Received December 28th, 2010, Revised March 7th, 2011, Accepted for publication April 26th, 2011.
*Permanent Address: Department of Electrical Engineering, Gorontalo State University, Jl. Jendral Sudirman no.6 Kota Gorontalo, Indonesia
An LMI Approach to H
∞Performance Analysis of Continuous-Time Systems with Two Additive Time-
Varying Delays
Wrastawa Ridwan* & Bambang Riyanto Trilaksono
School of Electrical Engineering and Informatics, Institute of Technology Bandung, Bandung, Indonesia
Email: r1space@yahoo.com
Abstract. This paper investigates the problem of H∞ performance analysis for continous–time systems with two additive time-varying delays in the state. Our objective is focused on stability analysis of a continuous system with two time- varying delays with an H∞ disturbance attenuation level γ. By exploiting Lyapunov-Krasovski functional and introducing free weighting matrix variables, LMI stability condition have been derived.
Keywords: H∞ performance analysis; Linear Matrix Inequality (LMI); time delay systems.
1 Introduction
Time delay is the property of a physical system by which the response to an applied force (action) is delayed in its effect. When information or energy is physically transmitted from one place to another, there is a delay associated with the transmission [1]. It is well known that the presence of time-delay is a source of instability [2]. Xia, et al. [3] presents some basic theories of stability and synthesis of systems with time-delay, in the form of
)) ( ( ) ( )
(t Axt Axt t
x d , where τ(t) represents time-varying delay. Wu, et al.
[4] presents a method referred to as the free-weighting-matrix (FWM) approach for the stability analysis and control synthesis of various classes of time-delay systems. In [5], a new model for time delay systems is proposed, that is
)) ( ) ( ( ) ( )
(t Axt Axt 1 t 2 t
x d . The new model is motivated by practical situation in Networked Control Systems (NCSs), where τ1(t) is the time-delay from sensor to the controller and τ2(t) is the time-delay from controller to the actuator.
Motivated by stability condition for system with two delays in the state, derived in [6], in this paper we investigate conditions under which the continuous system with two time-varying delays in the state is asymptotically stable with an H∞ disturbance attenuation level γ. It is well known in systems and control
community that H∞-norm constraint can be used to provide a prespecified disturbance attenuation level, and alternatively to analyze robust stability of dynamical system under unstructured uncertainty. By exploiting Lyapunov- Krasovski functional from [6] and introducing free weighting matrix variables, the stability condition for the system is derived by using linear matrix inequality (LMI) techniques.
Notation. The notation X > 0 denotes a symmetric positive definite, asterisk (*) represents the elements of symmetric term in the symmetric block matrix. The superscripts “T” and “-1” represent the transpose and inverse matrix, respectively. L2[0,) is the space of square integrable functions on [0,∞).
2 Problem definition
As stated previously, systems with two delays in the state can be found in the Networked Control System (NCS), shown in Figure 1 [5]. In NCS, the physical plant, sensor, controller and actuator are located at different locations and hence the signals among those components are transmitted over network media.
Figure 1 Networked control system [5].
We can see in Figure 1 that there are two delays, τs(t) represents the delay of data transmission from sensor to controller while τa(t) represents the delay from controller to actuator. The properties of these two delays may not be identical due to the network transmission condition hence it is not reasonable to combine them together [5]. Based on this observation, Lam, et al. [5] proposed new model for time delay systems, described by x(t)Ax(t)Adx(t1(t)2(t)). Based on such a system representation, Lam, et al. [5] derived the stability condition. Gao, et al. [7] presented a new stability condition and investigated
Physical Plant Sensor
Actuator
Controller
Network Induced Delay τs(t) Network Induced
Delay τa(t) Network Medium
the problem of H∞ performance analysis. Dey, et al. [6] constructed a new Lyapunov-Krasovskii functional in obtaining the stability condition for such system, and provided less conservative delay upper bound, as compared to the conditions in [5,7,8]. By using Lyapunov-Krasovskii functional in [6], in the present paper we investigate the problem of H∞ performance analysis for continuous–time systems with two additive time-varying delays in the state.
Consider the following system with two additive time varying delays in the state [7],
] 0 , [ ), ( ) (
), ( ) ( ) ( )
( ) (
), ( ) ( ) ( )
( ) (
2 1
2 1
t t
t x
t Fw t t t x C t Cx t y
t Ew t t t x A t Ax t x
d
d
(1)
where x(t)n is the state vector; y(t)p is the output vector;
) ( and )
( 2
1 t
t
represent two delays in the state; (t) is the initial condition on the segment
,0
.; w(t)l is the disturbance input which belongs to);
, 0
2[
L A, Ad, E, C, Cd, and F are known system matrices with appropriate dimension. For system in Eq. (1), it is assumed that [6,7],
, )
(
01 t 1 1d1, 02(t)2 , 2d2, (2) and 12, dd1d2
Our objective is to investigate whether the continuous-time system with two time-varying delays in the state is asymptotically stable with an H∞ disturbance attenuation level γ.
Definition 1 If there exist positive definite Lyapunov function V(x,t) such that the derivatives with respect to time t (w = 0) satisfies V(x,t)0, then system (1) is said to be asymptotically stable.
Lemma 1 For any z,yn and for any symmetric positive definite matrix
n
Xn [9],
Xy y z X z y
zT T T
2 1
Lemma 2 Schur Complement. Schur’s formula says that the following statements are equivalent [10]:
i. 0
22 12
12
11
T
ii. 0
0
12 1 22 12 11 22
T
3 Main Result
The Main result of the present paper is stated in the following theorem.
Theorem 1 The continuous-time system with two additive time-varying delays in the state (1) satisfying (2) either
(i) asymptotically stable with w = 0, or
(ii) stable with H∞ disturbance attenuation level γ (w ≠ 0)
if there exist matrices PPT 0, Q1Q1T 0, Q2 Q2T 0,Q3 Q3T 0, ,
1 0
1RT
R R2 R2T 0, R3R3T 0 and Gi, Li, Mi, Ni, i = 1,..., 4 are free matrices with Q2Q3 satisfying,
* 5 0
5 6 6 3 2 2
1
T T
(3) (4)
where,
I R R R Q Q d d Q
Q d
P Q
Q
2 3 2 2 1 1 3 1 2 1 3
2 1 2
1
1
*
*
*
*
0
*
*
*
0 0
1
*
*
0 0
0 1
*
0 0
0
22 21
2
, 21
A 0 Ad I E
T,
1 2 3 4 0
22
T T T
T G G G
G
,
0
*
*
*
*
0 0
*
*
*
0
*
*
0
*
0
4 4 3 3 3 3
4 4 3 2 3 2 2 2 2 2
4 4 1 3 3 1 1 2 1 2 1 1 1 1
3
T T T T
T T T T
T T
T T T
T T
T T T
N L N N L L
N M N N M L N N M M
M L N M L L N M M L M M L L
,
0 0 0
4 4 4
3 3 3
2 2 2
1 1 1
5
N M L
N M L
N M L
N M L
, 6
C 0 Cd 0 F
,
3
1 2 2 1 1 1 -1
5diag- R,-R ,-R
Proof.
Define a Lyapunov-Krasovski functional as in [6], )
( ) ( ) ( )
(t V1 t V2 t V3 t
V
), ( ) ( )
1(t x t Pxt
V T (4) (5)
) (
) ( ) (
3 )
(
2 )
( ) (
1 2
1
2 1 1
2 1
) ( ) ( )
( ) ( )
( ) ( )
(
t t
t t t
T t
t t
T t
t t t
T sQx sds x s Q x s ds x sQ x sds
x t
V
(5)
1
2 1 1
2 1
) ( ) ( )
( ) ( )
( ) ( )
( 1 2 3
3
t
t t
T t
t t
T t
t t
T s Rx s dsd x s Rx s dsd x s Rx s dsd
x t
V (6)
The time derivative of V(t) satisfying condition (2) is given by (as done in [5]) )
( ) ( 2 )
1(t x t Px t
V T , (7)
()
1
( ( ))
( ())) ( )
( 1 2 1 1 2 3 1
2 t x t Q Q xt d x t t Q Q xt t
V T T
1d1d2
xT(t(t))
Q1Q3
x(t(t)) (8)
t
t t
T
T t R R R xt x s Rx s ds
x t V
) (
1 3
2 2 1 1
3() () ( ) ( ) ( )
()
) (
3 )
(
2
1
1
) ( ) ( )
( ) (
t t
t t
T t
t t
T s Rx s ds x s Rx s ds
x
(9)
Now, introducing any free matrices Gi, i=1,2,3,4, one may write
( ) 1 ( 1()) 2 ( ()) 3 () 4
2xT t G xT t t G xT t t G xT t G
x(t)Ax(t)Adx(t(t))Ew(t)
0 (10) Simplifying Eq. (10), we get0 ) (
0
*
*
*
*
*
*
*
*
*
0
* ) (
4 4 4
3 4 3 3 3
2 2
2
1 4 1 3 1
2 1 1
t E G G G
E G G A G G A A G
E G G
A G
E G G A G G A A G G A G A A G
t
T T T d T
T d d
d
T T T
T d T T T T
T
(11)
where (t)
xT(t) xT(t1(t)) xT(t(t)) xT(t) w(t)
T.Now, to eliminate integral terms in Eq. (9), we can use the Newton-Leibniz formula, and introducing free matrices Li, i = 1, 2, 3, 4, we get
0 ) ( )) ( ( ) (
) ( ))
( ( ))
( ( )
( 2
) (
4 3
2 1 1
t
t t
T T
T T
ds s x t
t x t x
L t x L t t x L t t x L t x
(12)
Simplifying Eq. (12), we obtain
0 ) (
0 ) ( 2 )
(
0
*
*
*
*
0 0
*
*
*
0
*
*
0 0 0
*
0
) (
) (
4 3 2 1
4 3 3
2 4 3 1 2 1 1
ds s x L L L L
t L t
L L
L L L L L L L
t
t
t t T T
T T T T
T
T
(13)
Applying Lemma 1 on the last term of Eq. (13), we get
t
t t
T T
T t
t t
T t x s R xs ds
L L L L
R L L L L
t t ds s x L L L L
t
) (
1 4
3 2 1
1 1 4 3 2 1
) (
4 3 2 1
) ( ) ( ) (
0 0 ) ( ) ( ) (
0 ) ( 2
(14)
Substituting Eq. (14) in the last term of Eq. (13) and with little manipulation, we get
) ( 0 0 0
*
*
*
*
0 0
*
*
*
0
*
*
0 0 0
*
0 )
( ) ( ) (
4 3 2 1
1 1 4 3 2 1
4 3 3
2 4 3 1 2 1 1
) (
1 t
L L L L
R L L L L
L L L
L L L L L L L t ds s x R s x
T
T T
T T T
T
T t
t t
T
(15)We can remove the last two terms of Eq. (9) (integral terms) using similar way as done for Eqs. (12) – (15), to obtain
) (
0 0 0
*
*
*
*
0 0
*
*
*
0 0 0
*
*
0
*
0
) ( ) ( ) (
4 3 2 1
1 2 4 3 2 1
1 4 3 2 2
4 3 2 1 1 1
) (
2 1
t M M M M
R M M M M M
M M M
M M M M M M
t ds s x R s x
T
T T T
T T T T
T t
t t
T
(16)
) (
0 0 0
*
*
*
*
0 0
*
*
*
0
*
*
0
*
0 0 0
) ( ) ( ) (
4 3 2 1
1 3 4 3 2 1
2 4 3 3
4 3 2 2 2
1 1
) (
) (
3 1
t N N N N
R N N N N
N N N
N N N N N
N N
t ds s x R s x
T
T T
T T T
T t
t
t t
T
(17)where Mi, i = 1, 2, 3, 4 and Ni, i = 1, 2, 3, 4 are free matrices.
Substituting Eqs. (15), (16) and (17) into Eq. (9), we get
( ) ()
()) ( )
( 1 1 2 2 3 31 32 33 34
3 t x t R R R xt t V V V V t
V T T (18) where
0
*
*
*
*
0 0
*
*
*
0
*
*
0
*
0
4 4 3 3 3 3
4 4 3 2 3 2 2 2 2 2
4 4
` 3 3 1 1 2 1 2 1 1 1 1
31
T T T T
T T T T
T T
T T T
T T
T T T
N L N N L L
N M N N M L N N M M
M L N M L L N M M L M M L L V
T
L L L L
R L L L L
V
0 0
4 3 2 1
1 1 4 3 2 1
32
,
T
M M M M
R M M M M
V
0 0
4 3 2 1
1 2 4 3 2 1
1
33
,
T
N N N N
R N N N N
V
0 0
4 3 2 1
1 3 4 3 2 1
2
34
Then, from Eqs. (7), (8), (11) and (18) we have
() () () )
(t V1 t V2 t V3 t
V
) ( 0 0 0 0 0 0
*
*
*
*
*
*
*
*
*
* ) ( ) (
4 3 2 1
1 3 4 3 2 1
2 4 3 2 1
1 2 4 3 2 1
1 4 3 2 1
1 1 4 3 2 1
55 45 44
35 34 33
25 24 23 22
15 14 13 12 11
t N N N N
R N N N N
M M M M
R M M M M
L L L L
R L L L L
t t V
T T
T
T
(19)
where,
T T
T
TG L L M M
A A G Q
Q1 2 1 1 1 1 1 1
11
,12 ATG2TLT2M1M2TN1,
1 3 3 1 3 1
13GAd ATGT L LT MT N
,14 PG1ATG4T LT4M4T, E
G1
15
, 22
1d1
Q2Q3
M2M2TN2N2T,T T
d L M N N
A
G2 2 3 2 3
23
,24G2M4T N4T,25 G2E,
d1 d2
Q1 Q3
G3Ad AdTG3T L3 LT3 N3 N3T331
,
T T T T
dG L N
A
G3 4 4 4
34
,35G3E,44 R11R22R3G4G4T, E
G4
45
, 55 0.
Simplifying Eq. (19) we have
()) ( )
(t t 1 2 2 3 4 t
V T T (20)
where
0
*
*
*
*
0
*
*
*
0 0
1
*
*
0 0
0 1
*
0 0
0
3 2 2 1 1 3 1 2 1 3
2 1 2
1
1
R R R Q Q d d Q
Q d
P Q
Q
,
T 5 5 5
4
,
2 31
1 2 1 1 1
5diagR,R, R
, and 2, 3, 5 are given in Eq. (3).
Thus, we have
( )) (
) ( ) ( )
( ) ( ) (
4 6 6 3 2 2 1
2
t t
t w t w t y t y t V
T T
T
T T
(21)
where 1, 2, 3, 6 are given in (3) and 4 is given in (20).
First, we consider the asymptotic stability of system Eq. (1) satisfying Eq. (2) with w (t) = 0. For this case, from Eq. (21) we have
()) ( )
(t t 1 2 2 3 4 t
V T T (22)
where
xT t xT t t xT t t xT t
Tt) () ( ( )) ( ()) ()
( 1
