Robust observers for neutral jumping systems with uncertain information

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Robust observers for neutral jumping

systems with uncertain information

Magdi S. Mahmoud

a

, Peng Shi

b,*

, Jianqiang Yi

c

,

Jeng-Shyang Pan

d

a_{Canadian International College, El-Tagamoa El-Khamis, New Cairo City, Egypt} b_{School of Technology, Division of Mathematics and Statistics, University of Glamorgan,}

Pontypridd, Wales CF37 1DL, United Kingdom

c

Laboratory of Complex Systems and Intelligence Science, Institute of Automation, Chinese Academy of Sciences, China

d

Department of Electronic Engineering, National Kaohsiung University of Applied Science, 415 Chien-Kung Road, Kaohsiung 807, Taiwan, ROC

Received 8 December 2004; received in revised form 13 July 2005; accepted 17 July 2005

Abstract

In this paper, the problem of designing observer for a class of uncertain neutral sys-tems. The uncertainties are parametric and norm-bounded. Both robust observation and robust H1 observation methods are developed by using linear state-delayed

observers. In case of robust observation, suﬃcient conditions are established for asymp-totic stability of the system, which is independent of time delay. The results are then extended to robust H1 observation which renders the augmented system

asymptoti-cally stable independent of delay with a guaranteed performance measure. Furthermore, a memoryless state-estimate feedback is designed to stabilize the closed-loop neutral

* _{Corresponding author. Tel.: +44 1443 482147; fax: +44 1443 482711.}

E-mail addresses: pshi@glam.ac.uk (P. Shi), jianqiang.yi@mail.ia.ac.cn (J. Yi), jspan@cc. kuas.edu.tw(J.-S. Pan).

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system. In all cases, the gain matrices are determined by linear matrix inequality approach. Two numerical examples are presented to illustrate the validity of the theo-retical results.

Keywords: Time-delay systems; Linear neutral systems; Robust observation; Stochastic stability; State-delayed observers; Robust H1observation

1. Introduction

As is well known, dynamical models of many physical and engineering sys-tems incorporate time-delay factors, see for example, [11,13]. When using these models in the analysis and control design, there have been three basic approaches[1,20]: (1) using infinite-dimensional systems theory by embedding the class of time-delay systems into a larger class of dynamical systems for which the state evolution is described by appropriate operators in infi-nite-dimensional spaces; (2) applying algebraic systems theory in which the evolution of delay-differential systems is provided in terms of linear systems over rings and (3) employing functional differential systems (FDS) by incor-porating the influence of the hereditary effects of system dynamics on the change rate of the system and in this regards, it provides an appropriate mathematical structure. On the other hand, robust observation (or estima-tion) [5,26,9,28,30] is concerned with the state reconstruction when the plant model has uncertain parameters and is described by ordinary differential equations (ODE) or equivalent representation. Following the third approach based on FDS, results on estimating the state of uncertain system with state-delay are developed in[18]and related work can be found in[21]. An integral part of FDS[11–13,32]is the class of neutral-type systems which can be found in several applications including, but not limited to, chemical reactor, rolling mill, infeed grinding, lossless transmission lines and hydraulic systems. Stabil-ity analysis and feedback stabilization for neutral FDS have been studied in

[16,27] and other related work was reported in [20]. Recently H1 control

has been developed in [19] for a class of linear neutral systems with para-metric uncertainties.

On another research front line, there are a wide class of systems having a variable structure subject to random changes which may result from abrupt phenomena such as component and interconnection failures, parameters shifting, tracking and/or variation in the time frame of measurements. Sys-tems with this character may be modelled as hybrid ones; that is, the state space of the system contains both discrete and continuous states and is fre-quently termed jumping systems. Results on the stability and control of linear jumping systems can be found in[3,7,25,14,33,4,2,31,23,24]and the references

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therein. This paper build upon [20,19] and extends their results further by considering the state observation and stabilization problems for a class of lin-ear neutral systems with norm-bounded uncertainties. Note that the motiva-tion to investigate this kind of system is that some control systems depend not only on state delays but also on derivatives of delayed states, and this class of systems is referred as neutral delay systems. The system we studied in this paper is an extension of the standard neutral systems with jumps. Ini-tially, we address both problems of robust state observation and robust H1

observation and employ a new linear state-delayed observer such that the asymptotic stability of the combined neutral system and the proposed obser-ver is guaranteed for all admissible uncertainties. The main tool for solving the foregoing problems is the linear matrix inequality (LMI) approach. In this regard, it has been established that the solution of robust is expressed in terms of two LMIs involving scaling parameters. Looked at in this light, the developed methods provide new results which in some sense are the dual of [19]. Then, the robust stabilization problem is considered by designing memoryless state-estimate feedback such the asymptotic stability of the closed-loop stability is guaranteed. The analytical developments of this work are organized into theorems whereby the results are presented in a systematic and gradual build-up. Finally, two numerical examples are presented to illus-trate the validity of the theoretical results. It should be mentioned that in fact, our paper deals with a class of neutral systems with uncertainties and jumping parameters using the diﬀerence operator approach. The method of analysis is systematic and the developed results are new and delay-indepen-dent. There is a wide-class systems, such as fault-tolerant systems, satisfying these features.

1.1. Notations and facts

The notation in this paper is fairly standard. We use Wt, W1, k(W) and kWk to denote, respectively, the transpose, the inverse, the eigenvalues and the induced norm of any square matrix W. We use W > 0 (P, <, 6 0) to denote a symmetric positive definite (positive semidefinite, negative, nega-tive semidefinite) matrix W with km(W) and kM(W) being the minimum and

maximum eigenvalues of W and I denote the n· n identity matrix. The open left half of the complex plane is represented by C. The Lebesgue space L2½0; 1Þ consists of square-integrable functions on the interval [0, 1). Let

Cn;s¼ Cð½s; 0; RnÞ denote the Banach space of continuous vector functions

mapping the interval [s, 0] into Rn

with the topology of uniform conver-gence and equipped with the norm kxk,_sup

s6h60kxk where kÆk is the

Euclidean norm and s > 0 is termed the delay factor. Sometimes, the argu-ments of a function will be omitted in the analysis when no confusion can arise.

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Fact 1 (Schur Complement). Given constant matrices X1, X2, X3 where 0 < X1¼ Xt1 and 0 < X2¼ Xt2 then X1þ Xt3X 1 2 X3<0 if and only if X1 Xt3 X3 X2 <0 or X2 X3 Xt3 X1 <0.

Fact 2. Given any real matrices R1, R2, R3with appropriate dimensions, such

that 0 < R3¼ Rt3the following inequality holds:

Rt₁R2þ Rt2R16Rt1R3R1þ Rt2R 1 3 R2.

Fact 3. Let R1, R2, R3and 0 < R = R t

be real constant matrices of compatible dimensions and H(t) be a real matrix function satisfying Ht(t)H(t) 6 I. Then for any q > 0 satisfying qRt

2R2< R, the following matrix inequality holds:

ðR3þ R1HðtÞR2ÞR1ðRt3þ R t 2H t_ðtÞRt 1Þ 6 q1R1Rt1þ R3 R qRt2R2 1 Rt 3.

2. Class of neutral jumping systems

2.1. Model description

Given a probability spaceðX; F; PÞ where X is the sample space, F is the algebra of events and P is the probability measure deﬁned on F. Let the ran-dom processfgt; t2 ½0; Tg be a homogeneous, ﬁnite-state Markovian process

with right continuous trajectories and taking values in a ﬁnite set S ¼ f1; 2; . . . ; sg with generator I ¼ ðpijÞ, i; j 2 S with transition probability from

mode i at time t to mode j at time t + d, i; j2 S:

p_ij¼ Prðg_tþd¼ jjgt¼ iÞ ¼

aijdþ oðdÞ; if i6¼ j;

1þ aijdþ oðdÞ; if i¼ j

(

ð2:1Þ

with transition probability rates aijP0 for i; j2 S, i 5 j and

aii ¼

Xs m¼1;m6¼i

aim; ð2:2Þ

where d > 0 and limd#0o(d)/d = 0. The set S comprises the various operational

modes of the system under study.

We consider a class of stochastic uncertain neutral systems with Markovian jump parameters described over the spaceðX; F; PÞ by

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ðRDnÞ Mð_xtÞ, _xðtÞ BðgtÞ_xðt sÞ ¼ ½AoðgtÞ þ DAoðt; gtÞxðtÞ þ ½AdðgtÞ þ DAdðt; gtÞxðt sÞ þ N ðgtÞwðtÞ ¼ ADoðt; gtÞxðtÞ þ ADdðt;gtÞxðt sÞ þ N ðgtÞwðtÞ; ð2:3Þ xðgÞ ¼ /ðgÞ 2 Cð½s;0; Rn Þ; 8g 2 ½s; 0; ð2:4Þ yðtÞ ¼ ½CoðgtÞ þ DCoðt;gtÞxðtÞ þ ½CdðgtÞ þ DCdðt;gtÞxðt sÞ þ MðgtÞwðtÞ ¼ CDoðt;gtÞxðtÞ þ CDdðt; gtÞxðt sÞ þ MðgtÞwðtÞ; ð2:5Þ zðtÞ ¼ LðgtÞxðtÞ; ð2:6Þ

where xðtÞ 2 Rn _{is the system state, yðtÞ 2 R}m

is the measurement output, zðtÞ 2 Rp _{is the controlled output, wðtÞ 2 L}

2½0; 1Þ is the disturbance input,

zðtÞ 2 Rr

is the controlled output which belongs to L2½ðX; F; PÞ; ½0; 1Þ and

the factor s > 0 is a constant scalar representing the amount of time-lag in the state. Frequently the term MðxtÞ : C½s; 0 ! Rn,xðtÞ Bxðt sÞ is

called the diﬀerence operator and it oﬀers a fundamental role in the analytical development throughout the paper.

For each possible value gt= i, i2 S, we will denote the system matrices of

(RDn) associated with mode i by

AoðgtÞ , AoðiÞ; AdðgtÞ , AdðiÞ; NðgtÞ , N ðiÞ;

CoðgtÞ , CoðiÞ; CdðgtÞ , CdðiÞ; MðgtÞ , MðiÞ; BðgtÞ , BðiÞ;

ð2:7Þ where AoðiÞ 2 Rnn, AdðiÞ 2 Rnn, BðiÞ 2 Rnn, CoðiÞ 2 Rmn, CdðiÞ 2 Rmn,

NðiÞ 2 Rnr, MðiÞ 2 Rmr and LðiÞ 2 Rpn are known real constant matrices. Ao(i), Ad(i), Co(i), Cd(i), B(i), N(i) and M(i) are known real constant matrices

of appropriate dimensions which describe the nominal system of (RDn). The

matrices DAo(t, gt), DAd(t, gt), DCo(t, gt) and DCd(t, gt) are real, time-varying

matrix functions representing the norm-bounded parameter uncertainties. For gt= i, the admissible uncertainties are represented by

DAoðt; iÞ DAdðt; iÞ DCoðt; iÞ DCdðt; iÞ ¼ HaðiÞ HcðiÞ DðtÞ½ EoðiÞ EdðiÞ ; 8DtðtÞDðtÞ 6 I; 8t ð2:8Þ where HaðiÞ 2 Rna, HcðiÞ 2 Rpa, EoðiÞ 2 Rbn and EdðiÞ 2 Rbn, are known

real constant matrices and DðtÞ 2 Rab is an unknown matrix with Lebesgue measurable elements. The initial condition is speciﬁed as bo ,hx(0), x(s)i =

hxo, /(s)i, where /ðÞ 2 L2½s; 0.

In the absence of uncertainties (D(Æ) 0) and for each possible value gt= i,

i2 S, we obtain the nominal neutral system

ðRnÞ Mð_xtÞ, _xðtÞ BðiÞ_xðt sÞ ¼ AoðiÞxðtÞ þ AdðiÞxðt sÞ þ N ðiÞwðtÞ;

ð2:9Þ xðgÞ ¼ /ðgÞ 2 Cð½s; 0;Rn_{Þ; 8g 2 ½s; 0;} _ð2:10Þ

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yðtÞ ¼ CoðiÞxðtÞ þ CdðiÞxðt sÞ þ MðiÞwðtÞ; ð2:11Þ

zðtÞ ¼ LðiÞxðtÞ. ð2:12Þ

The following assumptions on systems (RDn) and (Rn) are recalled:

Assumption 2.1. kðAoðiÞÞ 2 C, i2 S.

Assumption 2.2. jk(B(i))j < 1, det[B(i)] 5 0, i 2 S.

Remark 2.1. Note that system(2.3)–(2.6)is a hybrid system in which one state x(t) takes values continuously, and another ‘‘state’’ g(t) takes values discretely. Being continuous in time and represents a wide class of physical systems thus Assumption 2.1 is quite standard. On the other hand, Assumption 2.2 provides a sufﬁcient condition on the eigenspectrum in the discrete space and its major role will be clariﬁed in the sequel. An alternative interpretation of Assumption 2.2 is that the difference operator MðxtÞ is delay-independently stable. The

kind of systems(2.3)–(2.6) can be used to represent many important physical systems subject to random failures and structure changes, such as electric power systems[34], control systems of a solar thermal central receiver, commu-nications systems, aircraft ﬂight control, and manufacturing systems

[3,25,14,33,6,8].

Our primary objective in this paper is to design robust state and robust H1

observers for the neutral system (RDn) with some desirable stability behavior

and then extend these designs to the neutral system (Rn). Towards our goal,

we let X(t, bo, go) denote the trajectory of the state x(t) from the initial state

(bo, go) and recall the following deﬁnition:

Deﬁnition 2.1. System RDn is said to be robustly stochastically stable

independent of delay (RSSID) if for all ﬁnite initial vector function /ðÞ 2 L2½s; 0 deﬁned on the interval [s, 0] and initial mode go2 S

Z 1 0 E kXðt; b0;goÞk 2 n o dt <þ1 ð2:13Þ

for all admissible uncertainties satisfying(2.8).

3. Robust observers

In the sequel, to derive the state estimate ^x2 Rn_{we will utilize the following}

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ðReÞ Mð_^xtÞ, _^xðtÞ B_^xðt sÞ ¼ AfðiÞ^xðtÞ þ AdðiÞ^xðt sÞ þ KfðiÞyðtÞ; ^xð0Þ ¼ 0;

^zðtÞ ¼ LðiÞ^xðtÞ; ð3:1Þ

where AfðiÞ 2 Rnn, KfðiÞ 2 Rnm, i2 S are the observer matrix gains to be

de-signed such that ^xreproduce x asymptotically for all admissible uncertainties satisfying(2.8).

3.1. The augmented system Let the state error be

~

x , xðtÞ ^xðtÞ. ð3:2Þ

From(2.3)–(2.5), (3.1) and (3.2), the state error dynamics can be represented by ðRDeÞ Mð_~xtÞ , _~xðtÞ BðiÞ_~xðt sÞ ¼ AfðiÞ~xðtÞ þ AdðiÞ~xðt sÞ

þ ½AoðiÞ KfðiÞCoðiÞ AfðiÞxðtÞ þ ½DAoðt; iÞ KfðiÞDCoðt; iÞxðtÞ

KfðiÞ½CdðiÞ þ DCdðt; iÞxðt sÞ þ ½N ðiÞ KfðiÞMðiÞwðtÞ. ð3:3Þ

A state-space augmented model of the observation error, ~zðtÞ ¼ zðtÞ ^zðtÞ, can then be constructed in terms of the augmented state vector and the extended matrix BðiÞ for each possible value gt= i, i2 S

fðtÞ , ~xðtÞ xðtÞ ; BðiÞ ¼ BðiÞ 0 0 BðiÞ ð3:4Þ by using(2.3)–(2.6), (3.3) and (3.4)as follows:

ðRDAÞ Mð_ftÞ , _fðtÞ BðiÞ_fðt sÞ

¼ ADoðt; iÞfðtÞ þ DDoðt; iÞfðt sÞ þ BðiÞwðtÞ; ð3:5Þ

~zðtÞ ¼ LfðiÞfðtÞ; ð3:6Þ

where

ADoðt;iÞ ¼ ½AðiÞ þ HAðiÞDðt;iÞEAðiÞ; DDoðt; iÞ ¼ ½DðiÞ þ HDðiÞDðt;iÞEDðiÞ;

LfðiÞ ¼ ½LðiÞ 0; AðiÞ ¼

AfðiÞ AoðiÞ KfðiÞCoðiÞ AfðiÞ

0 AoðiÞ ; fðt sÞ ¼ ~xðt sÞ xðt sÞ

; EAðiÞ ¼ ½ 0 EaðiÞ ; EDðiÞ ¼ ½0 EdðiÞ ; ð3:7Þ

DðiÞ ¼ AdðiÞ KfðiÞCdðiÞ 0 AdðiÞ

; BðiÞ ¼ NðiÞ KfðiÞMðiÞ NðiÞ

; ð3:8Þ HA¼

HaðiÞ KfðiÞHcðiÞ

HaðiÞ ; HDðiÞ ¼ KfðiÞHcðiÞ HcðiÞ . ð3:9Þ

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Had we followed another route and combined systems (Rn) and (Re), we would

have obtained the nominal augmented system

ðRAÞ Mð_ftÞ , _fðtÞ BðiÞ_fðt sÞ ¼ AðiÞfðtÞ þ DðiÞfðt sÞ þ BðiÞwðtÞ;

ð3:10Þ

~zðtÞ ¼ LfðiÞfðtÞ. ð3:11Þ

Remark 3.1. It should be stressed that system (RDA) describes a linear

uncertain jumping system of neutral-type the nominal version of which is represented by systems (RA). The matrices of both systems depend on the gains

Af(i), Kf(i), i2 S.

3.2. Stability analysis

The following theorems establish that the stability behavior of system (RDA) or (RA) is related to the existence of a positive deﬁnite solution of linear

matrix inequalities thereby providing a clear key to designing the state observers.

Theorem 3.1. Given gain matrices Af(i), Kf(i), i2 S and subject to Assumptions

2.1 and 2.2, system (RDA) with w 0 is (RSSID) if for given matrices

0 < LðiÞ ¼ Lt_{ðiÞ 2 R}2n2n_{, i}_{2 S, and letting}

LðiÞ ¼ LðiÞ n1ðiÞX

s

m¼1

aimLðmÞ; bLðiÞ ¼ LðiÞ þ nðiÞ

Xs m¼1

aimLðmÞ;

i2 S

for some scalars n(i) > 0, i2 S, there exist matrices 0 < PðiÞ ¼ Pt_{ðiÞ 2 R}2n2n_,

i2 S and scalars e(i) > 0, .(i) > 0, i 2 S satisfying the following linear matrix inequalities (LMIs): t KaðiÞ KnðiÞ Kt aðiÞ UaðiÞ 0 Kt nðiÞ 0 HaðiÞ 2 6 6 4 3 7 7 5 < 0; i 2 S; ð3:12Þ LðiÞ Bt_ðiÞb_{LðiÞ E}t

DðiÞ eðiÞB t_ðiÞEt

AðiÞ

bLðiÞBðiÞ bLðiÞ 0 0

EDðiÞ 0 I 0

eðiÞEAðiÞBðiÞ 0 0 eðiÞI

2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 <0; i2 S; ð3:13Þ

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t¼ PðiÞAðiÞ þ AtðiÞPðiÞ þ bLðiÞ þ Xs m¼1 aimPðmÞ þ .ðiÞE t AðiÞEAðiÞ;

Ka¼ PðiÞHAðiÞ PðiÞHAðiÞ PðiÞHDðiÞ

; Ua¼ diag .ðiÞI eðiÞI eðiÞI½ ;

Ha¼ LðiÞ BtðiÞbLðiÞBðiÞ eðiÞ BtðiÞE t AðiÞEAðiÞBðiÞ þ E t DðiÞEDðiÞ ;

KnðiÞ ¼ PðiÞ½AðiÞBðiÞ þ DðiÞ þ bLðiÞBðiÞ ð3:14Þ

for all admissible uncertainties satisfying(2.8).

Proof. For gt= i, i2 S and given 0 < LðiÞ ¼ LtðiÞ, let the Lyapunov

func-tional VðÞ : Rn_R þ S ! Rþ be selected as Vðt; f; gt¼ iÞ , V ðt; f; iÞ ¼ Mt_ðf tÞPðiÞMðftÞ þ Z 0 s ftðt þ hÞLðiÞfðt þ hÞ dh. ð3:15Þ The weak inﬁnitesimal operator If_a½ of the process {f(t), gt, t P 0} for system

(3.5)at the point {t, f, gt} is given by[15]

If_a½V ¼oV ot þ M t_{ð_f} tÞðtÞ oV of gt¼i þX s m¼1 aimVðt; f; i; mÞ. ð3:16Þ

Using(3.5)into Eqs.(3.15) and (3.16)and manipulating the terms we get If_a½V ¼ MtðftÞPðiÞ½ADoðt; iÞfðtÞ þ DDoðt; iÞfðt sÞ

þ ½ADoðt; iÞfðtÞ þ DDoðt; iÞfðt sÞ t PMðftÞ þ M t ðftÞ X s m¼1 aimPðmÞMðftÞ þ f t_{ðtÞLðiÞfðtÞ f}t_{ðt sÞLðiÞfðt sÞ} þX s m¼1 aim Z 0 s ftðt þ hÞLðmÞfðt þ hÞ dh

6_Mtðf_tÞPðiÞ½A_Doðt; iÞfðtÞ þ D_Doðt; iÞfðt sÞ þ ½ADoðt; iÞfðtÞ þ DDoðt; iÞfðt sÞ

t PðiÞMðftÞ þ M t_ðf tÞ X s m¼1 aimPðmÞMðftÞ þ ftðtÞbLðiÞfðtÞ ftðt sÞLðiÞfðt sÞ; ð3:17Þ where LðiÞ > 0, i 2 S by selection of nðiÞ, i 2 S. Applying the argument of ‘‘completing the squares’’ and over-bounding the result, it yields

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Ifa½V 6 M t_ðf

tÞ PðiÞADoðt;iÞ þ AtDoðt;iÞP þ bLðiÞ þ

Xs m¼1 aimPðmÞ " # MðftÞ þ Mt_ðf

tÞ PðiÞADoðt;iÞBðiÞ þ bLðiÞBðiÞ þ PðiÞDDoðt;iÞ

fðt sÞ þ ftðt sÞ PðiÞADoðt;iÞBðiÞ þ bLðiÞBðiÞ þ PðiÞDDoðt;iÞ

t

MðftÞ

ft_{ðt sÞ LðiÞ B}h t_ðiÞb_LðiÞBðiÞi_{fðt sÞ}

6_Mt_ðf_t_{Þ PðiÞA}_Do_{ðt;iÞ þ A}t Doðt;iÞPðiÞ þ bLðiÞ þ Xs m¼1 aimPðmÞ "

þ PðiÞADoðt;iÞBðiÞ þ bLðiÞBðiÞ þ PðiÞDDoðt;iÞ

L Bt_ðiÞb_LðiÞBðiÞ

h i1

PðiÞADoðt;iÞBðiÞ þ bLðiÞBðiÞ þ PðiÞDDoðt;iÞ

t#

MðftÞ

ftðt sÞ Mt_ðf

tÞ PðiÞADoðt;iÞBðiÞ þ bLðiÞBðiÞ þ PðiÞDDoðt;iÞ

h i

LðiÞ Bh t_ðiÞb_LðiÞBðiÞi1 _f

ðt sÞ ðPðiÞADoðt;iÞBðiÞ þ bLðiÞBðiÞ

h þ PðiÞDDoðt;iÞÞtMðftÞ

i 6_Mt_ðf

tÞ PðiÞADoðt;iÞ þ AtDoðt;iÞPðiÞ þ bLðiÞ

þX

s m¼1

aimPðmÞ þ PðiÞADoðt;iÞBðiÞ þ bLðiÞBðiÞ þ PðiÞDDoðt;iÞ

L Bh t_ðiÞb_L_ðiÞBðiÞi1 _P

ðiÞADoðt;iÞBðiÞ þ bLðiÞBðiÞ þ PðiÞDDoðt;iÞ

t

MðftÞ.

ð3:18Þ

Using Facts 2 and 3, it follows from(3.18)for some scalars e(i) > 0, .(i) > 0, i2 S that:

If_a½V 6 Mt_ðf

tÞ PðiÞAðiÞ þ AtðiÞPðiÞ þ bLðiÞ þ

Xs m¼1 aimPðmÞ ( þ .1_ðiÞPðiÞH AðiÞH t AðiÞPðiÞ þ .ðiÞE t AðiÞEAðiÞ þ e1ðiÞPðiÞ HAðiÞH t AðiÞ þ HDðiÞH t DðiÞ h i PðiÞ þ PðiÞ½AðiÞBðiÞ þ DðiÞ þ b LðiÞBðiÞ L Bt_ðiÞb_{LðiÞBðiÞ eðiÞ E}t

AðiÞEAðiÞ þ E t

DðiÞEDðiÞ

h i1

PðiÞ½AðiÞBðiÞ þ DðiÞ þ b LðiÞBðiÞt )

MðftÞ , M t_ðf

tÞPðiÞMðftÞ.

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By the Schur complements, inequality (3.19) is equivalent to LMIs (3.12) and (3.13)from which we conclude that for all admissible uncertainties satisfy-ing(2.8)If_a½V < 0 for all f 5 0 and Ifa½V 6 0 for all f.

Sincekx(t + b)k 6 ukx(t)k, "b 2 [s, 0] and some u > 0[17], it follows from

(3.15) that Vðt; f; iÞ 6 Mt_ðf tÞPðiÞMðftÞ þ lkfk 2 ; l¼ us max i kM½PðiÞ þ kM½LðiÞ . Therefore, for all f 5 0, we have

Ix_a½V Vðt; f; iÞ6 MtðftÞPðiÞMðftÞ MtðftÞPðiÞMðftÞ 6r , min i2S km½PðiÞ kM½PðiÞ . ð3:20Þ It is readily seen from(3.20)that r > 0 and hence we get If_a½V 6 rV ðt; f; iÞ. Then, it follows from[15], by using the Gronwall–Bellman lemma[20]and let-ting x(t = 0, /, i) = xo, that E½V ðt; f; iÞj/; go 6 ertVðt; fo;goÞ ð3:21Þ Therefore E½V ðt; f; iÞj/; go ¼ E M t_ðf tÞPðiÞMðftÞ þ Z 0 s ftðt þ hÞLðiÞfðt þ hÞdhj/; go ¼ E Mt_ðf tÞPðiÞMðftÞj/; go f g þ E Z 0 s ftðt þ hÞLðiÞfðt þ hÞdhj/; go 6ertVðt; fo;goÞ. ð3:22Þ Since E R_s0 ftðt þ hÞLðiÞfðt þ hÞ dhj/; go n o

P0; then some algebraic manipula-tion of(3.22)yields: EfMt_ðf tÞPðiÞMðftÞj/; gog 6 ertVðt; fo;goÞ ) E Z T 0 MtðftÞPðiÞMðftÞ dtj/; go 6 Z T 0 ertdt Vðxo;goÞ ¼ 1 r½e fT_{1V ðt; f} o;goÞ ) lim T!1E Z T 0 MtðftÞPðiÞMðftÞ dtj/; go 61 rf t oPðgoÞfoþ s r½LðgoÞkfðt þ hÞk 2 ; 8h 2 ½s; 0; ð3:23Þ

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wherekxðt þ hÞk2,_sup h2½s;0kfðt þ hÞk 2 2. Now, let PðiÞ ¼ max i2S kM½PðgoÞkfok 2 þ s½LðgoÞkxðt þ hÞk 2 r½PðgoÞkfok 2 ( ) ;

it follows from (3.23)for i2 S that lim T!1E Z T 0 ftðtÞfðtÞ dtj/; go 6_ft okMðPðiÞÞfo<þ1;

which, in the light of Deﬁnition 2.1, shows that system (RDA) is (RSSID). h

Theorem 3.2. Given gain matrices Af(i), Kf(i), i2 S and subject to Assumptions

2.1 and 2.2, system (RA) with w 0 is stochastically stable independent of delay

(SSID) if for given matrices 0 < LðiÞ ¼ Lt_{ðiÞ 2 R}2n2n_{, i}_{2 S, and letting}

LðiÞ ¼ LðiÞ n1ðiÞX

s

m¼1

aimLðmÞ; bLðiÞ ¼ LðiÞ þ nðiÞ

Xs m¼1

aimLðmÞ; i2 S

for some scalars n(i) > 0, i2 S, there exist matrices 0 < PðiÞ ¼ Pt_{ðiÞ 2 R}2n2n_,

i2 S satisfying the following LMIs: to KnðiÞ Kt nðiÞ HaoðiÞ <0; LðiÞ B t ðiÞbLðiÞ bLðiÞBðiÞ bLðiÞ " # <0; i2 S ð3:24Þ

to¼ PðiÞAðiÞ þ AtðiÞPðiÞ þ bLðiÞ þ

Xs m¼1

aimPðmÞ;

Hao¼ LðiÞ BtðiÞbLðiÞBðiÞ. ð3:25Þ

Proof. Follows from Theorem 3.1 by setting Et_A 0, Et_D 0, Ht_A 0. h Remark 3.2. It should be remarked that both Theorems 3.1 and 3.2 oﬀer new analytical results for the class of neutral-type dynamical systems under consid-eration. The results are cast in LMI format for which theMATLAB-LMIsoftware

is readily available [10]. More importantly, in the case BðiÞ 0 ) MðÞ ¼ xt,

systems (RDA) and (RA) become of retarded-type for which Theorems 3.1

and 3.2 retrieve the results of[29,22].

Remark 3.3. The need for Assumption 2.2 is evident from(3.14) and (3.25)in which case the conditions Ha> 0, Hao> 0 are required, respectively. In both

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Yð2Þ ¼ 2:9627 0:1527 0:1527 0:8405 ; Xð2Þ ¼ 1:1174 0:0109 0:0109 1:1657 ; Zð2Þ ¼ 1:1054 0:2104 0:2104 1:2132 ; Wð2Þ ¼ 0:9450 0:5117 0:4402 1:4204 ; .ð2Þ ¼ 3:4116; eð2Þ ¼ 5:5514; Kfð2Þ ¼ 0:1107 2:9167 1:1945 4:4627 ; Afð2Þ ¼ 0:0176 1:5031 3:0111 3:9213 ; Ksð2Þ ¼ 1:0578 0:8196 0:7414 1:8158 . 6. Conclusions

In this paper, the designs of robust observation, robust H1observation and

robust stabilization methods for a class of linear neutral-type continuous-time systems with norm-bounded parametric uncertainties have been presented. A linear state-delayed estimator is proposed such that the augmented system achieves desirable stability properties independent of delay. Then a memoryless state-estimate feedback control to stabilize the closed-loop system is designed. In all cases, the gain matrices are determined by solving linear matrix inequal-ities with scaling parameters. Two numerical examples are included to illustrate the validity of the theoretical results.

Acknowledgement

The authors would like to thank the referees for their valuable and helpful comments and suggestions which have improved the presentation and quality of the paper. Peng Shi also gratefully acknowledges the support of K.C. Wong Education Foundation, Hong Kong, and the Laboratory of Complex Sys-tems and Intelligence Science, Institute of Automation, Chinese Academy of Sciences, China.

References

[1] K. Boukas, Stabilization of stochastic nonlinear hybrid systems, Int. J. Innovat. Comput. Inform. Cont. 1 (1) (2005) 131–141.

[2] E. Boukas, Z.K. Liu, Deterministic and Stochastic Time-Delay Systems, Birkuhauzer, Boston, MA, 2002.

(14)

[3] O.L.V. Costa, M.D. Fragoso, Stability results for discrete-time linear systems with Markovian jumping parameters, J. Math. Anal. Appl. 179 (2) (1993) 154–178.

[4] M. Davis, Markov Models and Optimization, Chapman and Hall, London, 1992.

[5] V. Dragan, The linear quadratic optimization problem for a class of singularly perturbed stochastic systems, Int. J. Innovat. Comput. Inform. Cont. 1 (1) (2005) 53–63.

[6] R.J. Elliott, D.D. Sworder, Control of a hybrid conditionally linear Gaussian processes, J. Opt. Theory Appl. 74 (1992) 75–85.

[7] X. Feng, K.A. Loparo, Y. Ji, H.J. Chizeck, Stochastic stability properties of jump linear systems, IEEE Trans. Autom. Cont. 37 (1) (1992) 38–53.

[8] W. Fleming, S. Sethi, M. Soner, An optimal stochastic production planning problem with randomly ﬂuctuating demand, SIAM J. Cont. Opt. 25 (1987) 1494–1502.

[9] M. Fu, C.E. de Souza, L. Xie, H1-estimation for uncertain systems, Int. J. Robust Nonlinear

Cont. 2 (1992) 87–105.

[10] P. Gahinet, A. Nemirovski, A.L. Laub, M. Chilali, LMI Control Toolbox, The Math Works, Inc., Boston, MA, 1995.

[11] J. Hale, Functional Diﬀerential Equations, Springer-Verlag, New York, 1977.

[12] Y. Ji, H.J. Chizeck, X. Feng, K.A. Loparo, Stability and control of discrete-time jump linear systems, Cont. Theory Adv. Technol. 7 (2) (1991) 247–270.

[13] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Diﬀerential Equations, Kluwer Academic Publishers, London, 1992.

[14] N.N. Krasovskii, E.A. Lidskii, Analysis design of controllers in systems with random attributes, Part I, Autom. Remote Cont. 22 (1961) 1021–1025.

[15] H. Kushner, Stochastic Stability and Control, Academic, New York, 1967.

[16] H. Logemann, S. Townley, The eﬀect of small delays in the feedback loop in the stability of neutral systems, Syst. Cont. Lett. 27 (1996) 267–274.

[17] M.S. Mahmoud, N.F. Al-Muthairi, Design of robust controllers for time-delay systems, IEEE Trans. Autom. Cont. 39 (5) (1994) 995–999.

[18] M.S. Mahmoud, N.F. Al-Muthairi, S. Bingulac, Robust Kalman ﬁltering for continuous time-lag systems, Syst. Cont. Lett. 38 (1999) 309–319.

[19] M.S. Mahmoud, Robust H1control of linear neutral systems, Automatica 36 (5) (2000) 757–

764.

[20] M.S. Mahmoud, Robust Control and Filtering for Time-Delay Systems, Marcel-Dekker, New York, 2000.

[21] M.S. Mahmoud, M. Zribi, Stabilizing controllers using observers for uncertain systems with Delays, Int. J. Syst. Sci. 32 (6) (2001) 767–773.

[22] M.S. Mahmoud, P. Shi, Robust stability, stabilization and H1control of time-delay systems

with Markovian jump parameters, Int. J. Robust Nonlinear Cont. 13 (8) (2003) 755–784. [23] M.S. Mahmoud, P. Shi, Robust Kalman ﬁltering for continuous time-lag systems with

Markovian jump parameters, IEEE Trans. Circ. Syst.—I 50 (1) (2003) 98–105.

[24] M.S. Mahmoud, P. Shi, Methodologies for Control of Jump Time-Delay Systems, Kluwer Academic Publishers, Boston, 2003.

[25] T. Morozan, Stability and control for linear systems with jump Markov perturbations, Stochastic Anal. Appl. 13 (1) (1995) 91–110.

[26] K.M. Nagpal, P.P. Khargonekar, Filtering and smoothing in H1 setting, IEEE Trans.

Autom. Cont. 36 (1991) 152–166.

[27] D. OConner, T.J. Tarn, On the stabilization by state feedback for neutral diﬀerential diﬀerence equations, IEEE Trans. Autom. Cont. AC-28 (1983) 615–618.

[28] I.R. Petersen, D.C. McFarlane, Optimal guaranteed cost control and ﬁltering for uncertain linear systems, IEEE Trans. Autom. Cont. 39 (1994) 1971–1977.

[29] P. Shi, E.K. Boukas, H1 control for Markovian jumping linear systems with parametric

(15)

[30] P. Shi, Filtering on sampled-data systems with parametric uncertainty, IEEE Trans. Autom. Cont. 43 (7) (1998) 1022–1027.

[31] P. Shi, E.K. Boukas, R.K. Agarwal, Kalman ﬁltering for continuous-time uncertain systems with Markovian jumping parameters, IEEE Trans. Autom. Cont. 44 (8) (1999) 1592–1597. [32] R. Srichander, B.K. Walker, Stochastic analysis for continuous-time fault-tolerant control

systems, Int. J. Cont. 57 (2) (1989) 433–452.

[33] D.D. Sworder, Feedback control of a class of linear systems with jump parameters, IEEE Trans. Autom. Cont. 14 (1) (1969) 9–14.

[34] A.S. Willsky, A survey of design methods for failure detection in dynamic systems, Automatica 12 (5) (1976) 601–611.