Almost disturbance decoupling with internal stability: frequency domain conditions

IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. 35. NO. 6. JUNE I990 719 the point E, while B is the moment of inertia relative to the longitudinal link axis. The equations of motion of this mechanical system have the form (T + B + (mp’ + c - B ) sin’ pi ); . . +(mp’ + C - B) sin 2 9 , pi p = M 1 2 (mp’ + A ) & - -(mpZ + c - B ) sin 2pI $’ = M’ + mgp sin qI where g is acceleration of gravity. The following constraints are to be imposed on the control torques: At the initial moment of the time, the system is in the given configu- ration p(0) = 0, d o ) = 0, pi (0) = $71 1 $1 (0) = 0. It is required to find the control functions, assuring the transfer of the manipulator in minimal possible time T to the assigned final configuration dT) = 1 &T) = 0, PI ( T ) = $717 > $1 ( T ) = 0. Additionally, assume that 91 (0) = 9, (T). This condition is analo- gous to condition (1.9) and we can find the solution satisfying symmetry conditions ( I . 12). Investigations, similar to the above, have been carried out for a ma- nipulator with a rotating link. Simplified equations, obtained from the initial ones after neglecting the centrifugal force moment, were analyzed. Singular movement for this model consists of turning the base with a vertically positioned link. It is analytically proved that the movement with one switching of M ( t ) and a finite number of switchings of Mi ( 1 ) containing a singular mode cannot be optimal for the simplified equations and for the complete ones as well. The control and movement, satisfying Pontryagin’s maximum principle, have been numerically designed both for the initial and simplified systems with some values of dimensionless parameters. In this movement the link oscillates relative to the vertical and the movement M ( t ) switches over once. The comparison of the results obtained for two kinds of manipulators shows that their optimal movements have the same form. V . CONCLUSION In this note, the true structure of the time-optimal motions of a manip- ulator has been presented. True optimal motions of industrial robots are important from two points of view. First, if the optimal control is not too complex, it can be realized. Second, if the optimal control cannot be realized because of its complexity, it can be compared to a simple but perhaps nonoptimal one. REFERENCES [I] N. V. Banichuk and V. M. Mamalyga, “Optimal control in the nonlinear mechan- ical systems with the changeable inertia characteristic.” Izv. AN SSSR. I€€€ Tmns. Microwave Theory Techn., no. 2 , pp. 6-12, 1976. [2] E. I. Vorob’ev and A. N. Shchegoleva, “Time-optimization of the pneumatic ma- nipulator by the choice of the actuator’s switchings,” Maschinovedenie, no. 3. pp. 24-26, 1978. [3] L. D. Akulenko. N. N. Bolotnik, and A. A. Kaplunov. “Control optimization of the manipulators,” Preprint Inst. Problems of Mech. AS U S S R , no. 218. 72 pp.. 1983. [4] H. P. Geering. L. Guzzella. S. Hepner, and C . H. Onder. “Time-optimal motions of robots in assembly tasks,” IEEE Truns. Automat. Contr., vol. AC-31. pp. 512-518, 1986. [5J V. F. Borisov and M. I . Zelikin, “Chattering modes in the problem of robot’s control,” AS USSR, Appl. Mech. Math. submitted for publication. 161 L. S. Pontryagin. V . V. Boltyansky, R. V. Gdmkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes. [7] A. T. Fuller, “Study of an optimal nonlinear control system.” J. Electron. Contr.. vol. 15, no. I . pp. 63-71, 1963. 181 C . Marchal, “Chattering arcs and chattering controls.” J. Optimiz. Theory Appl.. vol. I I , pp. 4 4 - 4 6 8 . 1973. Moscow: Nauka, 1969. 191 R. Gabasov and F. M. Klrillovd, Singular Optima/ Controls. Moscow: Nauka. 1973, 156pp. [ 101 C . D. Johnson, “Singular solutions in problems of optimal control.” in Advances in Control Systems. Theory and Applications, C. J. Leondea. EA. New York: Academic, 1965. pp. 209-267. [ I I ] H. J . Kelly. R. E. Kopp, and H. G. Moyer, “Singular extremals,” in Topics in Optimizations. G . Leitmann, Ed. [I21 J. P. McDonnell and W. F. Powers. “Necessary conditions for joining optimal singular and nonsingular suharcs,” SIAM J. Contr.. vol. 9. no. 2, pp. 161-173. 1971. New York: Academic, 1967, pp. 63-101. Almost Disturbance Decoupling with Internal Stability: Frequency Domain Conditions A. BULENT OZCULER Absfmcf - This note considers the almost disturbance decoupling prob- lem with internal stability. The problem is that of determining a dynamic measurement feedback which makes the Hx-norm of the disturbance input-to-regulated output transfer matrix arbitrarily small while achiev- ing internal stability. It is shown that the solvability condition in fre- quency domain for this problem is a purely algebraic one and can be formulated in terms of a two-sided matrix matching equation involving polynomial system matrices. This is known to be a zero “ d a t i o n wn- difion. A synthesis procedure for the compensator in frequency domain is also given. I. INTRODUCTION A good deal of control theory literature has been concerned in recent years with modifying the behavior of a two-channel system in transfer matrix representation where the control input is U,, the disturbance input is u d , the measured output is y m , and the output to be controlled or regulated, the regulated output, is y r . The control scheme for the plant is a dynamic output feedback at the control channel U, to y,, . Thus, U, = -Z ,y , + U, where Z , is the transfer matrix representation of the compensator and U, is an external input. The control objective consists of various require- ments on the closed-loop transfer matrix between u d and y , denoted by Zdr . In the disturbance decoupling problem, the objective is to achieve Zdr = 0 by appropriate choice of the compensator. The regulator prob- lem aims at placing the poles of z d , in a given stability region. In the standard Hm-optimization problem, one of the objectives is to mini- mize the Hm-norm of Z d , by a suitable choice of the compensator. (See, e.g., Francis [ 2 ] . ) Finally, in the almost disturbance decoupling (AD) problem of Willems 191, the objective is to find, for every given positive t , a compensator Z,( t ) , possibly depending on E , such that the H m - norm of Zdr ( t ) 5 e . A second control objective in all these problems is to attain internal stability in the closed-loop system. Some of the four problems listed above with or without internal stabil- ity have been well investigated and solutions by many different approaches exist. We refer the reader to [ l ] , 13). [9], 121, respectively, and to the references therein, for a fairly up-to-date bibliography. The exceptions Manuscript received July 25, 1988: revised June 9. 1989. Paper recommended hy The author is with the Department of Electrical Engineering, Bilkent University, IEEE Log Number 9034502. Associate Editor, A. C. Antoulas. Ankara, Turkey. 0018-9286/90/0600-0719$01.00 @ 1990 IEEE 720 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. 15. NO. 6. JUNE 1990 are the regulator problem with internal stability (RIS), in the gener- ality posed above, and the almost decoupling problem with internal stability (ADIS). This paper is concerned with ADIS, where the sta- bility region is the closed left half complex plane. The problem ADIS has been formulated by Willems IS]. after a development of almost con- trolled and conditionally invariant subspaces, who also gave frequency domain and geometric solvability conditions for AD. Taking this as our model, we derive the frequency domain conditions for ADIS. However. unlike [SI, the synthesis procedure for the compensator is purely in the frequency domain. (The geometric conditions are obtained in 151. in an indirect manner "translating" the frequency domain condition via the use of polynomial models of Fuhrmann.) A . Notation We follow the notation and terminology of [6] closely. The reader is referred to [7] for various properties of HX-matrices that we use. Some of the special notation used in this note is listed below: R and C denote the real and complex numbers, R[z] and R ( z ) denote polynomials and rational functions of real coefficients in the indeterminate z , as usual. Further, I C ( : magnitude of c E C C - := { c EC: Re(c) < 0}, Cj, := { j w : w E R } C , := {c E C : Re(c) > 0) d e g ( a ) : = d e g ( ~ ) -deg(q) , a = p / q € R ( z ) , p , q t R [ z 1 R(z) - : strictly proper rational functions R(z)o: proper rational functions ( 1 , w : stability regions (or sets) R(z)!!: rational functions with all poles in 12 R(z)- IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL. 35. NO. 6. JUNE 1990 72 1 and matrices U, V over R(z), satisfying Proofi It is enough, by the problem definition, to produce a proper Y, satisfying Definition lii) and achieving Z,, E R(zE x’ . Then, Z , :=Y,(I - Z I Y c ) - ‘ is clearly a solution to the problem. Neglecting the constraint of properness for a moment, consider the choice Y, := - N 2 D - ‘ Q E - ‘ M 2 , where N 2 € R ( z ) E X r and M z ER(Z)I;’ are de- fined by the partitions M := [MI : M z ] , NI := [ N : : NS]. By straight- forward computations using (6), one obtains Z , Y , = - [ P - ( P N , - W , N 2 ) D - ‘ Q ] E - ‘ M 2 , Y,Zl = -NzD-‘ [QE-l(MI R+Mz Wl)-R], Zj -ZIY,ZI = Wl +(PNl -W~N~)D-’[R-QE-’(MIR+M~WI)]+ P E - ’ ( M 1 R + M 2 W l ) , which are all matrices over R(z)!,. Moreover, Zdr = W4 + T U - VS + V Q U , which is over R(z), . Consequently, the above choice of Y, satisfies all requirements save properness. It is now easy to show that given matrices-M,-N, U, V over the s-tated rings satisfy (6). thefe also exist matrices M, U such that M , N , U, V satis- fying (6) and Y, := - N z D - ’ Q E - ’ k 2 is proper. This is achieved by introducing a sufficiently large number of w-stable poles into M , U. We omit further details. e Lemma 2: Let E and D be Il-stable polynomial matrices and let & = II,Xn2 have a solution X inR(z) iJ+m’X‘rip’ , Then, the problem IUS (R, w ) is solvable for all w 12. Prooj? Let (7) be a partitioning of X such that X I is r x r . Let the polynomial matrices A , , Bi be as in ( 5 ) . Set MI :=Ao(QXI + R X 3 ) + A 2 ( T X I - W 3 X 3 ) , M2 :=Ao(QX2+RXI)+A2(TX2-W,X4)-AI, U : = A O S - A z W d - AiW2, N I : = ( X , Q - X2P)Bo + ( X I S +X2W2)B2, N2 : = ( X , Q - X 4 P ) B o + ( X 3 S + X 4 W 2 ) B 2 + B I , V : = - T B O + W 4 B 2 + W 3 B I and note that [MI : M2] := M and [Ni : N : ] := N’ are matrices over R(z)Q and U, V are polynomial matrices. Further, M , N , U , Y can be verified by direct substitution to satisfy (6). Since a polynomial matrix is w-stable rational for any w , it follows that MS(f2, w ) is solvable for all w in 12. a and W := 2 4 - z3xzz = 0, ( 8 ) W,. :=ZA - Z j Y , Z z tR(z )ZX“ (9) then for every real number t > 0, there exists a matrix X , (t) t R(zK:’ such that the matrices Z I X , ( t ) , X , ( t ) Z , , Z , - Z , X , ( t ) Z l are over R(z)!! and 1124 - Z3X,(t)ZZ )I I 6 . (10) Proof; Let p := deg ( X ) and 7 := 11 W , 11 %. If p 5 0, then one can set X , := X to prove the claim. So, let p > 0. Given any t > 0, if 1 := 11 W , 11 3L 5 t , then set X , := Y,. Otherwise, 7 > t and we proceed as follows to choose X , . Since W , is strictly proper, there exists a real number p > 0 for which sup O { W , ( j W ) } < t2-’ I w I ’ 0 Let a real number h be such that 0 < h < t2 - ’ / [p (? l2 - t22-2”)1’21 and consider the strictly proper w-stable rational function f ( z ) := ( zh + l ) - ’ . Note that l l f l l 3c = 1. We now claim that x , : = f ” X + ( I - f”)Y, (11) satisfies all the requirements. First note that deg ( X , ) 5 deg (f’ X ) + deg[(I - p ) Y c ] = -p +deg(X) +deg(Y,) < O by the choice of p and by properness of Y , . Hence, X , is a proper R-stable rational sta- bility followed by the stability of every term in its definition. Moreover, Z I - Z I X , Z I = ~ ” ( Z I - Z I X Z I ) + ( I - ~ ” ) ( Z ~ -ZIY,ZI) ,whereal lare matrices over R(z),! by the use of the hypothesis. Finally, we show ( I O ) byinductiononp. Le tp = I.NotethatT, : = Z 4 - Z 3 X c Z 2 =(I-f)W, by ( 8 ) and (1 I) . By exactly the same manipulations as in [IO], it can be shown that Z I X , = f ” Z , X + ( I - f ” ) Z , Y , , X , Z , = f ” X Z , + ( I - f ” ) Y c Z , , - ~i~~~ Q,; = 1, 2, 3 , 4 , let 0, P be unimodular polynomial matrices This establishes (10) for P = 1. Now let (10) hold for all 1 < p < 7 and consider the case p = r . In this case, T, = Z., - Z3 I f ‘ X + ( I - f‘)Yc]Z2 = ( 1 - f ’ ) W , = ( I - f ) W , + f ( l - f ’ - - ’ ) W c . Therefore, using the fact that llfll = 1 and r > I , by the induction hypothesis we such that A , A , n2P = [C 01, Un, = 0bv= [Ai A 4 ] have with B of full row rank and C of full column rank. Let B = &&, ~ ~ * ~ ~ ~ m z 5 and C , respectively, where B, and c, are nonsingular, with respect to R. - f ) w c I I x - f T - l ) W c / l x < € / 2 +€/2‘-’ < e . 0 We can now State and prove the main result. Theorem I : The problem ADIS(f2) is solvable if and only if the polynomial matrices E and D are 12-stable and there exists x E ~ ( Z ) : ; + m ) ~ ( r + p ) satisfying C = CsCu be stable-unstable and unstable-stable factorizations of B This proves (10) for arbitrary p . Lemma 3: The following are %$valent. a) ?ere exists x E !(z):Jfm’ ( r + p ) suchJhat IL = n 3 x r i 2 . b) A2 = 0, A3 = 0, A4 = 0, and h;’AIC;’ is polynomial. Proofi See 16, Remark 3.1 I] . e n, = I13XI12. (12) 111. A FREQUENCY DOMAIN SOLUTION TO ALMOST DECOUPLING In this section, we give a frequency domain solvability condition and a synthesis procedure for the solution. The solvability condition is what might have been expected from the results in [9] and [6]. There should be a stable rational solution to I& = X &, whereas the proof is somewhat more involved than the proofs of the results for the exact version and for almost decoupling without internal stability. Proof- “Only ff ’I: Let the problem be solvable so that, by prob- lem definition, for every real positive e , there exists a proper rat- ional matrix Z , (t) internally R-stabilizing the plant and achieving, with Yc(€) := Z,( t ) I f + ZIZc( t ) l - ’ , (13) Hence, by Definition I , E and D are Cl-stable and one has all the matrices Y c ( t ) , Z I Y , ( ~ ) , Y < ( E ) Z I , Z , - Z I Y r ( t ) Z ~ over R(z)!!. Consider the ( r + m ) x ( r + p ) matrix ~ ~ z d r ( ~ ) ~ ~ x = lIz4 -Z3Yr(t)Z2IIm 5 Throughout this section R : = C _ UC,, and w : = C - . The following result constitutes a crucial step in the construction of an almost decoupling compensator. The style of its proof is borrowed from solvability” of the equation Z4 = Z3X over R ( z ) . Lemma 4: Consider the transfer matrices Z , , Z 2 , Z 3 , Z4 of (1). If there exist Y , ~ R ( z ) l ? ’ and X E R ( z ) ; ~ ~ such that the matrices Z I Y , , Y , Z l , Z l - Z , Y , Z , , Z , X , X Z I , Z I - Z , X Z , areoverR(z)!!, 1 [ yc(t)PQ-’ Y C ( f ) an analogous result of Van Der Woude [ I O , Theorem (3.4)] on “almost Q-’ - Q-’RY,(E)PQ-’ -Q-’RY,(t) Y(t) := which is proper by properness of Y,(t) and by A). We now show that Y ( e ) is also 0-stable rational. By 12-stability of E = gcrf(P, Q ) and 122 IEEE TRANSACTIONS ON AUTOMATIC CONTROL. VOL 35, NO. 6. JUNE 1990 D = gclf(Q, R ) , there exist 12-stable rational K, , L, , i = I , 2 satisfying K , Q + K 2 P = I , Q L , + RL2 = I . Note that Q - ' R Y , ( t ) = ( K , R -KzWi)Y , . ( t ) + K z Z i Y , ( t ) , Y , ( t )PQ- ' = Y c ( t ) ( P L i - W I L ~ ) + Y , ( € ) Z i L z , Q- ' - Q - ' R Y , ( E ) P Q - ' = L , + K , - K I Q L , - K2WlL2 + K ? ( Z i - Z I Y , . ( E ) Z I ) L ~ + ( K i R - W I L ~ ) Y , ( E ) [ ( P L I - W I L ~ ) - K , Z , Y , ( t ) ( P L , - WIL2) - Z I L ~ ) which are all %stable rational. Thus, Y ( E ) is in R ( Z $ " " ' ~ ~ ~ ' ' ~ ' . Next note that Y ( E ) satisfies 0 [: Z d A € ) 1 ' ir, - n3Y( t ) r i z = In view of (13). we have that given any t > 0, there exists Y (t) t R(z)L,"' ' p i such that iin - rw(twiix I e . Ker 112 G K e r b , I m n GImII , ( 14) (15) It immediately follows by 19, Appendix] that over R ( z ) . Consequently, there exist unimodular polynomial matrices U and f' such that un, = [E' : O]', 112 P = [C : 01 where E is of fu l l row rank b and C is of full column rank c and OIL, i/ is in the form U V = [ A 0 ] 0 0 with A € R [ Z ] ~ ~ ' . Let E :=B,B1, C :=C,C', be unstable-stable and stable-unstable factorizations of B , C , respectively, with B,, and C, nonsingular. Let U be an w-stable polynomial of degree deg ( U ) > max {deg ( A ) , deg (b), deg (C), deg ( U ) , deg ( f')} and define the H"-matrices A := A u - * , B := B u - ' , C := C u - I , U := U 0 - I . V := Po-'. Note that, in view of (14), we have lIA - BY(t )C / I 3c I t 11 U11 1 1 VI1 %. Since / I U11 r y , / / VI[ 'x are both indepen- dent and E , it follows that if ADIS(Q) is solvable, then for every E > 0 there exists Y ( t ) t R ( z & " " l X ' r t p ) such that IIA -BY( t )CII , < E . (16) Consider the inner-outer factorizations [7] of B and C of the type B = B,B,, C = C , C , , where B, and C, are square inner matrices and Bo and CA are outer matrices. Note that E, = BUM, C, = N C , for some 0-bistable matrices M and N (i.e., B, and B, have the same zeros in C , ). We now show that B,-'AC,-' is 12-stable rational from which it immediately follows that (b-)lAC;' is f2-stable rational, or equivalently, polynomial. By the properties of inner matrices, we have B,- ' (z) = B:(z), C , - ' ( z ) = C:(z) for all z E C and, by (16), SUP IIK(jw) - U j w ) I I = IIA - BY(E)CII, IC (17) w tR where K :=B,?AC,*, L :=B,Y(t)C, . Thus, for all w E R , we have llK(jw)ll It + 1 I U j w ) l l i E + IILIIX (18) since by (17) it holds that IIIK(jw)II I IL( jw)II I i I lK ( jw ) - L ( j w ) l l I IIK - L I I x 5 6 . Now, if K has a pole zo t C , then consider a region R in C , having zo and a segment of jw-axis on its boundary and in which K is analytic. Such a region exists since K has only a finite number of poles. Since zii is a pole of K, there exists z , t R sufliciently close to ~0 for which liK(zl)ll > t + llLilx. By the maximum modulus theorem, it follows that for some j w on thc boundary of R, one has 1 1 K(jw)lI > E + 1 1 LII x , which clearly contradicts (18). Therefore, K = B,-'AC,-' is free o f C , poles as we set out to prove. By this, inclusions (14), and by Lemma 3, it follows that (12) admits an (2-stable rational solution. "If ": Let E and D be SI-stable polynomial matrices and let there exist an SI-stable rational X such that (12) holds. We first show that there exists Y E R(z):y x p such that Z , Y , Y Z , , Z , - Z , Y Z , are matrices over R ( z ) ! ) and Z4 = Z 7 Y Z 2 . By !?-stability of E and D and by A), there exist matrices Y , , i = I , 2 , 3 , 4 over R(z ) ! ! such that Q Y , + RY, = I , Y I Q - Y > P = I , QY2 +RY4 = 0 , Y 3Q - Y I P = 0, by [6, Lemma 4. I ] . Let X be partitioned as in (7), and define, as in [6,-p. 7621 (correcting a sign mistake in the definition of X 7 ) the matrices X , , i = I , 2 , 3, 4, by X I : = 2 Y , +XI - Y , Q X , - X , Q Y , - Y l R X , + X * P Y , , X Z : = Y z + X Z - Y I R X 4 - X I Q Y z - Y l Q X 2 + X 2 P Y 2 , X7 :=Y7 + X , - Y ~ R X I -X ,QYI + X ~ P Y I - Y . I Q X I , X , : = X 4 - X 3 Q Y 2 - Y , R X 4 + X 4 P Y 2 - Y 3 Q X 2 which are all matrices over R ( z ) ! , . It follows that x : = [ xi x , ] x , x, satisfies IL = n-,kIIz and further QX,.+RX4 = 0, k , Q - X 4 P = 0. Now let Y :=X4 and note by 1b = I13XI12 that Z4 = Z 3 Y Z 2 , and also by the last two equalities above, the matrices Z , Y , Y Z I , Z , - Z , Y Z , are all over R ( z ) , ) as can be verified by a routine computation. In addition to the existence of Y as above, our hypothesis also yields, by Lemma 2 in Section 11, the existence of Y , E R(Z) ," , :~ such that Z , Y , , Y , Z I , Z , - Z I Y , . Z l are matrices over R ( z ) { ) and such that 2, - Z3Y,Z2 is in R: ". Therefore, by Lemma 4 of this section, for every E > 0, there exists X,(E) tR(z)E:' such that Z , ( E ) : = X , ( t ) [ I - Z I X c ( t ) ] - ' , in- ternally 12-stabilizes the plant while achieving I[ Z4 - Z-, X , ( t )Z2 [ ( % < E . Therefore, ADIS (12) is solvable. 0 Comments: 1) After the submission of this note, the author became aware of an alternative solution by Wieland and Willems [8] who give geometric solvability conditions for the same problem. In the meantime, a paper by Linnemann, Postlethwaite, and Anderson [4] has appeared in which the authors give a solution to the problem for the more fundamental stability set 12 = C - . The frequency domain solvability condition of [4] is stated on the stable plant obtained after the application of an initial stabilizing feedback. 2) The technique of this note also easily yields conditions for the pole-placement version of ADIS (12). 3) It is possible to solve ADIS(12) with the more general internal stability constraint "the pair ( Z I , Z, . ( t ) ) is internally stable" [6]. 4) It is not true in general that " E and D are SI-stable and the equation Z4 = Z 3 X Z 2 has an %table solution implies the equation I& = II-,XII, has an SI-stable rational solution." The reason is that for unstable open-loop plants there may be unstable cancellations between ( T , Q ) or between ( Q , S ) . ACKNOWLEDCMENI The author would like to thank S. Wieland and J . C. Willems (and a reviewer) for pointing out a serious error in an earlier formulation of Theorem 1. REFERENCES V . Eldem and A. B. Ozguler, "Disturbance decoupling problems by measurement feedback: A characterization of all solutions and fixed modes." SIAM J . Confr. Oprimiz., vol. 26, no. I. pp. 168-185. 1988. B. A. Francis. "A guide to Hm-control theory," in Modelling Robusmess. and Sensirivity Reduction in Conrrol Sysrems (NATO AS1 Series. Vol. 34). R. F . Curtain, Ed. 1986. pp. 1-20. P. P. Khargonekar and A . B . Ozguler, "Regulator problem with internal stability: A frequency domain solution," IEEE Trans. Auromar. Conrr.. vol. AC-29. pp. 331-343, 1984. TRANSACTIONS ON AUTOMATIC CONTROL. VOL. 35. NO 6. JUNE 1990 A. Linnemann. 1. Postlethwaite, and B. D. 0. Anderaon. "Almost disturbance decoupling with stabilization by measurement feedback," preprint, Australian Nat. Univ., Canberra. ACT 2601. Australid. A. B. Ozguler. "Lecture notes on control of two-channel systems via a matrix fractional approach." Bilkent Univ. Rep.. P . 0 Box 8. Maltepe 06572 Ankara. Turkey. A. B. Ozguler and V. Eldem. "Disturbance d e c o u p h g problems via dynamic output feedback." IEEE Trans. Automal. Contr.. vol. AC-30, pp. 756-764. 1985. M. Vidyasagar. Control Synthesis: A Factorization Approach. Cambridge, MA: M.I.T. Press, 1985. S. Wieland and J. C . Willems. "Almost disturbance decoupling with internal stability," IEEE Trans. Automat. Contr.. vol. 34. pp. 277-286, 1989. J . C . Willems. "Almost invariant subspaces: An approach to high gain feed- back design- Part 11: Almost conditionally invariant subspaces," IEEE Trans. Automat. Contr., vol. AC-27, pp. 1071-1085. 1982. J . Van Der Woude, "Feedback decoupling and stabilizdtion for linear systems with exogenous variables." Ph.D. dissertation, Tech. Univ. Eindhoven, Eindhoven. The Netherlands, 1987. Pole Placement Direct Adaptive Control for Time-Varying Ill-Modeled Plants FOUAD GIN, MOHAMED M'SAAD, JEAN-MICHEL DION, A N D LUC DUGARD Abstract-This note addresses the problem of preserving stability of pole placement direct adaptive control in spite of output bounded distur- bances, time-varying plant model parameters, and unmodeled dynamics, assumed to be small in the mean. The controller parameler estimates are shown to track, in the mean, their true (time-varying) parameter values. Such a convergence property i s achieved using an ad hoc, internally generated, excitation sequence which ensures persistent excitation. Furthermore, in the ideal case the convergence of the parameter esti- mates is exponential, avoiding, in particular, possible chaotic phenom- ena. I. INTRODUCTION The fundamental practical feature which motivates the adaptive control theory is how to achieve acceptable performance vis-a-vis time-varying dynamics and plant model uncertainties. The latter are mainly due to external disturbances and unmodeled dynamics. However, most of the early investigations have been devoted to linear time-invariant systems, possibly subject to well-modeled disturbances, which will be referred to as the ideal case throughout this note. Robustness studies in adaptive control have been undertaken in the last few years, e.g., [13], [14], [IO], [ l ] , [SI, [6]. In the previous references, robustness issues in indirect adaptive control are addressed. On the other hand, the problem of robustness in direct adaptive control has been dealt with, based on minimum phase assumption, e.g., [2], [9], [ I l l . In this note, we propose a new solution to the closed-loop pole place- ment direct adaptive control. The controller parameters estimation is per- formed following the design philosophy proposed in [3] for the ideal case. The involved robust stability is achieved by using a least-squares-type algorithm with data normalization [ 131, parameter projection, and adap- tation gain monitoring. Furthermore, the adaptive control law is modified adding an internally generated impulse excitation sequence following a nonlinear feedback excitation approach [ 121. Manuscript received July 15. 1988; revised June 16. 1989. Paper recommended by Past Associate Editor, C . E. Rohrs. The work of F. Giri was supponed by the Ecole Nationale Superieure d'Electricite et d e Mecanique, Casablanca, Morocco. The authors are with the Laboratoire d'Automatique de Grenoble, ENSIEG, Saint- Martin-d'Heres, France. IEEE Log Number 9034503. 723 11. P R E L I M I N A R I E S Definition 2.1: Let a be a real number. A real sequence {s(/)} is said to be o1-asymptoticalIy small in /he mean ( a - ASM), if 1 . x I 1 I h - % I--3i v the set of all such sequences is denoted by So ( a ) . Proposition 2.1: a) Any a-ASM nonnegative real sequence is uniformly bounded. b) If ai 5 a?, then So(aI) is included in S,(a?). c) If { s ( t ) } E S.(a) and {s ' ( / ) } E S a ( a ' ) , then for any A, h' 2 0, v The proof of this proposition is evident from Definition 2. I . Proposition 2.2: Let { s ( t ) } be a real sequence. For any t > 0 and p E N - { O } , there exist an integer sequence { t x } and a finite integer T such that, if { s ( t ) } t &(a) . then for any k E N {As(t) +A's'(/)} t S,(Xa + X'a'). b)s( tk - j ) < p a + c , j = l , . . . , p . v The proof can be found in 151. such that for any t E N Proposition 2.3: Let { s ( t ) } and { r ( t ) } be nonnegative real sequences r (r + 1) 5 s ( t ) r ( t ) + K , (2.la) { s ( t ) } E S O ( o 1 ) (2.lb) where a and K , are nonnegative real constants. If 0 5 a < 1, then a) { r ( t ) } is uniformly bounded; v Part a) of the above proposition has been proved in [14]. Part b) can b) in addition, if K , = 0, then { r ( t ) } converges to zero. be established following closely the proof of part a). 111. THE PLANT REPRESE~TATION The plant to be controlled is assumed to be represented by the time- varying discrete-time model A(O*(t ) , q- l ) t (o = u ( t ) (3.la) ~ ( f ) = N O * ( [ ) , C 1 ) t ( 0 + q ( t ) (3 . lb) where {U(/)}, { y ( t ) } , and { ( ( t ) } are the input, output, and "partial state," respectively. {?(/)} is the plant unmodeled response. { 0' ( t ) } is a 2n-vector sequence ( n t N - {0}) and ( 3 . 1 ~ ) (3 . ld) where q-' is the backward shift operator and O,* ( t ) ( i = 1 , . . . ,2n) is the ith component of e * ( / ) . It is assumed that an integer n is known, such that the resulting se- quences { O * ( t ) } and {?(/)}, satisfy the following assumptions. A I : { ~ ~ O * ( f ) ~ ~ } is uniformly bounded by a known real R*. A2: There exists a positive real E< such that for any t E N: A(O"(t), q-1) = 1 + O;(t)q-l + ' ' ' + O,*(r)q-" B ( O * ( t ) , 4- l ) = O , * _ , ( t ) q - ' + . . . + O&(/)q-" Idet M,[A(O*(t) , q - ' ) , B ( O * ( t ) , q-')]1 2 c c , where M , [ ' , ' 1 de- notes the Sylvester matrix. A3: There exists a real v such that: { lq ( f ) i /m( t ) } E S , ( v ) , where {m( t ) } is defined by m ( / ) = um(t - 1) + max {mo , lu(t - 1)l + Iy(t - l)l} (3.2) and m(0) > 0, 0 < u < 1 and m, 2 0 are arbitrarily chosen Now. let U' be a real such that 0018-9286/90/0600-0723$01 .OO @ 1990 IEEE