control of mechanical systems

8/16/2019 Control of Mechanical Systems

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Control of Mechanical Systems

Giorgio Diana - Ferruccio Resta


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Contents

1 Introduction 3

2 Control elements of mechanical systems 92.1 Open and closed loop control systems . . . . . . . . . . . . . . . 192.2 Single Input Single Output (SISO) and Multi-Input Multi-Output

(MIMO) Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Classic control methodologies 25

3.1 State variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Transfer function . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.1 Laplace transform . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Representation of the system by means of block diagrams 383.2.3 The Frequency Response Function (FRF) . . . . . . . . . 403.2.4 Bode’s diagram for elementary systems . . . . . . . . . . 41

3.3 Stability analysis of dynamic system . . . . . . . . . . . . . . . . 47

3.4 Classical control . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4.1 Nyquist criterion, relative stability and Bode criterion . . 513.4.2 Root locus . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4.3 Closed loop transfer function . . . . . . . . . . . . . . . . 57

3.5 PID controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5.1 Proportional term . . . . . . . . . . . . . . . . . . . . . . 603.5.2 Derivative term . . . . . . . . . . . . . . . . . . . . . . . . 663.5.3 Integral term . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.6 Specifications for control systems design . . . . . . . . . . . . . . 91

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2 CONTENTS


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Chapter 1

Introduction

The aim of these notes is to provide the student of mechanics with the necessary notions not only to enable him/her to interface with system control experts but also to design or operate a mechanical system that is either wholly or partially equipped with a control system.

The main objective of our courses in applied mechanics is to schematize themechanical system in order to create a mathematical model capable of simu-lating the real behaviour of the mechanical system itself. These models areuseful both during the design as well as operating phase either to optimize aproject or to improve system functioning. These models can have one degree of freedom, several degrees of freedom or even infinite degrees of freedom. They

can be more or less sophisticated, linear or non-linear depending on the typeof problem under examination. The aim of a machine, intended in the broadsense of the word, is that of performing a given function. Consequently, themodel designed to represent it must be capable of defining the behaviour of themachine according to the instructions given or, in other words, of defining whatlinks exist between actions in input and the result of output machine motion.

Let us take, for example, a system consisting of a direct current motorcoupled with a centrifugal compressor; by providing the motor with a givenvoltage, the motor and compressor rotors begin to rotate. In addition to thecharacteristics of the motor and the compressor, the law of start-up and therunning speed depend on the value of the voltage applied to the motor. Thisrepresents the input variable that enables us to control the rotating speed of

the compressor rotor to which the pressures and the fluid flow rate, treated bythe compressor, are associated.

In this case, based on the tension applied, the model used to calculate theoutput angular speed consists of a simple system with one degree of freedomthat keeps account of the torque curves characteristic of the motor, the com-pressor and system inertia. This model is translated into a system of non linearequations due, for example, to the non linear link of the characteristic torque-

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4 CHAPTER 1. INTRODUCTION

Figure 1.1 An automatic machine

speed curve of the compressor. The linearization of the equations of motionin the neighbourhood of a given running speed enables us to analyse systemstability and to define, through linear approaches, the transfer functions, i.e.the links been input and output which in the non linear approach can only beobtained by means of numeric integration. Even in the case of a machine tool ormanufacturing machine which, thanks to the use of a suitable kinematic motion,has been designed to achieve the desired trajectory of the tool or the extremeof the kinematic chain (figure (1.1)), the model must reproduce the kinematicmotion and the dynamics of the machine and must be capable of simulating thelinks between the motor input variables and the output variables that producethe machine functions.

All courses in applied mechanics are targeted at creating these models andat the definition of the links between the actions in input and the law of motionin ouput, or rather they are aimed at defining which instruments are necessaryto govern and thus control the machine. The control of a mechanical system,intended as a closed loop action, is the final objective of the entire design pro-cess. Controlling a mechanical system means, for example, ensuring that the

system does not have undesired motions such as vibrations or instability. Inorder to achieve this, it is necessary to work on the parameters of the systemsuch as stiffness, mass and damping. In this particular case, we will be talkingabout passive control. The system does not always consist of one part whereaction takes place (the motor part) and one which represents output. It is morelikely for systems to have several inputs and several outputs, that are not alwayseasily identifiable. Take, for example, a motor car: the commands in input are


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Figure 1.2 Active control systems in a car: ABS (Antilock Braking System)

Figure 1.3 Active control systems in a car: TCS (Traction Control System)

the engine or braking actions but also the law with which the steering wheel isoperated. The irregularity of the road is seen as a disturbance or a magnitude

in input that basically influences the dynamics of the vehicle or the trajectoryand trim, which become the outputs of the system (figure 1.2 and 1.3).

It is thus essential to ensure optimum development of these models. Ingeneral, these are physical models, often with several degrees of freedom andnumerous inputs and outputs. In some cases, we use black box models whichonly represent the relationship between the input and output of the system for


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some components of the complete model, for which it is difficult to develop aphysical model. It goes without saying that, in these cases, the identificationprocess is fundamental. These black box models are developed both in a timedomain, in which it is usually necessary to work in order to reproduce thenon linearities present, as well as in a frequency domain. The forces that acton the system can be a function of the state of the system and thus be fieldforces that create a natural feedback. It thus becomes easy to introduce anactive control which, as we will see, defines an action depending in part onthe state of the system and, for the remainder, on a reference, which can beassimilated with a field force. From this point of view, in terms of approach,there is no difference in analysing the dynamic behaviour of a shaft supported byhydrodynamic lubricated bearings or by electromagnetic support bearings withactive control. In fact, in a lubricated bearing, in the meatus of the lubricantpresent between the journal and the bearing, pressure, which is a function of

the angular speed, builds up thus giving rise to a force on the journal whichbalances outside actions. In addition to the angular speed and the parametersof the bearing, this force is also a function of the relative position of the journalcentre with respect to that of the bearing. In other words, the force exchangedbetween the journal and the bearing is a function of vector z which defines thecoordinates x and y of the relative distance of the journal and bearing centres.

Given an outside load acting on the journal, the centre of the journal itself will assume a position of equilibrium so that the force produced by the lubricantis the same as that of the outside thrust. If, in addition to a displacement, aspeed is also associated at the centre of the journal, the force acting on the journal also becomes a function of this approach speed. Hence, the force F exchanged between the journal and the bearing is a function of the position

and speed with which the centre of the journal moves with respect to that of the bearing, i.e. F = f (z, ż). Once linearized in the neighbourhood of theequilibrium position, force F transmitted by the oil film can thus be seen asa proportional control action for the terms depending on z and as a derivativefor the terms depending on ż, since ż is a vector of two components whichdefines the displacement of the journal according to two orthogonal directionsin the plane. The proportional action is defined by an equivalent 2x2 matrix,referred to as an elastic matrix, while the derivative action is defined by a similardamping matrix.

The characteristics of these matrices have a considerable influence on thestability of the system within the neighbourhood of the equilibrium position.In other words, the force field created in the lubricated bearing spontaneouslygenerates a control system which tends to bring the journal into its equilibrium

position. Outside disturbances which tend to displace the journal from thisposition are contrasted by force field actions which generate a proportional andderivative action. In this situation, the derivative action is stabilizing becauselinearization results in a symmetrical matrix, definite positive in regard to speedterms. The matrix depending on positional terms, i.e. the one that provides theproportional action, is definite positive (hence it tends to restore the equilibriumposition), but is not symmetrical and can thus give rise to flutter-type instabil-


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ity. In this case, unlike that of a system with automatic control, it is not possibleto modify the gains, i.e. the matrices, to make the system stable because thematrices depend on the intrinsic characteristics of the bearing. However, bychanging the geometry of the bearing or operating conditions, it is possible toensure that the system is stable and thus design a bearing to optimize dynamicbehaviour (passive control).

In the case of a bearing with electromagnetic support, the support actionis achieved by the electromagnetic forces that are generated by controlling therelative journal-bearing position using a proportional action, similar, in everyway, to the above mentioned case. In this case, however, there is a real controllerwhich defines the gains and consequently the proportional and derivative ma-trices, or, in terms of state magnitude, only the gain matrix. The advantage isthat this matrix can be changed dynamically and adapted to system operating

conditions. It thus results, for example, that while in the case of the lubricatedbearing, by increasing the speed, one encounters a flexural critical speed definedby the elastic characteristics of the shaft but also by the equivalent elastic matrixof the bearing, in the case of a bearing with electromagnetic support, since it ispossible to dynamically change the gain matrix of the bearing, the eigenvaluesand, consequently, the natural frequencies can be modified in order to avoid aresonance condition between the angular speed and the natural frequency. Inother words, while the angular speed increases and one approaches resonanceconditions with a given gain matrix, it is possible to modify the control gainin order, for example, to decrease it and to lower the natural frequency belowthe actual rotation speed, thus avoiding the step through resonance (if not atthe instant due to the variation of the gain matrix parameters). Obviously, in

this case, the matrix gains must ensure that the system is stable and that sen-sitivity to disturbances are minimum, expressed, as a rule, by the stability index.

At this point, we insert all those control or identification techniques thatwork in space state domain which belong to a discipline known as the theory of systems . From the above, it must be clear that the analysis of a controlled sys-tem does not present any greater difficulties than the analysis of a non-controlledsystem since the biggest problem is managing to create a suitable model of themechanical system. Conversely, the difficulties involved in the synthesis phase of a controller remain. Furthermore, the insertion of control requires the suitableschematization of the electric, hydraulic or pneumatic actuators and the inser-tion of a control logic, performed in a more or less sophisticated manner usingproportional, derivative or integral actions, or resorting to various algorithms

to optimize the gain matrix. As previously mentioned, since these models aregenerally of the non-linear type, they work in the time domain and, only inspecific cases (or during the simplification stage) is it possible to use the moretraditional techniques of classic control, which operate with transfer functionsand thus in the Laplace domain.

It is therefore important to define and propose those space state control


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techniques that work in the time domain and that integrate suitably with themodelling of mechanical systems.


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Chapter 2

Control elements of mechanical systems

A machine is defined as a system designed by man to perform a given function.The concept of control and regulation is already implicit in this definition, inthe sense that the objective of a machine is to perform certain functions, accord-ing to specific instructions and thus, to follow predetermined laws of motion.Controlling a system means ensuring that it follows a desired law of motion.This control action is carried out either through self-controlled actuators (takefor example three phase asynchronous electric motors) or human actions (con-troller par excellence) on which all automatic control systems are based. Forthese reasons, in this text and more specifically in all texts on applied mechan-

ics, particular emphasis has been placed on how to control a machine, i.e. onhow one defines what actions of motion or resistance should be assigned to amachine in order to implement the desired law. In analytical terms, for example,the equations of a mechanical system can be written as:

mz̈ + rż + kz = f z (z, ż, z̈) + f c (t) (2.1)

where the linear terms are shown on the left, while in f z (z, ż, z̈) the nonlinear terms have been organized and finally f c (t) defines the action that onewishes to exercise on the system to obtain a desired law in z where z is the vectorof the independent coordinates of the system itself. In this context, definitionof the model, that one wishes to use in order to reproduce the real behaviourof the system, is essential. In this instance, we do not wish to dwell on the

sophistication of the model, developed either in the time or frequency domain,nor whether it is of the physical or black box type, linear or nonlinear. In orderto obtain a given law of motion z r (t) for the system in question, it is necessaryto impose an action f c (t) to ensure that the system responds according to thedesired law.

Were the system linear, this problem could easily be resolved: take, forexample, a vibrating system with only one degree of freedom, supported by the

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10 CHAPTER 2. CONTROL ELEMENTS OF MECHANICAL SYSTEMS

equation of motion:

mz̈ + rż + kz = f c (t) (2.2)

Having defined the law of motion that we wish to obtain as reference zr(t), thenecessary action is obtained by imposing:

f c (t) = mz̈r + rżr + kzr (2.3)

Take, for example, the case in which we wish to obtain a harmonic referencevalue:

zr = Z r0eiΩt (2.4)

the necessary excitation force is evidently:

f c =−mΩ2 + iΩr + k

Z r0e

iΩt = F c0eiΩt (2.5)

i.e. it must be a action equal in module to the module of F c0 and suitablyout-of-phase with respect to zr.

Another significant example is that of wanting to define the angular operat-ing speed of a pump whose characteristic curve, in terms of resistant torque M ris shown in figure 2.1.

Ω

M r

Ωr

Figure 2.1 Steady-state operating conditions of a pump and its characteristic curve

In this case we wish to obtain the desired reference speed Ω r: a first possi-bility consists in choosing a motor whose driving torque is equal to the resistantone at the desired speed. The availability of this torque depends on the type of motor used, since by choosing a three phase asynchronous motor, the charac-

teristic curve depends on the number of poles and on the presence of possiblereducers for synchronism speed positioning (figure 2.2).

In this case, the system self-regulates to a speed near to that of the syn-chronism but this is a regulation fixed at Ωr. If the application requires varyingthe angular speed Ωr, the choice must go to a type of motor capable of varyingthe driving torque: for example, as shown in figure 2.3, the choice of a direct


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Ω

M

M m

M r

Ωr

Figure 2.2 Steady-state operating conditions of a pump operated by a three phase asynchronous motor

current motor permits us to obtain various angular speeds Ωr: by varying inputvoltage V .

Ω

M

M m

V M r

Ωr

Figure 2.3 Steady-state operating conditions of a pump operated by a direct current motor

In this case too, the problem is solved by bearing in mind the equation of motion of the machine:

J t Ω̇ = M m (V, Ω)−M r (Ω) = 0 (2.6)

and by imposing Ω = Ωr it is possible to determine the value of the steady-state torque M m , having imposed the angular acceleration as nul. It is necessaryto analyse the dynamics of the system by means of the expression (2.6) so thatthis speed can be reached and the condition achieved stable.

In short, regulation or control are based on the knowledge of the model usedto reproduce the dynamic behaviour z of the mechanical system and on thereference definition zr with which the action of the actuator or motor element isdetermined. The actual behaviour of system z will generally be different fromzr due to model errors and outside disturbances which are always present. Theproblem associated with the presence of outside disturbances on the system will


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be analysed in paragraph 3.7. In order to ensure easy control, in addition tobeing stable, a system must have frequency response characteristics suited tothe type of reference law desired. In other words, a system responds to theaction of the excitation force through a specific integral, which, by referring tothe example shown in equations (2.2) and (2.3) is given by the equation (2.4).But if the integral of the associated homogenous equation:

mz̈ + rż + kz = 0 (2.7)

or rather:z = Z 0e

λt (2.8)

has at least one root λ with a real positive part (take, for example, a systemwith a negative damping r which gives rise to two complex conjugate solutionswith a real positive part), the unstable solution is added to the specific integral

zr (i.e. to the law of motion that we wish to obtain) thereby nullifying thecontroller’s attempt to obtain the desired law. To obviate this problem, thecontroller’s action must also regulate system stability. To achieve this, it mustperform actions that modify the intrinsic stability by using forces depending onthe state and motion of the system. Furthermore, to accurately control a dy-namic system, the transfer function or the transfer matrix must assume suitablevalues. Let us consider, for example, the vibrating system with one degree of freedom, shown in 2.4: this shows the transfer function between the force ininput and displacement z in output indicated by curve a in figure 2.4.

r

M

k

Z 0eiΩt

F 0eiΩt

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

[rad/s]

[ z 0

/ F 0

]

a)

Figure 2.4 Vibrating system with one degree of freedom: module of the transfer function between force and displacement resulting from the variation of the damping factor.

In fact, to control the system and obtain the desired law of motion zr (e.g.as in figure 2.5a), the spectrum of zr must have frequencies in the quasi staticzone (figure 2.5b).


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0 5 10 15 20−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

[s]

[ z r ]

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

[rad/s]

[ z r ]

Figure 2.5 Comparison of the desired law of motion zr (a) and relative spectrum with the transfer function of the system under examination (b).

It is difficult to control this system in the entire frequency field, in that it isdifficult to implement the desired law with a frequency contribution near to thesystem resonance (especially if damping is small and of an uncertain value) orin the zone above resonance (where there are very low transfer function values).One option might be to act on the system by modifying the transfer functionand enlarging the quasi static zone or increasing the damping factor (figure 2.6).

0 5 10 15 20 25 30 35 400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

[rad/s]

[ z 0

/ F 0

]

Figure 2.6 Transfer function with a high stiffness and damping value.

Given a certain law of motion zr and to ensure easy implementation, thesystem must have an appropriate transfer function: since the form of the transferfunction depends on the value of eigenvalues or transfer function poles, it isnecessary to position these eigenvalues appropriately:

λi = αi + iωi (2.9)


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which, in general, are a complex number (and can thus be represented asa point in the complex plane) where the imaginary part ω

i defines the value

of the frequency associated with the eigenvalue while the real part αi definesthe damping parameter. In particular the ratio αi/ωi is the non dimensioneddamping factor.

On the other hand, at times, the aim of control consists in ensuring that thesystem itself is capable of attenuating outside actions or rather, of responding,as little as possible, to outside excitations. In this case, the desired law of motionis zr = 0.

Take, for example, the reduction of vibrations transmitted by alternativemachines or attenuation of the disturbances present in the carbody of a vehiclemoving over rough ground (figure 2.7). In the latter case, the aim is to minimizethe transfer function (or the transfer matrices) between input (irregularity of theroad) and output (the motion in the carbody) in the frequency field in question.

In this case, we refer to the optimization of suspension parameters.

Figure 2.7 Car suspension

Figure 2.8 shows a simplified model (with one degree of freedom) of the ve-

hicle together with its relative transfer function, where displacement yter of theground, due to the irregularity profile, is in input and displacement of carbodyz in output. Figure 2.9 shows a typical continuous spectrum of road irregularitywhich constitutes the input while, in this case, the reference value zr is nul .The response of system z can be obtained as a product between the spectrumin input and the transfer function of the system itself: as can be noted in thesame figure, this response might not be satisfactory. Should we wish to limit,


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as far as is possible, system response to irregularity, it is necessary to increasethe adimensional damping of the system in order to obtain a reduced dynamicamplification in resonance or, on the contrary, in order to obtain a reduced re-sponse (figure 2.10), to displace the resonance to a lower frequency with respectto the frequency field in question.

r

M

k

z

yter

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

[rad/s]

[ y 0 t e r /

z 0

]

Figure 2.8 Equivalent diagram of a car suspension

0 5 10 15 20 25 30 35 400

5

10

15

20

[rad/s]

[ y 0 t e r /

z 0

]

0 5 10 15 20 25 30 35 400

5

10

15

20

25

[rad/s]

[ y 0 t e r /

z 0

]

Figure 2.9 Spectrum of road irregularity (above) and spectrum relative to system response (below)

This operation, which represents an important part of the design phase (pas-sive control), is carried out by acting on the suspension parameters and onsuitable shock absorbers.

As previously mentioned, to control a system in the above mentioned terms,i.e. to modify the position in the complex plane of the transfer function poles(eigenvalues), it is necessary to generate forces that depend on the state of thesystem itself and consequently on the displacement value and on the speed of parts of the system. As far as mechanical systems are concerned, the intro-duction of forces, depending on displacement and speed, is translated into amodification of the stiffness and damping parameters of the system and into the


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0 5 10 15 20 25 30 35 400

5

10

15

20

25

[rad/s]

[ y 0 t e r /

z 0

]

0 5 10 15 20 25 30 35 400

5

10

15

20

25

[rad/s]

[ y 0 t e r /

z 0

]

Figure 2.10 Passive suspension: comparison between different solution non optimised (left) and optimised (rigth)

subsequent variation of the system eigenvalues or poles. In the case of passivecontrol, this operation is carried out using suitable passive actuators (springsand shock absorbers), i.e. without there being any need to introduce energy.This operation is aimed at optimizing the parameters of the system.As an alternative, in order to achieve the same objective, it is possible, bymeans of control strategies, to use active electric, hydraulic or pneumatic actu-ators which generate actions depending on state variables.

By summarizing what has been said until now, control, by means of an ac-tuator, can produce an action obtained as a result of solving inverse dynamics,thus allowing us to obtain the desired law of motion of the system. On the con-

trary, by using the same or other components it can actuate forces depending onthe state of the system which, by modifying the eigenvalues, improve stability,sensitivity to disturbances and, in general, system response itself.

The first type of control, which solves inverse dynamics, is defined as open loop in that it does not necessarily use a measurement of the behaviour of thesystem or its state to exert a control action. This type of control is also definedby the word feed forward . The controller working in open loop must be awareof the link between z(t) and the control action f c(t) to be exerted. This canbe achieved, for example, by solving the inverse dynamics that links zr(t) tof c(t) or, however, through in-depth knowledge of the behaviour of the systembased on subsequent attempts. A measurement of system behaviour is often

used to decide how and when to exert an action in feed forward. In other word,in open loop control, it is useful to measure the output of a system in order,if necessary, to modify feed forward actions based on knowledge of its behaviour.

The second type of control involves a contribution that depends on its statethrough the use of an action proportional to its state, its derivative and its in-tegral (PID). These actions are referred to as closed loop actions because they


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require a measurement of the state of the system. This information, togetherwith the reference signal, i s given as input to the controller.

One specific way of exerting control actions is to implement a desired lawand consequently modify the state of the system consists in defining the actionsso that they will depend on the difference between the desired law zr and thestate measured z. Take, for example, a vibrating system with only one degreeof freedom z subjected to the action of an active actuator capable of providinga control force f c(t)proportional to the error between a desired law zr and theposition z (t) of the system:

mz̈ + rż + kz = f c(t) = k p(zr − z) (2.10)

The feedback control action is translated into a modification of the character-

istics of the system itself, in terms of stiffness k p, and into an action dependingon zr, similar to a feed forward action.

mz̈ + rż + (k + k p)z = f c(t) = k pzr (2.11)

The control action has thus modified the eigenvalues of the system (i.e. thenatural frequency and the damping factor) and its transfer function (figure 2.11).

If we now define a control action proportional not only to the error betweenthe desired value and its actual one but also to its derivative:

mz̈ + rż + kz = f c(t) = k p(zr − z) + r p(żr − ż) (2.12)

o rather:

mz̈ + (r + r p)ż + (k + k p)z = f c(t) = k pzr + r p żr (2.13)

we note how, in this case, the introduction of the law of control is translatednot only into an equivalent stiffness k p but also into an equivalent dampingr p, capable of modifying the transfer function so that the system is capable of reproducing the reference value up to higher frequencies (thus increasing thepassband) (figure 2.11).

As can be seen from the equation of motion (but this is also applicable toany system with control actions proportional to the error and to the derivativeof the error itself), the control action produces two effects: an action similarto a feed forward due to terms k pzr + r p żr , and one that is dependent on thestate of the system itself which modifies natural frequencies, damping factorsand, in general, modes of vibration. However, the feed forward action does notguarantee achievement of the desired law. In fact, if zr is constant, once thetransient is finished (steady state) z is given by the expression:

x = k pk + k p

zr (2.14)


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0 10 20 30 40 500

1

2

3

4

5

6

7

[rad/s]

[ x / x

r ]

Figure 2.11 Feedback controlled system with one degree of freedom

from which it results that, as long as z is equal to zr, it must be k p >> k1. Furthermore, in a closed loop action, the values of k p and r p must be chosenwith a view to positioning the eigenvalues in a suitable manner with respect tothe control strategy implemented. In other words, in this example, the naturalfrequency must be suitably higher than the frequencies present in the zr signaland the damping factor sufficiently high to ensure that resonance frequenciesdisturbances are not amplified. In order to evaluate the stability of the system,we consider the homogeneous solution, obtained by letting f c(t) = 0 ( (2.13)):

mz̈ + (r + r p)ż + (k + k p)z = f c(t) = 0 (2.15)

from which we obtain, for z = z0eλt:

λ1,2 = −r + r p

2m ± i

k + k p

m −

r + r p

2m

2= α ± iω (2.16)

by means of which it is necessary to evaluate the roots variation as a functionof the control parameters k p and r p. By plotting these eigenvalues in the complexplane, it is possible to obtain what is defined as a root locus which also shows howparameter α/ω, an indication of system stability, varies. Figure (2.12) showsthe root loci following variation of control gain k p: in the case of an action thatis only proportional (denoted by the symbol ”∆”) we note how the control onlymodifies the immaginary part, while in the case of a proportional action andderivative (denoted by the symbol ”O”), maintaining the ratio between kd andk p unvaried, we note how both the real and immaginary part of the eigenvalue is

modified. By comparing the characteristics of the open and closed loop controlstrategy it is useful to point out how it is often possible to achieve a mixedcontrol, i.e. to exert an open loop (feed forward) action and, how by usingclosed loop control it is possible to correct the errors between system responseand the desired law (this aspect of the problem will be dealt with further on).

1Actually control actions with kp >> k are generally not possible due to the limits of theactuating system. Thus it is difficult to reach z close to zr


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2.1. OPEN AND CLOSED LOOP CONTROL SYSTEMS 19

−10 −5 0 5 10−15

−10

−5

0

5

10

15

α

ω

Figure 2.12 Active control of single d.o.f. system: roots locus for a P (”∆”) and PD (”O”) control

2.1 Open and closed loop control systems

In the previous sections a first subdivision between two different types of controlstrategies was identified.

• Open loop control systems (block diagram in figure 2.13); in this diagramthe reference input zr(t), which represents the desired response, is givento the controller who thus defines a control force f c(t). This force is com-bined with a possible disturbance action f d(t) in order to give the totalforce acting on the system.The response of the system to the overall force f c(t) + f d(t) is z(t) whichassumes the role of a controlled variable. Open loop regulation means hav-

ing an accurate knowledge of the dynamics of the system, either througha mathematical model or through experimental measurements in order tohave an accurate knowledge of the links between input and output.

In this system, control force f c(t) does not depend on the actual state of the system but on an estimated one. Therefore, if outside disturbancesand uncertainties regarding knowledge of the model mean that the state


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of the system differs from the desired one, there is no way of correctingsuch behaviour.To improve this type of control, it is advisable to measure the behaviourof the system. This means that the necessary steps can be taken bymodifying the action in feed forward, in order, for example, to reducethe effect of the variation of some parameters on control action. Oncethe outside disturbance acting on the dynamics of the system have beenidentified and measured, it is then possible to insert the effects that thesehave on system behaviour.

C ontroller Actuator System

Disturbances and external actions

f c

+

zr(t) z(t)

f d

Figure 2.13 Block diagram of a open loop regulation system

• Closed loop control systems (figure 2.14); these require a sensor or trans-ducer that measures the controlled variable z (t) and transmits the corre-sponding information to the controller. The controller receives (or gener-ates) the reference signal zr(t) and processes the error signal thus givingrise to the control signal uc(t) to be transmitted to the actuator that pro-

vides control force f c(t). This type of system is referred to as a feedbackor closed loop control system. During the design phase of a closed loopcontroller, in-depth knowledge of the system to be controlled or develop-ment of a model, might not seem necessary. In actual fact, as previouslymentioned, feedback of part of the state involves a modification of the sys-tem characteristics, thus making a stability analysis, based on knowledgeof a numerical model, necessary.

Disturbances acting on the system are offset by control if the disturbance fre-quency is low with respect to the natural frequencies of the system. Conversely,it is possible to have amplifications and is therefore advisable to operate byfiltering the signal of the controlled variable prior to feedback.

In most cases, the most valid solution consists in carrying out a mixed con-

trol: by exploiting all the inherent information regarding the system, an openloop action evaluates a feed forward action by solving its inverse dynamics while,in the meantime, in parallel, a closed loop control action reduces the errors dueto uncertainties regarding the model used during the definition phase of theopen loop action and the inevitable unknown disturbances.

In order to modify the imaginary part of the eigenvalues of a mechanicalsystem by means of closed loop control, proportional action can often not be


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2.1. OPEN AND CLOSED LOOP CONTROL SYSTEMS 21

C ontroller Actuator

Sensor

System

Disturbances and external actions

+

−

f c

+

zr(t) z(t)

f d

Figure 2.14 Block diagram of a closed loop regulation system

achieved by the actuators on account of the load being too heavy (take, forexample, a structure). The case of a machine, for which actuators have alreadybeen sized in order to achieve the desired law of motion is different thus meaningthat they can be exploited to modify the natural frequencies.

Feedback can either be performed by an operator or automatically. To giveyou a clearer idea, take, for example, the case of a motorcar travelling in astraight line in which the driver, depending on the reading of the tachometerthat gives the controlled magnitude value (speed), decides whether to increaseor decrease pressure on the accelerator in order to regulate speed to the desiredvalue. In this case, the control loop is closed by the pilot who, after readingthe speed value, compares it with the desired value and adjusts it accordinglyby acting on the accelerator or the brake. In this case, the pilot acts as thecontroller.

Another example is the helmsman of a ship who has to follow an assignedroute: depending on the reading of the compass, which provides the controlledmagnitude value (the course), the helmsman will decide what adjustments tomake to the rudder (actuator) in order to ensure that the ship is not off-course.The operations which, in the case of manual regulation are performed by aman, can also, as previously mentioned, be performed automatically. Hencethe invention of automatic pilots which, by interfacing with instruments suchas compasses, transmit commands to the hydraulic or electric motors to ensureactuation of the bar.

Also worth mention is the example of an automatic textile machine which, in

order to ensure that the fabric is kept taut, regulates the torque on the windingreels according to the speed measurements of the xxx fabric itself (figure 2.15).

In most cases, the best solution is to perform a mixed control: an open loopaction, by exploiting knowledge of the machine itself, evaluates a feed forwardaction by solving its inverse dynamics. Meanwhile, in parallel, a closed loopcontrol action reduces the errors due to uncertainties regarding the model usedduring definition of the open loop action and inevitable, unknown disturbances.


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Figure 2.15 Control system for the tautening of an automatic machine

2.2 Single Input Single Output (SISO) and Multi-Input Multi-Output (MIMO) Systems

The systems to be controlled can be either single-input single-output (SISO),similar to those described in the simple examples given above, or they canbe multi-input and multi-output (MIMO). Thus, for example, a typical SISOsystem is the one described above, i.e. the automatic pilot of a ship, in whichthe feedback signal could be obtained from the indication given by the compass(in this case, electronic) and compared with the reference signal correspondingto the assigned course. From the difference signal (amplified and transformed)it is possible to obtain an action capable of making the rudder rotate either in

a clockwise or anti-clockwise direction according to whether the ship’s axis isdeviated in one or the other direction with respect to the prescribed direction.Another example is the positioning of a read and write head of a computer disk,in which a motor, on the basis of the position required with respect to the actualone measured, imposes the movement of the head itself (figure 2.16).

With reference to MIMO systems, we consider a robot, like those shown infigures 2.17 and 2.18, where the outputs are the displacement components in thespace of the end effector and possible grasping pressures; the control actions areperformed by a series of suitable actuators positioned in the articulated jointsof the robot’s arms which often receive inputs that are independent from eachother.

A final example of a MIMO system is a motor car whose controller (i.e. thedriver), while driving, avails himself of a series of input magnitudes (steering

lock, accelerator and brake) to control the trajectory and trim of the vehi-cle. In this case, the quantities to be controlled are the 6 degrees of freedomdefining vehicle motion (output magnitudes) which must be compared with thetrajectory plotted by the driver, who, by acting on the control equipment, triesto obtain the desired reference. The control actions performed by the drivercould be performed automatically in a vehicle that has all the necessary sensors(view, sensitivity to trajectory, accelerometers, gyroscopes,...). An expert driver


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2.2. SINGLE INPUT SINGLE OUTPUT (SISO) AND MULTI-INPUT MULTI-OUTPUT (MIMO) SYSTEMS 23

Figure 2.16 Control system for a floppy disk.

Figure 2.17 Robot sketch

(whose driving skills are still difficult to replace with automatic control) whois familiar both with the vehicle and the road, performs a feed forward actioninvolving definition of the trajectory and possible closed loop adjustments.


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Figure 2.18 Robot sketch


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Chapter 3

Classic control

methodologiesThe aim of the previous chapter was to give a broad outline of the problemsinvolved in a control system by preferably making reference to an analysis inthe time domain, more similar to the world of mechanics. On the contrary,as regards linear SISO systems, the techniques used traditionally are in thefrequency or Laplace domain. This chapter is dedicated to these techniques,better known as classic control techniques

The control techniques of linear systems can be divided into two categories:into the frequency domain (or classical control methods) and into state space (ormodern control theories). Classic control envisages a description of the systemby means of the system’s transfer function and its components and prevalentlydeals with SISO systems. SISO systems have a longstanding tradition and, inthis context, as previously mentioned, are analysed in the Laplace domain wherea lot of space is dedicated to closed loop systems with actions proportional tothe error between the desired law of motion and actual response. Hence, in theparagraphs that follow, we will briefly analyse the most important aspects of this classic treatment, doing our best when making comparisons with treatmentsin time and frequency domains, characteristic of the modelling of mechanicalsystems. We must also consider how even a system with several degrees of freedom can be seen and, therefore, analysed as a single input and single outputsystem. Take, for example, the system shown in figure 3.1 in which the aim isto control, by means of a force F k applied to point k, the position of point i ,

that is, zi. From a point of view of control, this system with several degrees of freedom (at most for a continous with infinite degrees of freedom) behaves like aSISO system and, thus, represents an important typology of dynamic systems.

A fundamental issue in engineering sciences consists in predicting the effectsthat specific actions will have on the real physical system under examination.This problem can be solved by means of the development of models capable of predicting the behaviour of the system itself. Hence, even the problem of the

25


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26 CHAPTER 3. CLASSIC CONTROL METHODOLOGIES

F k zi

Figure 3.1 SISO system with several d.o.f.

active control of industrial processes, mechanical systems, automatic plants,robots, etc. is closely linked to modelling problems, i.e. the ability to satis-factorily and efficiently deduce the system’s equations of motion, its structuraldynamics and the equations of the control system in order to ensure the ac-curate design of same. In particular, it is a question of accurately writing theequations that describe:

• the behaviour of the system to be controlled;

• the behaviour of the single elements of the control system (actuators,sensors...);

• the influence of outside disturbances (possibly known and measured);

• the behaviour that one wishes to impose on the process;

• strategy or type of control.

External disturbances, either measurable or unmeasurable, that disturb the

behaviour of the system are classified as disturbances. When writing the equa-tions of motion there are some uncertainties linked to the lack of knowledgeabout active external disturbances and imperfect knowledge of the model. Dif-ferent modellings of the physical mechanical system exist: from models withlumped parameters to models with distributed parameters, from modellingswith only a few degrees of freedom to continuous or discretized (FEM) mod-ellings.

A complete discussion and classification of the models of dynamic systemsis not the aim of this particular chapter.In any case, two different approaches can be highlighted to deal with this prob-lem, by means of:

• models based on state variables (time domain);

• models based on the transfer function (frequency domain).

The most important difference lies in the fact that the former highlight thebehaviour of the system by analysing it from the inside while the latter arelimited to describing the input-output connection.


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3.1. STATE VARIABLES 27

3.1 State variables

The equations of motion of a standard linear time invariant system are usuallywritten in the form:

[M ] z̈ + [R] ż + [K ] z = F z (3.1)

having indicated the matrices of the system’s mass, damping and stiffnessrespectively by means of [M ], [R], [K ] and the vectors that respectively containposition, speed and acceleration of the system’s independent variables by meansof z, z̈, z̈ . With, finally, F z being the vector containing the external forces actingon the system itself including any control actions. If the identity equation isadded to the equation of motion of the system:

[M ] z̈ = − [R] ż − [K ] z + F z

[M ] ż = [M ] ż (3.2)

it is possible to trace the equation of motion (3.1) to a differential equation of the first order known as a state equation :

z̈ż

=

[M ] [0]

[0] [M ]

−1 − [R] − [K ]

[M ] [0]

żz

+

[M ] [0]

[0] [M ]

−1 F z

0

(3.3)

that is:

ẋ =

z̈ż

= [A] x + [B] u (3.4)

having defined state matrix [A] as:

[A] =

[M ] [0][0] [M ]

−1 − [R] − [K ]

[M ] [0]

(3.5)

matrix [B] as:

[B] =

[M ] [0]

[0] [M ]

−1(3.6)

and finally the vector of the state variables and forces respectively as

x =

żz

(3.7)

u = F z0 (3.8)

Expression (3.4) represents the typical form used in automatic control dis-ciplines to write, in the time domain, the equations of a linear time invariantsystem in terms of state variables x. Take, for example, the vibrating systemwith one degree of freedom already analysed in the previous chapter:

mz̈ + rż + kz = f z (t) (3.9)


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3.2. TRANSFER FUNCTION 29

function plays a very important role in dynamic systems modelling and, in par-ticular, in the control theory.

For example, as regards asymptotically stable LTI systems, the harmonictransfer function is easy to measure: from an experimental point of view, it ispossible to excite the system in a harmonic way with an excitation force f (t) =F eiΩt, applied to the system (input) and to evaluate the harmonic responsez(t) = ZeiΩt of the system (output). Thus, the harmonic transfer function iseasily evaluated:

G(iΩ) = Z (iΩ)

F (iΩ) (3.17)

As an alternative, it is possible to resort to the evaluation of the transferfunction by means of the coherence function between input and output. In thefrequency domain, it is possible to define the coherence function between inputf and output z , which represents the index of linearity between the two signalsat a given frequency Ω. The coherence function γ 2xf (Ω) is the ratio of the squareof the absolute value of the cross-spectrual density function to the auto-spectraldensity functions of z and f :

γ 2xf (Ω) = |S zf (Ω)|

2

|S zz (Ω)| |S f f (Ω)| (3.18)

The auto-spectrum density functions S zz and S f f are the Fourier transformsof the auto-correlation functions Rzz and Rf f , which, for variables z(t), presents

itself as:

Rzz (τ ) = limT →∞

1

T

T 0

z(t)z(t + τ )dt (3.19)

The auto-correlation defines the characteristics of a signal in the time domainwhile the corresponding auto-spectral density function in the frequency domain.Furthermore, S zf , which represents the cross-spectral density function, i.e. theFourier transform of the cross-correlation function Rzf , has been introducedinto (3.18):

Rzf (τ ) = limT →∞

1

T

T

0

z(t)f (t + τ )dt (3.20)

The auto-spectral density function can easily be obtained as a product be-tween the spectrum of signal z and its complex conjugate (in the following with∗):

S zz (Ω) = Z ∗(iΩ) · Z (iΩ) (3.21)


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just as the cross-spectral density function proves to be the product of the spec-trum of a signal and of the complex conjugate of the other:

S zf (Ω) = Z ∗(iΩ) · F (iΩ) (3.22)

Coherence is a scalar function that can assume values included between 0and 1 in the entire frequency definition field; it measures the degree of linearcorrelation between two signals in correspondence to each Ω frequency value.Values of lesser coherence of the unit must be associated with:

• the presence of random noise in the input and output signals;

• non linear behaviour of the system under examination;

• presence of unmeasured excitations.

If we now consider the above mentioned system from an input f (t), based onthe assumption that there is no noise or other forms of excitation, the transferfunction of the system can be evaluated by means of relation:

S zz (iΩ) = |G(iΩ)|2 S ff (iΩ) (3.23)

being:|G(iΩ)|2 = G(iΩ)G(iΩ)∗ (3.24)

once the auto-spectral density function are known of system’s input andoutput.

In general excitation force f (t) is not the only one present due to the factthat the system is not insulated and noises and disturbances, that can oftennot be eliminated, are present. Should we wish to obtain the harmonic transferfunction of the system from these measurements, it is possible to take advantageof the fact that the coherence function γ 2xf between input and output providesthe only part of x(t) that is coherent with f(t). The system analysed responds toinput f (t) by constantly modulating and introducing a phase shift at the samefrequency: the output therefore proves to be coherent for the part caused by theforce f and not coherent for the part associated with noise and disturbances.

If the coherence function is high in the entire frequency range considered,the system response z is fully correlated with the input f (t) and the frequencyresponse is:

G(iΩ) = S f z(iΩ)S f f (iΩ)

(3.25)

By recalling the definition of the harmonic transfer function (3.17):

Z (iΩ) = G(iΩ)F (iΩ) (3.26)

and multiplying by th complex conjugate of the force spectrum F ∗(iΩ):


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F

∗

(iΩ)Z (iΩ) = G(iΩ)F

∗

(iΩ)F (iΩ) (3.27)we obtain:

S f z(iΩ) = G(iΩ)S f f (iΩ) (3.28)

and conseguentely the relation (3.25).Let us now consider the system with n degrees of freedom, already introduced

and shown in figure 3.2: this is supported by a state equation which is presentedin the form (3.4) with u = F k and x which defines the system’s state vector.

x =

żz

(3.29)

z =

z1

...zn

(3.30)

F k z5z1 zn

Figure 3.2 System SISO with several degrees of freedom

GjkzjF k

Figure 3.3 Transfer function: representation by means of a block diagram

If we wish to define output zj due to excitation force F k in the frequencydomain, it is possible to calculate (or measure) response zj = Z 0j eiΩt and obtainthe complex transfer function Gjk (iΩ) which represents the term of k − nthcolumn e of j − nth line of the harmonic transfer matrix G(iΩ) which can beobtained from the equation of motion:

[M ] z̈ + [R] ż + [K ] z = F (t) (3.31)

In fact, by imposing a harmonic excitation force F = F 0eiΩt, the steady-statesolution becomes: z = Z 0e

iΩt in which:

Z 0 =− [M ] Ω2 + iΩ [R] + [K ]

−1F 0 = [G(iΩ)] F 0 (3.32)

[G(iΩ)] is a matrix of the T.F. The harmonic transfer function, subsequentlyG(iΩ), is also defined by exciting the system with an impulsive excitation force


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and transforming system response in the frequency domain, according to Fourieranalysis. In fact, a unitary impulse amplitude at time t = a is defined as afunction in time:

δ (t − a) =

0 t = a∞ t = a

(3.33)

By remembering the Fourier transform expression of a signal:

S (iΩ) =

∞ −∞

s(t)e−iΩtdt (3.34)

the transform of a unit impulse at time t = 0 is:

∞ −∞

δ (t)e−iΩtdt = 1 (3.35)

i.e. unitary and constant in terms of frequency. This explains how the transformof the response of a system to impulse, being the product between the transferfunction and the spectrum of the excitation force is, in fact, the transfer function,of system:

G (iΩ) =

∞ −∞

g(t)e−iΩtdt (3.36)

In (3.36), g(t) represents the response to the unitary impulse while G(iΩ) isthe system’s transfer function. We here note that function G(iΩ) is defined onlyon condition that g(t) is limited and capable of being fully integrated. Thus,in other terms, the Fourier transform can only be defined if g (t) represents theresponse of a stable system to an impulse.To remove this limitation, in parallel with the Fourier transform, we use theLaplace transform:

G (s) =

∞ 0

g(t)e−stdt (3.37)

In (3.37), s represents a complex, so-called, Laplace variable, G(s) the trans-fer function (T.F.). according to Laplace and g(t) the impulse response. Theproperties and applications that the Laplace transform offer will be analysed in

detail further on.

From a theoretical point of view, it is possible to pass from a harmonic trans-fer function to that in the Laplace domain by replacing the imaginary variableiΩ with the complex one s. Vice-versa, from a practical experimental point of view, the response to impulse g (t), obtained experimentally, will be defined nu-merically according to the sampling time of the acquisition system. Expression


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(3.36) can, thus, be evaluated by means of numerical integration, thus definingthe harmonic transfer function G (iΩ). In order to obtain a subsequent ana-lytical expression, it will be necessary to define interpolating functions and toidentify the parameters (by defining, for example, the orders of the numeratorand denominator of G(s) in order to subsequently identify poles and zeros).This aspect of the problem or rather identification of parameters is of primeimportance if one wishes the model, expressed in terms of a transfer function,to be representative of reality.

3.2.1 Laplace transform

In regard to a description by means of a transfer function, in the case of linearsystems it is extremely convenient, as previously mentioned, to use the Laplacetransform which, from a methodological point of view, is similar to the Fourier

transform thanks to the substitution of iΩ with s, termed the Laplace domain.Thus, the Laplace transform allows for a transformation from the time domainto the complex domain of variable s, termed the Laplace domain. Let us considera real function x(t) of the real time variable t defined by t > 0. The function of the complex variable s is defined as the Laplace transform:

X (s) =

∞ 0

x(t)e−stdt (3.38)

and is indicated by L[x(t)]. The operation that allows us to associate thecorrespondent X (s) to function x(t) is referred to as the Laplace transform.In other words, the Laplace transform enables us to transform a given time

function to a complex variable function. In particular, given a real function x(t)of the real variable t defined by t > 0, if a real, positive number g exists so that:

limt→∞

x(t)e−gt (3.39)

tends to a finite value, then it exists and function F (s) of the complexvariable s is defined as a Laplace transform.

First and foremost, the Laplace transforms constitute a method to solve thelinear differential equations and, furthermore, this instrument is widely usedin the study of regulation and control systems in that it is fairly convenient tocharacterize the basic properties of linear systems by modelling them in terms of Laplace transform, thus avoidng the need to solve the differential equations that

regulate its behaviour. The Laplace operator is a linear operator and, as such,is possesses all the properties of linear operators. In particular, it is importantto remember that:

• the Laplace transform is a linear transformation so that the followingapplies:

L [A1x1(t) + A2x2(t)] = A1X 1(s) + A2X 2(s) (3.40)


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• the Laplace transform of the time derivative of a function x(t) is given by:

L

dx(t)

dt

= sX (s) − x(0+) (3.41)

where X (s) is L[x(t)] and where x(0+) represents the value of x(t) atinstant t = 0+. By repeatedly applying (3.41) we have:

dnx(t)

dtn = snX (s) −

nk=1

sn−kx(k−1)(0+) (3.42)

where x(k−1)(0+) represents the value of the derivative of order (k − 1)of x(t) with respect to the time calculated at instant 0+. At this point,it is important to remember that in the study of linear control systemsthe dynamic variations of the most important amplitudes are analysedstarting from initial conditions that are often null. In this case, functionsx(k−1)(0+) are null and therefore we have:

dnx(t)

dtn = snX (s) (3.43)

This last property means that, when passing in the Laplace domain, ageneric differential equation becomes algebraic: as will be seen further on,it is possible to work with the transfer functions in an algebraic way.

• the Laplace transform of the time integral of a function x(t) is given by:

L

x(t)dt

= X (s)

s (3.44)

• final value theorem that establishes the limit for t tending to the infiniteof a function x(t) is equal to the limit for s tending to zero of the productof variable s for the Laplace transform of function x(t), i.e.:

limt→∞

x(t) = lims→0

sX (s) (3.45)

• furthermore, the initial value theorem exists and establishes that the limitfor s tending to the infinite of product sX (s) is equal to the limit for ttending to zero of function x(t) of which X (s) is the transform, i.e.:

lims→∞

sX (s) = limt→0

x(t) = x(0+) (3.46)


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• the transform of a function x(t − a) is given by:

L[x(t− a)] = e−asX (s) (3.47)

Similarly, the transform of a function x(t + a) is given by:

L[x(t + a)] = e+asX (s) (3.48)

• with respect to variable s, the derivative of transform X (s) of functionx(t) is linked to function x(t) by:

L[tx(t)] = −dX (s)

ds (3.49)

and, furthermore

L[(−t)nx(t)] = −d

n

X (s)dsn (3.50)

• the integral of transform X (s) of function x(t) is given by:

s 0

X (s) ds = L

f (t)

t

(3.51)

Inverse Laplace transform

A times, in practice, we find ourselves performing an inverse operation to theLaplace transform. This is known as an anti-transform and is defined as:

L−1 [X (s)] = x(t) = 1

2πi

c+iΩ c−iΩ

X (s) estds (3.52)

Where i and c respectively represent the imaginary unit and a real constant.Let us consider, for example, a vibrating system of mass m and stiffness k,

supported by the equation of motion:

md2z(t)

dt2 + kz(t) = f z(t) (3.53)

where z(t) is the function that describes the behaviour of the output variableof the system subjected to a force f z having a constant direction and a variableintensity in time according to law f

z = f

z(t) based on the derivation theorem:

L

m

d2z(t)

dt2

= m

s2Z (s) − sZ (0)−

dz(t)

dt

t=0

(3.54)

we obtain:

ms2 + k

Z (s) − msZ (0)− m

dz(t)

dt

t=0

= F z(s) (3.55)


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limt→∞

z(t) = lims→0

sZ (s) = 0 (3.63)

which means that in the event of an impulse force, the system in questiontends to resume its initial equilibrium conditions. If, on the contrary, the systemwere subjected to a step force of amplitude f z0, the following would pertain:

limt→∞

z(t) = lims→0

sZ (s) = f z0

k (3.64)

which means that the system in question tends towards a displaced positionwith respect to the initial one of an amount equal to f z0/k.

The transfer function

By means of the Laplace transform it is, thus, possible to reduce the system’sdifferential equation of motion to an algebraic equation in the complex variabless. Take, for example, a vibrating system with one degree of freedom:

mz̈ + rż + kz = f z (t) (3.65)

By applying the Laplace transform to the equation of motion and by definingZ (s) = L[z(t)] and F z (s) = L[f z(t)], we obtain:

(ms2 + rs + k)Z (s) = F z(s) (3.66)

from which the system’s transfer function becomes:

G(s) = Z (s)F z (s)

= 1ms2 + rs + k

=

1

ms2 +

r

ms +

k

m

= N D

(3.67)

By now considering a generic case, the T.F. is thus the ratio between twopolynomials in s, where the degree of the numerator is small or equal to thatof the denominator 2. As will be seen further on, the concept of the transferfunction will be very useful to identify the system or a part of it within thecontrol loop.Let us now consider a generic T.F.:

G(s) = bmsm + bm−1sm

−1 + ... + b1s + b0ansn + an−1sn−1 + ... + a1s + a0

= N (s)

D(s) (3.68)

If we solve equation N (s) = 0, we will obtain m real or complex conjugate rootstermed zeroes and subsequently indicated by z1, z2, ..., zm. Similarly

for denominator D(s) one can find n roots indicated by p1, ..., pn, termedpoles . By splitting both the numerator and the denominator into factors, T.F.assumes the following form:

2For the physical system considered in this text, the degree of the numerator is alwayslower than that of the denominator.


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It is possible to even calculate the transfer function of a complex system con-sisting of several components represented by interconnected transfer function.First and foremost, let us consider a system constituted by two series compo-nents or rather a system in which the output of the former coincides with theinput of the latter. Under these conditions, the block representation or ratherrepresentation by means of a transfer function offers the important propertywhereby the overall transfer function coincides with the product of the singletransfer functions: for example, if G1(s) and G2(s) are the transfer functions of two series components (figure 3.5), the transfer function of the complete systemis equal to the product of the two transfer functions:

G(s) = G1(s)G2(s) (3.72)

G1(s) G2(s)

Z (s)F z (s)

Figure 3.5 Series system

Let us assume that the two components constituting the system under ex-amination are parallel, i.e. fed by the same input, whose outputs are summedto contribute to the overall output of the system. Under these conditions, theblock representation (figure 3.6) and the corresponding overall transfer functioncoincides with the sum of the single transfer functions:

G(s) = G1(s) + G2(s) (3.73)

G1(s)

G2(s)

Z (s)F z (s)

Figure 3.6 Parallel system

In conclusion, let us consider a generic system with feedback, i.e. a systemin which the output z (t) influences the input to the system itself.

This feedback can be intrinsic in the system itself or the result of a closed loopcontrol device.In this first analysis, let the transfer function of the system be G(s). The outputof the system, in terms of a signal transformed by means of Laplace, Z (s) isprocessed by means of a transfer function H (s) (this keeps account of possiblegains or phase lags with which the signal once again reaches the system itself)and then subtracted to the input U (s). Finally, the difference U (s)−H (s)Z (s)


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is fed into the system with transfer function G(s).Figure 3.7 represents the system under examination by means of a block dia-gram.

G(s)

H (s)

Z (s)U (s) U ′(s)+

−

Figure 3.7 Feedback system

Thus we have:

Z (s) = G(s)U ′(s) (3.74)

The product of the loop transfer functions are defined as an open loop transfer function : this often keeps account not only of the system under examination butalso of the control and actuating system, proving, in fact, to be the product of the transfer functions of the regulator, of the actuator and of the system underexamination..The ratio between the Laplace transform of output signal Z (s) and the Laplacetransform of the system input U (s) is defined as the closed loop transfer function .Being U ′(s) = U (s) −H (s)Z (s) and Z (s) = G(s)U ′(s), we obtain:

Z (s) = G(s) (U (s) −H (s)Z (s)) (3.75)

from which it is possible to obtain the closed-loop transfer function betweenthe output of the system Z (s) and input U (s):

Z (s)

U (s) =

G(s)

1 + H (s)G(s) (3.76)

3.2.3 The Frequency Response Function (FRF)

As previously mentioned, the frequency response is the steady-state response of the system to a harmonic input, as the frequency varies. In this context, theharmonic transfer function is nothing other than the frequency response to aharmonic input of unit amplitude and zero phase.In order to analyse a system and, in particular a controlled system, the fre-

quency response is capable of providing useful information. For this reason,historically, the first controller analysis methods are based on the study of thefrequency response of the system which, as will be seen further on, is of funda-mental importance even during the synthesis phase.Let us consider a time-invariant, linear dynamic system with transfer functionG(s). Let us recall that G(s) represents the ratio between the Laplace ouputtransform and the Laplace input transform. This function, on account of being


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independent from the type of input signal, is only characteristic of the systemitself and is nothing other than the Laplace transform of the system responseto an impulse.The complex function G(iΩ) of the real variable Ω obtained from transfer func-tion G(s) by substituting s = iΩ is defined as the frequency response.This function is traditionally represented through a pair of Cartesian diagramsin function of Ω or in polar diagrams.In Bode’s diagram , the module and the phase of the complex function G(iΩ) =|G(iΩ)| eiϕ as a function of the real variable Ω are represented in Cartesian di-agrams.Both these quantities can be represented by means of a linear scale as shown infigure 3.8 or by using a logarithmic scale of the frequency axis and indicating,for the ordinates, the value of the function module in decibels (dB) (figure 3.9):

|G(iΩ)|dB = 20 log |G(iΩ)| (3.77)

For example, considering a single-dof linear vibrating system, whose motionis described by the following 2nd order differential equation:

mz̈ + rż + kz = f z (t) (3.78)

and assuming that the system is subjected to a harmonic input force f z =F z0e

iΩt, the steady-state system’s response will be harmonic too, with the samefrequency (z = Z 0eiΩt). The system’s response can be obtained through thefollowing FRF:

Z 0F

z0

= 1

(iΩ)2m + iΩr + k (3.79)

that is the system’s transfer function (3.206), in which we substitute s = jΩ.A second possible representation consists in a polar or Nyquist diagram . The

polar diagram of a harmonic transfer function G(iΩ) is a representation of itsmodule in function of the phase in polar coordinates with the variation of Ωfrom zero to the infinite, i.e. vector locus G(iΩ) in function of Ω (tradition-ally speaking the phase is assumed to be positive in counterclockwise directionstarting from the real axis). In the polar diagram, the projections of G(iΩ) onthe real and imaginary axis thus respectively represent the real and imaginaryparts of the FRF itself. One of the advantages of this representation consists inhaving a single diagram for the whole FRF (figure 3.10).

3.2.4 Bode’s diagram for elementary systemsThe general frequency response G(iΩ) associated with the transfer functionexpressed by 3.70 can be represented in a factorized form as:

G(iΩ) = µΠ (1 + iΩτ i) Π

1 + 2 ξi

ωiiΩ− 1

ω2i

Ω2

(iΩ)nΠ (1 + iΩT i) Π

1 + 2 ξiωi

iΩ− 1ω2i

Ω2 (3.80)


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0 5 10 15 20 25 30 35 400

0.01

0.02

0.03

0.04

0.05

[rad/s]

m o d u l o

0 5 10 15 20 25 30 35 40−4

−3

−2

−1

0

[rad/s]

f a s e [ r a d ]

Figure 3.8 Frequency response of a second order system: linear scale

Both the numerator and denominator polynomialsare decomposed in prod-ucts of elementary binomials and trinomials: the first ones account for real

poles/zeros, while the second ones correspond to pairs of complex conjugatepoles/zeros. The term ( jΩ)n at the denominator polynomial represents bothpoles and zeros at origin.

By expressing the module of T.F. in dB, according to the definition givenby Bode’s diagram, the product of the single terms becomes the summation of the single elementary contributions:

|G(iΩ)|dB = 20 log |µ| − 20n log |iΩ|+

20 log |1 + iΩτ i| −

20 log |1 + iΩT i|+

+

20 log

1 + 2

ξ iωi

jΩ− 1

ω2iΩ2−

20log

1 + 2

ξ iωi

jΩ− 1

ω2iΩ2 (3.81)

and thus, plotting of the diagram can be performed by considering the singleterms separately and then adding them up. For a complete analysis, it is suffi-cient to consider the frequency response module of the terms shown hereinafter,in that the contributions to Bode’s diagram of the module of the factors of G(iΩ)corresponding to the zeroes of G(s) are obtained by means of a simple changeof sign starting from the pole contributions. In fact, for any complex numbers = 0 we have:


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100

101

102

−80

−60

−40

−20

[rad/s]

m o d u l o [ d B ]

Diagramma di Bode

100

101

102

−200

−150

−100

−50

0

[rad/s]

f a s e [ g r a

d i ]

Figure 3.9 Frequency response of a second order system: Bode’s diagram

1s

dB

= −|s|dB

For the gain, this is:

G(iΩ) = µ ⇒

|G(iΩ)|dB = 20 log |µ| (G(iΩ)) = 0◦/180◦ if µ > /


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Figure 3.10 Frequency response of a second order system: Nyquist’s diagram

10−1

100

101

−40

−20

0

20

40

[ d

B ]

10−1

100

101

−180

−90

0

[rad/sec]

[ d e g ]

1/(iΩ)2

1/(iΩ

)

1/(iΩ)

1/(iΩ)2

Figure 3.11 Harmonic transfer function for a simple pole and a multiple pole in the origin.


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G(iΩ) = 1

1 + iΩT ⇒

|G(iΩ)|dB = 20 log 11 + iΩT

(G(iΩ)) =

1

1 + iΩT

(3.84)

The equations 3.84 can be approximated as follows:

G(iΩ) = 1

1 + iΩT ⇒

|G(iΩ)|dB∼=

−20log1 = 0, Ω > 1/T

(G(iΩ)) ∼=

0◦, Ω > 1/T and T > 090◦, Ω >> 1/T and T


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G(iΩ) = 1

1 + j Ω

ωnς −

Ω2

ω2n

⇒

|G(iΩ)|dB = 20 log

1

1 + j Ω

ωnς −

Ω2

ω2n

(G(iΩ)) =

1

1 + j Ω

ωnς −

Ω2

ω2n

(3.86)

and the corresponding approximation:

G(iΩ) = 1

1 + j Ω

ωnς −

Ω2

ω2n

⇒

|G(iΩ)|dB∼=

−20log1 = 0, Ω > ωn

(G(iΩ)) ∼=

0◦, Ω > ωn

(3.87)

The graph of this function depends on the module but not on the dampingsign ς ; in correspondence to the resonance, the graph presents a maximum,termed resonance peak

100

101

102

−80

−60

−40

−20

0

[ d B ]

100

101

102

−180

−90

0

[rad/sec]

[ d e g ]

Figure 3.13 Harmonic transfer function for a pair (coppia) of complex conjugate poles ( ωn = 1)


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3.3. STABILITY ANALYSIS OF DYNAMIC SYSTEM 47

3.3 Stability analysis of dynamic system

It has been explained how the modelling of a mechanical system can be devel-oped both in the time domain as well as in the frequency domain by means of the transfer function. We will now deal with the stability analysis of a systemusing not only the equations in the time domain but also the transfer function inthe Laplace domain that has now been introduced. Starting from the equationsof motion of standard time-invariant linear mechanical system:

[M ] z̈ + [R] ż + [K ] z = F z (3.88)

the stability analysis is traced to the analysis of the roots of the characteristicequation:

det

[M ] λ2 + [R] λ + [K ]

= 0 (3.89)

A system is stable when the roots of equation (3.89) all have real negativeparts.Let us consider the equation of motion written in terms of a state equation :

ẋ =

z̈ż

= [A] x + [B] u (3.90)

in which the state matrix [A] assumes form (3.1), i.e.:

[A] =

[M ] [0]

[0] [M ]

−1 − [R] − [K ]

[M ] [0]

(3.91)

and the vector of the state variables:

x =

żz

(3.92)

The stability analysis is also performed by calculating the homogenous solutionof (3.90) and by imposing, as is known, solution:

x = X 0eλt (3.93)

Thus, the characteristic equation becomes:

det([I ] λ− [A]) = 0 (3.94)

whose solutions λ, which are nothing other than the eigenvalues of the statematrix [A], govern system stability: consequently, a system is stable if all theeigenvalues of the state matrix have negative real part. The above applies to

any time-invariant linear system supported by equation (3.90).

We now wish to demonstrate how the stability analysis can be conductedby means of the study of the matrix of the transfer functions in the Laplacedomain. By applying Laplace transform to the equations of motion of a me-chanical system, written in the state variables domain and assuming zero initialconditions, we obtain:


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L (ẋ) = L ([A] x + [B] u) ⇒ sX (s) = [A] X (s) + [B] U (s) (3.95)

or rather:X (s) = ([I ] s − [A])−1 [B] U (s) (3.96)

As can be noted, matrix [G] = ( [I ] s − [A])−1 [B] defines the matrix of thetransfer function between input U (s) and outputs X (s), where U (s) and X (s)are the two vectors, and represents the link between the Laplace transform of the input variables and the Laplace transform of the state of the system. Thismatrix [G] can be rewritten as:

[G] = ([I ] s − [A])−1 [B] =

1

det ([I ] s − [A]) [Aca] [B] (3.97)

having indicated the matrix of the algebraic complements of matrix ([I ] s − [A])by [Aca].In the case of SISO (single input-single output) systems, it can be noted thatmatrix [G] is nothing else but the transfer function of system G(s) and, in thecase of MIMO, represents the matrix of the transfer functions. All the rationalfunctions in [G(s)] have the same denominator polynomial, which is the samepolynomial of equation (3.94).Therefore, while equation (3.94) represents an equation in λ whose solutions(state matrix eigenvalues) define system stability, the denominator of transferfunctions (3.97) represents a polynomial in s whose solutions (transfer functionpoles) once again define system stability.This having been said, we reach the conclusion that the stability analysis of a time-invariant linear system is reduced to evaluation of the eigenvalues of the

state matrix or, similarly, or of the poles of the transfer function .

3.4 Classical control

In the previous chapters, we dealt with the problem of defining and represent-ing, or modelling, a dynamic system. During the design phase, there is alwaysa need for the system to behave in the desired way and to respond to a specificinstruction (command) in order to carry out the desired operation. In systemmodelling, the pre-set objective is to foresee the input to be provided for adesired output and, vice-versa, to estimate the output once that the input isknown. Due to the uncertainties of modelling and inevitable external distur-bances, the control action f c(t) does not always generate the desired output z(t).

For these reasons, control is necessary. A specific control strategy, as alreadydescribed in the introductory chapter, based on the feedback of output z (t)andon the comparison with the desired value zr(t). The feedback control diagramof a system can be envisaged as constituted by a controller which processes thereference and feedback signal (coming from the sensor), according to a controllogic, and which controls the actuator, i,e. the power organ acting on the sys-tem (figure 3.14). Now, let G(s) be the transfer function inclusive of controller


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3.4. CLASSICAL CONTROL 49

(G1(s)), actuator (G2(s)) and system (G3(s)) and H (s)that of the measurementchain.

ControlloreG1(s)

AttuatoreG2(s)

SistemaG3(s)

SensoreH (s)

+

−

f cucZ r(s) Z (s)

Figure 3.14 SDiagram of a closed loop regulation system

Function G3(s) of the mechanical system under control represents the T.F.between the input and output of a mechanical system with n d.o.f. supported

by the equation of motion:

[M ] z̈ + [R] ż + [K ] z = F z (3.98)

in which z represents the vector of the independent variables and F z thevector that contains the Langrangian components of the forces with respect tovariable z :

δL = F T δsF = F T [ΛF ] δz = F

T z δz (3.99)

where F is the vector of the external forces, δsF that of the physical virtualdisplacements and [ΛF ] the jacobian matrix that links the displacement of theapplication points of the forces to the variables z . For example, in the case of acontrol force we have:

δL = ucδsuc = uc [Λuc ] δz = U T z δz (3.100)

In reality, the actuator, whether hydraulic, electric or pneumatic, is a com-plex component which often, in turn, includes an internal control system (tobe outlined in detail in the chapters that follow). Further on in this text, wewill also make frequent reference to an ideal sensor and actuator, i.e. with aunitary and constant frequency transfer function. Conversely, as regards thecontroller, as anticipated in the previous chapter, this processes the differencebetween the desired law and the state of the system, to achieve an action that isproportional to this error or to its derivative or to its integral. These controllers,termed Proportional Integral Derivatives (PID) are used on an industrial leveland will be described, in detail, in the next section.

First and foremost, let us consider the problem of analysing the stability anddynamic response of a feedback system in the time domain. The equations of motion of the mechanical system with n degrees of freedom excited by a singlecontrol action f c(t), can be written in terms of state variables:

ẋ = [A] x + [B] u(t) (3.101)

control of mechanical systems

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