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Solution to Sylvester equation associated to linear descriptor systems
Systems & Control Letters 55 (2006) 835 – 838 www.elsevier.com/locate/sysconle
Solution to Sylvester equation associated to linear descriptor systems M. Darouach CRAN-CNRS (UMR 7039), Université Henri Poincaré, Nancy I IUT de Longwy, 186 Rue de Lorraine, 54400 Cosnes et Romain, France Received 30 December 2004; received in revised form 20 February 2006; accepted 1 April 2006 Available online 30 June 2006
Abstract This paper presents the solution of the constrained Sylvester equation associated to linear descriptor systems. This problem has been recently studied in Castelan and da Silva [On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117], 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 Castelan and da Silva [On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117] are necessary and sufficient. © 2006 Elsevier B.V. All rights reserved. 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. [2,4–6]). Sylvester equations associated with descriptor systems (or generalized Sylvester equations) have received wide attention in the literature, see [2,4–6]. Recently, sufficient conditions for the existence of the solution to these equations under a rank constraint have been given in [2]. The present paper considers the problem studied in [2]. A new approach to solve these equations is developed, necessary and sufficient conditions are presented. Consider the linear time-invariant multivariable system described by E x˙ = Ax + Bu,
(1a)
y = Cx,
(1b)
E-mail address: [email protected]. 0167-6911/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2006.04.004
where x ∈ Rn is the state vector, y ∈ Rp the output vector, and u ∈ Rm the input vector. The matrices E ∈ Rn×n , A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rp×n are known constant matrices, with rank(E) = q < n, rank(B) = m, and rank(C) = p < q. Consider Problem 1 studied in [2], which can be formulated as follows. Let D be a region in the open left half complex plane, D ⊆ C− , symmetric with respect to the real axis. The problem is to find matrices T ∈ R(q−p)×n , Z ∈ R(q−p)×n , and HT ∈ R(q−p)×(q−p) , such that T A − HT T E = −ZC, with (HT ) ⊂ D, under the rank constraint TE rank LA = n, C
(2)
(3)
where (HT ) is the spectrum of HT and L ∈ R(n−q)×n is any full row rank matrix satisfying 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 [6,8], and the state observer design. The observer design problem can be formulated as follows [2].
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M. Darouach / Systems & Control Letters 55 (2006) 835 – 838
For the descriptor system (1), consider a reduced-order observer of the form z˙ (t) = HT z(t) + T Bu(t) − Zy(t),
(4a)
x(t) ˆ = Sz(t) + N¯ y¯ + Ny(t),
(4b)
where z ∈ R(q−p) is the state of the observer and y¯ ∈ R(n−q) is a fictitious output. As shown in [2], if Problem 1 is solved for some matrices T, Z, and HT and if we compute the matrices S, N¯ , and N such that TE ¯ [S N N ] LA = I , C then, for observer (4) we have: (i) limt→∞ (z(t) − T Ex(t)) = 0, (ii) for y(t) ¯ = −LBu(t), the estimated state x(t) ˆ satisfies limt→∞ (x(t) − x(t)) ˆ = 0. Remark 1. Notice that for L = 0, Problem 1 is reduced to finding matrices T ∈ R(n−p)×n , Z ∈ R(n−p)×n , and HT ∈ R(n−p)×(n−p) , such that T A − HT T E = −ZC,
(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 ∈ Rn×n is singular and rank(E) = q < n, there exist two nonsingular matrices P and Q of appropriate dimensions such that I 0 A11 A12 Ec = P EQ = q , , Ac = P AQ = 0 0 A21 A22
(9)
T1 A11 − HT T1 = JC1 , T1 A12 = JC2 ,
(10a)
(10b) with (HT ) ⊂ D, and where J = −[T2 Z], C1 = AC211 , and C2 = AC222 .
Define the following matrices C = [C1 C2 ] and T = T EQ, then Eq. (10) becomes TAc − HT T = JC, and Eq. (8) can be written as T rank = n, C
(11)
(12)
−1
(13)
Remark 2. From the above results it is easy to see that condition rank LA C = n − q + p of Theorem 1 of [2] is equivalent to C being a full row rank matrix. Before proceeding, let us give the following definition which is useful for the sequel. Definition 1. System (1) is D-strongly detectable if and only if the following conditions are satisfied:
(2) rank
LP
0 with F = In−q . Then Problem 1 reduces to finding matrices T, J, and HT , with (HT ) ⊂ D, such that (11)–(13) are satisfied.
According to this partitioning, equation LE = 0 becomes [L1 L2 ]Ec = 0, which leads to L1 = 0 and L2 ∈ R(n−q)×(n−q) is an arbitrary nonsingular matrix, since L is of full row rank. Therefore, Eq. (3) can be written as TE T P −1 Ec rank LA = rank LP −1 Ac C CQ T1 0 (7) = rank L2 A21 L2 A22 = n C1 C2
and
(8)
then using the above results we obtain
= [L1 L2 ].
= [T1 T2 ]
= n,
T P −1 Ac − HT T P −1 Ec = −ZCQ,
(1) rank
TP
since L2 is a nonsingular matrix. Now Eq. (2) can be written as
TF = 0, (6)
These conditions are those required for the observer design with order (n − p), see [4,5,8].
CQ = [C1 C2 ],
0 A22 C2
where T = T EQ = [T1 0], which can also be written as
with (HT ) ⊂ D, under the rank constraint TE rank = n. C
−1
or equivalently TE T1 rank LA = rank A21 C1 C
A−E C E LA
C
= n, ∀ ∈ C, ∈ / D,
= n.
For the standard systems, where E=I , Definition 1 becomes. Definition 2. Let A ∈ Rn×n and C ∈ Rp×n be constant matrices, then the pair (C, A) is said to be D-detectable if and only if I − A rank = n ∀ ∈ C, ∈ / D. (14) C
M. Darouach / Systems & Control Letters 55 (2006) 835 – 838
837
Before we give our main results, we establish the following lemma.
On the other hand, Eq. (13) can be written as (R − KC)F = 0, or
Lemma 1. The following statements are equivalent:
KCF = RF .
(1) System detectable, (1) is D-strongly I −Ac F (2) rank = 2n − q, ∀ ∈ C 0 rank AC222 = n − q.
The necessary and sufficient condition forthe existence of a RF = rank F = n − q, solution to (21) is that rank CF = rank CF A22 which is equivalent to C2 being of full column rank. Under this condition, the solution to (21) is given by
C, ∈ / D, and
Proof. By using the above transformations we obtain A − E P (A − E)Q rank = rank C CQ I −A11 −A12 = rank −A21 −A22 C1 C2 and
Iq E P EQ −1 rank LA = rank LP P AQ = rank L2 A21 CQ C1 C A22 , = q + rank C2
which prove the lemma.
K = RF (CF )+ + Y (I − (CF )(CF )+ ),
(15)
0 L2 A22 C2
(16)
Theorem 1. Under the assumption that matrix C is of full row rank, or rank C = 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 T I −K R that T = R − KC and rank C = rank 0 I C = n, or R rank C = n. In this case (11) becomes equivalent to (17)
or HT R + K1 C + KCAc = RAc ,
(18)
where K1 = J − HT K. From Eq. (18) it is easy to see that the knowledge of matrices R, HT , K1 , and K is necessary and sufficient to determine the matrices T and Z of the initial problem. −1 Now let R = [M1 N1 ], then postmultiplying the two C sides of Eq. (18) by the nonsingular matrix [M1 N1 ] leads to HT = RAc M1 − KCAc M1
(19)
and K1 = RAc N1 − KCAc N1 .
(20)
(23)
where = RAc M1 − RF (CF )+ CAc M1 and = (I − (CF )(CF )+ )CAc M1 . ifthe pair (, ) is From (23), (HT ) ⊂ D if and only D-detectable, or equivalently rank I− = q − p, ∀ ∈ C, ∈ / D. Now, define the following full column matrices RF (CF )+ + , I − (CF )(CF ) (CF )+ and the nonsingular matrix S3 = M01 N01 I0 , then we have
The necessary and sufficient conditions for the existence of the solution to Problem 1 are then given by the following theorem.
(22)
where A+ is any generalized inverse of matrix A, satisfying AA+ A = A, and Y is an arbitrary matrix of appropriate dimension. Then substituting this expression in (19) yields HT = − Y ,
(R − KC)Ac − HT (R − KC) = JC
(21)
R S1 = C 0
0 0 , I
I S2 = 0 0
I − Ac F C 0 I − Ac = rank S1 C
rank
F S3 0 I − RAc M1 RF = n − q + p + rank S2 −CAc M1 CF I − = 2n − 2q + p + rank = 2n − q, if and only if the pair (, ) 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 1. 2.1. Algorithm Let the conditions of Theorem 1 be satisfied, then the following method computes the solution of Problem 1. Step 1: Compute matrices P and Q such that P EQ = I0 00 , 11 A12 then compute Ac = P AQ = A A21 A22 and CQ = [C1 C2 ].
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M. Darouach / Systems & Control Letters 55 (2006) 835 – 838
Step 2: Find a matrix R such that R C is nonsingular and let −1 [M1 N1 ] = R . Define D, and compute the gain matrix Y, C using the pole placement method (see [1,7]), such that HT = − Y , and (HT ) ⊂ D, then compute K1 and K from (20) and (22). Step 3: Compute J = K1 + HT K, T = R − KC, T1 = T I0q , 0 T2 = −J In−q , then deduce Z = −J Ip and T = [T1 T2 ]P . 0 We can make the following remarks. Remark 3. Contrary to the algorithm presented in [2], 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(C) = 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. Remark 5. In the above result, we have introduced a full row rank matrix R such that R is nonsingular. The choice of this C matrix is nonunique, one can see that the results are independent ¯ of this choice. ¯ In fact, let R be another full row rank matrix R such that C is nonsingular, then there exists a regular matrix ¯ = R. By premultiplying (17) by we obtain such that R ¯ ¯ KC)= ¯ where J=J, ¯ ¯ ¯ (R− KC)Ac − H¯ T (R− JC, H¯ T =HT −1 , ¯ and (HT )=(HT ), 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 [2] and define the LMI-type region D ⊆ C− by the existence of two matrices = T ∈ R(q−p)×(q−p) and ∈ R(q−p)×(q−p) such that D = { ∈ C : ¯ T < 0}, then we propose the following fD () = + + result. Theorem 2. Let D ⊆ C− be an LMI-type region, and assume that system (1) is D-strongly detectable, then there exists a solution to Problem 1, with (HT ) ⊂ D if and only if there exist a symmetric positive definite matrix X ∈ R(q−p)×(q−p)
and a matrix ∈ R(q−p)×(q−p) such that ⊗ X + ⊗ (X) − ⊗ () + T ⊗ (X)T − T ⊗ ()T < 0,
(24)
+
where =RAc M1 −RF (CF ) CAc M1 , =(I −(CF )(CF )+ ) CAc M1 , and ⊗ is the Kronecker’s product. In this case HT = − Y , with Y = X −1 . Proof. The proof is direct, in fact from [3], (HT ) ⊂ D if and only if there exists a symmetric positive definite matrix X ∈ R(q−p)×(q−p) such that ⊗ X + ⊗ (XH T ) + T ⊗ (XH T )T < 0.
(25)
From the above result we have HT =−Y , and by substituting this value in (25), and by putting = XY we obtain the desired result. 3. Conclusion 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 [2], 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. References [1] R. Byers, S.G. Nash, Approaches to robust pole assignment, Int. J. Control. 49 (1) (1989) 97–117. [2] E.B. Castelan, V.G. da Silva, On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117. [3] M. Chilali, P. Gahinet, H∞ design with pole placement constraint: an LMI approach, IEEE Trans. Automat. Control 41 (3) (1996) 358–366. [4] M. Darouach, M. Boutayeb, Design of observers for descriptor systems, IEEE Trans. Automat. Control 40 (7) (1995) 1323–1327. [5] M. Darouach, M. Zasadzinski, M. Hayar, Reduced-order observers design for descriptor systems with unknown inputs, IEEE Trans. Automat. Control 41 (1996) 1068–1072. [6] L. Fletcher, Eigenstructure assignment by output feedback in descriptor systems, IEEE Proc. 135 (4) (1988) 302–308. [7] A.L. Tits, Y. Yang, Globally convergent algorithms for robust pole assignment by state feedback, IEEE Trans. Automat. Control 41 (1996) 1432–1452. [8] A. Varga, On stabilization methods of descriptor systems, Systems Control Lett. 24 (2) (1995) 133–138.
Solution to Sylvester equation associated to linear descriptor systems M. Darouach CRAN-CNRS (UMR 7039), Université Henri Poincaré, Nancy I IUT de Longwy, 186 Rue de Lorraine, 54400 Cosnes et Romain, France Received 30 December 2004; received in revised form 20 February 2006; accepted 1 April 2006 Available online 30 June 2006
Abstract This paper presents the solution of the constrained Sylvester equation associated to linear descriptor systems. This problem has been recently studied in Castelan and da Silva [On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117], 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 Castelan and da Silva [On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117] are necessary and sufficient. © 2006 Elsevier B.V. All rights reserved. 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. [2,4–6]). Sylvester equations associated with descriptor systems (or generalized Sylvester equations) have received wide attention in the literature, see [2,4–6]. Recently, sufficient conditions for the existence of the solution to these equations under a rank constraint have been given in [2]. The present paper considers the problem studied in [2]. A new approach to solve these equations is developed, necessary and sufficient conditions are presented. Consider the linear time-invariant multivariable system described by E x˙ = Ax + Bu,
(1a)
y = Cx,
(1b)
E-mail address: [email protected]. 0167-6911/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2006.04.004
where x ∈ Rn is the state vector, y ∈ Rp the output vector, and u ∈ Rm the input vector. The matrices E ∈ Rn×n , A ∈ Rn×n , B ∈ Rn×m , and C ∈ Rp×n are known constant matrices, with rank(E) = q < n, rank(B) = m, and rank(C) = p < q. Consider Problem 1 studied in [2], which can be formulated as follows. Let D be a region in the open left half complex plane, D ⊆ C− , symmetric with respect to the real axis. The problem is to find matrices T ∈ R(q−p)×n , Z ∈ R(q−p)×n , and HT ∈ R(q−p)×(q−p) , such that T A − HT T E = −ZC, with (HT ) ⊂ D, under the rank constraint TE rank LA = n, C
(2)
(3)
where (HT ) is the spectrum of HT and L ∈ R(n−q)×n is any full row rank matrix satisfying 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 [6,8], and the state observer design. The observer design problem can be formulated as follows [2].
836
M. Darouach / Systems & Control Letters 55 (2006) 835 – 838
For the descriptor system (1), consider a reduced-order observer of the form z˙ (t) = HT z(t) + T Bu(t) − Zy(t),
(4a)
x(t) ˆ = Sz(t) + N¯ y¯ + Ny(t),
(4b)
where z ∈ R(q−p) is the state of the observer and y¯ ∈ R(n−q) is a fictitious output. As shown in [2], if Problem 1 is solved for some matrices T, Z, and HT and if we compute the matrices S, N¯ , and N such that TE ¯ [S N N ] LA = I , C then, for observer (4) we have: (i) limt→∞ (z(t) − T Ex(t)) = 0, (ii) for y(t) ¯ = −LBu(t), the estimated state x(t) ˆ satisfies limt→∞ (x(t) − x(t)) ˆ = 0. Remark 1. Notice that for L = 0, Problem 1 is reduced to finding matrices T ∈ R(n−p)×n , Z ∈ R(n−p)×n , and HT ∈ R(n−p)×(n−p) , such that T A − HT T E = −ZC,
(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 ∈ Rn×n is singular and rank(E) = q < n, there exist two nonsingular matrices P and Q of appropriate dimensions such that I 0 A11 A12 Ec = P EQ = q , , Ac = P AQ = 0 0 A21 A22
(9)
T1 A11 − HT T1 = JC1 , T1 A12 = JC2 ,
(10a)
(10b) with (HT ) ⊂ D, and where J = −[T2 Z], C1 = AC211 , and C2 = AC222 .
Define the following matrices C = [C1 C2 ] and T = T EQ, then Eq. (10) becomes TAc − HT T = JC, and Eq. (8) can be written as T rank = n, C
(11)
(12)
−1
(13)
Remark 2. From the above results it is easy to see that condition rank LA C = n − q + p of Theorem 1 of [2] is equivalent to C being a full row rank matrix. Before proceeding, let us give the following definition which is useful for the sequel. Definition 1. System (1) is D-strongly detectable if and only if the following conditions are satisfied:
(2) rank
LP
0 with F = In−q . Then Problem 1 reduces to finding matrices T, J, and HT , with (HT ) ⊂ D, such that (11)–(13) are satisfied.
According to this partitioning, equation LE = 0 becomes [L1 L2 ]Ec = 0, which leads to L1 = 0 and L2 ∈ R(n−q)×(n−q) is an arbitrary nonsingular matrix, since L is of full row rank. Therefore, Eq. (3) can be written as TE T P −1 Ec rank LA = rank LP −1 Ac C CQ T1 0 (7) = rank L2 A21 L2 A22 = n C1 C2
and
(8)
then using the above results we obtain
= [L1 L2 ].
= [T1 T2 ]
= n,
T P −1 Ac − HT T P −1 Ec = −ZCQ,
(1) rank
TP
since L2 is a nonsingular matrix. Now Eq. (2) can be written as
TF = 0, (6)
These conditions are those required for the observer design with order (n − p), see [4,5,8].
CQ = [C1 C2 ],
0 A22 C2
where T = T EQ = [T1 0], which can also be written as
with (HT ) ⊂ D, under the rank constraint TE rank = n. C
−1
or equivalently TE T1 rank LA = rank A21 C1 C
A−E C E LA
C
= n, ∀ ∈ C, ∈ / D,
= n.
For the standard systems, where E=I , Definition 1 becomes. Definition 2. Let A ∈ Rn×n and C ∈ Rp×n be constant matrices, then the pair (C, A) is said to be D-detectable if and only if I − A rank = n ∀ ∈ C, ∈ / D. (14) C
M. Darouach / Systems & Control Letters 55 (2006) 835 – 838
837
Before we give our main results, we establish the following lemma.
On the other hand, Eq. (13) can be written as (R − KC)F = 0, or
Lemma 1. The following statements are equivalent:
KCF = RF .
(1) System detectable, (1) is D-strongly I −Ac F (2) rank = 2n − q, ∀ ∈ C 0 rank AC222 = n − q.
The necessary and sufficient condition forthe existence of a RF = rank F = n − q, solution to (21) is that rank CF = rank CF A22 which is equivalent to C2 being of full column rank. Under this condition, the solution to (21) is given by
C, ∈ / D, and
Proof. By using the above transformations we obtain A − E P (A − E)Q rank = rank C CQ I −A11 −A12 = rank −A21 −A22 C1 C2 and
Iq E P EQ −1 rank LA = rank LP P AQ = rank L2 A21 CQ C1 C A22 , = q + rank C2
which prove the lemma.
K = RF (CF )+ + Y (I − (CF )(CF )+ ),
(15)
0 L2 A22 C2
(16)
Theorem 1. Under the assumption that matrix C is of full row rank, or rank C = 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 T I −K R that T = R − KC and rank C = rank 0 I C = n, or R rank C = n. In this case (11) becomes equivalent to (17)
or HT R + K1 C + KCAc = RAc ,
(18)
where K1 = J − HT K. From Eq. (18) it is easy to see that the knowledge of matrices R, HT , K1 , and K is necessary and sufficient to determine the matrices T and Z of the initial problem. −1 Now let R = [M1 N1 ], then postmultiplying the two C sides of Eq. (18) by the nonsingular matrix [M1 N1 ] leads to HT = RAc M1 − KCAc M1
(19)
and K1 = RAc N1 − KCAc N1 .
(20)
(23)
where = RAc M1 − RF (CF )+ CAc M1 and = (I − (CF )(CF )+ )CAc M1 . ifthe pair (, ) is From (23), (HT ) ⊂ D if and only D-detectable, or equivalently rank I− = q − p, ∀ ∈ C, ∈ / D. Now, define the following full column matrices RF (CF )+ + , I − (CF )(CF ) (CF )+ and the nonsingular matrix S3 = M01 N01 I0 , then we have
The necessary and sufficient conditions for the existence of the solution to Problem 1 are then given by the following theorem.
(22)
where A+ is any generalized inverse of matrix A, satisfying AA+ A = A, and Y is an arbitrary matrix of appropriate dimension. Then substituting this expression in (19) yields HT = − Y ,
(R − KC)Ac − HT (R − KC) = JC
(21)
R S1 = C 0
0 0 , I
I S2 = 0 0
I − Ac F C 0 I − Ac = rank S1 C
rank
F S3 0 I − RAc M1 RF = n − q + p + rank S2 −CAc M1 CF I − = 2n − 2q + p + rank = 2n − q, if and only if the pair (, ) 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 1. 2.1. Algorithm Let the conditions of Theorem 1 be satisfied, then the following method computes the solution of Problem 1. Step 1: Compute matrices P and Q such that P EQ = I0 00 , 11 A12 then compute Ac = P AQ = A A21 A22 and CQ = [C1 C2 ].
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M. Darouach / Systems & Control Letters 55 (2006) 835 – 838
Step 2: Find a matrix R such that R C is nonsingular and let −1 [M1 N1 ] = R . Define D, and compute the gain matrix Y, C using the pole placement method (see [1,7]), such that HT = − Y , and (HT ) ⊂ D, then compute K1 and K from (20) and (22). Step 3: Compute J = K1 + HT K, T = R − KC, T1 = T I0q , 0 T2 = −J In−q , then deduce Z = −J Ip and T = [T1 T2 ]P . 0 We can make the following remarks. Remark 3. Contrary to the algorithm presented in [2], 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(C) = 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. Remark 5. In the above result, we have introduced a full row rank matrix R such that R is nonsingular. The choice of this C matrix is nonunique, one can see that the results are independent ¯ of this choice. ¯ In fact, let R be another full row rank matrix R such that C is nonsingular, then there exists a regular matrix ¯ = R. By premultiplying (17) by we obtain such that R ¯ ¯ KC)= ¯ where J=J, ¯ ¯ ¯ (R− KC)Ac − H¯ T (R− JC, H¯ T =HT −1 , ¯ and (HT )=(HT ), 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 [2] and define the LMI-type region D ⊆ C− by the existence of two matrices = T ∈ R(q−p)×(q−p) and ∈ R(q−p)×(q−p) such that D = { ∈ C : ¯ T < 0}, then we propose the following fD () = + + result. Theorem 2. Let D ⊆ C− be an LMI-type region, and assume that system (1) is D-strongly detectable, then there exists a solution to Problem 1, with (HT ) ⊂ D if and only if there exist a symmetric positive definite matrix X ∈ R(q−p)×(q−p)
and a matrix ∈ R(q−p)×(q−p) such that ⊗ X + ⊗ (X) − ⊗ () + T ⊗ (X)T − T ⊗ ()T < 0,
(24)
+
where =RAc M1 −RF (CF ) CAc M1 , =(I −(CF )(CF )+ ) CAc M1 , and ⊗ is the Kronecker’s product. In this case HT = − Y , with Y = X −1 . Proof. The proof is direct, in fact from [3], (HT ) ⊂ D if and only if there exists a symmetric positive definite matrix X ∈ R(q−p)×(q−p) such that ⊗ X + ⊗ (XH T ) + T ⊗ (XH T )T < 0.
(25)
From the above result we have HT =−Y , and by substituting this value in (25), and by putting = XY we obtain the desired result. 3. Conclusion 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 [2], 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. References [1] R. Byers, S.G. Nash, Approaches to robust pole assignment, Int. J. Control. 49 (1) (1989) 97–117. [2] E.B. Castelan, V.G. da Silva, On the solution of a Sylvester equation appearing in descriptor systems control theory, Systems Control Lett. 54 (2005) 109–117. [3] M. Chilali, P. Gahinet, H∞ design with pole placement constraint: an LMI approach, IEEE Trans. Automat. Control 41 (3) (1996) 358–366. [4] M. Darouach, M. Boutayeb, Design of observers for descriptor systems, IEEE Trans. Automat. Control 40 (7) (1995) 1323–1327. [5] M. Darouach, M. Zasadzinski, M. Hayar, Reduced-order observers design for descriptor systems with unknown inputs, IEEE Trans. Automat. Control 41 (1996) 1068–1072. [6] L. Fletcher, Eigenstructure assignment by output feedback in descriptor systems, IEEE Proc. 135 (4) (1988) 302–308. [7] A.L. Tits, Y. Yang, Globally convergent algorithms for robust pole assignment by state feedback, IEEE Trans. Automat. Control 41 (1996) 1432–1452. [8] A. Varga, On stabilization methods of descriptor systems, Systems Control Lett. 24 (2) (1995) 133–138.