Chapter 3 Basic Concepts This chapter and the next are the most fundamental.We concentrate on the single-loop feedback system.Stability of this system is defined and characterized.Then the system is analyzed for its ability to track certain signals (i.e.,steps and ramps)asymptotically as time increases.Finally, tracking is addressed as a performance specification.Uncertainty is postponed until the next chapter. Now a word about notation.In the preceding chapter we used signals in the time and frequency domains;the notat ion was u(t)for a function of time and (s)for its Laplace transform.When the context is solely the frequency domain,it is convenient to drop the hat and write u(s);similarly for an impulse response G(t)and the corresponding transfer function G(s). 3.1 Basic Feedback Loop The most elementary feedback control system has three components:a plant (the object to be controlled,no matter what it is,is always called the plant),a sensor to measure the output of the plant,and a controller to generate the plant's input.Usually,actuators are lumped in with the plant.We begin with the block diagram in Figure 3.1.Notice that each of the three components controller plant sensor Figure 3.1:Elementary control system. has two inputs,one internal to the system and one coming from outside,and one output.These signals have the following interpretat ions: 27
Chapter Basic Concepts This chapter and the next are the most fundamental We concentrate on the singleloop feedback system Stability of this system is dened and characterized Then the system is analyzed for its ability to track certain signals ie steps and ramps asymptotically as time increases Finally tracking is addressed as a performance specication Uncertainty is postponed until the next chapter Nowaword about notation In the preceding chapter we used signals in the time and frequency domains the notation was ut for a function of time and us for its Laplace transform When the context is solely the frequency domain it is convenient to drop the hat and write us similarly for an impulse response Gt and the corresponding transfer function G s Basic Feedback Loop The most elementary feedback control system has three components a plant the ob ject to be controlled no matter what it is is always called the plant a sensor to measure the output of the plant and a controller to generate the plant s input Usually actuators are lumped in with the plant We begin with the block diagram in Figure Notice that each of the three components controller sensor plant r u y d n v Figure Elementary control system has two inputs one internal to the system and one coming from outside and one output These signals have the following interpretations
28 CHAPTER 3 BASICCONCEPTS rrq(cp(amaa ipt EPATEt u aataa fatmat d elaakTrbace anattaa measra sla EpICe ThE(elas(GiG atheaan-aC(aaeuns IhctmSyvesal (asaeaat IGGna(EGjetv(Sbuthe (abesimma by saa haty TiGha eOlaleschecesteCht farOr.aajticd acomn eetrEoteartorbare sIce,wrea Iela wenaaca Qftteeu.ua真，tm&sGesttselo1 eeeoaectem 88裙8a6 frG Lcc(ECIf MnS ECofetceaiarrsmn le sasma loema otsatptisaxa Imfsit msxeatcaladav(FRaTeteratguatn&s hCrI y=p() rattiate .2 tasematkp &s P=[P3P.]- wert y P31+P.u. wera taen cetoae aansncet tfeomptsotet ce (GIdEEaelarurasot esmSpanerreohe mastastefaIQ alatme(uctasaelaerIC E(TEChm y P(d+u)- v F(y+n)- u C(r v). Te.insirirteate.atarisanaeotaitmneccaaa otcceuaias SIkure.2.Our (aermstctsee IsetSimm DaiuI(CSaCGlea. The(OSwTi Te.igell-posedness.ThISmeaSctIgur e.2 ad (Earerultasekt tats a taserIasi G telirfeeclasnptsioa elasslasIey u.y.w aatcaltsotermm IajuIOSLaeteatats? efmmlajuItasesinsu e.3.FwIcearessitsi (Sloc terebase I(OSn 1Q3.x.,x1 rheCeTaeruIasaeolade a tce wrlle EQuctusetefimm DajuIOS x3 =r Fxh x.d+Cx3 x1 n+Px
CHAPTER BASIC CONCEPTS r reference or command input v sensor output u actuating signal plant input d external disturbance y plant output and measured signal n sensor noise The three signals coming from outsider d and nare called exogenous inputs In what follows we shall consider a variety of performance ob jectives but they can be summa rized by saying that y should approximate some prespecied function of r and it should do so in the presence of the disturbance d sensor noise n with uncertainty in the plant We may also want to limit the size of u Frequently it makes more sense to describe the performance ob jective in terms of the measurement v rather than y since often the only knowledge of y is obtained from v The analysis to follow is done in the frequency domain To simplify notation hats are omitted from Laplace transforms Each of the three components in Figure is assumed to be linear so its output is a linear function of its input in this case a twodimensional vector For example the plant equation has the form y P d u Partitioning the transfer matrix P as P P P we get y Pd Pu We shall take an even more specialized viewpoint and suppose that the outputs of the three components are linear functions of the sums or dierence of their inputs that is the plant sensor and controller equations are taken to be of the form y P d u v F y n u Cr v The minus sign in the last equation is a matter of tradition The block diagram for these equations is in Figure Our convention is that plus signs at summing junctions are omitted This section ends with the notion of wel lposedness This means that in Figure all closed loop transfer functions exist that is all transfer functions from the three exogenous inputs to all internal signals namely u y v and the outputs of the summing junctions Label the outputs of the summing junctions as in Figure For wellposedness it suces to look at the nine transfer functions from r d n to x x x The other transfer functions are obtainable from these Write the equations at the summing junctions x r F x x d Cx x n P x
3.1.BASIC FEEDBACK LOOP 29 d Figure 3.2:Basic feedback loop. d T2 3 Figure 3.3:Basic feedback loop. In matrix form these are [1)-( Thus,the system is well-posed iff the above 3 x 3 matrix is nonsingular,that is the determinant 1+PCF is not identically equal to zero.[For instance,the system with P(s)=1,C(s)=1, F(s)=-1 is not well-posed]Then the nine transfer functions are obtained from the equation )[() that is, 1 -PE -F 1 T2 1+PCF 1 CF (3.1) T3 PC P 1 A stronger notion of well-posedness that makes sense when P,C,and F are proper is that the nine transfer functions above are proper.A necessary and sufficient condition for this is that 1+PCF not be strictly proper [i.e.,PCF(oo)-1]. One might argue that the transfer functions of all physical systems are strictly proper:If a sinusoid of ever-increasing frequency is applied to a (linear,time-invariant)system,the amplit ude
BASIC FEEDBACK LOOP C P F r u y d v n Figure Basic feedback loop C P F r u y d v n x x x Figure Basic feedback loop In matrix form these are F C P x x x A r d n A Thus the system is wellposed i the above matrix is nonsingular that is the determinant PCF is not identically equal to zero For instance the system with P s Cs F s is not wellposed Then the nine transfer functions are obtained from the equation x x x A F C P r d n A that is x x x A PCF P F F C CF P C P r d n A A stronger notion of wellposedness that makes sense when P C and F are proper is that the nine transfer functions above are proper A necessary and sucient condition for this is that PCF not be strictly proper ie PCF One might argue that the transfer functions of all physical systems are strictly proper If a sinusoid of everincreasing frequency is applied to a linear timeinvariant system the amplitude
30 CHAPTER.,BASIC CONCEPTS of the output will go to zero.This is somewhat misleading because a real system will cease to behave linearly as the frequency of the input increases.Furt hermore,our transfer functions will be used to parametrize an uncertainty set,and as we shall see,it may be convenient to allow some of them to be only proper.A proportional-integral-derivative controller is very common in practice, especially in chemical engineering.It has the form 从+名+k3 This is not proper,but it can be approximated over any desired frequency range by a proper one, for example, k3 s1s+1 Notice that the feedback system is automatically well-posed,in the stronger sense,if P,C,and F are proper and one is strictly proper.For most of the book,we shall make the following standing assumption,under which the nine transfer functions in(3.1)are proper: P is strictly proper,C and F are proper. However,at times it will be convenient to require only that P be proper.In this case we shall always assume that |PCF1 at w=.,because such a controller would almost surely be unstable if implemented on a real system. 3.2 Internal Stability Consider a system with input u,output y,and transfer function G,assumed stable and proper. We can write G=G+G where G is a constant and G is strictly proper. Example: =13 In the time domain the equation is y()=G u(t)+G(t,1)u(1)d13 If u(t)<c for all t,then ≤1lc+ |G(1)川d1c3 一● The right-hand side is finite.Thus the output is bounded whenever the input is bounded.[This argument is the basis for entry (2,2)in Table 2.2.] If the nine transfer functions in(3.1)are stable,then the feedback system is said to be internally stable.As a consequence,if the exogenous inputs are bounded in magnit ude,so too are and x and hence u,y,and v.So internal stability guarantees bounded internal signals for all bounded exogenous signals
CHAPTER BASIC CONCEPTS of the output will go to zero This is somewhat misleading because a real system will cease to behave linearly as the frequency of the input increases Furthermore our transfer functions will be used to parametrize an uncertainty set and as we shall see it may be convenient to allow some of them to be only proper A proportionalintegralderivative controller is very common in practice especially in chemical engineering It has the form k k s ks This is not proper but it can be approximated over any desired frequency range by a proper one for example k k s ks s Notice that the feedback system is automatically wellposed in the stronger sense if P C and F are proper and one is strictly proper For most of the book we shall make the following standing assumption under which the nine transfer functions in are proper P is strictly proper C and F are proper However at times it will be convenient to require only that P be proper In this case we shall always assume that jPCF j at which ensures that PCF is not strictly proper Given that no model no matter how complex can approximate a real system at suciently high frequencies we should be very uncomfortable if jPCF j at because such a controller would almost surely be unstable if implemented on a real system Internal Stability Consider a system with input u output y and transfer function G assumed stable and proper We can write G G G where G is a constant and G is strictly proper Example s s s In the time domain the equation is yt Gut Z Gt u d If jutj c for all t then jytjjGjc Z jG j d c The righthand side is nite Thus the output is bounded whenever the input is bounded This argument is the basis for entry in Table If the nine transfer functions in are stable then the feedback system is said to be internal ly stable As a consequence if the exogenous inputs are bounded in magnitude so too are x x and x and hence u y and v So internal stability guarantees bounded internal signals for all bounded exogenous signals
32.INTERNAL STABILITY 31 The idea behind this definition of internal stability is that it is not enough to look only at input-output transfer functions,such as from r to y,for example.This transfer function could be stable,so that y is bounded when r is,and yet an internal signal could be unbounded,probably causing internal damage to the physical system. For the remainder of this section hats are dropped. Example In Figure 3.3 take -5 1 1 P(S-S:I F(S-1. Check that the transfer function fromr to y is stable,but that from d to y is not.The feedback sys tem is therefore not internally stable.As we will see later,this offense is caused by the cancellation of the controller zero and the plant pole at the point S-1. We shall develop aa test for internal stability which is easier than examining nine transfer func tions.Write P, and F as ratios of coprime polynomials (i.e.,polynomials with no common factors): P= Np. Nc.F-NE Mp Mc F The characteristic polymnomial of the feedback system is the one formed by taking the product of the three numerators plus the product of the three denominators: NPNCNF MPMCMF. The closed loop poles are the zeros of the characteristic polynomial. The orem∠ The feedback system is internally stable i there are no closed loop poles in Re Pro of For simplicity assume that F =1;the proof in the general case is similar,but a bit messier. From (3.1)we have :c(n Substitute in the ratios and clear fractions to get 1 MPMc NpMc MpMc MpNc MPMc MpNc (3.2) NPNc MpMc NPNC NpMc MPMo Note that the characteristic polynomial equals Np Nc+MpMc.Sufficiency is now evident;the feedback system is internally stable if the characterist ic poly nomial has no zeros in ReS0. Necessity involves a subtle point.Suppose that the feedback system is internally stable.Then all nine transfer functions in (3.2)are stable,that is,they have no poles in Re 0.But we cannot immediately conclude that the polynomial NpNc +MpMc has no zeros in ReS=0 because this polynomial may conceivably have a right half-plane zero which is also a zero of all nine numerators in (3.2),and hence is canceled to form nine stable transfer functions.However,the characteristic polynomial has no zero which is also a zero of all nine numerators,MpMC,NpMc,and so on
INTERNAL STABILITY The idea behind this denition of internal stability is that it is not enough to look only at inputoutput transfer functions such as from r to y for example This transfer function could be stable so that y is bounded when r is and yet an internal signal could be unbounded probably causing internal damage to the physical system For the remainder of this section hats are dropped Example In Figure take Cs s s P s s F s Check that the transfer function from r to y is stable but that from d to y is not The feedback sys tem is therefore not internally stable As we will see later this oense is caused by the cancellation of the controller zero and the plant pole at the point s We shall develop a test for internal stability which is easier than examining nine transfer func tions Write P C and F as ratios of coprime polynomials ie polynomials with no common factors P NP MP C NC MC F NF MF The characteristic polynomial of the feedback system is the one formed by taking the product of the three numerators plus the product of the three denominators NP NCNF MPMCMF The closedloop poles are the zeros of the characteristic polynomial Theorem The feedback system is internal ly stable i there are no closedloop poles in Res Proof For simplicity assume that F the proof in the general case is similar but a bit messier From we have x x x A P C P C C P C P r d n A Substitute in the ratios and clear fractions to get x x x A NP NC MPMC MPMC NPMC MPMC MP NC MPMC MP NC NP NC NPMC MPMC r d n A Note that the characteristic polynomial equals NP NC MPMC Suciency is now evident the feedback system is internally stable if the characteristic polynomial has no zeros in Res Necessity involves a subtle point Suppose that the feedback system is internally stable Then all nine transfer functions in are stable that is they have no poles in Re s But we cannot immediately conclude that the polynomial NP NC MPMC has no zeros in Res because this polynomial may conceivably have a right halfplane zero which is also a zero of all nine numerators in and hence is canceled to form nine stable transfer functions However the characteristic polynomial has no zero which is also a zero of all nine numerators MPMC NPMC and so on
32 GHAPTER.ST COM EPTS Plof of th sthatis ita a eisc3rtrollows fom terattratwetok coTimerat statwithtratis,Np ad Np aebopimea aetreternuma raGomincbrla. h6efaHetitt8eaf能Pweaeeaouapsuwma3 Theorem 2 The feedback system is internally stable i,the following two conditions hold: (a)The transfer function 1+PCF has no zeros in Res >0 (b)There is no pole zero cancellation in Res0 when the product PCF is formed P roof Rea tattereeabak systn is intehally stableif a ninetasrerrunctions 1 1·PF.F 1 1+PCF .CF PC P 1 ae ta3 egte86 O感惑&器” To Hoveb),whtepC-F a ltios of co imefolynomials: By THIGn 1 tecraallistic ply nomia NPNCNF+MPMCME gos in Re≥03 Ths terarne Mo)raeno common z(in R≥0，ad simily fortreorernumelard omincrTal3 2-)Assymea ad )3rabrPC-F a aove ad ietso beae of tecraatelistic plynomia tretis, NPNCNF+MPMcMF)2s0)=0. wemust kw thatRe(o:th wil pveinteastaiit by rkIGh 13suset te contiay tretRe≥03r 2MPMc MF)2s0)=0- te NPNCNF)0)=0. Butthis violate 3rhs 2 PMcMF)so)≠0- so weca divideby it ovet get 1+P3= thatis 1+2PCF)3o)=0-
CHAPTER BASIC CONCEPTS Proof of this statement is left as an exercise It follows from the fact that we took coprime factors to start with that is NP and MP are coprime as are the other numeratordenominator pairs By Theorem internal stability can be determined simply by checking the zeros of a polynomial There is another test that provides additional insight Theorem The feedback system is internal ly stable i the fol lowing two conditions hold a The transfer function PCF has no zeros in Res b There is no polezero cancel lation in Res when the product PCF is formed Proof Recall that the feedback system is internally stable i all nine transfer functions PCF P F F C CF P C P are stable Assume that the feedback system is internally stable Then in particular PCF is stable ie it has no poles in Res Hence PCF has no zeros there This proves a To prove b write P C F as ratios of coprime polynomials P NP MP C NC MC F NF MF By Theorem the characteristic polynomial NP NCNF MPMCMF has no zeros in Res Thus the pair NP MC have no common zero in Res and similarly for the other numeratordenominator pairs Assume a and b Factor P C F as above and let s be a zero of the characteristic polynomial that is NP NCNF MPMCMF s We must show that Res this will prove internal stability by Theorem Suppose to the contrary that Res If MPMCMF s then NP NCNF s But this violates b Thus MPMCMF s so we can divide by it above to get NP NCNF MPMCMF s that is PCF s
33 ASYMPTOTICTRA KING 33 which violates(a)-■ Finally,let us recall for lat er use the Nyquist stability crit erion-It can be derived from Theo- rem2 and the principle of the argument-Begin with the curve D in the compler plane It starts at the origin,goes up the imaginary aris,turns int o the right halfplane following a semicircle of infinite radius,and comes up the negative imagiary aris to the origin again: D As a point s makes me circuit around this curve,the point P(s)C(s)F(s)traces out a curve called the Nyquist plot of PCF-If PCF has a pole on the imaginary aris,then D mst hawe a small indentation to avoid it- NyquiSt CriteriOn Construct the Nyquist plot of PCF,indenting to the left around poles on the imaginary aris,Let n denote the total number cf poles of P,C,and F in Res >0 Then the feedback system is internally stable i,the Nyquist plot does not pass through the point 1 and encircles it exactly n times counterclockibise 3.3 Asym ptotic Tracking In this section we look at a typical performance specification,perfect asymptotic tracking of a reference signal-Both time domain and frequency domain occur,so the notation distinction is required- For the remainder of this chapter we specialize to the unity feedback case,F=1,so the block diagram is as in Figure 34-Here e is the tracking error;with h =d=0,e equals the reference input (ideal response),r,minus the plant output (actual resp anse),y- We wish to study this systems capability of tracking certain test inputs asympt otically as time tends to infinity-The two test inputs are the step r()= ft≥0 0 ift(0 and the ramp ct-ft≥0 r(t)= 0-ft(0 (c is a nonzero real number)-As an application of the former think of the temperature-control thermost at in aroom when you change the setting on the thermost at (step input),you would like
ASYMPTOTIC TRACKING which violates a Finally let us recall for later use the Nyquist stability criterion It can be derived from Theo rem and the principle of the argument Begin with the curve D in the complex plane It starts at the origin goes up the imaginary axis turns into the right halfplane following a semicircle of innite radius and comes up the negative imaginary axis to the origin again D As a point s makes one circuit around this curve the point P sCsF s traces out a curve called the Nyquist plot of PCF If PCF has a pole on the imaginary axis then D must have a small indentation to avoid it Nyquist Criterion Construct the Nyquist plot of PCF indenting to the left around poles on the imaginary axis Let n denote the total number of poles of P C and F in Res Then the feedback system is internal ly stable i the Nyquist plot does not pass through the point and encircles it exactly n times counterclockwise Asymptotic Tracking In this section we look at a typical performance specication perfect asymptotic tracking of a reference signal Both time domain and frequency domain occur so the notation distinction is required For the remainder of this chapter we specialize to the unityfeedback case F so the block diagram is as in Figure Here e is the tracking error with n d e equals the reference input ideal response r minus the plant output actual response y We wish to study this system s capability of tracking certain test inputs asymptotically as time tends to innity The two test inputs are the step rt c if t if t and the ramp rt ct if t if t c is a nonzero real number As an application of the former think of the temperaturecontrol thermostat in a room when you change the setting on the thermostat step input you would like
34 CHAPTER--BASIC C ONC EPTS Fue4:utrCcp feca Ieatcotay iccaelctee toa e,awaaietccane CC wJaedaufte a sato aa.tsaaaalacstla ictc Gss A.eoaradGactattasaaavecct sesata aetausfeOlaky aleafuraotejeaaD. Dleheoop traster funxction L:=PC.ThefaseictnGherelptr 1 KCIGOeS s:=,11 ,1+L eteensitinity fagtjon-mGEasrteletsetarEajit of ess ictc ●● sK[Saldra [Sem ftaly aetelasc.ebmbe(CCSs ats =0. Theorem Assume that the feedback system is irternally stable and n=d=0. (a)If r is astep thene(t).0 ast.i2 S has at least one zero at the origin (6)If r is aramp thene(t).0 ast.i2 has at least two zeros at the origin The(oSaEkatirofeira.vaue theorem Iy(s)saaidaL liaef C ttsicast港，oeqt Cbly a3iFCt6=o,Inyt)ersat Quasilso sy(s). Poof (athe iaefach oteceasCS(s)=Gs ThEtaeruItmG r 10e cuass.O es)-5(s) sietecac ss isieray saes isataetasesdmtaos a te Iaaelcce fctet)acesnaea (aes.aasIntsteeaueote fulm(s)ctt∈GG=o: -)=S(0)c ThektnaaSacQuaseOir s(0)=0. (b)smlay wlh f(S)=CS2. Mdetats sa.cats=p ir LnasarcetceTSG tetfce wexelicti fecss elay stacaagtep pon saicettccme aeen IeC),herteaaty (t)wl &mItaly kc aasecnntr
CHAPTER BASIC CONCEPTS C P r u y d n e Figure Unityfeedback loop the room temperature eventually to change to the new setting of course you would like the change to occur within a reasonable time A situation with a ramp input is a radar dish designed to track orbiting satellites A satellite moving in a circular orbit at constant angular velocity sweeps out an angle that is approximately a linear function of time ie a ramp Dene the loop transfer function L P C The transfer function from reference input r to tracking error e is S L called the sensitivity functionmore on this in the next section The ability of the system to track steps and ramps asymptotically depends on the number of zeros of S at s Theorem Assume that the feedback system is internal ly stable and n d a If r is a step then et as t i S has at least one zero at the origin b If r is a ramp then et as t i S has at least two zeros at the origin The proof is an application of the nalvalue theorem If y s is a rational Laplace transform that has no poles in Res except possibly a simple pole at s then limt yt exists and it equals lims sy s Proof a The Laplace transform of the foregoing step is r s cs The transfer function from r to e equals S so e s S s c s Since the feedback system is internally stable S is a stable transfer function It follows from the nalvalue theorem that et does indeed converge as t and its limit is the residue of the function e s at the pole s e S c The righthand side equals zero i S b Similarly with r s cs Note that S has a zero at s i L has a pole there Thus from the theorem we see that if the feedback system is internally stable and either P or C has a pole at the origin ie an inherent integrator then the output yt will asymptotically track any step input r
-,-PERFORMANCE 35 Ex ane To see hav this workstakethe simplet possibeexample C+13 Then thetransfer function from Ito eequals 1中=3 1 Sothe cpenloop pole at s =0 becomes a dosedloopzerocf theerror transfer fun(tion+then this zeo cancels the pale cf Is=>resulting in no unstable poles in eSimilar remarks apply for a ramp input. Theorem 3 is a special case cf an elemetary principle For perfect asymptctic tracking=the looptransfer function L must COntain an internal model cf theunstable pcles cf I. A similar,analysis can be done for the situation where I=n =0 and d is a sinusoid-say d(sin(,t(1 is the unit step You can shov this If the feedbad system is internally stablezthe y(ti0st、1,ir therP has azero at s=j,or Chas apale at s=方， (Exerase3→ 3.4 p erform anCe In this section we again lock at tradking a reference signal>but whereas in the preceding section weconsidered perfect aymptctictradking cf asingle signal-wewill now consider aset cf reference Signals and a bcund on the steadyktate error.This performance ckjective will be quanti/ed in terms cf aweighted nom bound. As beforeleL dencte the loop transrer functionL=P Thetrasrer function fron rEference input I'totradking error eis 1 S:=1+L calledthe sensitivity function.In the analysistofcllov>it will always be assumedthat thefeedbad System isinternally stable sos is astable-proper transfer function.Observethat sinceL isstrictly proper (since P is->S (j1. Thenamesensitivity function comnesfron thefcllowing idea.Let T denctethetransfer function from I'toy: Oneway toquantify hov sensitive T istovariations in P is totake the limiting ratiocf arelative peturbation in T(i.e≥△T;T-to a relative perturbation in P(i.e>△P:P→Thinking Cf P as a variable and T as afunction cf it>weget 票3 lim Theright hhand side is easily evaluated to bes.In this ways isthe sensitivity cf the clcsedlocp transfer fun(tiOn T to an in/nitesimal peturkation in P. Now wehaetodecide on aperformancespeci/cation-ameasurecf goodhess cf tracking.This decision depends on twothings what we know about I'and what measurewe choosetoassign to thetracking error.USually>Iis nct knovn in advance-few cOntrcl Systems are designed for Ce
PERFORMANCE Example To see how this works take the simplest possible example P s s C s Then the transfer function from r to e equals s s s So the openloop pole at s becomes a closedloop zero of the error transfer function then this zero cancels the pole of r s resulting in no unstable poles in e s Similar remarks apply for a ramp input Theorem is a special case of an elementary principle For perfect asymptotic tracking the loop transfer function L must contain an internal model of the unstable poles of r A similar analysis can be done for the situation where r n and d is a sinusoid say dt sint t is the unit step You can show this If the feedback system is internally stable then yt as t i either P has a zero at s j or C has a pole at s j Exercise Performance In this section we again look at tracking a reference signal but whereas in the preceding section we considered perfect asymptotic tracking of a single signal we will now consider a set of reference signals and a bound on the steadystate error This performance ob jective will be quantied in terms of a weighted norm bound As before let L denote the loop transfer function L P C The transfer function from reference input r to tracking error e is S L called the sensitivity function In the analysis to follow it will always be assumed that the feedback system is internally stable so S is a stable proper transfer function Observe that since L is strictly proper since P is Sj The name sensitivity function comes from the following idea Let T denote the transfer function from r to y T P C P C One way to quantify how sensitive T is to variations in P is to take the limiting ratio of a relative perturbation in T ie T T to a relative perturbation in P ie P P Thinking of P as a variable and T as a function of it we get lim P T T P P dT dP P T The righthand side is easily evaluated to be S In this way S is the sensitivity of the closedloop transfer function T to an innitesimal perturbation in P Now we have to decide on a performance specication a measure of goodness of tracking This decision depends on two things what we know about r and what measure we choose to assign to the tracking error Usually r is not known in advancefew control systems are designed for one
36 CHAPTER 3.BASIC CONCEPTS and only one input.Rat her,a set of possible rs will be known or at least post ulated for the purpose of design. Let's first consider sinusoidal inputs.Suppose that r can be any sinusoid of amplitude 1 and we want e to have amplitude e.Then the performance specification can be expressed succinctly as ‖Slo0 can be reflected into the left half-plane without changing the magnit ude.Let us consider four scenarios giving rise to an oo-norm bound on WiS.The first three require Wi to be stable. 1.Suppose that the family of reference inputs is all signals of the form r=Wirpf,where rpf,a pre-filtered input,is any sinusoid of amplitude <1.Thus the set of rs consists of sinusoids with frequency-dependent amplit udes.Then the maximum amplit ude of e equals WiS. 2.Recall from Chapter 2 that hB=会Kre and that r is a measure of the energy of r.Thus we may think of r(jw)2 as energy spectral density,or energy spectrum.Suppose that the set of all rs is {r:r=Wrpf,lrp时2≤1}， that is { rja)/w(GuPd≤1Y Thus,r has an energy constraint and its energy spectrum is weighted by 1/Wi(w)2.For example,if Wi were a bandpass filter,the energy spectrum of r would be confined to the passband.More generally,Wi could be used to shape the energy spectrum of the expected class of reference inputs.Now suppose that the tracking error measure is the 2-norm of e. Then from Table 2.2, sup llell2=sup{l‖Wirpf2:pfll2≤1}=IWISll oo, so IWiSlloo<1 means that llell2 1 for all rs in the set above. 3.This scenario is like the preceding one except for signals of finite power.We see from Table 2.2 that WiSlo equals the supremum of pow(e)over all rpf with pow(pf)<1.So Wi could be used to shape the power spectrum of the expected class of rs. 4.In several applications,for example aircraft flight-control design,designers have acquired through experience desired shapes for the Bode magnitude plot of S.In particular,suppose that good performance is known to be achieved if the plot of S(jw)lies under some curve. We could rewrite this as lS(0w川<IW(jw川-，w, or in ot her words WiSloo<1
CHAPTER BASIC CONCEPTS and only one input Rather a set of possible rs will be known or at least postulated for the purpose of design Let s rst consider sinusoidal inputs Suppose that r can be any sinusoid of amplitude and we want e to have amplitude Then the performance specication can be expressed succinctly as kSk Here we used Table the maximum amplitude of e equals the norm of the transfer function Or if we dene the trivial in this case weighting function Ws then the performance specication is kWSk The situation becomes more realistic and more interesting with a frequencydependent weighting function Assume that Ws is realrational you will see below that only the magnitude of Wj is relevant so any poles or zeros in Res can be re ected into the left halfplane without changing the magnitude Let us consider four scenarios giving rise to an norm bound on WS The rst three require W to be stable Suppose that the family of reference inputs is all signals of the form r Wrpf where rpf a preltered input is any sinusoid of amplitude Thus the set of rs consists of sinusoids with frequencydependent amplitudes Then the maximum amplitude of e equals kWSk Recall from Chapter that krk Z jrjj d and that krk is a measure of the energy of r Thus we may think of jrjj as energy spectral density or energy spectrum Suppose that the set of all rs is fr r Wrpf krpf k g that is r Z jrjW jj d Thus r has an energy constraint and its energy spectrum is weighted by jW jj For example if W were a bandpass lter the energy spectrum of r would be conned to the passband More generally W could be used to shape the energy spectrum of the expected class of reference inputs Now suppose that the tracking error measure is the norm of e Then from Table sup r kek supfkSWrpf k krpf k g kWSk so kWSk means that kek for all rs in the set above This scenario is like the preceding one except for signals of nite power We see from Table that kWSk equals the supremum of powe over all rpf with powrpf So W could be used to shape the power spectrum of the expected class of rs In several applications for example aircraft ightcontrol design designers have acquired through experience desired shapes for the Bode magnitude plot of S In particular suppose that good performance is known to be achieved if the plot of jSjj lies under some curve We could rewrite this as jSjj jWjj or in other words kWSk
