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IEEE TRANSACTIONS ON AUTOMATIC CONIXOL, VOL. AC-26, NO. 1, FEBRUARY 1981 253 A New Approach to the Perfect Regulation and the Bounded Peaking in Linear Multivariable Control Systems HIDENORI KIMURA, MEMBER, NEE Abstraet--This paper presents a new approach to the perfect regnlation (ps.) and the bounded peaking (bop.) in linear multivariable mntrd sys- tems, based on tbe pole and the eigenvector assignment technique. R-y spealdng, the ps. represents an ideal mntrod d o n which redoees the sett l ing~etoO,whiletheb.p.~~meanstheboundedovershootin this ideal dtuatiom Simple frequency domain chra&A&ions of tbe p.r. and the b.p. are derived. B y reveal some invariance properties d the pa. and the b.p. that provide a pow& tool for achieving the p.r. and the b.p. The egistenee condition for tbe p.r. is derived, wbich tuns out to be isextendedtoamoregeneralcontrd~~veofattainingttseps.forone o w while keeping the b.p. for another output. A condition on Mch the p.r. is realized by an output feedback is also derived. A simple design algorithm is proposed for arhiering the p.r. wb&b is essentially the eigenvector assignment promlure. Finally, an application to a real system kldisar9sed. identicaltotberesuHobtainedintheoplima!regolatoa~.Thisresult I. INTRODUCTTON I T is well known that the closed-loop poles of a control- lable plant can be assigned arbitrarily by a state feed- back [ I ] . Using this fact, we can make the closed-loop response as fast as desired (of course, at the cost of increasing feedback gain) by assigning poles with nega- tively large real parts. However, as the response is made faster in this respect, it usually exhibits an unboundedly growing overshoot or undershoot that is unacceptable for practical synthesis. To illustrate the situation, consider a plant described by The state feedback assigning the closed-loop poles { - p , -2p) is given by #=&x, 4 = [ -2p2, - 3 ~ 1 . (1.1) Fig. 1 shows the phase plane trajectories of the closed- loop system with initial state x(O)= [ 1 OlT. One observes that the peak value of x2 grows unboundedly as p in- creases. Thus, the unbounded overshoot is inevitable for all outputs except for y = xl. The above problem of unbounded output response was Manuscript received February 28, 1980; revised September 29, 1980. The author is with the Department of Control Engineering, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka, Japan. -1 -2 Fig. 1. Phase space trajectories of a second-order Closed-loOp system for various values of p. first treated theoretically by Mita for single-input single- output cases [2]. He showed that the presence of the zeros of the transfer function was the source of the unbounded overshoot. According to [2], if the open-loop system is zero-free, we can increase the response speed without causing excessive overshoot. An immediate consequence of this fact is that the pole-zero cancellation prevents excessive output response in speeding up the closed-loop dynamics. The extension of this observation to multivaria- ble cases is more or less the initial motivation of this paper. A one-parameter family of the state feedback control law like (1.1) is assumed to realize a prescribed set of root loci with respect to p. If the squared integral of the output converges -to 0 as p increases, we say that the control law attains the perfect regulation @.r.). It is almost a synonym to the “infinitely fast response.” Similarly, if the peak value of the normed output response curve remains bounded as p tends to infinity, we call that the control law satisfies the boundedpeaking (b.p.). The problem of the p.r. has been discussed so far mainly in connection with the so-called cheap control of 00 18-9286/8 1 /0200-0253$00.75 0 198 1 IEEE 254 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-26, NO. 1, FEBRUARY 1981 the optimal regulator [3]-[ll]. There, the parameter p in (1.1) appears as representing the relative weight on the output deviation to the weight on the input power in the definition of the cost functional. The p.r. in this context means that the optimal cost tends to 0 as p increases. The condition for the p.r. has been derived in its final form by Francis [ll]. The problem of the b.p. was treated in the same context in [9]. A s d a r type of problem has also been studied in the field of high gain control [12]-[16]. There, the parameter p in (1 . l) appears as a common multiplicative factor of the gain matrix. Some interesting results on the asymptotic structure of the closed-loop root loci have been obtained [12], [ 161, but the asymptotic behavior of the closed-loop response in the time domain, which is the main concern of the design objective, has been studied only for a special case [ 141. In this paper, we approach the problem of the p.r. and the b.p. from the viewpoint of pole and eigenvector as- signment [18], [27]. The two characteristic features of the asymptotic closed-loop eigenstructure of the p.r. will be demonstrated. First, all the finite transmission zeros should be cancelled out exactly or asymptotically. Second, a certain number of the closed-loop poles should be infinite, the eigensubspaces of which are determined according to the prime structure of the plant. In the synthesis of the p.r., we can assign all the asymptotes of the infinite root loci arbitrarily, as well as their order of growth to 00. This offers a considerable flexibility in the synthesis compared with the cheap optimal regulator, where the structure of the asymptotes is determined entirely by the plant. itself [SI. This flexibility demonstrates, in some sense, a su- periority of the pole and the eigenvector assignment tech- nique to the optimal regulator. As a synthesis method, the pole assignment is much more intuitive and computa- tionally easier than the optimal regulator. However, so far it has not been treated as a design tool. The primary reason for this seems to be that there exist many feedback gains attaining the same pole configuration in multivaria- ble cases. We have been unsuccessful in finding the proper way of selecting the one among them, just as with the optimal regulator, where we have no dnect way of de- termining the cost functional. This paper proposes a pro- cedure for the selection by specifying the asymptotic properties of the closed-loop system, just as was done for the optimal regulator in [8]. In Section I1 we give d e precise definitions of the p.r. and the b.p. In Section I11 we derive simple characteriza- tions of the p.r. and the b.p. in the frequency domain. They reveal some interesting invariance properties of the p.r. and the b.p. that provide a key tool in developing the computational procedure. Based on these results, we de- rive the existence condition of the p.r. by geometric ap- proach in Section IV. This condition turns out to be essentially identical to the result for the cheap optimal regulator derived in [ll]. The result is generalized in Section V, where we consider the p.r. and the b.p. for a pair of different outputs. The result obtained there con- tains, in a sense, the result of [9] as a special case. A condition is also derived under which the p.r. and the b.p. are realized by output feedback. Section VI discusses the computational aspects of the p.r. and the b.p., where the notion of interactor introduced in [17] plays an essential role. A numerical example is also shown to illustrate the procedure. In Section VII, an application to a real system is discussed, showing the effectiveness of the design pM- ciple developed in this paper. Notation The real field and the complex field are denoted by R and @, respectively. R" denotes the n-dimensional vector space on R, C - and @ + denote the open left half and the closed right half complex plain, respectively. Capital italic A denotes a map or a matrix and the corresponding sans serif letter A denotes its image. If AVCV for some sub- space V, the restriction of A on V is denoted by AIV. im A and ker A denote the image and the kernal of A , respec- tively. o ( A ) denotes the set of eigevalues of A . The sub- space spanned by vectors x,, - * . , x k is denoted by span {xI, x*,- . . , x&}. Finally, i denotes the set (1,2; - . , i ) . 11. PROBLEM FORMULATION Consider a linear plant described by i = A x + B u (2.la) y= cx (2.lb) where xEX(-R") is the state vector, UEU(-R') is the input vector and ~EY(-R") is the output vector. A : X+X, B: U+X and C: X+Y are constant maps (matrices). It is assumed that ( A , B ) is controllable. To rule out the trivialities, we assume that B is monic and C is epic. Let A,(p), i E n , be complex-valued functions defined for p > 0. Due to the well-known result in [ 11, we can find control laws parameterized by a scalar p > 0 u=F,x, 5: x+u, (2.2) satisfying 4 A +BF,) = { M P ) , A h ) , . * * 9 h m } . As p increases, A i ( p ) , i E n , describe a set of n root loci, whose choice is completely at our disposal. We restrict our attention entirely to those root loci satisfying either Ai(p)+yi(p+0O), Yi E@- (2.3) or - P Ihi(P)+Yi(P+W), Y;EQ=-- (2.4) The relation (2.3) implies that A,(p) is a finite root locus with the terminating point in C -. The set of all the root loci satisfying (2.3) is denoted by A,,. The relation (2.4) implies that A,(p) is a first-order mfinite root locus with the direction of the asymptote yi. The set of all the root loci satisfying (2.4) is denoted by Am. AS far as the synthesis is concerned, these two types of root loci are sufficient for the achievement of the p.r. Hence, we dis- ~ ~~ KIMURA: REGULATION AND BOUNDED PEAKING IN MULTIVARIABLE SYSTEMS 255 card all the infinite root loci of larger or smaller (than one) order that appear in the analysis of the high gain control [ 131, [ 161. Henceforth, we concentrate on < such that U(A+BF~)CA,UA, . (2.5) We denote by uo(A + B<) the subset of o(A + B<) in A, and by u,(A+BFp) that in A,. We sometimes use a slightly abused notation P - ' U ( A + B 5 ) = { p - ' A , ( p ) , . . . , p - ' A , ( p ) ) . Now we make another important assumption on F p : each entry of F p is a rational function of p. Due to [18], F, is chosen as < = [ g17 g2, ..., g , l [ f l ( ~ ) 7 f 2 ( P ) , " ' 7 f ( P ) 1 - ' 7 J ( ~ > = ( A , ( P ) z - A ) - ' B ~ , , where vectors gi are chosen such that J ( p ) are linearly independent. Hence, it is obvious that the assumption holds if A , ( p ) are rational functions. Therefore, this as- sumption on F, is by no means a strong restriction on assigning a set of root loci with a prescribed asymptotic structure. We denote the set of all feedback matrix 5 satisfying the above two assumptions by Fp(A7 B ) , i.e., ! F p ( A , B ) = { F p : a ( A + B F , ) ~ A o ~ A , , ~p is rational in p } . . If there is no possibility of confusion, we simply write as kP' From (2.1) and (2.2), the output of the closed-loop system with initial state x. is written as . Y P ( ~ ; x O ) = X ( t ; x(t; F,) A Cp+BFp)'. (2.6) Now we define the p.r. and the b.p. perfect regulation of the system (C, A , B ) if Definition I : A state feedback (2.2) in Fp attains the (2.7) where 1 1 denotes the Euclidian norm of a matrix. If (2.7) holds, we simply write < 5 (C , A , B) . Definition 2: A state feedback (2.2) in Fp satisfies the bounded peaking for the system ( C , A , B ) if, for some Po 0, sup SUP IIX(t; < 03. (2-8) P > P o f > Q If (2.8) holds, we simply write F p + (C , A , B) . Definition 1 corresponds to the p.r. in the cheap opti- mal regulator, in which the optimal cost converges to 0 as the penalty on input power tends to 0 [6]. The relation (2.7) requires that the closed-loop response becomes in- b.p . finitely fast as p+m. Definition 2 is precisely the same as the b.p. with respect to &,-norm in 191, except that in [9] C is considered to be the identity matrix. It requires that the peak value of the output response is uniformly bounded with respect to p. Of course, a stable closed-loop system with bounded poles only always satisfies the b.p. Therefore, the problem of the b.p. arises only when some of the closed-loop poles tend to infinity. 111. FREQUENCY DO- CHARACTERIZATION OF P.R. AND B.P. The most successful and established approach for at- tacking a group of problems including ours is no doubt the singular perturbation theory developed mainly in the time domain [ 141, [ 191. Based on the assumptions stated in the preceding section, we can derive a simple characteriza- tion of the p.r. and the b.p. in the frequency domain almost algebraically without having recourse to singular perturbation technique. Define the closed-loop transfer function matrix from the initial state to the output by T(s ; Fp) C(sI-A-BFp)-' . Note that, if < E Fp, T(s; Fp) is a rational matrix in p. This fact plays an important role throughout the section. Since T(s; Fp) is the Laplace transform of X( t ; Q, Parseval's identity yields Since the Laplace transform of p-*X(p-'t; 5) is T(ps; e), again appealing to Parseval's identity shows Introducing a new variable ;= p-'t yields Now we state two asymptotic properties of a para- meterized rational matrix, the first of which concerns a "regular" perturbation. Lemma 1: Let U(s; p ) be a strictly proper matrix in s, each pole of which converges to a finite point in C -. If U(s; p ) converges to a matrix U(s ) for p+m, then L-'U(s; p ) converges uniformly to L-'U(s), where L-' denotes the inverse Laplace transform. The proof is simply carried out by a standard analytical technique and is omitted here. Lemma 2: If F p E Fp and T ( p s ; F,)+O, then there exist strictly proper matrices Ul(s; p ) and U2(s; p ) in s such that (3 -3) where Ul(s; p ) converges to a strictly proper matrix U,(s) 256 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-26, NO. 1, FEBRUARY 1981 whose poles are all in 62 - and each pole of q ( s ; p), i= 1, 2, converges to a finite point in @ - . Proof: See Appendix. In (3.3), the first term represents the fast (boundary layer) mode of the closed-loop system, while the second term represents the slow mode. Now we are ready to state the main result of this section. Henceforth, we omit the statement (p+oo) in the descriptions of the limit process if it is clear from the context. Theorem I : F, E ffp satisfies F, "J' (Cy A , B ) if and only if, for each s, T(s; F,)+o, q p s ; F,)+o. (3 -4) Proof: Assume that E ffp and 5 2 (C , A , B) . Let det ( S I - A -BF,)=+,(s; ~ ) $ ~ ( s ; p), where $,(s; p) and 40(s; p) correspond to o,(A + BF,) and oo(A + BF,), re- spectively. The assumption (2.4) implies that, in the repre- sentation #,(s; p ) = s k +a , (p )sk - ' + * e - + a k ( p ) with k the number of infinite root loci, p-'a,(p) converges to a positive number for each j Ek. This implies that p) converges to a positive number for each s. Obviously, I,L~(~; p) converges to a fixed polynomial. Now we write q s ; F,)= a s ; F,)/+,(s; P)rco(s; P). Due to (2.7) and (3.1), U(jo; <)/+,(jw; p)+O for a.e. in w . From the above reasoning, we conclude that ~ - ~ U ( j o ; F,)+O for each w , and therefore, p-%(s; F,)+O for each s. Thus, we have T(s; F,)+O for each s. Similar argument applies to (3.2) and yields T(ps; F,)+O. Thus, the necessity has been proved. If (3.4) holds we can decompose T(s; F,) like (3.3). Due to Lemma 2, ~ , ( p - ' s ; p) converges to a finite matrix U,(O). Therefore, from the first relation of (3.4), we con- clude that Uz(s; p)+O. Since U,(s; p) is a rational matrix in p, the matrix U3(s; p) = pU2(s; p) converges to a finite strictly proper matrix U3(s). Hence, we can represent T ( s ; F , ) = - U , - ; p + - V 3 ( s ; p ) P l c a ) ; W s ; P)+Ul(S)Y Y(s; P)+U,(S) where both U,(s) and U,(s) are strictly proper matrices whose poles are in @-. Let X , ( t , p ) = L - ' q ( s ; p) and X,(t)=L-'U,(s), i = 1,3. Since X i ( t ) represents the output of an asymptotically stable system, we have l m l l X , ( t ) l 1 2 d t < M , i = l , 3 for some constant M . Due to Lemma 2, X , ( t ; p) converges to X i ( t ) uniformly. Therefore, for sufficiently large p, L m / l X i ( t ; p)l12dt po. Hence, for all Re [s] 2 a > 0, 2 M l m e - a ' d t = -. M a Letting a+@ verifies that T(s; 5 ) must converge to a strictly proper matrix. Thus, the first relation in (3.5) has been shown. Since X ( t ; F , ) represents the output of an asymptotically stable linear system, the boundedness (3.6) implies actually 1 1 X ( t ; < Mle --O' for sufficiently large p, where Q is a number satisfying 0 < CJ < min { - Reh,(p)}. Therefore, we conclude that, for sufficiently large p, for some number M>O. Due to (3.2), a similar argument used in the proof of 'Theorem 1 shows T(p jw; F,)+O for each w , and thus, T(ps; F,)+O for each s, which estab- lishes the second relation of (3.5). Assume that (3.5) holds. Since the condition of Lemma 2 holds, we consider the decomposition (3.3) again. The first term vanishes in the limit, so that, from the first condition of ( 3 3 , we conclude that U,(s; p)+T,(s). Due to Lemma 1, X2( t ; p) = L-'U2(s; p) converges uniformly to x,(t) = L -'T=(s). Since all the poles of T,(s) are in @-, sup,,,IIX,(t)ll < M for some M>O. Hence, SUP SUP II X2(C P)II < M (3 -7) p a p o r > O for sufficiently large po. Analogously, X , ( ? ; p) = L-W,(s; p) converges uniformly to x,(t> = L-'u,(s), which is also uniformly bounded. Therefore, we conclude KIMURA: REGULATION AND BOUNDED PEAKING IN MULTIVARIABLE SYSTEMS 257 for sufficiently large po. Taking the inverse Laplace trans- form of (3.3) yields x ( t ; < ) = X , ( p f ; p ) + X , ( t ; PI. The inequalities (3.7) and (3.8),imply that SUP SUP IIX(t; 5)ll P > P O t > O Q sup SUP II X,(p t ; P)II + SUP SUP IIX,(t; P)II P > P o t,o P > P o t > O Q 2 M . The proof is now complete. 0 Theorems 1 and 2 give complete characterizations of the p.r. and the b.p. in the frequency domain. Some analogous results are found in [9] in relation to the b.p. of all the state trajectories in the cheap optimal regulator. Now we discuss the implications of the above theorems. First, we note that the 0 matrix can be regarded as a strictly proper matrix. Therefore, (3.4) implies (3.5). This observation immediately leads to the following. CoroZZary 1: < + (Cy A , B ) always implies Fp + b.p. (Cy A, B ) . Mathematically, there exists a sequence of time func- tions whose L2-norm converge to 0, while their peak values grow unboundedly. The above corollary shows that such a pathological case never appears in the response curves of linear feedback systems. The p.r. always assures the “smooth” convergence of the response curve. Recall that the sufficiency proofs of the theorems heavily