Perfect and Subperfect Regulation in Linear Multivariable Control Systems

Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Kyot o, Japan , 1981

PERFECT AND SUB PERFECT REGULATION IN LINEAR MUL TIV ARIABLE CONTROL SYSTEMS H. Kimura Department of Control Engineering, Faculty of Engineering Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560, Japan

Abstract. The perfect ~egulation (p.r.) of linear multivariable systems with external signal is considered. The minimal phase property plays an essential role for the existence of the p.r. Some asymptotically ideal closed-loop properties of the p.r. are demonstrated, such as the complete desensitization, the complete servo performance with decoup1ing and the complete disturbance rejection, which characterize the mu1tivariab1e loop-tightness. An extension of the p.r. to non-minimum phase systems (the subperfect regulation) is derived, based on the cascade decomposition of the plant into the minimum phase part and the totally non-minimum phase part. It is a mu1tivariab1e extension of a well-known design technique for scalar systems to overcome the difficulty of phase non-minima1ity. Keywords. Mu1tivariab1e control systems; Feedback; Pole placement; Minimum phase systems; High gain control. INTRODUCTION There have been so far the two main streams in the field of high gain control. The one is the cheap control in the optimal LQ regulator (Kwakernaak et. al., 1972; Francis, 1979) and the other is the high gain output feedback (Kouvaritakis et. al., 1976; Young et. al., 1977). The former is mainly concerned with the asymptotic behavior of the cost performance in the time domain, while the latter is mainly concerned with the asymptotic structure of the closed-loop root loci in the frequency domain. Recently, a new approach to the high gain feedback problem was given in (Kimura, 1981) from the viewpoint of the eigenvalue-eigenvector assignment. His design principle aims to achieve asymptotically the infinitely fast response without excessive overshoot in the time domain. It allows a wider selection of the asymptotic structure of root loci in the frequency domain compared with the conventional approaches. This control strategy is called the perfect r eguZation (p.r.). In this paper we shall demonstrate that the p.r. can be applied to the output regulation of systems with external signal provided that the plant is of minimal phase and right invertible. Its realization via the dynamic observer is also discussed. We establish the multivariab1e tight-loop properties of the p.r., namely, the complete desensitization, the complete servo performance with decoupling and the complete disturbance rejection. These properties throw a new light on the fundamental closed-loop structure of multivariable control systems. Finally, we shall propose a new design strategy called t he subperfect reguZation, which is an extension of

eST 2 _ N

the p.r. to non-minimum phase systems. It starts with the cascade decomposition of the given plant into the minimum phase part and the totally non-minimum phase part. Then, the p.r. is applied to the minimum phase part, resulting in the improvement of the system performance as well as the simplification of the design. It is a mu1tivariable generalization of a widely accepted design tool for nonminimum phase plants in the classical control theory, whose origin dates back to the Smith's celebrated method of compensating the plant with dead time (Simth, 1959).

Notation : Rn denotes an n-dimensional euc1idian space. Subspaces of Rn are denoted by script letters, e.g., A, B, . The controllable subspace of (A, B) is denoted by
(lc)

1141

H. Kimura

1142

are the state of the plant, that of the external signal, input and the output, respectively. All the coefficient matrices are assumed to be compatibly dimensioned. We assume that (AI' Bl) is controllable and O(A 2 ) E C+. We sometimes use an abbreviated form of (1), z = Dx

x = Ax + Bu, x =

1'

A = [A:

1:1 , 2I

D = [D

(2)

,

A3] A2

B

Problem of Perfect Regulation with Internal Stability (PPRIS)

= [B:J

lim D exp{(A + BK )t} p

t+"-'

Consider a family of state feedback control laws parametrized by a scalar p, (3)

Define a performance index J J

Find an admissible control law (3) satisfying (1)

D21.

l

satisfies either (6) or (7), (ii) each entry of Kp is a rational function in P , is called admissible. Note that, from (6) and (7), Ai(P) E C- for sufficiently large P, and hence any admissible control law satisfies the internal stability. Our problem is now formulated as follows :

p

as

= fooll D exp(A + BK )t 11 2dt POP

(4)

(ii)

Jp

+

0

0,

p>PO,(8)

(p + (0),

(9)

where PO is some positive number. The condition (8) is equivalent to attaining the usual output regulation. The characteristic feature of the p.r. lies in (9). A slight modification of Theorem 2 in (Kimura, 1981) establishes a frequency domain characterization of the p.r. Let the Laplace transform of z(t) be z(s). Then,

where 11'11 denotes the norm of a matrix induced by the euclidian norm in Rn. The perfect regulation means an asymptotically ideal output regulation characterized by Jp + 0 (p z(s) = T(s ; Kp)x(O), (10) + (0). Roughly speaking, this is equivalent T(s ; Kp) = D(sI - A - BKp)-l. to the requirement that the closed-loop output response can be made infinitely fast by increasing p . We also require that all the Lemma 1 : An admissible Kp solves PPRIS if and only if all the poles of T(s ; K ) are in Ccontrollable part of the plant is stable for p sufficiently large p. This condition is usual- for sufficiently large p and ly called the internal stability (Wonham,1978). There are various ways of parametrizing the control law (3). In the usual high gain feedback systems, p appears as a multiplicative factor, i.e., u = pKx. In the cheap optimal regulator, it represents the relative weight on the output deviation to the input penalty. In this paper we parametrize the control law by specifying the asymptotic structure of the closed-loop root loci. Let o (A + BK

P

I
=

{A (p),

A

1

n

(p)}, (S)

l

where n = dim. Due to the pole-assignl ability of controllable systems we can always find a Kp satisfying (S) for any Ai(p). We restrict our attention to Kp for which each A. (p) satisfies either of the following two c6nditions (i) ( ii)

Ai( p) -1

p

+

A. (p) 1

Y E Ci +

Y. E C 1

(p

+ (0)

(6)

(p

(7)

+ (0)

The condition (6) characterizes a finite root locus, while (7) represents a first-order infinite root locus with Yi as the direction of its asymptote. We discard all the infinite root loci of other order, which simplifies the subsequent development considerably. Moreover, we assume that each entry of Kp is a rational function in p. It is easily seen that this condition always holds if Ai(P) (i E ~l) are rational functions. A parametrized control law (3) satisfying the above two conditions, namely, (i) each Ai(P) E o(A + BKpl
Based on this result, we can derive the basic solvability condition of PPRIS.

Th(wJte.m 1 rank

[AI

PPRIS is solvable if and only if

(12)

Dl

Re.maJtR : The condition (12) is identical to the existence condition of the p.r. to (Dl' AI' Bl ) (Kimura, 1981), that is, (D l , AI' Bl ) is of minimum phase (all the transmission zeros are in C-) and right invertible.

PJtoo6

Assume that the state feedback (13)

solves PPRIS.

Simple calculations yield

Lemma 7 implies that T . (s ; K ) + 0 (i = 1, 2). Therefore, Kpl must 1 solve ~PRIS for (Dl, AI, Bl)' From the above remark, the necessity of (12) has been established. The proof of the sufficiency goes as follows: Due to (12), we can find a Kpl that solves PPRIS for (D I , AI' Bl ), and then we choose Kp2 so that Kp = [K pl Kp21 is admissible and satisfies (8),

1143

Perfect and Subperfect Regulation the usual output regulation. This Kp is shown to satisfy (9) as well. Computation of Kp2 is carried out in a similar way to that in (Sebakhy et. al., 1976). Let O(A 2 ) = {AI' ... , Ak} with Vi being the degeneracy index of Ai and let tij' j £ ~i be the generalized eigenvectors of A2 corresponding to Ai. Due to (12) and Ai £ C+, there exist vectors f ij and g .. , j £ v., such that ~J

PERFECT REGULATION BY DYNAMIC OBSERVER In this section, we consider the implementation of the p.r. by the state observer based on the state measurement z. We assume that (D, A) is observable. The minimal order state observer is characterized by the relations

-~

WA = FW + GD O(F)

C-,

£

(17) rank[W' D'l = n

(18)

for some matrix triple (W, F, G), for j £ ~i' where we assume f iO Kp2 is given by

O.

A desired

U

-[f

V

-[gl1

f kV 1[t ll k gkV ][ tll k

ll

The state feedback (3) is then realized by the observercompensator

(14)

Kp2 = KplU + V,

where

F £ R(n-p)X(n- p ), G £ R(n-p) xp.

-1

u

tkv 1 k -1 tkv 1 . k

=

Ft,: + Gy + WEu

(19a)

Hp t,: + J p y,

(19b)

where H £ RrX(n-p) Jp £ Rrxp are determined p , by = lW' D'l-lK p ' . (20)

From these relations, we easily see that

I t is not difficult to see from (17) and (20)

that Taking into account that Kpl solves PPRIS for (Dl' AI' Bl), we see that Ti(s ; Kp) ~ 0, T.(ps : Kp) ~ 0, i = 1, 2. In view of Lemma 1, tfie sufficiency has been established. The following result is an immediate consequence of Lemma 1, which will be used repeatedly in what follows.

Lemma Z : If Kp solves PPRIS, T(s ; Kp)BKp

~

-D,

-1

p

T(ps; Kp)BKp ~O.l (16)

P~oo6 : The assertion follows immediately from Lemma 1 and the identity D + T(s ; Kp)BKp= T(s ; Kp)(sI - A). [J The meaning of the first relation of (13) is illustrated in Fig. 1. The dynamics represented in Fig. lea) is asymptotically reduced to a static transformation of Fig. l(b) as p ~ 00.

+

K.

B~

l

I

(a)

(22)

e = t,: - Wx.

Lemma 3 : If Kp is a solution of PPRIS, H and Jp determined by (20) satisfy p T (s ; K ) BH p p P~oo6

~

0, P

-1

T (ps; K ) BH p P

~

0

(23)

: Let lw' D'l -1 = rp' Q'l.

From (20), T(s ; Kp)BHp = T(s ; Kp)BKpP, T(s ; Kp)BJ p = T(s ; Kp)BKpQ. Due to Lemma 2, T(s ; Kp)BHp ~ -DP = 0 and P-lT(ps ; K )BH ~ 0, which proves the lemma. p p

z(s)

T(s

Kp )x(O) + TO(s ; Kp )e(O),

K)

T(s ; K )BH (sI - F)-I.

P

Asymptotic order-reduction of p.r.

P

P

Since Kp is a solution of PPRIS, T(s ; Kp) satisfies (11). Also, from (23), TO(s ; Kp) ~ 0 and TO(ps ; Kp) ~ O. Therefore, in view of Lemma 1, we have shown that the p.r. is achieved with respect to not only the initial state of the plant, but also that of the observer. Theo~em

1

e = Fe,

Due to the first relation of (18) , e (t) tends to 0 as t ~ 00. In this sense, the observercompensator (19) realizes the state feedback (3) asymptotically. In the present case, however, the gain Hp in (21) may grow infinitely as p increases. Therefore, careful examination on the asymptotic behavior of the closed-loop system is in order.

( p -,> 00)

(b)

Fig. 1.

(21)

By substituting (18) in (1), we have

0

-==

u = Kpx + Hpe,

2 : If PPRIS is solvable, then the p.r. is always achieved by the minimal order state observer.

Henceforth we omit the statement (p if it is evident from the context.

~

00)

H. Kimura

1144

TIGHT-LOOP PROPERTIES OF PERFECT REGULATION In the single- input single- output unity feedback system , the closed -loop transfe r functio n is given by ~ (s) = g(s)/( l + g(s)) with g(s) being the loop transm ission. If the loop gain is increas ed, it is expecte d that ~(s) '\,

(24)

1.

The feedbac k system satisfy ing (24) approx imately is usually called the ti ght-Zoo p system . The tight-l oop system , if it is attaine d retaining the loop stabil ity, exhibi ts some ideal perform ances such as desens itizatio n with respect to plant variati on, disturb ance rejecti on and ideal trackin g perform ance. In this section we shall discuss the multiv ariable version of the above tight-l oop proper ties demons trated by the p.r. The observ er-com pensato r (19) is represe nted as WB, (25a) (F + TH p ) ~ + (G + TJ p)z, T

t

u

= Hp ~ +

(25b)

Jpz,

The block diagram of the closed- loop system (11)(25 ) is illustr ated in Fig. 2, which is simpli fied in Fig. 3, where G(s)

= -D l (sI

U (s) P

- AI)

-1

Spes)

-Sp(S)D (sI l -1 , Rp(S)

Rp(s)

I + G(s)U (s). P

'I' (s) P

(26)

Bl ,

(sI - F - TH )-l(G + TJ P ) + J p' P -1 (27) D (sI -A 2 ) x 2 (0), 2 -1 A (sI - A ) x 2 (0). 2 3

=H

A )-1 1 (30)

Here, Rp(s) is the return differe nce matrix with the cut point 1 in Fig. 3 and Spes) is the associa ted sensit ivity matrix . n Now we shall exploi t a simple repres entatio (n-p)xn l of Sp(s). Let W = [W l W21, Wl E R (n-p) xn 2 and the corresp onding partiti on W2 E R be Kp = [Kpl Kp21. Then, from (17) and (20), we have (sI - F - THp)W l - (G + TJp)Dl Wl (sI - Al - Bl Kpl ) . After some manipu lations using the identit y I - BlHp(s I - F - TH p)-lWl -1 -1 -1 = (I + BlHp(S I - F) Wl ) and (sI - F) Wl -1 (sI - AI) = Wl - (sI - F) GD l , we finally obtain Spes) where Tl(s ; Kp)

= Dl(sI

- Al - BlK pl )

-1

.

Since Tl(s ; Kp)Bl = T(s ; Kp)B and ~ p (s) = I - Spes), we conclud e from Lemma 2 that Spes) -+ 0, ~p(s) -+ I, 'I'p(s) -+ O. (31)

P

The relatio ns (31) demon strate the tight-l oop res) proper ties of the p.r. The first one implie s that the sensit ivity of the output with respec t to the plant variati on tends to 0 as p indes) crease s. This proper ty is referre d to as the comp Zet e desens itiz ation. The second one is The plant output zl = -Dlx is represe nted by the multiv ariable genera lizatio n of (24), im(28) plying that the plant output reprodu ces the referen ce signal res) comple tely in the limit. where it implie s that the p.r. attains the Also, (29) (s), U (s)G(s) S ~ (s) = p p p comple te decoup ling asymp totical ly. This is referre d to as the compZe t e servo perform ance with decoup Zing. The third one implies that the rr=======~ A 31======d~s"==;-J --, of the disturb ance r ----' effect r - -- - - - --1 I ' , 1 the output is comon des) 1 "' 1 d in the rejecte pletely 1' 1 1' limit. This proper ty is referred to as the comp Zete " " disturb ance re jection . We " " summar ize the above discus ,L ___ _ ___ _ ______ _ _ _ " L __ ______ __ ___ _ sions. ~

Fig. 2.

Block diagram o f over-a ll closed -loop s ystem.

d(S)

r (5) _~"I.I"_--'"

Fig. 3.

Simpli fied r e presen tation of Fig. 2

~

3 : The p.r. achieve d by the observ ercompen sator posses ses asymp totical ly the following tight-l oop proper ties : (i) the comple te desens itizatio n (Sp(s) -+ 0), (n) the comple te servo perform ance with decoup ling (~p(s) -+ I), (Di) the comple te disturb ance rejecti on ('I' p (s) -+ 0).

Theo~em

An interes ting feature of the multiv ariable tight-l oop proper ties which has no counte rpart in single- input single- output cases is the asympt otic decoup ling in the proper ty It is worth noting that this proper ty (n). for system s in which the decoup ling even holds

114 5

Perfect and Subperfect Regulation cannot be attained in the usual sense. The connection between the loop-tightness and the decoupling was discussed in a different context in (Mayne, 1973). The multivariable closedloop properties of the cheap optimal control realized by the Kalman filter have been discussed in (Kwakernaak, 1972 ; Doyle, 1979), where the design objective was to maintain Sp (j w)* Sp(j w) ~ I which is attained by the state feedback. Here, we have derived a stronger result Sp(jw)*Sp(jw) + O.

L

dual of V*, the maximum (A, B)-invariant subspace contained in Ker D. A subspace V- of V* is defined by V- = max{V c V* : O(A + BKIV) c C- for some K}.

Lemma 4 : Assume that (D, A, B) is right invertible. If there exists a subspace I such that AI C I, X = I e (T* + V-), where X denotes the state space, (D, A, B) is similar to a system of the form

l- Am BD

n m

o l}V m

}V

A }v ,

}V

n

n

m n

(32)

D 'U [ D

m

V

m

V

BIn

- D n

1

Vn + p,

;\

C-. (35)

E

PItOOn : Omitted. The similarity relations (32) implies that the given plant (D, A, B) is the cascade connection of the system (D m, ~, Bm) whose t.z. are all in C- from (33) and the system (D , ~, Bn' I) whose t.z. are all in C+ from (35). Actually, a simple calculation yields n

It has been a well-known fact in the practice of control engineering that for non-minimum phase system the loop cannot be closed tightly without violating the loop stability. Theorem 1 demonstrates a multivariable generalization of this fact. In single-input single-output cases, we can always decompose the system into the minimum phase part and the non-minimum phase part in a cascade form. It is possible to close the loop for minimum phase part by adding a minor loop around the main controller, resulting in the improvement on the closed-loop performance. This is a common remedy to overcome the difficulty of the phase non-minimality, whose origin dates back to the Smith's celebrated design method for systems with large dead time (Smith, 1959). We extend this design procedure for general non-minimum phase mu1tivariable systems. The procedure is based on Lemma 4 below, which asserts that, under a mild condition, a plant can be represented in a cascade connection of the two sybsystems, one of which is of minimal phase and the other is of totally non-minimal phase. To state the lemma, we recall some geometric concepts developed in (Wonham, 1978). Let T* be the subspace defined by T* = min. This is the

'U

[

AI - A n

D(sI - A)-lB = C (s)G (s)

SUBPERFECT REGULATION FOR NON-MINIMUM PHASE SYSTEM

A

rank

n

Gm(s) =Dm(sI-Am)

m

-1

Bm' Cn(s) =Dn(sI-An )

-1

Bn+l.

(36) It is clear from the form of (32) that if (A, B) is controllable, both (~, Bm) and (Au, B ) are controllable. In that case, due to (~3) and Theorem 1, there exists a solution Kpm of PPRIS for (Dm, ~, Bm)' Using this, we characterize the state feedback control law by u = KPmxm + KPnxn' (37) K K + Kn' (38) pm n r xvn where Ku E R , and Kn E R are constant matrices. Lemmas 1 and 2 applied to Kpm and (Dm ~ Bm) yield K pn vmxvn

~ (s ; K )B K

l'm(s

K ) pm

~

K )B K + pm m pn

m

(s

+

0

'm

K n

pm

-D K

m n'

m pm

+

-D (39) m (40)

where

~

(s ; K

m

pm

)

= Dm(sI

- A - B K )-1.(41) m m pm

Since (A , Bn) is controllable, we can always find Ku ¥or which An + BnKu is stable. _Due to the condition_(34), we can always find Ku such ~hat Ku = -Dm~' Henceforth, we assume that Ku in 160) is chosen in this= way , i.e., O(A n - BnDmKu) C C-. The matrix ~ in (38) can be chosen arbitrarily and does not affect the asymptotic behavior. The control law (37)(38) described above will be called the subperfect regulation (s.p.r.). The asymptotic relations (39) are illustrated in Fig. 4. In the original block diagram Fig. 4(a), the block inside the broken lines asymptotically behaves as the static transformation -Dm, as was already shown in Fig. 1, thus, in the limit, the closed-loop system obeys the reduced order dynamics in Fig. 4(b). This asymptotic order reduction is the most significant characteristic feature of the s.p.r., which will be discussed more extensively.

having the following properties

rank [" ::m :m 1 rank D

m

p,

A E C+ ,

(33)

(34)

In what follows we assume that the problem of output regulation with internal stability (PRIS) is solvable for the system (1). If the plant (D l , AI' Bl ) is decomposed in a way described in Lemma 4, the system (1) is written as (42a) x m

1146

H. Kimura r - - - - - - - - - - - - - - - - - -, I

written as T2 (s; Kp) =Tm(s; Kp)U m + Tn (s ; Kp)U n . Letting p .... yi elds T200(s) = Tnoo (s)U n . Due to (45), this relation implies that the feedb ack control (47) attain the output regulation (8). Thus we have the following result. 0<>

I I

z .(s)

vIs)

~======~ ~ .~====~

I

(a)

( P -00 )

z .(s)

vIs)

Th eo~em

4 : The s.p.r. for the system (42) converges to the reduced-order closed-loop system (46)(47) which attains the output regulation. An analogous result can be derived f or the case where the state feedback is realized by the dynamic observer.

(b)

CONCLUSION

The design strategy based on the p.r. Asymptotic s tructure of s.p.r. has been established for the output regulation of linear mu1tivariab1e systems with external signal. The use of observers has been x A x + BD x + A x (42b) discussed for realizing the design objective. n n mm n n 3n 2 Some asymptotic closed-loop characteristics of Ax x (42c) the p.r. have been demonstrated, which estab2 2 2 lish the notion of the mu1tivariab1e loopz = D x + D x + D x (42d) tightness. The application of the p.r. to nonn n 2 2 mm minimum phase plant (s.p.r.) has been discussed A control law attaining the s.p.r. is given by based on the cascade decomposition of the plant into the minimum phase part and the non-minimum u = K x + K x + K x , (43) phase part. The asymptotic structure of the pm m pn n p2 2 s.p.r. has been examined. where Kpn is given by (32) and Kp2 is determined by the procedure in the proof of Theorem REFERENCES 1 for Kp = [Kpl Kp21. If we write U = [Urn' Un'l ' in (14), Kp2 is represented as Doy1e, J. C. and Stein, G. (1979). Robustness with observers. IEEE Trans . Autom. Con(44) trol, AC-24, 607-611. Francis, B.~1979). The optimal linear Simple but tedious calculations yield quadratic time-invariant regulator with cheap control. IEEE Trans. Autom. Con z p (s) = T (s ; K )x (0) +T (s ; K )x (0) trol, AC-24, 616-621. m P m n P n Kimura, H. (1981). A new approach to the + T (s; Kp )x (0), perfect regulation and the bounded peaking 2 2 in linear mu1tivariab1e control sys tems. IEEE Trans. Autom. Control ,AC-26. '" where T (s ; Kp) = G (s)(I - T (s ; K )B K m '" n m Pm m pn Kouvaritakis, B. and Shaked, U. (1976). Asymp(sI - An )-lB n )-l Tm (s ; Kpm) , Tn(s ; Kp) = Tm totic behavior of root loci of linear multivariable systems. Int. J. Control , (s ; Kp) BmKpn + Dn)(sI - ~)-l and T2(s ; Kp) Fig. 4.

Q, 297-340.

+ B K 2) + T (s ; K )A = (T (s ; K )(A mp n P 3n m p 3m + D )(sI - A )-1. In view of (39)(40), we 2 2 conclude that T (s ; Kp) .... 0, T (s ; K ) m n -l P .... T oo (s) (D + K ) (sI - A - B K) and n n n n n n T2 (s ; Kp) .... T2002~) = (D 2 + K2 + Tnoo(S) (Bn K2 + A3n »(sI - A2) where K2 = -DmUm + KnUn' Thus, in the limit, the closed-loop response is described as zoo(s) = Tnoo (s)xn(O) + T2oo (s)x (O). 2

(45)

In the time domain, it is written as x

n

z00

A x + A x + Bnu, 3n 2 n n Dx + Dx + u n n 2 2

u = Knxn + K2x 2 .

x

2

= A x (46a) 2 2 (46b) (47)

The relation (15) holds in this case, which is

Kwakernaak, H. and Sivan, P. (1972). The maximally achievable accuracy of linear optimal regulators and linear optimal filters. IEEE Trans. Autom. Control , AC-17, 79-86 . Kwakernaak, H. and Sivan, R. (1972), Linear Optimal Control . Wi1ey Interscience. Mayne, D. Q. (1973). The design of linear mu1tivariab1e systems. Automatica, 9, 201-207. Sebakhy, O. A. and Wonham, W. M. (1976). A design procedure for multiva riable regulators, Automatica, 12, 467-478. Smith, O. J. M. (1959). A controller to overcome dead time. I BA Journal, 6, 28-33. Wonham, W. M. (1978). Lir.ear Multivariable Control : A Geometric Approach, 2nd Ed. Springer. Young, K. K. D., Kokotovic, P. V. and Utkin, V. I. (1977). A singular perturbation analysis of high-gain feedback systems.

IEEE Trans. Autom. Control . MCll,931-938. For Discussion see page 1158