution to the limit cycling phenomenon is expected according to the calculated DF, to be found when an extra phase lead could be created for small signal levels (< 5 [inc]).

The non-linear network shown in figure 5.3 realizes this phase lead for small signals.

Saturation function

Fig. 5.3: Non-linear network providing phase lead at small signal levels

In literature [8] is proved that:

For convenience the non-linear network is referred to as NLF. Several configurations concerning the attachment of the NLF in the RNL controller are possible.

39


Figure 5.4 shows two possible configurations.

c _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ RNL controller I I I I

Concept 1

c _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ RNL controller I I I I

Fig. 5.4: Configurations for attaching the NLF to the RNL controller

Concept 2 is preferable above concept 1 because of the fact that the non-linear compensation function F,(x) is designed to react directly to the measured position. The parameters 6, í- and K determine the behaviour of the NLF. The specifications regarding these parameters are found by evaluating the Nyquist plot and determining:

1. which type of input signals to the RNL controller need a phase shift. In other words, for which amplitudes of the input signal is phase lead necessary. This determines the parameter S.

2 . which frequency needs the most phase lead. This frequency is referred t o as f2.

3 . the amount of phase lead necessary at frequency f2. This determines the value for the lead ratio X [7] , see also section 2.3.2;

According to the above mentioned system specification the following tuning rules regarding the parariieters 6, í- and K have been derived:

1. K = X - 1

2 . T =

3 . S is chosen in excess of the amplitudes limit cycling.

where f~ = (Z/x)-' - f 2 and where f1 is the starting frequency of the lead. 2nf1

The tuning rules are more detailed discussed in appendix N.

In case of the RNL controlled system the limit cycle characteristics are determined to be:

frequency = 133 [Hz] amplitude = 5e-6 [m] = 5 [inc]

The combination of this characteristics and the results of the evaluation of the DF yield the following choice for the NLF parameters:

5.2 Generating solutions to the limit cycle problem using the DF.

f2 = 133 [Hz] X = 8 S = 10e-6 [m] = 10 [inc]

which corresponds to a phase lead of approximately 48"

The behaviour of the resulting NLF in the frequency domain is visualised by giving the numerical determined DF. It is mentioned that the exact DF of these class of non-linear networks cannot be found in any analytical way. This is confirmed in literature [8] , page 153. The simulink file and simulink programm, used to determine the DF are given in appendix O. See figure 5.5 for the calculated DF.

o 45

0 4

o 35

03

O 25 o>

- E o 2

O 15

o 1

amplitude input signal = 20 O 05

>

O O 1 O2 0 3 0 4 O5 0 6 0 7 0 8 O9 1

Re

Fig. 5.5: DF of NLF visualized in Nyquist plot

The solid lines are the quasi linearized frequency responses with respect to a specific amplitude =€ the sinusoid amplitude. The frequeccy of the inpiit signd is logarithmically varied from 10 [Hz] to 1000 [Hz]. The frequency is increasing clockwise in the plot. The amplitude of the input signal is varied from le-6 [m] to 20e-6 [m]. The points with the same frequency are connected with dashed lines.

It is conchidecl that the Nyquist plot of the D F shows the expected behaviour of the non- linear network NLF. When the input signal is small in amplitude, the NLF applies phase lead to the signal. The amount of phase lead depends on the frequency of the input signal. When the amplitude becomes larger, the plot shows that the applied phase lead decreases as expected. When the amplitude becomes very large x >> 6, the signal is transferred by the NLF without adaptation.

To evaluate the influence of the application of NLF to the RNL controlled system, simulations are performed. A wide range of parameter settings as well as several implementation configurations are applied. The results still showed limit cycling for all applied settings, although influence on the size of the limit cycle is detected. The application of NLF to the

41


RNL controlled system therefore seems not to be a solution to the limit cycle problem.

This might be explained by the following consideration: the choice of the correction method (NLF) is based on the necessary phase shift to prevent limit cycling according to the derived DF of the RNL controlled system. In this linear approach it is assumed that the phase shifts according to the DF of both the NLF network and the RNL controlled system can be added according to the superposition principle. However, the superposition principle is not valid with respect to non-linear systems. Thus, to prevent limit cycling, choosing parameter settings based on the superposition principle seems not to be correct in this case.

It can be concluded that the most obvious way to get insight in the influence of NLF on the RNL controlled system behaviour in the frequency domain is deriving the DF of the total non-linear system (RNL controlled system including NLF adaption).

This suggests that if stabilizing parameter settings exists they can be found by an iteration method in which each setting is evaluated by deriving the corresponding DF of the NLF adapted RNL controlled system. However, due to laborious simulations, necessary to compute the DF, this method of finding the parameters can not expected to be feasible.

This research showed that the system information derived by the sinusoidal DF cannot be used to generate a solution to the limit cycle problem described in this research.

It can be generally stated that due to the approximating character of the sinusoidal DF method, the information obtained by this method is not suitable for non-linear control design.

42

Chapter 6

Conclusions and recommendations

The objective of this research is defined as the investigation of the applicability of a non-linear control concept with respect to high performance mechanical servo systems influenced by friction [26]. This non-linear control concept is referred to as the RNL control concept.

Theoretical analysis and simulation performed during this research of an RNL controlled one degree of freedom mass influenced by friction, showed a very good positioning behaviour. This confirmed the claims made by the designers of the RNL control concept in literature [26].

Experiments considering an industrial application, the LiMMS axis, which is generally assumed to approximate a one degree of freedom system fairly well, showed the occurrence of a limit cycle. Experiments in earlier research showed also the occurrence of limit cycle phenomena when applying the RNL control concept to industrial applications but no explanation was found.

This research proved that the limit cycle phenomenon is caused by the fact that the experimental application (LiMMS axis), essentially behaves as a two degrees of freedom system when RNL control is applied. This is confirmed by simulation results, which described the limit cycle behaviour of the RNL controlled LiMMS a g s accurately.

The main conclusion regarding the investigation of the applicability of the original RNE control concept to mechanical servo systems is given by the following statement:

o Since the LiMMS axis is considered to be one of the best available approximations of a one degree of freedom system concerning industrial applications, it is concluded that the proposed non-linear control concept is not applicable to this class of positioning devices due to the origination of the limit cycle phenomenon.

Since the RNE control concept shows an excellent settling behaviour in simulations, the limit cycle phenomenon was analysed in order to investigate the possibility of eliminating

43

Chapter 6 Conclusions and recommendations

the unstable behaviour of the RNL controlled industrial system. The method used to analyse the limit cycle behaviour is the sinusoidal input describing function (DF). The conclusions regarding the DF analysis of the RNL controlled system are:

a The DF method applied in this research to analyse and predict limit cycle behaviour, produced an accurate approxima,tion of the limit cycle Characteristic which occurred in simulation as well as in experiments.

a The method used is based on the sinusoidal describing function theory but in contrast to the analytical describing function, the method applied here is based on a numerical calculation of the DF of the total controlled system (open loop). The results obtained by applying this method proved the correctness of the numerical method. This means that the limit cycle behaviour of complete non-linear systems can be analysed by determining the DF of the complete non-linear system in a numerical way. This is in contrast with the conventional analytical method where splitting the system in a linear and a non-linear part is necessary to analyse the limit cycle behaviour. Furthermore, it is mentioned that by using the numerical method far more complex systems can be analysed compared to the complexity of the systems which can be analysed using the analytical method.

o A negative conclusion is that numerical determination of the describing function is very laborious due to the next two aspects. Firstly, it takes a lot of time to harmonize the simulation parameters with the system parameters in case of non-linear systems. Secondly, performing the actual simulations necessary to compute the DF of the complete controlled non linear system is very time consuming.

An attempt to find a solution to prevent the limit cycle behaviour is based on the frequency domain behaviour according to the sinusoid input describing function of the RNL controlled system. A non-linear phase shifting network (NLF) is applied in order to try to stabilize the system. The NLF adapted RNL controlled system is referred to as the extended RNL controlled system. The conclusions with respect to this attempt to prevent limit cycling are:

o No parameters settings are found to stabilize the extended RNL controlled system. The appearance of the limit cycle behaviour could not be suppressed.

o Furthermore, it is concluded that it is incorrect to generate a solution to the limit cycle problem which is only based on adapting the DF of the non linear system behaviour using the superposition principle.

6.2 Recommendations

This research showed that the result of non-linear system analysis by means of the sinusoidal input describing function is not enough for designing a stabilizing compensation method to prevent limit cycling behaviour. During this research it is mentioned that stabilizing parameters for the extended RNL controlled system may be found using an iterative process which includes the calculation of the DF with respect to the extended RNL controlled system (non linear system with the non linear adaptation in the controller). The

44

6.2 Recommendations

bottleneck in applying this method is the necessity of time consuming simulations. There- fore an investigation to the improvement of the numerical method to determine the DF is recommended.

If the numerical method is optimized it is recommended that the limit cycle problem concerning the extended RNL method is reviewed in order to find stabilizing parameters.

At last it is mentioned that the numerical friction model used during simulations is very time consuming. Therefore, it is recommended to perform an investigation to friction models which are more suited for simulation purposes than the friction model used during this simulation.

45

[i] Armstrong-Hlouvry, B., Friction modeling for control, Proc. of the American Control Conf. 1993, San Francisco, California, 1993, pp 1905-1907.

[2] Bosch, P.P.J., van de, Klauw, A.C., van der, Stochastische systeem theorie, dictaat nr: 5N060, Technische Universiteit Eindhoven, Februari 1994. Blz. 47

[3] Cool, J.C., Schijff, F.J., Viersma, T.J., Regeltechniek, Delft, Nederland, 1985.

[4] Cox, H., LIMMS Dynamic Analysis and Servo Performance: Modelling 63 Measure- ments, Philips CFT Internal Report CTB 88545-93-3013, March 1993.

[5] Darai, M., An alternative method for measuring frequency response function in closed loop, Philips CFT Internal Report CTB 595-95-3066, August 1995.

[6] Friedland, B., Park, Y.-J., On the Modeling and Simulation of friction, Journal of Dynamic Systems, Measurement and Control, Vol. 113, September 1991,pp. 354-362.

[7] Franklin, G.F., Powell, J.D., Emami-Naeini, A., Feedback Control of Dynamic Sys- tems, Addison-Wesley Publishing Company, 1994.

[8] Gelb, A. , Vander Velde, W.E., Multiple-Input Describing Functions and Nonlinear System Design, McGraw-Hill Electronic Sciences Series, 1968.

[9] Neidenhain, precision measurement systems catalog , March, 1990.

[lo] Hewlett and Packard, type 3567UA, Operating manual, 1995

[li] Hultermans, S.C., Toepassing wan de Clegg-integrator in een PID-regelaar, Philips CFT Internal Report CTR 595-95-0038, April, 1995.

[12] Hunink, H.J., Contrcl of a Mechanical Servo System with Friction, Philips CFT Internal Report CTR 595-94-0103, December, 1994.

[13] Hunink, H.J., Modelling, Analysis and the Control of Mechanical Servo-Systems With Friction, Philips Research Internal Report UR 010/94, 1994.

1141 Impex version 3.02., Impez user’s guide, Digital signal processing and control engineering GmbH, 1992.

[15] Karnopp, D,, Computer Simulation of Stick-Slip Friction in Mechanical Dynamic Systems, Journal of Dynamic Systems, Measurement and Control, Vol. 107, March 1985, pp.100-103.

47

[16] Konings, E.J.A., Modelling of Three Phase Brushless DC Motors, Philips Research Internal Report UR 013/92, 1992.

[17] Koster, M.P., Vibration of Cam Mechanisms, The Macmillan Press ltd, 1974.

[18] Kuo, B.C., Automatic Control Systems, Prentice-Hall International Editions, 1987.

[191 - - Laan, M. van de, Limit Cycles in Mechanical Servo-Systems, Philips Research Internal Report UR 015/95, 1995.

[20] O’Leary, T.M., Evaluation of the ”RNL” Friction Compensation Technique, report CTR 545-92-0070, December 1992.

[21] Ljung, L., System Identification: Theory For The User, 1987.

[22] Mathworks, The, MATLAB user’s guise,, March 1992.

[23] Rens, P.J., personal communication, 1995.

[24] Shaw, S., On the Dynamic Response of a System with Dry Friction, Journal of Sound and Vibration, Vol. 108(2), December 1986,pp. 305-325.

[25] Slotine, J.-J. E., Li, W., Applied non linear control, Englewood Cliffs, New Jersey, 1991.

[26] Southward, S.C.,Radcliffe, C.J., MacCluer, C.R., Robust Nonlinear Stick-Slip Friction Compensation, Journal of Dynamic Systems, Measurement and Control, Vol. 113, December 1991,pp. 639-645.

[27] Vervoordeldonk, M.J., personal communication, 1995.

48

Appendix A.

Bode plot of the PD controller

This appendix shows the PD controller with lowpass filter according to (A.1).

The parameters used in (A.l) are: The resulting Bode plot is given in figure A.l

I parameter value units hTP 2100 ["I

v s K d 28 k l Y 100 [-] B 0.7 [-I W n 754 pq

Table A.l: parameters 1-DOF system

frequency [Hz]

frequency [Hz]

Fig. A.1: Bode plot of the PD Lowpass controller

49

Appendix B

Application of the RNL compensation in a discrete environment

Application of the RNL compensation in a realistic environment includes a change in the characteristic of the non-linear friction compensation due to the quantized position feedback of the mass. For ease of notation, RNL control applied in a digital environment is referred to as discrete RNL.

B.l Influence of discrete RNL on system behaviour

Using a quantized position error as input to the controller introduces a zone around the desired position (error zero) where the quantized position error is zero. In this zone the RNL control force equals zero. This zone is referred to as deadzone. This is in contradiction with the original proposed non-linear control force where there is only one unique point where the position error is zero. The consequences of this deviation are investigated.

The deadzone Xdead is symmetric around error zero if an uniform quantizer is used (appendix E). If the monitored position of the mass is within a certain bound h around the desired position, the quantized error is zero. The interval h is defined as:

This implies a different behaviour of discrete RNL compared io the by Cotithward proposed method. Figure B.l illustrates the non-linear control force as function of the position (error). The deadzone is defined by:

Assuming the discrete RNL control is applied to a 1-DOF mass with friction as described in section 2.2.1, the following aspects regarding system behaviour can be mentioned:

%dead [zql,zqh] P . 2 )

1. If the RNL compensation is implemented correctly, the settled position error of the 1-DOF mass is within X d e a d . The quantization level determines the width of X d e a d

and therefore the settling accuracy of the system.

Appendix B Application of the RNL compensation in a discrete environment

t discrete RNL force ( + linear proprtional action)

position error

Fig. B.l: Discrete RNL with deadzone

2. The compensation force has to be chosen in excess of the maximum friction force level, so that the mass can be moved from standstill towards the desired position (error zero). The compensation force will switch off when the position error is between the limits Oql and Zqh, see also figure B.1. In the deadzone the mass has to come to standstill by the work of the friction force (the only energy dissipating action left in the system in the deadzone). If the friction force not succeeds in stopping the mass before the opposite limit ( Z q l or Oqh) of the deadzone is reached, overshoot occurs and the compensation force will reverse. The system now makes a new attempt to settle the mass in the opposite direction.

3. Due to the discrete nature of the controller a limit cycle will occur when the compensation force (duration and magnitude) is always able to let the mass cross the deadzone. In the next section the origination of limit cycling is described.

B.2 Limit cycling in discrete RNL

The cornpensation level in the original proposed RNL method [26] has to be chosen in excess of the maximum friction level. Southward et aZ. proved this method t o be Lyapunov stable regardless the size of the overcompensation.

In case of the discrete RNL method there is a definite limit to the amount of overcompensation. If the applied compensation force exceeds this limit, a limit cycle will originate as described in section B.1 item 3. In this section an analytical description regarding this limit cycle phenomenon is given. In the last section the derivation of the maximum permissible level of overcompensation will be described.

Suppose the one degree of freedom mass is at standstill just outside the deadzone. This

52

B.2 Limit cycling in discrete RNL

implies a quantisized position error of 1 [inc] (see encoder description in appendix E), see also figure B .2.

friction force: Fj

control force: F, Mass (ml ~l

Fig. B.2: Mass just outside RNL dead region

Furthermore, the friction force Fj is assumed to affect the mass. The position -0.5 [inc] is redefined as 2, = O. In this analytical description, the maximum static friction force level is assumed to equal the Coulomb friction force level and the friction phenomenon is assumed to be symmetric.

In the above described situation the discrete controller receives an error of 1 increment and generates a control force with the value of F$ in order to reduce the error. As of now this control force is referred to as p,. When the mass starts moving two forces influence the mass, the control force F, and an opposite directed friction force Fj. In the continuous case the controller force becomes zero the moment the measured error changes to zero increments. Due to the Zero Order Hold of the discrete controller the control force is kept constant during the sample time T,. The minimum duration of the control force is therefore the sample time of the discrete system (T,).

The movement of the mass can now be divided in two phases:

1. The first phase lasts from t = t o till t = t o + T,. During this phase the mass is influenced by two forces: the non-linear discrete control force F, and the friction force Fj. Due to these forces the mass moves a certain distance zn(T,) in the direction of Z q ~ .

2. During phase two the mass is only affected by the friction force Ff and the mass is therefore decelerated (when xn(T,) is within the deadzone. The distance the mass travels in this phase is added to the previous distance zn(T,). The total displacement is denoted as x,(t). Phase two begins at t > T, and ends when the mass is stopped within the deadzone or when the mass has reached the other boundary of the deadzone Z Y ~ .

53


A limit cycle arises if the total displacement x n ( t ) equals the length of the deadzone (quantization interval) h. From the description above becomes clear that a minimum magnitude of the control force F, is required to satisfy this requirement. In the next section this limit value of p, is derived to prevent limit cycling in discrete RNL.

-n n-r-r B.3 Maximum compensation force iri discrete JUY L

In this section the limit value of @, is derived based on the description in section B.2.

The movement x n ( t ) of the mass (m) during the period: t o 5 t 5 T,, representing the minimum duration of the control force, is described by:

@, + Ff m

Xn(t) =

A F m

&(t) = -t + .(to> A F 2m

xn ( t ) = -t2 + 2(to)t + xn(t0)

where t o = O. For ease of notation is defined:

Assuming & ( t o ) = xn(to) = O yields the distance the the mass travelled during phase one ( t = T,):

At t = T, the control force F, no longer affects the mass. The state of the mass at this point is described by:

A F 2m A F --TS m

-T,2

The equation for motion for t > T, is:

Ff &(t) = - m

which yields:

xn(tj = -(t Ff - t kn(t - Z) + ~ n ( T s ) 2m

(B.10)

(B.11)

The phenomenon of limit cycling occurs if the mass travels the complete distance h because in this case the process is repeated vice versa. So if (B.ll) equals h at a certain point in time the system is unstable. Substitution of (B.8) and (B.9) in (B. l l ) and simplifying yields:

(B.12)

54

B.4 Verification by simulations

Equate (B.12) to h and solving the new equation yields the value o f t at which the mass reaches the other boundary. If the equation is unsolvable the system is stable because the mass doesn't reach the other boundary of the deadzone. The equation to solve to determine if the system is stable is:

Ff F s E 2 -t2 +- -T,t - -Ts - h = O 2m m 2m

(B.13)

This equation is only solvable if the discriminant of this function D1 2 0. If Dl = O the system is unstable. So, the next equation has to be solved:

4F:T: t 4FjFST: t 8mhFf = O (B.14)

The solution to (B.14) is:

I -4FfT; f ,/l6F;.T,"- 128mhFfT;

8T: Fs(1,2) = (B.15)

If a negative friction force is applied the positive value of the set solutions Fs(l,21 determines the maximum allowable compensation to prevent limit cycling, and vice versa. As mentioned before the friction is assumed to be symmetric, so: FZ = -F;. In this case the value of the maximum allowable compensation force is defined Fs,maz. The direction of the control force depends on the sign of the error.

B .4 Verification by simulat ions

In order to verify the stability condition simulations are carried out. The simulations are executed using SIMULINK (see also section 2.3.3). For information about the numerical integration method and the accompanying parameters is referred to appendix E.

The simulations are performed using the 1-DOF system with friction as described in section 2.3.1 with a non-linear compensation force below the limit: @s,maz. The friction model used in this simulation is described in appendix C.1.4. The sample time T, is chosen to be 0.25e-3 [SI as described in section (2.2. The theoretical limit ~ s , s , m a z is determined using (B.15) to be 97 [NI. The relevant parameters of the system used in the simulations are listed in table B.1.

The first simulation is carried out with a compensation level lower than Fs,maz The initial position of the mass is set to increment (= 0.5e-6 [m]) in order to place the mass at the boundary of the dead zone X d e a d . The compensation level is set to 96 [NI which implies F,f to be 96 [NI and p; to be -96 [NI. The response of the controlled system is given in figure B.3.a and B.3.b. Figure B.3.a shows the quantized position (error) (= solid line) and the real position (=dash-dotted line). Figure B.3.b shows the control force and the friction force (=solid line). The following remarks can be made:

e The duration of the control force 9s T,.

e The real position does not reach the opposite boundary which is at - 0.5 [incl.

55


-0.81 , , , , , , 1 -1

indication quantity value units m Mass 10 [kl FS Static friction level 23 [NI Fc Coulomb friction level 23 [NI

A Amplification 66 1-1 N s D Viscous damping coefficient 20 [,i

- . _ I

Table B.l: parameters 1-DOF system

0.8

Fig. B.3: 1-DOF system controlled by discrete RNL with p$ = -p; = 96[N]. a: quantized position and real position b: control and friction force

o There is no origination of a limit cycle.

The second simulation is executed with equal parameter settings as used during the first simdation except fcr the compensatim leve! which is chosen to be iû5[N]. This ievel is above the limit level F,,,,, thus a limit cycle is expected. The response of the controlled system is given in figure B.4.

The response shows the expected limit cycle. The amplitude of the limit cycle varies be- tweer; 11-13 imrements and its frequency is i77 [Hz].

The final conclusion regarding discrete RNL and limit cycling due to overcompensation can be stated as:

The level of ps has to be chosen lower than the determined level of Fs,maz (B.15). In a earlier stage of research is mentioned that ps has to overcompensate the maximum friction level by io%. The recommendated value regarding the non-linear friction compensation p' has to be chosen as follows:

56

(B.16)

B.4 Verification by simulations

I O 0.01 0.02 0.03 0.04 0.05 0.06 0.07

- 1 S

time [SI

Fig. B.4: Discrete RNL controlled system response with with fl$ = --FF = 105[N]

It is mentioned that during the course of this research project the compensation level is determined at 110 % of the maximum static friction force.

57

Appendix C

x' = Ax+Bu +~emu, Reference Sum y = Cx+Dü -

Simulation model of 1-DOF system with friction

- -

This appendix describes the simulation setup used to simulate the behaviour of a 1-DOF system with friction controlled by a RNL adjusted PD controller. Figure (2.1 shows the simulink model used for simulation purposes. The setpoint used in the simulation equals

r*I mass oosition i X I I

I auantizer

Non linear friction compensation

I Friction model

Friction

Fig. C.í: Simuiink simulation model of i-DOF rnl controiieci system with friction

zero. The intension of the simulation is to show the nature of the response of the system to an initial position error. The next section describes each block in the simulation model with the accompanying s-function [22] if necessary. The last section in this appendix lists the parameters concerning the numerical integration routine.

'The s-function behaves like any other matlab function. S-functions are at the heart of how sirnulink works.

59

Appendix C Simulation model of 1-DOF system with friction

@.I simulation model in blocks

This section describes the blocks used in the simulation model. The discussed blocks are:

1. Lead lag;

2. Non-linear friction compensation;

3. Mechanics;

4. Friction;

5. Quantizer;

6. Zero Order Hold (ZOH). Each block is treated in one of the following subsections.

C.1.1 Lead lag

The block named lead lag comprehends the lead lag filter in combination with a lowpass filter described by:

The parameters used in simulation are given in section 2.3.2. The Transfer Fcn block in simulink [22] is used to represent the lead lag with low pass filter.

C.1.2 Non-linear friction compensation

This block contains the by Southward et al. proposed non-linear friction compensation (RNL compensation). For a description is referred to section 2.2.3. The s-function used to represent this Compensation is called frn1.m. the contents of this file is listed below.

function [sys, xo] = f r n l ( t , x , u , f l a % , F ~ , F - ~ , K ~ , F ~ )

X H = - ( F - / I C p ) ) ; X L = - ( F , /(%I);

if abs(flag) == 3

E = F e l K p ; “ H = X H + e; “ L = X L - E ;

if (u > 5~ 11 u < E L )

elseif (u > O & u <= 2~ elseif (u > E L & u < O)

else

end

sys(1) = o;

sys(1) = KP*(”H - u ) ;

sys(1) = KP*(j.L - u);

sys(1) = o;

elseif flag == O

sys = [0,0,1,1,0,0]; xo = [ I ;

s y s = [ 1; else

end

Input u stands for the error signal and output sys(í) for the RNL-force. The overcompensation force is defined as Fe. The rest of the variables are defined in chapter 2.

(7.1 simulation model in blocks

C.1.3 Mechanics

The state space model used to represent the 2-DOF system is given by:

O 0 O 0 A = [ o 1 1 .=[$ & ] c = 1; p l . = [ o O 0

L J

The states are chosen to be: 2 = [..IT where 5 is the position of mass. The input vector is chosen to be:

F , is the non-linear control force and F’ is the friction force. The parameter values are given in table 2.1.

C.1.4 Friction simulation model

The friction is modelled as described in chapter 2. A graphical representation of the numerical friction model is given in figure C.1.4.

I t Friction force

Fig. C.2: The numerical friction model

The discontinuity at zero velocity in the real friction phenomenon is approximated by introducing a very steep slope around zero velocity. The transition from the maximum sticking force level to the slipping force level is also modeled by a very steep function. The following program is used to simulate the friction according to the above mentioned simulation model. function [sys, a01 = f r i c t i o n ( t . x , u , f l a g , F ~ ~ ~ ~ ~ , F ~ ~ ~ ~ , ~ ~ ,ó,D)

ií abi(flag) == 3

61

Appendix C Simulation model of 1-DOF system with friction

elseif flag == O

sys = [0,0,1,1,0,0]; xo = [I;

else

sys = [ 1;

end

The parameters values have been chosen by iteration, i.e. the parameters (VI and 6) were optimized with regard to the simulation time and the behaviour of the friction force as function of time. This process included of coarse also the choice of the stepsizes in the numerical integration routine as discussed in section C.4. The chosen parameter values used in simulation are given in table C. l :

indication quantity value units Static friction limit

see figure C.1.4 see figure C.1.4 5e-6 [rnls] Damping coefficient

Table C. l : Numerical friction model parameters.

C.2 Zero Order Hold (ZOH)

The zero order hold keeps the output of the controller at a constant level during the sample time T,. During simulations performed in this research the sample time was set to:

T, = 0.2 (C.3)

C.3 Quantizer

An extension to a more realistic system description is the addition of a quantizer as mentioned in subsection 2.3.3. A graphical input-output relation of a uniform quantizer is given in figure C.3. As mentioned before the value of h is chosen to be le-6 [m]. The

62

(2.4 Numerical integration method and purumeters

output 1 h -

# Fig. C.3: Input-output

LI2 Input

elation of a uniform quantizer

block quantizer is realized using the block Matlab Fcn out of the SIMULINK library. The function used to realize the quantisation of the position is:

xi quantized = round(zi l h ) . h

where round(.) is a standard matlab command.

C.4 Numerical integration method and parameters

The numerical integration routine used to simulate the behaviour of the system is the Runge Kutta integration routine with variable step size. During simulation the step size is continually adjusted in order to meet the entered tolerance [La] . The values of the parameters of interest are:

parameter designation value unit

Atmin Min. step size le-7 [s] Atma, Max. step size le-5 [s] T o1 Tolerance ie-9 [ml

Table (3.2: Simulation parameters of the numerical integration method

63

Appendix D

Digital control environment

A schematical representation of the experimental setup is provided in figure D.l , including the signal flows.

control signal [A] control signal IVl

PC (AT386)

dSPACE

setpoint

v L 2

Li MM S- axis

Amplifier motor

I position [inc] Encoder

Fig. D.l: Schematical representation of the experimental setup

The three main components are identifiable in figure D.l. The PC communicates with the dSPACE in order to send a setpoint to the controller. The dSPACE environment pro- cesses the position data coming from the encoder and the setpoint data from the PC and generates a control signal in order to move the slider to the desired setpoint position. The amplifier transforms the control signal generated by dSPACE from a voltage to a current, which is the input signal to the (brushless AC synchrone) linear motor of the LiMMS axis. The next yaragraph describes the digital controller more detailed.

The digital control environment (dSPACE) For the realization of the control concepts the digital implementation environment dSPACE is used. The hardware component of the environment consists of:

analog to digital converters (ADC); digital to analog converters (DAC); encoder boards (for transforma.tion of the encoder signals); digital signal processor (DSP); dual ported memory (DPMEM);

65

Appendix D Digital control environment

The software component consists of:

the program impex. This program converts a state space model (A,B,C and D matrix) of a designed controller, to the necessary C-code which is used by the DSP.

Texas Instruments C-compiler. This program takes care of the implementation of the Dy impex generated C-code in the DCP environment.

A personal computer (AT386 processor) is used to interact with the DSP in order to be able to present start and stop signals or offering a setpoint to the controller. Communi- cation takes place through the DPMEM.

For the sake of completeness is mentioned that both, the PC (AT386) as well as the DSP can read and write in DPMEM.

66

Appendix E

Simulation model of 2-DOF system with friction

E.í Introduction

This appendix describes the simulation setup used to simulate the behaviour of a 2-DOF system with friction controlled by a RNL adjusted PD controller. Figure E.l shows the simulink model used for simulation purposes. The setpoint used in the simulation equals

x i

quantizer

position

RNL

Non linear friction compensation

Friction

Fig. E.l: Simulink simulation model of 2-DOF system with friction

zero. The intension of the simulation is to show the nature of the response of the system to an initial position error. The next section describes each block in the simulation model with the accompanying S-function [22] if necessary. The last section in this appendix Ests the parameters concerning the numerical integration routine.

'The S-function behaves like any other matlab function. S-functions are at the heart of how sirnulink works.

67'

Appendix E Simulation model of 2-DOF system with friction

E.2 simulation model in blocks

This section describes the blocks used in the simulation model. The discussed blocks are:

1. Lead lag;

2. Non-linear friction compensation;

3. Mechanics;

4. Friction;

5. Quantizer;

6. Zero Order Hold (ZOH).

Each block is treated in one of the following subsections.

E.2.1 Lead lag

The block named lead lag comprehends the lead lag filter in combination with a lowpass filter described by:

The parameters used in simulation are given in section 2.3.2.. The Transfer Fcn block in simulink [22] is used to represent the lead lag with low pass filter.

E.2.2 Non-linear friction compensation

This block contains the by Southward et al. proposed non-linear friction compensation (RNL compensation). For a description is referred to section 2.2.3. The S-function used to represent this compensation is called frn1.m. the contents of this file is listed below.

function [sys, X O ] = frnl(t,x,u,flag,F$,F-T,Kp,Fe)

X H = -(F-/Kp)); Z L = - (Fs 9 /(Kp));

if abs(flag) == 3

e = Fe/Iíp; Z H = X H + e; S L = X L - E ;

if (u > ZH 11 u < E L )

elseif (u > O & u <= ZH

elseif (u > EL & u < O)

else

end

sys(1) = o;

sys(1) = Kp*(”g - u);

sys(1) = KP*(ZL - u);

sys(1) = o;

elseif flag == O

sys = [0,0,1,1,0,0]; xo = [I;

sys = [ 1; else

end

Input u stands for the error signal and output sys(1) for the RNL-force. The overcompensation force is defined as Fe. The rest of the variables are defined in chapter 2.

68

E.2 simulation model in blocks

E.2.3 Mechanics

The state space model

A =

used to represen

O 0 1 O 0 0

_ _ - - -IC k -d y ?i y - - - TI2 “ 2 “ 2

.t the 2-DOF system is given by:

O 1 .- : i -- .>”’ -

The states are chosen to be: g = [2122k1k2lT where 21 is the position of the motor and 22 is the position of the mass. The input vector is chosen to be:

u= [ F;] F, is the non-linear control force and Fj is the friction force. The parameter values are given in table 2.1.

E.2.4 Friction simulation model

The friction is modeled as described in chapter 2. A graphical representation of the numerical friction model, referred to as the ”Classical Coulomb Friction Model for Simulation”, is given in figure E.2.

I 1 Friction force

I ‘ I f I

‘1 ‘1 + ‘ velocitg

Fig. E.2: The Classical Coulomb Friction Model for Simulation

The discontinuity at zero velocity in the real friction phenomenon is approximated by introducing a very steep slope around zero velocity [6]. The transition from the maximum

69

Appendia: E Simulation model of 2-DOF system with friction

sticking force level to the slipping force level is also modeled by a very steep function. The following program is used to simulate the friction according to the above mentioned simulation model.

function [syys, xO] = friction(t,x,u,flag,Fstick,FSlip,w1,6,D)

if abs(flag) == 3

"2 = "1 + 6; A= (Fstick - Fslip)/6;

elseif flag == O

sys = [0,0,1,1,0,0]; xo = [I;

else

sys = [ 1;

end

The parameters values have been chosen by iteration, i.e. the parameters ( V I and 6) were optimized with regard to the simulation time and the behaviour of the friction force as function of time. This process included of coarse also the choice of the stepsizes in the numerical integration routine as discussed in section E.5. The chosen parameter values used in simulation are given in table E.1:


E l i p Coulomb friction level 23 [NI VI see figure C.1.4 5e-6 [m/s] s see figure C.1.4 5e-6 [m/s] D Damping coefficient 20 [Ns/rnl

Fstick Static friction limit 27 [NI

Table E.l: Numerical friction model parameters.

E.3 Zero Order Hold (ZOH)

The zero order hold keeps the output of the controller at a constant level during the sample time Ts. During simulations performed in this research the sample time was set to:

Ts = 0.25e - 3[s] ( E 4

70

E.4 Q uantizer

E.4 Quantizer

An extension to a more realistic system description is the addition of a quantizer as mentioned in subsection 2.3.3. A graphical input-output relation of a uniform quantizer is given in figure E.3. As mentioned before the value of h is chosen to be le-6 [m]. The

Fig. E.3: Input-output relation of a uniform quantizer

block quantizer is realized using the block Matlab Fcn out of the SIMULINK library. The function used to realize the quantisation of the position is:

Xlquant i zed = round(xl/h) h where round(.) is a standard matlab command.

E.5 Numerical integration method and parameters

The numerical integration routine used to simulate the behaviour of the system is the Runge Kutta integration routine with variable step size. During simulation the step size is continually adjusted in order to meet the entered tolerance [La]. The values of the parameters of interest are:


atmin Min. step size le-7 [SI Atma, Max. step size le-5 [s] T 01 Tolerance íe-9 [m]

Table E.2: Simulation parameters of the numerical integration method

71

Appendix E Simulation model of 2-DOF system with friction

E.6 Overview most important system parameters


ml Motor mass 3.5 [bl 6.8 [kg]

d Damping connection 3e2 [ W m l F’ = -F‘ Static friction level (symmetric) 27 [NI Fd+(O) = -FL(O) Coulomb friction level (symmetric) 23 [NI D Viscous damping coefficient 20 [ W m I A Amplification 66 [WVI

T,.,.,J m, .nm uuau 111auu - / I C 2

ik Stiffness connection 1.2e7 [ N / m ]

inc Encoder resolution le-6 [m] Ts Sample time 0.25e-3 [SI

Table E.3: parameters l-DOF system

72

Appendix F

Identification

For the explanation of the identification procedure described in this appendix a block diagram of a linear controlled plant is used, see figure F.1.

Fig. F.l: Block diagram for system identification

In this figure C and P designate the transfer functions of the plant and the controller respectively. u and 2 are the measurable input and output of the plant. T and are setpoint signals, probing signals or disturbances. v is a disturbance signal. The input signal [(i) and the output signals u(t) and ~ ( t ) are used for the system identification method [5 ] .

F. l Spectral analysis theory

Before the actual theory is discussed some introductory theory is described. This theory is base on the hypothesis that the process under consideration is assumed to be strictly causal. Furthermore it is mentioned that the system’s signals are discrete time signals.

The output z ( k ) of a linear time invariant and strictly causal model is the convolution of the input to the model u(k) and the impulse response p ( k ) of the plant [a]. k is used to refer to the time instant. x(k) can now be described as:

2 ( k ) = .+yp(Z)u(k - i) 1=1

73

ADwendix F Identification

Defining the forward shift operator q as:

q *u(k) = u(k + i) and the backward shift operator q-' as:

4-1 . u ( k ) = u p - i)

With these definitions (F.l) can be written as:

1=1

The transfer function P(q) is now lefined as:

00

1=1

Hence, (F.4) can be written as:

At this point the necessary definitions are made in order to describe the identification theory used to determine the frequency response of the process (without controller) in a closed loop situation.

The closed loop situation can be characterized by:

where the sensitivity function S(q):

1 s(q) = 1 + C(q)P(q)

(F.9)

Assume the signals [ ( t ) and v ( t ) as well as the signals [ ( t ) and ~ ( t ) to be uncorrelated. Furthermore is mentioned that w ( t ) is a not measurable disturbance signal.

Define the following signals for ease of notation:

Substitution of (F.lO) and (F.ll) in (F.7) and (F.8) yields:

(F.lO) (F.l l)

(F.12)

(F.13)

F. 2 Determination of the frequency response of the LiMMS-axis

The cross spectra between x ( t ) and ( ( t ) on the one hand and between the signals u(t) and [ ( t ) on the other, can be calculated by [21]:

QZt(w) = S(ej") [P(ej")@C(w) + Qw,t(w) + P(ej")C(ej")@,~(w)] (F.14)

aut(") = S(e jw) [aT(o) + C(ejw)aT,t(w) - c ( e j W ) ~ , , t ( w ) ] (F.15)

As mentioned before the signals T and E as well as the signals u and ( are uncorïelated. Hence:

With the assumption that the spectral densities can be estimated exactly, it is found that the spectral analysis estimate of P(ej") is given by:

(F.16)

This means that identifying the true system dynamics is possible if the assumptions with regard to the uncorrelated signals are correct. It is also mentioned that the estimation becomes better as the number of data points grows.

F.2 Determination of the frequency response of the LiMMS- axis

When figure F.l is considered to represent the controlled LiMMS-axis, than plant P refers to the combination of the amplifier and the LiMMS-axis. By using (F.16) it is possible to calculate the spectral analysis estimate of the frequency response function concerning the plant P ( ejw ) .

Determination of the cross spectra needed for the identification, is realized by capturing the time signals in the experimental setup, by using a four channel Hewlett and Packard dynamic signal analyser [IO]. This analyser is used because of the possibility of measuring the three signals of interest synchronously. After capturing these time domain signals they are transformed to frequency domain signals by using the FFT method. The transformed signals are used for the calculation of the cross spectra. The frequency response of the plant is now calculated by dividing the cross spectra as shown in (F.16) through the mathematical functions utility of the dynamic signal analyser.

As already mentioned the measurements necessary to carry out the dynamic analysis, have to be done in a feedback controlled situation because the LIMMS axis' slider dynamics are by nature close to a double integrator. Furthermore the dynamics of the slider are non-linear due to the friction phenomenon. In case the relative velocity of the slider is constant and not equal to zero, a linear approximation of the slider's dynamics is possible because in this case the friction behaves as a linear damper.

The actual identification is performed by controlling the process with the PD controller described in section 2.3.2. The external signal ( is chosen to be a band limited white noise

75

Awwendia: F Identification

with a flat power spectrum between 10 [Hz] and 810 [Hz] during measurement. This choice is based on the frequency area of interest. The magnitude of the noise signal is chosen to be 2.5 Vpk. In order to minimize the effect of the friction a sine shaped setpoint T is offered to the system with a frequency of 1 [Hz] and a amplitude of 0.20 [m].

76

Appendix G

+

Determination of the dynamic friction component

X P

The experimental setup for determination of the disturbance forces is given by figure G

feedforward

Fig. G.l: Experimental setup for measuring disturbance forces

In the experimental setup a PD controller,C, with a bandwidth of 30 [Hz] is used to control the system. A setpoint, leading to approximate constant velocity is applied to the closed loop system. Furthermore a force feedforward is fed to the system to compensate for the inertia forces. The error which occurs during the trajectory with constant velocity is now only caused by the disturbance forces. The disturbance force Fd is assumed to consist of the friction force Ff and the cogging force Fcogg. Thus, the force generated by the controller exists only of the compensation for the disturbances. The information needed for identification of the disturbance forces can be obtained out of the measured control signal.

Feeding the setpoint given in figure G.2 to the controller results in a control force.

This control force is given in figure G.3.a and 6 .3 .b (zoomed).

It Is assumed that the mean of the control force necessary to keep the slider moving at a constant velocity represents the Coulomb friction force and the viscous friction force at

77

Appendix G Determination of the dynamic friction component

'0 O 1 O 2 03 0 4 0 5 0 6 0 7 O8 O9 1

1 lime [SI - r

I

20

O O 1 O2 03 0 4 0 5 0 6 O 7 O 8 O 9 1 time [SI

-1

O O 1 O2 03 0 4 0 5 O 6 0 7 O 8 O 9 1 m -0 2

time Is]

Fig. G.2: Setpoint signals

O o 1 O 15 o 2 O 25 03 0.35 time [SI

Fig. G.3: Measured control force a: measured control force in time b: zoomed plot

that velocity.

The equation for the friction force at a certain velocity is:

Fj = Fcoulomb + D - VebOCity ( G 4

Two measurements at different velocity are necessary to determine the variables Fcoulomb

and D. Analysis of the measurement given in figure G.3.a and G.3.b leads to an Fj of 34 [NI at a velocity of 0.6 [m/s]. Another measurement at a velocity of 1.11 [rn/s] yielded a Fj of 45 [Nl. Solving (G.l) for the two measurements yields:

Fcoulomb 23.4 [ra] D = 20 [Ns/m]

78

Appendix G Determination of the dynamic friction component

The periodical phenomenon, clearly visible at constant velocity, is caused by the cogging force. Analysis of this periodical phenomenon leads to the statement that the cogging force varies between -5[N] and 5 [NI. The shape of the cogging force force is sinusoidal with a period (p) of 16 [mm] (distance between the magnets. The frequency of the sinusoidal

where the velocity is the constant velocity of the slider.

79

Appendix H

Derivation of the theoretical model of the LiMMS-axis

The experimental setup concerning the LiMMS-axis is schematically represented by the block diagram given in figure H.l

Fig. H.l: Block diagram of the experimental setup

In the block diagram the process is modelled in block P , consisting of the amplifier and the LiMMS axis. Block C represents the digital controller (dSPACE). T refers to the setpoint position, c to the controller output voltage, f to the amplifier output current and x to the position of the slider mass. The signal e refers to the difference between setpoint and the encoder position.

In order to obtain a mathematical description of the slider mass’ dynamic behaviour, Newton’s law is applied to the 1-DOF mass system. This results in:

C F = m X (K.1)

where F is the sum of the external forces which affect the slider’s mass, x is the acceleration of the slider and m is the slider’s mass. The external forces affecting the l-DOF mass are assumed to be the control force F,, the friction force Ff and the cogging force K o g g e

As described in section 3.1 the amplifier is assumed to be an ideal voltage controlled current source. The linear amplifier gain is represented by Ka [A/V]. The motor gain is

81

Appendix H Derivation of the theoretical model of the LiMMS-axis

represented by K, [N/A]. The control (motor) force as function of the controller output c is mathematically described as:

Fc = Ka * Km . c

The friction force Ff is modelled by the static friction force Fstatic, a Coulomb friction force F ~ ~ ~ l ~ ~ b and a viscoiis friction force, Dk as described in section 2.2.1.

(H.2)

Substitution of (H.2), Fj en F,,,,in (H.1) yields the following differential equation regarding the dynamic behaviour of the slider:

(H.3) mx = K,Kmc - Ff t F,,,

In order to complete the model given in (H.3) the parameter values concerning the mass, m, the parameter Km and Ka, the friction force and the cogging force have to be determined.

The mass m of the slider measured by weighing is m = 10.3 [kg]. The parameter values concerning the amplifier gain and motor gain are: K, = 1 [A/V] and Km = 66 [N/A]. The maximum static friction force is measured by determining the maximum force needed to move the mass from standstill. This is carried out at a position where the cogging force equals zero. Six measurements at different positions are performed, three in both direc- tions. All measured values were 27 [NI f 1 [NI. The determination of the dynamic friction and cogging parameters is described in appendix G. The parameter values concerning the dynamic friction are: Fcoulomb = 23 [NI and the viscous damping coefficient D = 20 [Ns/m]. The periodic cogging varies between -5 [NI and 5 [NI with a period of 16 [mm].

82

Appendix I

Control software

1.1 C-source dSPACE

/* ---- IMPEX C Code Generator Vs. 2.0 ----

system file : D:\BAS\LIMMS\AUG95\RNLCONT.BBL

C source file : D:\BAS\LIMMS\AUG95\RNLCONT.C date : 24 Aug 95 10:03:22 am

rnl controller: lowpass 3in 5uit date: 23 aug. 95 INPUTSIGNALS:

SEP-scaled = POSITION SETPOINT ENC-scaled = MEASURED POSITION FEF-scaled = FORCE FEEDFORWARD

OUTPUTSIGNALS:

CON-scaled = CONTROLLER SIGNAL ERR-scaled = POSITION ERROR POS-scaled = NEASUXEC POSITIOM STP-scaled = POSTION SETPOINT FOF-scaled = FEEDFORWARD SIGNAL

Mefine IWPEX-CGEN-2-0

83

Appendix I Control software

#define SAMPLING-PERIOD 2.5OOOOOOE-O4 #define N-INPUTS 3 #define N-OUTPUTS 5 #define N-STATES 3

/* declara t ion of coef f ic ien ts */

/* dynamic matrix */

f l o a t a l -1 = 7.8598358E-01; f l o a t a i -2 = i. 6730222E-01; f l o a t al-3 = 1.4268285E-01; f l o a t a2-1 = -1.6730239E-01; f l o a t a2-2 = 9.8101881E-01; f l o a t a2-3 = -3.9591590E-02; f l o a t a3-1 = -1.4268273E-01; f l o a t a3-2 = -3.9591495E-02; f l o a t a3-3 = 1.1973663E-01;

/* input matrix */

f l o a t bi-1 = 7.8637026E+00; f l o a t bi-2 = -6.5965518E+00; f l o a t b2-1 = -7.0501446E-01; f l o a t b2-2 = 5.9140899E-01; f l o a t b3-1 = 7.5439839E-01; f l o a t b3-2 = -6.3283523E-01;

/* output mat r ix */

f l o a t cl-I = 3.3955537E+02; f l o a t c1-2 = -4.0096022E+Ol; f l o a t c1-3 = -3.3717993E+02;

/* direct links */

f l o a t d l - I = 9.3101678E+02; f l o a t d1-2 = -7.8099348E+02; f l o a t d1-3 = 1.0000000E+00; f l o a t d2-1 = 1.0000000E+00; f l o a t d2-2 = -8.3886080E-01; f l o a t d3-2 = 8.3886080E-01; f l o a t d4-1 = 1.0000000E+00; f l o a t d5-3 = 1.0000000E+00;

84

I. 1 C-source dSPACE

/* declara t ion of var iab les */

/* s t a t e var iab les */

f l o a t x i -d i sc re t e = 0.0000000E+00; f l o a t x2,discrete = Q.OOOOOOOE+OO; f l o a t x3-discrete = 0.0000000E+00; f l o a t xkl-1 = 0.0000000E+00; f l o a t xkl-2 = 0.0000000E+00; f l o a t xk1-3 = 0.0000000E+00;

/* input var iab les */

f l o a t sep-scaled = 0.0000000E+00; f l o a t enc-scaled = 0.0000000E+00; f l o a t fe f -sca led = 0.0000000E+00;

/* output var iab les */

f l o a t con-scaled = 0.0000000E+00; f l o a t err-scaled = 0.0000000E+00; f l o a t pos-scaled = 0.0000000E+00; f l o a t s tp-scaled = 0.0000000E+00; f l o a t fof-scaled = 0.0000000E+00;

/* temporary var iab les */

f l o a t temp-1 = 0.0000000E+00;

f l o a t execution-time;

/*----------------------------------------------- */ /* user defined var iab les */

f l o a t dpmem-0 = 0.0; f l o a t dpmem-1 = O . O ; f l o a t dpmem-2 = 0.0; f l o a t dpmem-6 = 0.0; f l o a t enable = 0.0; f l o a t off se t = 0.0; f l o a t cor rec t ion = O . O ; f l o a t Rnlforce = 0.0; /* Non-linear p a r t of t he cont ro l output*/ f l o a t Frn l = 0.0; /* Non-linear compensation fo rce */ f l o a t Kp = 2100 /* proport ional f a c t o r of PD-control*/ f l o a t nlwindow = 0.0; /* zone of ac t ive non-linear fo rce */ f l o a t deadwd = 0.0; /* Dead zone */ /*----------------------------------------------- */

85

Appendix i Control software

#include “impexc .h“

service-int errupt (1

r t atirq = 1; ++counter;

/* request a dsp -> host interrupt */

service-trace0 ;

/* update states */ xi-discrete = xkl-1; x2-discrete = xk1-2; x3-discrete = xkl-3;

/* read inputs */ start~ds2001(0x00000000); sep-scaled = O.l*dpmem-O; enc-scaled = ds3001(0x00000040, 0x00000004) - offset; fef-scaled = O.l*dpmem-i; user-input 0 ;

/* add input dependent part to outputs * / con-scaled = temp-i + dl-1 * sep-scaled + d1-2 * enc-scaled + d1-3 * fef-scaled * Rnlf orce ;

err-scaled = d2-1 * sep-scaled d2-2 * enc-scaled

d3-2 * enc-scaled

d4-1 * sep-scaled;

d5-3 * fef-scaled;

pos-scaled =

stp-scaled =

fof-scaled =

t

86

I.1 C-source dSPACE

/*------------------------------------------------------- */

/*------------------------------------------------------- * / con-scaled = enable* (con-scaled) ;

/* post outputs */ ds2101(0x00000080, 0x00000001, con-scaled); ds2101(0x00000080, 0x00000002, err-scaled); ds2101(0x00000080, 0x00000003, pos-scaled); ds2101(0x00000080, 0x00000004, stp-scaled); ds2101(0x00000080, 0x00000005, fof-scaled); user-output 0 ;

/*-ROBUST NONLINEAR STICKSLIP FRICTION COMPENSATION----*/

nlwindow = Frnl/(ôô*Kp);

/*FrnP in [NI */

if ((10 * err-scaled <= nlwindow) && (err-scaled > O))

else if ((10 * err-scaled >= -nlwindow) && (err-scaled < O))

else correction = 0.0;

correction = nlwindow - (10 * err-scaled);

correction = -nlwindow - (10 * err-scaled);

nodead = ((10 * err-scaled >= deadwd) I I (10 * err-scaled <= -deadwd));

Rnlforce = 0.1 * Kp * correction * nodead;

dpmem-2 = cvtdsp(dpmem c21) ; if (dgmem-2 < 0.5) < enable = 0.0;)

/* calculate new states */ xkl-1 = al-1 * xi-discrete + al-2 * x2-discrete + al-3 * x3-discrete + bl-1 * sep-scaled + bi-2 * enc-scaled;

a2-1 * xl-discrete + a2-2 * x2-discrete + a2-3 * x3-discrete d. b2-1 * sep-scaled + b2-2 * enc-scaled;

xk1-2 =

87

ADwendix I Control software

xkl-3 = a3-1 * xl-discrete + a3-2 * x2-discrete + a3-3 * x3-discrete + b3-1 * sep-scaled + b3-2 * enc-scaled;

/* calculate state dependent part of outputs */ temp-1 = cl-1 * xki-1 + c1-2 * xkl-2 + c1-3 * xk1-3;

dpmem- O = cvtdsp(dpmem col ; dpmem- 1 = cvtdsp(dpmem 611) ;

dpmem C3l dpmem C41 = cvtie3(10 * con-scaled); /*execution-time); */

= cvtie3(10 * err-scaled) ;/*exec-time) ; */

Frnl = lOO*cvtdsp(dpmem C6l) ; deadwd = cvtdsp(dpmem C71) ;

if (counter == 3800) c go = o; dpmem [SI = cvt ie3 ( 1. O) ;

3 /* end service-interrupt0 */

main0

c /*--------- user - initialisation---------------------------- */ offset = ds3001(0x00000040, 0x00000004); dpmem CO1 = cvtie3(0.0); dpmem 611 = cvt ie3 (0 0) ;

88

I.1 C-source dSPACE

dpmem [2] = cvtie3(1.0); dpmem C3l = cvtie3 (O. O) ; dpmem C41 = cvtie3(0.0); dpmem C5l = cvtie3(0ffset*8.388608); dpmem C6l = cvtie3(0.0); dpmem C9l = cvt ie3 (O. O) ; dpmem C 101 = cvtie3(0.0); enable = 1.0;

*/ /*-------------------------------------------------------

system-initialize(); enable-int errupts (SAMPLING-PERIOD) ;

if (cvtdsp(dpmem~lO]) > 0.5)

counter = O; dpmemC91 = cvtie3(0.0); dpmemClO] = cvtie3(0.0); go = 1;

1 */ /*-------------------------------------------------------

background (1 ;

system-terminate();

3 /* end main0 */

89

Appendix I Control soflware

1.2 Communication pascal source software

PROGRAM t e s t ;

<$F+ f o r c e f a r c a l l s 3

T T C C C --+ UdL3 LIL#, dos;

CONST

-( AT I n t e r f a c e Addresses 3

s t p = $308; sts = $309; i c r = $309; asr = $30a; p s r = $30b; w t r = $ 3 0 ~ ;

c B i t assignment of s t p 3

h ld = $04; < hold dsp I rst = $08; < r e s e t dsp 3 W d m = $10; { watchdog mode 3 wde = $20; { watchdog enable 3 bde = $40; { board enable 3 s h = $80; { small/huge memory mode 3

{ B i t assignment of sts 3

dspeoi = $10; a t i o e r r = $20; < PHS-Bus I/O-error l i n e 3

< AT -> DSP end of i n t e r rup t se rv ice f l a g 3

{ B i t assignment of i c r 3

a t i e n = $04; < enable DSP -> AT in t e r rup t s 3 dsp i rq = $10; { request AT -> DSP in t e r rup t 3 atack = $40; < acknowledge DSP -> AT in t e r rup t 3

< Host segment de f in i t i on 3

segment = $d000; c overlapped AT memory segment 3

I.2 Communication pascal source software

Host page de f in i t i ons 3

ps ize = $4000; { Page s i z e i n words 3 smempg = $800000 DIV ps ize ; C expansion memory page 3 iopg = $804000 DIV ps ize ; I/O page 3

CONST dpoffs = $4000; < dual-port memory o f f s e t t o s t a r t of iopg 3

VAR dpmem : ARRAY [O..$Offf] OF SINGLE ABSOLUTE segment:dpoffs;

CONST eo i = $20; end of i n t e r rup t command 3 i r q = 15; in t e r rup t request l i n e of AT 3 i s l c t = 3; in tno = $70 + ( i r q - 8) ; ( corresponding BIOS i n t e r r u p t 3 p ic1 = $20; ( address of AT i n t e r rup t c o n t r o l l e r 1 3 p ic2 = $aO; < address of AT i n t e r rup t c o n t r o l l e r 2 3

corresponding value t o be wr i t t en t o i c r >

VAR -imr2 : BYTE; - ivec t : POINTER;

(.backup of i n t mask of p ic2 3 ( backup of i n t e r r u t p vec tor 3

PROCEDURE init-ds1002;

BEGIN PORT [s tpl := sh < small address mode, 3

C board disabled, 3

91


{ watchdog disabled, 3 C watchdog mode r e t r y , 3 { dsp r e s e t 3

OR hld -( dsp not held 3 OR (iopg AND $03); ( l s b of i/o-page 3

PORT [ icr] := i s l c t ; s e l e c t i n t e r rup t l i n e 3 PORT [asrl := segment SHR 12; PORT [psr] := iopg SHR 2;

{ map dsp memory t o pc memory 3 ( msb of i/o-page 3

PORT [ w t r l := O; END ;

watchdog timing 3

TYPE IOvect = ARRAY [O. .lol OF SINGLE; spdata = ARRAY [0..37991 OF SINGLE; ( se tpoin t vector 3

VAR inout : IOvect; dspvect : IOvect ABSOLUTE dpmem; { define in/out vector i n dsp memory 3 se tpo in t : spdata; s e t p f f : spdata; e r r o r : spdata; out : spdata;

{ pos i t ion se tpoin t 3

VAR s t e p : SINGLE; s top : BOOLEAN; r e f r e s h : BOOLEAN;

( s tap groot te) {to wr i te values) ( to r e f r e s h screen every sample3

92


PROCEDURE in i t -hos t ; BEGIN

counter := 3800; go := f a l s e ; d i sab le := f a l s e ; rep := f a l s e ; setpno := o; Fstep := 0.025; Frnl := 0.0 ; deadwâ := 0.0; dstep := i e+; dspvect [O] := 0.0; dspvect Cl1 := 0 .0 ; dspvect[IO] := 0.0; dspvectC61 := 0.0; dspvectC71 := 0.0;

END ;

PROCEDURE se rve r ;

INTERRUPT; VAR i : i n t ege r ; j : i n t ege r ;

C procedure requi res f u l l context save 3

BEGIN dspvect [O] : = se tpo in t [counter] ; dspvect [I] := se tp f f [counter] ;

e r r o r [counter] : = dspvect e31 ; out [counter] : = dspvect E41 ;

dspvect C6l := Frn l ; (Definit ion of RNL cont ro l f o rce l dspvect C71 := deadwd; (Definit ion of dead zone)

counter : = count e r+ 1 ;

93

PORT [ i c r l := PORT [icr] AND $07

PORT [pic21 := eo i ; PORT [picl] := eo i ;

OR a tack;

END ;

PROCEDURE i n i t - i n t e r r u p t ;

BEGIN GetIntVec ( in tno , - ivec t ) ; Set ïn tvec ( in tno , @server) ; - i m 2 := PORT [p ic2+l l ; PORT [pic2+i] := -imr2 AND NOT

(i SHL ( i rq -8 ) ) ; END ;

PROCEDURE r e s to re - in t e r rup t ;

BEGIN INLINE ($fa) ; SetïntVec ( in tno , - ivec t ) ; PORT [pic2+i] := -imr2; INLINE ($fb) ;

END ;

( acknowledge DSP -> AT i n t e r r u p t 3 3

< c o n t r o l l e r s 3 ( eo i commands t o AT i n t e r r u p t

( save o ld in t e r rup t vec tor 3 ( s e t new in t e r rup t vec tor 1 ( save o ld i n t e r r u t p mask of p ic2 1

( enable AT in t e r rup t 1

< d i sab le in t e r rup t s during r e s t o r e 1 < r e s t o r e o ld in t e r rup t vec tor 3 ( r e s t o r e old in t e r rup t mask 3 C enable in t e r rup t s 3

PROCEDURE in i t - s e tpo in t ;

VAR i : INTEGER; F : t e x t ; G : t e x t ;

BEGIN ASSIGN(F, s e tpo in t . da t ; RESET(F) ;

BEGIN

END p

FOR i := O TO 3799 DO

READ(F se tpo in t Lil ;

94

1.2 Communication pascal source sofiware

CLOSE(F) ; ASSIGN(G, ’ se tpf f .da t ’ ) ; RESET (G) ; FOR i := O TO 3799 DO BEGIN

READ (G , setpf f [i] ; END ; CLOSE(G) ;

END ;

PROCEDURE i n i t - i nou t ;

BEGIN inoutCo1 := 1.0; inoutCl1 := 0 . 0 ; inoutC2I := 0.0; inoutCs1 := 0 . 0 ; inoutC41 := 0 . 0 ; counter := O ;

END ;

( movement allowed 1/0 : y/n 3

PROCEDURE save-error;

VAR i : in teger ; outp : TEXT; outputfilename : s t r i n g ;

BEGIN wri te ln( ’ saving e r ro r . dat ’ 1 ; outputfilename := ’ e r ro r .da t ’ ; ass ign(outp, outputf ilename) ; rewrite(outp1; f o r i := O TO 3799 DO BEGIN

END ; close(outp) ;

wri te in (outp , e r r o r Cil , ’ ’ 1 ;

END ;

PROCEDURE save-out;

95


VAR i : integer; outp : TEXT; outputf ilename : string;

BEGIN writeln( saving out .dat ' ; outputfilename := 'out.dat'; assign(outp ,outputf ilename) ; rewrite(outp1; for i := O TO 3799 DO BEGIN

END ; close(outp) ;

writein (outp,out Lil , ' ' 1 ;

END ;

PROCEDURE init-screen; BEGIN TEXTBACKGROUND(I);(BLUEI TEXTCOLOR(I4); (YELLOW) CLRSCR; WRITELN ; WRITE ( ' LiMMS CONTROL by: S.C. Hultermans TEXTCOLOR(4);(RED) WRITELN ( '27 april 1995 ' ;

WRITELN ( ' . . . . . . . . . . . . '1; WRITELN ( 'g - move '> ; WRITELN ('d - disable '1; WRITELN ( 'r - stop repeat' ) ; WRITELN ( ' s - save move-data'); WRITELN ( '9 - quit'); WRITELN ( ' + - Frnl := > ' ) ; WRITELN ( - - Frnl := ('1 ; WRITELN ('0 - Frnl := O ' ) ; WRITELN ('p - deadwd := > '1; WRITELN ('m - deadwd := < '1; WRITE ( > WRITE ( ' cntr : Frnl : deadwd : ' ;

WRITE ( j [NI : Lincr] : ' ;

WRITE ( ' '1;

WRITE ( 9 '1 9

96


TEXTCOLOR(I5); (WHITE) END ;

PROCEDURE write-values;

BEGIN if (counter = 3800) then write(’ no-go’); if (counter < 3800) then write(’ go’); WRITE ( ’ WRITE( ’ ’); WRITE( 66 * Frnl:3:3); {Frnl) WRITE ( ’ ’); WRITE(le6 * deadwd:4:1); (deadwd) if (disable) then write(’ disabld ’1; WRITE(chr(return));

’ ) ; WRITE (count er : 4) ;

END ;

PROCEDURE restore-screen; BEGIN TEXTBACKGROUND (O) ; (BLACK) TEXTCOLOR(7); (LGREY) CLRSCR ;

END ;

BEGIN CHECKBREAK:=FALSE; init-screen; WRITE(’ Initialising setpoint data, Start main . . . INLINE ($fa) ; hit-hQSt ( init hostparameters 3

( disable <CTRL> <BREAK> 3

’,chr(return));

init -ds1002 ; ( init DSP board 3 init-setpoint; { init setpoint data 3 init-inout; init-interrupt ; ( init interrupt system 3 WRITE(’ Start main 2 ... ’ , chr (return) ; PORT [stp] := PORT [stp] OR bde; dspvect := inout; ( send first vector to dsp 3 PORT [icr] := PORT [icr] AND $07 { enable DSP -> AT interrupt 3

OR atien OR atack, { acknowledge pending ints 1 PORT [stp] := PORT [stp] OR rst; { start DSP by releasing reset 3

{ enable board 3

97


INLINE ($fb) ; REPEAT

REPEAT write-values ; i f (go) then N counter := O ; go := f a l s e ;

dspvectC91 := O ; dspvect[lO] := 1 END :

i f ( ( r ep ) AND (dspvectC91 > 0.5)) then BEGIN go := t r u e END; i f d i sab le then dspvectC21 := O ; UNTIL KEYPRESSED;

key := READKEY; i f key = ’r’ then r ep := f a l s e ; i f key = ’g’ then go := t r u e ; i f key = ’d’ then d isab le := t r u e ; i f key = ’ s ’ then begin save-error; save-out end; i f key = ’+’ then Frn l := Frnl+Fstep; i f key = ’-’ then Frn l := Frnl-Fstep; i f key = ’0’ then Frn l := 0.0; i f key = ’p’ then deadwd := deadwd + dstep; i f key = ’my then deadwd := deadwd - dstep; UNTIL ( (key= y q’ ) o r (key= ’ Q ’ 1 ; dspvect [21 := O ; r e s to re - in t e r rup t ; res tore-screen; CHECKBREAK:=TRUE;

END. { enable <CTRL> <BREAK> 3

i

98

Appendix J

I I I I I I 1 I I '\ ' "

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :. . . . . . -

I I I I I I I I I

Third order setpoint

Figure J.l shows the complete setpoint information.

. . . . . . .

0.02 I I I I I I I I I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : c .-

x time [SI

Fig. J.l: Complete setpoint

99

Appendix K

Sinusoidal-input describing fun& ion (DF) theory

The describing function method is essentially a quasi linearization technique. Quasi linearization is a method where the output generated by a non-linear element due to a certain input signal of finit size is approximated by applying a linear operation to the input signal. This procedure results in a different linear approximation for the same non linearity when driven by inputs of different form, or even when driven by inputs of the same form but with a different magnitude or frequency. The approximating linear operation is called a describing function [8].

Calculating the describing function is done for specific input signal forms. In case of feedback control systems, the signal at the input to the non-linearity depends on the input to the system as well as on the feedback signal. The presence of the feedback loop complicates the determination of the signal form at the input of the non-linearity. In the RNL controlled system case all signals in the system are filtered by the lowpass character of the linear part of the loop. This leads to the assumption that three basic signal forms can occur at the input of the non-linearities:

1. constant (DC) signal;

2. sinusoidal signal;

3. gaussian signal.

The analysis described in this chapter is based on the assumption that the inputs to the non-linear elements in the closed loop system are sinusoidal. The describing function theory based on a sinusoidal input signal is called the sinusoidal-input describing function (DF) theory.

K . l Sinusoidal-input describing function (DF) theory

The sinusoidal-input describing function is an amplitude and frequency dependent ”transfer function” that describes the ”quasi linea.rized” relationship between the output of the non-Enear element and the sinusoidal input signal. In order to be able to determine the

101

Appendix K Sinusoidal-input describing function (DF) theory

DF, the non-linear element is exited by a sinusoidal input u(t):

u ( t ) = Asin(&) Y A, w > O A A, w # A(t) ,w(t)

see also figure K . l

Fig. K.1: Frequency and amplitude dependent non-linear element with sinusoid input signal

In the following description it is assumed that the non-linear element is part of a by feedback controlled system with a non-linear part as well as a linear part.

Certain conditions have to be satisfied by the non-linear system in order to generate a meaningful sinusoidal-input describing function [8] :

1. the non-linear element is time invariant;

2. the non-linear element is an odd function;

3. no subharmonics are generated by the non-linear element in response to the sinusoidal input;

4. the linear part of the system behaves as a lowpass filter.

These four conditions apply to a broad class of mechanical servo systems containing non- linear element s.

Condition 1 till 3 result in the statements that output YN,(t) is a periodic function with a period of % and that the Fourier expansion of YjvL(t) is:

co co

n=l co

n=l

n=l

where A,(A,w) and B,(A,w) are the Fourier coefficients and:

102

K. 2 Using the describing function to determine limit cycle characteristics

Remark that the static term in the Fourier expansion is absent. This is due t o the condition that the non-linearity is odd. This shows the importance of the concerned condition because in case the static term is present, it would pass the linear element of the system and be fed back to the non-linear element. This would cause the input to the non-linearity not to be a sinusoidal signal.

Since it is assumed that the system contains a lowpass character, the describing function theory claims that the output Y N L ( t ) of the non-linear system can be described by the fundamental component of the Fourier expansion.

The describing function (DF), N ( A , w ) is defined [8] as:

The DF is defined as the ratio of the fundamental component of the response of the non-linear element to a sinusoid input signal and the output signal. The first Fourier coefficients are obtained by multiplying both sides of equation (K.2) by either sin(wt) or cos(&) and integrating:

(K-4) M1cosû1 = 1 n Jin YNL(t)s in(wt)d(ot) Mlsin& = n Jin YN,(t)cos(wt)d(wt)

Now, multiplying the second equation of (K.4) by j , adding the two equations, and dividing both sides of the resultant equation by A, the following relationship is found:

where the relationship eJ" = cos(at) + j s i n ( a t ) is used. Comparing (K.5) and (K.3 the following expression for the sinusoidal-input describing function:

yield

For detailed information about the derivation of (K.6) is referred to literature [li]. With (K.6) a quasi linear transfer function of a non-linear frequency and/or amplitude dependent eiement can be determined. This aliows a restricted evaluation of the non-linearity in the frequency domain.

K.2 Using the describing function to determine limit cyde characteristics

This paragraph describes the use of the sinusoidal-input describing function with respect to predicting the limit cycle behaviour in non-linear systems. To explain the theory a hypothetical non-linear system is used. This system has no relation to earlier mentioned systems in this report. In order to use the sinusoid describing function theory to predict

103


Y G ( j w ) r ( t ) = o +, u(t c ( t )

-

element element

x ( t ) *

Fig. K.2: Single non-linearity closed loop system

limit cycles in non-linear control systems, the physical system is divided in a linear and a non-linear part, see figure K.2.

Subsequently the non-linear linear element is characterized by its DF, N ( A , w ) and the linear element by its transfer function G ( j w ) . The quasi linearization is based on the assumption that the signals in the system are sinusoidal. See figure K.3

Fig. K.3: Quasi-linearized system for limit cycle study using the DF.

When the input is set to zero and the DF is treated as a linearized transfer function, the following loop relationship is apparent:

z ( t ) = - N ( A , w ) G ( j w ) ~ ( t )

Excluding the trivial solution z ( t ) # O implies:

N ( A , w ) G ( j w ) + 1 = O

Suppose the rea! pair ( A 9 , w ~ ) satisfies (K.8), a sustained oscillation of ampïitiide AQ and frequency wo is possible without external excitation of the system.

For the solution of (K.8) an analytical and a graphical approach are possible. The approach used in this report is the graphical one because this method gives more insight. For the interested reader is referred to literature [8] for more information about the analytical met hod.

The graphical approach is based on the Nyquist criterion for linear systems. For linear systems as represented by figure K.4

104

K. 2 Using the describing function to determine limit cycle characteristics

Fig. K.4: Block diagram of a linear system configuration

the characteristic equation given by:

1 + KG(jw) = O

where: K = constant gain factor G ( j w ) = linear system dynamics

The roots of (K.9) give information about the stability of the system. Determining the stability of a linear system is often carried out by plotting the K G ( j w ) curve in the complex plane (Nyquist plot) where point (-1 ,O) is referred to as the critical point. If only G ( j w ) is used to generate the Nyquist plot, the critical point becomes ( -+ ,O) (rewriting F . 9 1 > * The existence of an intersection between the curve KG( jw) and the critical point, leads to an unforced oscillation in the controlled system when the system is excited. The frequency of the unforced oscillation can be calculated by using equation (K.9).

When the constant parameter gain K is replaced by a describing function of a non-linear element, N(A,w), linear techniques can be applied to analyse the stability of the system. Evaluating (KA) yields that the modification required for the non-linear case is a change of the critical point which becomes (-N~A;w) , O ) instead of ( -1 ,O) . The amplitude and frequency of the unforced oscillation are determined by solving' :

(K.10)

Remark that the critical point changes with the signal amplitude and frequency. The unforced oscillation with constant amplitude and frequency which rises in these situations is referred to as a limit cycle. So, when the curves of G ( j w ) and ---L.-- are plotted in the complex plane each intersection of the curves indicates the possible existence of a limit cycle. The amplitude and frequency of the possible limit cycles are determined by the values of A and w of the curves at the intersection point.

y44

'This equation is not to be viewed at as an analogy to t.he Nyquist criterion, it has been arrived at by other reasoning. The Nyquist criterion is only used for explanation [SI.

105


K.3 Stability of limit cycles

In figure K.5 an example is given. The example shows multiple intersection points between

Fig. K.5: Graphical limit cycle determination

the curve ---L..- and the curve of G ( j w ) . For simplicity the non-linearity is only assumed to be amplitude dependent. So, the describing function is represented by a single curve. In this case equation (K.10) has three solutions. In other words, three intersection points between the mentioned curves exist. The next paragraph discusses if and how these three solutions result in limit cycles.

N ( A )

When in a point of intersection either the frequency or the amplitude is disturbed and the disturbed oscillation returns to the same limit cycle (intersection point), the intersection point is called stable. In all other cases the intersection point is referred to as unstable. For determination of the stability of the intersection points 1 to 3 in figure K.5, the Nyquist criterion normally applied to linear systems is used. When for example in the equilibrium point the amplitude increases, the new operating point on the curve -1 is not encircled by the curve G ( j w ) and according to the Nyquist criterion this operating point is stable. When however the amplitude, with respect to equilibrium point 1, decreases the new operating point on the describing function curve, N ( A ) , is encircled by the curve G ( j o ) and according t o the Nyquist criterion this operating point is unstable which results in an increase of the amplitude and the system returns to intersection point 1. It is concluded that intersection point 1 is stable. A similar analysis can be made with respect to the intersection points 2 and 3 which are respectively unstable and stable.

W A )

Appendix L

DF calculation model

This appendix gives the required programms to calculate the DF. The method used is:

e calculation of sinusoidal time response;

e fitting the time responses (optimization);

o calculation of the quasi linear approximations.

The program used to calculate the time response of the open loop RNL controlled system to a sinusoidal input in order to determine the describing function is given by:

% Calculating the timeresponse of P D RNL L P F + LiMMS % Open loop in order to determine the DF. % T h e time vector t is internally generated by simulink

% S.C. Hultermans % date: 13-09-1995

clear

%loading system parameters sysnlf;

Amp=linspace( le-6,10e-6,10); f = logspace(2,3,100);

fnamel=’a’; fname2=’f’;

%Calculation of t h e open loop frequency response

for n=3:length(Amp); for m =9:16

clear t g Yg p p=20; om=2”pi3i(m); t=O; teind=(p*(2*pi/om)); Ampnu=Amp(n) freq=f(m) [tg,x,Ygl=rk45(’rnlo’,teind,[],[le-7 le-7 le-51);

evai([’save ’,fnamei,int2str(n),fname2,int2str(m),’.mat ’,’tg Yg p’]);

end % END O F FREQ. - LOOP end % END O F AMPL. - LOOP

The sirnulink model (rnlo) used in this program is given in figure L.I.

lo?

Appendix L DF calculation model

El- Input denpdlp(s)

PD + lowpacc

frnldead - Non linear compensation

-+J-+P zero order hold Amplifier

U Coulomb friction

Fig. L.l : Simulink model of open loop RNL controlled system: rn1o.m

The parameters necessary in the model are loaded by running the program sysn1f.m:

% sysn1f.m % parameters en s ta te space model limms-as % date: september 1995 % S.C. Hultermans

m=10.3 m2=6.8 ml=m-m2

k=1.2e7; d=3e2;

go************ 4 t h order a***************

A=[O O 1 O 0 0 0 1 - k / m l k / m l - d / m l d / m l k /m2 -k /m2 d / m 2 -d/m2];

B=[O O 0 0 l / m l O O -i /m2];

C=[l o o o

o o o 11;

0 1 0 0 O 0 1 0

D=[O O O 0 O 0 0 01;

%control parameters

fb=30; fd=12 % ( l / 3 ) * fb ; fe=4*fb; Kp=2.le3%le4; Kd=Kp/(2*pi*fd); gamma=100; %4 beta =0.707; we=2*pi*fe;

%Control s t ruc ture

numpd=[Kd Kp];

108

Appendix L DF calculation model

denpd=[(Kd/(gamma*Kp)) i] ;

numlp=[we'2]; denlp=[l 2*beta*we we'21;

numpdlp=conv(numpd,numlp); denpdlp=conv( denpd,denlp);

The simulation parameters and not mentioned models are the same as described in appendix E The actua.1 describing function is calculated by using an optimization routine. The program used is given below:

% Calculating t h e Descr.function van PD'RNL.LPF

% S.C. Hultermans % date: 15-09-1995

%options regarding the optimization routine clear qop = foptions; qop( 2)=le-8; qop( 3)=ie-8; qop( 14)=2000; qop( 16)=le-10;

teken=O;

fname='uyq'; fname3='fa'; fname-l='f';

%Fitt ing of t h e response by a fundamental component

Amp=linspace( le-6,10e-6,10); f = logspace(2,3,100);

for i=3:10 % amplitude

Ymag=[]; Yfase=[]; ftot=[];

for j=9:16 % Frequency

eval(['load ',fname3,intZstr(i),fnarne4,intZstr(j)]);

Tee=[O:5e-4:5e-4*length(yf)]; Tee=Tee'; T=Tee(250:length(yf)); yf=yf(25O:length(yf));

om=z*pi*f(j);

phi=O; Ndak=O; magn=O; amp=Amp(i) freq=f(j ) %start conditions of the optimization routine

bO=[O -pi]; x=fmins('simmin',bO,qop,[],yf,T,om) if x(1) i O

Ndak = -x(l); phase = x(2) - pi; teken = 1;

Ndak=x( 1); phase=x(Z);

else

end

nu=floor(abs(fd)/(Z*pi)); phi=(phase + abs(fd+nu*Z*pi))*lSO/pi magn=Ndak/amp

Ymag=[Ymag m a p ] ; Yfase=[Yfase phi]; ftot=[ftot freq];

plot(T,yf,T,Ndak*sin(Z*pi*f(j)*T + phase)) drawnow

clear Ndak phi x T Tee yf

end % END O F FREQ. - LOOP

evai([>save ',fname,intZstr(i),'.mat ','Ymag Yfase ftot']); clear Ymag Yfase ftot

109

Append ix L DF calculation model

end % END OF AMPL. - LOOP

The minimization criterium is given in the following m file: function [err]=errormin(x,yf,T,om) err=norm(yf-x(l)*sin(om*T + x(2))) The following files transforms the linearized data to complex numbers and plots the Nyquist curve. % nyq.m

fnamez’nyq’;

for h=3:10 eval([’load ’,fname,intistr(h)])

Im=[]; Re=[];

for k=l:length(Ymag);

Re(k)= Ymag(k)*cos(Yfase(k)*pi/(180)) ; Im(k)= Ymag(k)*sin(Yfase(k)*pi/(lSO)) ; com(k)=Re(k) + Im(k)*j;

end

eval([’Re’,intZstr(h),’= Re’]) eval([’Im’,int2str(h),’= Im’])

clear Re Im end

plot(Re3,Im3,Re3,Im3,>o>,Re4,Im4,Re5,lm5,Re6,Im6,Re7,Im7,Re8,lm8,Re9,Im9,RelO,ImlO,-l,O,‘o’) axis([-2.5 0.5 -0.5 O.l]),grid text(-2.1,-0.4, ’1’) text(-2,-0.3 ,’2’) text(-1.91,-0.25, ’3’) text(-1.8,-0.1346,’4’) text(-ï.55,-0.1387 , ’5’) text(-1.67,-0.065, ’6’) text(-1.57,-0.025 ,’7’) text(-ï.43,-0.0587, ’8’)h figure plot(Re4,Im4,Re5,Im5,Re5,Im5,’o’,Re6,Im6,-1,0,’0’) axis([-l.6 0.7 -0.3 O.l]),grid

The last program results in the given Nyquist plot.

110

Appendix M

Interpolation and filtering

Interpolation of the data sequence obtained by calculating the time responses of the open loop RNL controlled system is necessary to reduce the enormous amount of data. Filter- ing the data is necessary to eliminate undesirable signal components. Interpolation and filtering is realized by using the following program. % interpolation a n d filtering % % Date: 13-9-1995 % Author: S.C. Hultermans

fnamel=’a’; fnameZ=’f’; ext=’.mat’; fname3=’fa’; fname4=’f’;

Amp=linspace(le-6,lOe-6,10); f = logspace(2,3,100); fl=logspace(2,3,200); wl=2*pi*fl;

for i=3:10 % amplitude for j=9:16 % Frequency

% loading d a t a from earlier calculated time responses % of the RNL controlled system % using timeres.m eval([’load ’,fnamel,int2str(i),fname2,int2str(j),ext]);

amplitude = i frequency = j

% sample frequency = 2000 T=[O:5e-C:tg(lengt h(tg))];

% Interpolation (sample frequency) yi=interpl(tg,Yg,T);

%filteren % 0.5*2000 = Nyquist frequency [B,A]=butter(4,[(f(j)-ZO)/(O.5*2000) (f(j)+20)/(0.5*2000)]); yf=filter(B,A,yi); %Filtering the interpolated da ta

%using butterworth

% Calculation of t h e phase shift a t f( j) [ H a ] % Necessary for compensation in a later stage [mag fase]=dbode(B,A,5e-4,~1); fas=interpi(fl,fase,f(j)); fd=fas*(pi/ 180);

amp=Amp(i); freq=f(j );

% saving calculated data. eval([’save ’,fname3,int2str(i),fname4,int2str(j),’.mat ’,’amp freq yi yf fd’]);

clear a m p freq yi yf fd fas B A

end end

Appendix N

Tuning rules non-linear network LF

The non-linear filter (figure N. l ) is designed to generate a phase lead for input signals ~ ( t ) with smal amplitudes.

- I

Saturation function

Fig. N.l : Non-linear network providing phase lead at small signal levels

Out of the configuration given in figure N . l can be concluded that in case:

On the basis of parameter S is determined which signal levels is given a phase lead. The other parameters of interest are 7 and K . These parameters determine the amount of phase lead a t the required frequency if input level is smaller then S. The next paragraph is dedicated to the explanation of the tuning rules with respect to K and T .

In case ~ ( t > < 6, the transfer function of the NLF is given by:

Appendix N Tuning rules non-linear network NLF

The schematic magnitude-frequency part of the Bode plot regarding the transfer function given in (N.l) is given in figure N.2:

logarithmic scale -----

I t

I I 1 logarithmic scale

f fl f 2 f 3

__?c

Fig. N.2: Schematic magnitude-frequency part of the Bode plot

The lead of the filter starts at f1 and ends at f 3 . Frequency f 2 is the frequency where the most phase lead arises. It is defined that:

f 3 = * fl (N.2) where X is referred to as the lead ratio' It is easily derived that:

1 r = -

fl

Combination of (N.2), (N.3) and (N.4) yields:

K = X - l (N.5) Parameter X determines the maximum phase lead which occurs at frequency f i .

When it is determined at which frequency ( f 2 ) the maximum phase lead is required, parameter 7 can be tuned because the relationship between f1 and f;! is easily derived. This is described in the next paragraph.

When applying a logarithmic scale, frequency f 2 is in the middle between f1 and f 3 . Recall that f3 = X . f1. Transformation to linear axis yields:

{ log( j 3 ) ;log( f l +log ( fi ) } f 2 = 10

- - IO{ 402"L+~~dfd} - - 10{~~og(4+"9(fd}

- - ~ o ( ~ ~ 9 ~ ~ + l o s ( f l ) }

= dLf1

'A = 5 corresponds with a phase lead of approximately 42 degrees.

B 14

Appendix N Tuning rules non-linear network NLF

Now it can be stated that: fl = (JX)-l . f 2

At this point the necessary rules are available for tuning the NLF.

115

Appendix 0

DF calculation NLF

This appendix gives the required programs to calculate the DF of the NLF. The method used is:

e calculation of sinusoidal time responses;

e fitting the time responses (optimization);

s calculation of the quasi linear approximations.

The program used to calculate the time response of the open loop NLF controlled system to a sinusoidal input is given by:

% Program to calculate time responses to sinusoidal input signals % in order to derive the describing function of the non-linear % filter NLF. % author: S.C. Hultermans % date: sept 1995

%loading NLF parameters nlfpar;

Amp=linspace( le-6,2e-5,20); f = logspace(i,3,40);

fnamel=’nk’; fname2=’l’;

for n=lZ:length(Amp); for m =l:length(f);

clear t g Yg p

% Number of simulation periods p=10;

om=2*pi*f(m); t = O :

% creation of sinusoidal input signal to NLF

Amp(n) f (m) d t =( l/f(m))/400, teind=(p*(2*pi/ om)), T=[O le-5.teind]’, U=Amp(n)*sin(om*T); u t = [ T U],

% Calculation of response NLF to sinusoidal input [tg,x,Ygl=rk45(’dfnlfo’,teind,[],[le-9 O.l*dt dt],ut);

% Saving the results eval([’save ’,fnameï,int2str(n),fname2,int2str(m),>.t ’ , ’ t g Yg p’]);

end % END O F FREQ. - end % END O F AMPL. .

LOOP LOOP

Appendix O DF calculation NLF

K

Y

Input

Fig. 0.1: Simulink model of the NLF: cfn1fo.m

The simulink model (dfn1fo.m) used in this program is given in figure 0.1 The parameters necessary in the model are loaded by running the next program in MAT- LAB:

% parameters non-linear network NLF

% S.C. Hultermans

% parameters Non Linear Filter

f2=130.0 lca=5e-6 % Limit cycle Amplitude

lambda=8 % Phase lead

tau=l/(f2*(2*pi/( sqrt(1ambda)))); K d o m b d a - 1 ; delta=2*lca;

% Frequency of Limit Cycle

The actual describing function is calculated by using the same optimization routine (in- clilding minimization criteriam) as mentioned ir, appecdix L. But before the fitting roiitine is used, the time responses are first interpolated. This is described in appendix M. The last step is plotting the Nyquist curve. This is also described in appendix L

pure.tue.nl · report no: 95.166 s.c. hultermans an investigation to the applicability of a...

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