Research Article Generalized S C, A, B -Pairs for Uncertain Linear Infinite-Dimensional Systems

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1 Journal of Applied Mathematics Volume 2009, Article ID , 16 pages doi: /2009/ Research Article Generalized S C, A, B -Pairs for Uncertain Linear Infinite-Dimensional Systems Naohisa Otsuka 1 and Haruo Hinata 2 1 Division of Science, School of Science and Engineering, Tokyo Denki University, Hatoyama-Machi, Hiki-Gun, Saitama , Japan 2 Department of Management and Information Sciences, Nagasaki Institute of Applied Science, Aba-Machi 536, Nagasaki , Japan Correspondence should be addressed to Naohisa Otsuka, otsuka@mail.dendai.ac.jp Received 8 June 2009; Accepted 15 September 2009 Recommended by M. A. Petersen We introduce the concept of generalized S C, A, B -pairs which is related to generalized S A, B -invariant subspaces and generalized S C, A -invariant subspaces for infinite-dimensional systems. As an application the parameter-insensitive disturbance-rejection problem with dynamic compensator is formulated and its solvability conditions are presented. Further, an illustrative example is also examined. Copyright q 2009 N. Otsuka and H. Hinata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction In the framework of the so-called geometric approach, many control problems with state feedback and/or incomplete-state feedback e.g., controllability and observability problems, decoupling problems, and disturbance-rejection problems, etc. have been studied for finitedimensional systems see, e.g., 1, 2. Further, the concept of C, A, B -pairs was first introduced by Schumacher 3, and this concept has been used successfully to design dynamic compensators. After that Curtain extended the geometric concepts to infinitedimensional systems and various control problems have been studied see, e.g., On the other hand, from the practical viewpoint, Ghosh 12 and Otsuka 13 studied the concepts of simultaneous C, A, B -pairs and of generalized C, A, B -pairs, respectively, for finite-dimensional systems, and the parameter-insensitive disturbance-rejection problems for uncertain linear systems were studied. Then, Otsuka and Inaba extended the concepts of simultaneous invariant subspaces and simultaneous C, A, B -pairs to infinitedimensional systems. Further, Otsuka and Hinata 17 studied the concept of generalized invariant subspaces for infinite-dimensional systems.

2 2 Journal of Applied Mathematics The objective of this paper is to investigate the concept of generalized S C, A, B -pairs for infinite-dimensional systems and to study the parameter-insensitive disturbance-rejection problem with dynamic compensator. The paper is organized as follows. Section 2 gives the concept of generalized S C, A, B -pairs and its properties. In Section 3, the parameter-insensitive disturbancerejection problem with dynamic compensator is formulated and its solvability conditions are presented. Section 4 gives an example to illustrate our results. Finally, some concluding remarks are given in Section Generalized S C, A, B -pairs First, we give some notations used throughout this investigation. Let B X; Y denote the set of all bounded linear operators from a Hilbert space X into another Hilbert space Y; for notational simplicity, we write B X for B X; X. For a linear operator A the domain, the image, the kernel, and the C 0 -semigroup generated by A are denoted by D A, ImA, KerA, and {S A t ; t 0}, respectively. Further, the dimension and the orthogonal complement of a closed subspace V are denoted by dim V and V, respectively. Next, consider the following linear systems defined in a Hilbert space X: S ( α, β, γ ) d : dt x t A α x t B( β ) u t, y t C ( γ ) x t, 2.1 where x t X, u t U: R m, y t Y: R l are the state, the input, and the measurement output, respectively. Operators A α, B β, and C γ are unknown in the sense that they are represented as the forms: A α A 0 α 1 A 1 α p A p : A 0 ΔA α, B ( β ) B 0 β 1 B 1 β q B q : B 0 ΔB ( β ), 2.2 C ( γ ) C 0 γ 1 C 1 γ r C r : C 0 ΔC ( γ ), where α: α 1,...,α p R p, β: β 1,...,β q R q, γ: γ 1,...,γ r R r, A 0 is the infinitesimal generator of a C 0 -semigroup {S A0 t ; t 0} on X, A i B X i 1,...,p,B i B R m ; X i 0,...,q, and C i B X; R l i 0,...,r. Here, in the system S α, β, γ A 0,B 0,C 0 and ΔA α, ΔB β, ΔC γ mean the nominal system model and a specific uncertain perturbation, respectively. Since A i i 1,...,p are bounded linear operators, we remark that A α always generates a C 0 -semigroup and has the domain D A α D A 0 for all α R p. Further, from the practical viewpoint it is assumed that the dimensions of input and output are finite. Now, introduce a compensator K, L, M, N defined in a Hilbert space W of the form : d w t Nw t My t, Σ : dt u t Lw t Ky t, 2.3

3 Journal of Applied Mathematics 3 where N is the infinitesimal generator of a C 0 -semigroup {S N t ; t 0} on a Hilbert space W with the domain D N W, M B R l ; W,L B W; R m,andk B R l ; R m. If a compensator of the form Σ is applied to the system S α, β, γ, the resulting closedloop system S cl α, β, γ with the extended state space X e : X Wis easily seen to be [ ] d x t dt w t [ ( ) ( ) ( ) A α B β KC γ B β L MC ( γ ) N ][ ] x t, 2.4 w t where X Wmeans the direct sum of X and W. For the closed-loop system S cl α, β, γ, define [ ] [ ( ) ( ) ( ) ] x t A α B β KC γ B β L x e t :, A e αβγ w t : MC ( γ ) N 2.5 with domain D A e αβγ D A 0 W. For the system S α, β, γ, we give the following invariant subspaces. Definition 2.1. Let V be a closed subspace of X. i V is said to be a generalized A, B -invariant if there exists an F B X; R m such that ( A α B ( β ) F ) V D A0 V, α, β. 2.6 Also F V : {F B X; R m A α B β F V D A 0 Vfor all α, β}. ii V is said to be a generalized S A, B -invariant if there exists an F B X; R m such that S A α B β F t V V, t 0andallα, β. 2.7 Also V A, B; Λ : {V V is a generalized S A, B -invariant and is contained in a given closed subspace Λ.}. F s V : {F B X; R m S A α B β F t V Vforall t 0andall α, β}. iii V is said to be a generalized C, A -invariant if there exists a G B R l ; X such that ( ( )) A α GC γ V D A0 V, α, γ. 2.8 Also G V : {G B R l ; X A α GC γ V D A 0 Vfor all α, γ}. iv V is said to be a generalized S C, A -invariant if there exists a G B R l ; X such that S A α GC γ t V V, t 0andallα, γ. 2.9 Also V ε; C, A : {V V is a generalized S C, A -invariant and contains a given closed subspace ε}. G s V : {G B R l ; X S A α GC γ t V Vfor all t 0andallα, γ}.

4 4 Journal of Applied Mathematics Remark 2.2. i For the system S α, β, γ a generalized S A, B -invariant subspace V has the property that if an arbitrary initial state x 0 V, then there exists a state feedback u t Fx t which is independent of α and β such that the state trajectory x t Vfor all t 0. ii If A 0 is a bounded linear operator on X i.e., A 0 B X, then the statements i, ii and iii, iv in Definition 2.1 are equivalent, respectively. Further, in this case F s V F V and G s V G V. Theorem 2.3 see 17, 18. Suppose that p i,q i i 1,...,p,r i,s i i 1,...,q are arbitrary fixed real numbers such that p i <q i i 1,...,p and r i <s i i 1,...,q. Then, the following three statements are equivalent. i V is a generalized S A, B -invariant. ii There exists an F B X; R m such that S A0 B 0 F t V V t 0 and B i FV V i 1,...,q, and A i V V,...,p. iii There exists an F B X; R m such that S A α B β F t V V, t for all α i p i,q i i 1,...,p and β i r i,s i i 1,...,q. iv There exists an F B X; R m such that S A α B β F t V V, t for all α i {p i,q i } i 1,...,p and β i {r i,s i } i 1,...,q. The following theorem is the dual version of Theorem 2.3. Theorem 2.4 see 17, 18. Suppose that p i,q i i 1,...,p,t i,u i i 1,...,r are arbitrary fixed real numbers such that p i <q i i 1,...,p and t i <u i i 1,...,r. Then, the following three statements are equivalent. i V is a generalized S C, A -invariant. ii There exists a G B R l ; X such that S A0 GC 0 t V V t 0 and GC i V V i 1,...,r, and A i V V,...,p. iii There exists a G B R l ; X such that S A α GC γ t V V, t for all α i p i,q i i 1,...,p and γ i t i,u i i 1,...,r. iv There exists a G B R l ; X such that S A α GC γ t V V, t for all α i {p i,q i } i 1,...,p and γ i {t i,u i } i 1,...,r.

5 Journal of Applied Mathematics 5 For finite-dimensional systems, Schumacher 3 first introduced the concept of C, A, B -pair. The following definition is a generalized and infinite-dimensional version of C, A, B -pair. Definition 2.5. Let V 1 and V 2 be closed subspaces of X. Apair V 1, V 2 of subspaces is said to be a generalized S C, A, B -pair if the following three conditions hold. i V 1 is a generalized S C, A -invariant. ii V 2 is a generalized S A, B -invariant. iii V 1 V 2. For closed-loop system S cl α, β, γ, we give the following definition. Definition 2.6. Let V e be a closed subspace of X e. i V e is said to be a generalized A e -invariant if A e α,β,γ Ve D A e α,β,γ Ve for all α, β, γ. ii V e is said to be a generalized S A e t -invariant if S A e t Ve V e for all t 0andall α,β,γ α, β, γ. The following lemma was shown by Zwart. Lemma 2.7 see 11. Let V e be a closed subspace of X e and the following three subspaces are introduced: { [ ] } x S orth : x X V e for some w W w P X ( V e ), S 1 : S orth, { [ ] } x S 2 : x X V e for some w W w 2.14 P X V e, where P X is the projection operator from X e onto X along W. Then, the following statements hold. [ x ] i S 1 {x X V e }. 0 ii S 1 S 2. iii If dim(w <, then S 2 is a closed subspace of X and dim S 2 S 1 <. Lemma 2.8 see 4, 6, 11. Suppose that A is the infinitesimal generator of a C 0 -semigroup {S A t ; t 0} on X, and V is a closed subspace of X and Q 1 B X. Then, the following statements hold. i If S A t V Vfor all t 0, thena V D A V. ii If V D(A) and AV V,thenS A t V Vfor all t 0.

6 6 Journal of Applied Mathematics iii If S A Q1 t V Vfor all t 0, thenv D A V. iv If there exists a Q 2 B X such that S A Q2 t V Vfor all t 0 and Q 1 Q 2 V D A V, then S A Q1 t V Vfor all t 0. v If there exists a Q 2 B X such that S A Q2 t V Vfor all t 0 and Q 1 Q 2 V D A {0}, then S A Q1 t x S A Q2 t x for all t 0 and all x V. The following two lemmas are extensions of the results of Otsuka 13 to infinitedimensional systems. Lemma 2.9. If a pair V 1, V 2 of subspaces of X is a generalized S C, A, B -pair such that q r ImB i V 1 V 2 Ker C i, A i V 2 V 1, ( i 1,...,p ), 2.15 then there exist G G s V 1, G β B R l ; X, F γ F s V 2, F 0 B X; R m and K B R l ; R m such that G B ( β ) K G ( β ), ImG ( β ) V 2,F ( γ ) KC ( γ ) F 0, Ker F 0 V for all β, γ R q R r. Proof. Suppose that a pair V 1, V 2 is a generalized S C, A, B -pair satisfying the stated above conditions. Since q ImB i V 2, we remark that V 2 ImB β V 2 ImB 0. Claim 1. ĜC γ V 1 V 2 ImB 0 for all Ĝ G s V 1 and γ R r. To prove Claim 1, choose an arbitrary element x V 1. Then, by Lemma 2.8 iii there exists an x n V 1 D A o such that lim x n x. n 2.17 Now, noticing that A α ĜC γ V 1 D A 0 V 1 and V 1 V 2 we have ĜC ( γ ) x n ( A α ĜC ( γ )) x n A α x n V 1 V 2 ImB ( β ) V 2 ImB ( β ), 2.18 V 2 ImB 0. Since dimb β <, that V 2 ImB 0 is a closed subspace. Further, noticing that ĜC γ are bounded operators, ĜC ( γ ) x lim n ĜC ( γ ) x n V 2 ImB 0, 2.19

7 Journal of Applied Mathematics 7 which proves Claim 1. Next, Claims 2 and 3 hold as follows. Claim 2. There exists a G G s V 1 such that ImG V 2 ImB 0. To prove Claim 2, choose a Ĝ G s V 1 and x y z R l such that y r i 0 C iv 1 and z φ with r i 0 C iv 1 φ R l. Define a linear map G R n l by Gx : Ĝy. Then, for some x i V 1 Gx Ĝy r ĜC i x i i 0 V 2 ImB 0, ( by Claim 1 ), 2.20 which proves Claim 2. Claim 3. There exists a K B R l ; R m and G β B R l ; X such that G B β K G β, ImG β V 2 for all β R q. To prove Claim 3, let{y 1,...,y l } be a basis of R l. Then, it follows from Claim 2 that there exists an x i V 2 and u i R m such that Gy i x i B 0 u i x i ( q q β i B i u i B 0 u i β i B i u i x i ) q β i B i u i B ( β ) u i Define linear maps K B R l ; R m and G β B R l ; X by Ky i : u i, G ( β ) q y i : x i β i B i u i, respectively Then, Gy i G ( β ) y i B ( β ) Ky i, G ( β ) y i V 2, 2.23 which proves Claim 3. Claim 4. There exists an F B X; R m such that Im GC γ B β F V2 V 2. In fact, it follows from Claim 2 and hypotheses of this lemma that there exists a G G s V 1 such that ImGC 0 V2 ImG V 2 ImB

8 8 Journal of Applied Mathematics Let y V 2 be an arbitrary element. Then, there exist x V 2 and u R m such that GC 0 y x B 0 u. Define F B X; R m such that F y u, 2.25 Hence, GC 0 y x B 0 F y which implies GC 0 B 0 F y x V 2. Then, we can easily obtain ( Im GC ( γ ) B ( β ) F ) V2 V 2, 2.26 which proves Claim 4. Now, choose F F s V 2 and define F 0 B X; R m such that F 0 F F, on V 1, F 0 0, on V Then, the following claim holds. Claim 5. One has A α GC γ B β F 0 V 2 D A 0 V 2 for all α, β, γ R p R q R r and V 1 Ker F 0. In fact, at first, V 1 Ker F 0 is obvious. Therefore, we prove the first one. Since F F s V 2 implies S A α B β F t V 2 V 2, it follows from Lemma 2.8 v that it suffices to show B β F 0 GC γ B β F V 2 V 2 in order to prove S A α B β F0 GC γ t V 2 V 2. Now, let x V 2 be an arbitrary element. Then, there exists y V 1 and z V 1 V 2 such that x y z. Then, we have ( B ( β ) F0 GC ( γ ) B ( β ) F ) x ( B ( β ) F 0 GC ( γ ) B ( β ) F ) y ( B ( β ) F 0 GC ( γ ) B ( β ) F ) z ( GC ( γ ) B ( β ) F ) ( y B ( β ) F GC ( γ )) z Now, it follows from Lemma 2.8 iii that there exists a y n such that y n y. Hence, ( GC ( γ ) B ( β ) F ) y n ( A α GC ( γ )) y n ( A α B ( β ) F ) y n V Noticing that GC γ B β F is bounded linear operator, it follows from 2.28 and Claim 4 that ( B ( β ) F0 GC ( γ ) B ( β ) F ) x V 2, 2.30 which proves Claim 5.

9 Journal of Applied Mathematics 9 Finally, define a bounded linear operator F γ B X; R m such that F γ : KC γ F 0. Then, the following claim holds. Claim 6 F γ F V 2. In fact, ( A α B ( β ) F ( γ )) V2 D A 0 ( A α B ( β ) KC ( γ ) B ( β ) F 0 ) V2 D A 0 { A α ( B ( β ) K G ( β )) C ( γ ) G ( β ) C ( γ ) B ( β ) } F 0 V2 D A 0 { A α GC ( γ ) B ( β ) F 0 G ( β ) C ( γ )} ( ) V 2 D A 0, by Claim 3 ( A α GC ( γ ) B ( β ) ) F 0 V2 D A 0 ImG ( β ) 2.31 V 2, ( by Claim 3 and Claim 5 ), which proves Claim 6. This completes the proof of Lemma 2.9. Lemma If a pair V 1, V 2 of subspaces of X is a generalized S C, A, B -pair such that q r ImB i V 1 V 2 Ker C i, A i V 2 V 1, ( i 1,...,p ), V2 D A 0, 2.32 then there exist a compensator K, L, M, N on W : V 2 V 1 and a subspace Ve of X e such that V 1 S 1, V 2 S 2, and V e is generalized S A e t -invariant, where S 1 and S 2 are given in Lemma 2.7. Proof. Suppose that there exists a pair V 1, V 2 of subspaces satisfying the stated above conditions. Since V 1 and V 2 are closed subspaces and V 1 V 2, we have V 2 V 1 V 2 V 1. Define W : V 2 V 1 and Xe : X W. Let R : V 2 Wbe a bounded linear operator such that KerR V 1 and ImR W. Then, there exists a R B W; V 2 such that RR I, which implies R Rx 0 x V 1 see 2, p.95. Further, define {[ ] x V e : x V 2 } Rx Then, it follows from Lemma 2.7 that V 1 S 1 and V 2 S 2. Further, it follows from Lemma 2.9 that there exist G G s V 1, G β B R l ; X, F γ F s V 2, F 0 B X; R m,andk B R l ; R m such that G B ( β ) K G ( β ), ImG ( β ) V 2, F ( γ ) KC ( γ ) F 0, Ker F 0 V for all β, γ R q R r. Define L B W; R m and M B R p ; W such that L : F 0 R, M : RG 0. Noticing that V 1 V 2 D A 0, it is easily shown that ( A α B ( β ) F0 GC ( γ )) V i V i, i 1, 2, α, β, γ. 2.35

10 10 Journal of Applied Mathematics Hence, since A 0 B 0 F 0 GC 0 V 2 V 2 and V 1 Ker R, we have KerR KerR A 0 B 0 F 0 GC 0 V2 which is equivalent to that there exists an N B W such that NR R A 0 B 0 F 0 GC 0 V2 R A 0 B 0 F 0 G 0 C 0 V Then, the following two Claims hold. Claim 1. R A α B β F γ V 2 R A 0 B 0 F 0 V 2 for all α, β, γ. In fact, let x be an arbitrary element of V 2 : R ( A α B ( β ) F ( γ )) x R A 0 B 0 F 0 x R ( α 1 A 1 α p A p ) x RB ( β )( KC ( γ ) F0 ) x RB0 KC 0 F 0 x RB ( β ) KC ( γ ) x RB 0 KC 0 x R ( B ( β ) B 0 ) F0 x 2.37 R { B ( β ) KC 0 B 0 KC 0 } x 0 Claim 2. MC γ NR V 2 R A α B β F γ V 2 for all α, β, γ. In fact, ( MC ( γ ) NR ) V2 MC 0 NR V 2 RG 0 C 0 R A 0 B 0 F 0 G 0 C 0 V 2 R A 0 B 0 F 0 V R ( A α B ( β ) F ( γ )) V 2, ( by Claim 1 ). Finally, we have the following claim. Claim 3. A e α, β, γ V e V e for all α, [ β, γ. x ] Choose an arbitrary element of V e x V Rx 2. Since R Rx x Ker R V 1, we have LRx F 0 x. Then, [ ( ) ( ) ( ) ][ ] [( ( ) ( )) ( ) ] A α B β KC γ B β L x A α B β KC γ x B β LRx MC ( γ ) ( ( ) ) N Rx MC γ NR x [( ( )( ( ) )) ] A α B β KC γ F0 x R ( A α B ( β ) F ( γ )) ( ), by Claim 2 x [ ( ( ) ( )) ] A α B β F γ x R ( A α B ( β ) F ( γ )) x V e, ( α, β, γ ), 2.39 which proves that V e is generalized A e -invariant.

11 Journal of Applied Mathematics 11 Since V e D A e α,β,γ, it follows from Lemma 2.8 ii that Ve is generalized S A e t invariant. This completes the proof of this lemma. 3. Parameter-Insensitive Disturbance-Rejection by Dynamic Compensator In this section, the infinite-dimensional version of parameter insensitive disturbance-rejection problem for uncertain linear systems which was investigated by Otsuka 13 is studied. Consider the following uncertain linear system S α, β, γ, δ, σ defined in a Hilbert space X: d dt x t A α x t B( β ) u t E σ ξ t, y t C ( γ ) x t, 3.1 z t D δ x t, where x t X,u t U: R m,y t Y: R l,z t Z: R μ,andξ t L loc 1 0, ; Q are the state, the input, the measurement output, the controlled output, and the disturbance which is a Hilbert space Q valued locally integrable function, respectively. It is assumed that coefficient operators have the following unknown parameters: A α A 0 α 1 A 1 α p A p : A 0 ΔA α, B ( β ) B 0 β 1 B 1 β q B q : B 0 ΔB ( β ), C ( γ ) C 0 γ 1 C 1 γ r C r : C 0 ΔC ( γ ), 3.2 D δ D 0 δ 1 D 1 δ s D s : D 0 ΔD δ, E σ E 0 σ 1 E 1 σ t E t : E 0 ΔE σ, where A i,b i,c i are the same as system S α, β, γ in Section 2, D i B X; R μ,e i B Q; X, and α : α 1,...,α p,β : β 1,...,β q,γ : γ 1,...,γ r,δ : δ 1,...,δ s,σ : σ 1,...,σ t. Further, from the practical viewpoint, we assume that uncertain parameters satisfy α i [ ] p i,q i ( ) i 1,...,p, βi r i,s i ( ) i 1,...,q, γi t i,u i i 1,...,r, δ i [ ] h i,j i i 1,...,s, σ i k i,l i i 1,...,t 3.3 where p i,q i i 1,...,p, r i,s i i 1,...,q t i,u i i 1,...,r h i,j i i 1,...,s, andk i,l i i 1,...,t are given real numbers. In system S α, β, γ, δ, σ, A 0,B 0,C 0,D 0,E 0 and ΔA α,δb β,δc γ,δd δ,δe σ represent the nominal system model and a specific uncertain perturbation, respectively.

12 12 Journal of Applied Mathematics If a compensator of the form Σ is applied to system S α, β, γ, δ, σ, the resulting closedloop system with the extended state space X e : X Wis easily obtained as [ ] d x t dt w t z t [ D δ 0 ][ ] x t. w t [ ( ) ( ) ( ) ] A α B β KC γ B β L MC ( γ ) N [ ] [ ] x t E σ ξ t, w t For convenience, we set [ ] [ ( ) ( ) ( ) ] [ ] x t A α B β KC γ B β L E σ x e t :, A e α,β,γ w t : MC ( γ ), E e σ :, N D e δ : [ D δ 0 ]. Then, our disturbance-rejection problem with dynamic compensator is to find a compensator K, L, M, N of Σ such that D e δ t 0 S A e α,β,γ t τ Ee σ ξ τ dτ for all ξ L loc 1 0, ; Q which is a set of all locally square integrable functions on 0, 1, all t 0, and all parameters α, β, γ, δ, σ. This problem can be formulated as follows. Parameter Insensitive Disturbance-Rejection Problem with Dynamic Compensator (PIDRPDC) Given A i,b i,c i,d i,e i,find if possible a compensator K, L, M, N of 2.3 such that ( ) <S A e ImEe σ >: L S α,β,γ A e t ImEe σ Ker D e δ α,β,γ t for all parameters α, β, γ,δ, σ, where L Ω and the over bar indicate the linear subspace generated by the set Ω and the closure in X e, respectively. The following results are extensions of the results of Otsuka 13 to infinitedimensional systems. Theorem 3.1. If there exists a generalized S C, A, B -pair V 1, V 2 such that { q } { t r s ImB i ImE i V 1 V 2 Ker C i Ker D i }, i 0 i 0 ( ) A i V 2 V 1 i 1,...,p, V2 D A 0, 3.8 then the PIDRPDC is solvable.

13 Journal of Applied Mathematics 13 Proof. Suppose that the stated above conditions are satisfied. Then, it follows from Lemma 2.10 that there exist a compensator K, L, M, N on W : V 2 V 1 and a subspace V e of X e such that V 1 S 1, V 2 S 2 and V e is generalized S A e t -invariant. Further, it can be easily shown that ImE e σ V e Ker D e δ. Then, <S A e α,β,γ ImEe σ > <S A e α,β,γ Ve > V e Ker D e δ 3.9 for all parameters α, β, γ, δ, σ which implies the PIDRPDC is solvable. Corollary 3.2. Assume that V t i 0 ImE i; C, A and V A, B; r i 0 Ker D i have the minimal element V 1 and the maximal element V 2, respectively. If q ImB i V 1 V 2 r Ker C i, A i V 2 V 1 i 1,...,p, and V 2 D A 0, then the PIDRPDC is solvable. 4. An Illustrative Example Consider the following system with uncertain parameters α, β, γ R: x ( t, η ) t 2 x ( t, η ) η 2 { α 1 0 x ( t, μ )( φ 1 ( μ ) φ2 ( μ ) φ3 ( μ )) dμ } (φ1 ) φ2 ) φ3 )) ( φ 1 ) 2φ2 )) u t β { φ1 ) φ2 ) φ3 )} u t { φ 1 ) φ2 ) φ3 )} ξ t, y t z t 1 0x ( t, μ )( ( ) ( )) 2φ 1 μ φ2 μ dμ γ 1 x ( t, μ )( ( ) ( ) ( )) φ 1 μ φ2 μ 2φ3 μ dμ, x ( t, μ )( φ 1 ( μ ) φ2 ( μ ) 2φ3 ( μ )) dμ, 4.1 where x t, η is the temperature distribution of a bar of unit length at position η 0, 1 and time t 0. Moreover, u t R, ξ t R, and z t R are the input, the disturbance, and the controlled output, respectively, and φ k η 2sin kπη,forη 0, 1, k 1. Now, let a Hilbert space X : L 2 0, 1 which is a set of all square integrable functions on 0, 1. Then, we remark that {φ k ; k 1} is an orthonormal basis of X L 2 0, 1. Further, define the following operators as A α : A 0 A 1, α R, A 0 x ) : d2 x ) { }, x D A dη 2 0 : x x k φ k X λ k x k 2 <, A 1 x : x, φ 1 φ 2 φ 3 ( φ1 φ 2 φ 3 ), x D A1 : X, k 1 k 1

14 Eξ : ( φ 1 φ 2 φ 3 ) ξ, ξ D E : R, Journal of Applied Mathematics B ( β ) : B 0 βb 1, β R, B 0 u : ( ) φ 1 2φ 2 u, u D B0 : R, B 1 u : ( ) φ 1 φ 2 φ 3 u, u D B1 : R, C ( γ ) : C 0 γc 1, γ R, C 0 : x, 2φ 1 φ 2, x D C0 : X, C 1 : x, φ 1 φ 2 2φ 3, x D C1 : X, Dx : x, φ 1 φ 2 2φ 3, x D D : X, where, means the inner product in X. Then, it is easily seen that A 0 is the infinitesimal generator of a C 0 -semigroup {S A0 t ; t 0} on X and that for each k 1 φ k is an eigenvector of A 0 belonging to an eigenvalue λ k k 2 π 2. Moreover, the operators A 1, B 0,B 1,C 0,C 1,D,andE are all bounded. Then, by using these operators we can rewrite the above system as d dt x t A α x t B( β ) u t Eξ t, y t C ( γ ) x t, 4.3 z t Dx t. Since S A α t ImE/ Ker D for all t>0, one can see that the controlled output z of the original system 4.3 is influenced by disturbunces ξ. Let us define two closed subspaces V 1 and V 2 of X as V 1 : { a ( φ 1 φ 2 φ 3 ) a R }, V2 : { a ( φ 1 φ 2 φ 3 ) b ( φ1 φ 2 ) a, b R }. 4.4 If we introduce an operator G B R; X as Gs : { λ 2 λ 3 φ 1 λ 1 λ 3 φ 2 λ 1 λ 2 φ 3 } s, s R, 4.5 then the condition ii of Theorem 2.4 for the system 4.3 is satisfied, and hence it follows from Theorem 2.4 that V 1 is a generalized S C, A -invariant. Moreover, if we introduce an operator F B X; R as Fx : x, λ 1 φ 1 λ 2 φ 2 2λ 3 φ 3, x X, 4.6

15 Journal of Applied Mathematics 15 then it follows from Theorem 2.3 that V 2 is a generalized S A, B -invariant. Thus, we can see that the V 1, V 2 is a generalized S C, A, B -pair and it is easily shown that the pair V 1, V 2 satisfies the conditions of Theorem 3.1 for the system 4.3, thatis, {ImB 1 ImE} V 1 V 2 {Ker C 1 Ker D}, A 1 V 2 V 1, V 2 D A Therefore, it follows from Theorem 3.1 that the PIDRPDC is solvable by using a dynamic compensator which is constructed in terms of the proof of Lemma In fact, we can obtain the dynamic compensator Σ in a Hilbert space W described by d w t Nw t My t, Σ : dt u t Lw t Ky t, 4.8 where W : V 2 V 1 {aψ; ψ φ 1 φ 2,a R} and Nw : 4λ 1 4λ 2 9λ 3 w, Lw : w, λ 1 2λ 2 3λ 3 ψ, for w W, My : 2λ 1 λ 2 3λ 3 yψ, Ky : λ 1 λ 2 2λ 3 y, for y R, 4.9 which solves the PIDRPDC. 5. Concluding Remarks In this paper we studied the concept of generalized S C, A, B -pairs and its properties for infinite-dimensional systems. This concept is an extension of generalized C, A, B pairs investigated by Otsuka 13 to infinite-dimensional systems. After that a parameter insensitive disturbance-rejection problem with dynamic compensator was formulated and its solvability conditions were given. Further, an illustrative example was also examined. In the present investigation, it should be pointed out that the sufficient conditions of Theorem 3.1 is not easy to check. As future studies it is useful to investigate the existence conditions and computational algorithms of the minimal element V 1 and the maximal element V 2 of Corollary 3.2 in order to check easily the solvability conditions of the PIDRPDC. Further, we need to study stabilizability problems for the parameter insensitive disturbancerejected systems. References 1 G. Basile and G. Marro, Controlled and Conditioned Invariants in Linear System Theory, Prentice-Hall, Englewood Cliffs, NJ, USA, W. M. Wonham, Linear Multivariable Control: A Geometric Approach, vol. 10 of Applications of Mathematics, Springer, New York, NY, USA, 3rd edition, J. M. Schumacher, Compensator synthesis using C, A, B -pairs, IEEE Transactions on Automatic Control, vol. 25, no. 6, pp , R. F. Curtain, C, A, B -pairs in infinite dimensions, Systems & Control Letters, vol. 5, no. 1, pp , 1984.

16 16 Journal of Applied Mathematics 5 R. F. Curtain, Disturbance decoupling by measurement feedback with stability for infinitedimensional systems, International Journal of Control, vol. 43, no. 6, pp , R. F. Curtain, Invariance concepts in infinite dimensions, SIAM Journal on Control and Optimization, vol. 24, no. 5, pp , H. Inaba and N. Otsuka, Triangular decoupling and stabilization for linear control systems in Hilbert spaces, IMA Journal of Mathematical Control & Information, vol. 6, no. 3, pp , N. Otsuka, H. Inaba, and T. Oide, Decoupling by state feedback in infinite-dimensional systems, IMA Journal of Mathematical Control & Information, vol. 7, no. 2, pp , N. Otsuka, Simultaneous decoupling and disturbance-rejection problems for infinite-dimensional systems, IMA Journal of Mathematical Control & Information, vol. 8, no. 2, pp , N. Otsuka, H. Inaba, and K. Toraichi, Decoupling by incomplete state feedback for infinitedimensional systems, Japan Journal of Industrial and Applied Mathematics, vol. 11, no. 3, pp , H. Zwart, Geometric Theory for Infinite-Dimensional Systems, vol. 115 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, B. K. Ghosh, A geometric approach to simultaneous system design: parameter insensitive disturbance decoupling by state and output feedback, in Modelling, Identification and Robust Control, C. I. Byrnes and A. Lindquist, Eds., pp , North-Holland, Amsterdam, The Netherlands, N. Otsuka, Generalized C, A, B -pairs and parameter insensitive disturbance-rejection problems with dynamic compensator, IEEE Transactions on Automatic Control, vol. 44, no. 11, pp , N. Otsuka and H. Inaba, Parameter-insensitive disturbance rejection for infinite-dimensional systems, IMA Journal of Mathematical Control & Information, vol. 14, no. 4, pp , N. Otsuka and H. Inaba, A note on robust disturbance-rejection problems for infinite-dimensional systems, Systems & Control Letters, vol. 34, no. 1-2, pp , N. Otsuka and H. Inaba, Simultaneous C, A, B -pairs for infinite-dimensional systems, Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp , N. Otsuka and H. Hinata, Generalized invariant subspaces for infinite-dimensional systems, Journal of Mathematical Analysis and Applications, vol. 252, no. 1, pp , N. Otsuka, A note on generalized invariant subspaces for infinite-dimensional systems, IMA Journal of Mathematical Control & Information, vol. 21, no. 2, pp , 2004.

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