Theorem 1
Applied Mathematics and Computation 208 (2009) 58–68
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays O.M. Kwon a,*, Ju H. Park b a b
School of Electrical and Computer Engineering, Chungbuk National University, 410 SungBong-Ro, Heungduk-gu, Cheongju 361-763, Republic of Korea Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea
a r t i c l e
i n f o
Keywords: Stabilization Interval time-varying delays LMI Lyapunov’s method
a b s t r a c t In this paper, the problem of delay-range-dependent stabilization criterion for uncertain dynamic systems with time-varying delays is considered. The time-varying delays considered is assumed to be belong to a given interval in which lower bound of delay is not restricted to zero. By constructing a suitable augmented Lyapunov’s functional and utilizing free weight matrices, the criterion for stabilization is established in terms of linear matrix inequalities. Three numerical examples are given to show the effectiveness of proposed method. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Time delays often occur in many industrial systems such as chemical processes, biological systems, population dynamics, neural networks, large-scale systems, network control systems, and so on. The occurrence of the time delays may deteriorate system performance or even cause instability. Therefore, stability analysis and controller synthesis have been one of the most challenging issues [1–3]. Since delay-dependent stability criteria are less conservative than delay-independent ones when the size of time delays is small, many researchers have focused on delay-dependent stabilization and stability [4–7]. In delay-dependent stabilization and stability criteria, the main concern is to enlarge the feasible region of criteria for guaranteeing asymptotic stability of time-delay systems in a given time-delay interval. To do this, Park [8] proposed a new bounding lemma to reduce the conservatism of the stability criteria by introducing free variables in cross terms. Moon [9] proposed a stabilizing controller design method of uncertain time-delay systems by utilizing Park’s lemma [8]. Fridman and Shaked [10,11] presented a descriptor approach to design a stabilizing controller of time-delay systems. Yue [12] utilized neutral model transformation to obtain stability criteria for systems with discrete and distributed delays. Kwon and Park [13] proposed a method of designing controller for uncertain time-delay systems by introducing free weight variables in transformation operator. Recently, by adding zero equations with free variables, an improved stability or stabilization method has been investigated [6,14–17]. In Ref. [15], augmented Lyapunov’s functionals were proposed to increase the feasible region of stability criteria by taking integral terms of states as augmented ones. On the other hand, the stability analysis of dynamic systems with interval time-varying delays has been a focused topic of theoretical and practical importance [6,18,19] in very recent years. Interval time-varying delay means that a time delay varies in an interval in which the lower bound is not restricted to be zero. A typical example of dynamic systems with interval time-varying delays is networked control systems [20]. However, few results have been investigated to the problem of stabilization and stability for uncertain dynamic system with interval time-varying delays. * Corresponding author. E-mail address: [email protected] (O.M. Kwon). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.11.010
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
59
In this paper, we propose delay-range-dependent stabilization criterion for systems with interval time-varying delays and norm bounded parameter uncertainties. By constructing a suitable augmented Lyapunov’s functional, delay-range-dependent stabilization criterion is derived in terms of LMIs which can be solved efficiently by using the interior-point algorithms [21]. Also, to enlarge the feasibility region of criteria, zero equations with free weight matrices are utilized in time-derivative of the Lyapunov functional. Through three numerical examples, the effectiveness of the proposed methods are shown. In the sequel, the following notation will be used. Rn is the n-dimensional Euclidean space. Rmn denotes the set of m n real matrix. H denotes the symmetric part. X > 0ðX P 0Þ means that X is a real symmetric positive definitive matrix (positive semi-definite). I denotes the identity matrix with appropriate dimensions. k k refers to the induced matrix 2-norm. diagf g denotes the block diagonal matrix. Cn;h ¼ Cð½h; 0; Rn Þ denotes the Banach space of continuous functions mapping the interval ½h; 0 into Rn , with the topology of uniform convergence. 2. Problem statements Consider the following uncertain dynamic systems with time-varying delays
_ xðtÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðAd þ DAd ðtÞÞxðt hðtÞÞ þ ðB þ DBðtÞÞuðtÞ; xðsÞ ¼ /ðsÞ;
s 2 ½hU ; 0;
ð1Þ
where xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input, A; Ad , and B are known constant matrices with appropriate dimensions, /ðsÞ 2 Cn;h is a given continuous vector valued initial function, and DAðtÞ; DAd ðtÞ and DBðtÞ are the uncertainties of system matrices of the form
½ DAðtÞ DAd ðtÞ DBðtÞ ¼ DFðtÞ½ E1
E2
E3
ð2Þ
in which the time-varying nonlinear function FðtÞ satisfies
F T ðtÞFðtÞ 6 I
8t P 0:
ð3Þ
The delays, hðtÞ, are time-varying continuous functions that satisfies
_ hðtÞ 6 hD ;
0 6 hL 6 hðtÞ 6 hU ;
ð4Þ
where hL and hU are positive constants and hD is any constant value. For system (1), we design a memoryless state-feedback controller
uðtÞ ¼ KxðtÞ;
ð5Þ
mn
where K 2 R is a constant matrix to be designed later. With controller (5), system (1) can be rewritten as
_ xðtÞ ¼ ðA þ BKÞxðtÞ þ Ad xðt hðtÞÞ þ DpðtÞ; pðtÞ ¼ FðtÞqðtÞ;
ð6Þ
qðtÞ ¼ ðE1 þ E3 KÞxðtÞ þ E2 xðt hðtÞÞ: The purpose of this paper is to present a delay-dependent stabilization criterion for system (6). Before deriving our main results, we need the following facts and lemma. Fact 1. (Schur complement) Given constant symmetric matrices R1 ; R2 ; R3 where R1 ¼ RT1 and 0 < R2 ¼ RT2 , then R1 þ RT3 R1 2 R3 < 0 if and only if
"
#
R1 RT3 < 0 or R3 R2
R2
RT3
R3 < 0: R1
Fact 2. For any real vectors a; b and any matrix Q > 0 with appropriate dimensions, it follows that T
2aT b 6 aT Qa þ b Q 1 b: Lemma 1 [22]. For any constant matrix M 2 Rnn ; M ¼ M T > 0, scalar c > 0, vector function x : ½0; c ! Rn such that the integrations concerned are well defined, then
Z 0
c
T Z xðsÞ ds M 0
c
Z xðsÞ ds 6 c 0
c
xT ðsÞMxðsÞ ds:
ð7Þ
60
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
3. Main results In this section, we propose a new delay-dependent stabilization criterion for systems (6) with interval time-varying delays. For simplicity, the notations of several matrices are defined as follows:
R T t fT ðtÞ ¼ xT ðtÞ xT ðt hL Þ xT ðt hðtÞÞ xT ðt hU Þ x_ T ðtÞ xðsÞ ds thL fT1 ðtÞ ¼ ½ xT ðtÞ x_ T ðtÞ ;
R
thL thðtÞ
xðsÞ ds
T
R thðtÞ thU
T xðsÞ ds pT ðtÞ ;
fT2 ðtÞ ¼ ½ xT ðtÞ xT ðt hðtÞÞ x_ T ðtÞ pT ðtÞ ; fT3 ðtÞ ¼ ½ xT ðtÞ xT ðt hðtÞÞ l; U T1 ¼ ½ I U T2 U T3 U T4
0 0 0 0 0 0 0 0 ;
¼ ½0 0 0 0 0 I
0 0 0 ;
¼ ½0 0 0 0 0 0 I
0 0 ;
¼ ½0 0 0 0 0 0 0 I
0 ;
R ¼ Rði;jÞ ði; j ¼ 1; . . . ; 9Þ; Rð1;1Þ ¼ AP2 þ P2 AT þ BV þ V T BT þ R1 þ R2 þ R3 þ h2U M11 M22 þ ðhU hL Þ2 N11 þ PT3 þ P 3 þ S2 þ ST2 ; Rð1;3Þ ¼ Ad P2 þ M22 P3 þ PT4 P5 þ P7 S2 ;
Rð1;2Þ ¼ P5 þ S4 ; Rð1;4Þ ¼ P7 S4 ;
Rð1;5Þ ¼ P1 P2 þ P 2 AT þ V T BT þ h2U M12 þ ðhU hL Þ2 N12 þ S1 ; Rð1;6Þ ¼ MT12 þ S3 ;
Rð1;7Þ ¼ S5 ;
Rð2;2Þ ¼ R1 N22 ;
Rð2;3Þ ¼ N 22 þ P T6 ;
Rð2;7Þ ¼ NT12 þ S6 ;
Rð2;8Þ ¼ S6 ; Rð2;9Þ ¼ 0;
Rð1;8Þ ¼ S5 ;
Rð1;9Þ ¼ P 2 D;
Rð2;4Þ ¼ 0;
Rð2;5Þ ¼ 0;
Rð2;6Þ ¼ ST5 ;
Rð3;3Þ ¼ ð1 hD ÞR2 2M22 2N22 P4 PT4 P6 PT6 þ P8 þ PT8 þ hD ðG1 þ G2 þ G3 þ G4 Þ; Rð3;5Þ ¼ P2 ATd ;
Rð3;4Þ ¼ M22 þ N22 P8 ; Rð3;7Þ ¼ NT12 S5 ;
Rð3;8Þ ¼ MT12 NT12 S5 ;
Rð4;4Þ ¼ R3 M22 N22 ; Rð4;8Þ ¼
MT12
þ
NT12
Rð3;6Þ ¼ MT12 S3 ;
S6 ;
Rð3;9Þ ¼ 0;
Rð4;6Þ ¼ ST5 ;
Rð4;5Þ ¼ 0;
Rð4;7Þ ¼ S6 ;
Rð4;9Þ ¼ 0;
Rð5;5Þ ¼ 2P2 þ hU Q 1 þ ðhU hL ÞQ 2 þ h2U M22 þ ðhU hL Þ2 N22 ; Rð5;6Þ ¼ S2 ;
Rð5;7Þ ¼ S4 ;
Rð5;8Þ ¼ S4 ; Rð5;9Þ ¼ DP2 ;
Rð6;7Þ ¼ 0;
Rð6;6Þ ¼ M11 ;
Rð6;8Þ ¼ 0;
Rð6;9Þ ¼ 0;
Rð7;7Þ ¼ N11 ;
Rð7;8Þ ¼ 0; Rð7;9Þ ¼ 0; Rð8;8Þ ¼ M11 N11 ; Rð8;9Þ ¼ 0; W1 ¼ ½ E1 P2 þ E3 V 0 E2 P2 0 0 0 0 0 0 ; h
pffiffiffiffiffiffi
pffiffiffiffiffiffi
pffiffiffiffiffiffi pffiffiffiffiffiffi i hD ST5 hD ST5 ; U4
h D S2 h D S3 N1 ¼ U 1 U2 U3 N2 ¼ diagf G1 ; G2 ; G3 ; G4 g; "
T 1
P ¼
I
Rð9;9Þ ¼ aP2 ;
0 0 0 0 0 0 0 0 0 0 0 0 0
#
; 0 0 0 0 0 0 0 0 0 0 0 2 3 " # " # S1 S2 S4 M 11 M 12 N11 N 12 6 7 ; N¼ ; S ¼ 4 H S3 S5 5; M¼ H M 22 H N 22 H H S6 " # " # " # X1 X2 Y1 Y2 Z1 Z2 ; Y¼ ; Z¼ : X¼ H X2 H Y2 H Z2 0 0
I
ð8Þ Now, we have the following theorem. Theorem 1. For given positive scalars a; hL ; hU , and any scalar hD , system (6) under the control uðtÞ ¼ VP 1 2 xðtÞ is robustly stable for hL 6 hðtÞ 6 hU if there exist positive definite matrices P i ði ¼ 1; 2Þ; Ri ði ¼ 1; 2; 3Þ; Q i ði ¼ 1; 2Þ; Gi ði ¼ 1; . . . ; 4Þ; Si ði ¼ 1; 3; 6Þ; M ii ði ¼ 1; 2Þ; N ii ði ¼ 1; 2Þ; X i ði ¼ 1; 3Þ; Y i ði ¼ 1; 3Þ; Z i ði ¼ 1; 3Þ, and any matrices V; Pi ; ði ¼ 3; . . . ; 8Þ; M 12 ; N 12 ; S2 ; S4 ; S5 ; X 2 ; Y 2 ; Z 2 , satisfying the following LMIs
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
2
R
R
3
WT1
NT1
W1
NT2 3 NT1
6 4 H a1 P2 H H 2 T
7 T 0 5 þ P1 ðhL X þ ðhU hL ÞZÞP1 < 0;
ð9Þ
6 7 T 4 H a1 P2 0 5 þ P1 ðhU X þ ðhU hL ÞYÞP1 < 0; T H H N2 2 2 2 " #3 " #3 " # P3 P5 P7 6X 7 6Y 7 6Z 6 7 > 0; 6 7 > 0; 6 P P P8 4 6 4 5 4 5 4 Q1
H M > 0;
61
Q2
H N > 0;
H
ð10Þ 3 7 7 > 0; 5
ð11Þ
Q1 þ Q2
S > 0:
ð12Þ
Proof. For positive definite matrices P i ði ¼ 1; 2Þ; Ri ði ¼ 1; 2; 3Þ; Q i ði ¼ 1; 2Þ; M ii ði ¼ 1; 2Þ; N ii ði ¼ 1; 2Þ; Si ði ¼ 1; 3; 6Þ, and any matrices M 12 ; N 12 ; S2 ; S4 ; S5 ; S7 ; S8 ; S9 , let us consider the Lyapunov–Krasovskii functional candidate
V¼
6 X
Vi
ð13Þ
i¼1
where
V 1 ¼ fT1 ðtÞEPf1 ðtÞ; Z t Z V2 ¼ xT ðsÞR1 xðsÞ ds þ V3 ¼
Z
thL
Z
t thU
V 4 ¼ hU
Z
s t
t
xT ðsÞR2 xðsÞ ds þ
thðtÞ t
_ x_ T ðuÞQ 1 xðuÞ du ds þ
xT ðsÞR3 xðsÞ ds;
thU
Z
t
_ x_ T ðuÞQ 2 xðuÞ du ds;
s
fT1 ðuÞMf1 ðuÞ du ds;
Z
thL
Z
ð14Þ
t
s
thU
2
t
t s
V 5 ¼ ðhU hL Þ
thL
thU
Z
thU
Z
Z
fT1 ðuÞNf1 ðuÞ du ds;
3T 2 3 xðtÞ xðtÞ R R 6 t 7 6 t 7 xðsÞ ds 7 S6 thðtÞ xðsÞ ds 7 V6 ¼ 6 4 thðtÞ 5 4 5 R thL R thL xðsÞ ds xðsÞ ds thU thU and, E and P in V 1 are defined as
I 0 ; 0 0
E¼
P¼
P1 P2
0 : P2
ð15Þ
From V 1 , we have
_ xðtÞ V_ 1 ¼ 2fT1 ðtÞPT 0 _ xðtÞ ¼ 2fT1 ðtÞPT _ þ ðA þ BKÞxðtÞ þ Ad xðt hðtÞÞ þ DpðtÞ xðtÞ
ð16Þ
¼ 2fT2 ðtÞKT PT Cf2 ðtÞ ¼ fT2 ðtÞðKT PT C þ CT PKÞf2 ðtÞ; where
K¼
I
0 0 0
0 0
I
0
;
C¼
0
0
I
0
A þ BK
Ad
I
D
;
ð17Þ
and f1 ðtÞ and f2 ðtÞ are defined in (8). An upper bound of time-derivative of V 2 can be obtained as
V_ 2 6 xT ðtÞR1 xðtÞ xT ðt hL ÞR1 xðt hL Þ þ xT ðtÞR2 xðtÞ ð1 hD ÞxT ðt hðtÞÞR2 xðt hðtÞÞ þ xT ðtÞR3 xðtÞ xT ðt hU ÞR3 xðt hU Þ:
ð18Þ
62
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
We can obtain the results of calculation of V_ 3 as follows:
_ V_ 3 ¼ hU x_ T ðtÞQ 1 xðtÞ
t
_ ds x_ T ðsÞQ 1 xðsÞ
thðtÞ
Z
Z
thL
_ ds x_ T ðsÞQ 2 xðsÞ
thðtÞ
Z
Z
thðtÞ
_ ds þ ðhU hL Þx_ T ðtÞQ 2 xðtÞ _ x_ T ðsÞQ 1 xðsÞ
thU thðtÞ
_ ds: x_ T ðsÞQ 2 xðsÞ
ð19Þ
thU
Calculating V_ 4 leads to
" 2 V_ 4 ¼ hU
6
#T "
_ xðtÞ "
2 hU
xðtÞ
xðtÞ
M11 M12 H
#T "
_ xðtÞ
M11 H
2 R thðtÞ
#"
xðtÞ
#
t
"
xðsÞ
#T "
M11 M12
#"
xðsÞ
#
Z
ds hU _ _ H M22 xðsÞ xðsÞ 3T " 3 #" # 2Rt #2 R t xðsÞ ds xðsÞ ds M12 xðtÞ M11 M12 thðtÞ thðtÞ 4 5 4 5 R Rt t _ M 22 xðtÞ H M 22 _ ds _ ds xðsÞ xðsÞ
M22
hU
Z
_ xðtÞ
thðtÞ
thðtÞ
thðtÞ
thU
"
xðsÞ _ xðsÞ
#T "
M11 M12 H
M22
#"
xðsÞ _ xðsÞ
# ds
thðtÞ
3T "
3 #2 R thðtÞ #" " #T " # xðsÞ ds xðsÞ ds M11 M12 xðtÞ M11 M12 xðtÞ thU thU 2 4 5 4 5 ¼ hU R R thðtÞ thðtÞ _ _ H M22 H M 22 xðtÞ xðtÞ _ ds _ ds xðsÞ xðsÞ thU thU 2 R 3T " 3 #2 R t t M11 M12 xðsÞ ds xðsÞ ds thðtÞ thðtÞ 5 4 5 4 H M22 xðtÞ xðt hðtÞÞ xðtÞ xðt hðtÞÞ 2 3T " 3 #2 R thðtÞ R thðtÞ M11 M12 xðsÞ ds xðsÞ ds thU thU 4 5 4 5 ¼ fT ðtÞN1 fðtÞ H M22 xðt hðtÞÞ xðt hU Þ xðt hðtÞÞ xðt hU Þ ð20Þ where
2 6 6 6 6 6 6 6 6 N1 ¼ 6 6 6 6 6 6 6 6 6 4
2
2
hU M 11 M22
M 22
0
hU M 12
0
M T12
0
H
2M22
M 22
0
0
MT12
MT12
H
H
M 22
0
0
0
M T12
H
H
H
hU M 22
0
0
0
H
H
H
H
0
0
0
H
H
H
H
H M 11
H
H
H
H
H
H
M11
H
H
H
H
H
H
H
2
0
0
3
7 07 7 7 07 7 7 7 07 7; 7 07 7 7 07 7 07 5 0
and fðtÞ is defined in (8). Here, Lemma 1 was utilized in obtaining an upper bound of V_ 4 . Similarly, an upper bound of V_ 5 can be obtained as
V_ 5 6 fT ðtÞN2 fðtÞ; where
2 6 6 6 6 6 6 6 6 6 6 N2 ¼ 6 6 6 6 6 6 6 6 6 6 4
ð21Þ
ðhU hL Þ2 N11
0
0
0
ðhU hL Þ2 N12
0
0
0
H
N 22
N22
0
0
0
N T12
0
H
H
2N22
N22
0
0
N T12
N T12
H
H
H
N22
0
0
0
NT12
H
H
H
H
ðhU hL Þ2 N22
0
0
0
H
H
H
H
0
0
0
0 0
H
H
H
H
H
H N11
H
H
H
H
H
H
H
N11
H
H
H
H
H
H
H
H
0
3
7 07 7 7 07 7 7 07 7 7 7 0 7: 7 7 07 7 07 7 7 07 5 0
ð22Þ
63
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
Lastly, we have
2 6 6 V_ 6 ¼ 26 4
3T 2
xðtÞ
S1 7 6 7 6 xðsÞ ds thðtÞ 7 4H 5 R thðtÞ R thL H xðsÞ ds þ xðsÞ ds th thðtÞ
S4
S3
S6
H
U
_ ¼ f ðtÞN3 fðtÞ þ 2x ðtÞS2 hðtÞxðt hðtÞÞ þ 2 T
T
32
3 _ xðtÞ 76 7 _ 6 7 S5 7 54 xðtÞ xðt hðtÞÞ þ hðtÞxðt hðtÞÞ 5
S2
Rt
Z
xðt hL Þ xðt hU Þ !T
t
_ S3 hðtÞxðt hðtÞÞ þ 2
xðsÞ ds
Z
xðsÞ ds
T
6 f ðtÞN3 fðtÞ þ hD x
_ ST5 hðtÞxðt hðtÞÞ
T
_ ST5 hðtÞxðt hðtÞÞ
T ðtÞS2 G1 1 S2 xðtÞ
T
þ hD x ðt hðtÞÞG1 xðt hðtÞÞ þ hD Z
þ hD x ðt hðtÞÞG2 xðt hðtÞÞ þ hD
!T
thL
xðsÞ ds
thðtÞ
!T
thðtÞ
xðsÞ ds
thU
ST5 G1 4 S5
Z
!T
t
xðsÞ ds
thðtÞ
T
Z
xðsÞ ds
!T
thðtÞ
thU
þ hD
!T
thL thðtÞ
thðtÞ
þ2
Z
Z
Z
ST5 G1 3 S5
thL
S3 G1 2 S3
Z
!
t
xðsÞ ds
thðtÞ
! xðsÞ ds þ hD xT ðt hðtÞÞG3 xðt hðtÞÞ
thðtÞ
!
thðtÞ
xðsÞ ds þ hD xT ðt hðtÞÞG4 xðt hðtÞÞ;
thU
ð23Þ where
2 6 6 6 6 6 6 6 6 6 6 N3 ¼ 6 6 6 6 6 6 6 6 6 6 6 4
S2 þ ST2
S4
S2
S4
S1
S3
S5
S5
H
0
0
0
0
ST5
S6
S6
H
H
0
0
0
S3
S5
S5
0
ST5
S6
S6
H
H
H
0
H
H
H
H
0
S2
S4
S4
H
H
H
H
H
0
0
0
H
H
H
H
H
H
0
0
H
H
H
H
H
H
H
0
H
H
H
H
H
H
H
H
0
3
7 07 7 7 7 07 7 7 07 7 7 07 7 7 07 7 7 07 7 7 07 5 0
and Fact 2 was utilized in obtaining an upper bound of V_ 6 . As a tool to reduce the conservatism of stabilization criterion, we add the following zero equations with P i ði ¼ 3; . . . ; 8Þ to be chosen as
0 ¼ 2½xT ðtÞP3 þ xT ðt hðtÞÞP4 ½xðtÞ xðt hðtÞÞ 2fT3 ðtÞ
P3
Z
P4
0 ¼ 2½xT ðtÞP5 þ xT ðt hðtÞÞP6 ½xðt hL Þ xðt hðtÞÞ 2fT3 ðtÞ
t
_ ds; xðsÞ
ð24Þ
thðtÞ
P5
Z
P6
thL
_ ds xðsÞ
ð25Þ
thðtÞ
and
0 ¼ 2½xT ðtÞP 7 þ xT ðt hðtÞÞP 8 ½xðt hðtÞÞ xðt hU Þ 2fT3 ðtÞ
P7 P8
Z
thðtÞ
_ ds; xðsÞ
ð26Þ
thU
where f3 ðtÞ is defined in (8). The above three zero equations can be represented as
Z t Z thL Z thðtÞ P3 P P _ ds 2fT3 ðtÞ 5 _ ds 2fT3 ðtÞ 7 _ ds; 0 ¼ fT ðtÞN4 fðtÞ 2fT3 ðtÞ xðsÞ xðsÞ xðsÞ P 4 thðtÞ P 6 thðtÞ P8 thU
ð27Þ
64
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
where
2 T 6 P3 þ P3 6 6 6 6 H 6 6 6 6 6 H 6 6 N4 ¼ 6 6 6 H 6 6 6 H 6 6 H 6 6 H 6 6 4 H H
P 3 þ PT4 P 5 þ P7
P5 0 H H H H H H H
0
P T6 P 4 PT4
!
1
3 P7
0
0
0
0
0
0
0
0
0
0
0
0
0
B C @ P 6 PT6 A P8 þP 8 þ H H H H H H
PT8
0 H H H H H
07 7 7 7 07 7 7 7 7 07 7 7 7: 7 07 7 7 07 7 07 7 07 7 7 05
ð28Þ
0 0 0 0 0 0 0 0 H 0 0 0 H H 0 0 H H H 0 H H H H 0
Furthermore, for any positive matrices X; Y, and Z with appropriate dimensions, the following equations hold:
0¼
Z
t
thðtÞ
0¼
Z
thL
thðtÞ
0¼
Z
fT3 ðtÞXf3 ðtÞ ds fT3 ðtÞYf3 ðtÞ ds
Z
thðtÞ
Z
thðtÞ
thU
t
fT3 ðtÞZf3 ðtÞ ds
thL
thðtÞ
Z
fT3 ðtÞXf3 ðtÞ ds ¼ hðtÞfT3 ðtÞXf3 ðtÞ
Z
t
thðtÞ
fT3 ðtÞXf3 ðtÞ ds;
fT3 ðtÞYf3 ðtÞ ds ¼ ðhðtÞ hL ÞfT3 ðtÞYf3 ðtÞ
Z
thðtÞ
thðtÞ
thU
thL
fT3 ðtÞZf3 ðtÞ ds ¼ ðhU hðtÞÞfT3 ðtÞZf3 ðtÞ
Z
fT3 ðtÞYf3 ðtÞ ds;
ð29Þ ð30Þ
thðtÞ
thU
fT3 ðtÞZf3 ðtÞ ds:
ð31Þ
Since the following inequality holds from (3) and (6),
pT ðtÞpðtÞ 6 qT ðtÞqðtÞ;
ð32Þ
there exist positive matrix P2 and positive scalar a satisfying the following inequality
fT ðtÞWT ðaP2 ÞWfðtÞ pT ðtÞðaP2 ÞpðtÞ P 0;
ð33Þ
W ¼ ½ E1 þ E3 K 0 E2 0 0 0 0 0 0 :
ð34Þ
where
P 1 2 ;V
Let P2 ¼ ¼ KP 2 ; Pi ¼ P2 Pi P 2 ði ¼ 1; 3; 4; . . . ; 7; 8Þ; Ri ¼ P 2 Ri P 2 ; Q i ¼ P 2 Q i P2 ; Gi ¼ P2 Gi P 2 ; M ij ¼ P2 M ij P 2 ; N ij ¼ P2 N ij P 2 ; Si ¼ P P2 Si P 2 ; X i ¼ P 2 X i P2 ; Y i ¼ P2 Y i P 2 , Z i ¼ P2 Z i P 2 . From (16)–(36) and by applying S-procedure [21], the V_ ¼ 6i¼1 V_ i has a new upper bound as T T T T T 1 T 1 T 1 V_ 6 fT ðtÞ!1 ½R þ U T1 ðhD S2 G1 1 S2 ÞU 1 þ U 2 ðhD S3 G2 S3 ÞU 2 þ U 3 ðhD S5 G3 S5 ÞU 3 þ U 4 ðhD S5 G4 S5 ÞU 4 P2 ðhðtÞX 2 " #3 P3 Z t 7 f3 ðtÞ T 6 X f3 ðtÞ T 1 6 þ ðhðtÞ hL ÞY þ ðhU hðtÞÞZÞP2 þ W ðaP2 ÞW!1 fðtÞ !2 4 !2 ds P4 7 5 _ _ xðsÞ xðsÞ thðtÞ Q1 H 2 2 " #3 " # 3 P5 P7 T T Z thðtÞ Z thL 6Y 7 6Z 7 f3 ðtÞ f3 ðtÞ f3 ðtÞ f3 ðtÞ 7 6 7 !2 6 ! ! ! ds ds P P 24 6 5 2 8 4 5 2 xðsÞ _ _ _ _ xðsÞ xðsÞ xðsÞ thU thðtÞ Q2 H H Q1 þ Q2
ð35Þ
where
!1 ¼ diagfP2 ; P2 ; P2 ; P2 ; P2 ; P2 ; P2 ; P2 ; P2 ; g; !2 ¼ diagfP2 ; P2 ; P2 g;
PT2 ¼
I
0 0 0 0 0 0 0 0
0 0
I
0 0 0 0 0 0
:
ð36Þ
Let T T T T 1 T 1 T 1 T 1 X ¼ R þ U T1 ðhD S2 G1 1 S2 ÞU 1 þ U 2 ðhD S3 G2 S3 ÞU 2 þ U 3 ðhD S5 G3 S5 ÞU 3 þ U 4 ðhD S5 G4 S5 ÞU 4 þ W ðaP 2 ÞW:
ð37Þ
Note that for the delay intervals hL 6 hðtÞ 6 hU ,
X þ PT2 ½hðtÞX þ ðhðtÞ hL ÞY þ ðhU hðtÞÞZP2 < 0
ð38Þ
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O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
hold if and only if
X þ PT2 ½hL X þ ðhU hL ÞZP2 < 0
ð39Þ
X þ PT2 ½hU X þ ðhU hL ÞYP2 < 0:
ð40Þ
and
Using Fact 1, the inequalities (39) and (40) are equivalent to the LMIs (9) and (10), respectively. Therefore, if the LMIs (9)– (12) are satisfied, then the system (6) is guaranteed to be asymptotically stable and the corresponding controller gain is obh tained as K ¼ VP 1 2 . This completes our proof. Remark 1. The solutions of Theorem 1 can be obtained by solving the eigenvalue problem with respect to solution variables, which is a convex optimization problem [21]. In this paper, we utilize Matlab’s LMI Control Toolbox [23] which implements the interior-point algorithm. This algorithm is faster than classical convex optimization algorithms [21]. Remark 2. By iteratively solving the LMI given in Theorem 1 with respect to hU for fixed a; hL and hD , one can find the maximum upper bound of time delay hU for guaranteeing asymptotic stability of system (6). For the case uðtÞ ¼ 0, we can obtain the delay-dependent stability criteria for the system (6) with uðtÞ ¼ 0. For simplicity, we define the following notations before introducing stability criteria
R ¼ Rði;jÞ ði; j ¼ 1; . . . ; 9Þ; Rð1;1Þ ¼ PT2 A þ AT P2 þ R1 þ R2 þ R3 þ h2U M11 M22 þ ðhU hL Þ2 N 11 þ PT6 þ P6 þ S2 þ ST2 Rð1;2Þ ¼ P8 þ S4 ; Rð1;3Þ ¼ PT2 Ad þ AT P3 þ M22 P6 þ PT7 P 8 þ P10 S2 ; Rð1;4Þ ¼ P10 S4 ; Rð1;5Þ ¼ P1 PT2 þ AT P4 þ h2U M12 þ ðhU hL Þ2 N12 þ S1 ; Rð1;6Þ ¼ MT12 þ S3 ;
Rð1;7Þ ¼ S5 ;
Rð2;2Þ ¼ R1 N22 ;
Rð2;3Þ ¼ N22 þ PT9 ;
Rð2;7Þ ¼ NT12 þ S6 ;
Rð2;8Þ ¼ S6 ;
Rð3;3Þ ¼
PT3 Ad
þ
ATd P3
Rð2;5Þ ¼ 0;
Rð3;6Þ ¼ MT12 S3 ; Rð3;9Þ ¼ PT3 D þ ATd P5 ;
Rð3;8Þ ¼ MT12 N T12 S5 ;
Rð4;6Þ ¼ ST5 ;
Rð4;5Þ ¼ 0;
Rð4;7Þ ¼ S6 ;
Rð4;8Þ ¼
M T12
þ
Rð5;5Þ ¼
PT4
P4 þ hU Q 1 þ ðhU hL ÞQ 2 þ hU M 22 þ ðhU hL Þ2 N22 ;
Rð5;6Þ ¼ S2 ;
S6 ;
Rð4;9Þ ¼ 0;
ð41Þ 2
Rð5;7Þ ¼ S4 ;
Rð5;9Þ ¼ PT4 D P5 ;
Rð5;8Þ ¼ S4 ;
Rð6;7Þ ¼ 0;
Rð6;6Þ ¼ M11 ;
Rð6;8Þ ¼ 0;
Rð6;9Þ ¼ 0;
Rð7;8Þ ¼ 0; Rð7;9Þ ¼ 0; Rð8;8Þ ¼ M11 N 11 ; W ¼ ½ E1 0 E2 0 0 0 0 0 0 ; h
N1 ¼ U 1
N2 ¼ diagfG1 ; G2 ; G3 ; G4 g; M ¼
X¼ " Pc ¼
Rð7;7Þ ¼ N11 ;
Rð8;9Þ ¼ 0;
Rð9;9Þ ¼ PT5 D þ DT P5 H;
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi i h D S2 h D S3 hD ST5 hD ST5 ; U2 U3 U4 "
"
Rð2;6Þ ¼ ST5 ;
Rð2;9Þ ¼ 0;
Rð3;5Þ ¼ PT3 þ ATd P4 ;
Rð4;4Þ ¼ R3 M22 N22 ; N T12
Rð2;4Þ ¼ 0;
ð1 hD ÞR2 2M22 2N22 P7 P T7 P9 PT9 þ P 11 þ P T11 þ hD ðG1 þ G2 þ G3 þ G4 Þ;
Rð3;4Þ ¼ M22 þ N 22 P11 ; Rð3;7Þ ¼ NT12 S5 ;
Rð1;9Þ ¼ PT2 D þ AT P5 ;
Rð1;8Þ ¼ S5 ;
#
"
X1
X2
H
X2
P1
0
0
0
P2
P3
P4
P5
Y¼
;
# ;
Y1
Y2
H
Y2 ETc ¼
M 11
M 12
H
M 22
#
" Z¼
; "
I
#
Z1
Z2
H
Z2
0 0 0
0 0 0 0
#
" N¼
; #
N 11
N 12
H
N 22
2
# ;
3
S1 6 S¼4H
S2
S4
S3
7 S5 5 ;
H
H
S6
;
:
Corollary 1. For given positive scalars hL ; hU , and any scalar hD , system (6) with uðtÞ ¼ 0 is robustly stable if there exist positive definite matrices P1 ; H; Ri ði ¼ 1; 2; 3Þ; Q i ði ¼ 1; 2Þ; Gi ði ¼ 1; . . . ; 4Þ; Si ði ¼ 1; 3; 6Þ; M ii ði ¼ 1; 2Þ; N ii ði ¼ 1; 2Þ; X i ði ¼ 1; 3Þ; Y i ði ¼ 1; 3Þ; Z i ði ¼ 1; 3Þ, and any matrices V; P i ; ði ¼ 2; . . . ; 11Þ; M 12 ; N 12 ; S2 ; S4 ; S5 ; X 2 ; Y 2 ; Z 2 , satisfying the following LMIs:
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O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
2
R WT1 H NT1
6 6H 4
7 T 0 7 5 þ P1 ðhL X þ ðhU hL ÞZÞP1 < 0;
H
H 2
N
H T 1H
R W
6 6H 4 H
6X 6 4
P6
3
#3
T 2
6Y 6 4
Q1
M > 0;
"
2
7 P7 7 5 > 0;
H
ð42Þ
7 T 0 7 5 þ P1 ðhU X þ ðhU hL ÞYÞP1 < 0;
N
H "
T 2
NT1
H
2
3
H N > 0;
P8
#3
"
2
7 P9 7 5 > 0; Q2
6Z 6 4 H
ð43Þ
P 10
# 3
P11
7 7 > 0; 5
ð44Þ
Q1 þ Q2
S > 0:
ð45Þ
Proof. Let us consider the following Lyapunov–Krasovskii’s functional candidate
V¼
6 X
V i;
ð46Þ
V 1 ¼ fT2 ðtÞEc Pc f2 ðtÞ
ð47Þ
i¼1
where
and the same ones V i ði ¼ 2; . . . ; 6Þ in (14). Here, Pc and Ec are defined in (41). With the above Lyapunov–Krasovskii’s functional candidate, the proof of Corollary 1 can be carried out by using methods in the proof of Theorem 1. Hence, it is omitted. h Remark 3. In many industrial systems, hD is unknown. For this case, Corollary 1 can be extended if we do not consider V 6 Rt and the term thðtÞ xT ðsÞR1 xðsÞ ds of V 2 . 4. Numerical examples Example 1. Consider the uncertain dynamic systems with time-varying delays:
_ xðtÞ ¼ A þ DAðtÞxðtÞ þ ðAd þ DAd Þxðt hðtÞÞ þ BuðtÞ;
ð48Þ
where
A¼
0 0 0 1
;
Ad ¼
2 0:5 0
1
;
B¼
0 1
and
D ¼ I;
E1 ¼ E2 ¼ 0:2I:
ð49Þ
When hL ¼ 0, by applying Theorem 1 to the above system, the delay bounds for the conditions hD ¼ 0 and hD ¼ 0:5 are listed in Table 1. Furthermore, when hL ¼ 0:3 and hD ¼ 0, the results are listed in Table 2. From Tables 1 and 2, one can see that Theorem 1 provides significantly improved results. For example, the obtained stabilizing controller with hU ¼ 1:0 by applying Theorem 1 is uðtÞ ¼ ½30:0591 13:4944xðtÞ.
Table 1 Delay bounds with hL ¼ 0(Example 1, a ¼ 4). Methods
hD
hU
Moon et al. [9] Fridman et al. [10] Parlaklßi [15] Li et al. [6] Theorem 1
0
0.4500 0.5865 0.6900 0.84 1.4598
Fridman et al. [10] Parlaklßi [15] Theorem 1
0.5
0.4960 0.6000 1.4597
67
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68 Table 2 Delay bounds with hL ¼ 0:3 and hD ¼ 0(Example 1, a ¼ 4). Methods
hU
Li et al. [6] Theorem 1
0.9 1.4598
Table 3 MATIs with different methods (Example 2). Methods
MATI
Park et al. [24] Kim et al. [25] Yue et al. [26] Corollary 1
0.0538 0.7805 0.8695 1.0432
Table 4 Delay bounds with different values of hL and hD (Example 3). hL ¼ 0
Li et al. [6] Corollary 1
hL ¼ 0:3
hD ¼ 0:5
hD ¼ 0:9
hD ¼ 0:5
hD ¼ 0:9
0.31 0.56
0.31 0.50
0.55 0.65
0.55 0.62
Example 2. Consider the following Networked Control System
_ xðtÞ ¼
0
1
0 0:1
xðtÞ þ
0 0:1
uðtÞ:
ð50Þ
System (50) is assumed to be controlled by a state-feedback controller uðtÞ ¼ ½ 3:75 11:5 xðtÞ. By applying Corollary 1 and Remark 3 to the above system with unknown hD , we obtained maximum allowable transfer interval (MATI) as 1.0432. In Table 3, MATIs which guarantees the asymptotic stability of system (50) are listed. From Table 3, we can see our proposed method gives a larger MATI than the ones in other literature. Example 3. Consider the following uncertain systems with constant delays:
_ xðtÞ ¼ ðA þ DFðtÞE1 ÞxðtÞ þ ðAd þ DFðtÞE2 Þxðt hðtÞÞ 0:5 1 0:5 2 ; A¼ ; Ad ¼ 0 0:6 1 1 D ¼ I; E1 ¼ E2 ¼ 0:2I:
ð51Þ
In Table 4, the results for different condition of hD are compared with the results in Li [6]. From Table 4, it can be shown that our result for this example gives larger delay bounds than the ones in Li [6].
5. Conclusions In this paper, a new delay-range-dependent stabilization criterion for uncertain dynamic systems with interval timevarying delays is proposed. To obtain less conservative results, new augmented Lyapounv–Krasovskii’s functional and free weighting matrices are utilized by combining with the LMI framework for obtaining the stabilization of the system. The effectiveness of the proposed stabilization criterion is successfully verified by three numerical examples. References [1] [2] [3] [4]
J. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. S.I. Niculescu, Delay effects on stability: a robust approach, Lecture Notes in Control and Information Sciences, vol. 2, Springer-Verlag, New York, 2002. J.P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (2003) 1667–1694. Ju H. Park, O.M. Kwon, On new stability criterion for delay-differential systems of neutral type, Applied Mathematics and Computations 162 (2005) 627–637. [5] S. Xu, J. Lam, Y. Zou, Further results on delay-dependent robust stability conditions of uncertain neutral systems, International Journal of Robust and Nonlinear Control 15 (2005) 233–246. [6] T. Li, L. Guo, Y. Zhang, Delay-range-dependent robust stability and stabilization for uncertain systems with time-varying delay, International Journal of Robust and Nonlinear Control doi:10.1002/rnc.1280, 2007.
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[7] P.G. Park, J.W. Ko, Stability and robust stability for systems with a time-varying delay, Automatica 43 (2007) 738–743. [8] P.G. Park, A delay-dependent stability criterion for systems with uncertain linear state-delayed systems0, IEEE Transactions on Automatic Control 35 (1999) 876–877. [9] Y.S. Moon, P.G. Park, W.H. Kwon, Y.S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, International Journal of Control 74 (2001) 1447–1455. [10] E. Fridman, U. Shaked, An improved stabilization method for linear time-delay systems, IEEE Transactions on Automatic Control 47 (2002) 1931–1937. [11] E. Fridman, U. Shaked, Delay dependent stability and H1 control: constant and time-varying delays, International Journal of Control 76 (2003) 48–60. [12] D. Yue, S. Won, O. Kwon, Delay dependent stability of neutral systems with time delay: an LMI Approach, IEE Proceedings – Control Theory and Applications 150 (2003) 23–27. [13] O.M. Kwon, J.H. Park, On improved delay-dependent robust control for uncertain time-delay systems, IEEE Transactions on Automatic Control 49 (2004) 1991–1995. [14] S. Xu, J. Lam, Improved delay-dependent stability criteria for time-delay systems, IEEE Transactions on Automatic Control 50 (2005) 384–387. [15] M.N.A. Parlaklßi, Improved robust stability criteria and design of robust stabilizing controller for uncertain linear time-delay systems, International Journal of Robust and Nonlinear Control 16 (2006) 599–636. [16] M.N.A. Parlaklßi, Robust stability of uncertain time-varying state-delayed systems, IEE Proceedings – Control Theory and Applications 153 (2006) 469– 477. [17] O.M. Kwon, Ju H. Park, S.M. Lee, On stability criteria for uncertain delay-differential systems of neutral type with time-varying delays, Applied Mathematics and Computation 197 (2008) 864–873. [18] C.R. Knospe, M. Roozbehani, Stability of linear systems with interval time delay excluding zero, IEEE Transactions on Automatic Control 51 (2006) 1271–1288. [19] K.W. Yu, C.H. Lien, Stability criteria for uncertain neutral systems with interval time-varying delay, Chaos Solitons & Fractals 36 (2008) 920–927. [20] D. Yue, C. Peng, G.Y. Tang, Guaranteed cost control of linear systems over networks with state and input quantisation, IEE Proceedings – Control Theory and Applications 153 (2006) 658–664. [21] S. Boyd, L. Ghaoui, E. Feron, V. Balakrishanan, Linear matrix inequalities in system and control theory, Studies in Applied Mathematics, vol. 15, SIAM, Philadelphia, Pennsylvania, 1994. [22] K. Gu, An integral inequality in the stability problem of time-delay systems, in: Proceedings of 39th IEEE Conference on Decision Control, Sydney, Australia, 2000, pp. 2805–2810. [23] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox, The Mathworks, Natik, Massachusetts, 1995. [24] H.S. Park, Y.H. Kim, D.S. Kim, W.H. Kwon, A scheduling method for networked based control systems, IEEE Transactions on Control Systems Technology 10 (2002) 318–330. [25] D.S. Kim, Y.S. Lee, W.H. Kwon, H.S. Park, Maximum allowable delay bounds of networked control systems, Control Engineering Practice 11 (2003) 1301–1313. [26] D. Yue, Q.L. Han, C. Peng, State feedback controller design of networked control systems, IEEE Transactions on Circuits and Systems-II: Express Briefs 51 (2004) 640–644.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays O.M. Kwon a,*, Ju H. Park b a b
School of Electrical and Computer Engineering, Chungbuk National University, 410 SungBong-Ro, Heungduk-gu, Cheongju 361-763, Republic of Korea Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea
a r t i c l e
i n f o
Keywords: Stabilization Interval time-varying delays LMI Lyapunov’s method
a b s t r a c t In this paper, the problem of delay-range-dependent stabilization criterion for uncertain dynamic systems with time-varying delays is considered. The time-varying delays considered is assumed to be belong to a given interval in which lower bound of delay is not restricted to zero. By constructing a suitable augmented Lyapunov’s functional and utilizing free weight matrices, the criterion for stabilization is established in terms of linear matrix inequalities. Three numerical examples are given to show the effectiveness of proposed method. Ó 2008 Elsevier Inc. All rights reserved.
1. Introduction Time delays often occur in many industrial systems such as chemical processes, biological systems, population dynamics, neural networks, large-scale systems, network control systems, and so on. The occurrence of the time delays may deteriorate system performance or even cause instability. Therefore, stability analysis and controller synthesis have been one of the most challenging issues [1–3]. Since delay-dependent stability criteria are less conservative than delay-independent ones when the size of time delays is small, many researchers have focused on delay-dependent stabilization and stability [4–7]. In delay-dependent stabilization and stability criteria, the main concern is to enlarge the feasible region of criteria for guaranteeing asymptotic stability of time-delay systems in a given time-delay interval. To do this, Park [8] proposed a new bounding lemma to reduce the conservatism of the stability criteria by introducing free variables in cross terms. Moon [9] proposed a stabilizing controller design method of uncertain time-delay systems by utilizing Park’s lemma [8]. Fridman and Shaked [10,11] presented a descriptor approach to design a stabilizing controller of time-delay systems. Yue [12] utilized neutral model transformation to obtain stability criteria for systems with discrete and distributed delays. Kwon and Park [13] proposed a method of designing controller for uncertain time-delay systems by introducing free weight variables in transformation operator. Recently, by adding zero equations with free variables, an improved stability or stabilization method has been investigated [6,14–17]. In Ref. [15], augmented Lyapunov’s functionals were proposed to increase the feasible region of stability criteria by taking integral terms of states as augmented ones. On the other hand, the stability analysis of dynamic systems with interval time-varying delays has been a focused topic of theoretical and practical importance [6,18,19] in very recent years. Interval time-varying delay means that a time delay varies in an interval in which the lower bound is not restricted to be zero. A typical example of dynamic systems with interval time-varying delays is networked control systems [20]. However, few results have been investigated to the problem of stabilization and stability for uncertain dynamic system with interval time-varying delays. * Corresponding author. E-mail address: [email protected] (O.M. Kwon). 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.11.010
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
59
In this paper, we propose delay-range-dependent stabilization criterion for systems with interval time-varying delays and norm bounded parameter uncertainties. By constructing a suitable augmented Lyapunov’s functional, delay-range-dependent stabilization criterion is derived in terms of LMIs which can be solved efficiently by using the interior-point algorithms [21]. Also, to enlarge the feasibility region of criteria, zero equations with free weight matrices are utilized in time-derivative of the Lyapunov functional. Through three numerical examples, the effectiveness of the proposed methods are shown. In the sequel, the following notation will be used. Rn is the n-dimensional Euclidean space. Rmn denotes the set of m n real matrix. H denotes the symmetric part. X > 0ðX P 0Þ means that X is a real symmetric positive definitive matrix (positive semi-definite). I denotes the identity matrix with appropriate dimensions. k k refers to the induced matrix 2-norm. diagf g denotes the block diagonal matrix. Cn;h ¼ Cð½h; 0; Rn Þ denotes the Banach space of continuous functions mapping the interval ½h; 0 into Rn , with the topology of uniform convergence. 2. Problem statements Consider the following uncertain dynamic systems with time-varying delays
_ xðtÞ ¼ ðA þ DAðtÞÞxðtÞ þ ðAd þ DAd ðtÞÞxðt hðtÞÞ þ ðB þ DBðtÞÞuðtÞ; xðsÞ ¼ /ðsÞ;
s 2 ½hU ; 0;
ð1Þ
where xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the control input, A; Ad , and B are known constant matrices with appropriate dimensions, /ðsÞ 2 Cn;h is a given continuous vector valued initial function, and DAðtÞ; DAd ðtÞ and DBðtÞ are the uncertainties of system matrices of the form
½ DAðtÞ DAd ðtÞ DBðtÞ ¼ DFðtÞ½ E1
E2
E3
ð2Þ
in which the time-varying nonlinear function FðtÞ satisfies
F T ðtÞFðtÞ 6 I
8t P 0:
ð3Þ
The delays, hðtÞ, are time-varying continuous functions that satisfies
_ hðtÞ 6 hD ;
0 6 hL 6 hðtÞ 6 hU ;
ð4Þ
where hL and hU are positive constants and hD is any constant value. For system (1), we design a memoryless state-feedback controller
uðtÞ ¼ KxðtÞ;
ð5Þ
mn
where K 2 R is a constant matrix to be designed later. With controller (5), system (1) can be rewritten as
_ xðtÞ ¼ ðA þ BKÞxðtÞ þ Ad xðt hðtÞÞ þ DpðtÞ; pðtÞ ¼ FðtÞqðtÞ;
ð6Þ
qðtÞ ¼ ðE1 þ E3 KÞxðtÞ þ E2 xðt hðtÞÞ: The purpose of this paper is to present a delay-dependent stabilization criterion for system (6). Before deriving our main results, we need the following facts and lemma. Fact 1. (Schur complement) Given constant symmetric matrices R1 ; R2 ; R3 where R1 ¼ RT1 and 0 < R2 ¼ RT2 , then R1 þ RT3 R1 2 R3 < 0 if and only if
"
#
R1 RT3 < 0 or R3 R2
R2
RT3
R3 < 0: R1
Fact 2. For any real vectors a; b and any matrix Q > 0 with appropriate dimensions, it follows that T
2aT b 6 aT Qa þ b Q 1 b: Lemma 1 [22]. For any constant matrix M 2 Rnn ; M ¼ M T > 0, scalar c > 0, vector function x : ½0; c ! Rn such that the integrations concerned are well defined, then
Z 0
c
T Z xðsÞ ds M 0
c
Z xðsÞ ds 6 c 0
c
xT ðsÞMxðsÞ ds:
ð7Þ
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O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
3. Main results In this section, we propose a new delay-dependent stabilization criterion for systems (6) with interval time-varying delays. For simplicity, the notations of several matrices are defined as follows:
R T t fT ðtÞ ¼ xT ðtÞ xT ðt hL Þ xT ðt hðtÞÞ xT ðt hU Þ x_ T ðtÞ xðsÞ ds thL fT1 ðtÞ ¼ ½ xT ðtÞ x_ T ðtÞ ;
R
thL thðtÞ
xðsÞ ds
T
R thðtÞ thU
T xðsÞ ds pT ðtÞ ;
fT2 ðtÞ ¼ ½ xT ðtÞ xT ðt hðtÞÞ x_ T ðtÞ pT ðtÞ ; fT3 ðtÞ ¼ ½ xT ðtÞ xT ðt hðtÞÞ l; U T1 ¼ ½ I U T2 U T3 U T4
0 0 0 0 0 0 0 0 ;
¼ ½0 0 0 0 0 I
0 0 0 ;
¼ ½0 0 0 0 0 0 I
0 0 ;
¼ ½0 0 0 0 0 0 0 I
0 ;
R ¼ Rði;jÞ ði; j ¼ 1; . . . ; 9Þ; Rð1;1Þ ¼ AP2 þ P2 AT þ BV þ V T BT þ R1 þ R2 þ R3 þ h2U M11 M22 þ ðhU hL Þ2 N11 þ PT3 þ P 3 þ S2 þ ST2 ; Rð1;3Þ ¼ Ad P2 þ M22 P3 þ PT4 P5 þ P7 S2 ;
Rð1;2Þ ¼ P5 þ S4 ; Rð1;4Þ ¼ P7 S4 ;
Rð1;5Þ ¼ P1 P2 þ P 2 AT þ V T BT þ h2U M12 þ ðhU hL Þ2 N12 þ S1 ; Rð1;6Þ ¼ MT12 þ S3 ;
Rð1;7Þ ¼ S5 ;
Rð2;2Þ ¼ R1 N22 ;
Rð2;3Þ ¼ N 22 þ P T6 ;
Rð2;7Þ ¼ NT12 þ S6 ;
Rð2;8Þ ¼ S6 ; Rð2;9Þ ¼ 0;
Rð1;8Þ ¼ S5 ;
Rð1;9Þ ¼ P 2 D;
Rð2;4Þ ¼ 0;
Rð2;5Þ ¼ 0;
Rð2;6Þ ¼ ST5 ;
Rð3;3Þ ¼ ð1 hD ÞR2 2M22 2N22 P4 PT4 P6 PT6 þ P8 þ PT8 þ hD ðG1 þ G2 þ G3 þ G4 Þ; Rð3;5Þ ¼ P2 ATd ;
Rð3;4Þ ¼ M22 þ N22 P8 ; Rð3;7Þ ¼ NT12 S5 ;
Rð3;8Þ ¼ MT12 NT12 S5 ;
Rð4;4Þ ¼ R3 M22 N22 ; Rð4;8Þ ¼
MT12
þ
NT12
Rð3;6Þ ¼ MT12 S3 ;
S6 ;
Rð3;9Þ ¼ 0;
Rð4;6Þ ¼ ST5 ;
Rð4;5Þ ¼ 0;
Rð4;7Þ ¼ S6 ;
Rð4;9Þ ¼ 0;
Rð5;5Þ ¼ 2P2 þ hU Q 1 þ ðhU hL ÞQ 2 þ h2U M22 þ ðhU hL Þ2 N22 ; Rð5;6Þ ¼ S2 ;
Rð5;7Þ ¼ S4 ;
Rð5;8Þ ¼ S4 ; Rð5;9Þ ¼ DP2 ;
Rð6;7Þ ¼ 0;
Rð6;6Þ ¼ M11 ;
Rð6;8Þ ¼ 0;
Rð6;9Þ ¼ 0;
Rð7;7Þ ¼ N11 ;
Rð7;8Þ ¼ 0; Rð7;9Þ ¼ 0; Rð8;8Þ ¼ M11 N11 ; Rð8;9Þ ¼ 0; W1 ¼ ½ E1 P2 þ E3 V 0 E2 P2 0 0 0 0 0 0 ; h
pffiffiffiffiffiffi
pffiffiffiffiffiffi
pffiffiffiffiffiffi pffiffiffiffiffiffi i hD ST5 hD ST5 ; U4
h D S2 h D S3 N1 ¼ U 1 U2 U3 N2 ¼ diagf G1 ; G2 ; G3 ; G4 g; "
T 1
P ¼
I
Rð9;9Þ ¼ aP2 ;
0 0 0 0 0 0 0 0 0 0 0 0 0
#
; 0 0 0 0 0 0 0 0 0 0 0 2 3 " # " # S1 S2 S4 M 11 M 12 N11 N 12 6 7 ; N¼ ; S ¼ 4 H S3 S5 5; M¼ H M 22 H N 22 H H S6 " # " # " # X1 X2 Y1 Y2 Z1 Z2 ; Y¼ ; Z¼ : X¼ H X2 H Y2 H Z2 0 0
I
ð8Þ Now, we have the following theorem. Theorem 1. For given positive scalars a; hL ; hU , and any scalar hD , system (6) under the control uðtÞ ¼ VP 1 2 xðtÞ is robustly stable for hL 6 hðtÞ 6 hU if there exist positive definite matrices P i ði ¼ 1; 2Þ; Ri ði ¼ 1; 2; 3Þ; Q i ði ¼ 1; 2Þ; Gi ði ¼ 1; . . . ; 4Þ; Si ði ¼ 1; 3; 6Þ; M ii ði ¼ 1; 2Þ; N ii ði ¼ 1; 2Þ; X i ði ¼ 1; 3Þ; Y i ði ¼ 1; 3Þ; Z i ði ¼ 1; 3Þ, and any matrices V; Pi ; ði ¼ 3; . . . ; 8Þ; M 12 ; N 12 ; S2 ; S4 ; S5 ; X 2 ; Y 2 ; Z 2 , satisfying the following LMIs
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
2
R
R
3
WT1
NT1
W1
NT2 3 NT1
6 4 H a1 P2 H H 2 T
7 T 0 5 þ P1 ðhL X þ ðhU hL ÞZÞP1 < 0;
ð9Þ
6 7 T 4 H a1 P2 0 5 þ P1 ðhU X þ ðhU hL ÞYÞP1 < 0; T H H N2 2 2 2 " #3 " #3 " # P3 P5 P7 6X 7 6Y 7 6Z 6 7 > 0; 6 7 > 0; 6 P P P8 4 6 4 5 4 5 4 Q1
H M > 0;
61
Q2
H N > 0;
H
ð10Þ 3 7 7 > 0; 5
ð11Þ
Q1 þ Q2
S > 0:
ð12Þ
Proof. For positive definite matrices P i ði ¼ 1; 2Þ; Ri ði ¼ 1; 2; 3Þ; Q i ði ¼ 1; 2Þ; M ii ði ¼ 1; 2Þ; N ii ði ¼ 1; 2Þ; Si ði ¼ 1; 3; 6Þ, and any matrices M 12 ; N 12 ; S2 ; S4 ; S5 ; S7 ; S8 ; S9 , let us consider the Lyapunov–Krasovskii functional candidate
V¼
6 X
Vi
ð13Þ
i¼1
where
V 1 ¼ fT1 ðtÞEPf1 ðtÞ; Z t Z V2 ¼ xT ðsÞR1 xðsÞ ds þ V3 ¼
Z
thL
Z
t thU
V 4 ¼ hU
Z
s t
t
xT ðsÞR2 xðsÞ ds þ
thðtÞ t
_ x_ T ðuÞQ 1 xðuÞ du ds þ
xT ðsÞR3 xðsÞ ds;
thU
Z
t
_ x_ T ðuÞQ 2 xðuÞ du ds;
s
fT1 ðuÞMf1 ðuÞ du ds;
Z
thL
Z
ð14Þ
t
s
thU
2
t
t s
V 5 ¼ ðhU hL Þ
thL
thU
Z
thU
Z
Z
fT1 ðuÞNf1 ðuÞ du ds;
3T 2 3 xðtÞ xðtÞ R R 6 t 7 6 t 7 xðsÞ ds 7 S6 thðtÞ xðsÞ ds 7 V6 ¼ 6 4 thðtÞ 5 4 5 R thL R thL xðsÞ ds xðsÞ ds thU thU and, E and P in V 1 are defined as
I 0 ; 0 0
E¼
P¼
P1 P2
0 : P2
ð15Þ
From V 1 , we have
_ xðtÞ V_ 1 ¼ 2fT1 ðtÞPT 0 _ xðtÞ ¼ 2fT1 ðtÞPT _ þ ðA þ BKÞxðtÞ þ Ad xðt hðtÞÞ þ DpðtÞ xðtÞ
ð16Þ
¼ 2fT2 ðtÞKT PT Cf2 ðtÞ ¼ fT2 ðtÞðKT PT C þ CT PKÞf2 ðtÞ; where
K¼
I
0 0 0
0 0
I
0
;
C¼
0
0
I
0
A þ BK
Ad
I
D
;
ð17Þ
and f1 ðtÞ and f2 ðtÞ are defined in (8). An upper bound of time-derivative of V 2 can be obtained as
V_ 2 6 xT ðtÞR1 xðtÞ xT ðt hL ÞR1 xðt hL Þ þ xT ðtÞR2 xðtÞ ð1 hD ÞxT ðt hðtÞÞR2 xðt hðtÞÞ þ xT ðtÞR3 xðtÞ xT ðt hU ÞR3 xðt hU Þ:
ð18Þ
62
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
We can obtain the results of calculation of V_ 3 as follows:
_ V_ 3 ¼ hU x_ T ðtÞQ 1 xðtÞ
t
_ ds x_ T ðsÞQ 1 xðsÞ
thðtÞ
Z
Z
thL
_ ds x_ T ðsÞQ 2 xðsÞ
thðtÞ
Z
Z
thðtÞ
_ ds þ ðhU hL Þx_ T ðtÞQ 2 xðtÞ _ x_ T ðsÞQ 1 xðsÞ
thU thðtÞ
_ ds: x_ T ðsÞQ 2 xðsÞ
ð19Þ
thU
Calculating V_ 4 leads to
" 2 V_ 4 ¼ hU
6
#T "
_ xðtÞ "
2 hU
xðtÞ
xðtÞ
M11 M12 H
#T "
_ xðtÞ
M11 H
2 R thðtÞ
#"
xðtÞ
#
t
"
xðsÞ
#T "
M11 M12
#"
xðsÞ
#
Z
ds hU _ _ H M22 xðsÞ xðsÞ 3T " 3 #" # 2Rt #2 R t xðsÞ ds xðsÞ ds M12 xðtÞ M11 M12 thðtÞ thðtÞ 4 5 4 5 R Rt t _ M 22 xðtÞ H M 22 _ ds _ ds xðsÞ xðsÞ
M22
hU
Z
_ xðtÞ
thðtÞ
thðtÞ
thðtÞ
thU
"
xðsÞ _ xðsÞ
#T "
M11 M12 H
M22
#"
xðsÞ _ xðsÞ
# ds
thðtÞ
3T "
3 #2 R thðtÞ #" " #T " # xðsÞ ds xðsÞ ds M11 M12 xðtÞ M11 M12 xðtÞ thU thU 2 4 5 4 5 ¼ hU R R thðtÞ thðtÞ _ _ H M22 H M 22 xðtÞ xðtÞ _ ds _ ds xðsÞ xðsÞ thU thU 2 R 3T " 3 #2 R t t M11 M12 xðsÞ ds xðsÞ ds thðtÞ thðtÞ 5 4 5 4 H M22 xðtÞ xðt hðtÞÞ xðtÞ xðt hðtÞÞ 2 3T " 3 #2 R thðtÞ R thðtÞ M11 M12 xðsÞ ds xðsÞ ds thU thU 4 5 4 5 ¼ fT ðtÞN1 fðtÞ H M22 xðt hðtÞÞ xðt hU Þ xðt hðtÞÞ xðt hU Þ ð20Þ where
2 6 6 6 6 6 6 6 6 N1 ¼ 6 6 6 6 6 6 6 6 6 4
2
2
hU M 11 M22
M 22
0
hU M 12
0
M T12
0
H
2M22
M 22
0
0
MT12
MT12
H
H
M 22
0
0
0
M T12
H
H
H
hU M 22
0
0
0
H
H
H
H
0
0
0
H
H
H
H
H M 11
H
H
H
H
H
H
M11
H
H
H
H
H
H
H
2
0
0
3
7 07 7 7 07 7 7 7 07 7; 7 07 7 7 07 7 07 5 0
and fðtÞ is defined in (8). Here, Lemma 1 was utilized in obtaining an upper bound of V_ 4 . Similarly, an upper bound of V_ 5 can be obtained as
V_ 5 6 fT ðtÞN2 fðtÞ; where
2 6 6 6 6 6 6 6 6 6 6 N2 ¼ 6 6 6 6 6 6 6 6 6 6 4
ð21Þ
ðhU hL Þ2 N11
0
0
0
ðhU hL Þ2 N12
0
0
0
H
N 22
N22
0
0
0
N T12
0
H
H
2N22
N22
0
0
N T12
N T12
H
H
H
N22
0
0
0
NT12
H
H
H
H
ðhU hL Þ2 N22
0
0
0
H
H
H
H
0
0
0
0 0
H
H
H
H
H
H N11
H
H
H
H
H
H
H
N11
H
H
H
H
H
H
H
H
0
3
7 07 7 7 07 7 7 07 7 7 7 0 7: 7 7 07 7 07 7 7 07 5 0
ð22Þ
63
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
Lastly, we have
2 6 6 V_ 6 ¼ 26 4
3T 2
xðtÞ
S1 7 6 7 6 xðsÞ ds thðtÞ 7 4H 5 R thðtÞ R thL H xðsÞ ds þ xðsÞ ds th thðtÞ
S4
S3
S6
H
U
_ ¼ f ðtÞN3 fðtÞ þ 2x ðtÞS2 hðtÞxðt hðtÞÞ þ 2 T
T
32
3 _ xðtÞ 76 7 _ 6 7 S5 7 54 xðtÞ xðt hðtÞÞ þ hðtÞxðt hðtÞÞ 5
S2
Rt
Z
xðt hL Þ xðt hU Þ !T
t
_ S3 hðtÞxðt hðtÞÞ þ 2
xðsÞ ds
Z
xðsÞ ds
T
6 f ðtÞN3 fðtÞ þ hD x
_ ST5 hðtÞxðt hðtÞÞ
T
_ ST5 hðtÞxðt hðtÞÞ
T ðtÞS2 G1 1 S2 xðtÞ
T
þ hD x ðt hðtÞÞG1 xðt hðtÞÞ þ hD Z
þ hD x ðt hðtÞÞG2 xðt hðtÞÞ þ hD
!T
thL
xðsÞ ds
thðtÞ
!T
thðtÞ
xðsÞ ds
thU
ST5 G1 4 S5
Z
!T
t
xðsÞ ds
thðtÞ
T
Z
xðsÞ ds
!T
thðtÞ
thU
þ hD
!T
thL thðtÞ
thðtÞ
þ2
Z
Z
Z
ST5 G1 3 S5
thL
S3 G1 2 S3
Z
!
t
xðsÞ ds
thðtÞ
! xðsÞ ds þ hD xT ðt hðtÞÞG3 xðt hðtÞÞ
thðtÞ
!
thðtÞ
xðsÞ ds þ hD xT ðt hðtÞÞG4 xðt hðtÞÞ;
thU
ð23Þ where
2 6 6 6 6 6 6 6 6 6 6 N3 ¼ 6 6 6 6 6 6 6 6 6 6 6 4
S2 þ ST2
S4
S2
S4
S1
S3
S5
S5
H
0
0
0
0
ST5
S6
S6
H
H
0
0
0
S3
S5
S5
0
ST5
S6
S6
H
H
H
0
H
H
H
H
0
S2
S4
S4
H
H
H
H
H
0
0
0
H
H
H
H
H
H
0
0
H
H
H
H
H
H
H
0
H
H
H
H
H
H
H
H
0
3
7 07 7 7 7 07 7 7 07 7 7 07 7 7 07 7 7 07 7 7 07 5 0
and Fact 2 was utilized in obtaining an upper bound of V_ 6 . As a tool to reduce the conservatism of stabilization criterion, we add the following zero equations with P i ði ¼ 3; . . . ; 8Þ to be chosen as
0 ¼ 2½xT ðtÞP3 þ xT ðt hðtÞÞP4 ½xðtÞ xðt hðtÞÞ 2fT3 ðtÞ
P3
Z
P4
0 ¼ 2½xT ðtÞP5 þ xT ðt hðtÞÞP6 ½xðt hL Þ xðt hðtÞÞ 2fT3 ðtÞ
t
_ ds; xðsÞ
ð24Þ
thðtÞ
P5
Z
P6
thL
_ ds xðsÞ
ð25Þ
thðtÞ
and
0 ¼ 2½xT ðtÞP 7 þ xT ðt hðtÞÞP 8 ½xðt hðtÞÞ xðt hU Þ 2fT3 ðtÞ
P7 P8
Z
thðtÞ
_ ds; xðsÞ
ð26Þ
thU
where f3 ðtÞ is defined in (8). The above three zero equations can be represented as
Z t Z thL Z thðtÞ P3 P P _ ds 2fT3 ðtÞ 5 _ ds 2fT3 ðtÞ 7 _ ds; 0 ¼ fT ðtÞN4 fðtÞ 2fT3 ðtÞ xðsÞ xðsÞ xðsÞ P 4 thðtÞ P 6 thðtÞ P8 thU
ð27Þ
64
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
where
2 T 6 P3 þ P3 6 6 6 6 H 6 6 6 6 6 H 6 6 N4 ¼ 6 6 6 H 6 6 6 H 6 6 H 6 6 H 6 6 4 H H
P 3 þ PT4 P 5 þ P7
P5 0 H H H H H H H
0
P T6 P 4 PT4
!
1
3 P7
0
0
0
0
0
0
0
0
0
0
0
0
0
B C @ P 6 PT6 A P8 þP 8 þ H H H H H H
PT8
0 H H H H H
07 7 7 7 07 7 7 7 7 07 7 7 7: 7 07 7 7 07 7 07 7 07 7 7 05
ð28Þ
0 0 0 0 0 0 0 0 H 0 0 0 H H 0 0 H H H 0 H H H H 0
Furthermore, for any positive matrices X; Y, and Z with appropriate dimensions, the following equations hold:
0¼
Z
t
thðtÞ
0¼
Z
thL
thðtÞ
0¼
Z
fT3 ðtÞXf3 ðtÞ ds fT3 ðtÞYf3 ðtÞ ds
Z
thðtÞ
Z
thðtÞ
thU
t
fT3 ðtÞZf3 ðtÞ ds
thL
thðtÞ
Z
fT3 ðtÞXf3 ðtÞ ds ¼ hðtÞfT3 ðtÞXf3 ðtÞ
Z
t
thðtÞ
fT3 ðtÞXf3 ðtÞ ds;
fT3 ðtÞYf3 ðtÞ ds ¼ ðhðtÞ hL ÞfT3 ðtÞYf3 ðtÞ
Z
thðtÞ
thðtÞ
thU
thL
fT3 ðtÞZf3 ðtÞ ds ¼ ðhU hðtÞÞfT3 ðtÞZf3 ðtÞ
Z
fT3 ðtÞYf3 ðtÞ ds;
ð29Þ ð30Þ
thðtÞ
thU
fT3 ðtÞZf3 ðtÞ ds:
ð31Þ
Since the following inequality holds from (3) and (6),
pT ðtÞpðtÞ 6 qT ðtÞqðtÞ;
ð32Þ
there exist positive matrix P2 and positive scalar a satisfying the following inequality
fT ðtÞWT ðaP2 ÞWfðtÞ pT ðtÞðaP2 ÞpðtÞ P 0;
ð33Þ
W ¼ ½ E1 þ E3 K 0 E2 0 0 0 0 0 0 :
ð34Þ
where
P 1 2 ;V
Let P2 ¼ ¼ KP 2 ; Pi ¼ P2 Pi P 2 ði ¼ 1; 3; 4; . . . ; 7; 8Þ; Ri ¼ P 2 Ri P 2 ; Q i ¼ P 2 Q i P2 ; Gi ¼ P2 Gi P 2 ; M ij ¼ P2 M ij P 2 ; N ij ¼ P2 N ij P 2 ; Si ¼ P P2 Si P 2 ; X i ¼ P 2 X i P2 ; Y i ¼ P2 Y i P 2 , Z i ¼ P2 Z i P 2 . From (16)–(36) and by applying S-procedure [21], the V_ ¼ 6i¼1 V_ i has a new upper bound as T T T T T 1 T 1 T 1 V_ 6 fT ðtÞ!1 ½R þ U T1 ðhD S2 G1 1 S2 ÞU 1 þ U 2 ðhD S3 G2 S3 ÞU 2 þ U 3 ðhD S5 G3 S5 ÞU 3 þ U 4 ðhD S5 G4 S5 ÞU 4 P2 ðhðtÞX 2 " #3 P3 Z t 7 f3 ðtÞ T 6 X f3 ðtÞ T 1 6 þ ðhðtÞ hL ÞY þ ðhU hðtÞÞZÞP2 þ W ðaP2 ÞW!1 fðtÞ !2 4 !2 ds P4 7 5 _ _ xðsÞ xðsÞ thðtÞ Q1 H 2 2 " #3 " # 3 P5 P7 T T Z thðtÞ Z thL 6Y 7 6Z 7 f3 ðtÞ f3 ðtÞ f3 ðtÞ f3 ðtÞ 7 6 7 !2 6 ! ! ! ds ds P P 24 6 5 2 8 4 5 2 xðsÞ _ _ _ _ xðsÞ xðsÞ xðsÞ thU thðtÞ Q2 H H Q1 þ Q2
ð35Þ
where
!1 ¼ diagfP2 ; P2 ; P2 ; P2 ; P2 ; P2 ; P2 ; P2 ; P2 ; g; !2 ¼ diagfP2 ; P2 ; P2 g;
PT2 ¼
I
0 0 0 0 0 0 0 0
0 0
I
0 0 0 0 0 0
:
ð36Þ
Let T T T T 1 T 1 T 1 T 1 X ¼ R þ U T1 ðhD S2 G1 1 S2 ÞU 1 þ U 2 ðhD S3 G2 S3 ÞU 2 þ U 3 ðhD S5 G3 S5 ÞU 3 þ U 4 ðhD S5 G4 S5 ÞU 4 þ W ðaP 2 ÞW:
ð37Þ
Note that for the delay intervals hL 6 hðtÞ 6 hU ,
X þ PT2 ½hðtÞX þ ðhðtÞ hL ÞY þ ðhU hðtÞÞZP2 < 0
ð38Þ
65
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
hold if and only if
X þ PT2 ½hL X þ ðhU hL ÞZP2 < 0
ð39Þ
X þ PT2 ½hU X þ ðhU hL ÞYP2 < 0:
ð40Þ
and
Using Fact 1, the inequalities (39) and (40) are equivalent to the LMIs (9) and (10), respectively. Therefore, if the LMIs (9)– (12) are satisfied, then the system (6) is guaranteed to be asymptotically stable and the corresponding controller gain is obh tained as K ¼ VP 1 2 . This completes our proof. Remark 1. The solutions of Theorem 1 can be obtained by solving the eigenvalue problem with respect to solution variables, which is a convex optimization problem [21]. In this paper, we utilize Matlab’s LMI Control Toolbox [23] which implements the interior-point algorithm. This algorithm is faster than classical convex optimization algorithms [21]. Remark 2. By iteratively solving the LMI given in Theorem 1 with respect to hU for fixed a; hL and hD , one can find the maximum upper bound of time delay hU for guaranteeing asymptotic stability of system (6). For the case uðtÞ ¼ 0, we can obtain the delay-dependent stability criteria for the system (6) with uðtÞ ¼ 0. For simplicity, we define the following notations before introducing stability criteria
R ¼ Rði;jÞ ði; j ¼ 1; . . . ; 9Þ; Rð1;1Þ ¼ PT2 A þ AT P2 þ R1 þ R2 þ R3 þ h2U M11 M22 þ ðhU hL Þ2 N 11 þ PT6 þ P6 þ S2 þ ST2 Rð1;2Þ ¼ P8 þ S4 ; Rð1;3Þ ¼ PT2 Ad þ AT P3 þ M22 P6 þ PT7 P 8 þ P10 S2 ; Rð1;4Þ ¼ P10 S4 ; Rð1;5Þ ¼ P1 PT2 þ AT P4 þ h2U M12 þ ðhU hL Þ2 N12 þ S1 ; Rð1;6Þ ¼ MT12 þ S3 ;
Rð1;7Þ ¼ S5 ;
Rð2;2Þ ¼ R1 N22 ;
Rð2;3Þ ¼ N22 þ PT9 ;
Rð2;7Þ ¼ NT12 þ S6 ;
Rð2;8Þ ¼ S6 ;
Rð3;3Þ ¼
PT3 Ad
þ
ATd P3
Rð2;5Þ ¼ 0;
Rð3;6Þ ¼ MT12 S3 ; Rð3;9Þ ¼ PT3 D þ ATd P5 ;
Rð3;8Þ ¼ MT12 N T12 S5 ;
Rð4;6Þ ¼ ST5 ;
Rð4;5Þ ¼ 0;
Rð4;7Þ ¼ S6 ;
Rð4;8Þ ¼
M T12
þ
Rð5;5Þ ¼
PT4
P4 þ hU Q 1 þ ðhU hL ÞQ 2 þ hU M 22 þ ðhU hL Þ2 N22 ;
Rð5;6Þ ¼ S2 ;
S6 ;
Rð4;9Þ ¼ 0;
ð41Þ 2
Rð5;7Þ ¼ S4 ;
Rð5;9Þ ¼ PT4 D P5 ;
Rð5;8Þ ¼ S4 ;
Rð6;7Þ ¼ 0;
Rð6;6Þ ¼ M11 ;
Rð6;8Þ ¼ 0;
Rð6;9Þ ¼ 0;
Rð7;8Þ ¼ 0; Rð7;9Þ ¼ 0; Rð8;8Þ ¼ M11 N 11 ; W ¼ ½ E1 0 E2 0 0 0 0 0 0 ; h
N1 ¼ U 1
N2 ¼ diagfG1 ; G2 ; G3 ; G4 g; M ¼
X¼ " Pc ¼
Rð7;7Þ ¼ N11 ;
Rð8;9Þ ¼ 0;
Rð9;9Þ ¼ PT5 D þ DT P5 H;
pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi i h D S2 h D S3 hD ST5 hD ST5 ; U2 U3 U4 "
"
Rð2;6Þ ¼ ST5 ;
Rð2;9Þ ¼ 0;
Rð3;5Þ ¼ PT3 þ ATd P4 ;
Rð4;4Þ ¼ R3 M22 N22 ; N T12
Rð2;4Þ ¼ 0;
ð1 hD ÞR2 2M22 2N22 P7 P T7 P9 PT9 þ P 11 þ P T11 þ hD ðG1 þ G2 þ G3 þ G4 Þ;
Rð3;4Þ ¼ M22 þ N 22 P11 ; Rð3;7Þ ¼ NT12 S5 ;
Rð1;9Þ ¼ PT2 D þ AT P5 ;
Rð1;8Þ ¼ S5 ;
#
"
X1
X2
H
X2
P1
0
0
0
P2
P3
P4
P5
Y¼
;
# ;
Y1
Y2
H
Y2 ETc ¼
M 11
M 12
H
M 22
#
" Z¼
; "
I
#
Z1
Z2
H
Z2
0 0 0
0 0 0 0
#
" N¼
; #
N 11
N 12
H
N 22
2
# ;
3
S1 6 S¼4H
S2
S4
S3
7 S5 5 ;
H
H
S6
;
:
Corollary 1. For given positive scalars hL ; hU , and any scalar hD , system (6) with uðtÞ ¼ 0 is robustly stable if there exist positive definite matrices P1 ; H; Ri ði ¼ 1; 2; 3Þ; Q i ði ¼ 1; 2Þ; Gi ði ¼ 1; . . . ; 4Þ; Si ði ¼ 1; 3; 6Þ; M ii ði ¼ 1; 2Þ; N ii ði ¼ 1; 2Þ; X i ði ¼ 1; 3Þ; Y i ði ¼ 1; 3Þ; Z i ði ¼ 1; 3Þ, and any matrices V; P i ; ði ¼ 2; . . . ; 11Þ; M 12 ; N 12 ; S2 ; S4 ; S5 ; X 2 ; Y 2 ; Z 2 , satisfying the following LMIs:
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O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68
2
R WT1 H NT1
6 6H 4
7 T 0 7 5 þ P1 ðhL X þ ðhU hL ÞZÞP1 < 0;
H
H 2
N
H T 1H
R W
6 6H 4 H
6X 6 4
P6
3
#3
T 2
6Y 6 4
Q1
M > 0;
"
2
7 P7 7 5 > 0;
H
ð42Þ
7 T 0 7 5 þ P1 ðhU X þ ðhU hL ÞYÞP1 < 0;
N
H "
T 2
NT1
H
2
3
H N > 0;
P8
#3
"
2
7 P9 7 5 > 0; Q2
6Z 6 4 H
ð43Þ
P 10
# 3
P11
7 7 > 0; 5
ð44Þ
Q1 þ Q2
S > 0:
ð45Þ
Proof. Let us consider the following Lyapunov–Krasovskii’s functional candidate
V¼
6 X
V i;
ð46Þ
V 1 ¼ fT2 ðtÞEc Pc f2 ðtÞ
ð47Þ
i¼1
where
and the same ones V i ði ¼ 2; . . . ; 6Þ in (14). Here, Pc and Ec are defined in (41). With the above Lyapunov–Krasovskii’s functional candidate, the proof of Corollary 1 can be carried out by using methods in the proof of Theorem 1. Hence, it is omitted. h Remark 3. In many industrial systems, hD is unknown. For this case, Corollary 1 can be extended if we do not consider V 6 Rt and the term thðtÞ xT ðsÞR1 xðsÞ ds of V 2 . 4. Numerical examples Example 1. Consider the uncertain dynamic systems with time-varying delays:
_ xðtÞ ¼ A þ DAðtÞxðtÞ þ ðAd þ DAd Þxðt hðtÞÞ þ BuðtÞ;
ð48Þ
where
A¼
0 0 0 1
;
Ad ¼
2 0:5 0
1
;
B¼
0 1
and
D ¼ I;
E1 ¼ E2 ¼ 0:2I:
ð49Þ
When hL ¼ 0, by applying Theorem 1 to the above system, the delay bounds for the conditions hD ¼ 0 and hD ¼ 0:5 are listed in Table 1. Furthermore, when hL ¼ 0:3 and hD ¼ 0, the results are listed in Table 2. From Tables 1 and 2, one can see that Theorem 1 provides significantly improved results. For example, the obtained stabilizing controller with hU ¼ 1:0 by applying Theorem 1 is uðtÞ ¼ ½30:0591 13:4944xðtÞ.
Table 1 Delay bounds with hL ¼ 0(Example 1, a ¼ 4). Methods
hD
hU
Moon et al. [9] Fridman et al. [10] Parlaklßi [15] Li et al. [6] Theorem 1
0
0.4500 0.5865 0.6900 0.84 1.4598
Fridman et al. [10] Parlaklßi [15] Theorem 1
0.5
0.4960 0.6000 1.4597
67
O.M. Kwon, J.H. Park / Applied Mathematics and Computation 208 (2009) 58–68 Table 2 Delay bounds with hL ¼ 0:3 and hD ¼ 0(Example 1, a ¼ 4). Methods
hU
Li et al. [6] Theorem 1
0.9 1.4598
Table 3 MATIs with different methods (Example 2). Methods
MATI
Park et al. [24] Kim et al. [25] Yue et al. [26] Corollary 1
0.0538 0.7805 0.8695 1.0432
Table 4 Delay bounds with different values of hL and hD (Example 3). hL ¼ 0
Li et al. [6] Corollary 1
hL ¼ 0:3
hD ¼ 0:5
hD ¼ 0:9
hD ¼ 0:5
hD ¼ 0:9
0.31 0.56
0.31 0.50
0.55 0.65
0.55 0.62
Example 2. Consider the following Networked Control System
_ xðtÞ ¼
0
1
0 0:1
xðtÞ þ
0 0:1
uðtÞ:
ð50Þ
System (50) is assumed to be controlled by a state-feedback controller uðtÞ ¼ ½ 3:75 11:5 xðtÞ. By applying Corollary 1 and Remark 3 to the above system with unknown hD , we obtained maximum allowable transfer interval (MATI) as 1.0432. In Table 3, MATIs which guarantees the asymptotic stability of system (50) are listed. From Table 3, we can see our proposed method gives a larger MATI than the ones in other literature. Example 3. Consider the following uncertain systems with constant delays:
_ xðtÞ ¼ ðA þ DFðtÞE1 ÞxðtÞ þ ðAd þ DFðtÞE2 Þxðt hðtÞÞ 0:5 1 0:5 2 ; A¼ ; Ad ¼ 0 0:6 1 1 D ¼ I; E1 ¼ E2 ¼ 0:2I:
ð51Þ
In Table 4, the results for different condition of hD are compared with the results in Li [6]. From Table 4, it can be shown that our result for this example gives larger delay bounds than the ones in Li [6].
5. Conclusions In this paper, a new delay-range-dependent stabilization criterion for uncertain dynamic systems with interval timevarying delays is proposed. To obtain less conservative results, new augmented Lyapounv–Krasovskii’s functional and free weighting matrices are utilized by combining with the LMI framework for obtaining the stabilization of the system. The effectiveness of the proposed stabilization criterion is successfully verified by three numerical examples. References [1] [2] [3] [4]
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