Journal of the Franklin Institute 348 (2011) 568–588

0016-0032/$3

doi:10.1016/j

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Robust stabilization of polytopic discrete-timesystems with time-varying state delay:

A convex approach$

M�arcio F. Mirandaa, Valter J.S. Leiteb,�

aCOLTEC/UFMG, Av. Antonio Carlos 6627, 31270-010 Belo Horizonte, MG, BrazilbCEFET–MG/Campus Divinopolis, Rua �Alvares Azevedo 400, 35503-822 Divinopolis, MG, Brazil

Received 20 February 2009; received in revised form 5 January 2011; accepted 11 January 2011

Available online 19 January 2011

Abstract

Convex conditions, expressed as linear matrix inequalities (LMIs), for stability analysis and robust

design of uncertain discrete-time systems with time-varying delay are presented in this paper. Delay-

dependent and delay-independent convex conditions are given. This paper is particularly devoted to

the synthesis case where convex conditions are proposed to consider maximum allowed delay

interval. It is also presented some relaxed LMIs that yield less conservative conditions at the expense

of increasing the computational burden. Extensions to cope with decentralized control and output

feedback control are discussed. Numerical examples, including real world motivated models, are

presented to illustrate the effectiveness of the proposed approach.

& 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Time delays are unavoidable in digital controlled systems and are intrinsically presentedin many dynamical systems, such as robotics, informations, communications andnetworks. Metal cutting, chemical, thermal and rolling mill are examples of relevantprocesses that are affected by delays [4,17,29]. Also, discrete-time delays (or concentrateddelays) have been investigated in models of population dynamics, where they can yield

2.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

.jfranklin.2011.01.002

er has been partially supported by Brazilian agencies FAPEMIG and CNPq (308620/2009-7).

nding author.

dresses: fantini@coltec.ufmg.br (M.F. Miranda), valter@ieee.org, vjsleite@gmail.com (V.J.S. Leite).

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 569

destabilizing effects and a phenomenon of Hopf bifurcation [23]. It is well known that evensmall time delays can reduce the performance of systems and, in some cases, can lead themto instability. Due to their relevance, systems with delay have been widely studied in thelast two decades specially on continuous-time domain [17].

In general, the available methods for stability analysis or controller design for systemswith delay can be classified into delay-dependent or delay-independent in case theyexplicitly take into account or not, respectively, the value of the delay. Usually, delay-dependent conditions yield less conservative results, but for systems which stability doesnot depend on the time-delay value, the analysis performed through delay-dependentconditions can be very conservative. Also, it is interesting to note that delay-independentconditions cannot be obtained as a limit case of delay-dependent ones just by imposing themaximum delay value dmax-þ1, leading to a gap between these two types of delaystability conditions [29, p. 146; 20]. Also note that in real world applications, delays areoften found as time-varying and arise in systems in two main different ways: in the input [8]and in the state [28]. For systems with state delay, most of the used techniques for robuststability analysis and robust control design are based on Lyapunov–Krasovskii approach.This approach has been used to obtain convex optimization problems in terms of linearmatrix inequalities (LMIs) which can be solved efficiently by specialized numericalalgorithms [2]. Less attention has been paid to discrete-time systems with state delaymainly due to the possibility of study of an augmented delay-free system when the delay isknown and time-invariant [16,20].

In cases of time-varying delay no obvious technique is available in the literature to dealwith fundamental issues such as stability analysis or controller design, specially for convexdesign methods. In case of continuous-time systems, an extension of asymptotic stabilityanalysis coupled with strictly dissipative characteristics, where norm-bounded uncertain-ties are considered with time-varying delay is presented in [41]. An extension to couple withabsolute stability analysis of continuous-time systems with Lur’e type perturbations isfound in [18] — where the delay is considered time-invariant — and dissipative analysisand synthesis is proposed in [27] for nonlinear systems with time-varying delays by meansof nonconvex conditions. In special, see [1] for robust filter design depending on an upperbound on the time-derivative of the delay. However, none of these results have theirdiscrete-time counterpart found in the literature. This lack of results is even stronger forthe class of uncertain discrete-time system with time-varying state delay. This caserepresents a relevant challenge in control theory, since the complexity of the problem isaugmented by both the uncertainties and the presence of time-varying delay. Such class ofsystems is investigated in this paper.

In the literature, different aspects of discrete-time systems with delayed states have beenstudied. In [15] LMI conditions for robust stability analysis of discrete-time delayedsystems with saturation are proposed. In [37] bi-dimensional (2D) discrete-time systemswith delayed states are investigated, and delay-independent conditions for norm-boundeduncertainties and constant delay are given by means of nonconvex formulations. In [26]convex conditions have been proposed for discrete-time singular systems with time-invariant delay. Discrete-time switched systems with delayed states have been studiedin [13,14]. The former establishes the equivalence between the approach used here(Lyapunov–Krasovskii functions) and the one used, in general, for the stability of switchedsystems with time-varying delay. The latter gives nonconvex conditions for switchedsystems where each operation mode is subject to a norm-bounded uncertainty and with

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588570

constant delay. The problem of robust filtering for discrete-time uncertain systems withdelayed states is considered in some papers. Delayed state systems with norm-boundeduncertainties are studied in [39,3,38] and with polytopic uncertainties in [7]. In the last case thedelay is time-invariant. Recently, the problem of output feedback has attracted attentionand [11,12,24] can be cited as examples of on going research. In special, [12] presents results forprecisely known systems with time-varying delay including both static output feedback (SOF)and dynamic output feedback. The conditions, however, are presented as an interactivemethod that relax some matrix inequalities. In [11,24] time-varying delay is assumed and anonlinear algorithm is proposed to obtain a stabilizing controller. In [24] the results of [11] areextended, including polytopic uncertainties and constant Lyapunov–Krasovskii matrices. Aninteresting application can be found in the context of network control system: although mostof the studies in the literature on this subject deal with continuous-time models, nowadaysthere are some approaches using discrete-time models with delayed states [42]. See also [36]for a robust adaptive sliding mode control scheme applied to discrete-time systems withtime-varying delay in the state and subject to norm-bounded uncertainties. In the context ofdiscrete-time-varying systems with time-varying delay in the state, see [19,34] for convexapproaches to the dynamic output feedback controller design problem.Note that most of the problems mentioned up to here had their solutions obtained from

more fundamental problems: the stability analysis or the stabilizing controller design. Inthis sense, since most of the cited papers use constant Lyapunov–Krasovskii matrices, itcan be said that the research of these more fundamental problems is an important issue. Inthe literature, it is possible to observe that besides most of the conditions are structuredover quadratic stability and the majority consider norm-bounded uncertainties. See, forexample, [9,10,24]. Among these works, [10] presents the less conservative conditions, butwith formulations only for stability analysis of norm-bounded systems. It is worth tomention that norm-bounded uncertainty is not addressed in the present paper. Therefore,it is clear that there is room for investigation on robust stability analysis and robuststabilization of polytopic uncertain discrete-time systems with time-varying delayed state.Recently, convex delay-independent conditions for robust stabilization of polytopicsystems has been proposed in [20], where delay is time-varying and in [22], where delay isconstant and a H1 level of performance is assured to the closed-loop system.In this paper, both delay-dependent and delay-independent convex conditions for both

robust stability analysis and robust synthesis of state feedback gains for uncertain and time-invariant discrete-time systems with time-varying delay and polytopic uncertainties areproposed. The conditions employ parameter-dependent Lyapunov–Krasovskii functions andextra matrix variables. This issue is two fold: it achieves less conservative results anddecouples the system matrices from the function matrices allowing better results in bothanalysis and design. Also, some relaxed LMIs are given to illustrate how the proposedconditions can be extended to cope with recent results using other parameter-dependentformulations. The main advantage of the present proposal is the existence of a convexformulation to the synthesis of robust feedback gains. It is shown how these conditions can beused to cope with some interesting problems such as decentralized control, SOF, controllerfragility analysis and some special cases of input delay.This paper is organized as follows: In Section 2 the problems studied in this paper are

stated and some results employed here are presented. The main results are given in Section 3.Some physically motivated models are used as examples in Section 4. By last are given theconclusions.

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 571

Notation: RðRþÞ is the set of real (positive) numbers, N� stands for the set of the naturalnumbers excluded the 0 and I ½a; b� stands for the set of all integer n such that arnrb. xk 2 R

n

is the state vector at sample time k. I and 0 are the identity and null matrices of appropriatedimensions, respectively. M ¼ block-diagfM1;M2g stands for the block-diagonal matrix M

made up of the matrices M1 and M2 at the main diagonal. M40ðMo0Þ means that M ispositive (negative) definite. M 0 stands for the transpose of M. % is used to indicate diagonallysymmetric blocks in the LMIs.� stands for Kronecker product.Fd denotes the space of discrete

vector functions mapping the interval I ½�d; 0� into Rn with a finite d 2 N�. fdt ðkÞ 2 Fd denotes

a sequence of dþ1 vectors xk with k 2 I ½t�d; t�. The j-th term of this sequence is

xtþj�1�d ¼ fdt;jðkÞ 2 Rn. It is defined Jfd

t ðkÞJD ¼ maxj2I ½1;ðdþ1Þ�Jfdt;jðkÞJ where J � J stands

for the Euclidian vector norm. Fkd is the set defined by Fk

d ¼ ffdt ðkÞ : Jf

dt ðkÞJDokg, with

k 2 Rþ. fd stands for the null sequence fd ¼ f0; . . . ; 0g|fflfflfflfflfflffl{zfflfflfflfflfflffl}ðdþ1Þ terms

.

2. Preliminaries

Consider the uncertain discrete-time system with delayed state given by

xkþ1 ¼ ~AðaÞxk þ ~Ad ðaÞxk�dkþ ~BðaÞuk ð1Þ

with xk 2 fd0 ðkÞ for k 2 I ½�d ; 0� and k refers to the k-th sample. fd

0 ðkÞ is the initial

condition, necessary to assure existence and uniqueness for the solutions of Eq. (1), with

d ¼ maxðdkÞ. xk ¼ xðkÞ 2 Rn is the state vector, xk�dkis the state vector at dk past samples

and dk ¼ dðkÞ40 is the time-varying delay. uk ¼ uðkÞ 2 Rp stands for the control signal.

The system matrices are denoted by ½ ~AðaÞj ~AdðaÞj ~BðaÞ� ¼ ½ ~Aj ~Ad j ~B�ðaÞ ¼ ~UðaÞ 2 Rn�ð2nþpÞ

with the uncertain parameter a satisfying

O � a : a 2 RN ;XN

i¼1

ai ¼ 1; aiZ0

( )ð2Þ

The matrices ~UðaÞ belong to the polytope

~P � ~UðaÞ : ~UðaÞ ¼XN

i¼1

~Uiai; a 2 O

( )ð3Þ

where the vertices ~Ui ¼ ½ ~Aij ~Adij ~Bi� � ½ ~Aj ~Ad j ~B�i are known. Time delay is supposed to betime-varying subject to

dk 2 I ½d ; d � ð4Þ

The following state feedback control law is considered in this paper:

uk ¼ Kxk þ Kdxk�dkð5Þ

where ½K jKd � 2 Rp�2n are constant state feedback gains assuring the robust stability of theclosed-loop (1)–(4). That is, the robust stability of Eqs. (1) and (2) with Eqs. (4) and (5) isassured 8a 2 O. This yields an uncertain closed-loop system, with time-varying delay, given by

xkþ1 ¼ AðaÞxk þ AdðaÞxk�dkð6Þ

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588572

with xk 2 fd0 ðkÞ for k 2 I ½�d ; 0�, ½AðaÞjAdðaÞ� ¼ ½AjAd �ðaÞ ¼ UðaÞ belonging to the

polytope

P � UðaÞ : UðaÞ ¼XN

i¼1

Uiai; a 2 O

( )ð7Þ

where the vertices Ui ¼ ½AijAdi� ¼ ½AjAd �i are given by

Ui ¼ ½ ~Ai þ ~BiKj ~Adi þ ~BiKd � ð8Þ

Note that, if the delay dk is available at each sample time, then it is possible to use both gains, K

and Kd, to improve the closed-loop performance of Eq. (6). On the other hand, if the delay dk isnot available, the approach presented in this paper is still valid, being enough to make Kd ¼ 0

in Eq. (5).If xk 2 fd

t ðkÞ ¼ fdfor k 2 I ½t�d ; t�, then an equilibrium condition is achieved for the

closed-loop system (6), since xkþ1 ¼ xk ¼ 0, 8k4t and 8a 2 O.

Definition 1. For a given a 2 O, the trivial solution of Eq. (6) is said uniformly

asymptotically stable if for any k 2 Rþ such that for all initial conditions xk ¼ fd0 ðkÞ 2 Fk

d,

k 2 I ½�d ; 0�, it is verified

limt-1

fdt;jðkÞ ¼ 0; 8j 2 I ½t�d ; t�

This allows the following definition:

Definition 2. System (6) subject to Eqs. (2), (4) and (7) is said robustly stable if its respectivetrivial solution is uniformly asymptotically stable 8a 2 O.

The main objective of this work is to present convex conditions that can solve thefollowing fundamental problems:

Problem 1. Determine if system (6) subject to Eqs. (2), (4) and (7) is robustly stable.

Problem 2. Determine a pair of static feedback gains, K and Kd, such that Eqs. (1)–(4)controlled by Eq. (5) is robustly stable.

Sufficient convex formulations for solving Problems 1 and 2 are obtained by means of

parameter-dependent Lyapunov–Krasovskii functions as a counterpart of delay-depen-dent conditions for continuous-time systems with delayed states [17]. In this paper thefollowing Lyapunov–Krasovskii function candidate is considered

V ða; kÞ ¼X5v¼1

Vvða; kÞ40 ð9Þ

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 573

with

V1ða; kÞ ¼ x0kPðaÞxk; V2ða; kÞ ¼Xk�1

j¼k�dk

x0jQðaÞxj ð10Þ

V3ða; kÞ ¼X1�d

‘¼2�d

Xk�1j¼kþ‘�1

x0jQðaÞxj ð11Þ

V4ða; kÞ ¼X�1‘¼�d

Xk�1m¼kþ‘

y0mZðaÞym; V5ða; kÞ ¼Xk�1

j¼k�dk

y0jZðaÞyj ð12Þ

yj ¼ xjþ1�xj ð13Þ

In the Lyapunov–Krasovskii function above, if only the functions Vvða; kÞ, v=1,y,3 areused, as done for instance in [20], then delay-independent conditions are obtained. On theother hand, by adding functions V4ða; kÞ and V5ða; kÞ — see Eqs. (12) and (13) — delay-dependent convex formulation for both robust analysis stability and robust synthesisdesign are achieved. The parameter dependency of Eq. (9) is a relevant issue to deal withuncertain systems as it may reduce the conservatism of analysis and synthesis conditions.To be a Lyapunov–Krasovskii function, the candidate (9) must be positive definite andsatisfy

DV ða; kÞ ¼ V ða; k þ 1Þ�V ða; kÞo0 ð14Þ

8½xk0x0k�dk

�0a0 and 8a 2 O.The following result is employed in this paper to decouple the matrices of the parameter-

dependent Lyapunov–Krasovskii candidate from the matrices of the system.

Lemma 1 (Finsler’s Lemma). Let o 2 Rn, QðaÞ ¼ QðaÞ0 2 Rn�n and BðaÞ 2 Rm�n such that

rank ðBðaÞÞon. The following statements are equivalent:

(i)

o0QðaÞoo0; 8o : BðaÞo ¼ 0;oa0; (ii) (X ðaÞ 2 Rn�m : QðaÞ þ X ðaÞBðaÞ þ BðaÞ0X ðaÞ0o0.

Proof. The proof follows similar steps of the proof presented in [6] replacing the preciselyknown matrices by parameter-dependent matrices. &

For conciseness, in what follows, the dependency on a of terms Vvða; kÞ is omitted andthey are denoted by VvðkÞ, v=1,y,5.

3. Main results

In this section, the conditions for the robust stability analysis of Eq. (6) and for thedesign of robust state feedback gains to robust stabilization of Eq. (1).

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588574

3.1. Robust stability analysis

Lemma 2. If there exist symmetric matrices 0oPðaÞ 2 Rn�n, 0oQðaÞ 2 Rn�n, 0oZðaÞ 2Rn�n, a matrix X ðaÞ 2 R5n�n, dk 2 I ½d ; d � with d and d belonging to N�, such that

CðaÞ ¼ block-diagfMðaÞ;�QðaÞ;�ZðaÞ;�ZðaÞg þ X ðaÞBðaÞ þ BðaÞ0X ðaÞ0o0 ð15Þ

with

MðaÞ ¼PðaÞ 0

0 bQðaÞ�PðaÞ

" #þ

1 �1

�1 1

� �� ðd þ 1ÞZðaÞ ð16Þ

b ¼ d�d þ 1 ð17Þ

and

BðaÞ ¼ ½I �AðaÞ �Ad ðaÞ 0 0� ð18Þ

is verified 8a admissible, then system (6) subject to Eq. (4) is robustly stable. Besides,Eqs. (9)–(13) are a Lyapunov–Krasovskii function assuring the robust stability of the

considered system.

Proof. The positivity of the function (9) is clearly assured by PðaÞ ¼ PðaÞ040, QðaÞ ¼QðaÞ040, ZðaÞ ¼ ZðaÞ040. Eq. (14) is calculated taking into account the following terms:

DV1ðkÞ ¼ x0kþ1PðaÞxkþ1�x0kPðaÞxk ð19Þ

DV2ðkÞ ¼ x0kQðaÞxk�x0k�dkQðaÞxk�dk

þXk�1

i¼kþ1�dkþ1

x0iQðaÞxi�Xk�1

i¼kþ1�dk

x0iQðaÞxi

rx0kQðaÞxk�x0k�dkQðaÞxk�dk

þXk�d

i¼kþ1�d

x0iQðaÞxi ð20Þ

DV3ðkÞ ¼ ðd�d Þx0kQðaÞxk�Xk�d

i¼kþ1�d

x0iQðaÞxi ð21Þ

DV4ðkÞðykÞ ¼ d y0kZðaÞyk�X�1j¼�d

y0kþjZðaÞykþj

rdy0kZðaÞyk�y0k�d

ZðaÞyk�d�

Xk�d

i¼kþ1�d

y0iZðaÞyi ð22Þ

DV5ðkÞ ¼ y0kZðaÞyk�y0k�dkZðaÞyk�dk

þXk�1

i¼kþ1�dkþ1

y0iZðaÞyi�Xk�1

i¼kþ1�dk

y0iZðaÞyi

ry0kZðaÞyk�y0k�dkZðaÞyk�dk

þXk�d

i¼kþ1�d

y0iZðaÞyi ð23Þ

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 575

Therefore, Eq. (14) can be bounded using Eqs. (19)–(23) which yields

DV ðkÞrx0kþ1PðaÞxkþ1 þ x0k½bQðaÞ�PðaÞ�xk�x0k�dkQðaÞxk�dk

þy0kðd þ 1ÞZðaÞyk�y0k�d

ZðaÞyk�d�y0k�dk

ZðaÞyk�dko0 ð24Þ

with b given by Eq. (17). Replacing yk by the right side of Eq. (13) in Eq. (24) and takinginto account Eq. (6) it is possible to apply Lemma 1 with

o ¼ ½x0kþ1 x0k x0k�dky0

k�dy0k�dk�0

QðaÞ ¼ block-diagfMðaÞ;�QðaÞ;�ZðaÞ;�ZðaÞg

with X ðaÞ 2 R5n�n, MðaÞ and BðaÞ given by Eqs. (16) and (18), respectively, yielding CðaÞ ¼QðaÞ þ X ðaÞBðaÞ þ BðaÞ0X ðaÞ0o0 given by Eq. (15). Thus, this is equivalent to verifyEq. (24) subject to Eq. (6), which assures the robust stability of the considered system. &

An important issue in Lemma 2 is that there is no product between the matrices of thesystem and the matrices of the Lyapunov–Krasovskii proposed function, Eq. (9). This isexploited in this paper to reduce conservatism in both analysis and synthesis methods. Thedependency on d and d is discussed in this paper mainly for synthesis procedure.

It is worth of mentioning that conditions given in Lemma 2 are not convex in a whenthis parameter belongs to a continuous domain. To overcome this issue and to provide asolution to Problem 1, the same polytopic structure of the system is assumed for theLyapunov–Krasovskii matrices PðaÞ, QðaÞ and ZðaÞ:

PðaÞ ¼XN

i¼1

aiPi; QðaÞ ¼XN

i¼1

aiQi; ZðaÞ ¼XN

i¼1

aiZi; a 2 O ð25Þ

leading to the following result.

Theorem 1. If there exist symmetric matrices 0oPi 2 Rn�n, 0oQi 2 Rn�n, 0oZi 2 Rn�n,

i ¼ 1; . . . ;N, a matrix X 2 R5n�n, dk 2 I ½d ; d � with d and d belonging to N�, such that

Li ¼ block-diagfMi; �Qi; �Zi; �Zig þ XBi þ B0iX 0o0; i ¼ 1; . . . ;N ð26Þ

with

Mi ¼Pi 0

0 bQi�Pi

" #þ

1 �1

�1 1

� �� ðd þ 1ÞZi ð27Þ

Bi ¼ ½I �Ai �Adi 0 0� ð28Þ

and b is given by Eq. (17) is verified, then system (6) subject to Eqs. (4) and (3) is robustly

stable, characterizing a solution to Problem 1. Besides, Eqs. (9)–(13) with matrices PðaÞ, QðaÞand ZðaÞ given in Eq. (25) and a 2 O are a Lyapunov–Krasovskii function assuring the robust

stability of the considered system.

Proof. Condition (26) is sufficient for Eq. (15), given in Lemma 2. Note that, the positivityof the function (9)–(13) with Eq. (25) is clearly assured by Pi ¼ P0i40, Qi ¼ Q0i40,Zi ¼ Z0i40. Besides assuming Eq. (25), the extra matrix X ðaÞ given in Eq. (15) isconstrained to be parameter independent to obtain a convex formulation. Thus, X ðaÞ ¼X 2 R5n�n is a restriction on Finsler’s multiplier, leading to a sufficient condition for

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588576

CðaÞo0 given in Eq. (15). This allows to recover CðaÞ ¼PN

i¼1 Liaio0 for a 2 O, which isthe continuous domain of uncertainty assumed for Eq. (6). &

It is worth of mentioning that the presented approach does not introduce any modeltransformation neither extra dynamics. The use of Finsler’s Lemma allows the introductionof slack matrix variable X , which role is to lead to less conservative results by enlarging thesearch space. It can be found in the literature some similar results using the addition of a nullterm to Eq. (24) as done by [35]. However, the use of null term can be viewed as a specialcase of Finsler’s lemma, where some extra variables are set to zero.In this robust stability analysis some relaxed LMIs can be introduced to avoid the

constraint made over X ðaÞ. If this extra matrix is parameter dependent, the approachemployed in [21,31] can be used in a similar way to get less conservative conditions.However, the drawback of this approach is to increase the computational burden. Thus,considering

X ðaÞ ¼XN

i¼1

aiX i; a 2 O ð29Þ

the following condition can be stated.

Corollary 1. If there exist symmetric matrices 0oPi 2 Rn�n, 0oQi 2 Rn�n, 0oZi 2 Rn�n,

i ¼ 1; . . . ;N, matrices X i 2 R5n�n, dk 2 I ½d ; d � with d and d belonging to N�, such that

Li ¼ block-diagfMi;�Qi;�Zi;�Zig þ X iBi þ B0iX 0io0; i ¼ 1; . . . ;N ð30Þ

Lij ¼ block-diagfMij ;�Qi�Qj ;�Zi�Zj ;�Zi�Zjg þ X iBj þ B0jX 0iþX jBi þ B0iX 0jo0; i ¼ 1; . . . ;N�1; j ¼ i þ 1; . . . ;N ð31Þ

with Mi given in Eq. (27),

Mij ¼Pi þ Pj 0

0 bðQi þQjÞ�Pi�Pj

" #þ

1 �1

�1 1

� �� ðd þ 1ÞðZi þ ZjÞ ð32Þ

Bm ¼ ½I �Am �Ad;m 0 0�; m 2 fi; jg ð33Þ

and b given by Eq. (17) is verified, then system (6) subject to Eqs. (4) and (3) is robustly

stable, characterizing a solution to Problem 1. Besides, Eqs. (9)–(13) with matrices PðaÞ, QðaÞand ZðaÞ given in Eq. (25) and a 2 O, are a Lyapunov–Krasovskii function assuring the robust

stability of the considered system.

Proof. The proof follows from Eq. (15) by noting that it can be rewritten as CðaÞ ¼PNi¼1 a

2i Li þ

PN�1i¼1

PNj¼iþ1 aiajLijo0 and imposing Eqs. (30), (31) and (29). For details,

see [21] or [31]. &

Also notice that a quadratic stability based condition can be recovered from Theorem 1 bydefining Pi=P, Qi=Q and Zi=Z, i=1,y,N in Eq. (26). The resulting condition is a set ofLMIs with slack matrix variables and constant parameter-independent Lyapunov–Krasovskiimatrices. In this case, the system matrices can be time-varying, i.e., ai can be replaced byaðkÞi, with

PNi¼1 aðkÞi ¼ 1; aðkÞiZ0.

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 577

Another remark is that delay-independent conditions can be obtained by considering asimpler Lyapunov–Krasovskii function given by V ða; kÞ ¼

P3v¼1 VvðkÞ, with Vv(k) as in

Eqs. (10) and (11). In this case, the obtained conditions depend on the difference ðd�d Þ butnot on the delay itself. This condition is presented in the next corollary.

Corollary 2. If there exist symmetric matrices 0oPi 2 Rn�n, 0oQi 2 Rn�n, i ¼ 1; . . . ;N,�X 2 R3n�n, dk 2 I ½d ; d � with d and d belonging to N�, such that

�Li ¼ block-diagfPi;bQi�Pi;�Qig þ �X �B i þ �B 0i �X0o0; i ¼ 1; . . . ;N ð34Þ

with

�B i ¼ ½I �Ai �Adi� ð35Þ

and b given by Eq. (17) is verified, then system (6) subject to Eqs. (7), (4) and

jdkþ1�dkjrðd�d Þ is robustly stable, independently of the delay value. Besides, V ða; kÞ ¼P3v¼1 Vvða; kÞ40 with Vv, v ¼ 1; . . . ; 3, given by Eqs. (9)–(11) and matrices PðaÞ and QðaÞ

given in Eq. (25) with a 2 O, is a Lyapunov–Krasovskii function assuring the robust stability

of the considered system.

It is important to observe that the conditions of Corollary 2 cannot be obtained fromEq. (26) just by taking its limit as d-1. Note that, Corollary 2 is delay-independent and,thus, it does not depend on d itself, but it does depend on the difference d�d . Also notethat the same approach used in Corollary 1 can be applied in this case yielding thefollowing corollary.

Corollary 3. If there exist symmetric matrices 0oPi 2 Rn�n, 0oQi 2 Rn�n, �X i 2 R3n�n,

i ¼ 1; . . . ;N, integer constants d 2 N�, dk 2 I ½d ; d � with d and d belonging to N�, such that

�Li ¼ block-diagfPi; bQi�Pi;�Qig þ �X i�B i þ �B 0i �X 0io0; i ¼ 1; . . . ;N ð36Þ

�Lij ¼ block-diagfPi þ Pj ;bðQi þQjÞ�Pi�Pj ;�Qi�Qjg

þ �X j�B i þ �B 0i �X 0j þ �X i

�B j þ �B 0j �X 0io0; i ¼ 1; . . . ;N�1; j ¼ i þ 1; . . . ;N ð37Þ

with

�Bm ¼ ½I �Am �Adm�; m 2 fi; jg ð38Þ

and b given by Eq. (17) is verified, then system (6) subject to Eqs. (7), (4) and

jdkþ1�dkjrðd�d Þ is robustly stable, independently of the delay value. Besides, V ða; kÞ ¼P3v¼1 Vvða; kÞ40 with Vv, v ¼ 1; . . . ; 3, given by Eqs. (9)–(11) and matrices PðaÞ and QðaÞ

given in Eq. (25) with a 2 O, is a Lyapunov–Krasovskii function assuring the robust stability

of the considered system.

Proof. The proof follows the same ideas described in the proof of Corollary 1. &

Obviously, if d ¼ d , then the delay is taken as constant and uncertain. Finally, it is alsoworth of noting that the conditions proposed in Theorem 1 and Corollary 1 can be usedto verify the robust stability of the dual system obtained by replacing AðaÞ and AdðaÞ by

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588578

AðaÞ0 and AdðaÞ0, respectively. This leads to similar changes on the conditions of

Corollaries 2 and 3.Another relevant issue is that the technique proposed to obtain Corollaries 1 and 3 can

be improved by supposing that the matrices of the Lyapunov–Krasovskii function (9)depend polynomially on the parameter a. In this case, the results proposed by [30] can besimilarly applied at the expense of increasing computational effort. However this extensionis not dealt here, once the main objective in this paper is to obtain a convex condition forthe synthesis of robust feedback gain.It is interesting to note that the presence of the term V3ða; kÞ in the Lyapunov–Krasovskii

function, see Eq. (11), results on conditions that are dependent on the difference d�d . Thisdifference is related to the maximum variation rate allowed to the delay, which is such thatjdkþ1�dkj be equal to or less than ðd�d Þ. Therefore, although delay-independent, theconditions presented in Corollaries 2 and 3 depend on the maximum variation of the delaybetween two samples.

3.2. Robust feedback gain design

The robust analysis conditions given in Section 3.1 are exploited to obtain convexconditions for robust feedback gain design yielding a solution to Problem 2.

Theorem 2. If there exist matrices W 2 Rp�n, Wd 2 Rp�n, and matrices F 2 Rn�n,

0oPi ¼ P0i 2 Rn�n, 0oQi ¼ Q0i 2 Rn�n, 0oZi ¼ Z0i 2 Rn�n, i ¼ 1; . . . ;N, dk 2 I ½d ; d � with

d and d belonging to N�, such that

Pi þ ðd þ 1ÞZi þ F þ F 0 �ððd þ 1ÞZi þ F 0 ~A0i þW 0 ~B0iÞ �ðF0 ~A0di þW 0

d~B0iÞ

% bQi�Pi þ ðd þ 1ÞZi 0

% % �Qi

2664

3775

o0; i ¼ 1; . . . ;N ð39Þ

with b given by Eq. (17) is verified, then the uncertain system (1)–(4) subject to the control

law (5) with

K ¼WF�1 and Kd ¼WdF�1 ð40Þ

is robustly stable, thus providing a solution to Problem 2. Besides, Eqs. (9)–(13) with matrices

PðaÞ, QðaÞ and ZðaÞ given in Eq. (25) and a 2 O, are a Lyapunov–Krasovskii function

assuring the robust stability of the closed-loop system.

Proof. Firstly, note that xk1¼ Axk þ Adxk�dk

is stable if and only if xk1¼ A0xk þ A0dxk�dk

is stable. Then, from Theorem 1 by replacing Ai and Adi by ð ~Ai þ ~BiKÞ0 and ð ~Adi þ ~BiKd Þ

0,

respectively, defining X ¼ ½F 0n�4n�0, KF ¼W and KdF ¼Wd and noting that the resulting

condition can be simplified once the required positiveness of Zi is already included in thehypothesis of the theorem. Also note that the regularity of matrix F is assured by block (1,1)

of Eq. (39), once it yields F þ F 0o�ðPi þ ðd þ 1ÞZiÞo0, i=1,y,N. &

Similarly what has been done in the analysis case, for the synthesis condition it is alsopossible to present a delay-independent condition that allows a delay variation given byjdkþ1�dkjrd�d . This is presented in the next corollary.

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 579

Corollary 4. If there exist matrices W 2 Rp�n and Wd 2 Rp�n, F 2 Rn�n, 0oPi ¼ P0i,0oQi ¼ Q0i, i ¼ 1; . . . ;N, jdkþ1�dkjrd�d with d and d belonging to N�, such that

Pi þ F þ F 0 �ðF 0 ~A0i þW 0 ~B0iÞ �ðF0 ~A0di þW 0

d~B0iÞ

% bQi�Pi 0

% % �Qi

264

375o0; i ¼ 1; . . . ;N ð41Þ

with b given by Eq. (17) is verified, then system (1) subject to Eqs. (2)–(4) is robustly stabilizable,for any value of the delay dk, by means of static robust feedback gains K and Kd given in Eq. (40).

Besides, V ða; kÞ ¼P3

v¼1 Vvða; kÞ40 with Vv, v ¼ 1; . . . ; 3, given by Eqs. (9)–(11) and matrices

PðaÞ and QðaÞ given in Eq. (25) with a 2 O, is a Lyapunov–Krasovskii function assuring the

robust stability of the closed-loop.

Proof. The proof can be obtained from the conditions of Corollary 2, replacing Ai and Adi

by ð ~Ai þ ~BiKÞ0 and ð ~Adi þ ~BiKd Þ

0, respectively, and defining X ¼ ½F 0n�2n�0, KF ¼W and

KdF ¼Wd . &

Quadratic stability conditions can be obtained from both Eqs. (39) and (41) simplyimposing Pi ¼ P, Qi ¼ Q and Zi ¼ Z, i ¼ 1; . . . ;N (Z ¼ 0 in Eq. (41)). In this case, systemmatrices can be time-varying encompassing, for example, actuators faults.

Note that conditions presented in Corollary 1 or 3 are not suitable for a convex synthesisprocedure once the parameter dependency on X i does not allow, by convex means, thedesign of constant gains K and Kd.

The proposed conditions can be specialized to deal with some constrained control designsuch as decentralized control, SOF and input delay. This is discussed in the sequence.

3.2.1. Decentralized control

This issue refers to decentralized control, employed when interconnected systems mustbe controlled by means of local information only. In this case, decentralized control gainsK ¼ KD and Kd ¼ KdD can be obtained by imposing block-diagonal structure to matricesW, Wd and F as follows:

W ¼WD ¼ block-diagfW 1; . . . ;W Rg

Wd ¼WdD ¼ block-diagfW 1d ; . . . ;W

Rd g

F ¼ FD ¼ block-diagfF1; . . . ;F Rg

where R denote the number of defined subsystems. In this case, one gets the robust block-diagonal state feedback gains KD ¼WDF�1D and KdD ¼WdDF�1D . It is worth to mentionthat the matrices of the Lyapunov–Krasovskii function, PðaÞ, QðaÞ and ZðaÞ, do not haveany restrictions in their structures, which leads to less conservative designs.

3.2.2. Static output feedback

This is the case when only a linear combination of the states is available for feedbackand the output signal is given by yk ¼ ~Cxk. If ~C is full row rank, then it is always possibleto find a regular matrix T such that ~CT�1 ¼ ½Iq 0�. Using such matrix T in a similarity

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588580

transformation applied to Eq. (1) it leads to

xkþ1 ¼ AðaÞxk þ AdðaÞxk�dkþ BðaÞuk ð42Þ

where AðaÞ ¼ T ~AðaÞT�1, AdðaÞ ¼ T ~AdðaÞT�1 and BðaÞ ¼ T ~BðaÞ, xk ¼ Txk and theoutput signal is given by yk ¼ ½Iq 0�xk. Thus, the objective here is to find robust staticfeedback gains K 2 Rp�q and Kd 2 Rp�q such that Eq. (42) is robustly stabilizable by thecontrol law

uk ¼ Kyk þKdyk�dkð43Þ

These gains can be determined by using the conditions of Theorem 2 or Corollary 4 andimposing the following structures:

with F 11o 2 R

q�q, F21o 2 Rðn�qÞ�q, F22

o 2 Rðn�qÞ�ðn�qÞ, WK 2 Rp�q, WKd2 Rp�q which yields

K ¼ ½Kj0� and Kd ¼ ½Kd j0�

Note that, similar to the decentralized case, no constraint is taken over the Lyapunov–Krasovskii function matrices leading to less conservative conditions.

3.2.3. Input delay

A relevant issue in Control Theory is the study of stability and stabilization of inputdelay systems, which is quite frequent in many real systems [39,3]. The approach presentedin this paper can be used to handle input delays as summarized in the sequel. Consider thecontrolled system given by

xkþ1 ¼ ~AðaÞxk þ ~BðaÞuk�dkð44Þ

with ~AðaÞ and ~BðaÞ belonging to polytope (3), ~Adi ¼ 0 and a 2 O. This type of system isdetailed investigated in [40], where the problem is converted into an optimization problemin Krein space with a stochastic model associated. Here, the delayed input control signal isconsidered as

uk�dk¼ Kdxk�dk

ð45Þ

The closed-loop system is given by

xkþ1 ¼ ~AðaÞxk þ ~BðaÞKdxk�dkð46Þ

Thus, with known Kd, closed-loop system (46) is similar to Eq. (6) with AðaÞ replaced by~AðaÞ and Ad ðaÞ by ~BðaÞKd . This leads to simple analysis stability conditions obtainedfrom Theorem 1, replacing Ai by ~Ai and Adi by ~BiKd , i ¼ 1; . . . ;N. Besides, similarreplacements can be used with conditions presented in Corollaries 1– 3. It is interesting tonote that this issue is two fold once both controller fragility and input delay can beaddressed by this formulation. In the former it is required that no uncertainty affects theinput matrix, i.e., ~BðaÞ ¼ ~B, 8a 2 O, while the latter can be used to investigate, forexample, the bounds of stability of a closed-loop system where there exists delay due todigital processing or information propagation.In case of Kd design, it is possible to take similar steps with conditions of Theorem 2

and Corollary 4. In this case, it is necessary to impose Ai ¼ ~Ai, Adi ¼ 0, i ¼ 1; . . . ;N andW ¼ 0 leading to K ¼ 0. Also, the Schur stability of ~AðaÞ is required in this case. Although

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 581

possible, the design of a stabilizing Kd for Eq. (46) usually leads to a gain with very smallentries modulus. On the other hand, since Theorem 2 gives a convex way to determine K

and Kd, it is expected that an approach similar to that used in this paper can be used toderive convex conditions to deal with some performance specification, such as guaranteedH1 index. In this case, the design of K and Kd or only Kd certainly becomes even moreinteresting. This also can be the case when dealing with saturating actuators as done in thecontinuous-time domain by [33].

Finally, observe that static delayed output feedback control can be additionallyaddressed here by considering what is pointed out in Section 3.2.2.

3.3. Numerical complexity

The numerical complexity of the proposed conditions depend on the number of scalarvariables, V, and on the number of rows, G, in the LMIs. In this paper it has been usedMatlab with the interface YALMIP [25] and the solver SeDuMi [32] whose numberfloating point operations performed to the same problems has an order given byV2G5=2 þ G7=2. It has been verified by the authors that SeDuMi usually presents animportant time reduction in solving the proposed optimization problems comparatively toother commercially available softwares, for the tests performed in this paper. For theconditions present here, it is possible to determine the following relationships:VT1 ¼ ðn2ð3N þ 10Þ þ 3NnÞ=2, GT1 ¼ 8Nn, VT2 ¼ ðn2ð3N þ 2Þ þ nð3N þ 4pÞÞ=2, GT2 ¼5Nn, VC2 ¼ ðn2ð2N þ 6Þ þ 2NnÞ=2, GC2 ¼ 6Nn, VC4 ¼ n2ðN þ 1Þ þ nðN þ 2pÞ, GC4 ¼ 3Nn,VC1 ¼ ð13nþ 3ÞNn=2, GC1 ¼ ðN2 þ 6NÞn=2, VC3 ¼ ð6nþ 1ÞNn, GC3 ¼ ðN2 þ 4NÞn=2.Thus, it is evident that the number of scalar variables and the number of LMI linesincrease significantly with the conditions of Corollaries 1 and 3 w.r.t. Theorem 1and Corollary 2, respectively.

4. Numerical examples

In all examples it has been used a Intel Core 2 Duo, T 7600, 2.33 GHz, with 2 Gb ofRAM, using MatLab, the parser YALMIP [25] and SeDuMi toolbox [32].

Example 1 (Stabilization design). A physically motivated problem is considered in thisexample. It consists of a fifth order state space model of an industrial electric heaterinvestigated in [4]. This furnace is divided into five zones, each of them with athermocouple and a electric heater as indicated in Fig. 1. The state variables are thetemperatures in each zone (x1; . . . ; x5), measured by thermocouples, and the control inputsare the electrical power signals (u1; . . . ; u5) applied to each electric heater. The temperature

Fig. 1. Schematic diagram of the industrial electric heater.

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588582

of each zone of the process must be regulated around its respective nominal operationalconditions (see [4] for details). The dynamics of this system is slow and can be subject toseveral load disturbances. Also, a time-varying delay can be expected, since the velocity ofthe displacement of the mass across the furnace may vary. A discrete-time with delayedstate model for this system has been obtained as given by Eq. (1) with d ¼ 15, where

~A ¼ ~A0 ¼

0:97421 0:15116 0:19667 �0:05870 0:07144

�0:01455 0:88914 0:26953 0:11866 �0:22047

0:06376 0:12056 1:00049 �0:03491 �0:02766

�0:05084 0:09254 0:28774 0:82569 0:02570

0:01723 0:01939 0:29285 0:03544 0:87111

26666664

37777775 ð47Þ

~Ad ¼ ~Ad0 ¼

�0:01000 �0:08837 �0:06989 0:18874 0:20505

0:02363 0:03384 0:05282 �0:09906 �0:00191

�0:04468 �0:00798 0:05618 0:00157 0:03593

�0:04082 0:01153 �0:07116 0:16472 0:00083

�0:02537 0:03878 �0:04683 0:05665 �0:03130

26666664

37777775 ð48Þ

and

~B ¼ ~B0 ¼

0:53706 �0:11185 0:09978 0:04652 0:25867

�0:51718 0:73519 0:57518 0:40668 �0:12472

0:29469 0:31528 1:16420 �0:29922 0:23883

�0:20191 0:19739 0:41686 0:66551 0:11366

�0:11835 0:16287 0:20378 0:23261 0:36525

26666664

37777775 ð49Þ

where ~A0, ~Ad0, and ~B0 are called here the nominal matrices of this system. Note that, thisnominal system has unstable modes. The design of a stabilizing state feedback gain for thissystem has been considered in [4] by using optimal control theory. In this case, anaugmented delay-free system with order equal to 85 has to be used and the control gain hasbeen determined for the time-invariant delay d ¼ 15 by means of a Riccati equation.

Table 1

Feasible structures (|) achieved by conditions of Theorem 2 (T2), Corollary 4 (C4), and quadratic stability

approach (QS) obtained from Theorem 2 and Corollary 4, respectively QS-T2 and QS-C4. The symbol ‘–’ means

that no feasible solution has been obtained.

Structure K and Kd Only K

T2 C4 QS-T2 QS-C4 T2 C4 QS-T2 QS-C4

{1,1,3} – – – – – – – –

{1,3,1} – – – – – – – –

{3,1,1} | | – – – – – –

{2,3} | | – – – – – –

{3,2} | | – – – – – –

{1,4} | | – – – – – –

{4,1} | | | | | | | |{5} | | | | | | | |

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 583

In [22] an uncertain system given by

~AðrÞ ¼ ð1þ rÞ ~A0; ~AdðZÞ ¼ ð1þ ZÞ ~Ad0 and ~BðsÞ ¼ BdðsÞ ¼ ð1þ sÞ ~B0 ð50Þ

has been considered in the context of H1 control with uncertain and time-invariant delayand delay-independent conditions. In this case, the parameters r, Z and s are theuncertainties. This set of uncertainties defines a polytope with eight vertices, obtained bycombination of the extremum values of uncertain parameters.

Here the stabilization of this uncertain system with

jrjr0:3; jZjr0:3 and jsjr0:08 ð51Þ

is investigated under time-varying delay assumption 10rdkr20. Both, delay-dependentand delay-independent conditions have been used. Different gain structures, as discussedin Section 3.2.1, have been investigated. The obtained results are summarized in Table 1.The size of each square block in the decentralized control structure is indicated in the firstcolumn.

Note that quadratic stability conditions (QS) identified in Table 1 refer to the conditionsof Theorem 2 and of Corollary 4 with Pi ¼ P, Qi ¼ Q and Zi ¼ Z, i ¼ 1; . . . ;N. In Table 1,the notation fs1; s2; s3g used in the first column means that the controller is structured into ablock-diagonal matrix where the submatrices have dimensions si � si, i ¼ 1; 2; 3. Theseconditions can be obtained directly from the conditions of Theorem 2 or Corollary 4 and, inthis case, they will have extra matrices decoupling system matrices from the Lyapunov–Krasovskii function ones. This leads to a less conservative formulation in the sense thatstructure constraints to achieve robust decentralized state feedback gains are not imposed tothe matrices of the Lyapunov–Krasovskii function. This is not verified if quadratic stabilityconditions are obtained in a standard way, for example, by replacing in Eq. (24) xkþ1 by theright-hand side of Eq. (1) and yj, as given in Eq. (13). These conditions are not shown in thispaper. Also note that if time-invariant delay is considered, i.e. setting d ¼ d , the samestructures achieved in [22] are recovered here.

It is clear by this example that some decentralized control structures are possible only byusing K and Kd, or equivalently, xk and xk�dk

in the feedback loop. These structures,namely f3; 1; 1g, f2; 3g, f3; 2g, f4; 1g and f1; 4g, can be of interest in real applications sincetheir respective control implementations can less expensive than using full controllers. Inspecial, note that for structures f3; 1; 1g, f2; 3g, f3; 2g the quadratic stability based approachcannot achieve a feasible solution, while the present proposal does.

In special, consider the control structure given by f3; 1; 1g and suppose that the delayvalue (10rdkr20) is available at each control instant. In this case the following gains areobtained with Theorem 2:

K ¼ block-diag

�1:3616 �0:9610 �0:2231

�1:1835 �2:3621 0:1941

0:3637 0:7230 �0:9838

264

375;�1:1373;�1:9104

8><>:

9>=>;

Kd ¼ block-diag

�0:0847 0:1932 �0:2354

�0:1451 0:1269 �0:2713

0:1243 �0:0573 0:1432

264

375;�0:2156;�0:0546

8><>:

9>=>;

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588584

Suppose that system (1) with Eqs. (47)–(51) is subject to an initial condition given byf200 ðkÞ, where each vector belonging to f20

0 ðkÞ is equal to ½10;�10; 5;�5; 8�0. These values

correspond to typical deviations in the real furnace zones temperature studied in [4].In Fig. 2 it is shown the states behavior of the closed-loop system. At each subplot is showneight curves, one for each vertex of the polytope describing Eqs. (47)–(51). The time-varying delay employed in these simulations is given by dk ¼ 15þ roundð5sin0:4pkÞ,as shown in Fig. 3. It is interesting to note that the majority of the time-responsesreach the zero between samples 5 and 8. However, the delay, typically about 15, acts as

0 10 20 30 40 50 60 70 80−10

0

10

0 10 20 30 40 50 60 70 80−10

0

10

0 10 20 30 40 50 60 70 80−20

0

20

0 10 20 30 40 50 60 70 80−5

0

5

0 10 20 30 40 50 60 70 80−10

0

10

k

x 1x 2

x 3x 4

x 5

Fig. 2. State response for each vertex of Eqs. (47)–(51).

0 10 20 30 40 50 60 70 8010

12

14

16

18

20

k

d k

Fig. 3. Time-varying delay used in the simulations.

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 585

a perturbation for almost time-response about samples between 10 and 20. For the curvespresented in Fig. 3 it is possible to say that they reach the equilibrium condition after 40samples.

Example 2 (Robust stability analysis). Consider Eq. (6) with

A ¼0:6 0

0:35 0:7

� �; Ad ¼

0:1 0

0:2 0:1

� �ð52Þ

This system has been investigated in [24]. Using Theorems 1, 2 and 5 of [24] with d ¼ 1 it isachieved, respectively, d ¼ 9, 12 and 8. Using Corollary 2 of the present proposal thestability is assured for 1rdkr8, the same bound achieved by [24, Theorem 5], which is theonly one from [24] that can deal with uncertain system. Once Corollary 2 and [24, Theorem5] are delay-independent, their conditions can assure the stability of the considered systemfor dk 2 ½d; d þ 7� with 1rdr1. Besides, the delay-dependent conditions provide inTheorems 1 and 2 of [24] cannot be used to deal with uncertain systems.

Consider a slight modified system where matrices A and Ad are both affected by amultiplicative uncertainty given by ð1þ rÞ. This yields a polytopic representation with twovertices given by Vi ¼ ½AijAdi�, with Ai ¼ Að1þ ð�1ÞirÞ, Adi ¼ Ad ð1þ ð�1Þ

irÞ, i ¼ 1; 2 andrZ0. Fig. 4 has been obtained by varying r from 0 and searching with conditionsfrom Corollary 2 and from [24, Theorem 5] the maximum value of d that this system isverified to be stable, for d ¼ 1. Both conditions achieve the same bounds on d and r.However, the numerical complexity of condition [24, Theorem 5] calculated as indicated inSection 3.3 is about 67.8% greater than the numerical complexity of Corollary 2. Thus, inthis example, the information achieved by [24, Theorem 5] can be obtained by Corollary 2in a less expensive way. Also note that, due to the polytopic representation of theuncertainties, norm-bounded based approaches, such as [9,10], does not apply to this case.

Example 3 (Input delay and controller fragility analysis). In this example, it is considered aphysically motivated model of a cold rolling mill system found in [5], where a sample

1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

�

d

Fig. 4. r� d , with d ¼ 1.

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588586

period of Ts ¼ 0:1 s has been used. The nominal delay-free model is assumed for thissystem with matrices ~A0 and ~B0 given by

~A0 ¼01�9 x0I9�9 09�1

" #and ~B0 ¼

2:76 �1:35 �0:46

09�3

� �ð53Þ

with x0 ¼ 0:112. Suppose there is a time-varying delay, dk, in the feedback loop and that xis uncertain and given by x ¼ x075%. Thus, it is possible to obtain a polytopicrepresentation of the corresponding ~AðaÞ and ~BðaÞ given by

~Ui ¼ ½ ~Aij ~Bi�; i ¼ 1; 2 ð54Þ

where ~A1 and ~A2 are computed using the extremum values of x, ~B1 ¼ ~B2 ¼ B0, n ¼ 10 andp ¼ 3. Suppose that this system is controlled by a static feedback gain

Kd ¼

�5:1784 0:6114 �1:6754 4:1266 �6:8142

0 0 0 0 0

�4:0239 �1:9537 �0:8677 �0:4145 �0:2618

264�1:6566 3:4657 4:3534 1:9838 �0:6713

0 0 0 0 0

�0:2294 �0:2557 �0:3091 �0:3361 7:0238

375� 10�2 ð55Þ

By using Theorem 1 the closed-loop system is robustly stable for 1rdkr3. That is, thestate used in the feedback action can be delayed up to 0.3 s from the actual instant.Consider that there is an implementation error on the value of Kd up to B%. In this case,

the gain Kd is replaced by an uncertain gain ~K d given by ~K d ¼ Kdð1þ eÞ, with0rerB=100%. This yields a 4 vertex polytope, with vertices computed by combining theextremum values of ~K d and ~A. By means of conditions of Theorem 1 it is made a search onthe maximum value of p such that the closed-loop system remains stable. It is found thatBT1 ¼ 0:3%. If d is reduced from 3 to 2, then it is found that the uncertainty allowed in ~K d

can be admitted up to BT1 ¼ 22:9%. This shows that, in this example, the fragility of thecontroller is very sensitive to the maximum value of dk.

5. Conclusions

Delay-dependent and delay-independent convex conditions for both robust stability androbust stabilizability of uncertain discrete-time systems with time-varying delay in the stateare given. All system matrices are supposed to be affected by polytopic-type uncertainties.The proposed conditions employ parameter-dependent Lyapunov–Krasovskii functionsand extra-variables yielding less conservative results than other conditions available in theliterature. Also, these slack variables decouple the system matrices from the functionmatrices, allowing a convex formulation in the synthesis case. All conditions encompassquadratic stability based ones as special cases. Relaxed LMI conditions have beenpresented for the robust stability case, as a way to reduce the conservatism of theapproach. It has been shown that robust stabilizability conditions can be easily used tocope with decentralized control, static output feedback, input delay and control fragilityanalysis. Numerical examples are presented, including models based on two distinct realprocesses, to illustrate the efficiency of the proposed conditions.

M.F. Miranda, V.J.S. Leite / Journal of the Franklin Institute 348 (2011) 568–588 587

Acknowledgement

The authors are thankful to the anonymous reviewers for their contributions that haveimproved the presentation of this paper.

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