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Generalized Invariant Subspaces for Infinite-Dimensional Systems
Journal of Mathematical Analysis and Applications 252, 325᎐341 Ž2000. doi:10.1006rjmaa.2000.7011, available online at http:rrwww.idealibrary.com on
Generalized Invariant Subspaces for Infinite-Dimensional Systems Naohisa Otsuka Department of Information Sciences, Tokyo Denki Uni¨ ersity, Hatoyama-Machi, Hiki-Gun, Saitama 350-0394, Japan E-mail: [email protected]
and Haruo Hinata Nagasaki Institute of Applied Science, Aba-Machi 536, Nagasaki 851-0193, Japan E-mail: [email protected] Submitted by William F. Ames Received December 3, 1999
In this paper some generalized invariant subspaces for infinite-dimensional systems are investigated, and then some sufficient conditions for parameter-insensitive disturbance-rejection problems with state feedback and with measurement output feedback to be solvable are studied. 䊚 2000 Academic Press Key Words: generalized invariant subspaces; uncertain systems; parameter-insensitive disturbance-rejection; geometric approach; infinite-dimensional systems.
1. INTRODUCTION The notion of invariant subspaces has been used successfully to study various control problems for linear finite-dimensional systems by Basile and Marro w2x and Wonham w17x. This notion has been extended to systems defined in Hilbert spaces, and some of the corresponding problems in Hilbert spaces have also been studied by Curtain w4, 5x, Zwart w18x, and Otsuka et al. w8, 14, 15x. After that, the simultaneous versions of those invariant subspaces for both finite- and infinite-dimensional linear systems have also been studied by Ghosh w7x and Otsuka et al. w11᎐13, 16x in order to investigate uncertain linear systems whose systems operators are repre325 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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OTSUKA AND HINATA
sented as convex combinations of given operators, and then parameter-insensitive disturbance-rejection problems have been studied. Further, Bhattacharyya w3x introduced the notion of generalized controlled Ž A, B .-invariant subspaces in order to study uncertain linear finite-dimensional systems whose systems matrices are represented as the nominal system matrices and specific uncertain perturbations, and then a parameter-insensitive disturbance-rejection problem with state feedback, which means robust with respect to the specific uncertain perturbations, was studied. Further, for such systems, Otsuka w9, 10x introduced the notions of generalized conditioned Ž C, A.-invariant subspaces, generalized Ž A, B, C .invariant subspaces, and generalized Ž C, A, B .-pairs, and then parameterinsensitive disturbance-rejection problems with static output feedback and with dynamic compensator were also investigated. In such a historical background, from the theoretical viewpoint, it is meaningful to investigate the infinite-dimensional version of those generalized invariant subspaces and their properties. Therefore, in this paper the infinite-dimensional versions of generalized invariant subspaces are first studied. Further, the parameter-insensitive disturbance-rejection problems with state feedback and with static output feedback for infinite-dimensional systems are formulated, and then their solvability conditions are given. Finally, an illustrative example is given.
2. GENERALIZED INVARIANT SUBSPACES First, some notations used throughout this investigation are given. Let BŽ X; Y . denote the set of all bounded linear operators from a Hilbert space X into another Hilbert space Y, and for notational simplicity, BŽ X; X . is written as BŽ X .. For a linear operator A, the domain, the image, the kernel, and the C0-semigroup generated by A are denoted by DŽ A., Im A, Ker A, and SAŽ t .; t G 04 , respectively. The notation R n denotes the n-dimensional Euclidean space. Next, consider the following linear systems defined in a Hilbert space X,
SŽ ␣ ,  , ␥ . :
¡d ¢y Ž t . s C Ž ␥ . x Ž t . ,
~ dt x Ž t . s AŽ ␣ . x Ž t . q B Ž  . u Ž t . ,
where x Ž t . g X, uŽ t . g U [ R m , y Ž t . g Y [ R l are the state, the input, and the measurement output, respectively. And operators AŽ ␣ ., B Ž  .,
GENERALIZED INVARIANT SUBSPACES
327
and C Ž␥ . are unknown in the sense that they are represented as the forms A Ž ␣ . s A 0 q ␣ 1 A1 q ⭈⭈⭈ q␣ p A p [ A 0 q ⌬ A Ž ␣ . , B Ž  . s B0 q  1 B1 q ⭈⭈⭈ q q Bq [ B0 q ⌬ B Ž  . , C Ž ␥ . s C0 q ␥ 1C1 q ⭈⭈⭈ q␥r Cr [ C0 q ⌬C Ž ␥ . , where ␣ [ Ž ␣ 1 , . . . , ␣ p . g R p ,  [ Ž  1 , . . . ,  q . g R q, ␥ [ Ž␥ 1 , . . . , ␥r . g R r , A 0 is the infinitesimal generator of a C0-semigroup SA 0Ž t .; t G 04 on X, A i g BŽ X . Ž i s 1, . . . , p ., Bi g BŽR m ; X . Ž i s 0, 1, . . . , q ., and Ci g BŽ X; R l . Ž i s 0, 1, . . . , r .. Here, in system SŽ ␣ ,  , ␥ ., Ž A 0 , B0 , C0 . and Ž ⌬ AŽ ␣ ., ⌬ B Ž  ., ⌬C Ž␥ .. mean the nominal system model and a specific uncertain perturbation, respectively. Now, since A i Ž i s 1, . . . , p . are bounded linear operators, one can say that AŽ ␣ . always generates a C0-semigroup and has the domain DŽ AŽ ␣ .. s DŽ A 0 . for all ␣ g R p. Further, from the practical viewpoint, one can say that the dimensions of input and output are finite dimensions. DEFINITION 2.1. Let V Ž; X . be a closed subspace. Ži. V is said to be a generalized controlled SŽ A, B .-invariant subspace if there exists an F g BŽ X; R m . such that SAŽ ␣ .qBŽ  . F Ž t . V ; V
Ž t G 0.
for all Ž ␣ ,  . g R p = R q. For a closed subspace ⌳, define V Ž A, B; ⌳ . [ V N V is a generalized controlled S Ž A, B . -invariant subspace and V ; ⌳ 4 . Žii. V is said to be a generalized conditioned SŽ C, A.-invariant subspace if there exists a G g BŽR l ; X . such that SAŽ ␣ .qG CŽ ␥ . Ž t . V ; V
Ž t G 0.
for all Ž ␣ , ␥ . g R p = R r. For a closed subspace , define V Ž ; C, A . [ V N V is a generalized conditioned S Ž C, A . -invariant subspace and ; V 4 . Žiii. V is said to be a generalized SŽ A, B, C .-invariant subspace if there exists an H g BŽR l ; R m . such that SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž t . V ; V for all Ž ␣ ,  , ␥ . g R p = R q = R r.
Ž t G 0.
328
OTSUKA AND HINATA
For system SŽ ␣ ,  , ␥ ., a generalized SŽ A, B, C .-invariant subspace V has the property that, if an arbitrary initial state x Ž0. stays in V, then there exists a measurement feedback H g BŽR l ; R m . which is independent of all Ž ␣ ,  , ␥ . g R p = R q = R r such that the state trajectory x Ž t . stays in V for all t G 0. The following lemma was given by Curtain w4x. LEMMA 2.2 Žsee w4x.. Let V be a closed subspace of X and let A be an infinitesimal generator with C0-semigroup SAŽ t .; t G 04 on X, and let Q1 g BŽ X .. Ži. If SAq Q Ž t .V ; V for all t G 0, then Ž A q Q1 .Ž V l DŽ A.. ; V. 1 Žii. If there exists a Q 2 g BŽ X . such that SAqQ Ž t .V ; V for all t G 0 2 and Ž Q1 y Q 2 .Ž V l DŽ A.. ; V, then SAqQ 1Ž t .V ; V for all t G 0. The following lemma is used to define two projection operators which play important roles to state the main results in this section. LEMMA 2.3. ⌫V by
Let V be a closed subspace. Define two subspaces R V and q
RV [
F By1 i V,
m where By1 i V [ u g R N Bi u g V 4
is1
and r
⌫V [
Ý Ci V . is1
Then, the following two statements hold. Ži.
There exists a subspace ⍀ such that R m s RV [ ⍀ .
Žii. There exist closed subspaces V1 , V2 of X and subspaces ⌳ 1 , ⌳ 2 of R l such that R l s C0 V1 [ C0 V2 [ ⌳ 1 [ ⌳ 2 , ⌫V s C0 V2 [ ⌳ 1 ,
and
C0 V2 s C0 V l ⌫V ,
V s V1 [ V2 [ Ž Ker C0 l V . .
Proof. Ži. The proof is obvious. Žii. Since V is a closed subspace, there exists a closed subspace ˜ Further, since V˜ ; V such that V s V˜ [ ŽKer C0 l V .. Then, C0 V s C0 V. ˜ there exists a closed subspace V2 ; V˜ such C0 V l ⌫V s C0 V˜ l ⌫V ; C0 V, that C0 V2 s C0 V l ⌫V . Then, there exists a closed subspace V1 ; V˜ such
GENERALIZED INVARIANT SUBSPACES
329
that V˜ s V1 [ V2 . Thus, V s V1 [ V2 [ ŽKer C0 l V . and C0 V s C0 V1 [ C0 V2 . Moreover, since C0 V l ⌫V ; ⌫V , there exists a subspace ⌳ 1 ; ⌫V such that ⌫V s Ž C0 V l ⌫V . [ ⌳ 1. Further, since C0 V1 [ C0 V2 [ ⌳ 1 ; R l , there exists a subspace ⌳ 2 such that R l s C0 V1 [ C0 V2 [ ⌳ 1 [ ⌳ 2 . By Lemma 2.3, the following operators can be defined. DEFINITION 2.4. For a closed subspace V of X, define the following two projection operators. Ži. Žii.
Q V : R m ª R m , projection operator onto R V along ⍀. PV : R l ª R l , projection operator onto C0 V1 [ ⌳ 2 along ⌫V .
Then, the following lemma is used to prove Theorem 2.6. LEMMA 2.5. The following two statements are equi¨ alent. Ži. There exists an F g BŽ X; R m . such that SA qB F Ž t .V ; V Ž t G 0. 0 0 and Bi FV ; V Ž i s 1, . . . , q .. Žii. There exists an F˜ g BŽ X; R m . such that SA qB Q F˜Ž t .V ; V Ž t G 0 0 V 0.. Proof. Ži. « Žii. Suppose that there exists an F g BŽ X; R m . such that SA 0qB 0 F Ž t .V ; V Ž t G 0. and Bi FV ; V Ž i s 1, . . . , q .. Then, FV ; q Ž B0 Q V F y B0 F .V s 04 . F is1 By1 i V s R V . Hence, Q V FV s FV. Thus, Then, it follows from Lemma 2.2Žii. that SA 0qB 0 Q V F Ž t .V ; V Ž t G 0.. Žii. « Ži. Suppose that there exists an F˜ g BŽ X; R m . such that ˜ Then, SA 0qB 0 F Ž t .V ; V Ž t SA 0qB 0 Q V F˜Ž t .V ; V Ž t G 0.. Define F [ Q V F. ˜ G 0.. Further, Bi FV s Bi Q V FV ; Bi R V ; V Ž i s 1, . . . , q .. This completes the proof. The following theorem is the infinite-dimensional version of the results of Bhattacharyya w3x. THEOREM 2.6. Let V be a closed subspace of X. Then, the following three statements are equi¨ alent. Ži. V is a generalized controlled SŽ A, B .-in¨ ariant subspace. Žii. There exists an F g BŽ X; R m . such that SA qB F Ž t .V ; V Ž t G 0. 0 0 and Bi FV ; V Ž i s 1, . . . , q ., and A i V ; V Ž i s 1, . . . , p .. Žiii. There exists an F˜ g BŽ X; R m . such that SA qB Q F˜Ž t .V ; V Ž t G 0 0 V 0., and A i V ; V Ž i s 1, . . . , p .. Proof. Ži. « Žii. Suppose that V is a generalized controlled SŽ A, B .invariant subspace. Then, there exists an F g BŽ X; R m . such that SAŽ ␣ .qBŽ  . F Ž t . V ; V for all Ž ␣ ,  . g R = R . p
q
Ž t G 0.
Ž 1.
330
OTSUKA AND HINATA
First, suppose that ␣ 1 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s 0 in Ž1.. Then, SA 0qB 0 F Ž t . V ; V
Ž t G 0. ,
which with Lemma 2.2Ži. implies
Ž A 0 q B0 F . Ž V l D Ž A 0 . . ; V .
Ž 2.
Further, suppose that ␣ 1 s 1 and ␣ 2 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s 0 in Ž1.. Then, SA 0qA 1qB 0 F Ž t . V ; V
Ž t G 0. ,
which with Lemma 2.2Ži. implies
Ž A 0 q A1 q B0 F . Ž V l D Ž A 0 . . ; V .
Ž 3.
Hence, it follows from Ž2. and Ž3. that A1Ž V l DŽ A 0 .. ; V. Since A1 is a bounded linear operator, A1V ; V. Similarly, one can prove A i V ; V Ž i s 2, . . . , p .. Next, suppose that  1 s 1 and ␣ 1 s ⭈⭈⭈ s ␣ p s  2 s ⭈⭈⭈ s  q s 0 in Ž1.. Then,
Ž A 0 q Ž B0 q B1 . F .Ž V l D Ž A 0 . . ; V .
Ž 4.
Hence, it follows from Ž2., Ž4., and boundedness of B1 F that B1 FV ; V. Similarly, one can prove Bi FV ; V Ž i s 2, . . . , q .. Žii. « Ži. Suppose that there exists an F g BŽ X; R m . such that SA 0qB 0 F Ž t .V ; V Ž t G 0. and Bi FV ; V Ž i s 1, . . . , q ., and A i V ; V Ž i s 1, . . . , p .. Now,
Ž AŽ ␣ . q B Ž  . F . s Ž A 0 q ␣ 1 A1 q ⭈⭈⭈ q␣ p A p . q Ž B0 q  1 B1 q ⭈⭈⭈ q q Bq . F 4 p
s Ž A 0 q B0 F . q
q
Ý ␣ i A i q Ý i Bi F. is1
Since S A 0qB 0 F Ž t .V ; V and from Lemma 2.2Žii. that
p ŽÝ is1
is1
␣ i A i q Ý qis1 i Bi F .V ; V, it follows
SAŽ ␣ .qBŽ  . F Ž t . V ; V
Ž t G 0.
for all Ž ␣ ,  . g R p = R q, which implies V is a generalized controlled SŽ A, B .-invariant subspace. Žii. m Žiii. The proof follows from Lemma 2.5.
331
GENERALIZED INVARIANT SUBSPACES
The following lemma is used to prove Theorem 2.8. LEMMA 2.7. The following two statements are equi¨ alent. Ži. There exists a G g BŽR l ; X . such that SA qG C Ž t .V ; V Ž t G 0. 0 0 and GC i V ; V Ž i s 1, . . . , r .. ˜ g BŽR l ; X . such that SA 0qG˜ PV C 0Ž t .V ; V Ž t G 0.. Žii. There exists a G Proof. Ži. « Žii. Suppose that there exists a G g BŽR l ; X . such that SA 0qG C 0Ž t .V ; V Ž t G 0. and GC i V ; V Ž i s 1, . . . , r .. Then, from Lemma 2.2Žii., it suffices to show Ž GPV C0 y GC 0 .V ; V. Choose an arbitrary element x g V. Then, by Lemma 2.3Žii., x can be decomposed as x s y q z q u Ž y g V1 , z g V2 , u g Ker C0 l V .. Hence, r
Ž GPV C0 y GC0 . x s yGC0 z g GC0 V2 ; G⌫V s
Ý GCiV ; V . is1
It follows from Lemma 2.2Žii. that SA 0qG P V C 0Ž t .V ; V Ž t G 0.. ˜ g BŽR l ; X . such that Žii. « Ži. Suppose that there exists a G ˜ V . Then, SA 0qG C 0Ž t .V ; V Ž t SA 0qG˜ P V C 0Ž t .V ; V Ž t G 0.. Define G [ GP ˜ V Ci V ; GP ˜ V ŽKer PV . s 04 ; V Ž i s 1, . . . , r .. G 0.. Further, GC i V s GP This completes the proof. The following results are the infinite-dimensional versions of Otsuka w9x. THEOREM 2.8. Let V be a closed subspace of X. Then, the following three statements are equi¨ alent. Ži. V is a generalized conditioned SŽ C, A.-in¨ ariant subspace. Žii. There exists a G g BŽR l ; X . such that SA qG C Ž t .V ; V Ž t G 0. 0 0 and GC i V ; V Ž i s 1, . . . , r ., and A i V ; V Ž i s 1, . . . , p .. ˜ g BŽR l ; X . such that SA 0qG˜ PV C 0Ž t .V ; V Ž t G 0., Žiii. There exists a G and A i V ; V Ž i s 1, . . . , p .. Proof. Ži. « Žii. Suppose that V is a generalized conditioned S Ž C, A.-invariant subspace. Then, there exists a G g BŽR l ; X . such that SAŽ ␣ .qG CŽ ␥ . Ž t . V ; V
Ž t G 0.
Ž 5.
for all Ž ␣ , ␥ . g R p = R r. First, suppose that ␣ 1 s ⭈⭈⭈ s ␣ p s 0 s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž5.. Then, S A 0qG C 0Ž t . V ; V
Ž t G 0. ,
which with Lemma 2.2Ži. implies
Ž A 0 q GC0 . Ž V l D Ž A 0 . . ; V .
Ž 6.
332
OTSUKA AND HINATA
Further, suppose that ␣ 1 s 1 and ␣ 2 s ⭈⭈⭈ s ␣ p s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž5.. Then, SA 0qA 1qG C 0Ž t . V ; V
Ž t G 0.
which with Lemma 2.2Ži. implies
Ž A 0 q A1 q GC0 . Ž V l D Ž A 0 . . ; V .
Ž 7.
Hence, it follows from Ž6. and Ž7. that A1Ž V l DŽ A 0 .. ; V. Since A1 is a bounded linear operator, A1V ; V. Similarly, one can prove A i V ; V Ž i s 2, . . . , p .. Next, suppose that ␥ 1 s 1 and ␣ 1 s ⭈⭈⭈ s ␣ p s ␥ 2 s ⭈⭈⭈ s ␥r s 0 in Ž5.. Then,
Ž A 0 q G Ž C0 q C1 . .Ž V l D Ž A 0 . . ; V .
Ž 8.
Hence, it follows from Ž6., Ž8., and boundedness of GC1 that GC1V ; V. Similarly, one can prove GCi V ; V Ž i s 2, . . . , r .. Žii. « Ži. Suppose that there exists a G g BŽR l ; X . such that SA 0qG C 0V ; V Ž t G 0. and GC i V ; V Ž i s 1, . . . , r ., and A i V ; V Ž i s 1, . . . , p .. Now,
Ž AŽ ␣ . q GC Ž ␥ . . s Ž A 0 q ␣ 1 A1 q ⭈⭈⭈ q␣ p A p . q G Ž C0 q ␥ 1C1 q ⭈⭈⭈ q␥r Cr . 4 p
s Ž A 0 q GC 0 . q
r
Ý ␣i Ai q
Ý ␥ i GCi .
is1
is1
p Since SA 0qG C 0Ž t .V ; V and ŽÝ is1 ␣ i A i q Ý ris1 ␥ i GCi .V ; V, it follows from Lemma 2.2Žii. that
SAŽ ␣ .qG CŽ ␥ . Ž t . V ; V
Ž t G 0.
for all Ž ␣ , ␥ . g R p = R r , which implies V is a generalized conditioned SŽ C, A.-invariant subspace. Žii. m Žiii. The proof follows from Lemma 2.7. The following lemma is used to prove Theorem 2.10. LEMMA 2.9. The following two statements are equi¨ alent. Ži. There exists an H g BŽR l ; R m . such that SA qB H C Ž t .V ; V Ž t G 0 0 0 0., Bi HC j V ; V Ž i s 0, . . . , q, j s 0, . . . , r; Ž i, j . / Ž0, 0.., and A i V ; V Ž i s 1, . . . , p .. Žii. There exists an H˜ g BŽR l ; R m . such that SA qB Q H˜P C V ; V Ž t G 0 0 V V 0 0., and A i V ; V Ž i s 1, . . . , p ..
GENERALIZED INVARIANT SUBSPACES
333
Proof. Ži. « Žii. Suppose that there exists an H g BŽR l ; R m . such that SA 0qB 0 H C 0Ž t .V ; V Ž t G 0., Bi HC j V ; V Ž i s 0, . . . , q, j s 0, . . . , r; Ž i, j . / Ž0, 0.., and A i V ; V Ž i s 1, . . . , p .. Then, it follows from Lemmas 2.5 and 2.7 that there exists an F˜ g BŽ X; R m . such that SA 0qB 0 Q V F˜Ž t .V ; V ˜ g BŽR l ; X . such that SA 0qG˜ PV C 0Ž t .V ; V Ž t G 0. and there exists a G Ž t G 0.. From Remark in w18, p. 106x, there exists an H˜ g BŽR l ; R m . such that S A 0qB 0 Q V H˜P V C 0Ž t .V ; V Ž t G 0.. Žii. « Ži. Suppose that there exists an H˜ g BŽR l ; R m . such that SA 0qB 0 Q V H˜P V C 0V ; V Ž t G 0. and A i V ; V Ž i s 1, . . . , p .. Define H [ ˜ V . Then, SA 0qB 0 H C 0Ž t .V ; V Ž t G 0.. And, one obtains Bi HC j V s Q V HP ˜ V C j V ; Bi Im Q V ; Bi R V ; V Ž i s 1, . . . , q, j s 0, . . . , r .. FurBi Q V HP ˜ V C j V ; B0 Q V HP ˜ V ⌫V s 04 ; V Ž j s 1, . . . , r .. ther, B0 HC j V s B0 Q V HP This completes the proof. Finally, the following theorem can be obtained. THEOREM 2.10. Let V be a closed subspace of X. Then, the following three statements are equi¨ alent. Ži. V is a generalized SŽ A, B, C .-in¨ ariant subspace. Žii. There exists an H g BŽR l ; R m . such that SA qB H C Ž t .V ; V Ž t G 0 0 0 0., Bi HC j V ; V Ž i s 0, . . . , q, j s 0, . . . , r; Ž i, j . / Ž0, 0.., and A i V ; V Ž i s 1, . . . , p .. Žiii. There exists an H˜ g BŽR l ; R m . such that SA qB Q H˜P C V ; V 0 0 V V 0 Ž t G 0., and A i V ; V Ž i s 1, . . . , p .. Proof. Ži. « Žii. Suppose that V is a generalized S Ž A, B, C .-invariant subspace. Then, there exists an H g BŽR l ; R m . such that SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž t . V ; V
Ž t G 0.
Ž 9.
for all Ž ␣ ,  , ␥ . g R p = R q = R r. First, suppose that ␣ 1 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž9.. Then, SA 0qB 0 H C 0Ž t . V ; V
Ž t G 0. ,
which with Lemma 2.2Ži. implies
Ž A 0 q B0 HC0 . Ž V l D Ž A 0 . . ; V .
Ž 10 .
Further, suppose that ␣ 1 s 1 and ␣ 2 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž9.. Then, SA 0qA 1qB 0 H C 0Ž t . V ; V
Ž t G 0. ,
334
OTSUKA AND HINATA
which with Lemma 2.2Ži. implies
Ž A 0 q A1 q B0 HC0 . Ž V l D Ž A 0 . . ; V .
Ž 11 .
Hence, it follows from Ž10. and Ž11. that A1Ž V l DŽ A 0 .. ; V. Since A1 is a bounded linear operator, A1V ; V. Similarly, one can prove A i V ; V Ž i s 2, . . . , p .. Next, suppose that  1 s 1 and ␣ 1 s ⭈⭈⭈ s ␣ p s  2 s ⭈⭈⭈ s  q s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž9.. Then,
Ž A 0 q Ž B0 q B1 . HC0 .Ž V l D Ž A 0 . . ; V .
Ž 12 .
Hence, it follows from Ž10., Ž12., and boundedness of B1 HC0 that B1 HC0 V ; V. Further, suppose that ␥ 1 s 1 and ␣ 1 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s ␥ 2 s ⭈⭈⭈ s ␥r s 0 in Ž9.. Then,
Ž A 0 q B0 H Ž C0 q C1 . .Ž V l D Ž A 0 . . ; V .
Ž 13 .
From Ž10., Ž13., and boundedness of B0 HC1 , one obtains B0 HC1V ; V. Similarly, one can prove Bi HC j V ; V Ž i s 1, . . . , q, j s 1, . . . , r .. Žii. « Ži. Suppose that there exists an H g BŽR l ; R m . such that SA 0qB 0 H C 0Ž t .V ; V, Bi HC j V ; V Ž i s 0, . . . , q, j s 0, . . . , r; Ž i, j . / Ž0, 0.., and A i V ; V Ž i s 1, . . . , p .. Now, p
r
Ž AŽ ␣ . q B Ž  . HC Ž ␥ . . s Ž A 0 q B0 HC0 . q Ý ␣ i A i q Ý ␥ i B0 HCi is1 q
q
q Ý i Bi HC0 q is1
is1 r
Ý Ý i ␥ j Bi HC j . is1 js1
Since SA 0qB 0 H C 0Ž t .V ; V and p
ž
q
r
Ý ␣ i A i q Ý ␥ i B0 HCi q Ý i Bi HC0 is1
is1
is1 q
qÝ
r
Ý i ␥ j Bi HC j
is1 js1
/
V ; V,
it follows from Lemma 2.2Žii. that SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž t . V ; V
Ž t G 0.
for all Ž ␣ ,  , ␥ . g R p = R q = R r , which implies V is a generalized SŽ A, B, C .-invariant subspace. Žii. m Žiii. The proof follows from Lemma 2.9.
GENERALIZED INVARIANT SUBSPACES
335
Concerning the three generalized invariant subspaces, the following corollary can be obtained. COROLLARY 2.11. V is a generalized SŽ A, B, C .-in¨ ariant subspace if and only if V is a generalized controlled SŽ A, B .-in¨ ariant and generalized conditioned SŽ C, A.-in¨ ariant subspace. Proof. ŽNecessity. Suppose that V is a generalized S Ž A, B, C .-invariant subspace. Then, it follows from Theorem 2.10 that there exists an H˜ g BŽR l ; R m . such that SA 0qB 0 Q V H˜P V C 0Ž t .V ; V Ž t G 0., and A i V ; V Ž i s 1, . . . , p .. Thus, from Theorems 2.6 and 2.8, V is a generalized controlled SŽ A, B .-invariant and generalized conditioned SŽ C, A.-invariant subspace. ŽSufficiency. Suppose that V is a generalized controlled SŽ A, B .-invariant and generalized conditioned SŽ C, A.-invariant subspace. Then, it follows from Theorems 2.6 and 2.8 that there exists an F˜ g BŽ X; R m . such ˜ g BŽR l ; X . such that that SA 0qB 0 Q V F˜Ž t .V ; V Ž t G 0. and there exists a G SA 0qG˜ P V C 0Ž t .V ; V Ž t G 0., and A i V ; V Ž i s 1, . . . , p .. Thus, from Remark in w18, p. 106x, there exists an H˜ g BŽR l ; R m . such that SA 0qB 0 Q V H˜P V C 0Ž t .V ; V Ž t G 0.. Hence, it follows from Theorem 2.10 that V is a generalized SŽ A, B, C .-invariant subspace. Theorems 2.6, 2.8, and 2.10 say that generalized invariant subspaces are connected with the invariances of a finite number of conditions. Therefore, we can check whether a given subspace V is a generalized invariant subspace or not from these theorems.
3. PARAMETER-INSENSITIVE DISTURBANCE-REJECTION In this section, the infinite-dimensional versions of parameter-insensitive disturbance-rejection problems for uncertain linear systems which were investigated by Bhattacharyya w3x and Otsuka w9x are studied. Consider the following uncertain linear system SŽ ␣ ,  , ␥ , ␦ , . defined in a Hilbert space X,
¡ d x Ž t . s AŽ ␣ . x Ž t . q B Ž  . u Ž t . q E Ž . Ž t . ,
SŽ ␣ ,  , ␥ , ␦ , . :
~ dt
y t sC ␥ x t ,
¢z ŽŽ t .. s D ŽŽ ␦ .. x ŽŽ t .. ,
where x Ž t . g X, uŽ t . g U [ R m , y Ž t . g Y [ R l , z Ž t . g Z [ R , and ŽŽ0, ⬁.; Q . are the state, the input, the measurement output, the Ž t . g Lloc 1 controlled output, and the disturbance which is a Hilbert space Q valued
336
OTSUKA AND HINATA
locally integrable function, respectively. It is assumed that coefficient operators have the following unknown parameters, A Ž ␣ . s A 0 q ␣ 1 A1 q ⭈⭈⭈ q␣ p A p [ A 0 q ⌬ A Ž ␣ . , B Ž  . s B0 q  1 B1 q ⭈⭈⭈ q q Bq [ B0 q ⌬ B Ž  . , C Ž ␥ . s C0 q ␥ 1C1 q ⭈⭈⭈ q␥r Cr [ C0 q ⌬C Ž ␥ . , D Ž ␦ . s D 0 q ␦ 1 D 1 q ⭈⭈⭈ q␦ s Ds [ D 0 q ⌬ D Ž ␦ . , E Ž . s E0 q 1 E1 q ⭈⭈⭈ qt Et [ E0 q ⌬ E Ž . , where A i , Bi , Ci are the same as system SŽ ␣ ,  , ␥ . in Section 2, Di g BŽ X; R ., Ei g BŽ Q; X ., and ␣ [ Ž ␣ 1 , . . . , ␣ p . g R p ,  [ Ž  1 , . . . ,  q . g R q, ␥ [ Ž␥ 1 , . . . , ␥r . g R r , ␦ [ Ž ␦ 1 , . . . , ␦ s . g R s, [ Ž 1 , . . . , t . g R t. In system SŽ ␣ ,  , ␥ , ␦ , ., Ž A 0 , B0 , C0 , D 0 , E0 . and Ž ⌬ AŽ ␣ ., ⌬ B Ž  ., ⌬C Ž␥ ., ⌬ DŽ ␦ ., ⌬ E Ž .. represent the nominal system model and a specific uncertain perturbation, respectively. Now, we apply to system SŽ ␣ ,  , ␥ , ␦ , . a measurement output feedback of the form u Ž t . s Hy Ž t .
Ž s HC Ž ␥ . x Ž t . . ,
where H g BŽR l ; R m .. Then, the resulting closed-loop system is given as SH Ž ␣ ,  , ␥ , ␦ , . :
¡d ¢z Ž t . s D Ž ␦ . x Ž t . .
~ dt x Ž t . s Ž AŽ ␣ . q B Ž  . HC Ž ␥ . . x Ž t . q E Ž . Ž t . , Our parameter-insensitive disturbance-rejection problem for system SŽ ␣ ,  , ␥ , ␦ , . is stated as follows: Given operators A i , Bi , Ci , Di , Ei for system SŽ ␣ ,  , ␥ , ␦ , ., find, if possible, a measurement output feedback gain H g BŽR l ; R m . such that the closed-loop system SH Ž ␣ ,  , ␥ , ␦ , . rejects the disturbances from the controlled output z for all parameters Ž ␣ ,  , ␥ , ␦ , . g R p = R q = R r = R s = R t. To achieve this control requirement, we must solve the following problem: Given operators A i , Bi , Ci , Di , Ei for system SŽ ␣ ,  , ␥ , ␦ , ., find, if possible, a measurement output feedback gain H g BŽR l ; R m . such that DŽ ␦ .
t
H0 S
AŽ ␣ .qBŽ  . H CŽ ␥ .
Ž t y . E Ž . Ž . d s 0 Ž t G 0 . for all g Lloc 1 Ž 0, ⬁; Q . ,
GENERALIZED INVARIANT SUBSPACES
337
or equivalently Žsee, e.g., w1, p. 234x. ² S AŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N Im E Ž . : [L
ž DS
AŽ ␣ .qBŽ  . H CŽ ␥ .
Ž t . Ž Im E Ž . . ; Ker D Ž ␦ .
/
tG0
for all parameters Ž ␣ ,  , ␥ , ␦ , . g R p = R q = R r = R s = R t , where LŽ ⍀ . and overbar mean the linear subspace generated by the set ⍀ and the closure in X, respectively. This problem can be rephrased as follows. Problem 3.1 ŽParameter-Insensitive Disturbance-Rejection Problem with Measurement Output Feedback ŽPIDRPMOF... Given operators A i , Bi , Ci , Di , Ei for system SŽ ␣ ,  , ␥ , ␦ , ., find, if possible, a measurement output feedback gain H g BŽR l ; R m . such that ² SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N Im E Ž . : ; Ker D Ž ␦ . for all parameters Ž ␣ ,  , ␥ , ␦ , . g R p = R q = R r = R s = R t. Remark 3.2. If C Ž␥ . s I Židentity operator., then Problem 3.1 is called the parameter-insensitive disturbance-rejection problem with state feedback ŽPIDRPSF.. Now, some sufficient conditions for the problems to be solvable are given. THEOREM 3.3. V such that
If there exists a generalized SŽ A, B, C .-in¨ ariant subspace t
Ý
s
Im Ei ; V ;
is0
F Ker Di , is0
then the PIDRPMOF is sol¨ able. Proof. Suppose that there exists a generalized SŽ A, B, C .-invariant subspace V such that t
Ý is0
s
Im Ei ; V ;
F Ker Di . is0
Then, there exists an H g BŽR l ; R m . such that SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž t . V ; V
Ž t G 0.
338
OTSUKA AND HINATA
for all Ž ␣ ,  , ␥ . g R p = R q = R r. Hence,
¦
² SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N Im E Ž . : ; S AŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N
t
Ý is0
;
Im Ei
; ² SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N V : sV s
;
F Ker Di is0
; Ker D Ž ␦ . for all parameters Ž ␣ ,  , ␥ , ␦ , . g R p = R q = R r = R s = R t. Thus, the PIDRPMOF is solvable. The following corollaries follow from Theorem 3.3. s COROLLARY 3.4. If there exists a V g VŽ A, B; F is0 Ker Di . l t Ž .. V Ý is0 Im Ei ; C, A , then the PIDRPMOF is sol¨ able. s COROLLARY 3.5. Assume that VŽ A, B; F is0 Ker Di . has a maximal t element V *. If V * g VŽÝ is0 Im Ei ; C, A., then the PIDRPMOF is sol¨ able.
COROLLARY 3.6. Assume that VŽÝtis0 Im Ei ; C, A. has a minimal eles ment V#. If V# g VŽ A, B; F is0 Ker Di ., then the PIDRPMOF is sol¨ able. THEOREM 3.7. Assume that C Ž␥ . s I. If there exists a V g s VŽ A, B; F is0 Ker Di . satisfying Ýtis0 Im Ei ; V, then the PIDRPSF is sol¨ able. s Ker Di . has COROLLARY 3.8. Assume that C Ž␥ . s I and VŽ A, B; F is0 t a maximal element V *. If Ý is0 Im Ei ; V *, then the PIDRPSF is sol¨ able.
COROLLARY 3.9. Assume that C Ž␥ . s I and VŽÝtis0 Im Ei ; C, A. has a s minimal element V#. If V# ; F is0 Ker Di , then the PIDRPSF is sol¨ able. A maximal element V * and a minimal element V# are important to check the solvability conditions for the problems. However, one can say that a maximal element V * and a minimal element V# do not always exist in general for infinite-dimensional systems Žsee, e.g., w5, 18x..
GENERALIZED INVARIANT SUBSPACES
339
4. AN ILLUSTRATIVE EXAMPLE Consider the following mathematical model,
⭸ xŽ t, . ⭸t
s
⭸ 2 xŽ t, . ⭸ 2
q ␣ x Ž t , . q u1 Ž t . 1 Ž . q u 2 Ž t . 2 Ž .
q  u1 Ž t . Ž 2 1 Ž . y 2 Ž . . q u 2 Ž t . 3 Ž . 4 q Ž t . Ž 1Ž . y 2 Ž . . , x Ž t , 0. s 0 s x Ž t , 1. , zŽ t. s
1
H0 Ž Ž . q 1
2
Ž . . x Ž t , . d ,
where x Ž t, . is the temperature distribution of a bar of unit length at position and time t, u i Ž t . g R are the inputs. Ž t . g R is the disturbance and z Ž t . g R is the controlled output. And i Ž . s '2 sinŽ i .; i G 14 is an orthonormal basis of L2 w0, 1x, which are a set of eigenvectors with eigenvalues i s yi 2 2 ; i G 14 . Now, it is assumed that ␣ and  are uncertain parameters. We desire that the temperature at a certain measurement point be independent of the disturbances for the uncertain system. Let X [ L2 w0, 1x and define various operators as follows, A0 [
⭸2 ⭸ 2
,
A1 [ identity operator,
A Ž ␣ . [ A 0 q ␣ A1 ,
D Ž A Ž ␣ . . s D Ž A 0 . s x g X N x⬙ g X , x Ž 0 . s x Ž 1 . s 0 4 , B0 [ w 1 , 2 x ,
B1 [ w 2 1 y 2 , 3 x ,
B Ž  . [ B0 q  B1 g B Ž R 2 ; X . ,
E [ 1 y 2 g B Ž R; X . ,
D [ ² ⭈ , 1 q 2 : g BŽ X ; R. , where ² ⭈ , ⭈ : means the inner product in X. Then, the above model can be described by d dt
x Ž t . s A Ž ␣ . x Ž t . q B Ž  . u Ž t . q E Ž t . , z Ž t . s Dx Ž t . .
Since DS AŽ ␣ .Ž t .Im E s e ␣ t Ž e 1 t y 2 t . / 0, one can say that the controlled output of the original system is influenced by disturbances.
340
OTSUKA AND HINATA
Let us define the following orthonormal basis of X:
1 [
1
'2
Ž 1 y 2 . ,
2 [
1
'2 Ž 1 q 2 . ,
i [ i
Ž i G 3. .
By using these bases, define a state feedback F g BŽ X; R 2 . and a closed subspace V as follows,
¡
F i [
ž ~ ž
0,
2 y 1
'2
2 y 1
'2
¢Ž 0, 0.
T
,0
T
/ /
Ž i s 1. , T
Ž i s 2. ,
V [ span 1 , 3 4 .
Ž i G 3. ,
Then, it can be shown that Ž A 0 q B0 F . i s i i Ž i G 1., Im E ; V, and V ; DŽ A 0 .. It follows from Lemma 4.5 in w6x that SA 0qB 0 F V ; V. Since B1 F 1 s wŽ 2 y 1 .r '2 x 3 g V and B1 F 3 s 0 g V, we have B1 FV ; V. Further, it is obvious that A1V ; V and V ; Ker D. Thus, it follows from Theorems 2.6 and 3.7 that V is a generalized controlled S Ž A, B .-invariant subspace and the PIDRPSF is solvable.
5. CONCLUDING REMARKS In this paper some generalized invariant subspaces for infinite-dimensional linear systems were studied, and then their properties were investigated. Especially, it is useful that Theorems 2.6, 2.8, and 2.10 say infinitely many conditions are equivalent to a finite number of conditions. Further, the infinite-dimensional versions of the parameter-insensitive disturbancerejection problems with state feedback and with constant measurement output feedback for uncertain linear systems which were investigated by Bhattacharyya w3x and Otsuka w9x were formulated, and then some sufficient conditions for those problems to be solvable were presented. As further study, it is important to investigate the existence conditions of a maximal element V * and a minimal element V# in order to test practically the solvability conditions.
GENERALIZED INVARIANT SUBSPACES
341
REFERENCES 1. A. V. Balakrishnan, ‘‘Applied Functional Analysis,’’ Springer-Verlag, BerlinrNew York, 1981. 2. G. Basile and G. Marro, Controlled and conditioned invariant subspaces in linear system theory, J. Optim. Theory Appl. 3 Ž1969., 306᎐315. 3. S. P. Bhattacharyya, Generalized controllability, Ž A, B .-invariant subspaces and parameter invariant control, SIAM J. Algebraic Discrete Meth. 4 Ž1983., 529᎐533. 4. R. F. Curtain, Ž C, A, B .-pairs in infinite dimensions, Syst. Control Lett. 5 Ž1984., 59᎐65. 5. R. F. Curtain, Invariance concepts in infinite dimensions, SIAM J. Control Optim. 24 Ž1986., 1009᎐1030. 6. R. F. Curtain and A. J. Pritchard, ‘‘Infinite Dimensional Linear Systems,’’ Lecture Notes in Control and Information Sciences, Springer-Verlag, BerlinrNew York, 1978. 7. B. K. Ghosh, A geometric approach to simultaneous system design: Parameter insensitive disturbance decoupling by state and output feedback, in ‘‘Modeling, Identification and Robust Control’’ ŽC. I. Byrnes and A. Lindquist, eds.., North-Holland, Amsterdam, 1986, pp. 476᎐484. 8. N. Otsuka, Simultaneous decoupling and disturbance-rejection problems for infinite-dimensional systems, IMA J. Math. Control Inf. 8 Ž1991., 165᎐178. 9. N. Otsuka, Generalized invariant subspaces and parameter insensitive disturbance-rejection problems with static output feedback, IEEE Trans. Automat. Control 45 Ž2000., to appear. 10. N. Otsuka, Generalized Ž C, A, B .-pairs and parameter insensitive disturbance-rejection problems with dynamic compensator, IEEE Trans. Automat. Control 44 Ž1999., 2195᎐2200. 11. N. Otsuka and H. Inaba, Parameter-insensitive disturbance-rejection for infinite-dimensional systems, IMA J. Math. Control Inf. 14 Ž1997., 401᎐413. 12. N. Otsuka and H. Inaba, A note on robust disturbance-rejection problems for infinite-dimensional systems, Syst. Control Lett. 34 Ž1998., 33᎐41. 13. N. Otsuka and H. Inaba, Simultaneous Ž C, A, B .-pairs for infinite-dimensional systems, J. Math. Anal. Appl. 236 Ž1999., 415᎐437. 14. N. Otsuka, H. Inaba, and T. Oide, Decoupling by state feedback in infinite-dimensional systems, IMA J. Math. Control Inf. 7 Ž1990., 125᎐141. 15. N. Otsuka et al., Decoupling by incomplete-state feedback for infinite dimensional systems, Japan J. Industrial Appl. Math. 11 Ž1994., 363᎐377. 16. N. Otsuka et al., Parameter insensitive disturbance-rejection problem with incompletestate feedback, IEICE Trans. Fund. Electron. Comm. Computer Sci. E78-A Ž1995., 1589᎐1594. 17. W. M. Wonham, ‘‘Linear Multivariable Control: A Geometric Approach,’’ SpringerVerlag, BerlinrNew York, 1984. 18. H. Zwart, ‘‘Geometric Theory for Infinite-Dimensional Systems,’’ Lecture Notes in Control and Information Sciences, Springer-Verlag, BerlinrNew York, 1989.
Generalized Invariant Subspaces for Infinite-Dimensional Systems Naohisa Otsuka Department of Information Sciences, Tokyo Denki Uni¨ ersity, Hatoyama-Machi, Hiki-Gun, Saitama 350-0394, Japan E-mail: [email protected]
and Haruo Hinata Nagasaki Institute of Applied Science, Aba-Machi 536, Nagasaki 851-0193, Japan E-mail: [email protected] Submitted by William F. Ames Received December 3, 1999
In this paper some generalized invariant subspaces for infinite-dimensional systems are investigated, and then some sufficient conditions for parameter-insensitive disturbance-rejection problems with state feedback and with measurement output feedback to be solvable are studied. 䊚 2000 Academic Press Key Words: generalized invariant subspaces; uncertain systems; parameter-insensitive disturbance-rejection; geometric approach; infinite-dimensional systems.
1. INTRODUCTION The notion of invariant subspaces has been used successfully to study various control problems for linear finite-dimensional systems by Basile and Marro w2x and Wonham w17x. This notion has been extended to systems defined in Hilbert spaces, and some of the corresponding problems in Hilbert spaces have also been studied by Curtain w4, 5x, Zwart w18x, and Otsuka et al. w8, 14, 15x. After that, the simultaneous versions of those invariant subspaces for both finite- and infinite-dimensional linear systems have also been studied by Ghosh w7x and Otsuka et al. w11᎐13, 16x in order to investigate uncertain linear systems whose systems operators are repre325 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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OTSUKA AND HINATA
sented as convex combinations of given operators, and then parameter-insensitive disturbance-rejection problems have been studied. Further, Bhattacharyya w3x introduced the notion of generalized controlled Ž A, B .-invariant subspaces in order to study uncertain linear finite-dimensional systems whose systems matrices are represented as the nominal system matrices and specific uncertain perturbations, and then a parameter-insensitive disturbance-rejection problem with state feedback, which means robust with respect to the specific uncertain perturbations, was studied. Further, for such systems, Otsuka w9, 10x introduced the notions of generalized conditioned Ž C, A.-invariant subspaces, generalized Ž A, B, C .invariant subspaces, and generalized Ž C, A, B .-pairs, and then parameterinsensitive disturbance-rejection problems with static output feedback and with dynamic compensator were also investigated. In such a historical background, from the theoretical viewpoint, it is meaningful to investigate the infinite-dimensional version of those generalized invariant subspaces and their properties. Therefore, in this paper the infinite-dimensional versions of generalized invariant subspaces are first studied. Further, the parameter-insensitive disturbance-rejection problems with state feedback and with static output feedback for infinite-dimensional systems are formulated, and then their solvability conditions are given. Finally, an illustrative example is given.
2. GENERALIZED INVARIANT SUBSPACES First, some notations used throughout this investigation are given. Let BŽ X; Y . denote the set of all bounded linear operators from a Hilbert space X into another Hilbert space Y, and for notational simplicity, BŽ X; X . is written as BŽ X .. For a linear operator A, the domain, the image, the kernel, and the C0-semigroup generated by A are denoted by DŽ A., Im A, Ker A, and SAŽ t .; t G 04 , respectively. The notation R n denotes the n-dimensional Euclidean space. Next, consider the following linear systems defined in a Hilbert space X,
SŽ ␣ ,  , ␥ . :
¡d ¢y Ž t . s C Ž ␥ . x Ž t . ,
~ dt x Ž t . s AŽ ␣ . x Ž t . q B Ž  . u Ž t . ,
where x Ž t . g X, uŽ t . g U [ R m , y Ž t . g Y [ R l are the state, the input, and the measurement output, respectively. And operators AŽ ␣ ., B Ž  .,
GENERALIZED INVARIANT SUBSPACES
327
and C Ž␥ . are unknown in the sense that they are represented as the forms A Ž ␣ . s A 0 q ␣ 1 A1 q ⭈⭈⭈ q␣ p A p [ A 0 q ⌬ A Ž ␣ . , B Ž  . s B0 q  1 B1 q ⭈⭈⭈ q q Bq [ B0 q ⌬ B Ž  . , C Ž ␥ . s C0 q ␥ 1C1 q ⭈⭈⭈ q␥r Cr [ C0 q ⌬C Ž ␥ . , where ␣ [ Ž ␣ 1 , . . . , ␣ p . g R p ,  [ Ž  1 , . . . ,  q . g R q, ␥ [ Ž␥ 1 , . . . , ␥r . g R r , A 0 is the infinitesimal generator of a C0-semigroup SA 0Ž t .; t G 04 on X, A i g BŽ X . Ž i s 1, . . . , p ., Bi g BŽR m ; X . Ž i s 0, 1, . . . , q ., and Ci g BŽ X; R l . Ž i s 0, 1, . . . , r .. Here, in system SŽ ␣ ,  , ␥ ., Ž A 0 , B0 , C0 . and Ž ⌬ AŽ ␣ ., ⌬ B Ž  ., ⌬C Ž␥ .. mean the nominal system model and a specific uncertain perturbation, respectively. Now, since A i Ž i s 1, . . . , p . are bounded linear operators, one can say that AŽ ␣ . always generates a C0-semigroup and has the domain DŽ AŽ ␣ .. s DŽ A 0 . for all ␣ g R p. Further, from the practical viewpoint, one can say that the dimensions of input and output are finite dimensions. DEFINITION 2.1. Let V Ž; X . be a closed subspace. Ži. V is said to be a generalized controlled SŽ A, B .-invariant subspace if there exists an F g BŽ X; R m . such that SAŽ ␣ .qBŽ  . F Ž t . V ; V
Ž t G 0.
for all Ž ␣ ,  . g R p = R q. For a closed subspace ⌳, define V Ž A, B; ⌳ . [ V N V is a generalized controlled S Ž A, B . -invariant subspace and V ; ⌳ 4 . Žii. V is said to be a generalized conditioned SŽ C, A.-invariant subspace if there exists a G g BŽR l ; X . such that SAŽ ␣ .qG CŽ ␥ . Ž t . V ; V
Ž t G 0.
for all Ž ␣ , ␥ . g R p = R r. For a closed subspace , define V Ž ; C, A . [ V N V is a generalized conditioned S Ž C, A . -invariant subspace and ; V 4 . Žiii. V is said to be a generalized SŽ A, B, C .-invariant subspace if there exists an H g BŽR l ; R m . such that SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž t . V ; V for all Ž ␣ ,  , ␥ . g R p = R q = R r.
Ž t G 0.
328
OTSUKA AND HINATA
For system SŽ ␣ ,  , ␥ ., a generalized SŽ A, B, C .-invariant subspace V has the property that, if an arbitrary initial state x Ž0. stays in V, then there exists a measurement feedback H g BŽR l ; R m . which is independent of all Ž ␣ ,  , ␥ . g R p = R q = R r such that the state trajectory x Ž t . stays in V for all t G 0. The following lemma was given by Curtain w4x. LEMMA 2.2 Žsee w4x.. Let V be a closed subspace of X and let A be an infinitesimal generator with C0-semigroup SAŽ t .; t G 04 on X, and let Q1 g BŽ X .. Ži. If SAq Q Ž t .V ; V for all t G 0, then Ž A q Q1 .Ž V l DŽ A.. ; V. 1 Žii. If there exists a Q 2 g BŽ X . such that SAqQ Ž t .V ; V for all t G 0 2 and Ž Q1 y Q 2 .Ž V l DŽ A.. ; V, then SAqQ 1Ž t .V ; V for all t G 0. The following lemma is used to define two projection operators which play important roles to state the main results in this section. LEMMA 2.3. ⌫V by
Let V be a closed subspace. Define two subspaces R V and q
RV [
F By1 i V,
m where By1 i V [ u g R N Bi u g V 4
is1
and r
⌫V [
Ý Ci V . is1
Then, the following two statements hold. Ži.
There exists a subspace ⍀ such that R m s RV [ ⍀ .
Žii. There exist closed subspaces V1 , V2 of X and subspaces ⌳ 1 , ⌳ 2 of R l such that R l s C0 V1 [ C0 V2 [ ⌳ 1 [ ⌳ 2 , ⌫V s C0 V2 [ ⌳ 1 ,
and
C0 V2 s C0 V l ⌫V ,
V s V1 [ V2 [ Ž Ker C0 l V . .
Proof. Ži. The proof is obvious. Žii. Since V is a closed subspace, there exists a closed subspace ˜ Further, since V˜ ; V such that V s V˜ [ ŽKer C0 l V .. Then, C0 V s C0 V. ˜ there exists a closed subspace V2 ; V˜ such C0 V l ⌫V s C0 V˜ l ⌫V ; C0 V, that C0 V2 s C0 V l ⌫V . Then, there exists a closed subspace V1 ; V˜ such
GENERALIZED INVARIANT SUBSPACES
329
that V˜ s V1 [ V2 . Thus, V s V1 [ V2 [ ŽKer C0 l V . and C0 V s C0 V1 [ C0 V2 . Moreover, since C0 V l ⌫V ; ⌫V , there exists a subspace ⌳ 1 ; ⌫V such that ⌫V s Ž C0 V l ⌫V . [ ⌳ 1. Further, since C0 V1 [ C0 V2 [ ⌳ 1 ; R l , there exists a subspace ⌳ 2 such that R l s C0 V1 [ C0 V2 [ ⌳ 1 [ ⌳ 2 . By Lemma 2.3, the following operators can be defined. DEFINITION 2.4. For a closed subspace V of X, define the following two projection operators. Ži. Žii.
Q V : R m ª R m , projection operator onto R V along ⍀. PV : R l ª R l , projection operator onto C0 V1 [ ⌳ 2 along ⌫V .
Then, the following lemma is used to prove Theorem 2.6. LEMMA 2.5. The following two statements are equi¨ alent. Ži. There exists an F g BŽ X; R m . such that SA qB F Ž t .V ; V Ž t G 0. 0 0 and Bi FV ; V Ž i s 1, . . . , q .. Žii. There exists an F˜ g BŽ X; R m . such that SA qB Q F˜Ž t .V ; V Ž t G 0 0 V 0.. Proof. Ži. « Žii. Suppose that there exists an F g BŽ X; R m . such that SA 0qB 0 F Ž t .V ; V Ž t G 0. and Bi FV ; V Ž i s 1, . . . , q .. Then, FV ; q Ž B0 Q V F y B0 F .V s 04 . F is1 By1 i V s R V . Hence, Q V FV s FV. Thus, Then, it follows from Lemma 2.2Žii. that SA 0qB 0 Q V F Ž t .V ; V Ž t G 0.. Žii. « Ži. Suppose that there exists an F˜ g BŽ X; R m . such that ˜ Then, SA 0qB 0 F Ž t .V ; V Ž t SA 0qB 0 Q V F˜Ž t .V ; V Ž t G 0.. Define F [ Q V F. ˜ G 0.. Further, Bi FV s Bi Q V FV ; Bi R V ; V Ž i s 1, . . . , q .. This completes the proof. The following theorem is the infinite-dimensional version of the results of Bhattacharyya w3x. THEOREM 2.6. Let V be a closed subspace of X. Then, the following three statements are equi¨ alent. Ži. V is a generalized controlled SŽ A, B .-in¨ ariant subspace. Žii. There exists an F g BŽ X; R m . such that SA qB F Ž t .V ; V Ž t G 0. 0 0 and Bi FV ; V Ž i s 1, . . . , q ., and A i V ; V Ž i s 1, . . . , p .. Žiii. There exists an F˜ g BŽ X; R m . such that SA qB Q F˜Ž t .V ; V Ž t G 0 0 V 0., and A i V ; V Ž i s 1, . . . , p .. Proof. Ži. « Žii. Suppose that V is a generalized controlled SŽ A, B .invariant subspace. Then, there exists an F g BŽ X; R m . such that SAŽ ␣ .qBŽ  . F Ž t . V ; V for all Ž ␣ ,  . g R = R . p
q
Ž t G 0.
Ž 1.
330
OTSUKA AND HINATA
First, suppose that ␣ 1 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s 0 in Ž1.. Then, SA 0qB 0 F Ž t . V ; V
Ž t G 0. ,
which with Lemma 2.2Ži. implies
Ž A 0 q B0 F . Ž V l D Ž A 0 . . ; V .
Ž 2.
Further, suppose that ␣ 1 s 1 and ␣ 2 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s 0 in Ž1.. Then, SA 0qA 1qB 0 F Ž t . V ; V
Ž t G 0. ,
which with Lemma 2.2Ži. implies
Ž A 0 q A1 q B0 F . Ž V l D Ž A 0 . . ; V .
Ž 3.
Hence, it follows from Ž2. and Ž3. that A1Ž V l DŽ A 0 .. ; V. Since A1 is a bounded linear operator, A1V ; V. Similarly, one can prove A i V ; V Ž i s 2, . . . , p .. Next, suppose that  1 s 1 and ␣ 1 s ⭈⭈⭈ s ␣ p s  2 s ⭈⭈⭈ s  q s 0 in Ž1.. Then,
Ž A 0 q Ž B0 q B1 . F .Ž V l D Ž A 0 . . ; V .
Ž 4.
Hence, it follows from Ž2., Ž4., and boundedness of B1 F that B1 FV ; V. Similarly, one can prove Bi FV ; V Ž i s 2, . . . , q .. Žii. « Ži. Suppose that there exists an F g BŽ X; R m . such that SA 0qB 0 F Ž t .V ; V Ž t G 0. and Bi FV ; V Ž i s 1, . . . , q ., and A i V ; V Ž i s 1, . . . , p .. Now,
Ž AŽ ␣ . q B Ž  . F . s Ž A 0 q ␣ 1 A1 q ⭈⭈⭈ q␣ p A p . q Ž B0 q  1 B1 q ⭈⭈⭈ q q Bq . F 4 p
s Ž A 0 q B0 F . q
q
Ý ␣ i A i q Ý i Bi F. is1
Since S A 0qB 0 F Ž t .V ; V and from Lemma 2.2Žii. that
p ŽÝ is1
is1
␣ i A i q Ý qis1 i Bi F .V ; V, it follows
SAŽ ␣ .qBŽ  . F Ž t . V ; V
Ž t G 0.
for all Ž ␣ ,  . g R p = R q, which implies V is a generalized controlled SŽ A, B .-invariant subspace. Žii. m Žiii. The proof follows from Lemma 2.5.
331
GENERALIZED INVARIANT SUBSPACES
The following lemma is used to prove Theorem 2.8. LEMMA 2.7. The following two statements are equi¨ alent. Ži. There exists a G g BŽR l ; X . such that SA qG C Ž t .V ; V Ž t G 0. 0 0 and GC i V ; V Ž i s 1, . . . , r .. ˜ g BŽR l ; X . such that SA 0qG˜ PV C 0Ž t .V ; V Ž t G 0.. Žii. There exists a G Proof. Ži. « Žii. Suppose that there exists a G g BŽR l ; X . such that SA 0qG C 0Ž t .V ; V Ž t G 0. and GC i V ; V Ž i s 1, . . . , r .. Then, from Lemma 2.2Žii., it suffices to show Ž GPV C0 y GC 0 .V ; V. Choose an arbitrary element x g V. Then, by Lemma 2.3Žii., x can be decomposed as x s y q z q u Ž y g V1 , z g V2 , u g Ker C0 l V .. Hence, r
Ž GPV C0 y GC0 . x s yGC0 z g GC0 V2 ; G⌫V s
Ý GCiV ; V . is1
It follows from Lemma 2.2Žii. that SA 0qG P V C 0Ž t .V ; V Ž t G 0.. ˜ g BŽR l ; X . such that Žii. « Ži. Suppose that there exists a G ˜ V . Then, SA 0qG C 0Ž t .V ; V Ž t SA 0qG˜ P V C 0Ž t .V ; V Ž t G 0.. Define G [ GP ˜ V Ci V ; GP ˜ V ŽKer PV . s 04 ; V Ž i s 1, . . . , r .. G 0.. Further, GC i V s GP This completes the proof. The following results are the infinite-dimensional versions of Otsuka w9x. THEOREM 2.8. Let V be a closed subspace of X. Then, the following three statements are equi¨ alent. Ži. V is a generalized conditioned SŽ C, A.-in¨ ariant subspace. Žii. There exists a G g BŽR l ; X . such that SA qG C Ž t .V ; V Ž t G 0. 0 0 and GC i V ; V Ž i s 1, . . . , r ., and A i V ; V Ž i s 1, . . . , p .. ˜ g BŽR l ; X . such that SA 0qG˜ PV C 0Ž t .V ; V Ž t G 0., Žiii. There exists a G and A i V ; V Ž i s 1, . . . , p .. Proof. Ži. « Žii. Suppose that V is a generalized conditioned S Ž C, A.-invariant subspace. Then, there exists a G g BŽR l ; X . such that SAŽ ␣ .qG CŽ ␥ . Ž t . V ; V
Ž t G 0.
Ž 5.
for all Ž ␣ , ␥ . g R p = R r. First, suppose that ␣ 1 s ⭈⭈⭈ s ␣ p s 0 s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž5.. Then, S A 0qG C 0Ž t . V ; V
Ž t G 0. ,
which with Lemma 2.2Ži. implies
Ž A 0 q GC0 . Ž V l D Ž A 0 . . ; V .
Ž 6.
332
OTSUKA AND HINATA
Further, suppose that ␣ 1 s 1 and ␣ 2 s ⭈⭈⭈ s ␣ p s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž5.. Then, SA 0qA 1qG C 0Ž t . V ; V
Ž t G 0.
which with Lemma 2.2Ži. implies
Ž A 0 q A1 q GC0 . Ž V l D Ž A 0 . . ; V .
Ž 7.
Hence, it follows from Ž6. and Ž7. that A1Ž V l DŽ A 0 .. ; V. Since A1 is a bounded linear operator, A1V ; V. Similarly, one can prove A i V ; V Ž i s 2, . . . , p .. Next, suppose that ␥ 1 s 1 and ␣ 1 s ⭈⭈⭈ s ␣ p s ␥ 2 s ⭈⭈⭈ s ␥r s 0 in Ž5.. Then,
Ž A 0 q G Ž C0 q C1 . .Ž V l D Ž A 0 . . ; V .
Ž 8.
Hence, it follows from Ž6., Ž8., and boundedness of GC1 that GC1V ; V. Similarly, one can prove GCi V ; V Ž i s 2, . . . , r .. Žii. « Ži. Suppose that there exists a G g BŽR l ; X . such that SA 0qG C 0V ; V Ž t G 0. and GC i V ; V Ž i s 1, . . . , r ., and A i V ; V Ž i s 1, . . . , p .. Now,
Ž AŽ ␣ . q GC Ž ␥ . . s Ž A 0 q ␣ 1 A1 q ⭈⭈⭈ q␣ p A p . q G Ž C0 q ␥ 1C1 q ⭈⭈⭈ q␥r Cr . 4 p
s Ž A 0 q GC 0 . q
r
Ý ␣i Ai q
Ý ␥ i GCi .
is1
is1
p Since SA 0qG C 0Ž t .V ; V and ŽÝ is1 ␣ i A i q Ý ris1 ␥ i GCi .V ; V, it follows from Lemma 2.2Žii. that
SAŽ ␣ .qG CŽ ␥ . Ž t . V ; V
Ž t G 0.
for all Ž ␣ , ␥ . g R p = R r , which implies V is a generalized conditioned SŽ C, A.-invariant subspace. Žii. m Žiii. The proof follows from Lemma 2.7. The following lemma is used to prove Theorem 2.10. LEMMA 2.9. The following two statements are equi¨ alent. Ži. There exists an H g BŽR l ; R m . such that SA qB H C Ž t .V ; V Ž t G 0 0 0 0., Bi HC j V ; V Ž i s 0, . . . , q, j s 0, . . . , r; Ž i, j . / Ž0, 0.., and A i V ; V Ž i s 1, . . . , p .. Žii. There exists an H˜ g BŽR l ; R m . such that SA qB Q H˜P C V ; V Ž t G 0 0 V V 0 0., and A i V ; V Ž i s 1, . . . , p ..
GENERALIZED INVARIANT SUBSPACES
333
Proof. Ži. « Žii. Suppose that there exists an H g BŽR l ; R m . such that SA 0qB 0 H C 0Ž t .V ; V Ž t G 0., Bi HC j V ; V Ž i s 0, . . . , q, j s 0, . . . , r; Ž i, j . / Ž0, 0.., and A i V ; V Ž i s 1, . . . , p .. Then, it follows from Lemmas 2.5 and 2.7 that there exists an F˜ g BŽ X; R m . such that SA 0qB 0 Q V F˜Ž t .V ; V ˜ g BŽR l ; X . such that SA 0qG˜ PV C 0Ž t .V ; V Ž t G 0. and there exists a G Ž t G 0.. From Remark in w18, p. 106x, there exists an H˜ g BŽR l ; R m . such that S A 0qB 0 Q V H˜P V C 0Ž t .V ; V Ž t G 0.. Žii. « Ži. Suppose that there exists an H˜ g BŽR l ; R m . such that SA 0qB 0 Q V H˜P V C 0V ; V Ž t G 0. and A i V ; V Ž i s 1, . . . , p .. Define H [ ˜ V . Then, SA 0qB 0 H C 0Ž t .V ; V Ž t G 0.. And, one obtains Bi HC j V s Q V HP ˜ V C j V ; Bi Im Q V ; Bi R V ; V Ž i s 1, . . . , q, j s 0, . . . , r .. FurBi Q V HP ˜ V C j V ; B0 Q V HP ˜ V ⌫V s 04 ; V Ž j s 1, . . . , r .. ther, B0 HC j V s B0 Q V HP This completes the proof. Finally, the following theorem can be obtained. THEOREM 2.10. Let V be a closed subspace of X. Then, the following three statements are equi¨ alent. Ži. V is a generalized SŽ A, B, C .-in¨ ariant subspace. Žii. There exists an H g BŽR l ; R m . such that SA qB H C Ž t .V ; V Ž t G 0 0 0 0., Bi HC j V ; V Ž i s 0, . . . , q, j s 0, . . . , r; Ž i, j . / Ž0, 0.., and A i V ; V Ž i s 1, . . . , p .. Žiii. There exists an H˜ g BŽR l ; R m . such that SA qB Q H˜P C V ; V 0 0 V V 0 Ž t G 0., and A i V ; V Ž i s 1, . . . , p .. Proof. Ži. « Žii. Suppose that V is a generalized S Ž A, B, C .-invariant subspace. Then, there exists an H g BŽR l ; R m . such that SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž t . V ; V
Ž t G 0.
Ž 9.
for all Ž ␣ ,  , ␥ . g R p = R q = R r. First, suppose that ␣ 1 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž9.. Then, SA 0qB 0 H C 0Ž t . V ; V
Ž t G 0. ,
which with Lemma 2.2Ži. implies
Ž A 0 q B0 HC0 . Ž V l D Ž A 0 . . ; V .
Ž 10 .
Further, suppose that ␣ 1 s 1 and ␣ 2 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž9.. Then, SA 0qA 1qB 0 H C 0Ž t . V ; V
Ž t G 0. ,
334
OTSUKA AND HINATA
which with Lemma 2.2Ži. implies
Ž A 0 q A1 q B0 HC0 . Ž V l D Ž A 0 . . ; V .
Ž 11 .
Hence, it follows from Ž10. and Ž11. that A1Ž V l DŽ A 0 .. ; V. Since A1 is a bounded linear operator, A1V ; V. Similarly, one can prove A i V ; V Ž i s 2, . . . , p .. Next, suppose that  1 s 1 and ␣ 1 s ⭈⭈⭈ s ␣ p s  2 s ⭈⭈⭈ s  q s ␥ 1 s ⭈⭈⭈ s ␥r s 0 in Ž9.. Then,
Ž A 0 q Ž B0 q B1 . HC0 .Ž V l D Ž A 0 . . ; V .
Ž 12 .
Hence, it follows from Ž10., Ž12., and boundedness of B1 HC0 that B1 HC0 V ; V. Further, suppose that ␥ 1 s 1 and ␣ 1 s ⭈⭈⭈ s ␣ p s  1 s ⭈⭈⭈ s  q s ␥ 2 s ⭈⭈⭈ s ␥r s 0 in Ž9.. Then,
Ž A 0 q B0 H Ž C0 q C1 . .Ž V l D Ž A 0 . . ; V .
Ž 13 .
From Ž10., Ž13., and boundedness of B0 HC1 , one obtains B0 HC1V ; V. Similarly, one can prove Bi HC j V ; V Ž i s 1, . . . , q, j s 1, . . . , r .. Žii. « Ži. Suppose that there exists an H g BŽR l ; R m . such that SA 0qB 0 H C 0Ž t .V ; V, Bi HC j V ; V Ž i s 0, . . . , q, j s 0, . . . , r; Ž i, j . / Ž0, 0.., and A i V ; V Ž i s 1, . . . , p .. Now, p
r
Ž AŽ ␣ . q B Ž  . HC Ž ␥ . . s Ž A 0 q B0 HC0 . q Ý ␣ i A i q Ý ␥ i B0 HCi is1 q
q
q Ý i Bi HC0 q is1
is1 r
Ý Ý i ␥ j Bi HC j . is1 js1
Since SA 0qB 0 H C 0Ž t .V ; V and p
ž
q
r
Ý ␣ i A i q Ý ␥ i B0 HCi q Ý i Bi HC0 is1
is1
is1 q
qÝ
r
Ý i ␥ j Bi HC j
is1 js1
/
V ; V,
it follows from Lemma 2.2Žii. that SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž t . V ; V
Ž t G 0.
for all Ž ␣ ,  , ␥ . g R p = R q = R r , which implies V is a generalized SŽ A, B, C .-invariant subspace. Žii. m Žiii. The proof follows from Lemma 2.9.
GENERALIZED INVARIANT SUBSPACES
335
Concerning the three generalized invariant subspaces, the following corollary can be obtained. COROLLARY 2.11. V is a generalized SŽ A, B, C .-in¨ ariant subspace if and only if V is a generalized controlled SŽ A, B .-in¨ ariant and generalized conditioned SŽ C, A.-in¨ ariant subspace. Proof. ŽNecessity. Suppose that V is a generalized S Ž A, B, C .-invariant subspace. Then, it follows from Theorem 2.10 that there exists an H˜ g BŽR l ; R m . such that SA 0qB 0 Q V H˜P V C 0Ž t .V ; V Ž t G 0., and A i V ; V Ž i s 1, . . . , p .. Thus, from Theorems 2.6 and 2.8, V is a generalized controlled SŽ A, B .-invariant and generalized conditioned SŽ C, A.-invariant subspace. ŽSufficiency. Suppose that V is a generalized controlled SŽ A, B .-invariant and generalized conditioned SŽ C, A.-invariant subspace. Then, it follows from Theorems 2.6 and 2.8 that there exists an F˜ g BŽ X; R m . such ˜ g BŽR l ; X . such that that SA 0qB 0 Q V F˜Ž t .V ; V Ž t G 0. and there exists a G SA 0qG˜ P V C 0Ž t .V ; V Ž t G 0., and A i V ; V Ž i s 1, . . . , p .. Thus, from Remark in w18, p. 106x, there exists an H˜ g BŽR l ; R m . such that SA 0qB 0 Q V H˜P V C 0Ž t .V ; V Ž t G 0.. Hence, it follows from Theorem 2.10 that V is a generalized SŽ A, B, C .-invariant subspace. Theorems 2.6, 2.8, and 2.10 say that generalized invariant subspaces are connected with the invariances of a finite number of conditions. Therefore, we can check whether a given subspace V is a generalized invariant subspace or not from these theorems.
3. PARAMETER-INSENSITIVE DISTURBANCE-REJECTION In this section, the infinite-dimensional versions of parameter-insensitive disturbance-rejection problems for uncertain linear systems which were investigated by Bhattacharyya w3x and Otsuka w9x are studied. Consider the following uncertain linear system SŽ ␣ ,  , ␥ , ␦ , . defined in a Hilbert space X,
¡ d x Ž t . s AŽ ␣ . x Ž t . q B Ž  . u Ž t . q E Ž . Ž t . ,
SŽ ␣ ,  , ␥ , ␦ , . :
~ dt
y t sC ␥ x t ,
¢z ŽŽ t .. s D ŽŽ ␦ .. x ŽŽ t .. ,
where x Ž t . g X, uŽ t . g U [ R m , y Ž t . g Y [ R l , z Ž t . g Z [ R , and ŽŽ0, ⬁.; Q . are the state, the input, the measurement output, the Ž t . g Lloc 1 controlled output, and the disturbance which is a Hilbert space Q valued
336
OTSUKA AND HINATA
locally integrable function, respectively. It is assumed that coefficient operators have the following unknown parameters, A Ž ␣ . s A 0 q ␣ 1 A1 q ⭈⭈⭈ q␣ p A p [ A 0 q ⌬ A Ž ␣ . , B Ž  . s B0 q  1 B1 q ⭈⭈⭈ q q Bq [ B0 q ⌬ B Ž  . , C Ž ␥ . s C0 q ␥ 1C1 q ⭈⭈⭈ q␥r Cr [ C0 q ⌬C Ž ␥ . , D Ž ␦ . s D 0 q ␦ 1 D 1 q ⭈⭈⭈ q␦ s Ds [ D 0 q ⌬ D Ž ␦ . , E Ž . s E0 q 1 E1 q ⭈⭈⭈ qt Et [ E0 q ⌬ E Ž . , where A i , Bi , Ci are the same as system SŽ ␣ ,  , ␥ . in Section 2, Di g BŽ X; R ., Ei g BŽ Q; X ., and ␣ [ Ž ␣ 1 , . . . , ␣ p . g R p ,  [ Ž  1 , . . . ,  q . g R q, ␥ [ Ž␥ 1 , . . . , ␥r . g R r , ␦ [ Ž ␦ 1 , . . . , ␦ s . g R s, [ Ž 1 , . . . , t . g R t. In system SŽ ␣ ,  , ␥ , ␦ , ., Ž A 0 , B0 , C0 , D 0 , E0 . and Ž ⌬ AŽ ␣ ., ⌬ B Ž  ., ⌬C Ž␥ ., ⌬ DŽ ␦ ., ⌬ E Ž .. represent the nominal system model and a specific uncertain perturbation, respectively. Now, we apply to system SŽ ␣ ,  , ␥ , ␦ , . a measurement output feedback of the form u Ž t . s Hy Ž t .
Ž s HC Ž ␥ . x Ž t . . ,
where H g BŽR l ; R m .. Then, the resulting closed-loop system is given as SH Ž ␣ ,  , ␥ , ␦ , . :
¡d ¢z Ž t . s D Ž ␦ . x Ž t . .
~ dt x Ž t . s Ž AŽ ␣ . q B Ž  . HC Ž ␥ . . x Ž t . q E Ž . Ž t . , Our parameter-insensitive disturbance-rejection problem for system SŽ ␣ ,  , ␥ , ␦ , . is stated as follows: Given operators A i , Bi , Ci , Di , Ei for system SŽ ␣ ,  , ␥ , ␦ , ., find, if possible, a measurement output feedback gain H g BŽR l ; R m . such that the closed-loop system SH Ž ␣ ,  , ␥ , ␦ , . rejects the disturbances from the controlled output z for all parameters Ž ␣ ,  , ␥ , ␦ , . g R p = R q = R r = R s = R t. To achieve this control requirement, we must solve the following problem: Given operators A i , Bi , Ci , Di , Ei for system SŽ ␣ ,  , ␥ , ␦ , ., find, if possible, a measurement output feedback gain H g BŽR l ; R m . such that DŽ ␦ .
t
H0 S
AŽ ␣ .qBŽ  . H CŽ ␥ .
Ž t y . E Ž . Ž . d s 0 Ž t G 0 . for all g Lloc 1 Ž 0, ⬁; Q . ,
GENERALIZED INVARIANT SUBSPACES
337
or equivalently Žsee, e.g., w1, p. 234x. ² S AŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N Im E Ž . : [L
ž DS
AŽ ␣ .qBŽ  . H CŽ ␥ .
Ž t . Ž Im E Ž . . ; Ker D Ž ␦ .
/
tG0
for all parameters Ž ␣ ,  , ␥ , ␦ , . g R p = R q = R r = R s = R t , where LŽ ⍀ . and overbar mean the linear subspace generated by the set ⍀ and the closure in X, respectively. This problem can be rephrased as follows. Problem 3.1 ŽParameter-Insensitive Disturbance-Rejection Problem with Measurement Output Feedback ŽPIDRPMOF... Given operators A i , Bi , Ci , Di , Ei for system SŽ ␣ ,  , ␥ , ␦ , ., find, if possible, a measurement output feedback gain H g BŽR l ; R m . such that ² SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N Im E Ž . : ; Ker D Ž ␦ . for all parameters Ž ␣ ,  , ␥ , ␦ , . g R p = R q = R r = R s = R t. Remark 3.2. If C Ž␥ . s I Židentity operator., then Problem 3.1 is called the parameter-insensitive disturbance-rejection problem with state feedback ŽPIDRPSF.. Now, some sufficient conditions for the problems to be solvable are given. THEOREM 3.3. V such that
If there exists a generalized SŽ A, B, C .-in¨ ariant subspace t
Ý
s
Im Ei ; V ;
is0
F Ker Di , is0
then the PIDRPMOF is sol¨ able. Proof. Suppose that there exists a generalized SŽ A, B, C .-invariant subspace V such that t
Ý is0
s
Im Ei ; V ;
F Ker Di . is0
Then, there exists an H g BŽR l ; R m . such that SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž t . V ; V
Ž t G 0.
338
OTSUKA AND HINATA
for all Ž ␣ ,  , ␥ . g R p = R q = R r. Hence,
¦
² SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N Im E Ž . : ; S AŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N
t
Ý is0
;
Im Ei
; ² SAŽ ␣ .qBŽ  . H CŽ ␥ . Ž ⭈ . N V : sV s
;
F Ker Di is0
; Ker D Ž ␦ . for all parameters Ž ␣ ,  , ␥ , ␦ , . g R p = R q = R r = R s = R t. Thus, the PIDRPMOF is solvable. The following corollaries follow from Theorem 3.3. s COROLLARY 3.4. If there exists a V g VŽ A, B; F is0 Ker Di . l t Ž .. V Ý is0 Im Ei ; C, A , then the PIDRPMOF is sol¨ able. s COROLLARY 3.5. Assume that VŽ A, B; F is0 Ker Di . has a maximal t element V *. If V * g VŽÝ is0 Im Ei ; C, A., then the PIDRPMOF is sol¨ able.
COROLLARY 3.6. Assume that VŽÝtis0 Im Ei ; C, A. has a minimal eles ment V#. If V# g VŽ A, B; F is0 Ker Di ., then the PIDRPMOF is sol¨ able. THEOREM 3.7. Assume that C Ž␥ . s I. If there exists a V g s VŽ A, B; F is0 Ker Di . satisfying Ýtis0 Im Ei ; V, then the PIDRPSF is sol¨ able. s Ker Di . has COROLLARY 3.8. Assume that C Ž␥ . s I and VŽ A, B; F is0 t a maximal element V *. If Ý is0 Im Ei ; V *, then the PIDRPSF is sol¨ able.
COROLLARY 3.9. Assume that C Ž␥ . s I and VŽÝtis0 Im Ei ; C, A. has a s minimal element V#. If V# ; F is0 Ker Di , then the PIDRPSF is sol¨ able. A maximal element V * and a minimal element V# are important to check the solvability conditions for the problems. However, one can say that a maximal element V * and a minimal element V# do not always exist in general for infinite-dimensional systems Žsee, e.g., w5, 18x..
GENERALIZED INVARIANT SUBSPACES
339
4. AN ILLUSTRATIVE EXAMPLE Consider the following mathematical model,
⭸ xŽ t, . ⭸t
s
⭸ 2 xŽ t, . ⭸ 2
q ␣ x Ž t , . q u1 Ž t . 1 Ž . q u 2 Ž t . 2 Ž .
q  u1 Ž t . Ž 2 1 Ž . y 2 Ž . . q u 2 Ž t . 3 Ž . 4 q Ž t . Ž 1Ž . y 2 Ž . . , x Ž t , 0. s 0 s x Ž t , 1. , zŽ t. s
1
H0 Ž Ž . q 1
2
Ž . . x Ž t , . d ,
where x Ž t, . is the temperature distribution of a bar of unit length at position and time t, u i Ž t . g R are the inputs. Ž t . g R is the disturbance and z Ž t . g R is the controlled output. And i Ž . s '2 sinŽ i .; i G 14 is an orthonormal basis of L2 w0, 1x, which are a set of eigenvectors with eigenvalues i s yi 2 2 ; i G 14 . Now, it is assumed that ␣ and  are uncertain parameters. We desire that the temperature at a certain measurement point be independent of the disturbances for the uncertain system. Let X [ L2 w0, 1x and define various operators as follows, A0 [
⭸2 ⭸ 2
,
A1 [ identity operator,
A Ž ␣ . [ A 0 q ␣ A1 ,
D Ž A Ž ␣ . . s D Ž A 0 . s x g X N x⬙ g X , x Ž 0 . s x Ž 1 . s 0 4 , B0 [ w 1 , 2 x ,
B1 [ w 2 1 y 2 , 3 x ,
B Ž  . [ B0 q  B1 g B Ž R 2 ; X . ,
E [ 1 y 2 g B Ž R; X . ,
D [ ² ⭈ , 1 q 2 : g BŽ X ; R. , where ² ⭈ , ⭈ : means the inner product in X. Then, the above model can be described by d dt
x Ž t . s A Ž ␣ . x Ž t . q B Ž  . u Ž t . q E Ž t . , z Ž t . s Dx Ž t . .
Since DS AŽ ␣ .Ž t .Im E s e ␣ t Ž e 1 t y 2 t . / 0, one can say that the controlled output of the original system is influenced by disturbances.
340
OTSUKA AND HINATA
Let us define the following orthonormal basis of X:
1 [
1
'2
Ž 1 y 2 . ,
2 [
1
'2 Ž 1 q 2 . ,
i [ i
Ž i G 3. .
By using these bases, define a state feedback F g BŽ X; R 2 . and a closed subspace V as follows,
¡
F i [
ž ~ ž
0,
2 y 1
'2
2 y 1
'2
¢Ž 0, 0.
T
,0
T
/ /
Ž i s 1. , T
Ž i s 2. ,
V [ span 1 , 3 4 .
Ž i G 3. ,
Then, it can be shown that Ž A 0 q B0 F . i s i i Ž i G 1., Im E ; V, and V ; DŽ A 0 .. It follows from Lemma 4.5 in w6x that SA 0qB 0 F V ; V. Since B1 F 1 s wŽ 2 y 1 .r '2 x 3 g V and B1 F 3 s 0 g V, we have B1 FV ; V. Further, it is obvious that A1V ; V and V ; Ker D. Thus, it follows from Theorems 2.6 and 3.7 that V is a generalized controlled S Ž A, B .-invariant subspace and the PIDRPSF is solvable.
5. CONCLUDING REMARKS In this paper some generalized invariant subspaces for infinite-dimensional linear systems were studied, and then their properties were investigated. Especially, it is useful that Theorems 2.6, 2.8, and 2.10 say infinitely many conditions are equivalent to a finite number of conditions. Further, the infinite-dimensional versions of the parameter-insensitive disturbancerejection problems with state feedback and with constant measurement output feedback for uncertain linear systems which were investigated by Bhattacharyya w3x and Otsuka w9x were formulated, and then some sufficient conditions for those problems to be solvable were presented. As further study, it is important to investigate the existence conditions of a maximal element V * and a minimal element V# in order to test practically the solvability conditions.
GENERALIZED INVARIANT SUBSPACES
341
REFERENCES 1. A. V. Balakrishnan, ‘‘Applied Functional Analysis,’’ Springer-Verlag, BerlinrNew York, 1981. 2. G. Basile and G. Marro, Controlled and conditioned invariant subspaces in linear system theory, J. Optim. Theory Appl. 3 Ž1969., 306᎐315. 3. S. P. Bhattacharyya, Generalized controllability, Ž A, B .-invariant subspaces and parameter invariant control, SIAM J. Algebraic Discrete Meth. 4 Ž1983., 529᎐533. 4. R. F. Curtain, Ž C, A, B .-pairs in infinite dimensions, Syst. Control Lett. 5 Ž1984., 59᎐65. 5. R. F. Curtain, Invariance concepts in infinite dimensions, SIAM J. Control Optim. 24 Ž1986., 1009᎐1030. 6. R. F. Curtain and A. J. Pritchard, ‘‘Infinite Dimensional Linear Systems,’’ Lecture Notes in Control and Information Sciences, Springer-Verlag, BerlinrNew York, 1978. 7. B. K. Ghosh, A geometric approach to simultaneous system design: Parameter insensitive disturbance decoupling by state and output feedback, in ‘‘Modeling, Identification and Robust Control’’ ŽC. I. Byrnes and A. Lindquist, eds.., North-Holland, Amsterdam, 1986, pp. 476᎐484. 8. N. Otsuka, Simultaneous decoupling and disturbance-rejection problems for infinite-dimensional systems, IMA J. Math. Control Inf. 8 Ž1991., 165᎐178. 9. N. Otsuka, Generalized invariant subspaces and parameter insensitive disturbance-rejection problems with static output feedback, IEEE Trans. Automat. Control 45 Ž2000., to appear. 10. N. Otsuka, Generalized Ž C, A, B .-pairs and parameter insensitive disturbance-rejection problems with dynamic compensator, IEEE Trans. Automat. Control 44 Ž1999., 2195᎐2200. 11. N. Otsuka and H. Inaba, Parameter-insensitive disturbance-rejection for infinite-dimensional systems, IMA J. Math. Control Inf. 14 Ž1997., 401᎐413. 12. N. Otsuka and H. Inaba, A note on robust disturbance-rejection problems for infinite-dimensional systems, Syst. Control Lett. 34 Ž1998., 33᎐41. 13. N. Otsuka and H. Inaba, Simultaneous Ž C, A, B .-pairs for infinite-dimensional systems, J. Math. Anal. Appl. 236 Ž1999., 415᎐437. 14. N. Otsuka, H. Inaba, and T. Oide, Decoupling by state feedback in infinite-dimensional systems, IMA J. Math. Control Inf. 7 Ž1990., 125᎐141. 15. N. Otsuka et al., Decoupling by incomplete-state feedback for infinite dimensional systems, Japan J. Industrial Appl. Math. 11 Ž1994., 363᎐377. 16. N. Otsuka et al., Parameter insensitive disturbance-rejection problem with incompletestate feedback, IEICE Trans. Fund. Electron. Comm. Computer Sci. E78-A Ž1995., 1589᎐1594. 17. W. M. Wonham, ‘‘Linear Multivariable Control: A Geometric Approach,’’ SpringerVerlag, BerlinrNew York, 1984. 18. H. Zwart, ‘‘Geometric Theory for Infinite-Dimensional Systems,’’ Lecture Notes in Control and Information Sciences, Springer-Verlag, BerlinrNew York, 1989.