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Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities
Chaos, Solitons and Fractals 42 (2009) 1251–1257
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities Shyh-Feng Chen * Department of Electrical Engineering, China Institute of Technology, Taipei 11581, Taiwan, ROC
a r t i c l e
i n f o
Article history: Accepted 10 March 2009
a b s t r a c t The asymptotic stability problem for discrete-time systems with time-varying delay subject to saturation nonlinearities is addressed in this paper. In terms of linear matrix inequalities (LMIs), a delay-dependent sufficient condition is derived to ensure the asymptotic stability. A numerical example is given to demonstrate the theoretical results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In practical implementations, the values of transfer-function coefficients are stored in registers with the finite wordlength. It is well known that discrete-time systems implemented in finite wordlength format are intrinsic saturation nonlinearities [1]. As a result, the stable discrete-time systems may become the unstable discrete-time systems due to such nonlinearities. During the last decades, the stability problem for discrete-time systems subject to saturation nonlinearities has been investigated in the literature [2–13]. As is well known, time delays are frequently encountered in various practical engineering systems such as chemical systems, power systems, and networked control systems. Delays are often a source of instability and poor performance. Therefore, various issues of time-delay systems have been reported in the literature. See, e.g. [14–20], and references cited therein. It is worth noting that most of the existing results for this topic only deal with discrete-time state-delayed systems without saturation nonlinearity. In many practical applications, however, saturation nonlinearities may appear in discrete-time state-delayed systems when its transfer function is implemented by a state-space model with the finite wordlength format. Recently, the problems of stability analysis for discrete-time systems with constant delay subject to saturation nonlinearities has been addressed [21]. However, the proposed condition in [21] was delay-independent results, rather than delay-dependent results. It is well known that the delay-dependent condition may offer less conservative results than the delay-independent counterparts, especially when the delays are small. To the best of the author’s knowledge, the delay-dependent stability problems for discrete-time systems with time-varying delays subject to saturation nonlinearities has not been adequately investigated and still remains challenging task. The purpose of the paper is as follows. New LMI-based [22] sufficient condition is derived to ensure the asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities. The obtained condition depends not only on the difference between the upper and lower delay bounds but also on the upper delay bounds of the time-varying delays, which is less conservative than that of the delay-independent stability condition for discrete-time systems with constant delay subject to saturation nonlinearities. The notation used throughout the paper is quite standard. Rn is the n-dimensional Euclidean space, and Rnm is the set of n m real matrices. PT stands for the transpose of a matrix P, and P > 0 (<0) means that the symmetric matrix P is positive (negative) definite. The boldface characters represent matrix variables, and H is used as an ellipsis for the terms that are implied by symmetry. * Fax: +886 2 26534518. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.026
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S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257
2. Problem formulation Consider the discrete-time systems with time-varying delay subject to saturation nonlinearities described by
xðk þ 1Þ ¼ gðfðkÞÞ ¼ ½ g 1 ðf1 ðkÞÞ g 2 ðf2 ðkÞÞ g n ðfn ðkÞÞ
T
T
fðkÞ ¼ ½ f1 ðkÞ f2 ðkÞ fn ðkÞ ¼ AxðkÞ þ Ad xðk dðkÞÞ xðkÞ ¼ uðkÞ; n
nn
where x 2 R is the state vector, A; Ad 2 R varying delay satisfying
8 k ¼ dH ; dH þ 1; . . . ; 0;
ð1Þ
are system matrix with compatible dimensions. The state delay dðkÞ is a time-
dL 6 dðkÞ 6 dH ;
ð2Þ
where dL and dH denote the lower and upper delay bounds, respectively. uðkÞ; k ¼ dH ; dH þ 1; . . . ; 0, is a given initial condition sequence. We assume the saturation nonlinearities given by
9 8 fi ðkÞ > 1 > > = < 1; g i ðfi ðkÞÞ ¼ fi ðkÞ; 1 6 fi ðkÞ 6 1 ; > > ; : 1; fi ðkÞ < 1
8 i ¼ 1; 2; . . . ; n:
ð3Þ
The following definition will be used later. Definition 1 [23,24]. A square matrix P ¼
pii P
n X
jpij j;
pij 2 Rnm is called diagonally dominant matrix if
8 ı ¼ 1; 2; . . . ; n:
j–i
3. Main result In this section, a delay-dependent sufficient condition for the asymptotic stability of systems (1)–(3) is presented. Theorem 2. Given positive integers dL and dH , if there exist matrices 0 < P 2 Rnn , 0 < Q 2 Rnn , 0 < W 2 Rnn , R1 R2 2 R2n2n and diagonally dominant matrices M ¼ MT ¼ mij 2 Rnn , N ¼ nij 2 Rnn such that 0
2
3
W11
H
H
H
H
H
6 AT M 6 d
Q
H
H
H
H 7 7 7 H 7 7 < 0; H 7 7 7 H 5
6 6 R3 W¼6 6 NT A 6 6 4 W51
0
W R3
H
H
N T Ad
0
P ðN þ NT Þ
H
MAd
0
0
dH R3 2M
R2
0
0
0 R2 n X mii P jmij j;
2
ð4Þ
R1 ð5Þ
j–i
nii P
n X
jnij j;
ð6Þ
i–j
hold for i ¼ 1; 2; . . . ; n, where 2
W11 ¼ P þ ð1 þ dH dL ÞQ þ W þ dH R1 R3 þ ðA In ÞT M þ MðA In Þ; W51 ¼ MðA 2In Þ þ
2 dH RT2 ;
ð7Þ
then system (1)–(3) is asymptotically stable for any time-varying delay dðkÞ 2 ½dL ; dH . Proof. Define the following Lyapunov functional candidate as
mðkÞ ¼
5 X
ms ðkÞ;
s¼1
where
m1 ðkÞ ¼ xT ðkÞPxðkÞ; m2 ðkÞ ¼
k1 X t¼kdðkÞ
xT ðtÞQ xðtÞ;
ð8Þ
1253
S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257 k1 X
m3 ðkÞ ¼
xT ðtÞWxðtÞ;
t¼kdH
m4 ðkÞ ¼
dL k1 X X
xT ðtÞQ xðtÞ;
s¼dH t¼kþs 1 k1 X X
dT ðtÞRdðtÞ; s¼dH t¼kþs dT ðtÞ ¼ xT ðtÞ gT ðtÞ ;
ð10Þ
gðtÞ ¼ xðt þ 1Þ xðtÞ;
ð11Þ
m5 ðkÞ ¼ dH
and P, Q , W, R ¼
R1 RT2
R2 R3
ð9Þ
are all positive definite matrices. By evaluating the first-forward difference Dmi ðkÞ ¼ mi ðk þ 1Þ
mi ðkÞði ¼ 1; 2; . . . ; 5Þ along the trajectories of system (1), one has Dm1 ðkÞ ¼ g T ðfðkÞÞPgðfðkÞÞ xT ðkÞPxðkÞ; k X
Dm2 ðkÞ ¼
k1 X
xT ðtÞQ xðtÞ
t¼kþ1dðkþ1Þ
xT ðtÞQ xðtÞ;
t¼kdðkÞ k1 X
¼ xT ðkÞQ xðkÞ xT ðk dðkÞÞQ xðk dðkÞÞ þ
t¼kþ1dðkþ1Þ k1 X
6 xT ðkÞQ xðkÞ xT ðk dðkÞÞQ xðk dðkÞÞ þ
xT ðtÞQ xðtÞ
kd XL
T
¼ x ðkÞQ xðkÞ x ðk dðkÞÞQ xðk dðkÞÞ þ
xT ðtÞQ xðtÞ;
t¼kþ1dðkÞ
t¼kþ1dH T
k1 X
xT ðtÞQ xðtÞ k1 X
xT ðtÞQ xðtÞ;
t¼kþ1dL
ð12Þ
T
x ðtÞQ xðtÞ;
t¼kþ1dH
Dm3 ðkÞ ¼ xT ðkÞWxðkÞ xT ðk dH ÞWxðk dH Þ; kd XL
Dm4 ðkÞ ¼ ðdH dL ÞxT ðkÞQ xðkÞ
xT ðtÞQ xðtÞ;
t¼kþ1dH k1 X
2
Dm5 ðkÞ ¼ dH dT ðkÞRdðkÞ dH
dT ðtÞRdðtÞ:
t¼kdH
For any vectors y 2 Rn ; z 2 Rn and matrix 0 < R 2 Rnn , we have
yT Ry zT Rz 6 yT Rz zT Ry:
ð13Þ
Then, it is easy to verify that
dH
k1 X
k1 X
dT ðtÞRdðtÞ ¼
t¼kdH
dT ðtÞRdðtÞ ðdH 1Þ
t¼kdH k1 X
6
k1 X
dT ðtÞRdðtÞ
t¼kdH k2 k1 X X
T
d ðtÞRdðtÞ
t¼kdH
T
"
T
d ðlÞRdðrÞ þ d ðrÞRdðlÞ ¼
l¼kdH r¼lþ1
k1 X
t¼kdH
# " T
d ðtÞ R
k1 X
# dðtÞ :
ð14Þ
t¼kdH
Hence,
"
Dm5 ðkÞ 6
2 dH dT ðkÞRdðkÞ
k1 X
# " T
d ðtÞ R
t¼kdH
¼
2 dH xT ðkÞR1 xðkÞ
þ
# dðtÞ
t¼kdH
2 2dH xT ðkÞR2
" 2½xðkÞ xðk dH ÞT RT2
k1 X
gðkÞ þ
k1 X
t¼kdH
2 dH
#
" T
g ðkÞR3 gðkÞ
k1 X t¼kdH
# T
x ðtÞ R1
"
k1 X
# xðtÞ
t¼kdH
xðtÞ ½xðkÞ xðk dH ÞT R3 ½xðkÞ xðk dH Þ:
ð15Þ
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S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257
For the sake of simplicity, we give the following definitions and identities:
a1 ¼ fT ðkÞNgðfðkÞÞ þ g T ðfðkÞÞNT fðkÞ g T ðfðkÞÞðN þ NT ÞgðfðkÞÞ
T ¼ ½AxðkÞ þ Ad xðk dðkÞÞ NgðfðkÞÞ þ g T ðfðkÞÞNT ½AxðkÞ þ Ad xðk dðkÞÞ g T ðfðkÞÞ N þ NT gðfðkÞÞ
b1 ¼ ½fðkÞ xðkÞT MgðfðkÞÞ þ g T ðfðkÞÞM½fðkÞ xðkÞ gT ðkÞð2MÞgðfðkÞÞ b2 ¼ ½fðkÞ xðkÞT MxðkÞ xT ðkÞM½fðkÞ xðkÞ þ gT ðkÞð2MÞxðkÞ ¼ ½ðA IÞxðkÞ þ Ad xðk dðkÞÞT MxðkÞ xT ðkÞM½ðA IÞxðkÞ þ Ad xðk dðkÞÞ þ gT ðkÞð2MÞxðkÞ
a2 ¼ b1 þ b2 ¼ ½fðkÞ xðkÞT MgðkÞ þ gT ðkÞM½fðkÞ xðkÞ gT ðkÞð2MÞgðkÞ ¼ ½ðA IÞxðkÞ þ Ad xðk dðkÞÞT MgðkÞ þ gT ðkÞM½ðA IÞxðkÞ þ Ad xðk dðkÞÞ gT ðkÞð2MÞgðkÞ;
ð16Þ
where M is a symmetric diagonally dominant matrix and N is a diagonally dominant matrix. It follows from (12) and (15) and (16) that
DmðkÞ ¼ m1 ðkÞ þ m2 ðkÞ þ m3 ðkÞ þ m4 ðkÞ þ m5 ðkÞ 6 g T ðfðkÞÞPgðfðkÞÞ xT ðkÞPxðkÞ þ xT ðkÞQ xðkÞ xT ðk dðkÞÞQ xðk dðkÞÞ þ xT ðkÞWxðkÞ 2
2
xT ðk dH ÞWxðk dH Þ þ ðdH dL ÞxT ðkÞQ xðkÞ þ dH xT ðkÞR1 xðkÞ þ 2dH xT ðkÞR2 gðkÞ " # " # " # k1 k1 k1 X X X 2 T T T T þ dH g ðkÞR3 gðkÞ x ðtÞ R1 xðtÞ 2½xðkÞ xðk dH Þ R2 xðtÞ t¼kdH
t¼kdH
t¼kdH
½xðkÞ xðk dH ÞT R3 ½xðkÞ xðk dH Þ ¼ hT ðkÞUhðkÞ ¼ hT ðkÞUhðkÞ þ a1 a1 þ a2 b1 b2 ¼ hT ðkÞWhðkÞ a1 b1 ;
ð17Þ
where
" 2
k1 P
xT ðkÞ xT ðk dðkÞÞ xT ðk dH Þ g T ðfðkÞÞ gT ðkÞ
hðkÞ ¼
U11
6 0 6 6 6 R3 U¼6 6 0 6 6 2 T 4 dH R2 R2
H
H
H
H
Q
H
H
H
H 7 7 7 H 7 7; H 7 7 7 H 5
0
W R3
H
H
0 0
0 0
P 0
H 2 dH R3
0
R2
0
U11 ¼ P þ ð1 þ dH dL ÞQ þ W þ
0 2 dH R1
;
t¼kdH
3
H
#T xT ðtÞ
R1 R3 :
ð18Þ
Note that a1 and b1 can be expressed as follows:
a1 ¼
n X i¼1
b1 ¼
n X i¼1
(
"
2½fi ðkÞ g i ðfi ðkÞÞ nii g i ðfi ðkÞÞ þ (
"
n X
nij g j fj ðkÞ
#)
j–i
2½fi ðkÞ g i ðfi ðkÞÞ mii g i ðfi ðkÞÞ þ
n X
mij g j fj ðkÞ
; #) :
ð19Þ
j–i
By conditions (5) and (6) and the saturation nonlinearities (3), a1 and b1 are nonnegative. Therefore, it can be established that DmðkÞ < 0 for all hðkÞ – 0 if conditions (4)–(6) are satisfied. Then, the system (1)–(3) is asymptotically stable. This completes the proof of the Theorem 2. Remark 3. Note that conditions (4) depends not only on the difference between the upper and lower delay bounds but also on the upper delay bounds. Thus, Theorem 2 presents a delay-dependent stability condition for discrete-time systems with time-varying delay subject to saturation nonlinearities. Moreover, Theorem 2 is a standard LMI form, which can be easily solved by Matlab LMI toolbox [25]. Remark 4. With lower delay bounds dL given, the upper delay bounds dH can be obtained by iteratively solving the LMIs given in Theorem 2 with respect to dH . For constant delay case, the upper delay bounds are the same as lower delay bounds (i.e., dH ¼ dL ¼ d). In this case, the system (1) reduce to the following system:
S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257
1255
xðk þ 1Þ ¼ gðAxðkÞ þ Ad xðk dÞÞ; xðkÞ ¼ uðkÞ;
8 k ¼ d; d þ 1; . . . ; 0;
ð20Þ
where d is positive integer denoting time delay. gðÞ are defined in (3). Then Theorem 2 can be reduced to the following Corollary. if there exist matrices 0 < P 2 Rnn , 0 < W 2 Rnn , 0 < R ¼ R1T R2 2 R2n2n , and Corollary 5. Given positive integer d, R2 R3 diagonally dominant matrices M ¼ MT ¼ mij 2 Rnn , N ¼ nij 2 Rnn such that
2
11
H
6R þ 6 3 6 6 NT A 6 6 6 4 41
ATd M
H
W R3
T
N Ad
H
P NþN
T
H
H
H 7 7 7 7 7 < 0; 7 7 H 5
H
MAd
0
2 R3 2M d
R2
0
0
R2 n X mii P jmij j;
3
H
ð21Þ
R1 ð22Þ
j–i
nii P
n X
jnij j;
ð23Þ
j–i
hold for i ¼ 1; 2; . . . ; n; where
2 R1 R3 þ ðA In ÞT M þ MðA In Þ; 11 ¼ P þ W þ d 2 RT ; 41 ¼ MðA 2In Þ þ d
ð24Þ
2
then system (20) subject to saturation nonlinearities (3) is asymptotically stable for any delay d 2 ½0; d. Proof. Define the following Lyapunov functional candidate as
mðkÞ ¼ xT ðkÞPxðkÞ þ
k1 X
xT ðtÞWxðtÞ þ d
t¼kd
where dT ðtÞ ¼ xT ðtÞ
1 X k1 X t¼kþs s¼d
dT ðtÞRdðtÞ;
ð25Þ
R
R
gT ðtÞ , gðtÞ ¼ xðt þ 1Þ xðtÞ, P, W, and R ¼ R1T R2 are all positive definite matrices. Then the Cor3 2
ollary can be obtained by using similar lines as in the proof of Theorem 2.
In the delay-independent case, Corollary 5 may be reduced to following simpler results. Corollary 6. If there exist matrices 0 < P 2 Rnn , 0 < W 2 Rnn and diagonally dominant matrix N ¼ nij 2 Rnn such that
2
P þ W 6 0 4 T
H W T
3
H H T
7 5 < 0;
N Ad P ðN þ N Þ N A n X jnij j; 8 i ¼ 1; 2; . . . ; n; nii P
ð26Þ
ð27Þ
j–i
then system (20) subject to saturation nonlinearities (3) is asymptotically stable. Remark 7. Corollary 6 provides a delay-independent stability conditions for discrete-time systems with constant delay subject to saturation nonlinearity. Note that the diagonally dominant matrix N is not required to be positive definite matrix in Corollary 6. When the matrix N is set to be positive definite and diagonally dominant matrix and choose P ¼ N, Corollary 6 can reduced to Theorem 1 of [21] with single delay and no uncertainties in the system. With Ad ¼ 0 in (20), the system (20) reduce to the following system:
xðk þ 1Þ ¼ gðAxðkÞÞ: The saturation nonlinearities gðÞ are defined in (3). Then, we have the following Corollary. Corollary 8. If there exist matrix 0 < P 2 Rnn and diagonally dominant matrix N ¼ nij 2 Rnn such that
ð28Þ
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S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257
P
H
< 0; NT A P ðN þ NT Þ n X jnij j; 8 i ¼ 1; 2; . . . ; n; nii P
ð29Þ ð30Þ
j–i
then system (28) subject to saturation nonlinearity (3) is asymptotically stable. Remark 9. It should be noted that Corollary 8 is a special case of [2,3].
4. A numerical example Consider a discrete-time systems with time-varying state delays subject to saturation nonlinearities, where
A¼
0:92 0:5 0:21 0:9
;
Ad ¼
0:06
0:1
0:04
0:1
:
We assume that the lower delay bounds of dðkÞ is dL ¼ 2. Our purpose is to determine the upper delay bound dH such that the system (1)–(3) is asymptotically stable for any delay dðkÞ 2 ½dL ; dH . By using the LMI control toolbox to solve Theorem 2 as follows:
P¼
607:2355
95:6886
;
10:6583
3:2214
;
3:2214 17:8518 888:8193 888:0410 N¼ ; M¼ ; 52:4310 60:1377607 888:0410 888:5045 2 3 2:0502 1:9492 5:9034 5:9093 6 7 37:4312 33:4267 6 1:9492 1:8922 5:2180 5:2230 7 W¼ ; R¼6 7; 4 5:9034 5:2180 6:7030 6:7072 5 33:4267 31:6243
95:6886 562:9530
26:8607 128:9789
Q¼
5:9093 5:2230 6:7072 6:7170 we can find the upper delay bounds dH ¼ 12. It is noted that the delay-independent result in Corollary 6 and [21] is not feasible.
5. Conclusions This paper deals with the stability analysis for discrete-time systems with time-varying delay subject to saturation nonlinearities. Sufficient conditions in term of LMIs are derived to ensure that the system is asymptotically stable, which depends not only on the difference between the upper and lower delay bounds but also on the upper delay bounds of the time-varying state delays. An example is given to illustrate the usage of the proposed approach. Acknowledgment This research is supported by the National Science Council of Taiwan under Grant NSC 96-2221-E-157-009. References [1] Liu D, Michel AN. Dynamical systems with saturation nonlinearities. London: Springer-Verlag; 1994. [2] Singh V. Modified LMI condition for the realization of limit cycle-free digital filters using saturation arithmetic. Chaos, Solitions & Fractals 2007;32:1448–53. [3] Kar H. An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. Digital Signal Process 2007;17:685–9. [4] Kar H, Singh V. Stability analysis of 1-D and 2-D fixed-point state-space digital filters using any combination of overflow and quantization nonlinearities. IEEE Trans Signal Process 2001;49:1097–105. [5] Kar H, Singh V. Stability analysis of discrete-time systems in a state-space realization with partial state saturation nonlinearities. IEE Proc Control Theory Appl 2003;150:205–8. [6] Singh V. Stability analysis of discrete-time systems in a state-space realization with state saturation nonlinearities: linear matrix inequality approach. IEE Proc Control Theory Appl 2005;152:9–12. [7] Kar H, Singh V. Elimination of overflow oscillations in fixed-point state-space digital filters with saturation arithmetic: an LMI approach. IEEE Trans Circuits Syst II 2004;51:40–2. [8] Liu D, Michel AN. Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters. IEEE Trans Circuits Syst I 1992;39:798–807. [9] Liu D, Michel AN. Stability analysis of state-space realizations for two-dimensional filters with overflow nonlinearities. IEEE Trans Circuits Syst I 1994;41:127–37. [10] Liu D, Molchanov A. Asymptotic stability of a class of linear discrete systems with multiple independent variables. Circuits Syst Signal Process 2003;22:307–24.
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities Shyh-Feng Chen * Department of Electrical Engineering, China Institute of Technology, Taipei 11581, Taiwan, ROC
a r t i c l e
i n f o
Article history: Accepted 10 March 2009
a b s t r a c t The asymptotic stability problem for discrete-time systems with time-varying delay subject to saturation nonlinearities is addressed in this paper. In terms of linear matrix inequalities (LMIs), a delay-dependent sufficient condition is derived to ensure the asymptotic stability. A numerical example is given to demonstrate the theoretical results. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction In practical implementations, the values of transfer-function coefficients are stored in registers with the finite wordlength. It is well known that discrete-time systems implemented in finite wordlength format are intrinsic saturation nonlinearities [1]. As a result, the stable discrete-time systems may become the unstable discrete-time systems due to such nonlinearities. During the last decades, the stability problem for discrete-time systems subject to saturation nonlinearities has been investigated in the literature [2–13]. As is well known, time delays are frequently encountered in various practical engineering systems such as chemical systems, power systems, and networked control systems. Delays are often a source of instability and poor performance. Therefore, various issues of time-delay systems have been reported in the literature. See, e.g. [14–20], and references cited therein. It is worth noting that most of the existing results for this topic only deal with discrete-time state-delayed systems without saturation nonlinearity. In many practical applications, however, saturation nonlinearities may appear in discrete-time state-delayed systems when its transfer function is implemented by a state-space model with the finite wordlength format. Recently, the problems of stability analysis for discrete-time systems with constant delay subject to saturation nonlinearities has been addressed [21]. However, the proposed condition in [21] was delay-independent results, rather than delay-dependent results. It is well known that the delay-dependent condition may offer less conservative results than the delay-independent counterparts, especially when the delays are small. To the best of the author’s knowledge, the delay-dependent stability problems for discrete-time systems with time-varying delays subject to saturation nonlinearities has not been adequately investigated and still remains challenging task. The purpose of the paper is as follows. New LMI-based [22] sufficient condition is derived to ensure the asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities. The obtained condition depends not only on the difference between the upper and lower delay bounds but also on the upper delay bounds of the time-varying delays, which is less conservative than that of the delay-independent stability condition for discrete-time systems with constant delay subject to saturation nonlinearities. The notation used throughout the paper is quite standard. Rn is the n-dimensional Euclidean space, and Rnm is the set of n m real matrices. PT stands for the transpose of a matrix P, and P > 0 (<0) means that the symmetric matrix P is positive (negative) definite. The boldface characters represent matrix variables, and H is used as an ellipsis for the terms that are implied by symmetry. * Fax: +886 2 26534518. E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.026
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S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257
2. Problem formulation Consider the discrete-time systems with time-varying delay subject to saturation nonlinearities described by
xðk þ 1Þ ¼ gðfðkÞÞ ¼ ½ g 1 ðf1 ðkÞÞ g 2 ðf2 ðkÞÞ g n ðfn ðkÞÞ
T
T
fðkÞ ¼ ½ f1 ðkÞ f2 ðkÞ fn ðkÞ ¼ AxðkÞ þ Ad xðk dðkÞÞ xðkÞ ¼ uðkÞ; n
nn
where x 2 R is the state vector, A; Ad 2 R varying delay satisfying
8 k ¼ dH ; dH þ 1; . . . ; 0;
ð1Þ
are system matrix with compatible dimensions. The state delay dðkÞ is a time-
dL 6 dðkÞ 6 dH ;
ð2Þ
where dL and dH denote the lower and upper delay bounds, respectively. uðkÞ; k ¼ dH ; dH þ 1; . . . ; 0, is a given initial condition sequence. We assume the saturation nonlinearities given by
9 8 fi ðkÞ > 1 > > = < 1; g i ðfi ðkÞÞ ¼ fi ðkÞ; 1 6 fi ðkÞ 6 1 ; > > ; : 1; fi ðkÞ < 1
8 i ¼ 1; 2; . . . ; n:
ð3Þ
The following definition will be used later. Definition 1 [23,24]. A square matrix P ¼
pii P
n X
jpij j;
pij 2 Rnm is called diagonally dominant matrix if
8 ı ¼ 1; 2; . . . ; n:
j–i
3. Main result In this section, a delay-dependent sufficient condition for the asymptotic stability of systems (1)–(3) is presented. Theorem 2. Given positive integers dL and dH , if there exist matrices 0 < P 2 Rnn , 0 < Q 2 Rnn , 0 < W 2 Rnn , R1 R2 2 R2n2n and diagonally dominant matrices M ¼ MT ¼ mij 2 Rnn , N ¼ nij 2 Rnn such that 0
2
3
W11
H
H
H
H
H
6 AT M 6 d
Q
H
H
H
H 7 7 7 H 7 7 < 0; H 7 7 7 H 5
6 6 R3 W¼6 6 NT A 6 6 4 W51
0
W R3
H
H
N T Ad
0
P ðN þ NT Þ
H
MAd
0
0
dH R3 2M
R2
0
0
0 R2 n X mii P jmij j;
2
ð4Þ
R1 ð5Þ
j–i
nii P
n X
jnij j;
ð6Þ
i–j
hold for i ¼ 1; 2; . . . ; n, where 2
W11 ¼ P þ ð1 þ dH dL ÞQ þ W þ dH R1 R3 þ ðA In ÞT M þ MðA In Þ; W51 ¼ MðA 2In Þ þ
2 dH RT2 ;
ð7Þ
then system (1)–(3) is asymptotically stable for any time-varying delay dðkÞ 2 ½dL ; dH . Proof. Define the following Lyapunov functional candidate as
mðkÞ ¼
5 X
ms ðkÞ;
s¼1
where
m1 ðkÞ ¼ xT ðkÞPxðkÞ; m2 ðkÞ ¼
k1 X t¼kdðkÞ
xT ðtÞQ xðtÞ;
ð8Þ
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S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257 k1 X
m3 ðkÞ ¼
xT ðtÞWxðtÞ;
t¼kdH
m4 ðkÞ ¼
dL k1 X X
xT ðtÞQ xðtÞ;
s¼dH t¼kþs 1 k1 X X
dT ðtÞRdðtÞ; s¼dH t¼kþs dT ðtÞ ¼ xT ðtÞ gT ðtÞ ;
ð10Þ
gðtÞ ¼ xðt þ 1Þ xðtÞ;
ð11Þ
m5 ðkÞ ¼ dH
and P, Q , W, R ¼
R1 RT2
R2 R3
ð9Þ
are all positive definite matrices. By evaluating the first-forward difference Dmi ðkÞ ¼ mi ðk þ 1Þ
mi ðkÞði ¼ 1; 2; . . . ; 5Þ along the trajectories of system (1), one has Dm1 ðkÞ ¼ g T ðfðkÞÞPgðfðkÞÞ xT ðkÞPxðkÞ; k X
Dm2 ðkÞ ¼
k1 X
xT ðtÞQ xðtÞ
t¼kþ1dðkþ1Þ
xT ðtÞQ xðtÞ;
t¼kdðkÞ k1 X
¼ xT ðkÞQ xðkÞ xT ðk dðkÞÞQ xðk dðkÞÞ þ
t¼kþ1dðkþ1Þ k1 X
6 xT ðkÞQ xðkÞ xT ðk dðkÞÞQ xðk dðkÞÞ þ
xT ðtÞQ xðtÞ
kd XL
T
¼ x ðkÞQ xðkÞ x ðk dðkÞÞQ xðk dðkÞÞ þ
xT ðtÞQ xðtÞ;
t¼kþ1dðkÞ
t¼kþ1dH T
k1 X
xT ðtÞQ xðtÞ k1 X
xT ðtÞQ xðtÞ;
t¼kþ1dL
ð12Þ
T
x ðtÞQ xðtÞ;
t¼kþ1dH
Dm3 ðkÞ ¼ xT ðkÞWxðkÞ xT ðk dH ÞWxðk dH Þ; kd XL
Dm4 ðkÞ ¼ ðdH dL ÞxT ðkÞQ xðkÞ
xT ðtÞQ xðtÞ;
t¼kþ1dH k1 X
2
Dm5 ðkÞ ¼ dH dT ðkÞRdðkÞ dH
dT ðtÞRdðtÞ:
t¼kdH
For any vectors y 2 Rn ; z 2 Rn and matrix 0 < R 2 Rnn , we have
yT Ry zT Rz 6 yT Rz zT Ry:
ð13Þ
Then, it is easy to verify that
dH
k1 X
k1 X
dT ðtÞRdðtÞ ¼
t¼kdH
dT ðtÞRdðtÞ ðdH 1Þ
t¼kdH k1 X
6
k1 X
dT ðtÞRdðtÞ
t¼kdH k2 k1 X X
T
d ðtÞRdðtÞ
t¼kdH
T
"
T
d ðlÞRdðrÞ þ d ðrÞRdðlÞ ¼
l¼kdH r¼lþ1
k1 X
t¼kdH
# " T
d ðtÞ R
k1 X
# dðtÞ :
ð14Þ
t¼kdH
Hence,
"
Dm5 ðkÞ 6
2 dH dT ðkÞRdðkÞ
k1 X
# " T
d ðtÞ R
t¼kdH
¼
2 dH xT ðkÞR1 xðkÞ
þ
# dðtÞ
t¼kdH
2 2dH xT ðkÞR2
" 2½xðkÞ xðk dH ÞT RT2
k1 X
gðkÞ þ
k1 X
t¼kdH
2 dH
#
" T
g ðkÞR3 gðkÞ
k1 X t¼kdH
# T
x ðtÞ R1
"
k1 X
# xðtÞ
t¼kdH
xðtÞ ½xðkÞ xðk dH ÞT R3 ½xðkÞ xðk dH Þ:
ð15Þ
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S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257
For the sake of simplicity, we give the following definitions and identities:
a1 ¼ fT ðkÞNgðfðkÞÞ þ g T ðfðkÞÞNT fðkÞ g T ðfðkÞÞðN þ NT ÞgðfðkÞÞ
T ¼ ½AxðkÞ þ Ad xðk dðkÞÞ NgðfðkÞÞ þ g T ðfðkÞÞNT ½AxðkÞ þ Ad xðk dðkÞÞ g T ðfðkÞÞ N þ NT gðfðkÞÞ
b1 ¼ ½fðkÞ xðkÞT MgðfðkÞÞ þ g T ðfðkÞÞM½fðkÞ xðkÞ gT ðkÞð2MÞgðfðkÞÞ b2 ¼ ½fðkÞ xðkÞT MxðkÞ xT ðkÞM½fðkÞ xðkÞ þ gT ðkÞð2MÞxðkÞ ¼ ½ðA IÞxðkÞ þ Ad xðk dðkÞÞT MxðkÞ xT ðkÞM½ðA IÞxðkÞ þ Ad xðk dðkÞÞ þ gT ðkÞð2MÞxðkÞ
a2 ¼ b1 þ b2 ¼ ½fðkÞ xðkÞT MgðkÞ þ gT ðkÞM½fðkÞ xðkÞ gT ðkÞð2MÞgðkÞ ¼ ½ðA IÞxðkÞ þ Ad xðk dðkÞÞT MgðkÞ þ gT ðkÞM½ðA IÞxðkÞ þ Ad xðk dðkÞÞ gT ðkÞð2MÞgðkÞ;
ð16Þ
where M is a symmetric diagonally dominant matrix and N is a diagonally dominant matrix. It follows from (12) and (15) and (16) that
DmðkÞ ¼ m1 ðkÞ þ m2 ðkÞ þ m3 ðkÞ þ m4 ðkÞ þ m5 ðkÞ 6 g T ðfðkÞÞPgðfðkÞÞ xT ðkÞPxðkÞ þ xT ðkÞQ xðkÞ xT ðk dðkÞÞQ xðk dðkÞÞ þ xT ðkÞWxðkÞ 2
2
xT ðk dH ÞWxðk dH Þ þ ðdH dL ÞxT ðkÞQ xðkÞ þ dH xT ðkÞR1 xðkÞ þ 2dH xT ðkÞR2 gðkÞ " # " # " # k1 k1 k1 X X X 2 T T T T þ dH g ðkÞR3 gðkÞ x ðtÞ R1 xðtÞ 2½xðkÞ xðk dH Þ R2 xðtÞ t¼kdH
t¼kdH
t¼kdH
½xðkÞ xðk dH ÞT R3 ½xðkÞ xðk dH Þ ¼ hT ðkÞUhðkÞ ¼ hT ðkÞUhðkÞ þ a1 a1 þ a2 b1 b2 ¼ hT ðkÞWhðkÞ a1 b1 ;
ð17Þ
where
" 2
k1 P
xT ðkÞ xT ðk dðkÞÞ xT ðk dH Þ g T ðfðkÞÞ gT ðkÞ
hðkÞ ¼
U11
6 0 6 6 6 R3 U¼6 6 0 6 6 2 T 4 dH R2 R2
H
H
H
H
Q
H
H
H
H 7 7 7 H 7 7; H 7 7 7 H 5
0
W R3
H
H
0 0
0 0
P 0
H 2 dH R3
0
R2
0
U11 ¼ P þ ð1 þ dH dL ÞQ þ W þ
0 2 dH R1
;
t¼kdH
3
H
#T xT ðtÞ
R1 R3 :
ð18Þ
Note that a1 and b1 can be expressed as follows:
a1 ¼
n X i¼1
b1 ¼
n X i¼1
(
"
2½fi ðkÞ g i ðfi ðkÞÞ nii g i ðfi ðkÞÞ þ (
"
n X
nij g j fj ðkÞ
#)
j–i
2½fi ðkÞ g i ðfi ðkÞÞ mii g i ðfi ðkÞÞ þ
n X
mij g j fj ðkÞ
; #) :
ð19Þ
j–i
By conditions (5) and (6) and the saturation nonlinearities (3), a1 and b1 are nonnegative. Therefore, it can be established that DmðkÞ < 0 for all hðkÞ – 0 if conditions (4)–(6) are satisfied. Then, the system (1)–(3) is asymptotically stable. This completes the proof of the Theorem 2. Remark 3. Note that conditions (4) depends not only on the difference between the upper and lower delay bounds but also on the upper delay bounds. Thus, Theorem 2 presents a delay-dependent stability condition for discrete-time systems with time-varying delay subject to saturation nonlinearities. Moreover, Theorem 2 is a standard LMI form, which can be easily solved by Matlab LMI toolbox [25]. Remark 4. With lower delay bounds dL given, the upper delay bounds dH can be obtained by iteratively solving the LMIs given in Theorem 2 with respect to dH . For constant delay case, the upper delay bounds are the same as lower delay bounds (i.e., dH ¼ dL ¼ d). In this case, the system (1) reduce to the following system:
S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257
1255
xðk þ 1Þ ¼ gðAxðkÞ þ Ad xðk dÞÞ; xðkÞ ¼ uðkÞ;
8 k ¼ d; d þ 1; . . . ; 0;
ð20Þ
where d is positive integer denoting time delay. gðÞ are defined in (3). Then Theorem 2 can be reduced to the following Corollary. if there exist matrices 0 < P 2 Rnn , 0 < W 2 Rnn , 0 < R ¼ R1T R2 2 R2n2n , and Corollary 5. Given positive integer d, R2 R3 diagonally dominant matrices M ¼ MT ¼ mij 2 Rnn , N ¼ nij 2 Rnn such that
2
11
H
6R þ 6 3 6 6 NT A 6 6 6 4 41
ATd M
H
W R3
T
N Ad
H
P NþN
T
H
H
H 7 7 7 7 7 < 0; 7 7 H 5
H
MAd
0
2 R3 2M d
R2
0
0
R2 n X mii P jmij j;
3
H
ð21Þ
R1 ð22Þ
j–i
nii P
n X
jnij j;
ð23Þ
j–i
hold for i ¼ 1; 2; . . . ; n; where
2 R1 R3 þ ðA In ÞT M þ MðA In Þ; 11 ¼ P þ W þ d 2 RT ; 41 ¼ MðA 2In Þ þ d
ð24Þ
2
then system (20) subject to saturation nonlinearities (3) is asymptotically stable for any delay d 2 ½0; d. Proof. Define the following Lyapunov functional candidate as
mðkÞ ¼ xT ðkÞPxðkÞ þ
k1 X
xT ðtÞWxðtÞ þ d
t¼kd
where dT ðtÞ ¼ xT ðtÞ
1 X k1 X t¼kþs s¼d
dT ðtÞRdðtÞ;
ð25Þ
R
R
gT ðtÞ , gðtÞ ¼ xðt þ 1Þ xðtÞ, P, W, and R ¼ R1T R2 are all positive definite matrices. Then the Cor3 2
ollary can be obtained by using similar lines as in the proof of Theorem 2.
In the delay-independent case, Corollary 5 may be reduced to following simpler results. Corollary 6. If there exist matrices 0 < P 2 Rnn , 0 < W 2 Rnn and diagonally dominant matrix N ¼ nij 2 Rnn such that
2
P þ W 6 0 4 T
H W T
3
H H T
7 5 < 0;
N Ad P ðN þ N Þ N A n X jnij j; 8 i ¼ 1; 2; . . . ; n; nii P
ð26Þ
ð27Þ
j–i
then system (20) subject to saturation nonlinearities (3) is asymptotically stable. Remark 7. Corollary 6 provides a delay-independent stability conditions for discrete-time systems with constant delay subject to saturation nonlinearity. Note that the diagonally dominant matrix N is not required to be positive definite matrix in Corollary 6. When the matrix N is set to be positive definite and diagonally dominant matrix and choose P ¼ N, Corollary 6 can reduced to Theorem 1 of [21] with single delay and no uncertainties in the system. With Ad ¼ 0 in (20), the system (20) reduce to the following system:
xðk þ 1Þ ¼ gðAxðkÞÞ: The saturation nonlinearities gðÞ are defined in (3). Then, we have the following Corollary. Corollary 8. If there exist matrix 0 < P 2 Rnn and diagonally dominant matrix N ¼ nij 2 Rnn such that
ð28Þ
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S.-F. Chen / Chaos, Solitons and Fractals 42 (2009) 1251–1257
P
H
< 0; NT A P ðN þ NT Þ n X jnij j; 8 i ¼ 1; 2; . . . ; n; nii P
ð29Þ ð30Þ
j–i
then system (28) subject to saturation nonlinearity (3) is asymptotically stable. Remark 9. It should be noted that Corollary 8 is a special case of [2,3].
4. A numerical example Consider a discrete-time systems with time-varying state delays subject to saturation nonlinearities, where
A¼
0:92 0:5 0:21 0:9
;
Ad ¼
0:06
0:1
0:04
0:1
:
We assume that the lower delay bounds of dðkÞ is dL ¼ 2. Our purpose is to determine the upper delay bound dH such that the system (1)–(3) is asymptotically stable for any delay dðkÞ 2 ½dL ; dH . By using the LMI control toolbox to solve Theorem 2 as follows:
P¼
607:2355
95:6886
;
10:6583
3:2214
;
3:2214 17:8518 888:8193 888:0410 N¼ ; M¼ ; 52:4310 60:1377607 888:0410 888:5045 2 3 2:0502 1:9492 5:9034 5:9093 6 7 37:4312 33:4267 6 1:9492 1:8922 5:2180 5:2230 7 W¼ ; R¼6 7; 4 5:9034 5:2180 6:7030 6:7072 5 33:4267 31:6243
95:6886 562:9530
26:8607 128:9789
Q¼
5:9093 5:2230 6:7072 6:7170 we can find the upper delay bounds dH ¼ 12. It is noted that the delay-independent result in Corollary 6 and [21] is not feasible.
5. Conclusions This paper deals with the stability analysis for discrete-time systems with time-varying delay subject to saturation nonlinearities. Sufficient conditions in term of LMIs are derived to ensure that the system is asymptotically stable, which depends not only on the difference between the upper and lower delay bounds but also on the upper delay bounds of the time-varying state delays. An example is given to illustrate the usage of the proposed approach. Acknowledgment This research is supported by the National Science Council of Taiwan under Grant NSC 96-2221-E-157-009. References [1] Liu D, Michel AN. Dynamical systems with saturation nonlinearities. London: Springer-Verlag; 1994. [2] Singh V. Modified LMI condition for the realization of limit cycle-free digital filters using saturation arithmetic. Chaos, Solitions & Fractals 2007;32:1448–53. [3] Kar H. An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. Digital Signal Process 2007;17:685–9. [4] Kar H, Singh V. Stability analysis of 1-D and 2-D fixed-point state-space digital filters using any combination of overflow and quantization nonlinearities. IEEE Trans Signal Process 2001;49:1097–105. [5] Kar H, Singh V. Stability analysis of discrete-time systems in a state-space realization with partial state saturation nonlinearities. IEE Proc Control Theory Appl 2003;150:205–8. [6] Singh V. Stability analysis of discrete-time systems in a state-space realization with state saturation nonlinearities: linear matrix inequality approach. IEE Proc Control Theory Appl 2005;152:9–12. [7] Kar H, Singh V. Elimination of overflow oscillations in fixed-point state-space digital filters with saturation arithmetic: an LMI approach. IEEE Trans Circuits Syst II 2004;51:40–2. [8] Liu D, Michel AN. Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters. IEEE Trans Circuits Syst I 1992;39:798–807. [9] Liu D, Michel AN. Stability analysis of state-space realizations for two-dimensional filters with overflow nonlinearities. IEEE Trans Circuits Syst I 1994;41:127–37. [10] Liu D, Molchanov A. Asymptotic stability of a class of linear discrete systems with multiple independent variables. Circuits Syst Signal Process 2003;22:307–24.
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