Stability of discrete-time uncertain systems with a
time-varying state delay
K-F Chen and I-K Fong*
Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, Republic of China
The manuscript was received on 5 December 2007 and was accepted after revision for publication on 25 April 2008. DOI: 10.1243/09596518JSCE534
Abstract: The stability analysis problem for unforced discrete-time systems with a time-varying state delay is considered. Stability criteria for nominal and uncertain system models are derived such that the upper and lower bounds on the delay time can be determined to ensure asymptotic stability. There is no need to assume that the system is stable when the delay vanishes. Adopting the linear matrix inequality approach enables the computations to be implemented conveniently. Compared with existing results in the literature, the proposed method is less conservative for many cases.
Keywords: linear matrix inequality, discrete-time system, time-varying delay, stability
1 INTRODUCTION
Time delay occurs in electrical, mechanical, eco-nomic, and biological systems, and is a well-known cause of instability and performance degradation. In recent years, increasing attention has been paid to the problems of stability analysis and controller design for time delay systems. This is particularly necessary in the field of chemical process control, where time delay is almost inevitable, and control-lers are sought to enhance system stability and performance [1–4].
After controllers are designed for nominal systems, one often has to check the robustness with respect to all parameter uncertainties in the system model. If the time delay is also uncertain, with its true value being different from its nominal value, then the system’s robust stability should be confirmed for a range of time delay values [5]. The stability problem has been investigated in the literature for systems with an interval time delay [6, 7], and in certain cases it has been found that some controllers can tolerate less time delay variation than others [8, 9].
Although there are already many stability analy-sis methods proposed for time delay systems, including delay-independent methods [10, 11] and delay-dependent methods [12–23], most start with the assumption that the system is stable when the time delay is zero, and try to find out the full extent of the tolerance to the time delay. The assumption of stability with no time delay is even adopted implicitly in some results that discuss the stability with respect to intervals of time delay that exclude zero [21–23]. Only a few studies have assumed that the system to be analysed is known to be stable for a non-zero time delay [5]. However, as stated above, after a controller is designed to stabilize a system, the closed-loop system is usually known to be stable for the specific nominal time delay. Furthermore, the controller may not be robust enough to cover such a large time delay range as to ensure that the closed-loop system is stable when the time delay is shortened to zero. Thus, assuming stability at a zero-valued time delay may not be practical.
In this paper, the stability analysis problem for uncertain and nominal unforced discrete-time systems with a time-varying state delay is studied. Stability conditions are derived based on a Lyapu-nov function defined for an augmented system. It is assumed that the system is asymptotically stable with a specific time delay, and upper bounds on the
*Corresponding author: Department of Electrical Engineering, National Taiwan University, 1 Roosevelt Road Sec. 4, Taipei 10617, Taiwan, Republic of China. email: ikfong@cc.ee.ntu. edu.tw
allowable additional time delay are derived. The stability criteria are formulated using the linear matrix inequality (LMI) approach [24], which is convenient to check. Numerical examples are provided to illustrate the application of the developed results, and compar-isons with results from existing methods are made to show that the proposed method is not only less conservative for the considered examples, but also applicable to systems which are not stable when the time delay is zero.
2 PROBLEM FORMULATION
Consider the following uncertain discrete-time delay system
x kz1ð Þ~ AzDA kð ð ÞÞx kð Þz Að dzDAdð Þk Þx k{h kð ð ÞÞ x kð Þ~w kð Þ, for {h2¡k¡0
ð1Þ where k is the discrete-time index, x [ Rnis the state vector, A and Adare real n6n parameter matrices,
and h(k) is a non-negative time delay depending on k. It is assumed that
0¡h1¡h kð Þ¡h2 ð2Þ
where h1and h2are positive integers, and the initial
conditions of the system are specified by the given vector function w(?) in equation (1). The two real n6n matrix functions DA(?) and DAd(?) represent
time-varying parameter uncertainties. It is further assumed that the uncertainties are norm-bounded and can be described as
DA kð Þ~DF kð ÞEa, DAdð Þ~DF kk ð ÞEd ð3Þ where D, Ea, and Edare real constant matrices with
appropriate dimensions and F(k) is an unknown matrix satisfying
FTð ÞF kk ð Þ¡I ð4Þ
When the uncertainties in the system matrices are ignored, a nominal system of equation (1) is
x kz1ð Þ~Ax kð ÞzAdx k{h kð ð ÞÞ x kð Þ~w kð Þ, for {h2¡k¡0
ð5Þ To find the delay-dependent stability criteria for discrete-time delay systems such as (5), it is often assumed that the system is asymptotically stable when
h(k) 5 0. This is explicitly expressed in [25], and also in [26] and [27] in order to discuss the continuous-time case, but it is often not mentioned in other related works, although the assumption is actually necessary therein. Here for a broader investigation purpose, the system (5) is assumed to be asymptotically stable when h(k) 5 h1. In order to handle this kind of situation, an
augmented state vector
z kð Þ~ x kð Þ x k{1ð Þ .. . x k{hð 1Þ 2 6 6 6 6 4 3 7 7 7 7 5 ð6Þ
and its delayed term
z kzhð 1{h kð ÞÞ~ x kzhð 1{h kð ÞÞ x k{1zhð 1{h kð ÞÞ .. . x k{h kð ð ÞÞ 2 6 6 6 6 4 3 7 7 7 7 5
are defined, and the augmented system
z kz1ð Þ~~AAz kð Þz~AAdz kzhð 1{h kð ÞÞ ð7Þ is discussed, where ~ A A~ A 0 0 0 I 0 0 0 0 I 0 0 .. . .. . P .. . .. . 0 0 I 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 , AA~d~ 0 0 0 Ad 0 0 0 0 0 0 0 0 .. . .. . P .. . .. . 0 0 0 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5
are matrices of compatible dimensions. The problem here is to find h2 to guarantee that equation (5)
remains asymptotically stable.
In the subsequent development, the following lemma will play an important role, and is presented here. Note in matrix inequalities, the notation ‘‘. (>) 0’’ means positive (semi-)definiteness, ‘‘, 0’’ means negative definiteness, and ‘‘A ( B’’ means A 2 B ( 0. Lemma 1 [17]. Given a symmetric matrix M and matrices H, E with appropriate dimensions, then
MzHFEzETFTHTv0
for all F satisfying FTF ( I, if and only if there exists an e . 0 such that
3 STABILITY ANALYSIS
In this section, the Lyapunov stability method will be adopted to find an h2 for a given h1 to ensure the
stability of the augmented system (7), and thus the stability of system (5). The main result is presented below. For the sake of simplicity, * is used as an ellipsis for the terms that are clear by the symmetry in a block matrix, and a special font such as ‘‘R’’ is used to represent matrix variables.
Theorem 1. The system (7) is asymptotically stable for 0 ( h1( h(k) ( h2, if there exist matrices
Pw0, Qw0, Rw0, Y, and Z, all with the size (h1+ 1)n6(h1+ 1)n, such that w11z ~AATP ~AztQzt^A AATR ^AA w12z ~AATP ~AAdzt^AATR ~AAd tY 1 w22z ~AATdP ~AAdzt~AATdR ~AAd tZ 1 1 {tR 2 6 4 3 7 5v0 ð8Þ where ^AA~~AA{I, t~h2{h1, w11~{PzQzYzYT, w12~{YzZT,and w
22~{Q{Z{Z T
Proof. By assumption, equations (5) and (7) are asymptotically stable when h(k) 5 h1for all k,
there-fore only the case with h2.h1needs to be discussed.
Let y(k) 5 z(k + 1) 2 z(k). Then
y kð Þ~ ~AA{Iz kð Þz~AAdz kzhð 1{h kð ÞÞ ð9Þ and z kzhð 1{h kð ÞÞ~z kð Þ{ X k{1 i~kzh1{h kð Þ y ið Þ ð10Þ
Choose a Lyapunov function candidate as
V kð Þ~V1ð ÞzVk 2ð ÞzVk 3ð ÞzVk 4ð Þk ð11Þ where V1ð Þ~zk Tð ÞPz kk ð Þ, V2ð Þ~k X k{1 i~kzh1{h kð Þ zTð ÞQz ii ð Þ, V3ð Þ~k X0 i~1zh1{h2 X k{1 m~kzi zTð ÞQz mm ð Þ, V4ð Þ~k X{1 i~h1{h2 X k{1 m~kzi yTð ÞRy mm ð Þ
and P, Q, R are positive definite matrices to be determined. Consider the Lyapunov difference
DVi(k) 5 Vi(k + 1) 2 Vi(k) for i 5 1,2,3,4. Firstly, DV1ð Þ~zk Tðkz1ÞPz kz1ð Þ{zTð ÞPz kk ð Þ ~zTð Þ ~k AATP ~AA{Pz kð Þ z2zTð Þ ~k AATP ~AAdz kzhð 1{h kð ÞÞ zzTðkzh1{h kð ÞÞ ~AAT dP ~AAd z kzhð 1{h kð ÞÞ ð12Þ Secondly DV2ð Þ~k Xk i~kz1zh1{h kz1ð Þ zTð ÞQz ii ð Þ { X k{1 i~kzh1{h kð Þ zTð ÞQz ii ð Þ ~zTð ÞQz kk ð Þ {zTðkzh1{h kð ÞÞQz kzhð 1{h kð ÞÞ z X k{1 i~kz1zh1{h kz1ð Þ zTð ÞQz ii ð Þ { X k{1 i~kz1zh1{h kð Þ zTð ÞQz ii ð Þ ð13Þ where X k{1 i~kz1zh1{h kz1ð Þ zTð ÞQz ii ð Þ¡ X k{1 i~kz1zh1{h2 zTð ÞQz ii ð Þ ~ X kzh1{h kð Þ i~kz1zh1{h2 zTð ÞQz ii ð Þz X k{1 i~kz1zh1{h kð Þ zTð ÞQz ii ð Þ ð14Þ Hence DV2ð Þ¡k X kzh1{h kð Þ i~kz1zh1{h2 zTð ÞQz ii ð ÞzzTð ÞQz kk ð Þ {zTðkzh1{h kð ÞÞQz kzhð 1{h kð ÞÞ ð15Þ Thirdly DV3ð Þ~k X0 i~1zh1{h2 Pk m~kz1zi zTð ÞQz mm ð Þ { P k{1 m~kzi zTð ÞQz mm ð Þ 0 B B B B @ 1 C C C C A ~tzTð ÞQz kk ð Þ{ X k i~kz1zh1{h2 zTð ÞQz ii ð Þ ð16Þ (8)
where Xk i~kz1zh1{h2 zTð ÞQz ii ð Þ~ X kzh1{h1 i~kz1zh1{h2 zTð ÞQz ii ð Þo X kzh1{h kð Þ i~kz1zh1{h2 zTð ÞQz ii ð Þ ð17Þ Thus DV3ð Þ¡tzk Tð ÞQz kk ð Þ{ X kzh1{h kð Þ i~kz1zh1{h2 zTð ÞQz ii ð Þ ð18Þ Finally DV4ð Þ~k X{1 i~h1{h2 Pk m~kz1zi yTð ÞRy mm ð Þ { P k{1 m~kzi yTð ÞRy mm ð Þ 0 B B B B @ 1 C C C C A ~tyTð ÞRy kk ð Þ{ X k{1 i~kzh1{h2 yTð ÞRy ii ð Þ ð19Þ where X k{1 i~kzh1{h kð Þ yTð ÞRy ii ð Þ¡ X k{1 i~kzh1{h2 yTð ÞRy ii ð Þ ð20Þ
Thus, with the relation in equation (9) DV4ð Þ¡zk Tð Þ t^k AATR ^AA z kð Þ z2zTð Þ t^k AATR ~AAdz kzhð 1{h kð ÞÞ zzTðkzh1{h kð ÞÞ t~AAT dR ~AAd z kzhð 1{h kð ÞÞ { X k{1 i~kzh1{h kð Þ yTð ÞRy ii ð Þ ð21Þ
From equations (12), (15), (18), and (21) DV kð Þ~DV1ð ÞzDVk 2ð ÞzDVk 3ð ÞzDVk 4ð Þk ¡jTð ÞVj kk ð Þ{ X k{1 i~kzh1{h kð Þ yTð ÞRy ii ð Þ ð22Þ where jT(k) 5 [zT(k) zT(k + h12h(k))] and V~ {Pz tz1ð ÞQz~AA TP ~AAzt^AATR ^AA AA~TP ~AA dzt^AATR ~AAd 1 AA~T dP ~AAd{Qzt~AATdR ~AAd " #
Define zT(k) 5 [zT(k) zT(k + h12h(k)) yT(i)] and
J~ X11 X12 Y 1 X22 Z 1 1 R 2 6 4 3 7 5o0 ð23Þ
where J is partitioned in accordance with z(k), and
X~ X11 X12 1 X22 ~ Y Z R{1 Y Z T ð24Þ By equation (10) X k{1 i~kzh1{h kð Þ zTð ÞJz kk ð Þ¡zTð Þ tXk 11zYTzYz kð Þ z2zTð Þ tXk 12{YzZTz kzhð 1{h kð ÞÞ zzTðkzh1{h kð ÞÞ tX 22{Z{ZTz kzhð 1{h kð ÞÞ z X k{1 i~kzh1{h kð Þ yTð ÞRy ii ð Þ ð25Þ
From equations (22) and (25)
DV kð Þ¡jTð ÞVj kk ð Þ{ X k{1 i~kzh1{h kð Þ yTð ÞRy ii ð Þ z X k{1 i~kzh1{h kð Þ zTð ÞJz kk ð Þ{ X k{1 i~kzh1{h kð Þ zTð ÞJz kk ð Þ ~jTð ÞWj kk ð ÞztjTð ÞXj kk ð Þ { X k{1 i~kzh1{h kð Þ zTð ÞJz kk ð Þ ð26Þ where W~ w11z ~AATP ~AAztQzt^AATR ^AA w12z ~AATP ~AAdzt^AATR ~AAd w22z ~AATdP ~AAdzt~AATdR ~AAd " #
Clearly in WztXv0 implies DV(k) , 0 for all non-zero j(k), and thus the asymptotic stability of the augmen-ted system (7) and (5) with a time-varying delay 0 ( h1( h(k) ( h2. Furthermore, by equation (24)
WztXv0 is equivalent to equation (8), according to the Schur complement. This completes the proof. Remark 1. Note in the proof of Theorem 1, no commonly used [20–22] inequalities, such as the simple inequality 22aTb ( aTW21a + bTWb for all n61 vectors a, b and n6n matrix W . 0 and some more complicated inequalities, are applied. This
reduces potential conservatism of the derived stability condition.
Remark 2. To determine the largest h2 for the
stability of equation (5), a straightforward but slow method is to check the feasibility of LMI (8) for various integer values of t one by one until the largest t is found. Alternatively the feasibility of equation (8) with t as a variable can be cast into a generalized eigenvalue problem (GEVP)
Qz ^AATR ^AA AA^TR ~AAd Y 1 AA~T dR ~AAd Z 1 1 {R 2 6 4 3 7 5vl Y1 00 ð27Þ
where l 5 1/t is a real variable
Y~ {w11{ ~AA
TP ~AA {w
12{ ~AATP ~AAd 1 {w22{ ~AATdP ~AAd
" #
and the following optimization problem can be formed to search for the smallest l
min
lw0,Pw0,Qw0,Rw0,Y,Zl subject to equation 27ð Þ ð28Þ
However, to utilize existing software such as the LMI Control Toolbox of Matlab [28], some adjustment is needed to match the problem formulation therein. Let Tw0 be an additional symmetric matrix variable of proper dimension, and change equation (27) to
Qz ^AATR ^AA AA^TR ~AAd Y 1 AA~T dR ~AAd Z 1 1 {R 2 6 4 3 7 5v T1 00 ð29Þ TvlY ð30Þ
Note that equation (29) and equation (30) together are basically equivalent to equation (27), except now with Tw0 as an additional condition, but due to Rw0 and Qw0 in the left-hand side of equation (27), the condition is not supposed to cause too much conservativeness. Therefore, the following GEVP optimization problem is adopted as a substitute for equation (28)
min
lw0,Pw0,Qw0,Rw0,Y,Z,Tw0l subject to equations 29ð Þ and 30ð Þ
ð31Þ When the time-varying delay h(k) reduces to a constant h satisfying 0 ( h1( h ( h2, Theorem 1 can
be specialized to the following corollary by modifying the Lyapunov function V(k) with V3(k) dismissed and
V2(k) changed to V2ð Þ~k Pk{1i~kzh1{h2z
Tð ÞQz ii ð Þ. The proof is a reduced version of that for Theorem 1. Corollary 1. The system (5) with h(k) 5 h is asymp-totically stable for 0 ( h1( h ( h2, if there exist
matrices Pw0, Qw0, Rw0, Y, and Z, all with the size (h1+ 1)n6(h1+ 1)n, such that
w11z ~AATP ~Azt^A AATR ^AA w12z ~AATP ~AAdzt^AATR ~AAd tY w22z ~AAdTP ~AAdzt~AATdR ~AAd tZ {tR 2 6 6 4 3 7 7 5 <0 ð32Þ
For the uncertain system (1) with a time-varying delay satisfying equation (2), the augmented system z kz1ð Þ~Az kð ÞzAdz kzhð 1{h kð ÞÞ ð33Þ will be discussed, where
A~ AzDA kð Þ 0 0 0 I 0 0 0 0 I 0 0 .. . .. . P .. . .. . 0 0 I 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ~ A 0 0 0 I 0 0 0 0 I 0 0 .. . .. . P .. . .. . 0 0 I 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 z D 0 0 .. . 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 F kð Þ E½ a 0 0 0~~AAz ~DDF kð Þ~EEa Ad~ 0 0 0 AdzDAdð Þk 0 0 0 0 0 0 0 0 .. . .. . P .. . .. . 0 0 0 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ~ 0 0 0 Ad 0 0 0 0 0 0 0 0 .. . .. . P .. . .. . 0 0 0 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 z D 0 0 .. . 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 F kð Þ 0 0½ 0 Ed ~ ~AAdz ~DDF kð Þ~EEd
The problem is still to find how large h2 can be for
equation (1) to remain asymptotically stable. By adapting Theorem 1 to the system (33) and with the help of Lemma 1, the following result holds.
Theorem 2. The system (1) with parametric un-certainties (3) and (4) is asymptotically stable for 0 ( h1( h(k) ( h2 if there exist matrices
Pw0, Qw0, Rw0, Y, and Z, all with the size (h1+ 1)n6(h1+ 1)n, such that {P 0 0 P ~AA P ~AAd 0 P ~DD 1 {t{1R 0 R ^AA R ~AA d 0 R ~DD 1 1 {R 0 0 0 0 1 1 1 w11ztQz~EETaEE~a w12z~EETaEE~d tY 0 1 1 1 1 w22z~EET dEE~d tZ 0 1 1 1 1 1 {tR 0 1 1 1 1 1 1 {I 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 v0
Proof. By the Schur complement, equation (8) is equivalent to S~ {P 0 0 P ~AA P ~AAd 0 1 {t{1R 0 R ^AA R ~AA d 0 1 1 {R 0 0 0 1 1 1 w11ztQ w12 tY 1 1 1 1 w22 tZ 1 1 1 1 1 {tR 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 v0 ð35Þ By replacing A,in equation (35) with A¯ 5 A,+ D,F(k)E,a
and A,dwith A¯d5A
,
d+ D
,
F(k)E,d, a sufficient condition
for the stability of the augmented system (33) with uncertainties is found to be Sz P ~DD R ~DD 0 0 0 0 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 F kð Þ 0 0 0 ~EEa EE~d 0 z 0 0 0 ~ E ET a ~ E ETd 0 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 FTð Þ ~k DDTP DD~TR 0 0 0 0v0 ð36Þ
By Lemma 1, equation (36) is guaranteed for equation (4) if there exists a scalar e . 0 such that
eSze2 P ~DD R ~DD 0 0 0 0 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 ~ D DTP DD~TR 0 0 0 0 z 0 0 0 ~ E ETa ~ E ETd 0 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 0 0 0 EE~a EE~d 0 v0 ð37Þ
With eP, eQ, eR, eY, and eZ replaced by P, Q, R, Y, and Z, respectively, it is seen that equation (37) is equivalent to equation (34) by the Schur complement. This completes the proof.
Table 1 Interval time delay stability of equation (5) 0 ( h1( h(k) ( h2 By Theorem 1 0 ( h(k) ( 10, 5 ( h(k) ( 11, 7 ( h(k) ( 12, 9 ( h(k) ( 13 By Lemma 2 of [20] 0 ( h(k) ( 8, 5 ( h(k) ( 11, 8 ( h(k) ( 12, 10 ( h(k) ( 13 By Corollary 1 of [20] 0 ( h(k) ( 8 By Lemma 3 of [20] 3 ( h(k) ( 10, 5 ( h(k) ( 11, 7 ( h(k) ( 12, 9 ( h(k) ( 13
Table 2 Interval time delay stability of equation (1) 0 ( h1( h(k) ( h2 By Theorem 2 0 ( h(k) ( 5, 4 ( h(k) ( 6, 6 ( h(k) ( 7 By Theorem 1 of [20] 0 ( h(k) ( 4, 3 ( h(k) ( 5, 5 ( h(k) ( 6 By Theorem 2 of [20] 2 ( h(k) ( 5, 4 ( h(k) ( 6 By Theorem 3 of [20] 3 ( h(k) ( 5, 4 ( h(k) ( 6 (34)
4 NUMERICAL EXAMPLES
Example 1 [20]. Consider the uncertain discrete-time system (1) with a discrete-time-varying delay (2) and
A~ 0:8 0 0 0:97 " # ,Ad~ {0:1 0 {0:1 {0:1 " # ,D~ 0:1 0 0 0:02 " # , Ea~ 1 0 0 1 " # ,Ed~ 0:5 0 0 0:5 " #
Note that the nominal system (5) with the above A and Adis asymptotically stable when the time delay
is zero. For comparison purposes, several values of h1 are found which make all eigenvalues of
equation (7) with h(k) ; h1 be located inside the
unit circle of the complex plane. To each h1 the
corresponding h2 is determined to guarantee the
asymptotic stability of the above system. By utiliz-ing the LMI Control Toolbox of Matlab [28] to implement equation (31), several time delay inter-vals for the case without parametric uncertainties (F(k) ; 0) are computed and compared with those from Lemma 2, Corollary 1, and Lemma 3 of [20]. The comparison is shown in Table 1, where it is seen that less conservative results are obtained here. Then by applying Theorem 2 to the case with parametric uncertainties, several time delay inter-vals are computed and compared with those from Theorem 1, Theorem 2, and Theorem 3 of [20]. The comparison is shown in Table 2, and less conser-vative results are also obtained here. Finally, consider the nominal system (5) with a constant time delay h satisfying 0 ( h1( h ( h2. By setting
h150 and applying Corollary 1, a time delay range
0 ( h ( 16 for asymptotic stability is obtained. Example 2 [21]. Consider the nominal discrete-time system (5) with discrete-time-varying delay (2) and
A~ 0:6 0 0:35 0:7 ,Ad~ 0:1 0 0:2 0:1
The system (5) with the above A and Ad is also
asymptotically stable when the time delay is zero. For h152, equation (31) gives an upper bound
h2515 on the time delay to ensure the system
stability, whereas Theorem 1 and Theorem 2 of [21] give h2510 and h2513, respectively. Therefore,
Theorem 1 is less conservative for this example than the existing method.
Example 3. Consider the nominal discrete-time system (5) with time-varying delay (2) and
A~ 0:8 {0:1 0:8 0:7 ,Ad~ 0:3 0 0:8 0:1
Note that the system (5) is unstable with the above A and Ad when the time delay is zero. However, for
h155, equation (31) gives a time delay interval
5 ( h(k) ( 6, and for h156 another interval
6 ( h(k) ( 7 that guarantee the asymptotic stability. Furthermore, for the system with a time-invariant delay h satisfying 0 ( h1( h ( h2 and various h1,
Corollary 1 generates the corresponding time delay intervals 3 ( h ( 4, 4 ( h ( 6, 5 ( h ( 8, 6 ( h ( 10, and 8 ( h ( 11, which together imply the system is asymptotically stable for 3 ( h ( 11.
5 CONCLUSIONS
Using the Lyapunov method and the LMI frame-work, this paper has proposed stability criteria for the asymptotic stability of uncertain and nominal unforced discrete-time systems with a time-varying state delay. Among other methods in the literature assuming the stability of time delay systems for zero delay, the basic assumption of this paper is the system is asymptotically stable for a possibly non-zero time delay. This makes the application more flexible and in some cases produces less conserva-tive results. The numerical examples provided illustrate the application and efficacy of the devel-oped results.
ACKNOWLEDGEMENT
This research is supported by the National Science Council of Taiwan, the Republic of China, under grant NSC 95-2221-E-002-130-MY3.
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