A GENERALIZATION OF THE YOULA-KUČERA PARAMETRIZATION FOR MIMO STABILIZABLE SYSTEMS. Alban Quadrat

Transcription

1 A GENERALIZATION OF THE YOULA-KUČERA ARAMETRIZATION FOR MIMO STABILIZABLE SYSTEMS Alban Quadrat INRIA Sophia Antipolis, CAFE project, 2004 Route des Lucioles, B 93, Sophia Antipolis cedex, France. Alban.Quadrat@sophia.inria.fr. Abstract: The purpose of this paper is to ive new necessary and sufficient conditions for internal stabilizability and existence of left/riht/doubly coprime factorizations for linear MIMO systems. In particular, we eneralize the Youla- Kučera parametrization of all stabilizin controllers for every internally stabilizable MIMO plant which does not necessarily admit doubly coprime factorizations. Keywords: Youla-Kučera parametrization of all stabilizin controllers, internal stabilizability, left/riht/doubly coprime factorizations, lattices. 1. INTRODUCTION For finite-dimensional linear systems i.e. ordinary differential equations), it is well nown that a transfer matrix is internally stabilizable if and only if it admits a doubly coprime factorization Vidyasaar, 1985). However, this result is enerally not true for infinite-dimensional linear systems e.. differential time-delay systems, partial differential equations) Quadrat, 2003; Vidyasaar, 1985) or for multidimensional linear systems Mori, 2002; Sule, 1994). The Youla-Kučera parametrization of all stabilizin controllers was developed for transfer matrices which admit doubly coprime factorizations Vidyasaar, 1985). The fact that this parametrization is affine in a matrix of free parameters hihly simplifies the research of all optimal stabilizin controllers. Indeed, this parametrization allows us to transform this non-linear optimal problem with constraints into a free affine, and thus, convex optimal problem. The purpose of this paper is to ive new eneral necessary and sufficient conditions for the existence of left/riht/doubly coprime factorizations and internal stabilizability. In order to do that, we shall introduce the concept of lattices on vector spaces Bourbai, 1989) into the fractional representation approach to analysis and synthesis problems Vidyasaar, 1985). Usin these results, we shall exhibit the eneral parametrization of all the stabilizin controllers for a stabilizable plant which does not necessarily admit doubly coprime factorizations. In particular, if the transfer matrix admits a doubly coprime factorization, then the previous parametrization becomes the Youla- Kučera one. These results eneralize for MIMO systems the results obtained in Quadrat, 2003). 2. FRACTIONAL RERESENTATION AROACH Let us recall the fractional representation approach to synthesis problems Vidyasaar, 1985). Let us consider a commutative interal domain A of proper) stable SISO plants. Example 1. For instance, we have the followin examples of interal domains of stable systems

2 RH n/d n, d R[s], de n de d ds ) 0 Re s < 0}, H C + ) f HC + ) A ft) + f sup s C + fs) < + }, a i δt t i ) f L 1 R + ), i0 a i ) i 0 l 1 Z + ), 0 t 0 t 1...}, where C + s C Re s > 0}, HC + ) is the rin of holomorphic functions in C + Quadrat, 2003; Vidyasaar, 1985). A transfer function p belons to RH resp. H C + ), Â Lf) f A}, where Lf) denotes the Laplace transform) iff p is the transfer function of an exponentially stable resp. L 2 R + )- stable, L R + )-stable) linear time-invariant finitedimensional resp. infinite-dimensional) system. Let us define the quotient field of A, namely: K QA) n/d 0 d, n A}. K QA) corresponds to the class of A-stable and A-unstable SISO plants. For instance, we have p e s /s 1) / H C + ) because p has an unstable pole in C +. But, p QH C + )) because we have p n/d, where: n e s /s + 1), d s 1)/s + 1) H C + ). More enerally, we can consider the class of MIMO plants defined by transfer matrices with entries in K QA). If we have K q ), then we can always write as D 1 N Ñ D 1, where: R D : N) A q p, R Ñ T : DT ) T A p ). Definition 1. Vidyasaar, 1985) Let A be an interal domain of stable SISO plants and K QA) its quotient field. A transfer matrix K q ) is internally stabilizable if there exists a stabilizin controller C K ) q of, namely a controller C K ) q such that all the entries of the followin matrix belon to A: 1 Iq H, C) C ) I q C) 1 I q C) 1 1) C I q C) 1 + C I q C) 1 ) I q + C ) 1 C C ) 1 C ) 1 C C ) 1 A transfer matrix K q ) admits a left-coprime factorization if there exist 2) R D : N) A q p, S X T : Y T ) T A p q, such that det D 0, D 1 N and: R S D X N Y I q. A transfer matrix K q ) admits a riht-coprime factorization if there exist R Ñ T : DT ) T A p ), S Ỹ : X) A ) p, such that det D 0, Ñ D 1 and: S R Ỹ Ñ + X D. A transfer matrix K q ) admits a doubly coprime factorization iff admits a left and riht-coprime factorization. 3. CORIME FACTORIZATIONS Definition 2. Bourbai, 1989) Let V be a finitedimensional K QA)-vector space. An A- submodule M of V is called a lattice on V if there exist free A-submodules L 1 and L 2 of V such that: L1 M L 2, r A L 1 ) dim K V ). roposition 1. Bourbai, 1989) An A-submodule M of V is a lattice on V iff the K-vector space K M m K, m M} V and M is contained in a finitely enerated A- submodule of V. Example 2. If K q ), then the A-module I q : ) A p is a lattice on the K-vector space K q. Similarly, the A-module A 1 p is a lattice on the K-vector space K 1 ). Definition 3. Bourbai, 1989) Let V and W be two finite-dimensional K-vector spaces and M resp. N) a lattice on V resp. W ). Then, we denote by N : M the A-submodule of hom K V, W ) formed by the K-linear maps f : V W which satisfy fm) N. roposition 2. Bourbai, 1989) Let V and W be two finite-dimensional K-vector spaces and M resp. N) a lattice on V resp. W ). Then, we have: 1) N : M is a lattice on hom K V, W ) f : V W f is a K linear map}, 2) the canonical map N : M hom A M, N), which maps f N : M into f M, is bijective.

3 Example 3. If K q ), then we have: A : I q : ) A p f hom K K q, K) fi q : ) A p ) A} λ K 1 q λ I q : ) A p A} λ K 1 q λ A 1 q, λ A 1 ) } λ A 1 q λ A 1 ) }. Theorem 1. 1) K q ) admits a leftcoprime factorization iff there exists a square matrix D A q q such that det D 0 and I q : ) A p D 1 A q, 3) i.e. I q : ) A p is a free A-module of ran q. Then, D 1 N N D A q ) ) is a left-coprime factorization of. 2) K q ) admits a riht-coprime factorization iff there exists a square matrix D A ) ) such that det D 0 and A 1 p A 1 ) I D 1, 4) is a free A-module of ran p q. Then, Ñ D 1 Ñ D A q ) ) is a riht-coprime factorization of. roof. 1. If admits a left-coprime factorization D 1 N, with D X N Y I q, then we have I q : D) A p D 1 D : N)) A p D 1 D : N) A p and D : N) A p A q, because D : N) A p A q and µ A q, we have µ D : N) λ D : N) A p, where: X λ µ A p. Y Therefore, we have I q : D) A p D 1 A q, and thus, I q : D) A p A q is a free A-module of ran q because we have D 1 A q A q. 1. Let us denote by e i } 1 i p resp. f j } 1 j q ) the canonical basis of A p resp. A q ), namely e i resp. f i ) is the vector defined by 1 in the ith position and 0 elsewhere. Let us also denote 1 :... : ), i K q, and D 1 D1 1 :... : Dq 1 ), where D 1 i A q. Now, if there exists D A q q such that det D 0 and I q : ) A p D 1 A q, then: f i I q : ) e i I q : ) A p D 1 A q λ i A q : f i D 1 λ i, 1 i q, j I q : ) e q+j I q : ) A p D 1 A q µ i A q : j D 1 µ i, 1 j p q, D 1 D 1 f D 1 A q I q : ) A p ν A p : D 1 I q : ) ν, 1 q, D λ 1 :... : λ q ) A q q : I q D 1 D D D 1, N µ 1 :... : µ ) A q ) : D 1 N, S ν 1 :... : ν q ) X T : Y T ) T A p q : I q : ) S D 1 D X N Y I q, which shows that admits a left-coprime factorization D 1 N, D X N Y I q. 2 can be proved similarly. 4. INTERNAL STABILIZABILITY Theorem 2. A plant, defined by a transfer matrix K q ), is internally stabilizable iff one of the followin equivalent assertions is satisfied: 1) There exists S U T : V T ) T A p q which satisfies det U 0) and: U a) S A V p ), b) I q : ) S U V I q. Then, the controller C V U 1 internally stabilizes the plant. 2) There exists T X : Y ) A ) p which satisfies det Y 0 and: a) T X : Y ) A q p, b) T X + Y I. Then, the controller C Y 1 X internally stabilizes the plant. 3) There exist S U T : V T ) T K p q and T X : Y ) K ) p which satisfy det U 0, det Y 0 and: Iq U X Y V U Iq I V X Y p, 5) U I V q : ) A p p, 6) X : Y ) A p p. 7) Then, the controller C V U 1 Y 1 X internally stabilizes the plant. roof. 1 Let us suppose that K q ) is internally stabilizable, i.e. there exists a controller C K ) q such that we have: A 1 I q C) 1 A q q, A 2 I q C) 1 A q ), A 3 C I q C) 1 A ) q, A 4 + C I q C) 1 A ) ). 8)

4 From 8), we obtain C A 3 A 1 1. If we define S A T 1 : A T 3 ) T A p q, then we have: A1 A S I q : ) 1 A 3 A 3 A1 A 2 A p p, A 3 A 4 I q : ) S A 1 A 3 I q C) 1 C I q C) 1 I q. 1 Let us suppose that there exists a matrix S U T : V T ) T A p q satisfyin det U 0, and 1.a and 1.b. If we define C V U 1 K ) q, then, usin point 1.b, we obtain: I q C U V ) U 1 U 1 I q C) 1 U A q q. Hence, usin point 1.a and 1), we obtain H, C) Iq C) 1 I q C) 1 ) C I q C) 1 + C I q C) 1 U U A p p, V + V i.e. C V U 1 internally stabilizes the plant. 2 can be proved similarly. 3 Let us suppose that K q ) is internally stabilized by C K ) q. Followin the proofs of 1 and 2, we obtain that S A T 1 : A T 3 ) T resp. T B 3 : B 4 ), where B 3 C ) 1 C and B 4 C ) 1 ) satisfies 6) resp. 7)) and Iq A1 B 3 B 4 A 3 because we have: ) I q 0 I B 3 A 1 + B 4 A 3 I p, B 3 A 1 C ) 1 C) I q C) 1 C ) 1 C I q C) 1 ) B 4 A 3. Moreover, from 9), we obtain 1 Iq B 3 B 4 which proves 5). A1 A 3 A1 Iq I A 3 B 3 B p, 4 3 Let us suppose that there exist S U T : V T ) T K p q, T X : Y ) K ) p, 9) which satisfy 5), 6) and 7). Hence, S resp. T ) satisfies 1.a and 1.b resp. 2.a and 2.b), and thus, by point 1 resp. point 2), C 1 V U 1 resp. C 2 Y 1 X) is a stabilizin controller of. From 5), we have X U Y V, and thus, C 1 V U 1 Y 1 X C 2 is a stabilizin controller of. Definition 4. 1) 0 M f M M 0 is a short exact sequence of A-modules if the A-linear maps f and satisfy that f is injective, is surjective and er im f. 2) Bourbai, 1989) A short exact sequence is a split exact sequence if one of the followin equivalent assertions is satisfied: there exists h : M M, A-linear map, which satisfies h id M, there exists : M M, A-linear map, which satisfies f id M, there exist two A-linear maps φ : M M M, ψ h : f) : M M M which satisfy: φ ψ h : f) id M id M, ψ φ h : f) id M. In this case, we say that M is isomorphic to M M, denoted by M M M. We shall denote a split exact sequence by: 0 M M f M 0. h 10) 3) Bourbai, 1989) An A-module M is projective if there exist an A-module and r Z + such that we have M A r, i.e. M is a summand of a finite free A-module namely, a finite product of A). Corollary 1. A plant K q ) is internally stabilizable iff one of the followin equivalent assertions is satisfied: 1) The A-module I q : ) A p ) is projective. 2) The A-module A 1 p is projective. roof. 1. Let us define the followin A-linear map: : A p I q : ) A p, λ I q : ) λ. The A-linear map is surjective and: er λ λ T 1 : λ T 2 ) T A p λ 1 λ 2 } λ 2 ) T : λ T 2 ) T A p λ 2 A : λ 2 A q } λ I 2 A λ 2 A q } ) A : A 1 p.

5 Therefore, we have the followin exact sequence 0 I q : ) A p A p ) f A : A 1 p 0, 11) where f is defined by fλ) T : I) T T λ. Now, from 1 of Theorem 2, we now that is internally stabilizable iff there exists a matrix S U T : V T ) T A p q such that S Iq : ) A p p, I q : ) S I q, i.e. iff there exists an A-linear map h : I q : ) A p A p, µ S µ, 12) satisfyin h id Iq: ) Ap, i.e. iff 11) is a split exact sequence see 2 of Definition 4). However, if the exact sequence 11) splits, then we have A p Iq : ) A p A : A 1 p ))), which shows that I q : ) A p is a projective A-module. Conversely, if I q : ) A p is a projective A-module, then 11) is a split exact sequence because 11) ends by a projective A- module Bourbai, 1989), and thus, is internally stabilizable iff I q : ) A p is projective. 2 can be proved similarly. Corollary 2. 1) If K q ) admits a leftcoprime factorization D 1 N, where D X N Y I q, det X 0, and X T : Y T ) T A p q, then the matrix S X D) T : Y D) T ) T A p q satisfies 1.a and 1.b of Theorem 2 and C Y X 1 is a stabilizin controller of. 2) If K q ) admits a riht-coprime factorization Ñ D 1, where Ỹ Ñ + X D, det X 0, and Ỹ : X) A ) p, then the matrix T D Ỹ : D X) A ) p satisfies 2.a and 2.b of Theorem 2 and C X 1 Ỹ is a stabilizin controller of. 5. A GENERALIZATION OF THE YOULA-KUČERA ARAMETRIZATION Lemma 1. Let us consider the split exact sequence 10). Then, we have: 1) All the A-linear maps h : M M satisfyin h id M are of the form h h + f l, where l is any element of hom A M, M ), namely any A-linear map from M to M. 2) All the A-linear maps : M M satisfyin f id M are of the form + l, where l is any element of hom A M, M ), namely any A-linear map from M to M. 3) For every l hom A M, M ), we have: h + f l : f) id l M id M, h + f l : f) id l M. Theorem 3. If K q ) is internally stabilizable and S U T : V T ) T A p q resp. T X : Y ) A ) p ) is a matrix satisfyin det U 0 resp. det Y 0) and 1.a and 1.b resp. 2.a and 2.b) of Theorem 2. Then, all stabilizin controllers of are iven by CQ) V + Q) U + Q) 1 Y + Q ) 1 X + Q), where Q is every matrix which belons to 14) )) Ω A : A 1 p : I I q : ) A p ) 15) L A ) q L A ) ), L A q q, L A q ) }, 16) detu + Q) 0, and satisfies dety + Q ) 0. roof. Usin the fact that is internally stabilizable, then, by 3 of Theorem 2, there exist S U T : V T ) T A p q, T X : Y ) A ) p, which satisfy 5), 6) and 7). Then, the A-linear map h : I q : ) A p A p, defined by 12), satisfies f h id Iq: ) Ap, and thus, 11) becomes the split exact sequence defined )) by 13), where : A p A : A 1 p is defined by λ) T λ, λ A p. By Lemma 1, we obtain h + f l : f) l id Iq: ) A p id A:A 1 p T : I T ), )T h + f l : f) id l A p, where l belons to see 2 of roposition 2): )) hom A I q : ) A p, A : A 1 p )) A : A 1 p : I I q : ) A p ). Therefore, every riht inverse of has the form h + f l, whereas every left-inverse of f has the form l, where l Ω. Hence, we have

6 0 I q : ) A p A p h f A : ) A 1 p 0, 13) I q : ) A p h+f l A p, U + Q ν ν, V + Q ) A p l A : A 1 p, µ X + Q) : Y + Q ) µ, for every Q Ω, and thus, by 3 of Theorem 2, we obtain that every controller of has the form 14), where Q Ω is such that detu + Q) 0 and dety + Q ) 0. Usin the fact that I q : ) A p is a lattice on K q and A : A 1 p T : I T ) T )) is a lattice on K, we obtain that: Ω L K ) q L I q : ) A p λ A λ A q }} L K ) q L A q, L A λ A λ A q }} L K ) q L A q A, L A A, L A q A q, L A A q } L A ) q L A ) ), L A q q, L A q ) }. Corollary 3. If K q ) admits a doubly coprime factorization D 1 N Ñ D 1, D N X Ñ Ỹ X Y D I p, 17) then all stabilizin controllers of are of the form CΛ) Y + D Λ) X + Ñ Λ) 1 X + Λ N) 1 Ỹ + Λ D), where Λ A ) q is every matrix such that detx + Ñ Λ) 0 and det X + Λ N) 0. roof. If admits a doubly coprime factorization D 1 N Ñ D 1, then, by 1 and 2 of Theorem 1, we have I q : ) A p D 1 A q, A 1 p A 1 ) I D 1, ) and thus, A : A 1 p I D A and: Ω D A : D 1 A q T K ) q T D 1 A p D A }. Let us denote D 1 D1 1 :... : Dq 1 ), where D 1 i A q. If T Ω, then T D 1 i D A, i.e. there exists λ i A such that T D 1 i D λ i, 1 i q. Now, if we denote Λ λ 1 :... : λ q ) A ) q, then we have T D 1 D Λ T D Λ D. Conversely, if T D Λ D, with Λ A ) q, then T D 1 D Λ, and thus, we have: T D 1 A q D Λ A q D A T Ω, Ω D Λ D Λ A ) q } D A ) q D. By Corollary 2, S X D) T : Y D) T ) T satisfies 1.a and 1.b of Theorem 2, and thus, by 1 of Theorem 2, C Y D) X D) 1 Y X 1 is a stabilizin controller of. Moreover, by Corollary 2, T D Ỹ : D X) satisfies 2.a and 2.b of Theorem 2, and thus, by 2 of Theorem 2, C D X) 1 D Ỹ ) X 1 Ỹ is a stabilizin controller of. Usin 17), we obtain that Ỹ X + X Y 0, and thus, we have C C. By Theorem 3, we obtain that all the stabilizin controllers of are defined by CΛ) Y D + D Λ D) X D + D Λ D) 1 Y + D Λ) D D 1 X + Ñ Λ) 1 Y + D Λ) X + Ñ Λ) 1, CΛ) D X + D Λ D ) 1 D Ỹ + D Λ D) X + Λ N) 1 D 1 D Ỹ + Λ D) X + Λ N) 1 Ỹ + Λ D), where Λ A ) q is every matrix which satisfies detx + Ñ Λ) 0 and det X + Λ N) 0. REFERENCES Bourbai, N. 1989). Commutative Alebra Chap Spriner Verla. Mori, K. 2002). arameterization of stabilizin controllers over commutative rins with application to multidimensional systems. IEEE Trans. Circ. Sys. 49, Quadrat, A. 2003). On a eneralization of the Youla-Kučera parametrization. art I: The fractional ideal approach to SISO systems. To appear in Systems and Control Letters. Sule, V. R. 1994). Feedbac stabilization over commutative rins: the matrix case. SIAM J. Control Optim. 32, Vidyasaar, M. 1985). Control System Synthesis: A Factorization Approach. MIT ress.