ABSOLUTE STABILITY AND APPLICATION TO DESIGN OF OBSERVER-BASED CONTROLLER FOR NONLINEAR TIME-DELAY SYSTEMS

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  • 362 Asian Journal of Control, Vol. 9, No. 3, pp. 362-371, September 2007

    Manuscript received December 3, 2005; revised July 6, 2006; accepted October 2, 2006.

    Bassem Ben Hamed is with the Facult des Sciences de Gabs, Dpartement de Mathmatiques, Cit Riadh, Zirig 6072, Gabs, Tunisia (e-mail: bassem@picard.ups-tlse.fr).

    Abdallah Ben Abdallah and Mohamed Chaabane are with the Institut Prparatoire aux Etudes dIngnieur de Sfax, Route Menzel Chaker, BP 805 Sfax 3000, Tunisia.

    This work was partially supported by 03/UR/15-04 Stabili-ty and Control Systems Laboratory of the Faculty of Sciences of Sfax, Tunisia.

    Brief Paper

    ABSOLUTE STABILITY AND APPLICATION TO DESIGN OF OBSERVER-BASED CONTROLLER FOR NONLINEAR

    TIME-DELAY SYSTEMS

    Bassem Ben Hamed, Abdallah Ben Abdallah, and Mohamed Chaabane

    ABSTRACT

    In this paper we present a new sufficient condition for absolute stability of delay Lure system. This condition improves the one given in [1]. We use this new criterion to construct an observer-based con-trol for a class of nonlinear time-delay systems. Some examples are given to illustrate the results of this paper.

    KeyWords: Delay system, absolute stability, Lure system, LMI, S-procedure, Lyapunov-Krasovskii functional.

    I. INTRODUCTION

    Time-delay systems constitute basic mathematical models of real phenomena such as nuclear reactors, chem-ical engineering systems, biological systems, and popula-tion dynamics models. They are often a source of instabili-ty and degradation in control performance in many control problems, see [2].

    During the last two decades, the problem of stability of linear time-delay systems has been subject of considera-ble research efforts. Many significant results have been reported in the literature. For the recent progress, the reader is referred to [3] and the references therein.

    Since, the introduction of absolute stability by Lure (1957), the problem of absolute stability of a class of non-linear control systems with a fixed matrix in the linear part of the system and one or multiple uncertain nonlinearities satisfying the sector constraints has been studied in [4-9].

    Due to time-delay occurring in practical systems, the problem of absolute stability for systems with delay has been also studied in [1,10-15]. However, the results men-tioned above are delay-independent. When the time-delay is small, these results are often overly conservative; espe-cially, they are not applicable to closed-loop systems which are open-loop unstable and are stabilized using delayed inputs, due to either delayed measurements or delayed ac-tuator action in the input channels. Recently, in [16-18], the authors has studied the delay-dependent absolute stability for uncertain time-delay systems, which motivates the present work.

    In this paper, we will deal with the problem of abso-lute stability for a class of time-delay systems which can be represented as a feedback connection of a linear dynamical system and a nonlinearity satisfying a sector constraint. Delay-dependent absolute stability criteria will be derived with employing some integral inequalities, and a new crite-ria are obtained and formulated in the form of linear matrix inequalities (LMI) which improve the one given in [16]. We use this new criterion to construct a delay-dependent dynamic observer-based output feedback control such that the observer error system is presented as the feedback in-terconnection of a linear system and a state-dependent mul-tivariable sector-bounded nonlinearity. In the observer de-sign, we extend the works of [19-21] in the case without delay. Finally, some numerical examples are given to illu-strate the applicability of the results. Notation. Throughout this paper, R is the set of real numbers, Rn denotes the n dimensional Euclidean space, and Rn m is

  • B.B. Hamed et al.: Absolute Stability and Application to Design of Observer-Based Controller 363

    the set of all n m real matrices. I is the identity matrix ma-trix. The set Cn, h := C ([h, 0], Rn) is the space of continuous functions mapping the interval [h, 0] to Rn, where h is a positive constant. The notation A > 0 denote that the matrix A is positive definite.

    II. ABSOLUTE STABILITY ANALYSIS

    We consider the following delay-system

    0 1( ) ( ) ( ) ( )x t A x t A x t h B t= + + , 0 1( ) ( ) ( )y t C x t C x t h= + ,

    ( ) ( ( ))t t y t = , , (1) where x(t) Rn is the system state, y(t) Rp the measured output, and the nonlinear function ( ) p p+ ., . : R R R is assumed to be continuous and belongs to sector [0, K], i.e. (., .) satisfies

    ( )[ ( ) ] 0 ( ) pt y t y Ky t y + , , , , ,R RF (2) where K is a positive definite matrix. The matrices A0, A1, B, C0, and C1 are real matrices with appropriate dimen-sions.

    The initial condition of (1) is given by

    ( ) ( ) [ 0] n hx t t t h ,= , , , .C It is assumed that the right-hand side of (1) is conti-

    nuous and satisfies enough smoothness conditions to en-sure the existence and uniqueness of the solution through every initial condition .

    We first introduce the following definition. Definition 1. The system (1) is said to be absolutely stable in the sector [0, K] if the system is globally uniformly asymptotically stable for any nonlinear function (t, y(t)) satisfying (2).

    The development of the work in this paper requires the following lemma which is Proposition 3 in Ref. [18].

    Proposition 1. Let x(t) Rn be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices

    1 2n nM M , R and 0X X= >F , and a scalar h > 0: ( ) ( ) ( ) ( )tt h s X x s ds t tx FF

    1( ) ( )h t X t+ ,F F (3) where

    1 1 1 2

    2 2

    M M M MM M

    + + := , F F

    F1

    2

    ( )( )

    ( )x tM

    tx t hM

    := , := . F

    FF

    Proof. See [18].

    Remark 1. The inequality (3) is called an integral inequa-lity. It plays a key role in the derivation of a criterion for absolute stability in this paper. Note that the free matrices M1 and M2 make the LMI (4) proposed in Theorem 1 more feasible. Using the inequality (3), we will generalize the absolute stability criterion given in [16].

    Under the sector condition (2), we will give a suffi-cient condition for absolute stability of system (1). We have the following theorem.

    Theorem 1. For given scalar h > 0, the system (1) with nonlinear function satisfying (2) is absolutely stable if there exist a scalar > 0, a positive definite matrices P > 0, Q > 0, R > 0, and real matrices M1, M2 Rnn such that the LMI

    11 112 13 0

    22 223 1

    02 00

    hA R hMhA R hM

    I hB RhR

    hR

    = < ,

    FF

    FF

    F (4)

    where

    11 0 0 1 1A P PA Q M M = + + + +F F , 12 1 1 2PA M M = +F , 13 0PB C K = F , 22 2 2Q M M = F , 23 1C K = F ,

    holds.

    Proof. We consider the Lyapunov-Krasovskii functional candidate

    ( ) ( ) ( ) ( ) ( )tt t hV t x x t Px t x s Qx s ds, = + F F 0 ( ) ( )th t s Rx s ds dx ++ , F (5)

    where the matrices P, Q, and R are positive definite. The derivative of V along the trajectories of system (1) is given by

    ( ) 2 ( ) ( ) ( ) ( ) ( ) ( )tV t x t Px t x t Qx t x t h Qx t hx, = + F FF ( ) ( ) ( ) ( )tt hh t Rx t s Rx s dsx x+ . F F (6)

    Applying the integral inequality (3) to the term of the right-hand side of (6) for any matrices M1, M2 Rnn yields the following inequality

    ( ) ( ) ( )V t t t , F (7) where

    11 12 13

    22 23

    33

    ( )( ) ( )

    ( )

    x tt x t h

    t

    = , := ,

  • 364 Asian Journal of Control, Vol. 9, No. 3, September 2007

    with

    111 0 0 1 1 0 0 1 1A P PA Q M M hA RA hM R M = + + + + + +F F F F ,

    112 1 1 2 0 1 1 2PA M M hA RA hM R M = + + +F F F , 13 0PB hA RB = + F ,

    122 2 2 1 1 2 2Q M M hA RA hM R M = + +F F F , 23 1hA RB = F 33 hB RB = F .

    A sufficient condition for absolute stability of the system (1) is that there exist real matrices P > 0, Q > 0, and R > 0 such that

    ( ) ( ) ( ) 0V t t t < , F (8) for all (t) 0. The sector condition (2) implies

    0 1( ) ( ) ( ) [ ( ) ( )] 0u t u t u t KC x t KC x t h+ + .F F (9) Using the S-procedure, we can find > 0 such that

    ( ) ( ) 2 ( ) ( )t t t t F F 0 12 ( ) [ ( ) ( )] 0t KC x t KC x t h + < ,F (10)

    for all (t) 0. Rewrite (10) as

    ( ) ( ) 0t t < ,F (11) where

    11 12 13

    22 23

    33

    = ,

    with

    ij ij = , ( 1 2i j, = , ), 13 13 0C K = F , 23 23 1C K = F , 33 33 2 I = .

    Using Shur complement, the LMI ( < 0) is equivalent to the LMI (4). This completes the proof.

    Remark 2. In [17], the authors considered the absolute stability of system (1) for the case of C1 = 0. The idea in [17] was to first transform the system to a system with a distributed delay, then to apply an inequality to some cross terms. However, from the proof process of Theorem 1, one can clearly see that neither model transformation nor bounding technique for cross terms is involved.

    III. STABILIZATION OF NONLINEAR DELAY SYSTEM

    This section present the delay-dependent stabilization condition obtained by using the absolute stability proposed in section 2.

    Consider the following nonlinear control delay system

    0 1( ) ( ) ( ) ( ) ( )x t A x t A x t h B t Gu t= + + + , 0 1( ) ( ) ( )y t C x t C x t h= + ,

    ( ) ( ( ))t t y t = , , (12) where A0, A1, B, G, C0, C1 are real matrices with appropri-ate dimensions, and the nonlinearity (t, y) belong in the sector [0, K], K > 0.

    The initial condition of (12) is given by

    ( ) ( ) [ 0] n hx t t t h ,= , , , .C It is assumed that the right-hand side of (12) is continuous and satisfies enough smoothness conditions to ensure the existence and uniqueness of the solution through every initial condition .

    The closed-loop system with the state feedback con-trol

    ( ) ( )u t N x t= , (13) is given by

    0 1( ) ( ) ( ) ( ) ( )x t A GN x t A x t h B t= + + + . (14) The following theorem gives a sufficient condition for sta-bilization of the system by means a state feedback when the nonlinearity ( )t y , belong in the sector [0, K]. Theorem 2. For given numbers 0 1 2ih i> , , = ,R , if there exist a scalar 0> , a positive definite matrices

    0 0 0P Q R> , > , > , and a matrix n rY R such that the LMI

    11 12 13 14 1

    22 23 24 2

    34 02 0

    0

    h R

    h RI

    hR

    hR

    = < ,

    (15)

    where

    11 0 1 0 1( ) ( )P A I A I P GY Y G Q = + + + + + +F F F , 12 1 2 1( )A P P = + , 13 0B P C K = F , 14 0hPA hY G = +F F F ,

  • B.B. Hamed et al.: Absolute Stability and Application to Design of Observer-Based Controller 365

    22 22Q P = , 23 1P C K = F , 24 1hPA = F , 34 hB R = F ,

    holds. Then the origin of the controlled system (12) is sta-bilizable by the linear state feedback (13) where

    1N Y P

    = . Proof. Let h > 0, and 1, 2 are fixed reals. Using Theorem 1, the closed-loop system is stable if there exist a positive definite matrices P > 0, Q > 0, R > 0, and M1, M2 Rnn such that the LMI (4) with replacing A0 by A0 + GN holds, then the origin of system (14) is globally uniformly asymptotically stable. This is equivalent to the feasibility of the following LMI

    0TT T = < ,F where T = diag{P1, P1, I, R1, R1}. Denoting 1P P= ,

    1 1,Q P QP = 1,R R= 1 ,NP Y = and picking Mi = iP, i = 1, 2, we obtain the desired LMI (15). Remark 3. The criterion in Theorem 2 does not require any assumptions about the systems matrices, e.g., the pairs (A0, G) and (A0 + A1, G) need not stabilizable.

    IV. OBSERVER-BASED CONTROL FOR A CLASS OF NONLINEAR DELAY SYSTEM

    In this section, observer-based control approach, is developed to perform the stabilization of the system pro-posed in section 3 where the state information is not avail-able.

    4.1 System description

    Now, we consider the time delay system of the form

    0 1( ) ( ) ( ) ( )x t A x t A x t h B t= + + , 0 1( ) ( ) ( )y t C x t C x t h= + ,

    0 1( ) ( ( ) ( ))t H x t H x t h = + , (16) where x(t) Rn is the state, y(t) Rr is the measured out-put, h > 0 is the constant delay, and the multivariable non-linearity (.) : Rp Rp is decentralized in the sense that i(v), v Rp, 1 i p, depends only on vi; that is

    1 1( )( )

    ( )p p

    vv

    v

    = . # (17)

    We suppose that the nonlinearities (.) satisfy ( ) ( ) 1i ii it ta b t t t t i pt t

    , , , , . R (18)

    If ( ) 1i i p . , , is continuous differentiable, then its slope is restricted by

    ( ) [ ] for all 1i i i i ii

    d v a b i p vdv , , .R

    The scalar function i staisfies (18) with 0ia = , 1 i p . (If 0ia , we can define a new functions

    ( ) ( )i i it t a t := which satisfies (18) with 0ia = , 1i i ib b a i p= , , and we rewrite (16) as i i i

    i

    0 1

    0 1

    0 1

    ( ) ( ) ( ) ( )

    ( ) ( ) ( )

    ( ) ( ( ) ( ))

    x t A x t A x t h B t

    y t C x t C x t h

    t H x t H x t h

    = + + ,= + ,

    = + ,

    with

    ii

    0 1 00

    1 1 11

    diag{ ... }

    diag{ ... } .

    p

    p

    A B a a HA

    A B a a HA

    = , , ,= , ,

    The matrices n niA R , r niC R , n pB R and p n

    iH R , 0 2i , are constant matrices. The initial condition of (16) is given by

    ( ) ( ) [ 0] n hx t t t h ,= , , , .C It is assumed that the right-hand side of (16) is continuous and satisfies enough smoothness conditions to ensure the existence and uniqueness of the solution through every initial condition .

    4.2 Observer design

    In this subsection, observer is designed for a class of delay-dependent nonlinear systems. The approach is to represent the observer error system as the feedback inter-connection of a linear system and a state-dependent multi-variable sector-bounded nonlinearity. We extend a design proposed in [19-21] in the case without delay.

    With the assumption (18), our observer has the fol-lowing form

    0 1 ( ) ( ) ( ) ( ( ) ( )) ( )x t A x t A x t h L y t y t B t= + + + , 0 1 ( ) ( ) ( )y t C x t C x t h= + ,

    0 1 ( ) ( ( ( ) ( )) ( ) ( ))t S y t y t H x t H x t h = + + , (19)

  • 366 Asian Journal of Control, Vol. 9, No. 3, September 2007

    Our task is to design the matrices p rS R and n rL R to make the observer error ( ) ( ) ( )e t x t x t=

    approach zero. At this point we assume that the solution of (16) does not escape to infinity in finite time.

    From (16) and (18), the dynamics of the observer error e(t) is governed by

    0 0 1 1( ) ( ) ( ) ( ) ( )e t A LC e t A LC e t h= + + + [ ( ( )) ( ( ))]B v t w t , (20)

    where

    0 1( ) ( ) ( )v t H x t H x t h= + , 0 1 ( ) ( ( ) ( )) ( ) ( )w t S y t y t H x t H x t h= + + .

    To represent the observer error system (20) as the feedback interconnection of a linear system and multivariable sector nonlinearity, we view ( ( )) ( ( ))v t w t as a function of t and

    0 0 1 1( ) ( ) ( ) ( ) ( ) ( ) ( )z t v t w t H SC e t H SC e t h:= = + + + .

    That is, a state-dependent multivariable nonlinearity in t and z(t):

    ( ( )) ( ( )) ( ( ))t z t v t w t , := . (21) We rewrite the observer error (20) as

    0 0 1 1( ) ( ) ( ) ( ) ( ) ( )e t A LC e t A LC e t h B t= + + + + , 0 0 1 1( ) ( ) ( ) ( ) ( )z t H SC e t H SC e t h= + + + ,

    ( ) ( ( ))t t z t = , . (22) First, we start showing under the condition given in (18) that ( ( ))t z t , satisfies a multivariable sector property. Indeed, since

    ( ( )) ( ( )) ( ( ))i i it z t v t w t , = ( ( )) ( ( )) ( ( ))i i i i i iv t w t t z t= = ,

    10

    ( ( ) ( ))

    ( ( ) ( )) 1i i i

    i is v w t v t

    v t w t d i ps = +

    = , .

    Thus, from (18), each component ( )i ., . satisfies the following sector condition

    ( ( )) [ ( ( )) ( )] 0 1i i i i i it z t t z t b z t i p , , , . (23) Taking 1diag{ ... } 0pK b b= , , > , it can be seen that (see [4])

    ( ( )) [ ( ( )) ( )] 0t z t t z t Kz t , , .F (24) The convergence of observer is a consequence of Theorem 1. We have the following result.

    Theorem 3. For given numbers 1 2i i , = , , if there exist a positive scalar 0> , a positive definite matrices 0P > ,

    0 0Q R> , > , n rY R , and p rS R such that the LMI

    11 12 13 14 1

    22 23 24 2

    34 02 0

    0

    h R

    h RI

    hR

    hR

    = < ,

    (25)

    with

    11 0 1 0 1 0 0( ) ( )A I P P A I YC C Y Q = + + + + + +F F , 12 1 2 1 1( ( ) )P A I YC = + + , 13 0 0( )PB H SC K = + F , 14 0 0hA P hC Y = +F F F , 22 22Q P = , 23 1 1( )H SC K = + F , 24 1 1hA P hC Y = +F F F , 34 hB P = F ,

    holds. Then the system (19) is an observer of system (16). The gain matrix S is given by the resolution of the previous LMI and

    1L P Y= . Proof. A sufficient condition for the convergence of the observer is given by Theorem 1 with replacing Ai by Ai + LCi, and Hi by Hi + SCi, i = 1,2. That is the existence of > 0, a positive definite matrices 0 0 0P Q R> , > , > , and real matrices 1 2

    n nM M , R such that 11 12 13 14 1

    22 23 24 2

    34 02 00

    hMhM

    IhR

    hR

    :=

  • B.B. Hamed et al.: Absolute Stability and Application to Design of Observer-Based Controller 367

    This LMI is equivalent to

    0T ZZ Z = < , where Z = diag 1 1( )I I I R P R P , , , , . If we choose

    1 2i iM P i= , = , , Y PL= , and 1R PR P= , we obtain the desired LMI (25).

    4.3 Observer-based control feedback Our goal is to find a sufficient condition for to the ex-

    istence of an observer-based control feedback witch robust stabilize the following system

    0 1( ) ( ) ( ) ( ) ( )x t A x t A x t h B t Gu t= + + + , 0 1( ) ( ) ( )y t C x t C x t h= + ,

    0 1( ) ( ( ) ( ))t H x t H x t h = + , (26) where ( ) mu t R is the control input, and the nonlinearity (.) satisfies the condition (17)-(18) with ai = 0. Consider the closed-loop system (26) with the control feedback

    ( ) ( )u t N x t= , (27) and the dynamic of observer is

    0 1 ( ) ( ) ( ) ( ) ( ( ) ( )) ( )x t A GN x t A x t h L y t y t B t= + + + + , 0 1 ( ) ( ) ( )y t C x t C x t h= + ,

    0 1 ( ) ( ( ) ( ) ( ( ) ( )))t H x t H x t h S y t y t = + + . (28) It follows that

    0 0 1 1( ) ( ) ( ) ( ) ( ) ( )e t A LC e t A LC e t h B t= + + + + , ( ) ( ( ))t t z t = , , (29)

    with

    ( ( )) ( ( )) ( ( ))t z t v t w t , = , 0 1( ) ( ) ( )v t H x t H x t h= + ,

    0 1 ( ) ( ( ) ( )) ( ) ( )w t S y t y t H x t H x t h= + + , 0 0 1 1( ) ( ) ( ) ( ) ( ) ( ) ( )z t v t w t H SC e t H SC e t h= = + + + .

    Here, the objective of this subsection is to develop a pro-cedure to design an observer-based controller for the time- delay system (26), such that the resulting closed subsystem given by

    0 1( ) ( ) ( ) ( ) ( ( )) ( )x t A GN x t A x t h B v t GNe t= + + , (30)

    is globally asymptotically stable.

    Theorem 4. For a given numbers 1 2 , R , if there exist a positive scalars 1 20 0> , > , a positive definite matric-es x x x e e eP Q R P Q R, , , , , and two matrices N1, 2 n nN R such that

    11 12 13 0 1

    22 23 1 2

    22 0 0

    0

    x x

    x x

    T T T

    T T

    T

    x

    x

    hP A hY G h RhP A h R

    I hBT T

    hR

    hR

    + := = < ,

    F F F

    F

    FF

    (31) and

    11 012 1 0 0 1

    22 11 1 1 2

    1

    ( )( )

    02 00

    N ee

    e

    e

    e

    e

    P B H SC K hA R hNH SC K hA R hN

    I hB RhR

    hR

    + +

    = < ,

    FF F

    FF F

    F

    (32) where

    11 0 0 0 0 1 1e e eP A A P Q YC C Y N N = + + + + + +FF F F 12 1 1 1 2eP A YC N N = + + ,F 22 2 2eQ N N = ,F 11 0 1 0 1( ) ( )x x xA I P P Q GY Y GA I = + + + + ++ F F F 12 1 2 1( )x xT A P P = + , 13 2 0xT B P H K = ,F 22 22 xT xQ P = , 23 2 1xT P H K = .F

    Then the system (26) is stabilizable by the dynamic feed-back (27)-(28), where the matrices gain is given by

    11xeL P Y N P Y= , = .

    Proof. Define the Lyapunov-Krasovskii functional V as follows

    ( ) ( ) ( )x t e tV t V t x a V t e:= , + , , (33) where

    ( ) ( ) ( ) ( ) ( )tx t x xt hV t x x t P x t x s Q x s ds, = + F F 0 ( ) ( )t xh t s R x s ds dx ++ , F (34)

    and

    ( ) ( ) ( ) ( ) ( )te t e et hV t e e t P e t e s Q e s ds, = + F F 0 ( ) ( )t eh t s R e s ds de ++ , F (35)

    with t t n hx e ,, C , and a is a postive real will be chosen. The Lyapunov-Krasovskii functional (33) is legitimate

    functional, see [1] and [22].

  • 368 Asian Journal of Control, Vol. 9, No. 3, September 2007

    The corresponding derivate of the functional (34) is given by

    ( ) 2 ( ) ( ) ( ) ( )t x xx t x t P x t x t Q x txV , = + FF ( ) ( )xx t h Q x t h F ( ) ( ) ( ) ( )tx xt hh t R x t s R x s dsx x+ . F F (36)

    Applying the integral inequality (3) to the term of the right- hand side of (36) and the inequalities

    12 ( )( ) ( ) ( ) ( )x xe t GN P x t x t P x tb F F F

    ( )( ) ( )xbe t GN P GN e t+ ,F F and

    2 ( )( ) ( ) ( )( ) ( )x xhe t GN R t hbe t GN R GN e t F F F F ( ) ( )x

    h t R tb

    + ,F F

    we obtain, for any matrices 1 2n nM M , R and any b > 0,

    the following inequality

    1( ) ( ) ( ) ( ) ( )t x xxht x t R t x t P x tV b b

    , + + F F F

    ( )( ) ( (1 ) ) ( )x xe t GN bP h b R GN e t+ + + ,F F (37) where

    11 12 13

    22 23

    33

    = ,

    0

    1

    ( )( ) ( )

    ( ( ))

    Nx t At x t h A

    v t B

    := , := ,

    F

    F F

    F

    with 0 0NA A GN:= + , and

    0 011 1N x x N xA P P A Q M = + + +F F

    0 0

    11 1 1N x N xM hA R A hM R M+ + +F F ,

    0

    112 1 1 2 1 1 2x N x xP A M M hA R A hM R M = + + +F F F ,

    013 x N xP B hA R B = + F , 1

    22 2 2 1 1 2 2x x xQ M M hA R A hM R M = + +F F F , 23 1 xhA R B = F , 33 xhB R B = F .

    Now, the corresponding derivate of functional (35) is given by

    ( ) 2 ( ) ( ) ( ) ( ) ( ) ( )t e e ee t e t P e t e t Q e t e t h Q e t heV , = + F FF ( ) ( ) ( ) ( )te et hh t R e t s R e s dse e+ . F F (38)

    Applying the integral inequality (3) to the term of the right- hand side of (38) for any 1 2

    n nN N , R yields the fol-lowing inequality

    ( ) ( ) ( )te t e t tV , , F (39) where

    11 12 13

    22 23

    33

    ( )( ) ( )

    ( )

    e tt e t h

    t z

    = , = , ,

    with

    11 0 0 0 0 1 1( ) ( )e e eP A LC A LC P Q N N = + + + + + +F F 11 1 0 0 0 0( ) ( )e ehN R N h A LC R A LC+ + + + ,F F

    12 1 1 1 2( )eP A LC N N = + +F 10 0 1 1 1 2( ) ( )e eh A LC R A LC hN R N+ + + + ,F F

    13 0 0( )e eP B h A LC R B = + + ,F 1

    22 2 2 2 2e eQ N N hN R N = +F F 1 1 1 1( ) ( )eh A LC R A LC+ + + ,F

    23 1 1( ) eh A LC R B = + ,F 33 ehB R B = .F

    Using (37) and (39), we obtain

    ( ) ( ) ( ) ( )( ) ( ( 1) ) ( )x xV t t t e t GN abP a b hR GNe t + + + F F F ( ) ( ) ( ) ( )x x

    h aa t R t x t P x tb b

    + + + . F F F

    A sufficient condition for stability is

    ( ) ( ) ( ) 0 and ( ) ( ) 0eV t t t t t < , < ,F F Indeed, in this case by applying the S-procedure we can choose b > 0 large enough such that

    1( ) ( ) ( ) ( ) 0x xht R t x t P x tb b

    + + < . F F F

    Then we chose a > 0 small such that

    ( ) ( ) ( )( ) ( ( 1) ) ( ) 0x xt t e t GN abP a b hR GNe t + + + < .F F F This implies the negativity of ( )V t , and hence the stabili-ty.

    Using Theorem 3, ( ) ( ) ( ) 0e tV t e t t,

  • B.B. Hamed et al.: Absolute Stability and Application to Design of Observer-Based Controller 369

    11 12 13 0 1

    22 23 1 2

    33 000

    e

    e

    e

    e

    e

    hA R hNhA R hNhB R

    hRhR

    = < ,

    F F

    F F

    F

    where

    11 0 0 0 0 1 1

    12 1 1 1 2

    13 1 0 0

    22 2 2

    23 1 1 1

    33 1

    ( ) ( )

    ( )

    ( )

    ( )

    2

    e e e

    e

    e

    e

    P A LC A LC P Q N N

    P A LC N N

    P B H SC K

    Q N N

    H SC K

    I

    = + + + + + + = + + , = + , = , = + , = .

    F F

    F

    F

    F

    F

    Suppose that we have determine the matrix N. In this case and for a given 1 , we can transform as in proof of Theo-rem 3, < 0 into LMI in Pe > 0, Qe > 0, 0eR > ,

    1 2eY P L S N N= , , , . To establish a sufficient condition in term of LMI, we

    use the sector condition

    ( ( )) [ ( ( )) ( )] 0v t v t Kv t ,F (40) and the S-procedure implies

    ( ) ( ) 2 ( ( )) ( ( ))t t v t v t F F 0 12 ( ( )) [ ( ) ( )] 0v t K H x t H x t h + < ,F (41)

    for all ( ) 0t . This give the following LMI 11 12 13 0 1

    22 23 1 2

    33 000

    N x

    x

    x

    x

    x

    hA R hMhA R hMhB R

    hRhR

    = < ,

    F F

    F F

    F

    where

    11 0 0 1 1x N N x xP A A P Q M M = + + + +F F 12 1 1 2xP A M M = + ,F 13 0xP B H K = ,F 22 2 2xQ M M = ,F 23 1H K = ,F 33 2 I = .

    As in the proof of Theorem 2, for a given 1 2 , R , this LMI is equivalent to the existence of 2 0> such that the following LMI in 0 0 0 n px x xP Q R Y > , > , > , R :

    11 12 13 0 1

    22 23 1 2

    33 0 0

    0

    x x

    x x

    T T T

    T T

    TT

    x

    x

    hP A hY G h RhP A h R

    hBT T

    hR

    hR

    + := = < ,

    F F F

    F

    F F

    where

    11 0 1 0 1( ) ( )x x xA I P P Q GY Y GA I = + + + + ++ F F F 12 1 2 1( )x xA P P = + , 13 2 0xB P H K = ,F 22 22 xxQ P = , 23 2 1xP H K = ,F 33 22 I = .

    holds. The gain matrix N is given by

    1xN P Y= . (42)

    This completes the proof. Remark 4. Two steps to solve the LMIs in Theorem 4. The first step is the resolution of LMI T < 0 and we de-termine the gain matrix N. The second step is the resolution of the LMI < 0 and determine the gain matrices S and L.

    V. NUMERICAL EXAMPLES

    Example 1.

    We consider the time-delay system (1) where the non-linearity satisfy condition (2) with

    0 12 02 1 0 49 1 0 5 00 5 0 22 0 1 0 68 0 0 2

    A A B . . . = , = , = , . . . . .

    0 10 4 0 0 2 0 0 3 00 0 5 0 0 3 0 0 3

    C C K. . . = , = , = . . . .

    (43)

    This example has been studied in [16] where the nonli-nearity belong the sector [K1, K2]. However, one can easily formulate a criterion along the idea in [16] in the sector [0, K]. By new criterion, the numerical example result is hmax = 2.023.

    Applying Theorem 1, the maximum allowed time- delay is computed as hmax = 2.615. The solutions of LMI (4) are

  • 370 Asian Journal of Control, Vol. 9, No. 3, September 2007

    114 126 89 035 112 482 30 26689 035 277 363 30 266 114 814

    P Q. . . . = , = , . . . .

    41 825 100 196100 196 368 121

    R. . = , . .

    1 28 139 33 797 13 901 35 81914 423 127 778 37 239 136 017

    M M . . . . = , = , . . . .

    and 545 12 0= . > . It is a gain to show that the criteria developed in

    Theorem 1 is less conservative than the corresponding cri-teria in [16].

    Example 2. We consider the time-delay system (12) where the

    nonlinearity ( ) ., . satisfy condition (2), the matrices 0 1 0 1A A B C C, , , , are given by (43), and the matrix G is

    given by

    0 81 2

    G. = . .

    If we consider the Example 1 with max3 3 2 615h h= . > = . , then the open-loop system is unstable.

    Using Theorem 2, we find a control state feedback such that the closed-loop system is stable. For 1 0 07 = . and 2 0 09 = . , the solutions of LMI (15) are

    0 178 0 01 0 194 0 0180 01 0 084 0 018 0 237

    P Q. . . . = , = , . . . .

    1 58 0 451[ 0 021 0 278]

    0 451 1 976R Y

    . . = , = . . , . .

    and 1 2= . . So, the origin of controlled system (12) is stabilizable by the linear state feedback (13) where

    [ 0 322 3 348]N = . . .

    VI. CONCLUSIONS

    The problem of absolute stability of a class of time-delay systems with sector-bounded nonlinearity have been considered. New delay-dependent stability criteria with the sector condition have been proposed. This criteria have been formulated in the form of linear matrix inequali-ties (LMI), and derived with employing some integral in-equalities. We used this new criterion to construct a delay- dependent dynamic observer-based output feedback control such that the observer error system have been presented as the feedback interconnection of a linear system and a state-dependent multivariable sector-bounded nonlinearity.

    Finally, some numerical examples are given to illustrate the applicability of the results.

    REFERENCES

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    3. Gu, K., V.L. Kharitonov, and J. Chen, Stability of Time Delay Systems, Birkauser, Boston (2003).

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