Delay-dependent Robust Stabilization for Uncertain

Singular Systems with State Delay

WU Zheng-Guang1 ZHOU Wu-Neng2
Abstract This paper considers the problem of delay-dependent robust stabilization for uncertain singular delay systems. In terms
of linear matrix inequality (LMI) approach, a delay-dependent stability criterion is given to ensure that the nominal system is
regular, impulse free, and stable. Based on the criterion, the problem is solved via state feedback controller, which guarantees that
the resultant closed-loop system is regular, impulse free, and stable for all admissible uncertainties. An explicit expression for the
desired controller is also given. Some numerical examples are provided to illustrate the validity of the proposed methods.
Key words Singular time-delay system, delay-dependent, robust stabilization, linear matrix inequality

1 Introduction 2 Problem formulation

Consider the uncertain singular system with state delay
Over the past decades, much attention has been fo-
described by
cused on the problems of stability analysis and stabiliza-
⎧
tion for singular delay systems. Especially, with the de- ⎪ x(t) = (A + ∆A)x x(t − d)+
x(t) + (Ad + ∆Ad )x
velopment of the robust control theory, many robust sta- ⎨ Eẋ
(B + ∆B)u u(t), (1)
bilization methods have been proposed for uncertain sin- ⎪
⎩
gular delay systems. The existing results can be classified ¯ 0]
x (t) = φ (t), t ∈ [−d,
into two types: delay-independent stabilization and delay-
dependent stabilization. Generally, the delay-independent where x (t) ∈ Rn is the state, u (t) ∈ Rm is the control
case is more conservative than the delay-dependent case, input. d is an unknown but constant time delay and d¯
especially when the time delay is comparatively small. The is a constant satisfying 0 ≤ d ≤ d. ¯ φ (t) is a compati-
delay-independent case has been extensively studied (see, ble vector valued initial function. The matrix E ∈ Rn×n
e.g. [1, 2] and the references therein); however, there may be singular and rank E = r ≤ n is assumed. A, Ad ,
are only few papers on the delay-dependent case[3,4] . [3] and B are known real constant matrices with appropriate
discussed the problem of delay-dependent robust stability dimensions. ∆A, ∆Ad , and ∆B are unknown matrices rep-
analysis, and a delay-dependent robust stability criterion resenting norm-bounded parametric uncertainties and are
was obtained. But the considered system was assumed to assumed to be of the form:
be necessarily regular and impulse free; moreover, a matrix    
∆A ∆Ad ∆B = M F (t) N1 N2 N3 (2)
describing the relationship between fast and slow subsys-
tems was needed and an improper choice of the matrix where M , N1 , N2 , and N3 are known real constant matri-
would make the results unreliable. In [4], the problem of ces with appropriate dimensions, and F (t) ∈ Rq×k is an
delay-dependent robust stabilization was solved via state unknown real and possibly time-varying matrix satisfying
feedback controller, and an expression for the desired con-
troller was given by solving a set of nonlinear matrix in- F T (t)F (t) ≤ I (3)
equalities with an equation constraint, which would result
in some numerical problems and make the design procedure The parametric uncertainties ∆A, ∆Ad , and ∆B are said
complex and unreliable. To the best of our knowledge, the to be admissible if both (2) and (3) hold.
problem of delay-dependent robust stabilization for uncer- The nominal unforced singular delay system of (1) can
tain singular delay systems has not been fully studied in be written as
the literature and still remains open. x(t) + Adx (t − d)
x(t) = Ax
Eẋ (4)
In this paper, we investigate the problem of delay-
dependent robust stabilization for uncertain singular sys- Definition 1[5] .
tems with state delay. The considered systems are not 1. The pair (E, A) is said to be regular if det(sE − A) is
assumed to be necessarily regular and impulse free. The not identically zero.
considered problem is to design a state feedback controller 2. The pair (E, A) is said to be impulse free if
such that the resultant closed-loop system is robustly sta- deg(det(sE − A)) = rank E.
ble. In terms of two linear matrix inequalities (LMIs), a Definition 2. For a given scalar d¯ > 0, the singular
sufficient condition for the solvability of the problem is de- delay system (4) is said to be regular and impulse free for
rived. When this condition is satisfied, the desired state any constant time delay d satisfying 0 ≤ d ≤ d, ¯ if the pairs
feedback controller is obtained. (E, A) and (E, A + Ad ) are regular and impulse free.
Remark 1. The regularity and the absence of impulses
of the pair (E, A) ensures the system (4) with time delay
d = 0 to be regular and impulse free, while the fact that
Received March 7, 2006; in revised form June 7, 2006 the pair (E, A + Ad ) is regular and impulse free ensures the
Supported by National Natural Science Foundation of P. R. China
(60503027) system (4) with time delay d = 0 to be regular and impulse
1. Mathematics Institute, Zhejiang Normal University, Jinhua free.
321004, P. R. China 2. College of Information Science and Tech-
nology, Donghua University, Shanghai 201620, P. R. China Definition 3. The uncertain singular delay system (1) is
DOI: 10.1360/aas-007-0714 said to be robustly stable, if the system with u (t) = 0 is
No. 7 WU Zheng-Guang and ZHOU Wu-Neng: Delay-dependent Robust Stabilization for · · · 715

regular, impulse free, and stable for all admissible uncer- By Lemma 3, it follows from (10) that
tainties ∆A and ∆Ad .
In this paper, we shall address the following problem. 0 > ÃT P̃ + P̃ T Ã + d¯X̃ + Q̃ + Ỹ Ẽ + Ẽ T Ỹ T +
Delay-dependent robust stabilization problem.
P̃ T Ãd − Ỹ Ẽ + (P̃ T Ãd − Ỹ Ẽ)T − Q̃ ≥ (11)
For a given scalar d¯ > 0, design a state feedback controller
T T
u (t) = Kx x(t), K ∈ Rm×n for system (1) such that the (Ã + Ãd ) P̃ + P̃ (Ã + Ãd )
resultant closed-loop system is robustly stable for any con-
stant time delay d satisfying 0 ≤ d ≤ d. ¯ In this case, the According to Lemma 1, we can deduce from (9) and (11)
system is said to be robustly stabilizable. that the pair (Ẽ, Ã + Ãd ) is regular and impulse free.
We conclude this section by presenting several prelimi- Since rank Ẽ = rank E = r ≤ n, there exist nonsingular
nary results, which will be used in the proof of our main matrices M and N such that
results.
Lemma 1[6] . The singular system Eẋ x(t) = Ax x(t) is Ir 0
Ê = M ẼN = (12)
regular, impulse free, and stable, if and only if there exists 0 0
a matrix P such that
Denote
P TE = ETP ≥ 0 (5)
A11 A12
P T A + AT P < 0 (6) Â = M ÃN =
A21 A22
P11 P12
Lemma 2[7] . Given matrices Ω, Γ, and Ξ with appro- P̂ = M −T P̃ N = (13)
P21 P22
priate dimensions and with Ω symmetrical, Ω + ΓF Ξ +
Ξ T F T ΓT < 0 for any F satisfying F T F ≤ I, if and only if Y11 Y12
there exists a scalar ε > 0 such that Ω + εΓΓT + ε−1 Ξ T Ξ < Ŷ = N T Ỹ M −1 =
Y21 Y22
0.
Lemma 3[8] . For symmetric positive-definite matrix Q From (9) and using the expressions of Ê and P̂ in (12) and
and matrices P and R with appropriate dimensions, matrix (13), it is easy to obtain P12 = 0, P11 ≥ 0; therefore,
inequality P T R + RT P ≤ RT QR + P T Q−1 P holds.
P11 0
3 Main results P̂ =
P21 P22
Initially, we present the following theorem for the singu-
From (10), we get
lar delay system (4), which will play a key role in the proof
of our main results.
ÃT P̃ + P̃ T Ã + Ỹ Ẽ + Ẽ T Ỹ T < 0 (14)
Theorem 1. For a prescribed scalar d¯ > 0, the singular
delay system (4) is regular, impulse free, and stable for any
¯ if there exist Now, pre-multiplying and post-multiplying (14) by N T and
constant time delay d satisfying 0 ≤ d ≤ d,
symmetric positive-definite matrices P1 , Q, Z and matrices N , respectively, we can obtain AT T
22 P22 + P22 A22 < 0. This

S, P2 , P3 , X11 , X12 , X22 , Y1 , and Y2 such that implies that A22 is nonsingular, and thus the pair (Ẽ, Ã) is
regular and impulse free.
⎡ ⎤
Ω11 Ω12 P2T Ad − Y1 E Noting the fact that det (sE − A) = det (sẼ − Ã),
Ω =⎣ ∗ −P3 − P3T + dX ¯ 22 + dZ
¯ P3T Ad − Y2 E ⎦ deg(det (sE − A))) = deg(det (sẼ − Ã))) = rank Ẽ =
∗ ∗ −Q rank E, det (sE − (A + Ad )) = det (sẼ − (Ã + Ãd )), and
<0 (7) deg(det (sE − (A + Ad ))) = deg(det (sẼ − (Ã + Ãd ))) =
rank Ẽ = rank E, we can easily see that the pairs (E, A)
⎡ ⎤ and (E, A + Ad ) are regular and impulse free, and thus
X11 X12 Y1 system (4) is regular and impulse free.
⎣ ∗ X22 Y2 ⎦ > 0 (8) Next, we will show that system (4) is stable. To the end,
∗ ∗ Z we propose the following function.

where R ∈ Rn×l is any matrix satisfying E T R = 0 and xt ) = V1 (x

V (x xt ) + V2 (x
xt ) + V3 (x
xt )
Ω11 = P2T A + AT P2 + dX
¯ 11 + Q + Y1 E + E T Y1T
Ω12 = E P1 + SR − P2T + AT P3 + E T Y2T + dX
T T ¯ 12 where
Proof. From (7), it is easy to show that xt ) = x (t)T E T P1 Ex
V1 (x x(t),
0
x(α)T E T ZEẋ
t
xt ) = −d t+β ẋ
V2 (x x(α)dαdβ
Ẽ T P̃ = P̃ T Ẽ ≥ 0 (9) xt ) = t−d x (α)T Qx
V3 (x
t
x(α)dα
ÃT P̃ + P̃ T Ã + d¯X̃ + Q̃ + Ỹ Ẽ + Ẽ T Ỹ T + Differentiating V (xxt ) with respect to t, we have
(P̃ T Ãd − Ỹ Ẽ)Q̃−1 (P̃ T Ãd − Ỹ Ẽ)T < 0 (10) x (t)
T
0 I x (t)
xt ) = 2
V̇1 (x P̃ T +
x(t)
Eẋ A −I x(t)
Eẋ
where
E 0 0 I 0 0 T
x (t) 0
Ẽ =
0 0
, Ã =
A −I
, Ãd =
Ad 0 2 P̃ T x (t − d)
x(t)
Eẋ Ad
P1 E + RS T 0 X11 X12  t
P̃ = , X̃ =
P2 P3 ∗ X22 V̇2 (x x(t)T E T ZEẋ
xt ) ≤ d¯ẋ x(t) − x(α)T E T ZEẋ
ẋ x(α)dα
Y1 0 Q 0 t−d
Ỹ = , Q̃ =
Y2 0 0 ¯
dZ xt ) = x (t)T Qx
V̇3 (x x(t) − x (t − d) Qx T
x(t − d)
716 ACTA AUTOMATICA SINICA Vol. 33

It is clear that S, P2 , P3 , X11 , X12 , X22 , Y1 , and Y2 such that both (8)
⎡ ⎤T ⎡ ⎤⎡ ⎤ and the following linear matrix inequality hold.
 t x (t) X11 X12 Y1 x (t) ⎡ ⎤
⎣ Eẋx(t) ⎦ ⎣ ∗ X22 Y2 ⎦ ⎣ Eẋ
x(t) ⎦ dα ≥ 0 Ω11 Ω12 P2T AT
d − Y1 E
T

t−d x(α)
Eẋ ∗ ∗ Z x(α)
Eẋ ⎣ ∗ T
−P3 − P3 + dX ¯ 22 + dZ¯ P3 Ad − Y2 E ⎦ < 0
T T T

∗ ∗ −Q
Thus, (17)
 t where R ∈ Rn×l is any matrix satisfying ER = 0 and
x(t)T E T ZEẋ
xt ) ≤ d¯ẋ
V̇2 (x x(t) − x(α)T E T ZEẋ
ẋ x(α)dα+ Ω11 = P2T AT + AP2 + dX ¯ 11 + Q + Y1 E T + EY1T
t−d Ω12 = EP1 + SRT − P2T + AP3 + EY2T + dX ¯ 12
⎡ ⎤T ⎡ ⎤
 t x (t) X11 X12 Y1 Remark 4. Sufficient conditions of Theorem 1 and The-
⎣ Eẋ x(t) ⎦ ⎣ ∗ X22 Y2 ⎦ × orem 2 may lead to different results. Hence, we can sepa-
t−d x(α)
Eẋ ∗ ∗ Z rately apply Theorem 1 and Theorem 2 and then choose
⎡ ⎤ the less conservative one.
x (t) Next, based on Theorem 1, we give the following delay-
⎣ Eẋx(t) ⎦ dα ≤ dependent robust stability criterion.
x(α)
Eẋ Theorem 3. For a prescribed scalar d¯ > 0, the un-
T certain singular delay system (1) is robustly stable for any
x (t) X11 X12 x (t) constant time delay d satisfying 0 ≤ d ≤ d, ¯ if there exist
d¯ +
x(t)
Eẋ X21 X22 x(t)
Eẋ scalar ε > 0 and symmetric positive-definite matrices P1 ,
T Q, Z and matrices S, P2 , P3 , X11 , X12 , X22 , Y1 , and Y2
x (t) Y1 E
2 x(t) − x (t − d)]+
[x such that the linear matrix inequalities (8) and (18) hold.
x(t)
Eẋ Y2 E
⎡ ⎤
x(t)T E T ZEẋ
d¯ẋ x(t). Ξ11 Ξ12 P2T Ad − Y1 E + εN1T N2 P2T M
⎢ ∗ Ξ22 P3T Ad − Y2 E P3T M ⎥
⎢ ⎥<0
Hence, ⎣ ∗ ∗ −Q + εN2 N2T
0 ⎦
xt ) ≤ ξ (t)T Ωξξ (t)
V̇ (x ∗ ∗ ∗ −εI
(18)
where ξ (t) = [xx(t)T (Eẋ
x(t))T x (t − d)T ]T . From (7), we get where R ∈ Rn×l is any matrix satisfying E T R = 0 and
xt ) < 0, and thus system (4) is stable.
V̇ (x
Ξ11 = P2T A + AT P2 + dX ¯ 11 + Q + Y1 E + E T Y1T + εN1T N1
Remark 2. In the proof of Theorem 1, it is noted that Ξ12 = E P1 + SR − P2T + AT P3 + E T Y2T + dX
T T ¯ 12
neither model transformation nor bounding technique for Ξ22 = −P3 − P3T + dX ¯ 22 + dZ¯
cross terms, which are usually used in the existing results, Proof. Using Schur complement, we obtain from (18)
is required. Hence, the derivation procedure is simpler and that
the condition of Theorem 1 is less conservative than those Ω + ε−1 M M T + εN T N < 0 (19)
of existing ones, which will be demonstrated by examples.
If the matrices, in (8), Y1 = 0, Y2 = 0, X12 = 0 and where Ω is same as that on the left side of (7) and M =
X11 = X22 = Z = εI/d¯ (ε → 0), then Theorem 1 provides [ M T P2 M T P3 0 ]T , N = [N1 0 N2 ].
the result on delay-independent stability analysis, which is By Lemma 2, it follows from (19) that
stated as follows. ⎡ ⎤
Corollary 1. The singular delay system (4) is regular, Ψ11 Ψ12 P2T (Ad + ∆Ad ) − Y1 E
impulse free, and stable, if there exist symmetric positive- ⎣ ∗ Ξ22 P3 (Ad + ∆Ad ) − Y2 E ⎦ < 0
T
(20)
definite matrices P1 , Q, Z and matrices S, P2 , and P3 such ∗ ∗ −Q
that
⎡ T ⎤ where
P2 A + AT P2 + Q Π12 P2T Ad Ψ11 = P2T (A + ∆A) + (A + ∆A)T P2 + dX ¯ 11 + Q+
⎣ ∗ −P3 − P3T P3T Ad ⎦ < 0 (15) T T
Y1 E + E Y1
∗ ∗ −Q Ψ12 = E T P1 + SRT − P2T + (A + ∆A)T P3 + E T Y2T + dX ¯ 12
According to Theorem 1 and Definition 3, we have the de-
where R follows the same definition as that in Theorem 1 sired result immediately.

and Π12 = E T P1 + SRT − P2T + AT P3 . Now, we are in the position to present the result on the
Remark 3. As we have seen above, the result of The- problem of delay-dependent robust stabilization.
orem 1 is powerful in the sense that it provides suffi- Theorem 4. For a prescribed scalar d¯ > 0, the uncer-
cient conditions for both the delay-dependent and delay- tain singular delay system (1) is robustly stabilizable for
independent cases. any constant time delay d satisfying 0 ≤ d ≤ d, ¯ if there
Since the solution of det (sE − A − e−sd Ad ) = 0 is same exist scalars ε > 0 and , symmetric positive-definite ma-
as that of det (sE T − AT − e−sd AT d ) = 0, system (4) is trices P1 , Q, Z and matrices S, P2 , X, X11 , X12 , X22 , Y1
regular, impulse free, and stable, if and only if the system and Y2 such that the linear matrices inequalities (8) and
(21) hold,
E Tζ̇ζ (t) = ATζ (t) + AT
d ζ (t − d) (16) ⎡ ⎤
Π11 Π12 P2T AT
d − Y1 E
T
(N1 P2 + N3 X)T (N2 P2 )T
⎢ ∗ Π22
P2T AT T

(N1 P2 + N3 X)T
(N2 P2 )T ⎥
is regular, impulse free, and stable. Hence, using ⎢ d − Y2 E ⎥
⎢ ∗ ∗ −Q + εM M T ⎥ <0
Theorem 1 for system (16) leads to the following theorem: ⎢ 0 0 ⎥
⎣ ∗ ∗ ∗ −εI 0 ⎦
Theorem 2. For a prescribed scalar d¯ > 0, the singular ∗ ∗ ∗ ∗ −εI
delay system (4) is regular, impulse free, and stable for any (21)
constant time delay d satisfying 0 ≤ d ≤ d, ¯ if there exist where R ∈ Rn×l is any matrix satisfying ER = 0 and
symmetric positive-definite matrices P1 , Q, Z and matrices Π11 = P2T AT + X T B T + AP2 + BX + dX
¯ 11 + Q+
No. 7 WU Zheng-Guang and ZHOU Wu-Neng: Delay-dependent Robust Stabilization for · · · 717

Y1 E T + EY1T + εM M T Example 2. Consider the following uncertain singular

Π12 = EP1 + SRT − P2T + (AP2 + BX) + EY2T + dX ¯ 12 delay system.
T ¯ ¯
Π22 = −P2 − P2 + dX22 + dZ
In this case, a desired state feedback controller is given by x(t) = (A + A)x
Eẋ x(t) + (Ad + Ad )x
x(t − d)

u (t) = XP2−1x (t) (22) where

2 0 1 0 −2.4 2
Proof. Setting P2 = P3 and X = KP2 and using Schur E= ,A= , Ad =
0 0 0 −2 0 1
complement, we obtain from (21) that
and the uncertain matrices A and Ad satisfy A ≤
Ξ + εM M T + ε−1 N T N < 0 (23) λ, Ad ≤ λ(λ > 0). This system is of the form in system
(1) with u (t) = 0. Then, we can write M = λI, N1 =
⎡
where ⎤ N2 = 0.5I.
Ψ11 Ψ12 P2T Ad − Y1 E T
In this example, we choose R = [0 1]T . Table 2 gives
Ξ=⎣ ∗ Ψ22 P3T Ad − Y2 E T ⎦
the comparison of the maximum allowed delay d¯ for various
∗ ∗ −Q parameter λ. It is clear that the conditions in this paper
T
MT 0 0 gives better results than those in [12, 13].
M=
0 0 MT
N1 P2 + N3 KP2 N1 P3 + N3 KP3 0
N= Table 2 Comparison of delay-dependent stability conditions of
N2 P2 N2 P3 0 Example 2
and
Ψ11 = P2T (A + BK)T + (A + BK)P2 + dX ¯ 11 + Q+
λ 0.25 0.30 0.35 0.40 0.45 0.50
T T
Y1 E + EY1 [12] 0.4209 0.3939 0.3637 0.3279 0.2817 0.2106
Ψ12 = EP1 + SRT − P2T + (A + BK)P3 + EY2T + dX ¯ 12
[13] 0.8087 0.7942 0.7689 0.7262 0.6521 0.5054
Ψ22 = −P3 − P3T + dX
¯ 22 + dZ
¯
Theorem 3 0.8514 0.8249 0.7924 0.7438 0.6641 0.5110
By Lemma 2, it follows from (23) that
⎡ ⎤
Θ11 Θ12 P2T (Ad + Ad )T − Y1 E T
⎣ ∗ Example 3. Consider the uncertain singular delay sys-
Ψ22 P3T (Ad + Ad )T − Y2 E T ⎦ < 0 (24)
tem (1)
⎡ with parameters
⎤ as⎡follows. ⎤
∗ ∗ −Q
1 1 0 2 1 1
where E = ⎣ 1 −1 1 ⎦ , A = ⎣ −1 0 1 ⎦
Θ11 = P2T AT ¯ T
K + AK P2 + dX11 + Q + Y1 E + EY1
T
⎡
2 0 1
⎤
0.5 0 1
⎡ ⎤
Θ12 = EP + SRT − P2T + AK P3 + EY2T + dX ¯ 12 −1.5 0.5 −0.8 1 2
AK = A + BK + A + BK Ad = ⎣ 1 1 0.5 ⎦ , B = ⎣ 1.5 0 ⎦
According to Theorem 2 and Definition 3, the desired result 0.7 0.5 1 0 1
follows immediately.
 T  
M = 0.4 0.3 0.1 , N 1 = 0.2 0.4 0.5
   
4 Numerical examples N 2 = 0.3 0.7 0.5 , N 3 = 0.4 0.5
In this example, we choose R = [−1 1 2]T . For  =
In this section, some examples are provided to illustrate 0.5, Theorem 4 yields d¯ = 3.1, and the corresponding state
the effectiveness and the less conservatism of the obtained feedback gain is
results.
Example 1. Consider the following singular delay −2.3593 0.7100 4.9681
system[10] K=
−2.3048 −1.4295 −5.5923

1 0 0.5 0 −1.1 1 Also, for  = 1, d¯ = 1.32 and

x(t) =
ẋ x (t) + ·
0 0 0 −1 0 0.5
x (t − d) −0.1183 0.6422 1.2963
K=
−3.7556 −2.2408 −4.5535
In this example, we choose R = [0 1]T . The upper
bounds on the time delay from Theorem 1 and Theorem 2 5 Conclusion
are shown in Table 1. For comparison, the table also lists In this paper, the problem of delay-dependent robust sta-
the upper bounds obtained from the criteria in [3, 4, 9∼16]. bilization for singular delay systems with norm bounded
It can be seen that our methods are less conservative. parametric uncertainties has been studied. A delay-
dependent robust stability condition is presented and a
Table 1 Comparison of delay-dependent stability conditions of design procedure of the desired state feedback controller
Example 1 is given. All the obtained results are formulated in terms
of strict LMIs involving no decomposition of the system
Theorem matrices, which makes the design procedure relatively sim-
Methods [3,9,11] [4,12] [13] [14] [15,16] [10] ple and reliable. Neither model transformation nor bound-
1&2
Maximum
ing technique is needed in the development of the results.
d¯ allowed
– 0.5567 0.8708 0.9091 0.9680 1.0423 1.0660 Numerical examples show that the results of the proposed
methods are less conservative than those of the existing
methods.
718 ACTA AUTOMATICA SINICA Vol. 33

References 12 Zhong Ren-Xin, Yang Zhi. Delay-dependent robust control

of descriptor systems with time delay. Asian Journal of Con-
1 Xu S Y, Dooren P V, Stefan R, Lam J. Robust stability trol, 2006, 8(1): 36∼44
analysis and stabilization for singular systems with state de-
lay and parameter uncertainty. IEEE Transactions on Auto- 13 Gao H L, Z S Q, Cheng Z L, Xu B G. Delay-dependent state
matic Control, 2002, 47(7): 1122∼1128 feedback guaranteed cost control uncertain singular time-
delay systems. In: Proceedings of IEEE Conference on Deci-
2 Zhu S Q, Cheng Z L, Feng J. Robust output feedback sion and Control, and European Control Conference. IEEE,
stabilization for uncertain singular time delay systems. In: 2005. 4354∼4359
Proceedings of 2004 American Control Conference. Boston,
USA: AACC, 2004. 5416∼5421 14 Fridman E. Stability of linear descriptor systems with de-
lay: a Lyapunov-based approach. Journal of Mathematical
3 Zhong Ren-Xin, Yang Zhi. Robust stability analysis of sin- Analysis and Applications, 2002, 273(1): 24∼44
gular linear system with delay and parameter uncertainty.
Journal of Control Theory and Applications, 2005, 3(2): 15 Fridman E. A Lyapunov-based approach to stability of de-
195∼199 scriptor systems with delay. In: Proceedings of IEEE Con-
ference on Decision and Control, Orlando, USA: IEEE, 2001.
4 Zhu S Q, Cheng Z L, Feng J. Delay-dependent robust stabil- 2850∼2855
ity criterion and robust stabilization for uncertain singualr
time-delay systems. In: Proceedings of American Control 16 Fridman E, Shaked U. H∞ control of linear state-delay de-
Conference. Portland, USA: AACC, 2005. 2839∼2844 scriptor systems: an LMI approach. Linear Algebra and Its
Applications, 2002, 351(1): 271∼302
5 Dai L. Singular Control Systems. Berlin: Springer-Verlag,
1989
6 Masubuchi I, Kamitane Y, Ohara A, Suda N. H∞ control WU Zheng-Guang Master student at
for descriptor systems: a matrix inequalities approach. Au- Zhejiang Normal University. He received
tomatica, 1997, 33(4): 159∼162 his bachelor degree from Zhejiang Normal
7 Petersen I R. A stabilization algorithm for a class of uncer- University in 2004. His research interest
tain linear systems. Systems & Control Letters, 1987, 8(4): covers robust control and system theory.
351∼357 E-mail: nashwzhg@126.com
8 Xie L, de Souza C E. Robust H∞ control for linear systems
with norm-bounded time-varying uncertainty. IEEE Trans-
actions on Automatic Control, 1992, 37(8): 1188∼1191
9 Boukas E K, Almuthairi N F. Delay-dependent stabilization
of singular linear systems with delays. International Journal
of Innovative Computing, Information and Control, 2006, ZHOU Wu-Neng Professor at
2(2): 283∼291 Donghua University. He received his
Ph. D. degree from Zhejiang University in
10 Yang Fan, Zhang Qing-Ling. Delay-dependent H∞ control 2005. His research interest covers robust
for linear descriptor systems with delay in state. Journal of control and system theory. Corresponding
Control Theory and Applications, 2005, 3(1): 76∼84 author of this paper.
11 Boukas E K, Liu Z K. Delay-dependent stability analysis E-mail: zhouwuneng@163.com
of singular linear continuous-time system. IEE Proceedings
Control Theory & Applications , 2003, 150(4): 325∼330