Solution to Sylvester equation associated to linear descriptor systems

Transcription

1 Solution to Sylvester equation associated to linear descriptor systems Mohamed Darouach To cite this version: Mohamed Darouach. Solution to Sylvester equation associated to linear descriptor systems. Systems and ontrol Letters, Elsevier, 2006, 55 (10), pp <hal > HAL Id: hal Submitted on 10 Oct 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Solution to Sylvester equation associated to linear descriptor systems M. Darouach RAN-NRS (UMR 7039), Université Henri Poincaré, Nancy I IUT de Longwy, 186 Rue de Lorraine OSNES et ROMAIN FRANE mohamed.darouach@iut-longwy.uhp-nancy.fr Abstract This paper presents the solution of the constrained Sylvester equation associated to linear descriptor systems. This problem has been recently studied in 1, where sufficient conditions for the existence of the solution are given. In the present paper, a simple and direct method is developed to solve this problem. This method shows that the conditions given in 1 are necessary and sufficient. Keywords: Descriptor systems; Sylvester equation; Existence conditions; Observers; Regional pole placement 1 Introduction Many problems in control and systems theory are solved by computing the solution of Sylvester equations. As it is well known, these equations have important applications in stability analysis, in observers design, in output regulation with internal stability, and in the eigenvalue assignment (see, e.g 1, 2, 3, 4). Sylvester equations associated with descriptor systems (or generalized Sylvester equations) have received wide attention in the literature, see 1, 2, 3, 4. Recently, sufficient conditions for the existence of the solution to these equations under a rank constraint have been given in 1. The present paper considers the problem studied in 1. A new approach to solve these equations is developed, necessary and sufficient conditions are presented. onsider the linear time-invariant multivariable system described by Eẋ = Ax + Bu, y = x, (1a) (1b) where x IR n is the state vector, y IR p the output vector, and u IR m the input vector. The matrices E IR n n, A IR n n, B IR n m, and IR p n are known constant matrices, with rank(e) = q < n, rank(b) = m and rank() = p < q. onsider the Problem 1 studied in 1, which can be formulated as follows: Let D be a region in the open left half complex plane, D, symmetric with respect to the real axis. The problem is to find matrices T IR (q p) n, Z IR (q p) n, and H T IR (q p) (q p), such that 1

3 TA H T TE = Z, (2) with σ(h T ) D, under the rank constraint TE rank LA = n, (3) where σ(h T is the spectrum of H T and L IR (n q) n is any full row rank matrix satsfying LE = 0. The Sylvester equation (2) has a close relation with many problems in linear control theory of descriptor systems, such as the eigenstructure assignment 4, 5, and the state observer design. The observer design problem can be formulated as follows 1: For the descriptor system (1), consider a reduced-order observer of the form ż(t) = H T z(t) + TBu(t) Zy(t), ˆx(t) = Sz(t) + Nȳ + Ny(t), (4a) (4b) where z IR (q p) is the state of the observer and ȳ IR (n q) is a fictitious output. As shown in 1, if Problem 1 is solved for some matrices T, Z and H T and if we compute the matrices S, N and N such that S N N TE LA = I, then, for observer (4) we have: i) lim t (z(t) TEx(t)) = 0, ii) for ȳ(t) = LBu(t), the estimated state ˆx(t) satisfies lim t (x(t) ˆx(t)) = 0. Remark 1 Notice that for L = 0, Problem 1 is reduced to finding matrices T IR (n p) n, Z IR (n p) n, and H T IR (n p) (n p), such that with σ(h T ) D, under the rank constraint TA H T TE = Z, (5) rank TE = n. (6) These conditions are those required for the observer design with order (n p), see 5. 2 Main results In this section we will present a new and simple solution to Problem 1. Necessary and sufficient conditions to solve this problem are also given. Since matrix E IR n n is singular and rank(e) = q < n, there exist two nonsingular matrices P and Q of appropriate dimensions such that Iq 0 A11 A E c = PEQ =, A 0 0 c = PAQ = 12, Q = A 21 A 1 2, TP 1 = T 1 T 2, 22 and LP 1 = L 1 L 2. According to this partitioning, equation LE = 0 becomes L 1 L 2 Ec = 0, which leads to L 1 = 0 and L 2 IR (n q) (n q) is an arbitrary nonsingular matrix, since L is of full row rank. 2

4 Therefore, equation (3) can be written as TE TP 1 E c T 1 0 rank LA = rank LP 1 A c = rank L 2 A 21 L 2 A 22 = n, (7) Q 1 2 or equivalently since L 2 is a nonsingular matrix. Now equation (2) can be written as then using the above results we obtain TE T 1 0 rank LA = rank A 21 A 22 = n, (8) 1 2 TP 1 A c H T TP 1 E c = ZQ, (9) T 1 A 11 H T T 1 = J 1, T 1 A 12 = J 2, with σ(h T ) D, and where J = T 2 Z, 1 = Define the following matrices = 1 A21 1, and 2 = A and T = TEQ, then equation (10) becomes. (10a) (10b) TA c H T T = J, (11) and equation (8) can be written as T rank = n, (12) where T = TEQ = T 1 0, which can also be written as 0 with F = I n q. TF = 0, (13) Then Problem 1 reduces to finding matrices T, J, and H T, with σ(h T ) D, such that (11) - (13) are satisfied. Remark 2 From the above results it is easy to see that condition rank theorem 1 of 1 is equivalent to being a full row rank matrix. LA Before proceeding, let us give the following definition which is useful for the sequel. = n q + p of Definition 1. System (1) is D-strongly detectable if and only if the following conditions are satisfied A λe 1) rank = n, λ, λ / D, E 2) rank LA = n. 3

5 For the standard systems, where E = I, Definition 1 becomes. Definition 2. Let A IR n n and IR p n be constant matrices, then the pair (, A) is said to be D-detectable if and only if λi A rank = n, λ, λ / D. (14) Before we give our main results, we establish the following lemma. Lemma 1. The following statements are equivalent 1) System (1) is D-strongly detectable, λi Ac F 2) rank = 2n q, λ, λ / D, and rank 0 A22 2 = n q. Proof. By using the above transformations we obtain: λi A A λe P(A λe)q 11 A 12 rank = rank = rank A Q 21 A 22, (15) 1 2 and E PEQ I q 0 rank LA = rank LP 1 PAQ = rank L 2 A 21 L 2 A 22 A22 = q + rank, (16) Q which prove the lemma. The necessary and sufficient conditions for the existence of the solution to Problem 1 are then given by the following theorem. Theorem 1. Under the assumption that matrix is of full row rank, or rank = n q + p, the necessary and sufficient condition for the existence of the solution to Problem 1 is that system (1) must be D-strongly detectable. Proof. Under condition (12), there exists a matrix R, such that T = R K and rank I K R R rank = n, or rank = n. In this case (11) becomes equivalent to 0 I or T = (R K)A c H T (R K) = J, (17) H T R + K 1 + KA c = RA c, (18) where K 1 = J H T K. From equation (18) it is easy to see that the knowledge of matrices R, H T, K 1, and K is necessary and sufficient to determine the matrices T and Z of the initial problem. 1 R Now let = M 1 N 1, then post multiplying the two sides of equation (18) by the nonsingular matrix M 1 N 1 leads to and H T = RA c M 1 KA c M 1, (19) K 1 = RA c N 1 KA c N 1. (20) 4

6 On the other hand, equation (13) can be written as (R K)F = 0, or KF = RF. (21) The necessary and sufficient condition for the existence of a solution to (21) is that rankf = RF A22 rank = rankf = n q, which is equivalent to being of full column rank. Under this F 2 condition, the solution to (21) is given by K = RF(F) + + Y (I (F)(F) + ), (22) where A + is any generalised inverse of matrix A, satisfying AA + A = A, and Y is an arbitrary matrix of appropriate dimension. Then substituting this expression in (19) yields H T = Λ Y Ω, (23) where Λ = RA c M 1 RF(F) + A c M 1 and Ω = (I (F)(F) + )A c M 1. From (23), σ(h T ) D if and only if the pair (Ω, Λ) is D-detectable, or equivalently rank q p, λ, λ / D. R 0 Now, define the following full column matrices S 1 = 0 0 I M1 N and the nonsingular matrix S 3 = 1 0, then we have 0 0 I rank λi Ac F 0 λi Λ = Ω I RF(F) +, S 2 = 0 I (F)(F) +, 0 F) + λi Ac F = ranks 1 S 0 3 λi RAc M = n q + p + ranks 1 RF 2 A c M 1 F λi Λ = 2n 2q + p + rank = 2n q, Ω if and only if the pair (Ω, Λ) is D-detectable, and using lemma 1, we obtain the result of the theorem. From the above results we can give the algorithm to solve Problem Algorithm Let the conditions of theorem 1 be satisfied, then the following method computes the solution of Problem 1. I 0 Step1: ompute matrices P and Q such that PEQ =, then compute A 0 0 c = PAQ = A11 A 12 and Q = A 21 A R Step2: Find a matrix R such that is nonsingular and let 1 R M 1 N 1 =. Define D, and compute the gain matrix Y, using the pole placement method (see 7 and 8), such that H T = Λ Y Ω, and σ(h T ) D, then compute K 1 and K from (20) and (22). 5

7 Step3: ompute J = K 1 + H T K, T = R K, T 1 = T 0 Z = J and T = T 1 T 2 P. I p We can make the following remarks. Iq, T 0 2 = J In q, then deduce 0 Remark 3 ontrary to the algorithm presented in 1, where many transformations are used in addition to the resolution of a Sylvester equation, in our paper the problem is reduced to a pole placement one which can be solved by the existing robust methods. Remark 4 When rank() = n q + p d, for some integer 0 < d n q, we can apply the above approach to this case, if and only if system (1) is D-detectable. R Remark 5 In the above result, we have introduced a full row rank matrix R such that is nonsingular. The choice of this matrix is non unique, one can see that the results are independent R of this choice. In fact, let R be another full row rank matrix such that is nonsingular, then there exists a regular matrix θ such that θ R = R. By premultiplying (17) by θ we obtain ( R K)A c H T ( R K) = J, where J = θj, HT = θh T θ 1, and σ( H T ) = σ(h T ), which shows that the solution given above is independent of the choice of matrix R. 2.2 Regional pole placement in LMI regions This section discusses the solution of Problem 1 by regional pole placement in LMI regions of the complex plane. We shall use the same notations as in 1 and define the LMI type region D by the existence of two matrices = T IR (q p) (q p) and Γ IR (q p) (q p) such that D = {λ : f D (λ) = + λγ + λγ T < 0}, then we propose the following result. Theorem 2. Let D be an LMI type region, and assume that system (1) is D-strongly detectable, then there exists a solution to Problem 1, with σ(h T ) D if and only if there exist a symmetric positive definite matrix X IR (q p) (q p) and a matrix Ψ IR (q p) (q p) such that X + Γ (XΛ) Γ (ΨΩ) + Γ T (XΛ) T Γ T (ΨΩ) T < 0, (24) where Λ = RA c M 1 RF(F) + A c M 1, Ω = (I (F)(F) + )A c M 1, and is the Kronecker s product. In this case H T = Λ Y Ω, with Y = X 1 Ψ. Proof. The proof is direct, in fact from 6, σ(h T ) D if and only if there exists a symmetric positive definite matrix X IR (q p) (q p) such that X + Γ (XH T ) + Γ T (XH T ) T < 0. (25) From the above result we have H T = Λ Y Ω, and by substituting this value in (25), and by putting Ψ = XY we obtain the desired result. 3 onclusion In this paper, we have presented a simple method to solve the generalized Sylvester equation associated to descriptor systems observers and control design. This problem was considered in 1, where only sufficient conditions were given. We have also given the necessary and sufficient conditions for the existence of the solution. Solution by LMI regional pole placement was also presented. 6

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