D-admissibility for Rectangular Descriptor Systems Chia-Po Wei and Li Lee Abstract— The problem of D-admissibility for rectangular descriptor systems, which is a natural generalization of the D-admissibility problem for square descriptor systems, is considered in this paper. Necessary and sufficient LMI-based conditions are derived to characterize the D-admissibility for the more general descriptor system setups.

I. INTRODUCTION It is well-known that, comparing to the state-space model, the descriptor model [1] is a more natural and powerful mathematical representation for many practical systems. However, in addition to the stability issue commonly addressed for state-space systems, the properties of regularity and impulse immunity (or causality) are two important special issues in studying the descriptor systems [2], [3]. A descriptor system, usually described by the (E, A) pair with matrices E and A of the same size, is called admissible if it possesses the all three properties of regularity, impulse immunity, and stability [4], [5]. Since stability of (E, A) is determined by the location of all finite eigenvalues of the pair, a more general admissibility concept of (E, A) can be investigated by choosing different regions for those finite eigenvalues to lie inside. When the standard stability region, i.e. the open left half plane for continuous systems or the interior of unit disk for discrete systems, is generalized to a specified subset D of the complex plane, the associated problem is named the D-admissibility problem. For square descriptor systems, i.e. both E and A are square matrices, the admissibility problem is studied in [4],[6],[7] for continuous case and in [5],[8] for discrete case, respectively, while the generalized D-admissibility problem is addressed in [9]. When matrices E and A are allowed to be nonsquare, the LMI-based admissibility verification result of [4] is generalized in [10], while the generalized Lyapunov equation characterization for admissibility is extended from [6] to [11]. The result of characterizing admissibility via the Kronecker form in [10] is further extended to solve the problem of stabilization via interconnection for rectangular descriptor systems [12]. Other research work about rectangular descriptor systems may refer to [13] and [14]. The purpose of the paper is to extend the D-admissibility This work was supported by National Science Council of Taiwan, R.O.C. under grant no. NSC 95-2221-E-110-098. The authors are with the Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung 804, TAIWAN

[email protected]

problem in the direction of allowing matrices E and A to be rectangular. In view of the nonsquare E and A, the direct extension of the definition for D-admissibility from the square descriptor systems to the rectangular cases is not applicable. To cope with this generalization, Iwasaki and Shibata [15] find that the admissibility problem of rectangular continuous-time descriptor systems is equivalent to checking the causality and whether all unobservable hidden modes of (F, H) are in the open left half plane, where matrices F and H are constructed from system matrices of the descriptor system. But in [15] there has no numerically verifiable necessary and sufficient condition been derived. By knowing that if a problem can be characterized in terms of a set of LMI conditions, then the problem can be solved efficiently by numerical algorithms created from the interior point methods [16], this paper aims to derive necessary and sufficient LMI conditions to characterize the D-admissibility problem for rectangular descriptor systems. The following notations are used in the sequel. For a matrix D, its Hermitian is denoted as D ∗ and, when it is full-column rank, D † is used to denote any left inverse of it. For matrices M and N having the same number of columns, [M ; N ] is used to mean [M ∗ N ∗ ]∗ , and for a square matrix A, herm (A) stands for the Hermitian matrix A+A∗ . Finally, the symbol ⊗ denotes the Kronecker product between two matrices. II. PRELIMINARIES Consider the continuous-time descriptor system described by ˙ = A ζ(t) E ζ(t) (1) where both E and A are (m + p) × (n + p) matrices with rank(E) = p and ζ is the descriptor variable. By the singular value decomposition, there always exist nonsingular matrices M and N such that   Ip 0 . M EN = 0 0 Let 

A M AN = C

 B , D

N

−1



x(t) ζ(t) = ξ(t)



where A ∈ Cp×p , B ∈ Cp×n , C ∈ Cm×p , D ∈ Cm×n , x and ξ are the descriptor variables in the changed coordinate

and, for any t ≥ 0, x(t) ∈ Cp and ξ(t) ∈ Cn . Then system (1) can be written as ˙ = M AN N −1 ζ(t) M EN N −1 ζ(t) or x(t) ˙ = Ax(t) + Bξ(t) 0 = Cx(t) + Dξ(t).

(2)

Note that if A is not square, so is D. In contrast to state-space systems, a descriptor system as in (2) may not have solution for an arbitrary initial condition [x(0− ); ξ(0− )], or may have nonunique solution when it does exist, and moreover the solution may contain impulsive terms. Neither nonunique solution nor impulsive response is desired in engineering practice. We call a descriptor system causal if the system does not have these two features. Definition 1 (Definition 1 of [15]): The descriptor system (2) is said to be causal if the following conditions hold: a) There is no impulsive solution, i.e., the system is impulse free. b) For each x(0− ), the solution, if any, is unique.

Lemma 2: Consider the descriptor system (2) and let F and H be as defined in (3). The following statements are equivalent: i) The descriptor system (2) is admissible. ii) D has full-column rank and for each x(0− ), the solution of x(t) ˙ = Fx(t),

0 = Hx(t)

(4)

if any, approaches zero as time goes to infinity. iii) D has full-column rank and for any x(0− ) in the unobservable subspace of (F, H), x(t) = eF t x(0− ) approaches zero as time goes to infinity. iv) D has full-column rank and all unobservable hidden modes of (F, H) are in the open left half plane; that is, (F, H) is detectable. Proof: i) ⇔ ii): In view of Lemma 1, system (2) is causal if and only if D has full-column rank. Thus, under the assumption of full-column rank of D or equivalently the existence of D † , equivalence between i) and ii) is established if the following two sets {(x, ξ) : (x, ξ) satisfies (2)}

The causality of the descriptor system (2) is completely determined by the column rank of D, as shown in the following lemma. Lemma 1 (Lemma 1 of [15]): The descriptor system (2) is causal if and only if D has full-column rank. Therefore, when descriptor system (2) is square, Lemma 1 becomes that the system is causal if and only if D is nonsingular, which is a well-known result. Besides causality, stability of the system solution is another feature that needs to be guaranteed. If a descriptor system is causal and all system solutions are stable, then the descriptor system is said to be admissible, as defined below. Definition 2 (Definition 2 of [15]): The descriptor system (2) is said to be admissible if it is causal and, for each x(0− ), the solution, if any, approaches zero as time goes to infinity. It has been mentioned without proof in [15] that the admissibility of descriptor system (2) can be equivalently verified by checking the column rank of D and locations of the unobservable hidden modes of (F, H), with
F , A − BD† C, H , I − DD† C (3) where D† denotes the Moore-Penrose inverse of D. For the sake of completeness, a detailed proof of the result is provided in the next lemma. However, different from [15], the matrix D† in (3) denotes any left inverse of D when it is full-column rank.

n

(x, ξ) : x satisfies (4), ξ , −D † Cx

can be shown to be equal. By noting that
Cx+Dξ = 0 ⇔ I − DD† Cx(t) = 0,

o

ξ = −D † Cx

where above ⇒ is due to D has full-column rank, the equality holds. ii) ⇔ iii): The solution of (4) is x(t) = eF t x(0− )

with HeF t x(0− ) = 0,

∀t ≥ 0.

The right equality implies that x(0− ) is in the unobservable subspace of (F, H). iii) ⇒ iv): Let λ ∈ C be an unobservable hidden mode of (F, H). Then there exists a nonzero vector z such that Fz = λz,

Hz = 0.

Obviously, z is in the unobservable subspace of (F, H). By iii), we have
x(t) = eF t z = I + Ft + (Ft)2 /2! + · · · z
= I + λt + (λt)2 /2! + · · · z = eλt z

approaches zero as time goes to infinity. So λ is in the open left half plane. iv) ⇒ iii): It is straightforward to show that, without loss of generality, we may assume (F, H) is in the observable canonical form. That is     F1 0  (F, H) = , H1 0 F2 F3

where F1 ∈ C(p−q)×(p−q) , F3 ∈ Cq×q , H1 ∈ Cm×(p−q) , q is the dimension of the unobservable subspace of (F, H), (F1 , H1 ) is observable, and the unobservable hidden modes of (F, H) are eigenvalues of F3 . Let’s denote the unobservable subspace of (F, H) by Munob , {[0; x2 ] ∈ Cp : x2 ∈ Cq } . Then for any z ∈ Munob ,
eF t z = I + Ft + (Ft)2 /2! + · · · z     I 0 F1 0 = + t 0 I F2 F3  2    F 0 2 0 + 1 t /2! + · · · ∗ F32 x2     0 0
= = F3 t . I + F3 t + (F3 t)2 /2! + · · · x2 e x2

Since all unobservable hidden modes of (F, H) are in the open left half plane, F3 is Hurwitz stable. Hence, eF t z approaches zero as time goes to infinity. If the descriptor system (2) is square, then the pair (F, H)
defined as in (3) becomes A − BD−1 C, 0 and unobservable hidden modes of the pair are exactly the eigenvalues of A − BD−1 C. Thus the equivalence i) ⇔ iv) of Lemma 2 coincides with the well-known result that descriptor system (2) is admissible if and only if D is nonsingular and all eigenvalues of A − BD −1 C are in the open left half plane. By Lemma 1, the causality of the descriptor system (2) ensures the existence of D † . However, since D is not square, its left inverse D† is not unique. According to (3), different choices of D† will generate different pairs (F, H). It is natural to ask whether the unobservable hidden modes of (F, H) vary with the choice of D † . The next proposition shows that the answer is no. Proposition 1: Consider the descriptor system (2) with D being assumed full-column rank. Let F and H be given by (3). Then the unobservable hidden modes of (F, H) do not vary with the choice of D † . Proof: For any nonsingular matrix Q, it is straightforward to verify that the unobservable hidden modes of (F, H) are identical to those of (F, QH) and, moreover, D† Q−1 = (QD)† ; that is, the multiplication of the left inverse of D and Q−1 is the left inverse of QD. So F = A − BD† Q−1 QC = A − B(QD)† QC
QH = I − QDD† Q−1 QC
= I − QD(QD)† QC.

(5) (6)

ˆ be nonsingular Since D has full-column rank, let matrix Q † ˆ ˆ such that QD = [I; 0], then (QD) = [I Y ], where Y ˆ = [C1 ; C2 ]. Let λ ∈ C be an is any matrix. Define QC

ˆ unobservable hidden mode of (F, QH). Then there exists a nonzero vector z such that ˆ QHz = 0.

(F − λI) z = 0, ˆ From (6), QHz = 0 gives     I  I 0 − I 0 0 I

Y





 C1 z=0 C2

which implies C2 z = 0. From (5), (F − λI) z = 0 gives       C1 − λI z = (A − BC1 − λI) z = 0. A−B I Y C2 In summary, the (λ, z) pair has to satisfy C2 z = 0,

(A − BC1 − λI) z = 0.

Thus λ is an unobservable hidden modes of (A − BC1 , C2 ), which is independent of Y . Hence, the unobservable hidden modes of (F, H) do not vary with the choice of D † . III. D-ADMISSIBILITY FOR RECTANGULAR DESCRIPTOR SYSTEMS We know from Lemma 2 that the admissibility of descriptor system (2) is related to the location of unobservable hidden modes of (F, H). Also, from iii) of Lemma 2, the decay rate of x(t) is controlled by the real parts of unobservable hidden modes of (F, H). The two facts motivate us to consider the D-admissibility problem for descriptor systems described by δx = Ax + Bξ 0 = Cx + Dξ

(7)

where A ∈ Cp×p , B ∈ Cp×n , C ∈ Cm×p , D ∈ Cm×n are given matrices, δx = x(t) ˙ when the system dynamics is continuous and δx = x(k + 1) when the system dynamics is discrete, respectively. That is to determine whether all unobservable hidden modes of the associated (F, H) pair are in a specified region in the complex plane described by ) (    ∗  r s λ λ <0 (8) D, λ∈C: s¯ q 1 1 where r ∈ R, s ∈ C, q ∈ R are given. The most important two D regions are ) (    ∗  0 1 λ λ <0 open left half plane: λ ∈ C : 1 0 1 1 for the continuous-time case and ( )  ∗    λ 1 0 λ open unit disk: λ ∈ C : <0 1 0 −1 1 for the discrete-time case. In fact, D can represent any open half planes and any open disks in the complex plane by choosing r, s, and q appropriately.

In view of Lemma 2, the definition of D-admissibility for the descriptor system (7) is given below. Definition 3 (D-admissibility): The descriptor system (7) is said to be D-admissible if D has full-column rank and all unobservable hidden modes of (F, H), as defined previously in (3), are in the D-region defined in (8). The following well-known Finsler’s Lemma plays a key role in the development of the main result of this paper. Lemma 3 (Finsler’s Lemma): Let G ∈ Cn×n , G = G∗ , and F ∈ Cm×n . The following statements are equivalent: i) x∗ Gx < 0 for all x ∈ Cn \ {0}, F x = 0. ii) There exists a scalar µ ∈ R such that G − µF ∗ F < 0. iii) There exists a matrix X ∈ Cm×n such that G + X ∗ F + F ∗ X < 0. iv) There exists a Hermitian matrix Θ ∈ Cm×m such that G + F ∗ ΘF < 0. Proof: For i) ⇔ ii), see p. 26 of [18]. iii) ⇒ i) is obvious. ii) ⇒ iii) is established by letting X = −1/2µF . iv) ⇒ i) is obvious. ii) ⇒ iv) is established by letting Θ = −µI.

The following theorem gives necessary and sufficient LMI conditions for the D-admissibility of descriptor system (7). Theorem 1: Consider the descriptor system (7). Let the D-region be given by (8) and define
F , A − BD† C, H , I − DD† C. The following statements are equivalent:

a) The descriptor system (7) is D-admissible. b) D has full-column rank, and there exist a positive definite matrix P and a matrix L such that  ∗     F + LH r s F + LH ⊗P < 0. (9) I s¯ q I c) D has full-column rank, and there exists a positive definite matrix P such that  ∗     r s F ∗ F ⊗P x<0 x I s¯ q I (10) ∀x ∈ Cp \ {0},

Hx = 0.

d) D has full-column rank, and there exist a positive definite matrix P and matrices R, S such that  ∗     A B r s A B ⊗P I 0 s¯ q I 0 (11)  ∗  ∗    R C  + ∗ C D + R S < 0. D∗ S

If r ≥ 0, the requirement that D has full-column rank can be removed. e) D has full-column rank, and there exist a positive definite matrix P and a Hermitian matrix Θ such that    ∗   A B r s A B ⊗P I 0 s¯ q I 0 (12)  ∗   C + Θ C D < 0. D∗ If r ≥ 0, the requirement that D has full-column rank can be removed.

Proof: a) ⇒ b): As mentioned previously, without loss of generality, we may assume (F, H) is in the observable canonical form. That is     F1 0  , H1 0 (F, H) = F2 F3

where (F1 , H1 ) is observable and the unobservable hidden modes of (F, H) are eigenvalues of F3 . Since system (7) is D-admissible, all eigenvalues of F3 are in the D-region (8). Since (F1 , H1 ) is observable, there exists a matrix L1 such that all eigenvalues of F1 + L1 H1 are in the D-region. Therefore, the matrix L , [L1 ; 0] makes all eigenvalues of      L  F1 0 + 1 H1 0 F + LH = F2 F3 0   F 1 + L 1 H1 0 = F2 F3 are in the D-region. Since all eigenvalues of F + LH are in the D-region, Theorem 1 of [17] ensures the existence of a positive definite matrix P such that (9) holds. b) ⇒ c) is obvious. c) ⇒ a): Let λ ∈ C be an unobservable hidden mode of (F, H). Then there exists a nonzero vector z such that Fz = λz,

Hz = 0.

Replacing x in the inequality in (10) with z leads to  ∗     r s F ∗ F z ⊗P z I s¯ q I   ∗       λ r s λ = ⊗z ⊗P ⊗z 1 s¯ q 1  ∗    ! λ r s λ = ⊗ (z ∗ P z) 1 s¯ q 1 < 0. Since P is positive definite, above implies λ is in the Dregion. c) ⇔ d): Suppose D has full-column rank and there exists a positive definite matrix P such that (10) holds. Condition

iii) There exist a positive definite matrix P and matrices R, S such that   ∗   0 P A B A B P 0 I 0 I 0 (14)  ∗  ∗   C  R  + ∗ C D + R S < 0. D∗ S

(10) can be written as    ∗   † A − BD† C r s ∗ A − BD C x<0 ⊗P x I s¯ q I
∀x ∈ Cp \ {0}, I − DD† Cx = 0

or  ∗     Ax + Bξ r s Ax + Bξ ⊗P <0 x s¯ q x
∀x ∈ Cp \ {0}, I − DD† Cx = 0, ξ , −D† Cx.

iv) There exists a matrix X such that X E = E ∗ X ∗ ≥ 0,

Since D has full-column rank, above condition is equivalent to    ∗   Ax + Bξ r s Ax + Bξ <0 ⊗P x s¯ q x (13) p

∀x ∈ C \ {0},



I E, 0

Cx + Dξ = 0.

⇔

[x; ξ] = 0.

Hence, condition (13) can be written as  ∗  ∗      x A B r s A B x ⊗P <0 ξ I 0 s q I 0 ξ       x x n+p ∈C \ {0}, = 0. ∀ C D ξ ξ

In view of the equivalence between i) and iii) in Lemma 3, above condition is equivalent to the existence of a matrix [R S] such that (11) holds. If r ≥ 0, the (2,2)-block of (11) reads rB ∗ P B + S ∗ D + D∗ S < 0

⇒

S ∗ D + D∗ S < 0

which implies D has full-column rank. d) ⇔ e) is established by the equivalence between iii) and iv) in Lemma 3.

(15)

where

Since D has full-column rank, we have that for all [x; ξ] satisfying Cx + Dξ = 0 x=0

A∗ X ∗ + X A < 0

or

 0 , 0



A A, C

 B . D

Proof: i) ⇔ ii) ⇔ iii) is a direct result of Theorem 1. iii) ⇒ iv): (14) can be written as   ∗   ∗   R  A P + PA PB + herm C D <0 S∗ B∗P 0   ∗     R  P  herm A B + ∗ C D S 0    ∗ P R A B = herm <0 0 S∗ C D

which is the second inequality in (15) if we define   P R∗ X , . 0 S∗

(16)

It can be seen that X satisfies X E = E ∗ X ∗ ≥ 0. iv) ⇒ iii): Any matrix X satisfying X E = E ∗ X ∗ ≥ 0 must have structure (16) with P = P ∗ ≥ 0. Since A∗ X ∗ + X A < 0 is strict, we may perturb P so that P > 0 while the feasibility of the strict inequality is maintained. Substituting X defined as in (16) into A∗ X ∗ + X A < 0, we have (14).

The following corollary shows that the Lyapunov theorem for square continuous-time descriptor systems [4],[7] is a special case of Theorem 1.

The following corollary shows that a version of the Lyapunov theorem for square discrete-time descriptor systems [19],[20] is a special case of Theorem 1.

Corollary 1: Consider the descriptor system (7) with square D and let the D-region be ( )  ∗    λ 0 1 λ D= λ∈C: <0 1 1 0 1

Corollary 2: Consider the descriptor system (7) with square D and let the D-region be ) (    ∗  λ 1 0 λ <0 D= λ∈C: 0 −1 1 1

that is, D is the open left half plane. The following statements are equivalent:

that is, D is the open unit disk. The following statements are equivalent:

i) The descriptor system (7) is D-admissible. ii) D is nonsingular and there exists a positive definite matrix P such that
∗
A − BD−1 C P + P A − BD−1 C < 0.

i) The descriptor system (7) is D-admissible. ii) D is nonsingular and there exists a positive definite matrix P such that
∗
A − BD−1 C P A − BD−1 C − P < 0.

iii) There exist a positive definite matrix P and a Hermitian matrix Θ such that   ∗   A B P 0 A B 0 −P I 0 I 0 (17)  ∗   C Θ C D < 0. + D∗ iv) There exists a Hermitian matrix X = diag(X1 , X4 ) such that X1 > 0,

X4 < 0,

∗

∗

A XA − E XE < 0

(18)

where 

I E, 0

 0 , 0



A A, C

 B . D

Proof: i) ⇔ ii) ⇔ iii) is a direct result of Theorem 1. iii) ⇒ iv): First we show Θ in (17) is negative definite. The (2,2)-block of (17) reads ∗

∗

B P B + D ΘD < 0

∗

⇒

D ΘD < 0

IV. EXAMPLE

 3 0 0 C = 0 3 0 , 0 0 3 

and

Since D has full column rank and all unobservable hidden modes of (F, H) lie in the D-region, system (7) is Dadmissible. Solving (11), the LMI solver of Matlab 6.5 returns " # 1.24

P = 0.36

0.09

"

0.30 R = −226.09 −169.58

225.71 −0.75 −226.54

0.36 1.56 0.60

0.09 0.60 , 0.76

#

"

168.94 150.73 224.63 , S = 225.97 −0.19 18.68

Solving (12), the LMI solver returns " # " 9.45 P = −2.58 −2.90

−2.58 3.84 0.90

−2.90 2.26 0.90 , Θ = 0.09 1.02 0.15

0.09 −2.32 −0.46

#

−357.49 −225.45 . 188.19

#

0.15 −0.46 . −0.71

If we change the D-region to

then system (1) is not D-admissible, since the unobservable hidden mode −1.1 is not in the D-region. In this case, solving either (11) or (12), the LMI solver returns no solutions. (In fact, the LMI solver reports that the LMI problem is marginally infeasible, that is, the given LMI constraints may be feasible but are not strictly feasible. Since the given LMI constraints, (11) or (12), are strict, the LMI problem is determined to be infeasible.) V. CONCLUSIONS AND FUTURE WORKS





−2/3 1/3 B =  5/3 −10/3 , −1 −1  1 −2 D = −1 −1 , 4 −5 

 −2/3 −1/3 1/3 D = −4/3 0 1/3 †

{λ ∈ C : 2
{λ ∈ C : 2
which implies D is nonsingular, and hence Θ is negative definite. Let X1 = P and X4 = Θ. Then the proof is finished if we can show (17) is the same as the third inequality in (18), which is straightforward. iv) ⇒ iii) is established by letting P = X1 and Θ = X4 .

Consider system (7) with   −2.1 1 0 A =  −1 −2.1 2 , 3 5 −2.1

which can be expressed as



is a left inverse of D. (The choice of D † is not unique.) Then we have     −3 1 1 −2.1 1/3 1/3 F =  −11 −13/30 11/3 , H =  −6 2 2 . −12 4 4 −3 4 −0.1

By the Popov-Belevitch-Hautus (PBH) test, the unobservable hidden modes of (F, H) are −4.1 and −1.1. Let the Dregion be given by ( )  ∗    λ 0 1 λ λ∈C: <0 1 1 2.1 1

The D-admissibility problem of rectangular descriptor systems is analyzed in this paper. For this problem, necessary and sufficient LMI conditions are derived by the Finsler’s Lemma. The controller design for the D-admissibility problem of rectangular descriptor systems is the future research topic. R EFERENCES [1] L. Dai, Singular Control Systems. Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 1989. [2] F.L. Lewis, “A survey of linear singular systems,” J. Circuits, Systems, and Signal Processing, vol. 5, no. 1, pp. 3-36, 1986. [3] D.J. Bender and A.J. Laub, “The linear-quadratic optimal regulator for descriptor systems,” IEEE Trans. Automat. Contr., vol. 32, no. 8, pp. 672-688, 1987. [4] I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, “The H ∞ control for descriptor systems: A matrix inequalities approach,” Automatica, vol. 33, pp. 669-673, 1997. [5] S. Xu and C. Yang, “Stabilization of discrete-time singular systems: A matrix inequalities approach,” Automatica, vol. 35, pp. 1613-1617, 1999. [6] K. Takaba, N. Morihira, and T. Katayama, “A generalized Lyapunov theorem for descriptor system,” Syst. Contr. Lett., vol. 24, pp. 49-51, 1995. [7] J.Y. Ishihara and M.H. Terra, “On the Lyapunov theorem for singular systems,” IEEE Trans. Automat. Contr., vol. 47, no. 11, pp. 1926-1930, 2002.

[8] K.L. Hsiung and L. Lee, “Lyapunov inequality and bounded real lemma for discrete-time descriptor systems,” IEE Proc. Control Theory and Applications, vol. 146, no. 4, pp. 327-331, 1999. [9] C.H. Kuo and L. Lee, “Robust D-admissibility in generalized LMI regions for descriptor systems,” Proceedings of the 5th Asian Control Conference, pp. 1058-1065, 2004. [10] I. Masubuchi and E. Shimemura, “An LMI condition for stability of implicit systems,” Proceedings of the 36th Conference on Decision and Control, pp. 779-780, 1997. [11] J.Y. Ishihara and M.H. Terra, “Generalized Lyapunov theorems for rectangular descriptor systems,” Proceedings of the 40th Conference on Decision and Control, pp. 2858-2859, 2001. [12] I. Masubuchi, “Stability and stabilization of implicit systems,” Proceedings of the 39th Conference on Decision and Control, pp. 36363641, 2000. [13] J.Y. Ishihara and M.H. Terra, “Impulsive controllability and observability of rectangular descriptor systems,” IEEE Trans. Automat. Contr., vol. 46, no. 6, pp. 991-994, 2001. [14] J.D. Zhu, S.P. Ma, and Z.L. Cheng, “Singular LQ problem for nonregular descriptor systems,” IEEE Trans. Automat. Contr., vol. 47, no. 7, pp. 1128-1133, 2002. [15] T. Iwasaki and G. Shibata, “LPV system analysis via quadratic separator for uncertain implicit systems,” IEEE Trans. Automat. Contr., vol. 46, no. 8, pp. 1195-1208, 2001. [16] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, 1994. [17] D. Henrion and G. Meinsma, “Rank-one LMIs and Lyapunov’s inequality,” IEEE Trans. Automat. Contr., vol. 46, no. 8, pp. 1285-1288, 2001. [18] R.E. Skelton, T. Iwasaki, and K.M. Grigoriadis, A Unified Algebraic Approach to Linear Control Design. London; Bristol, PA: Taylor & Francis, 1998. [19] A. Rehm and F. Allg¨ower, “An LMI approach towards stabilization of discrete-time descriptor systems,” Proceedings of the 15th World Congress of IFAC, Barcelona, Spain, 2002. [20] L. Lee, J.L. Chen, and C.H. Fang, “On LMI approach to admissibilization of discrete-time descriptor systems,” Proceedings of the 5th Asian Control Conference, pp. 1001-1008, 2004.