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2 ANALYSIS OF LINEARONTROL SYSTEMS

2 1 INTRODUCTION

I n this introduc tion we give a brief description of control problems a nd of theconte nts of this chapter.A control system is a dynamic system which as time evolves behaves in acertain prescribed way, generally without human interference. Controltheory deals with the analysis and synthesis of control systems.The essential compon ents of a contro l system (Fig. 2.1) are : 1) theplaizt,which is the system to be controlled; 2) one or more sensors, which giveinformation abou t the plant; and (3) th e controller, the heart of the co nt ro lsystem, which compares the measured values to their desired values andadjusts the inp ut variables to the plant.A n example of a c ontrol system is a self-regulating hom e heating system,which maintains at all times a fairly constant temperature inside the homeeven though the outside temperature may vary considerably. The systemoperates without human intervention, except that the desired temperaturemust b e set . In th is control system the plant is the hom e and the heatingequipment. T he sensor generally consists of a temperature transdu cer insidethe home, sometimes complemented by an outside temperature transducer.The coi~froller s usually combined with the inside temp erature sensor in thetherm ostat, which switches the heating equipm ent o n a nd off as necessary.An oth er example of a control system is a trackin g antenna, w hich witho uthuman aid points at all times at a moving object, for example, a satellite.Here the pla nt is the antenna and the motor tha t drives it . The sensor con-sists of a potentiometer or other transducer which measures the antennadisplacement, possibly augmented with a tachometer for measuring theangular velocity of the antenna shaft. The controller consists of electronicequipment w hich supplies the appropria te in put voltage to th e driving mo tor.Mthough at first glance these two control problems seem different, uponfurther study they have much in comm on. First, i n bo th cases the plant a nd

the c ontroller a re described by difTerential equation s. C onsequ ently, themathematical tool needed to analyze the behavior of the control system in119

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12 Analysis of inear Contrul Syslcms

Fig 2 1 Schemalic rcprcscntation of a conlrol systemboth cases consists of the collection of methods usually referred to as systemtheory. Second , both con trol systems exhibit the feature of feedback that is ,the actual operation o f the c ontro l system is compared to th e desired opera-t ion an d the inpu t to the p lant is adjusted o n the basis of this comparison.Feedback has several attractive properties. Since the actual operation iscontinuously compared to the desired operation, feedback control systemsare able to operate satisfactorily despite adverse conditions, such as dis-t ~ s b a n c e s hat act upon the system, or uariations ill pla~lt roperties. I n ahom e heating system , disturbances are caused by fluctuations in the outsidetemperature and wlnd speed, and variations in plant properties may occurbecause the heating equipment in parts of the home may be connected ordisconnected. I n a tracking anten na disturbances in the f orm of wind gustsact up on the system, an d plan t variations occur because of different frictioncoefficients at different temp eratures.I n this chap ter we introduce c ontro l problems, describe possible solutionsto these problems, analyze those solutions, and present basic design objec-tives. I n the chapters tha t follow, we formu late con trol problems as mathe-matical optimization problems and use the results to synthesize controlsystems.The basic design objectives discussed are stated mainly for time-invariantlinear control systems. Usually, they are developed in terms of frequencydom ain characterist ics, since in t h ~ s oma in the most acute insight can b egained. We also extensively discuss the time domain description of controlsystems via state equations, however, since numerical computations areoften mo re conveniently performed in the time doma in.Th is chapter is organized as follows. In Section 2.2 a general description isgiven of tracking problems, regulator problems, and terminal controlproblems. In Section 2 3 closed-loop controllers are introduced. In the re-maining sections various properties of control systems are discussed, such

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2 2 The Formulation of Control Problems 121as stabilily, steady-state tracking properties, transient tracking properties,effects of disturbances and observation noise, and the influence of plantvariations. Both single-input single-output and mu ltivariable con trol systemsar e considered.

2 2 THE FORMUL TION OF CONTROL PROBLEMS2 2 1 IntroductionIn this section the following two types of con trol problems are introduc ed:1) tracking probleriis and, as special cases, r e g ~ ~ l a t o rrablen7s; and 2)ter~ninal or~tralproblerns.

I n later sections we give detailed descriptions of possible con trol schemesan d discuss a t length bow to analyze these schemes. I n particular, thefollowing topics are emphasized: ro ot mean squ are rms) tracking error, rmsinput, stability, transmission, transient behavior, disturbance suppression,observation noise suppression, and plant p aram eter variation com pensation.23 2 The Formulation of Tracking and Regulator ProblemsW e now describe in general terms a n imp ortant class of control problems-tracliir~gproblems.Given is a system, usually called the plant, which cannotbe altered by the designer, with the following variables associated with itsee Fig. 2.2).

disturbonce vorioblecontrolled vorioble zinput varimble

re fe ren e vorioble ohservotion n o i s er v

Fig 2 2 The plant1 A n illput variable n t ) which influences the plant and which can bemanipulated;2. A distlubarlce variable v , t ) which influences the plant but which cann otbe man ipulated;

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22 Analysis of Linear Control Systems3. An obserued uariable y t) which is measured by means of sensors andwhich is used t o ob tain information a bo ut the state of the plant; this observedvariable is usually contaminated with obseruation noise u,, t);4. controlled uariable z t) which is the variable we wish to con trol;5 reference uarioble r t) which represents the prescribed value of thecontrolled variable z t).T he tracking problem roughly is the following. F o r a given referencevariable, find an appropriate input so tha t the controlled variable tracks thereference variable, that is,

z t) r f) , t to, 2 1where to is the time at which con trol starts. Typically, th e reference variableis no t kno wn in advance. practical constraint is th at the range of valuesover which the input u t) is allowed to vary is limited. Increasing this rangeusually involves replacement of the plant by a larger a nd thu s m ore expen-sive one. A s will be seen, this con straint is of m ajor importance an d preventsus from obtaining systems that track perfectly.In designing tracking systems so as to satisfy the basic requirement 2;1the following aspects must be take n in to account.

1 Th e disturba nce influences the plant in a n unpredictable way.2. Th e plan t param eters may not be know n precisely an d may vary.3. Th e initial state of the plant may n ot be known.4. The observed variable may not directly give information about thestate of the plant a nd m oreover may be contam inated with observation noise.Th e inpu t to th e plant is to be generated by a piece of equipment th at willbe called the controller. We distinguish between two types of controllers:open-loop and closed-loop controllers. O pen-loop co ntrollers generate n t) o nth e basis of past an d present values of the reference variable only seeFig. 2.3), that is,

11 t) ~ o L [ ~ T ) ,o 11 to. 2 2

r e f e r e n c e i n p u t c o n t r o l l e dv r i b l e

Eig 2 3 An open-loop control system,

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2.2 The Eormulntion of ontrol Problems 123

observation n o i s ev m

c o n t r o l l e d

Fig. 2.4. closed-loop control system.Closed-loop controllers take advantage of the information about the plantthat comes with the observed variable; this operalion can be represented by(see Fig. 2.4

No te that neither in 2 2 nor in 2 3 are fu ture values of the reference variableo r the observed variable used in gen erating the inpu t variable since they a reunknown. The plant and the controller will be referred to as the controlSJJStelJlAlready a t this stage we note that closed-loop controllers are m uch m orepow erful than o pen-loop controllers. Closed-loop controllers can accum ulateinformation about the plant during operation and thus are able to collectinformation ab ou t the initial state of the plan t, reduce the effects of the dis-turbance, and compensate for plant parameter uncertainty and variations.Open-loop controllers obviously have no access to any inform ation ab ou t theplan t except fo r what is available before co ntrol starts. The fact that open-loopcontrollers are not afflicted by observation noise since they do not use theobserved variable does no t m ake up fo r this.An important class of tracking problems consists of those problems wherethe reference variable is consta nt over lo ng periods of time. I n such cases it iscustomary t o refer to the reference variable as the se tpo int of the system andto speak of regt~latorproblems ere the main problem usually is to m aintainthe controlled variable a t the set point in spite of disturbances th al act u ponthe system. In this chapter tracking and regulator problems are dealt with

inputv o r i o b l er e f e r e n c ev a r i b L e p l a n t

simultaneously.This section concluded with two examples.

c o n t r o l l e rn r i o b l e z

s nsors o b s e r v e d

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24 nnlysis o Lincnr Control SystemsExample 2 1 A position servo sjutmuI n this example we describe a c ontro l problem tha t is analyzed extensivelylater. Imagine an object moving in a plane. At the origin of the plane is arotating antenna which is supposed to p oint in the direction of the object atall times. T he an ten na is driven by a n electric motor. Th e contro l problem isto comm and the m otor such that

1 ) t to, 2-4where O t) denotes the angular position of the antenna and B, t) the angularposition of the object. W e assume that Or [) s ma de available as a mech anicalangle by manually pointing binoculars in the direction of the object.The plant consists of the antenna and the motor. The disturbance is thetorque exerted by wind on the antenna. The observed variable is the outputof a potentiometer or o ther transducer mounted on the shaft of the anten nagiven by

v t ) o t ) l j t ), 2-5where v t ) is the mea surement noise. In this example the angle O t) is to becontrolled and therefore is the co~ltrolled ariable. T h e reference variable isthe direction of the object Or /) .The input to the plant is tlie input voltageto t l ie m otor pA possible method of forcing the antenna to point toward the object is asfollows. Both the angle of the antenna O t ) and the angle of the object OJt)are converted to electrical variables using potentiometers or other trans-ducers mounted on the shafts of the ante nna an d the binoculars. Then O t) ssubtracted from B, t); the direrence is amplified and serves as the inputvoltage to the motor. As a result when Or / ) B t ) is positive a positiveinput voltage is produced that makes the antenna rotate in a positive direc-tion so that the difference between B, t) a n d B t) is reduced. Figure 2 5 givesa representation of this control scheme.Th is scheme obviously represents a closed-loop controller. An open-loopcon troller would generate the driving voltage p t ) on the basis of the referenceangle O, t) alone. Intuitively we immed iately see th at such a contro ller ha sn o way to com pensate fo r external disturbances such as wind torqu es o rplant parameter variations such as different friction coefficients at differenttem peratu res. As we shall see the closed-loop controller do es offer pro-tection against such phenomena.Th is problem is a typical tracking problem.Example 2 2 A stirred tarllc regulator system

T h e preceding example is relatively simple since th e plan t ha s only a singleinput and a single controlled variable. M~rllivariable ontro l problems wherethk plan t has several inputs an d several controlled variables are usually

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o i t u o t o r

f e e d 2

I

r e f e r e n c e f o rf l o w

f l o w s t r e a ms e n s o rc o n t r o l l e rr e f e r e n c e f o r c o n c e n tr o t io n

s e n s o r

Fig 2 6 he stirred tank cont rol system.

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28 nalysis of Linear Control Syslcms

2.3 CLOSED LOOP CONTROLLERS; THE BASICDESIGN OBJECTIVE

I n this section we present detailed descriptions of the plant an d of closed-loopcontrollers. These descriptions constitute the framework for the discussionof the remainder of this chapter. Furthermore, we define the mean squaretracking error and the mean square input and show how these quantities canbe computed.Throughout this chapter and, indeed, throughout most of this book, itis assumed that the plant can be described as a linear differential systemwith som e of its inputs stochastic processes. Th e slate differential equ ation ofthe system is i t ) A f ) .z t ) B t ) l ~ t ) v , , f ) , 2 6( I , ) 2 .Here x t ) is the state of Ule plant and tc t) the i ~ ~ p u tariable. The initial sro tex is a stochastic variable, and the disturbance variable u s , [ ) is assumed t o bea stochastic process. The observed variable y t) is given by

where the obseruarion r~oise , ,, t) s also assumed to be a stochastic process.The cor~trolled ariable is z t ) D t ) x t ) . 2 8Finally, the reference variable r t) s assumed to be a stochastic process of thesame dimension as the controlled variable z t ) .Th e general closed-loop controller will also be taken to be a linear differen-tial system, with the reference variable r f )and t he observed variable y t ) asinputs, and the plant input ~ t )s output. Th e state differential equation ofthe closed-loop controller will have the fo rm

while Llie output equation of the controller is of the form

Here the index r refers to the reference variable an d the index to feedback.The qua ntity q t ) is the state of the controller. T he initial state q, is either agiven vector or a stochastic variable. Figure 2.7 clarifies the interconnectionof plant an d controller, which is referred lo as the c o~ t f r a lJ , S ~ ~ I I I .f K f t )and H, t ) the closed-loop controller reduces to an open-loop con-troller see Fig. 2 .8 ) . We refer to a control system with a closed-loop

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2 3 Closcd Loop Controllers 131controller as a closed-loop control system, and to a control system with a nopen-loop controller as an open-loop control system .We now define two measures of control system performance that willserve as our m ain tools in evaluating how well a c ontro l system performs itstask:Definition 2 1 The mean sqnare tracking ewor C, t) m d the mean squareinprrt C,, t) are defined as:

Here the tracking error e f ) s giuen by

and W , f) and W,, t), t t o , are giuerl nonnegative-definite syrl~rnerriciaeiglriii~gmatrices.When W J t ) is diagonal as it usually is C, t ) is the weighted sum of themean square tracking errors of each of the components of the controlledvariable. When the error e t ) is a scalar variable an d Wa 1 then 4is the rr m tracking error. Similarly when the input i f / ) s scalar and W 1then JC,, t) is the r im input.O ur aim in designing a control system is t o reduce the m ean square trackingerror C. t) as much as possible. Decreasing C, t ) usually implies increasingthe mean square input C,, t). Since the maximally permissible value of themean s quar e input is determined by the capacity of the pla nt a compromisemust be found between the requirement of a small mean square trackingerror and the need to keep the mean square input down to a reasonablelevel. W e a re thus led to the following statement.Basic Design Objective. In the design o control sjrstems, //re lo ~ve st ossiblemeall square tracking error shorrld be acl~ieuedwithout le tting the m ean squarei r p t exceed its n ~a xim ally ernrissible ualire.I n late r sections we derive from the basic design objective m ore speciEcdesign rules in particu lar fo r time-invariant co ntro l systems.W e now describe how the mean square tracking error C, t ) and the meansquare input C,, t) can be computed. Firs t we use the sta te augm entationtechnique of Section 1 5 4 to obtain the state differential equation of the

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132 nnlysis of Linear Control Systemscontrol system. Combining the various state and output equations we find

2 13

Fo r the tracking error an d the input we write

The computation of C , ( t ) and C , , ( t ) s performed in two stages. First, wedetermine the mean o r deteuiiinisticpart of e ( t ) and u ( t ) , denoted by~ ( t ) E {e ( t ) } , ~ ( t ) E { ~r ( t )} , t 2 to . 2 15

These means are computed by using the augmented state equation 2 13 andthe output relations 2 14 where the stochastic processes ~ ( t ) , , ( t ) , andu,,,(t)ar e replaced with their means, a nd th e initial state is taken as the mea nof col [x(to),4( 0)1.Ne xt we denote by Z t ) , ? ( I ) ,and so on , the variables ~ ( t ) ,( t ) , and so on,with their means Z(t ) , ?(I) , and so o n, subtracted:

. (I) x ( t ) Z ( t ), j ( t ) q ( t ) ( t ) , and so on, t o . 2-16With this notation we write for the mean square tracking error and themean square input

C . ( t ) ~ { a * ( t ) M / , , ( t ) ~ ( t ) }17~( t )W, , ( t ) f i ( t )E{tiT(t)Wu(t)i i( t)}. 2 17Th e terms E { Z Z ( f ) ~ . ( t )Z ( t ) }nd E{tiZ (t)W,,( t ) i i ( t ) } an easily be foun d whenthe variance matrix of col [. (t), q(t)] is known. In order to determine thisvariance matrix, we must model the zero mean parts of r ( t ) , v , ( t ) , andu,,,( t)as o utpu t variables o f linear differential systems driven by white noisesee Section 1.11.4 . Then col [ j . t) , q( t ) ] is augmented with the state of themodels generating the various stochastic processes, an d the variance m atrixof the resulting augmented state can be computed using the differentialequation for the variance matrix of Section 1 11 2 The entire procedure isillustraled in the examples.

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2 3 Closed Loop Controllers 133Example 2.4. Tlre position servo wit11 tliree rlierent conb.ollersW e continue Example 2.1 (Section 2.2.2). Th e motion of the anten na canbe described by the differential equation

Here J i s the mom ent of inertia of all the rotating parts, including the antenna.Furthermore, is the coefficient of viscous friction, ~ ( f )s the torque appliedby the mo to r , and ~ , , ( t ) s the disturbing torque caused by the wind. Them otor torque is assumed to be proportional to p(t ), the inpu t voltage to them otor , so that ~ ( 1 ) I v ( t ) .Defining the sta te variables fl(t) = O(t) and 6 ) t), th e sta te differentialequation of the system is

The controlled variable [ t ) is the angu lar position of the ante nna :

W hen a ppropriate , the following numerical values are used:

Design I Position feedbacli via a zero-order co r~frollerIn a first atlempt to design a control system, we consider the controlscheme outlined in Example 2.1. The only variable measured is the angularposition O(r), so th at we write for th e observed variable

where ~ ( t )s the measurement noise. The controller proposed can be de-scribed by the relationp ( t) v , ( t ) l(t)l, 2 24

where O,(t) is the reference angle an d a gain con sta nt. Figu re 2.9 gives asimplified block diagram of the control scheme. H ere it is seen how a n inp utvoltage to the motor is generated that is proportional to the differencebetween th e reference ang le O,(t) and the observed angu lar position il(f).

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134 Annlysis of Lincnr Control Systemsd i s t u r b i n g torque

Td

r iv ing

observedv o r i b l e

Fig 2 9 Simplified block diagram of a position feedback con trol system via a zero-ordercontroller.The signs are so chosen that a positive value of BJt) t) results in apositive torque u po n the shaft of the antenna. T he question wh at to choose fo r1 is left open for the time being; we return to it in the examples of latersections.Th e state differential equation of the closed-loop system is obtained fro m2-19 223 and 2-24:

We note that the controller 2-24 does not increase the dimension of theclosed-loop system as compared to th e plant since it does no t contain anydynamics. W e refer t o controllers of this type as zero-order coi~trollers.In later examples it is seen how the mean square tracking error and themean square input can be computed when specific models are assumed forthe stochastic processes B, t), ~ , t ) , nd r t ) entering into the closed-loopsystem equation.esign II Position and vel oc it ~~eedback uia a zero-order controllerAs we shall see in considerable detail in later chapters the m ore informa-tion the control system has abo ut the state of the system the better it can b emad e to perform. Let us therefore introduce in addition to th e potentiometertha t measures the angular position a tachometer mou nted on the shaft ofthe an tenna which measures the angu lar velocity. Th us we observe th ecomplete state although contaminated with observation noise of course.W e write for th e observed variable

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2 3 Closed Loop Controllers 35d i s t u r b i n g t o r q u e

I d

0 g Lor oo5 t i on

Fig 2.10. Simplified block diagram of a position and velocity feedback control sy stem viaa zero-order controller.where y t ) = col [ ? l l f ) , / . t )] and where u t )= col [v1 t ) , , t ) ] is theobservation noise.We now suggest the following simple contro l scheme (see Fig. 2.10 :

W = w x t ) M I p ~ b t ) . 2-27This time the m otor receives as input a voltage tha t is not only proportionalto the tracking error O , t) Q t ) but which also contains a contribution pro-portional to the angular velocity d /). This serves the following purpose.Let us assume t hat a t a given instant B, t) B t) is positive, and that d t ) sand large. This means th at the antenn a moves in the right directionbut with great speed. Therefore it is probably advisable not to continuedriving the anten na hut to start decelerating and thus avoid overshootingthe desired position. W hen is correctly chosen, the scheme 2-27 canaccomplish this, in co ntrast to th e scheme 2-24. We see later tha t the presentscheme can achieve much better performance than that of Design I.esign 111 Positiortfeedbock via afirst-order controllerIn this design app roac h it is assumed , as in Design I that only the angularposition B t) is measured. If the observation did not contain any noise, wecould use a differentiator t o obtain 8 t ) from O / ) an d continue as in Design

11 Since observation noise is always present, however, we cannot dilferen-tiate since this greatly increases the noise level. We therefore attempt touse an approximate dilferentialor (see Fig. 2.1 I ) , which has the property offiltering the noise to som e extent. Such an approxim ate differentiator canbe realized as a system with transfer function

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136 Analysis of Linear ontrol Syrlemsd i r tu i i n g torque

Fig.2.11. Simplified blockdingrum olnpositionfeedbuckcontrol systemusinga first-ordercontroller.where T is a (small) positive time constant. The larger T is the less accuratethe differ entiato r is, bu t th e less the noise is amplified.The inpu t to the plan t can now be represented as

where ?I( ) is the observed angu lar position as in 2 23 an d where S(t) is the"approxim ate derivative," th at is, a( ) satisfies the differential equat ion

This time the controller is dynamic, of order one. Again, we defer to latersections the detailed analysis of the performance of this control system;this leads to a proper choice of the time constant T an d the gains r and pA s we shall see, the performa nce of this design is in between those of Designand Design 11; better performance can be achieved than with Design Ialth ou gl~ ot as good as with Design 11.

2.4 T H E S T A B IL I T Y O F C O N T R O L S Y S T E M SIn the preceding section we introduced the control system performancemeasures C, f) nd C,,(t). Since generally we expect tha t the c ontrol systemwill operate over long periods of time, the least we require is that both

C ) and C,(t) remain b oun ded as t increases. This leads us directly to aninvestigation of the stability of the control system.If the control system is not stable, sooner or later some variables willstart to grow indefinitely, which is of course unacceptable in any controlsystem that operates for some length of time (i.e., during a period largerthan the time constant of the growing exponential). If the control system is

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2.4 The Stability of ontrol Systems 137unstable, usually CJt) o r C,, t), or both , will also grow indefmitely. We thu sarrive a t the following design objective.Design Objective 2.1. The co i~ t ro l s~~s temlrould be asj ~i np fo tic al ~table.Under the assumption that the control system is time-invariant, DesignObjective 2.1 is equivalent to the requirem ent tha t all characteristic values ofthe augmented system 2-13 th at is, the characteristic values of the matrix

have strictly negative real parts. By referring back to Section 1.5.4, Theorem1.21, the characteristic polynomial of 2-31 can be w ritten asdet sI A) det sI -L det [ I H s)G s)], 2-32where we have denoted by H s) = C s I -A)-lBthe transfer matrix of the plant from the input u to b e the observed variable?I nd by G s) =F s I )- K, Hf 2-34the transfer matrix o f the controller from I to -11.One of the functions of the controller is to move the poles of the plantto better locations in the left-hand complex plane so as to achieve an im-proved system performance. If the plant by itself is unstable, sfabilizing thesystem by moving the closed-loop poles to proper locations in the left-halfcomplex plan e is the mait1 function of the controller see Example 2.6 .Exnmple 2.5. Position servoLet us analyze the stability of the zero-order position feedback controlsystem proposed for the antenna drive system of Example 2.4, Design ITh e plan t transfer function the transfer function from the driving voltageto the a nten na position) is given by

IH s) = s U) The con troller transfer function is

Thu s by 2-32 the closed-loop poles are the roots of

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138 Analysis of Linenr Control Systems

Fig. 2.12. Roo t loci for posilion servo . Solid lines, loci for second-order system; dashedlines, modifications of loci due to the presence of the pol nt 10 s- .Fig ure 2.12 shows th e loci of the closed-loop poles with A as a parameter forthe numerical values 2 22.I t is seen tha t ideally th e con trol system is stable fo r aU positive values ofA . In practice however the system becomes unstable fo r large A. The reasonis tha t am ong other things we have neglected the electrical time constan tT of the motor. T akin g this into account the transfer function of mo to rplus antenna is

A s a result the closed-loop characteristic polynom ial is

Figure 2.12 shows the modification of the root loci that results forF o r A where

the closed-loop system is unstable. I n the present case A 85.3 Vlrad.

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2 4 The Stnbility of Control Systems 39Example 2.6. The stabilization of the i~ ~ u e r t e d p e r t ~ r l ~ i~ ~ zAs an example of an unstable plant, we consider the inverted pendulum ofExample 1.1 (Section 1.2.3). In Example 1.16 (Section 1.5.4), we saw that byfeeding back the angle' (f) via a zero-order controller of the formit is not possible to stabilize the system for any value of the gain A. I t ispossible, however, to stabilize the system by feeding back the completestate x(t) as follows p (t ) -Icx(t). 2-43Here ic is a c onstan t row vector t o be determined. W e note th at implementa-tion of this controller requires measurement of all four state variables.

I n Exam ple 1.1 we gave the linearized state differential equ ation o f thesystem, w hich is of the form

where b is a colum n vector. Substitution of 2-43 yields

The stability of this system is determined by the characteristic values of thematrix A k. In Chapter 3 we discuss methods for determining optinralcontrollers of the for m 2-4 3 th at stabilize the system. By using those m ethod s,an d using the num erical values of Exam ple 1.1 i t can be foun d, for example,tha tk (86.81 , 12.21, -118.4, -33.44) 2-47

stabilizes the linearized system. With this value for li, the closed-loopcharacteristic values are -4.706 l.3 82 an d -1.902 3.420.T o determine the stability of the actu al (non linear) closed-loop system, weconsider the nonlinear state differential equation

y 1 [ os L 1~

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140 Analysis of Linear Control Systemswhere the definitions of the components h t and 5 are the same asfo r the linearized equations. Sub stitution of the expression 2-43 for p( t ) into2-48 yields the closed-loop state differential equation. Figure 2.13 gives theclosed-loop response of the angle $ ( t ) for differen t initial values (O ) whileall other initial conditions are zero. F o r $(O) 10' the motion is indistin-guishable from the motion that would be found for the linearized system.F o r ( O ) 20 som e deviations occur, while fo r $(0) 30 the system is nolonger stabilized by 2-47.

Rig. 2.13. hc behavior of t he angle $ ( t ) for the stabilized inverled pendulum: a)#(O)10'; b) $(O) 20 ; c) #(O) 30 .This example also illustrates Theorem 1.16 Section 1.4.4), where it isstated that when a linearized system is asymptotically stable the nonlinear

system fro m w hich i t is derived is also asymptotically stable. W e see th at inthe present case the range over which linearization gives useful results isquite large.2.5 T H E S T E A D Y S T A T E A N A L Y S I S OF T H ET R A C K I N G P R O P E R T I E S2.5.1 The Steady State Mean Square Tracking Error and Inputn Section 2 3 we introduced the mean squa re trackin g error C nd the meansquare input C . Fr om the control system equations 2-13 and 2-14 i t can beseen that all three processes r (t ) , u,,(t), and v,,(t), that is, the reference

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2 5 Steady State Trucking Properties 141variable, the disturbance variable, and the observation noise, have an effecton C and C . Fro m now until the end of the chapter, we assume tha t r t ) ,v , t ) , and v,,, t)are statistically inlcorrelated stochastic processes so th at theircontributions to C and C can be investigated separately. In the presentand the following section, we consider the contribution of the referencevariahle r t ) to C , t ) and C , t ) alone. The effect of the disturbance and theobservation noise are investigated in later sections.W e divide the du ration of a control process into two periods: the transientand the steadjt-state period. These two periods can be characterized asfollows. T he transient perio d starts a t the beginning of the process an d ter-minates when the quantities we are interested in usually the mean sq uaretracking erro r an d input) approxim ately reach their steady-state values. Fro mtha t time o n we say that t he process is in its steady-state period. W e assume,of course, that the quantities of interest converge to a certain limit as timeincreases. The duration of the transient period will be referred to as thesettling lime.In thk design of control systems, we must take into account the perfor-mance of the system during both the transient period and the steady-stateperiod. The present section is devoted to the analysis of the steady-stateproperties of tracking systems. In the next section the transient analysis isdiscussed. I n this section and the next, the following assumptions a re made.

I Design Objective 2.1 is sotis fed , that is, the corftroi system is asjm p-totically stable;2 . The control sjutenl is tinle-inuariant and t l ~ e seighting natrices WeandW are constant;

3 . The disturbance v , t ) arld the observation noise u,,, t)are identical to zero;4 . The reference variable r t ) car1 be represerlted asr 0 r , t J t ) , 2 49

wlme r, is a stochastic vector and r, t) is o zero-1nea11wide-sense stationarj~vector stocliasticprocess, zu ~correlatedn~ itfi,.Here the stochastic vector r , is the constant part of the reference variablean d is in fact the set point for the controlled variable. The zero-mean processr , t ) is the variablepart of the reference variable. We assume t ha t th e second-order mom ent matrix of r , is given by

E { r , rU T}= R 2-50while the variable part r , t ) will be assumed to have the power spectraldensity matrix Z, w).Under the assumptions stated the mean square tracking error and themean square input converge to constant values as t increases. We thus d e h e

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142 Analysis o Linear Control Systemsthe steady-state mean square traclcing error

C,, lim C.(t), 2 51i m

and the steadjwtate mean square inputC = im C,,(t). 2 52

t m

In order to compute C,, and C let us denote by T ( s ) he trarts~nission fth e closed-loop control system tha t is the transfer matrix from the referencevariable r to the controlled variable a. We furthermore denote by N ( s ) thetransfer matrix of the closed-loop system from the reference variable r t othe inpu t variable 11.In order to derive expressions for the steady-state mean square trackingerror and input we consider the contributions of the constant pa rt r , andthe variable part r,(t) of the reference variable separately. The c onsta nt pa rtof the reference variable yields a steady-state response of the controlledvariable and input as follows

lim z( t ) T(O)r, 2 53I mandlim tr(t)= N(O)r,,

I ,respectively. The corresponding contributions to the steady-state squaretracking error and input are[T(O)r, , ]TWo[T(0)r , ? ,] t r {ro ra TIT (0 ) ] ~ w J T ( O ) ] } 2 55and

[N(0)r,]TW,[N(O)r,l = r [r,rOTNZ (0)W,,N(O)]. 2 56I t follows that the contributions of the constarlt pa rt of the reference variableto the steady-state mean square tracking error and input respectively are

t r {R,[T(O) ITW6[T(O) ] and t r [R,NT(0)W,,N(O)]. 2 57Th e contributions of the variable pa rt of the reference variable t o th e steady-state mean square tracking error and input are easily found by using theresults of Section 1 10 4 and Section 1.10.3. The steady-state mean squaretracking error turns out to beC = r R ,[T (O ) ] T W J T ( 0 ) ]

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2 5 Steady State Tracking Prapcrties 143while the steady-state mean square in pu t is

These form ulas are th e s tarting po int fo r deriving specific design objectives.I n th e next subsec tion we confine ourselves to th e single-input single-outputcase, where both the input v and the controlled variable z are scalar andwhere the interpretation of the formulas 2-58 and 2-59 is straightforward. InSection 2 5 3 we turn t o the m ore general multiinput multioutput case.In conclusion we obtain expressions for T s) and N s) in terms of thevarious transfer matrices of the plant a nd the controller. Let us denote thetransfe r matrix of the pla nt 2-6-2-8 now assumed to be time-invariant)from the input t to the controlled variable z by K s) and th at from the inputt o the observed variable y by H s). Also, let us den ote the transfer m atrixof the con troller 2-9, 2-10 also time-invariant) from the reference variab ler to t l by P s), and from the plan t observed variable y t o u by G s). Thuswe have:

K s) = D ( d A)- B, H s) C s1- A)-lB, 2-60P s) =F s1- L)- K, H,, G s) F s1 - L)-IK, H .Th e block diagram o f Fig. 2.14 gives the relations between the several systemvariables in terms of transfer matrices. From this diagram we see that, if

IIc l o s e d l o o p c n t ro l l e rL _I

Fig 2 14 Th e transfer matrix block diagram of a linear time-invariant closed-loop controlsystem.

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44 Analysis of Linear Control Systemsr t )has a Laplace transform R s ) , in terms of Laplace transforms the severalvariables are related by U s ) P s ) R s ) G s ) Y s ) ,

Y s ) H s ) ~ s ) , 2 61Z s )= K s )U s ) .

Eliminalion of the appropriale variables yields

T s ) and N s ) are of course related by

2.5.2 The Single-Input Single-Output CaseIn this section it is assumed tha t both the inpu t 11 and the controlled variablez and therefore also the reference variable r , are scalar variables. Withoutloss of generality we take both W and IV = 1. A s a result the steady-state mea n square tracking error and the steady-state mean square in put canhe expressed as

F ro m the first of these expressions we see th at since we wish to designtracking systems with a small steady-state mean square tracking error thefollowing advice mus t be given.Design Objective 2.2. In order to obtain a sriloll steady-state ri~ean p o r efroclcing error, tlie tror~sii~issiori s ) of a time-inuoriont li~ tea r ontrol sy st emsl~o~tlr le designed such that

Z, fu) T j w ) l 2 66is s~i iol lfo r ll real w . 111porticulor, 114en iorizero sefpo iri ts ore liliely to occur,T 0 ) lrould be iiiaclr close to 1.

The remark about T 0 ) can be clarified as follows. I n certain app licationsit is importanl that the set point of the control system be maintained veryaccurately. In particular this is the case in regulator problems where the

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2 5 Steady-State Tracking Properties 145variable part of the reference variable is altogether absent. In such a caseit may be necessary that T 0)very precisely equa l 1We now examine the contributions to the integral in 2-65a from variousfrequency regions. Typically, as w increases, (w) decreases to zero. Itthus follows from 2 4 that it is sufficient to make I T ( j w ) small forthose frequencies where S , ( w ) assumes significant values.In order to emphasize these remarks, we introduce two notions: thefreqrreltcy band o f th e corttrol systeiit and the freqaencj~band of tlre referencevariable. The frequency band of the control system is roughly the range offrequencies over which T ( j w ) s close to :Definition 2 2 Let T s ) be the scalar iransiitissiorr of on asyittptotically stabletiitie-iiluoriant liltear control system with scalar i t p t and scalar controlleduariable. Tlrerl thefieqrrency band of the co~ltrol ystein is d e fi ~ ~ e ds the seto f f r e q ~ ~ e n c i e s, w 0 for idtich

ivlrere E is a given nuiiiber that is small with respect to 1 I f the reql~eitcybandis an iittervol [w, , w,] , we coll w: w , the bandwidth of the coittrol systeitt. fthe freyueitcy ba nd is an iriterual [O w ] ise refer to w , as th e crrtoff fi.eqrrcncyof the system.Figure 2.15 illustrates the notions of frequency band , b andw idth, and cutofffrequency.

bondwidth o fh e c on t r o l sy s t e mFig 2 15 Illustration of the definition of the frequency band, bandwidth, and cutofffrequency of a single-input single-output time-invariant control system. It is assumed thatT j w ) - O a s w - a .

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46 Analysis of Linear ontrol SystemsIn this book we usually deal with loivpass transmissions where the fre-quency ba nd is the interval from the zero frequency to th e cutoff frequencyw,. Th e precise v a p e of the cutoff frequency is of course very muc h depen dent

upon the number E. When E = 0.01, we refer to w , as the cutofffreq~fefzcy.W e use a similar terminology f or different values of E . Frequently, how ever,we find it convenient to speak of the break fregl~erzcyof the c ontro l system,which we define as tha t corner frequency where the asym ptotic Bode p lot ofIT jw)l breaks away from unity. Th us the break frequency of the first-ordertransmission

is a while the break frequency of the second-order transmission

is w,. N ote , however, tha t in bo th cases the cutoff frequency is considerablysmaller than the break frequency, dependent upon E and, in tile second-ordercase, dependent upon the relative damping . Tab le 2.1 lists th e 1 an d1 0 cut-off frequencies for various cases.Table 2 1 Relation between Break Frequency and Cutoff Frequency for F irst- andSecond-Order Scalar Transmissions

Second-order systemwith break frequency O,First-order systemwith break frequency a 5 0.4 = 0.707 5 = 1.51 cutoff freq. 0.01~ 0.0120, 0.00710~ 0.00330,

10 cutoR freq. 0 . 1 ~ 0.120, 0.0710, 0.0330,

Next we define the frequency hand of the reference variable, which isthe range of frequencies over which X, w) is significantly different fro m zero:Definition 2.3 . Let r be a scalor wide-sense stationafy stochastic process isitlzpower spectral defzsi t j~f i~nct io~~, w ). Tlzefi.eqrfency band of r t ) is defitzeda s t l ~ eet of freqfrefzcies w , , for wl~iclf

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2.5 Stendy-Stnlc Tracking Properties 147Here a. is so clfo sen flrat tire frequency band contains a given fractionwhere E is s~l ia ll vitlr respect to 1, of hal fof thepower of theprocess that is

df = 1 &m b df 2 71

I f the freqrrer~cyband is an interval [w w, ] , we defiirle w , w , as the band-ividtb of tlieproce ss. Iftlrefre qrre ~ic y and is an interual [O w , ] , ive refer to w ,as the cutofffi.cqrrency of the process.Figure 2.16 illustrates th e notions of frequency ba nd , ban dw idth, an d cutofffrequency of a stochastic process.

Is t o c h a s t i c p r o c e s s

Rig. 2.16. Illustration of the definition of the frequency band bandwidlh and cutofffrequency of scalar stodm stic process r

Usually we deal with low-pass-type stochastic processes that have aninterval of the form [0, w,] as a frequency band. The precise value of thecutoff frequency is of course very much dependenl upon the value of E .When E = 0.01, we speak of the 1 cr~tofffrequencywhich means that theinterval [0, w J contains 99 of haK the power of the process. A similartermino logy is used for othe r values of E Often, how ever, we find it convenientto speak of the breakfrequency of the process, which we d e h e as the cornerfrequency where the asymptotic Bode plot of &(w) breaks away from itslow-frequency asymptote, that is, from X, O). Let us take as an exampleexponentially correlated noise with rms value a and time constant 0 This

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148 Analysis of Linear Control Systemsprocess has the power spectral density function

so that its break frequency is 110. Since this power spectral density Cunctiondecreases very slowly with w he 1 and 1 0 cutoff frequencies are mu chlarger than 110; in fact, they are 63.6610 and 6.31410, respectively.Let us now reconsider the integral in 2 65a. Using the notions just intro-duced, we see that the main contribution to this integral comes from thosefrequencies which are in the frequency band of the reference variable butnot in the frequency band of the system (see Fig. 2.17). We thus rephraseDesign Objective 2 2 as follows.

I I-I f r e q u e n c y b o n d ofcontrol s y s t e m I

f r e q u e n c y b o n d ofr e f e r e n c e

f requency .range that 15 responsible fo r theg r e o t e r p o r t o f the meon squore t r a c k i n g errorFig 2 17 Illustration of Design Objective 2 2 A

Design Objective 2 2A iz order to obtain a small stead jwta te ntean squaretrackirg error the frequertcj~baud of t l ~ eontrol sj~ ste m liauld co~rtairl s rzluchas possible of tlre freq~rerzcybond of the uariable part of the referr~zceuorioble.If ~l on ze ro et points are Iilcely to occur T(0) sliould be mode close to 1An important aspect of this design rule is that it is also useful when verylittle is known about the reference variable except for a rough idea of itsfrequency ba nd.

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2 5 Stendy State Tracking Properties 49Le t us now consider the second aspec t of the design-the steady-statemean square input. A consideration of 2 65b leads us to fo rmu late our nextdesign objective.

Design Objective 2 3 ii order to obtaiu a sriioN stear/y-state mean squareii~prrt in an asy ~i~ pto tico llytable sin gl e-i rp t sirz gle -oo p~ t fi171eiiiuaria11tlinear control sjuteii7,

should be made sriiaN for all real w . This can be achieued by n ~ a k i i ~ gN jw)Imtflcientty sniall over tlrefreqoencjr baiid of the reference variable.I t should be noted that this objective does no t contain the advice to keepN 0 ) small, such as would follow from considering the first term of 2 65b.This term represents the contribution of the constant part of the referencevariable, that is, the set point, to the input. The set point determines thedesired level of the controlled variable and therefore also that of the input.It must be assumed th at th e plant is so designed th at it is capable of sustainingthis level. T he second term in 2 65b is impo rtant for the dynamic range of theinput, that is, the variations in the input about the set point that are per-missible. Since this dynam ic range is restricted, the m agnitude of the seco ndterm in 2 65b m ust be limited.

It is not difficult to design a control system so th at one of the DesignObjectives 2.2A or 2 3 is completely satisfied. Since T s ) and N s ) are relatedby T s ) K s ) N s ) , 2 74however, the design of T s ) affects N s ) , an d vice-versa. We elabo rate a littleon this point a nd show how Objectives 2 2 and 2.3 may conflict. The plantfrequency response function IK jw)I usually decreases beyond a certainfrequency, say w,. If lT jw)I is to stay close to 1 beyond this frequency, itis seen from 2 74 tha t IN jw)l must iiicrease beyond o , . The fact thatIT jw)l is not allowed to decrease beyond w , implies that the referencevariable frequency band extends beyond a, , . As a result, IN jw)I will belarge over a frequency range where Z , w ) is not small, which may mean animportant contribution to the mean square input. If this results in over-loading the plant, either the bandw idth of the control system m ust be reducedat the expense of a larger tracking error), o r the plant m ust be replaced by amore powerful one.The designer must h d a technically sound compromise between therequirements of a small mean square tracking error a nd a mean sq uare inputthat matches the dynamic range of the plant. This compromise should bebased on the specifications of the control system such as the maximal

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15 Annlysis of Linear Cantrol Systemsallowable rms tracking error or the maximal power of the plant. In laterchapters, where we are concerned with the synthesis problem, optimalcompromises t o this dilemma are found.

At this point a brief comment on computational aspects is in order. InSection 2.3 we outlined how time do main m ethods can be used to calculatethe mean sq uare tracking error and mean square input. In the lime-invariantcase, th e integral expressions 2-65a and 2-6513 offer an alternative computa-tional approach. Explicit solutions of the resulting integrals have beentabulated fo r low-order cases (see, e.g., N ewto n, Go uld, an d Kaise r 1957),Appendix E; Seifert and Steeg (1960), Appendix). F o r num erical computa-tions we usually prefer the time-domain approach, however, since this isbetter suited for digital computation. Nevertheless, the frequency domainexpressions as given are extremely importan t since they allow us to form ulatedesign objectives that can no t be easily seen, if at all, fro m the time d om ainapproach.Example 2.7. The track ingproper ties of the position servoL et us consider the position servo problem of Examples 2.1 (Section 2.2.2)and 2.4 (Section 2.3), and let us assume that the reference variable is ade-quately represented as zero-mean exponentially correlated noise with rmsvalue and time constan t T . W e use the num erical values

rad,T,= 10s.I t follows from the value of the time con stant an d from 2-72 th at th e referencevariable break frequency is 0.1 rad /s, its 1 0 cutoff frequency 0.63 rad /s,and its 1 cut off frequency 6.4 rad/s.esign I Let us first consider Design I of Example 2.4 where zero-orderfeedback of the position has been assumed. I t is easily found th at the trans-mission T(s) and the transfe r function N s ) are given by

We rewrite the transmission as

where

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152 Analysis of Linear anhol Systcmsof the control system and the reference variable are from 2-19 2-24 and2-80:

With this equation as a starting point, it is easy to set up and solve theLyapunov equation fo r the steady-state variance matrix of the augmen tedsta te col [[,(t), C2(t), B,(t)] (Theorem 1.53, Sec tion 1.11.3). Th e result is

A s a result, we o btain for the steady-state m ean sq uare tracking erro r:C,, = im E{[e(t) ,(r)ln} ql, qI3

t m

1 0 2-83K T~

Trwhere the are the entries of Q plot of the steady-state rms tracking e rrorI.is given in F lg. 2.19. We no te th at increasing r beyond 15-25 V/rad decreasesthe rms tracking error only very little. Th e fact tha t C,, does no t decrease tozero as - is attributa ble to th e peaking effect in the transmission w hichbecomes more an d more pronounced a s J becomes larger.The steady-state rms in pu t voltage can be foun d to b e given by

C,, = E{ t)} E{P[O(t) O,(t)l3}= PC,,. 2-84

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2.5 Steady Stnte Tracking roperties 153

Fig 2.19. Rms racking error and rms input vollage as functions of the gain ? for theposition servo Design I

Figure 2.19 shows tha t according to what one would intuitively feel the rmsinput keeps increasing with the gain A Comparing the behavior of the rmstracking error an d the rm s inp ut voltage confirms the opinion that there isvery little point in increasing th e gain beyond 15-25 V /ra d since the increasein rms in pu t voltage does not result in any appreciable redu ction in the rm strackin g error. We observe however th at the resulting design is no t verygoo d since the rms tracking error achieved is ab ou t 0.2 tad which is no tvery small as compared t o the rms value of th e reference variable of 1 rad.Design 11 Th e second design suggested in Ex ample 2 4 gives bet ter resultssince in th is case the tachometer feedback gain facto r p can be so chosen tha tthe closed-loop system is well-damped for eac h desired bandw idth whicheliminates the peaking effect. In this design we find for thd transmission

,

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54 nnlysis of incnr Control Systemsthe undamped natural frequency of the system is

0 J iand the relative damp ing

Th e break frequency of the system is w which can be mad e arbitrarily largeby choosing large eno ugh . By choosing p such that the relative dampingis in the neighborhood of 0.7 the cu tof ffrequency of the con trol system canbe m ade correspondingly large. Th e steady-state rms tracking e rror is

while the steady-state me an sq uare inp ut voltage is given by

C can be made arbitrarily small by choosing an d p large enough. For agiven rm s input voltage it is possible to achieve an rm s tracking error th atis less than for Design I The problem of how t o choose the gains ? and psuch that for a given rms input a minimal rms tracking error is obtainedis a mathematical optimization problem.In Chapter 3 we see how this optimization problem can be solved. Atpres ent we confine ourselves to an intuitive argum ent as fouows. Le t us sup-pose th at fo r each value of he tachometer gain p is so chosen tha t the rel-ative dam ping is 0.7. Let us furth erm ore suppose tha t it is given th atthe steady-state rms input voltage should not exceed 3 V. Then by trialand error it can be found using the formulas 2-88 and 2-89 tha t for

500 V/rad p 0.06 s 2-90the steady -state rms tracking er ror is 0.1031 rad while the steady-state rmsinput voltage is 30.64 V. These values of the gain yield a near-minimal rmstracking error fo r the given rms input. W e observe th at this design is betterthan Design I where we achieved an rms tracking error o f ab ou t 0.2 rad .Still Design I1 is not very g ood since the rms tracking erro r of 0.1 rad is no tvery small as compared to the rms value of the reference variable of 1 rad.

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156 nalysis o Linear Control Systems2 . Tlte constant part of the reference variable is a stoclrastic uariable iviflttalcorrelated conrporIeirts, so that its second-order nzoiitent matrix cart beexpressed as

R = diag (R,,,, ROac R , 1 2-94Fro m a practical point of view these assump tions are no t very restrictive.By using 2-93 and 2-94 it is easily found th at the steady-state mean squa retracking error can be expressed as

/ n ~ . j ( w ) { [ ~ ( - j w ) IT W .[ T( jw ) ] }. d f, 2-95i m

where{ [ T ( - j w ) I T W J T ( j w ) ll . 2-96

denotes the i-thdiagonal element of the matrix IT ( - jo ) ] T W o [ ~ ( j ~ )]Let us now consider one of the te rn s on the right-hand side of 2-95:

This expression describes the contribution of the i-th component of thereference variable to the tracking error as transmitted through the system.I t is therefore appropriate to introduce the following notion.Definition 2.4. Let T ( s ) be the nt x fransntission of an asyntptotical[ystable tirite-inuariar~tinear control system. Then we define flte requency bandof the i-tlt l nk of the cortfrol system as the set of freq~reirciesw , w 0 orwlticl~

{ [T (- jw ) ITW .[T(jw) W ~ , ~ ~ 2-98Here E is a given nuniber ivlticl~ s sinall ivitlt respect to 1 W s the weiglrfingma trix for the nrean square tracking error, and W0 ; enotes the i-th diagonalelenrent of We.Once the frequency band of the i-th link is established we can of coursedefine the bandisidtlr and the cutofffreqtrei~cyof the i-th link if they exist asin Definition 2.2. I t is noted tha t Definition 2.4 also holds for nondiagonalweighting matrices W a .The reason th at the magnitude of

is compared to W s , i j s that it is reasonable to compare the contribution2-97 of the i-th component of the reference variable to the mean squaretracking error to its contribution when no control is present tha t is when

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2 5 Stcndy Stnte Tracking Propertics 57T s )= 0 . This latt er contribution is given by

2 99We refer to the normalized function { [ T - j w ) T W a [ ~ j w ) } i i /W c , i ias the d~er ence f r nz c t io~zf the i-th link. In the single-input single-outputcase this func tion is ) T j w )- 1) .W e a re now in a position to extend ~ e s i ~ nbjective 2.2A as follows.Design Objective 2.2B. Le t T s ) be tile nl x nz transnzissiarz of an asjmzp-totically stable time-iczvaria~ztirzear control sjtstenz for i~~ lzicl toth the constatltpart and the uariable part of the reference uariable have zcncorrel~ tedconz-ponents. Th en in order to obtain a s~jzall stead^ state mean sylcare trackingerror, the freqtiency band o f each o f ,>he 11 links shorrld contain as nztrch aspossible o f t h e reqrrencj~band of the carresponditzg component of the referenceuariable. f the i-111 conlponelzt, i = 1 , 2 , - . z, of the reference variable islikebr to have a nonzero setpo int , {[T O ) jT W,[T O)- llii sl~otdd e modesmall as compared to W a,{ +As an am endment to this rule we observe that if the contribution to C ofone particular term in the expression 2 95 is much larger than those of theremaining terms then the advice of the objective should be applied moreseverely to the co rrespondin g link tha n to th e oth er links.

I n view of the assumptions 1 and 2, it is not unreasonable to suppose th atthe w eighting matrix W , is diagonal tha t isw diag We,11, O , ~ , , . W , 1. 2 100

The n we can w rite{ [ T - j o ) - ITW,[ T jw) l l i i

where { T j w ) -or

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58 nalysis uf Linear Control Systemsof the reference variable are uncorrelated assumptions an d 2), we canwrite

where { ~ ~ - j w ) ~ , ~ j r o ) } , ~s the i-th d iagonal element of NT -ju ).W,,,V jw). Th is immediately leads to the following design objective.Design Objective 2.3A. In order to obtain o small sieadj-state mean squareinpnt in an as j m ~p to t i c a l ~tab le time-inuariant linea r con trol system with arlm-diniensional reference uariable wit11 uncorre lated contponents,

INT -jw)W,,N jo)},, 2 103slrod d be made sm all ouer the fre qt~ en cj~and of tlre i-th component of tlrereference uariable, for i 1 , 2 , 7Again, as in Objective 2.3, we impose no special reslrictions on{NT 0) W,,N O)};; even if the i-th c om po ne nt of t he reference v aria ble islikely to have a nonzero set point, since only the fluctuations about the setpoint of the inpu t need be restricted.Example 2.8. The con trol of a st ir re d t0111cLet us take up the problem of controlling a stirred tank, as described inExa mp le 2.2 Section 2.2.2). Th e linearized state differential equa tion isgiven in Exam ple 1.2 Section 1.2.3); it is

As the components of the controlled variable z t) we choose the outgoingflow and the outgoing concentration so th at we write

The reference variable r t) thus ha s as its components p, t) an d p, t), th edesired outgoing flow a nd t he desired outgoing co nce ntratio n, respectively.W e now pr opos e the following simple controller. If the outgoing flow is to osmall, we a djust the flow of feed 1 prop ortion ally to the difference betweenthe actu al flow an d the desired flow; thus we let

However, if the outgoing concentration differs from the desired value, theflow of feed 2 is adjusted as follows:,d l ) k d p d t) Cs t)l. 2 107Figure 2.20 gives a block diagram of this control scheme. The reason that

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r e f e r e n c e f o r f l o w i n co m i n g f l o w 1 o u t g o i n g f l o wP l l t l I l ltl 11 It = 6 It~ L o n t

r e f e r e n c e f o r c o n c e n t r o t i o n i n c o m i n g f l o w 2 o u t g o i n g c o n c e n t r o t i o nP Z I ~ I v ~ ( t 1 1 2 i t ~ = 5 2 ( t 1

Fig 2 20 A closed-loop control scheme for the stirred tnnk.

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16 nalysis of Linear ontrol Systemsthis simple scheme is expected to wo rk is th at feed 2 has a higher concentra-tion than feed 1; thus the concentration is more sensitive to adjustments ofth e second flow. As a result, th e first flow is mor e suitable for regulating th eoutgo ing flow. H owever, since the second flow also affects the outgo ing flow,an d the f i s t flow the concentration, a certain am ou nt of interaction seemsunav oidable i n this scheme.For this control system the various transrer matrices occurring in Fig.2.14 can be expressed as follows:

In Examp le 1.17 Section 1.5.4), we fou nd tha t the characteristic polyno mialof the closed-loop system is given by

from which we see that the closed-loop system is asymptotically stable forall positive values of t he gains ic and lc:.I t can be fo und tha t the transmission of the system is given byT s )= K s)[ l G s)H s)]-lP s)

O.Olkl s It 0.02) O.Olk, s 0.02) 2 110-0.25k l s 0.01) k, 0.75 ~ O.O1kl 0.0075)As a result, we find that

I t is easy to see th at if k , and k simultaneously approach infinity then[ T s ) ] - so th at perfect tracking is obtained.

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2 5 StendyStntc Tracking roperties 161Th e transfer matrix N(s) can be found to be

W hen Ic, a n d 1 simultaneously ap proa ch infinity,

which means that the steady-state mean square input C will be infiniteunless the enlries of X, w) decrease fast eno ugh with wIn order t o find suita ble values fo r the gains lc, a n d lc,, we now applyDesign Objective 2 2B an d determine k, and k o tha t the frequency bandsof the two links of th e system contain the frequency b ands of the comp onen tsof the reference variable. This is a complicated problem, however, and there-fore we prefer to use a trial-and-error approach that is quite typical of theway multivariable control problems are commonly solved. This approach isas follows. T o determine lc, we assume th at th e second feedback link has n otyet been connected. Similarly, in o rder t o determine k? we assume th at thefirst feedback link is disconnected. Thus we obtain two single-input single-ou tpu t problems which ar e m uc h easier to solve. Finally, the control systemwith both feedback links connected is analyzed and if necessary thedesign is revised.When the second feedback link is disconnected, the transfer functionfro m the first inp ut t o the first controlled variable is

Proportional feedback according to 2 106 results in the following closed-loo p transfer function from p ,(t) t o [,(t):

We immediately observe that the zero-frequency transmission is differentfrom 1 this can be remedied by inserting an e xtra gainf nto the connectionfrom the first compo nenl of the reference variable as follows:

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62 nalysis o Linear Control SystemsWith this 2 115 is modified to

0.0111. f

For each value of li it is possible to choose l o that the zero-frequencytransmission is 1. Now the value of li depends upon the c utoff frequencydesired. F o r k , 10 the 10 cu tof f frequency is 0.011 rad /s (see Table 2.1).Let u s assume th at this is sufficient fo r the pu rpose of the control system.The corresponding value that should be chosen forf, is 1.1.When studying the second link in a similar manner, it can be found thatthe feedback scheme

results in the following closed-loop transfer function from p,(t) to

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64 nalysis of inenr Control Systemschosen arbitrarily small since the left-hand side of 2-128 is bounded frombelow. For E 0.1 the cutoff frequency is about 0.01 rad/s. The horizontalpart of the curve a t low frequencies is mainly attributable to th e second termin the numerator of 2-128 which originates from the off-diagonal entry inthe first colum n of T (j w ) This entry represents pa rt of the interactionpresent in the system.We now consider the se ond link (the concentration link). Its frequencyband follows from the inequality

.02~'. 2-129By dividing by 0.02 a nd rearranging, it follows for this inequality,

2-130Th e Bode plot of the left-hand side of this inequality, which is the d ifferenceFunction of the second lin k, is also show n in Fig. 2.21. I n this case as well,the horizontal p art of the curve a t low frequencies is caused by the interactionin the system. If the requirements on E are not t oo severe, th e cutoff frequencyof the second link is somewhere near 0.01 rad/s.Th e cutoff frequencies obtained are reasonably close to the 1 0 cutoffFrequencies of 0.011 rad/s and 0.0095 rad/s of the single-loop designs.Moreover, the interaction in the system seems to be limited. In conclusion,Fig. 2.22 pictures the step response matrix of the control system. The plotsconfirm th at the contro l system exhibits m ode rate interaction (both dyn amicand static). Each link has the step response of a first-order system with atime constant of approximately 10 s.

roug h idea of the resulting input amp litudes can be obtained a s follows.F r o m 2-116we see tha t a step of 0.002 m vs in th e flow (assuming that this isa typical value) results in an initial flow change in feed of k,1,0.002 =0.022 m3/s. Similarly, a step of 0.1 km ol/m 3 in the con cen tration results inan initial flow change in feed 2 of k,&O.l = 0.01267 m3/s. Compared to thenom inal values of the incomin g flows (0.015 m3/s an d 0.005 m 3/s, respec-tively), these values are far too large, which means that either smaller stepinput amplitudes must be chosen or the desired transition must be mademo re gradually. T he latte r can be achieved by redesigning the contro l systemwith smaller bandwidths.In Problem 2.2 a m ore sophisticated design of a controller fo r the stirredtan k is considered.

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2 6 Transient Tracking Properties 165

Fig 2 22 Step response matrix of thestirred tank control system. Left column: Responseso f the outgoing Row and con centration to a step of 0 002 mJ/s in the set point o f the flow.Right column: Responses of the outgoing flow and concentration to a step of 0.1 kmol/mJin the set point of the concentration.

2 6 T H E T R A N S I E N T A N A L Y S IS O F T H ET R A C K I N G P R O P E R T I E SIn the previous section we quite extensively discussed the steady-stateproperties of tracking systems. This section is devoted to the frar~sientbehavior of tracking systems, in particular t ha t of the mean s quare trackingerror and the mean square input. We define the settling rime of a certainquantity be it the mean square tracking error, the mean square inpu t, or anyother variable) as the time it takes the variable to reach its steady-state valueto within a specified accuracy. When this accuracy is, say, 1 o f the maximaldeviation from the steady-state value, we speak of th e setf/ ing t ime Forothe r percentages similar tern~ inolog ys used.Usually, when a c ontrol system is started the initial tracking error, and asa result the initial inpu t also , is large. Obviously, it is desirable th at the m eansquare tracking error settles down to its steady-state value as quickly aspossible after starting up or after upsets. We thus formulate the followingdirective.

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66 nnlysis o Linenr onhd ystemsDesign Objective 2.4. A control s j ~ s t e ~ ? ~l~ould e so designed that the settlirlgt h e of the mean square tracking error i s as short asposs ible.

As we have seen in Section2.5.1, the m ean sq uare trackin g error attribu tableto the reference variable consists of two contributions. On e originates fromthe constant part of the reference variable and the other from the variablepart. T he transient behavior of the contribu tion of the variable part must befound by solving the matrix differential equation for th e variance matrix ofthe state of the control system, which is fairly laborious. The transientbehavior of the co ntribution of the consta nt par t of the reference variable tothe mean square tracking error is much simpler to f ind; this can be donesimply by ev aluating the response of the con trol system t o nonz ero initial con-ditions a nd t o steps in the reference variable. As a rule, co m pu ting theseresponses gives a very good impression of the transient behavior of thecontrol system, and this is what we usually do.For asymptotically stable time-invariant linear control systems, someinformation concerning settling times can often be derived from the locationsof the closed-loop poles. This follows by no ting th at all responses are exponen-tially dam ped m otions with time constants th at are the negative reciprocalsof the real parts of th e closed-loop c haracteristic values of the system. Sincethe settling time of

e- lo, t 0, 2-131is 4.60, a bo un d for the settling time t , of any variable is

where &, i 1 , 2 , n are the closed-loop characteristic values. Notethat for squared variables such as the mean square tracking error and themean square input, the settling time is half that of the variable itself.The hound 2-132 sometimes gives misleading results, since it may easilyhappen that the response of a given variable does not depend upon certaincharacteristic values. Later Section 3.8) we mee t instances, for example,where the settling time of the rm s tracking er ror is determined by th e closed-loop poles furthest from the origin and not by the nearby poles, while thesettling time of the rms input derives from the nearby closed-loop poles.Ex am ple 2.9. Tlte settling time of the tracking error of the position seruoLe t us consider Design I of Examp le 2.4 Section 2.3) for the position servo.Fr om the steady-state analysis in Exam ple 2.7 Section 2.5.2), we learnedth at as the gain increases the rms steady-state tracking error keeps de-creasing, althou gh beyond a certain value 15-25 V /rad ) very little imp rove-ment in the rms tracking error is obtained, while the rms input voltage

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2 7 Effecls of Disturbances 167becomes larger and larger. We now consider the settling time of the trackingerror. To this end, in Fig. 2.23 the response of the controlled variable to astep in the reference variable is plotted for various values of A from zeroinitial conditions. As can be seen, the settling time of the step responsehence also tha t of the tracking error) first decreases rapidly as increases,hut beyond a value of A of ab out I5 V/rad the settling time fails to improvebecause of the increasingly oscillatory behavior of the response. In this case

Rig 2 23 Response of Design I of theposition servo to a step of 0 1 rad in therefercncc variable for various values ofthe gain i

as well, the most favorable value of A seems to he ab out 15 V/rad, whichcorresponds to a relative dam ping see Example 2.7) of ab ou t 0.7. Fr om theplots of [T[jo lf Fig. 2.18, we see th at fo r this value of the gain th e largestbandwidth is achieved without undesirable peaking of the transmission.

2.7 T H E E F F E C T S O F D I S T U R B A N C E S IN T H ES I N G L E I N P U T S I N G L E O U T P U T C A S E

I n Section 2.3 we saw tha t very often disturbances a ct up on a con tro l system,adversely affecting its tracking o r regulating performance. In this section wederive expressions for the increases in the steady-state mean square trackingerror and the steady-state mean square input attributable to disturbances,and formulate design objectives which may serve as a guide in designingcon trol systems c apable of counteracting disturbances.

Throughout this section the following assumptions are made.1. Tlte disturba tlce uariable u, t) s a stocl~asticrocess that is ~~tlcar relate dwith the reference uariable r t) a n d the abseruatio tl noise u,,, t).

As a result, we can obtain the increase in the mean squ are tracking erro r andthe mean squa re inpu t simply by setting r t ) an d u,,, t) identical t o zero.2. The controlled uariable is also the abserued variable, that is, C D .

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68 Annlysis o Linear Control Systems

and that in the time-i~~uorianfase

The assumption that the controlled variable is also the observed variable isquite reasonable, since it is intuitively clear that feedback is most effectivewhen the controlled variable itself is directly fed hack.3. The control sj~stent s as yn lp to ti ca l~ table and tinie-inuariant.4. Th e i r p t variable and the controlled uariable, 11ence also the referencevariable, are scalars. FVo oaltd W,,are both 1.The analysis of this section can he extended to multivariable systems butdoing so adds very little to th e conclusions of this a n d the following sections.5. Th e distwbonce variable u, t) cart be written as

1~11erehe c o n s t a ~ ~ tart u,,, of the disturbance variable is a stochastic vectorlvith giuen seco nho rder m on te ~i t larrix, and where the uariable part u,, t) ofthe disturbance uoriable is a wide-sense station arj~ ero mean stoclrasticprocessivith power spectral density n iat ris C,Jo , zuzcorreloted ivit11 v,,,.The transfer matrix from the disturbance variable u, t) to the controlledvariable z t ) can be fou nd from the relation see Fig. 2.24)

where Z s ) and V ) denote the Laplace transforms of z f ) and u, t),

Fig 2 24 Transfer matrix block diagram of closed loop conlrol system with plantdisturbance v .

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2 7 EUccts of isturbances 69respectively so tha t

Here we have used the fact that the controlled variable is a scalar so thatH (s)G (s ) s also a scalar function. We now introduce the function

which we call the sensitiuitj~frmctio~lf the contro l system for reasons to beexplained later.We comp ute the con tribution of the disturbance variable to the steady-slate mea n square tracking error as the sum of two terms one originalingfrom the constant part and one from the variable part of the disturbance.Since Z ( s ) S ( s ) D ( s I A ) - l V , ( s ) , 2-139the steady-state response of the controlled variable to the constant part ofthe disturbance is given by

lim ~ ( t ) S ( 0 ) D ( - A ) - l~ n , S(O)u,,. 2-140i m

Here we have assumed that the matrix A is nonsingular-the case where Ais singular is treated in Problem 2 4 Furth erm ore we have abbreviateduuo=D(-A) u,. 2-141

A s a result of 2-140 the contribution of the constant par t of the disturbanceto the steady-state mean square tracking error is

where V , is the second-order moment of u that is V ,=E{u&}. Further-more it follows from 2-139 with the methods of Sections 1 10 4 and 1.10.3that the contribution of the variable part of the disturbance to the steady-state mean square tracking error can be expressed as

rn= L I s ( ~ ~ ) I % & ) 2-143Here we have abbreviated

&(w) = D ( j w I A ) - l X u l , ( w ) ( - j w I A T ) - l ~ z ' . 2144Consequently the increase in the steady-state mean squa re track ing error

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170 Annlysis o f Linear olltrol Systemsattributa ble to the disturbance is given byC (with disturbance) C (without disturbance)

= 1S(O)( V lmS ( j ~ ) ~ ( w 45mBefore discussing how to make this expression small, we give an interpreta-tion. C onsider the situation of Fig. 2.25 where a variable u, t) acts upon theclosed-loop system. This variable is added to the controlled variable. It is

Fig. 2.25. Transfer matrix block diagram or a closed-loop control syslem with theequivalent disturbance q at the controlled variable.easily found th at in term s of Laplace transforms w ith the reference variableand the initial conditions identical to zero the controlled variable is given by

where V, s) denotes the Laplace transform of u, t). W e immediately see tha tif u, t) is a stochastic process with as constant part a stochastic variablewith second-order mo men t fo and as variable part a zero-mean wide-sensestat ionary stochastic process with power spectra l density S , , w ) , the increasein the steady-state mean squa re tracking e rror is exactly given by 2 145. W etherefore call the process u, t) with these properties th e eqtrivalent disturbanceat the controlled uariable.An examination of 2 145 leads to the following design rule.Design Objective2.5. Z rder to reduce the illcrease of th e st ea e-s ta te riieaizsquare traclci~~grror attributable to distlrbmlces in an asjmtptotically stablelinear tiiiie-iiiuariont coritrol sjuter i~ eit11 a scalar controlled variable, ivhich isalso the obserued uariable, the absolute ual~re f the se~isitiuiQ~firi~ctio~i( jw)s l ~ a ~ i l de made sri~all ver the freq~rericy b a d of rlie e q~ ~iu ale ntist~rrbarice t

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2 7 Effects of isturbances 7the controlled uarioble. If corlstalrt errors are of special concern, S 0) s l ~ o l ~ l dbe mode ssia ll, preferably zero.Th e last sentence of this design rule is no t valid w ithout furthe r qu alificationfo r con trol systems where the matrix of the plan t is singular; this case isdiscussed in Prob lem 2.4. I t is noted that since S jw ) is given by

a small S jw) generally m ust be achieved by malting the loop gain H jw)G jw)of the control system large over a suitable frequency range. This easilyconflicts with Design Objective 2.1 Section 2.4) concerning th e stability of.the con trol system see Exam ple 2.5, Section 2.4jm

. . A compromise must beround.Reduction of constant errors is of special importance for regulator andtracking systems where the set point of the controlled variable must bemaintained with great precision. Constant disturbances occur very easily incon trol systems, especially because of e rrors ma de in establishing th e nom inalinput. C on stan t errors can often be completely eliminated by mak ing S 0) =0, which is usually achieved by introducing integrating action, that is, byletting the controller transfer function G s ) have a pole at the origin seeProblem 2.3).Let us now turn to a consideration of the steady-state mean square input.I t is easily found th at in terms of Laplace transforms we can write seeFig. 2.24)

where U s) is the Laplace transform of u t ) . I t follows fo r the increase in thesteady-state mean square input, using the notation introduced earlier in thissection,

C,,, with disturb ance ) C,,, witho ut disturbance)

This expression results in the following directive.Design bjective 2 6 IIIorder to obtain a small increase in the steady-statememz square input attributable to the dist~rrborlce r~an asj~ iilptotically tablelinear time-i~tuarionf ontrol system ndth a scalar c o~ ~ tr ol le dariable that is

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72 Annlysis of Linear Control Systemsalso ihe obserurd variable mrd a scalar input

sltould be niade snlall over tl~erequency band of tlte eyrriualerrt disttirbonce a tthe car~trolled arioble.I n t is directive n o attention is paid to th e constant pa rt of the inp ut since,as assumed in the discussion of Objective 2.3, the plant must be able tosustain these constant deviations.Design Objective 2.6 conflicts with Objective 2.5. Making the loop gainH j w ) G j w ) large, as required by Objective 2.5, usually does not result insmall values of 2 150. Again a compromise must be found.Example 2.10. Tlte effect of disturbarrces o1r the position servoI n this example we study the effect of disturbances on Design of th eposition servo of Example 2.4 Section 2.3). It is easily found that the sen-sitivity function of the control system as proposed is given by

I n Fig. 2.26 Bode plots of IS jw) l are given for several values of the gain AI t is seen tha t by choosing larger the frequency band over which disturbancesuppression is obtained also becomes larger. If the equivalent disturbance atthe controlled variable, however, has much power near the frequency whereIS jw) l has its peak, then perhaps a smaller gain is advisable.

Fig. 2.26. Bodep lots of thesensitivity function of the position control system Design I asfunction of the gain A

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2 7 Kects of isturbances 173In Example 2.4 we assumed that the disturbance enters as a disturbingtorque r, t) acting on the shaft of the motor. If the variable part of thisdisturb ing torq ue has the power spectral density function X,, w), the variable

part of the equivalent disturbance at the controlled variable has the powerspectral density func tion2-152j d j w 4

Th e pow er spectral density of the contribution of the disturbing torque to thecontrolled variable is found by multiplying 2-152 by IS jw)12 an d thus isgiven by

Le t us sup pose that the variable p art of the disturbing torque can be repre-sented as exp onentially corre lated noise with r m s value u7 and time constantT o that

The increase in the steady-state mean square tracking error attributableto the disturbing torqu e can be computed by integrating 2-153 or by m odelingthe disturbance, augm enting the state differential e qua tion, an d solving forthe steady-state variance matrix of the augmented state. Either way we findC with disturbing torque) Camwithout disturbing torque)

From this we see that the addition to C monotonically decreases to zerowith increasing 1.Thus the larger 1 he less the disturbing torque ar ec ts thetracking properties.In the absence of the reference variable, we have p t) = -1q t ) so thatthe increase in the mean square input voltage attributable to the disturbingtorque is 1 imes th e increase in the mean square tracking error:C with disturbing torque) C without disturbing torque)

F o r m , C monotonically increases to

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174 nalysis of Lioenr Control SystemsI t is easily found from 2 25 that a constant disturbing torqu e 7 results in asteady-state displacement o f the controlled variable o f

0 2 158K ?Clearly, this displacement can also be made arbitrarily small by makingthe gain ,? sufficiently arge.

2.8 THE EFFECTS OF OBSERVATION NOISE IN THESINGLE INPUT SINGLE OUTPUT CASE

In any closed-loop scheme, the effect of o bservation noise is to som e extentfelt. n this section the contribution of the observation noise to the meansquare tracking error and the mean square input is analyzed. To this end,the following assumptions are made.1. The observation noise u,(t) is a stoclrastic process ~ ~ h i c hs iotcorrelatedi ~ ~ i f l ihe reference uoriable r ( t ) and the plant disturbartce u ).As a result, the increase in the mean square tracking error and the meansquare input attributable to the observation noise may be com puted simplyby setting r ( t ) and v,(t) identical to zero.2. Tlre confroller variable is also the observed variable, tliat is, C D ,so tlrat

? l ( t) z ( f ) urn@), 2 159an d, in the tiiiie-iitvariaitt case,

H ( s ) K ( s ) .3. Tlre coittrol systeni is a syrtrpto tical~ table and time-inuariaitt.4. The iitpirt variable and the coirfralled variable, hence also tlie referenceuariable, are scalars. +reand W are both 1.

Here also the analysis can be extended to multivariable systems but againvery little additional insight is gained.5. The abservatioit noise is a zero-mean wide-sense stationary stocltasticprocess with power spectral deiisify fiotction X,,, w).Figure 2 27 gives a transfer function block diagram of the situation thatresults fro m these assump tions. It is seen tha t in terms of Laplace transform s

zb) -H(s)G(s)[V,(s) Z(s) l , 2 161so that

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2 8 Effccts of Observntion Noise 175

Fig. 2 27 Transfer matrix block diagram of a closed loop control system with observationnoise.

Consequently, the increase in the steady-state mean square tracking errorattributable to the observation noise can be written asC w ith o bservation noise) C without observation noise)

O ur n ext design objective can thu s be formulated as follows.Design Objective 2 7 In order to redtrce the iircrease in tlre st ea dj wta te inearlspare tracking error attriblttable to obseruation noise in all os~rn~ptoticallystable linear time-i~tuariantcorttral systertt ivitli a scalar controlled variablethat is also the obserued variable, the sj~stent hotdd be designed so that

is smaN ooer the freyrmtcy b a d of tlre observation noise.Obviously, this objective is in conflict with O bjective 2.5 since making the

loop gain H jw) G jw) large, as required by Objective 2.5 results in a valueof. 2-164 that is near I which means that the observation noise appearsunattenuated in the tracking error. This is a result of the fact that if a largeloop gain H jw)G jw) is used the system is so controlled that z t ) u,,, t)instead of z t ) tracks the reference variable.A simple compu tation shows that the transfer function from the observa-tion noise to the p lant inpu t is given by

which results in the following increase in the steady-state m ean squ are inp ut

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76 Analysis of Linen Control ystemsattributable to observation noise:C with observation noise) C without observation noise)

This yields the design rule tha t to m ake th e increase in th e steady-state me ansquare in put attributable to the observation noise small,

should be made small over the frequency band of the observation noise.Clearly, this rule is also in conflict with Objective 2.5.Example 2 11 The positio~z eruo ivitl~ osition feedback onlyLet us once again consider the position servo of Example 2.4 Section 2.3)with the three different designs proposed. In Examples 2.7 Section 2.5.2) an d2.9 Section 2.6), we ana lyze d Design I and chose 15 V/rad as the bestvalue of the gain. In Example 2.7 it was found that Design I1 gives bette rperformance because of the additional feedback link from the angularvelocity. Le t us now suppose, however, tha t fo r some reason financial o rtechnical) a tachometer cannot be installed. We then resort to Design 111,which attemp ts to approximate Design I1 by using an approxim ate differenti-ator with time constant Td. f no observation noise were present, we couldchoose T = 0 and Design 111would reduce to Design II Let us suppose thatobservation noise is present, however, and that is can be represented asexponentially correlated noise with time constan t

T 0.02 san d rms valuea = 0.001 tad.

The presence of the observation noise forces us to choose T 0. In orderto determine a suitable value of T we first assume th at T will turn ou t to besmall enoug h so that the gains and 2 can be chosen as in Design 11. Thenwe see how large T can be made withou t spoiling the performance of Design11, while a t the same time sufficiently reducing the effect of th e observationnoise.I t is easily foun d tha t the transmission of the con trol system according toDesign I11 is given by

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2 8 Etfcels of Observation Noise 77To determine a suitably small value of T e argu e as follows. T he closed-loop system according to Design 11 with the numerical values obtained inExample 2.7 fo r an d p has an undamped natural frequency w of about20 rad/s with a relative damping of 0.707. Now in order not to impede thebehavior of the system the time constant T f the differentiator should bechosen small with respect to the inverse na tur al frequency tha t is small withrespect t o 0.05 s. In Fig. 2.28 we have plotted the transmission 2 170 for

Fig 2 28 The effect of T o n the transmission of Design of the position servo

various values of T . t is seen that for T 0.01 s the transmission is hardlyaffected by the approximate derivative ope ration bu t tha t for T 0.1 sdiscrepancies occur.Let us now consider the effect of the observation noise. Modeling u,, t)in the usual way the additions to the steady-state mean square tracking erro rand input attributable to the observation noise can be computed from thevariance m atrix of the augm ented state. The numerical results ar e plotted inFig. 2.29. These plots show that for small T he steady-state mean squareinp ut is greatly increased. An acceptable value of T eems to be a bo ut 0.01 s.F o r this value the squ are ro ot of the increase in the steady-state mean squ areinput is only about 2 V the square root of the increase in the steady-statemean squ are tracking erro r o f ab ou t 0.0008 rad is very small an d the trans-mission of the con trol system is hardly affected.

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78 Analysis of inear Control Systems0.001 2

rm 0.0010trackingerror 0.00080.0006Od' 0.0004P

F i g 2.29. The quare roots of the additions to the steady statemean square tracking errorand input voltage due to observation noise as a function of T or Design III of the positionservo.

2.9 THE EFFECT OF PLANT PARAMETERUNCERTAINTY I N THE SINGLE INPUTSINGL