Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993 INFINITE COUPLED RICCATI EQUATIONS ARISING IN CERTAIN OPTIMAL CONTROL PROBLEM O.L.V. Costa* and M.D. Fragoso** *University of sao Paulo, Department of Electronic Engineering, 05508900 sao Paulo, Brazjl **Nariollal Laboratory for Scientific Computation - LNCc/CNPq, Rua Lauro Muller 455, 22290 160, Rio de Janeiro, RJ, Brazil Alwtnd. The optimal control problem for ditcrete time linear 'Jltem. subject. to Marltovian jump. in the paramekn w cOIaidered. The Itate space of the Markov chain w usumed to be infinite countable and the colt functional to be minimised w the infinite time horizon quadratic colt. It is .hoWD that thia optimiution problem w equivalent to the .olution of an infinite countable set of coupled algebraic Riccati equatioIa (ICARE) . Sufficient condition. for exiltence and uniqueness of a positive semi-definite stabililable .olution to the ICARE are pre.ented. The •• condition. are Itated in terms of the concept. of stochastic stabilizability (SS) and Itochastic detectability (SO). K"J Worda. optimal control, Itochastic control. Marltov procelSel. dynamic procrammi.n&. wcret. time 'Jltem.. 'Jltem failure and recovery. Itahility . 1. lNTRODUCTION The subject matter of thil paper i. the optimal control problem for the class of ditcrete-time linear systems with Marltovian jumping parameter. . The modeling novelty here. regarding previouo works (lee for inltance. Chi.eclt et aL. 1986; Colta and Frago.o; Fragolo. 1988; Griffitho and Loparo. 1985; Ji and Chiseclt. 1990; Mariton. 1985; Moro.an. 1983; Sworder. 1969) is that the Itate space of the Marltov chain. which modelo the jump •• is ... sumed to be infinite countoble. It turn. out that the main hindrance regarding the exiltence of an optimal control policy for the above problem lie. on the behavior of an infinite countable set of algebraic Riccati equation. (ICARE) . Ipso 'octo. interest is centered mainly on the Itudy of thia .et of equation •. We pre.ent sufficient conditioIa for exiltence and uniqueness of a positive semi-definite stabilizable .olution to the ICARE . Thio i. achieved via the concepts of Itoch ... tic .tabililability (SS) and Itochastic detectability (SO) . The convergence of an infinite countable .et of coupled Riccati difference equation. to the ICARE i. allO analyzed. Several dynamic systems are inherently vulnerable to abrupt change. in their .tructures due. for iIatance. to component and/ or interconnections failure •• inter lil . Thia is to be found. for inltance. in robotic manipulator systems. in aircraft control .yltem •• large scale flexible structure. for space stations (such as antenna. solar arrays. etc). on which an actuator or a .ensor failure is a quite common occurrence. In thia paper we con.ider the following cl .... of wcrete-time dynamical system. . which in many c ... e. repre.ent the situation de.cribed above: x(It+1) = A 8 (It)x(lt) + B 8 (It)u(lt) • k = 0.1.... (1) where x(lt) denotes the Itate vector. u(k) is the control input .equence and the abrupt change. i. appended into the model via the Marltov chain 8(1t) taking value. in {1.2 •. .. } with .tationary transition probabilities matrix P = [pJ. We assume that both x(lt) and .(It) are acce.sible at each time t. Furthermore . consider the followinc colt functional ((x(0) .8(0)) = inf E.".( 11 112 + 11 R 112) (2) u E 'U. k=O '\ where 'U. . the set of admiuible control policie •• i. such that for each u = (u(O).u(l) •.. .) E 'U. the above sequence is convergent and model (1) is mean square Itable (ie. E( 11 x(k) 11 ..... 0 as k ..... 00 ). 2. NOTATION AND ASSUMPTIONS Throughout this paper R and C stands for the set of real and complex numbers respectively. C n is the n-dimenlional complex Euclidean space and "" = {0.1 •... }. N = {1 .2 •... }. We denote by M(Cm.C n ) the normed linear space of all n by m complex matrices and for simplicity we .hall write M(C n ) whenever n = m. We shall write -. ' and " for complex conjugate. transpose and conjugate 731 transpole respectively. We Ihall \lie the notation L 0 aud L > 0 if a .elf-adjoint matrix is positive .emi-definite or positive definite respectively and write for the ,1Io eicenvalue of L . We denote M(C n )+ = {L E M(cD); L = L" O} and Ihall write 11. 11 L for the norm in C n induced by the inner product < X'J > L = x"Ly whenever L > O. Either the uniform induced norm in M( cD) or the standard norm in C n are repreoented by 11.11. Let respectively) be the linear space made up of all infinite .equencel of complex matrices H = (H1.H, .... ). E M(Cm.C n ). such that J:I 11 H; 11 converg .. (sup{ 11 11; i = 1.2 •... } < 00) . For H E (H E respectively) we define a norm in lGf,a by 11 H 1I1= .EII H; 11 (11 H lloo=sup{1I H; lI;i=l.l •... }) . .=1 We shall write and whenever n = m. = {H E H; E M(C n )+. i = 1.l •... } and rimilarly for It is euy to verify that and col are Banach spacel . For H = (HI .... ). L = (L 1 .... ) in we uoe the notation H $ L to indicate that H; $ LI fOl' each i in N. It is clear that if H $ L. IIHIl 1 $IILllt. Finally for any complex Banach space X we deno\<: by B(X) the Banach space of all bounded linear tramformatlon. of X into X with the uniform induced norm repre.ented by 11. 11. and for L E B(X) we denote by r ,,(L) the spectral radiuo of L. We conclude thia section with the followinc usumptlon. recardinc eq. (1) and colt criterion AI) A = .... ) E B = .... ) E Al) x(O) is a second order random variAble . A3) Q E M(C n )+ and R E M(C m )+. R > O. 3. STABILITY RESULTS For H E G E .et Fi = " - F = (FI .. .. ) md define the following operator L(H) = (L1(H).···). L .(H) = lim E p .. F· R. (3) ) N-+oo i=l 1) I""} 1 Propo.RioD 1 : L E For any j = 1.l •...• and urwninc N, NI' we get