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 |PCF<1 at w =.which ensures that 1+PCF is not strictly proper. Given that no model,no matter how complex,can approximate a real system at sufficiently high frequencies,we should be very uncomfortable if PCF>1 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