Gain scheduled control for discrete-time systems depending on bounded rate parameters

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2005; 15:473–494Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/rnc.1001

Gain scheduled control for discrete-time systems dependingon bounded rate parameters

Francesco Amato1,n,y, Massimiliano Mattei2,z and Alfredo Pironti3,}

1Corso di Laurea in Ingegneria Informatica e Biomedica, Dipartimento di Medicina Sperimentale e Clinica,

Universita degli Studi Magna Græcia di Catanzaro, Via T. Campanella 115, 88100 Catanzaro, Italy2Dipartimento di Informatica, Matematica, Elettronica e Trasporti, Universita degli studi Mediterranea di Reggio

Calabria, Via Graziella, Localita Feo De Vito, 89100 Reggio Calabria, Italy3Dipartimento di Informatica e Sistemistica, Universita degli Studi di Napoli Federico II,

Via Claudio 21, 80125 Napoli, Italy

SUMMARY

In this paper we consider a linear, discrete-time system depending multi-affinely on uncertain, real time-varying parameters. A new sufficient condition for the stability of this class of systems, in terms of afeasibility problem involving linear matrix inequalities (LMIs), is obtained under the hypothesis that abound on the rate of variation of the parameters is known. This condition, obtained by the aid ofparameter dependent Lyapunov functions, obviously turns out to be less restrictive than that one obtainedvia the classical quadratic stability (QS) approach, which guarantees stability in presence of arbitrary time-varying parameters. An important point is that the methodology proposed in this paper may result in beingless conservative than the classical QS approach even in the absence of an explicit bound on the parametersrate of variation. Concerning the synthesis context, the design of a gain scheduled compensator based onthe above approach is also proposed. It is shown that, if a suitable LMI problem is feasible, the solution ofsuch problem allows to design an output feedback gain scheduled dynamic compensator in a controller-observer form stabilizing the class of systems which is dealt with. The stability conditions are then extendedto take into account L2 performance requirements. Some numerical examples are carried out to show theeffectiveness and to investigate the computational burden required by the proposed approach. Copyright# 2005 John Wiley & Sons, Ltd.

KEY WORDS: discrete-time systems; uncertain systems; time-varying parameters; robust stabilization

1. INTRODUCTION

In the past years, two major approaches have emerged to investigate the stability properties ofdiscrete-time linear systems containing uncertain parameters. The first one assumes that the

Received 3 December 2002Accepted 4 May 2005Copyright # 2005 John Wiley & Sons, Ltd.

yE-mail: [email protected]

nCorrespondence to: Francesco Amato, Corso di Laurea in Ingegneria Informatica e Biomedica, Dipartimento diMedicina Sperimentale e Clinica, Universita degli Studi Magna Græcia di Catanzaro, Via T. Campanella 115, 88100Catanzaro, Italy.

zE-mail: [email protected]}E-mail: [email protected]

uncertainties are time-invariant and establishes whether or not the eigenvalues of the systemmatrix are in the unit disk for all the allowable values of the parameters (see References[1, Chapter 2, 2]). However, when the uncertainties are time-varying, this approach may fail.Indeed robust stability with respect to constant parameters (i.e. stability for each fixed value ofthe uncertain parameters), does not guarantee stability when the parameters are time-varying(see References [3, 4]).

For time-varying parameters, another approach, called the quadratic stability (QS) approach,is commonly used (see References [5–8]). In this approach one looks for a quadratic Lyapunovfunction which does not depend on the uncertainties and which guarantees stability for all theallowable values of the parameters. Roughly speaking, this approach guarantees uniformasymptotic stability when the parameters are time-varying and there is no restriction on howfast the parameters can vary. In this paper we consider the situation in which, as it oftenhappens in real world applications (see for example Reference [9]), the parameters are time-varying with a bounded rate of variation.

A first result of the paper is a novel sufficient condition to test the robust stability of adiscrete-time linear system depending on uncertain time-varying parameters with a knownbound on their rate of variation. This result turns out to be less conservative than the classicalQS approach.

As in the continuous-time context (see for example References [10–12]), thecondition guaranteeing uniform asymptotic stability of the discrete-time uncertain systemis found with the aid of quadratic parameter dependent Lyapunov functions. In particular,the Lyapunov function to prove robust stability is chosen in the class of the piecewiseconstant mappings of the parameters [13–15]. The problem of finding such a parameterdependent Lyapunov function is then converted into a feasibility problem involvinga set of interlaced Lyapunov like linear matrix inequalities (LMIs). In Reference [16]parameter dependent Lyapunov functions depending linearly on the parameter vector areused; however, the parameter rate of variation is not taken into account and the parameterbounds are given in terms of the vector 1-norm instead of the vector1-norm that we consider inthis paper.

The novel methodology proposed in our paper can also be applied when the rate of variationof the parameters is unbounded; the interesting point is that, also in this case, the new approachmay result in being less conservative than the classical QS approach. This fact represents apeculiarity of discrete-time systems; indeed, in the continuous-time case, the approach viaquadratic parameter dependent Lyapunov functions cannot be applied when no bound on therate of variation is available. This shows, according to Reference [17], that there is a deepdifference between the continuous and the discrete-time framework for what concerns robuststability versus time-varying parameters.

A second result carried out in the paper is a sufficient condition for the solution of thestabilizability problem via state feedback. This condition, obtained combining some of theresults appeared in Reference [18] with our novel approach, converts the problem of designingthe state feedback compensator into a standard LMI feasibility problem.

The more general problem of stabilizing the plant via a dynamic output feedbackcompensator, when the full state is not available for feedback, is also discussed. A sufficientcondition for the existence of such a compensator is still obtained in terms of a feasibilityproblem involving LMIs. The dynamic compensator turns out to be in the standard controller–observer form.

Copyright # 2005 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2005; 15:473–494

F. AMATO, M. MATTEI AND A. PIRONTI474

Both the state and output feedback compensators are gain scheduled [19] with the plantparameters. For both the analysis and synthesis problems, by following the approach ofReference [20], we consider the companion problems involving L2 performances. However, theoutput feedback performance problem cannot be reduced to an LMI feasibility problem.

It is worth to notice that, in the hybrid piecewise affine (PWA) systems context, similarapproaches have been proposed in the discrete-time domain [21, 22]. In these papers sufficientconditions for the stability and stabilizability with H2=H1 performance via state feedback ofPWA systems are provided. The main difference between the robust control context, consideredin this paper, and the hybrid system context is that in the first case the focus is on the vector ofuncertain parameters, whose jumps in a discrete time interval are bounded by its rate ofvariation, while in the second case the focus is on the events that possibly drive the state from acertain region to another at each time-step.

The paper is organized as follows. In Section 2 the problem we deal with is precisely statedand some preliminary definitions are given. In Section 3 the analysis problem is considered:sufficient conditions for robust stability and performances of an autonomous discrete-timeuncertain linear system depending on bounded rate parameters are provided together with anumber of considerations. Finally, Section 4 deals with the design problem: sufficient conditionsfor the synthesis of state and output feedback compensators are provided. Some numerical non-trivial examples are provided to show the effectiveness and to investigate the computationalburden required by the proposed technique.

2. PROBLEM STATEMENT

Consider the discrete-time autonomous uncertain system described by the following state spaceequation:

xðkþ 1Þ ¼ AðpÞxðkÞ ð1Þ

where the system matrix is assumed to depend multi-affinely on the parameters. The multi-affinedependence on the parameters can be written as

AðpÞ ¼X1

i1;i2;...;ir¼0

Ai1;i2;...;ir pi11 p

i22 � � � p

irr

with Ai1;i2;...;ir 2 Rn�n and p ¼ ½p1 p2 � � � pr�T 2 Rr is the vector of the uncertain parameters.In most part of the literature on robustness of linear systems, it is standard to assume that the

dependence on parameters be affine, or equivalently that the system matrices belong to suitablepolytopes in the matrix space (see among others References [10, 11, 17, 18, 23], and thebibliography therein). This is due to the fact that the affine dependence is of a quite generalnature and, at the same time, it allows to state a number of sufficient conditions guaranteeingrobust stability. In this paper we consider the more general multi-affine dependence onparameters which allows to deal with a larger class of control problems and practicalapplications (see for example Reference [9]). It is worth to notice that, when the dependence onparameters is more complex than multi-affine, one can resort to the so-called polytopic setcovering technique proposed in Reference [24] to convert a more general nonlinear dependence


GAIN SCHEDULED CONTROL FOR DISCRETE-TIME SYSTEMS 475

in a multi-affine one. This technique however introduces some conservatism in the stabilityconditions.

In our treatment the parameters pi are assumed to be time-varying and to satisfy the followingassumptions.

Assumption 1Each parameter belongs to a known interval

pi 2 ½%pi; %pi�; for i ¼ 1; 2; . . . ; r ð2Þ

therefore p 2}; where

} ¼ ½%p1; %p1� � ½

%p2; %p2� � � � � � ½

%pr; %pr�

Assumption 2The rate of variation of each parameter is bounded and satisfies the following conditionfor all k:

Dpi ¼ jpiðkþ 1Þ � piðkÞj4%hi for i ¼ 1; 2; . . . ; r ð3Þ

The first objective of the paper is to find a solution to the following stability analysis problem.

Problem 1 (Robust stability analysis)Given system (1) with pð�Þ being any vector of unknown time-varying parameters satisfyingAssumptions 1 and 2, find a sufficient condition guaranteeing its uniform asymptotic stability.

Remark 1If the discrete-time system we are dealing with is obtained by the discretization of a continuous-time process (see Reference [25]) subject to bounded rate parameters piðtÞ; with j’pij4hci ; i ¼1; . . . ; r; the value of %hi in Assumption 2 depends on the sampling period. Indeed, if the samplingperiod Ts is constant, we have (letting tkþ1 ¼ tk þ Ts and, according with the notation used inthe paper, k ¼ tk)

jpiðkþ 1Þ � piðkÞj ¼jpiðkþ 1Þ � piðkÞj

TsTs ¼ j’pið*t Þj Ts

where *t 2 ½tk; tkþ1� and the last equality follows from the Mean Value Theorem; therefore

jpiðkþ 1Þ � piðkÞj4hci Ts ¼: %hi ð4Þ

Now let us consider the forced system described by the following state space equations:

xðkþ 1Þ ¼ AðpÞxðkÞ þ BðpÞuðkÞ ð5aÞ

yðkÞ ¼ CðpÞxðkÞ ð5bÞ



where

BðpÞ ¼X1

i1;i2;...;ir¼0

Bi1;i2;...;ir pi11 p

i22 � � � p

irr ; CðpÞ ¼

X1i1;i2;...;ir¼0

Ci1;i2;...;ir pi11 p

i22 � � � p

irr

Bi1;i2;...;ir 2 Rn�m; Ci1;i2;...;ir 2 Rp�n; i1; i2; . . . ; ir ¼ 0; 1

The second goal of the paper is to find a solution to the following state and output feedbackcontroller synthesis problems.

Problem 2 (State feedback } SF)Given system (5a), pð�Þ being any vector of time-varying parameters which satisfies Assumptions1 and 2, find a state feedback gain scheduled memoryless linear compensator in the form

u ¼ *KðpÞx ð6Þ

such that the closed loop connection of (6) with (5a) is uniformly asymptotically stable.

Problem 3 (Output feedback } OF)Given system (5), pð�Þ being any vector of time-varying parameters which satisfies Assumptions 1and 2, find an output feedback, gain scheduled, linear dynamic controller in the form

xcðkþ 1Þ ¼ *AcðpÞxcðkÞ þ *BcðpÞyðkÞ ð7aÞ

uðkÞ ¼ *CcðpÞxcðkÞ þ *DcðpÞyðkÞ ð7bÞ

with xcðkÞ 2 Rn; such that the closed loop connection of (7) with (5) is uniformly asymptoticallystable.

Throughout the paper we will give a solution to Problems 1–3 in terms of optimizationproblems involving LMIs. Moreover, we shall show that our approach also provides a lessconservative alternative to the QS approach both for the analysis and the synthesis problems,even if the parameters in systems (1) and (5) have no bounds on their rate of variation.

It is worth to notice that, based on the results provided in Reference [20], also L2 performancerequirements} can be considered both for the analysis and the synthesis problem. In this casesystem (1) should be replaced with

xðkþ 1Þ ¼ AðpÞxðkÞ þ B1ðpÞwðkÞ ð8aÞ

zðkÞ ¼ C1ðpÞxðkÞ þD11ðpÞwðkÞ ð8bÞ

where wðkÞ 2 Rq is the disturbance vector, zðkÞ 2 Rs is the controlled output and the matrixvalued functions B1ðpÞ; C1ðpÞ and D11ðpÞ depend multi-affinely on parameters. In the same waysystem (5) should be replaced with

xðkþ 1Þ ¼ AðpÞxðkÞ þ B1ðpÞwðkÞ þ B2ðpÞuðkÞ ð9aÞ

zðkÞ ¼ C1ðpÞxðkÞ þD11ðpÞwðkÞ þD12ðpÞuðkÞ ð9bÞ

yðkÞ ¼ C2ðpÞxðkÞ þD21ðpÞwðkÞ ð9cÞ

} In a similar way H2 performance requirements can be introduced.



With respect to systems (8) and (9), Problems 1–3 have to be reformulated requiring L2

performance on the w-z channel.However, as it will be shown in Section 4.2, the performance requirement leads to a non-

convex optimization problem for what concerns the solution of the OF Problem 3.

3. A SUFFICIENT CONDITION FOR THE ROBUST STABILITY PROBLEM

3.1. Main result

Before proving our main results, we need to state two technical lemmas. The first lemma is asufficient condition for the stability of a linear time-varying discrete-time system in terms of aclassical difference Lyapunov equation. It can be easily proven by evaluating the first differenceof the Lyapunov function Vðx; kÞ ¼ xTðkÞPðkÞxðkÞ along the system solutions, where PðkÞ is anypositive definite matrix-valued sequence.

Lemma 1If there exists a positive number a and a bounded positive definite matrix sequence PðkÞ whichsatisfies the following Lyapunov inequality:

ATðkÞPðkþ 1ÞAðkÞ � PðkÞ5� a � I for all k50 ð10Þ

then the discrete-time linear time-varying system

xðkþ 1Þ ¼ AðkÞxðkÞ ð11Þ

is uniformly asymptotically stable.

Note that the above lemma requires the construction of a bounded matrix sequence PðkÞ: Thenext result is taken from Reference [8].

Lemma 2Let M 2 Rn�n be a positive definite matrix and N 2 Rn�n an arbitrary matrix; then thematrix-valued function ATðpÞMAðpÞ �N is negative definite for all p belonging to the hyper-rectangle ½a1; b1� � ½a2; b2� � � � � � ½ar; br� if and only if ATðpÞMAðpÞ �N is negative definitefor all p belonging to the set of vertices of the hyper-rectangle, namely the set fa1; b1g �fa2; b2g � � � � � far; brg:

Now let us partition each interval ½%pi; %pi� into ni disjoint sub-intervals of equal length qi

(see Figure 1)

½%pi; %pi� ¼

[nil¼1

½pi;l; pi;lþ1�

¼:[nil¼1

Ji;l pi;lþ1 � pi;l ¼ qi > 0 i ¼ 1; . . . ; r; l ¼ 1; . . . ; ni ð12Þ

note that pi;1 ¼%pi and pi;niþ1 ¼ %pi: Now define the following function

j : p 2}! jðpÞ ¼ ½ j1ðp1Þ j2ðp2Þ � � � jrðprÞ�T 2 Nr



where N denotes the set of integer numbers and (see Figure 2)

pi 2 ½pi;l; pi;lþ1½) ji ¼ l

Note that, at a given instant k; jðpðkÞÞ univocally individuates the sub-hyper-rectangle

J1; j1 � J2; j2 � � � � � Jr; jr

to which pðkÞ belongs.Now define the integers mi; i ¼ 1; 2; . . . ; r; as the maximum number of sub-intervals that the

parameter pi can jump in one discrete-time step compatibly with Assumption 2. In other wordsthe mi are defined in such a way that the following inequalities are satisfied:

jiðpiðkÞÞ_�ð�miÞ4jiðpiðkþ 1ÞÞ4jiðpiðkÞÞ

_�mi; i ¼ 1; 2; . . . ; r ð13Þ

where for any relative number x

ji_�x :¼

ji þ x if ji þ x 2 ½1; ni�

ni if ji þ x > ni

1 if ji þ x51

8>><>>: ð14Þ

From the definition of mi and Assumptions 1 and 2 it follows that, in one discrete-time step, pican move to a sub-interval which is at a distance (measured in number of sub-intervals of ½

%pi; %pi�)

which is less than or equal to mi (see Figure 3 in the case mi ¼ 2).Therefore, if the parameter vector p belongs to the following hyper-interval at the time step k

J1; ji � J2; j2 � � � � � Jr; jr ð15Þ

at the time step kþ 1; by virtue of the above-mentioned assumptions, it will belong to thefollowing set: [

l12I1;...;lr2Ir

J1; j1_�l1� J2; j2

_�l2� � � � � Jr; jr

_�lr

ð16Þ

where Ii ¼ f�mi; . . . ;�1; 0;þ1; . . .þmig:

iq

1,iipp ip……

1, +liplip ,……

2,ip i+1iip v, ip ,v

Figure 1. Partition of the interval ½%pi; %pi�:

lpj ii =)( iii pj v=)(1)( =ii pj

1,iip 1,ip 1+1,ip p vi,ip ……2,ip ip

i+1ip ,v……

Figure 2. Definition of the function jiðpiÞ:



In order to compact the notation we also define the following vectorization of the_�

operation

j_�l :¼ ½ j1

_�l1 j2

_�l2 � � � jr

_�lr�T ð17Þ

We have the following theorem.

Theorem 1If there exist n1n2 . . . nr positive definite matrices PðjÞ ¼ Pð j1; j2; . . . ; jrÞ; such that the followingLMIs are satisfied for ji ¼ 1; 2; . . . ; ni and li ¼ �mi; . . . ;�1; 0; 1; . . . ;mi; i ¼ 1; 2; . . . ; r :

ATðpÞPðj_�lÞAðpÞ � PðjÞ50 8p : pi 2 fpi;ji ; pi;ji_�1g ð18Þ

then system (1) is uniformly asymptotically stable for all time-varying realizations of theparameter vector pð�Þ satisfying Assumptions 1 and 2.

ProofSince (18) are a finite number of inequalities, there exists a positive number a such that for allji ¼ 1; 2; . . . ; ni and li ¼ �mi; . . . ;�1; 0; 1; . . . ;mi; i ¼ 1; 2; . . . ; r

ATðpÞPðj_�lÞAðpÞ � PðjÞ5� a � I 8p : pi 2 fpi;ji ; pi;ji_�1g

By virtue of Lemma 2, we obtain that the last inequality also holds for all p; that is we can write

ATðpÞPðj_�lÞAðpÞ � PðjÞ5� a � I 8p 2 J1; j1 � J2; j2 � � � � � Jr; jr ð19Þ

for ji ¼ 1; 2; . . . ; ni and li ¼ �mi; . . . ;�1; 0; 1; . . . ;mi; i ¼ 1; 2; . . . ; r:Now let pð�Þ be any realization of the parameter vector satisfying Assumptions 1 and 2.

According to the above notation jðpðkþ 1ÞÞ coincides, for some r-tuple ðl1; . . . ; lrÞ; with j_�l:

Therefore, condition (19) implies that

ATðpðkÞÞPðjðpðkþ 1ÞÞÞAðpðkÞÞ � PðjðpðkÞÞÞ5� a � I; 8k50

which in turn, by virtue of Lemma 1, implies that the time-varying system

xðkþ 1Þ ¼ AðpðkÞÞxðkÞ

is uniformly asymptotically stable. The proof follows from the arbitrariness of pð�Þ: &

Based on Theorem 1, we conclude that Problem 1 is solvable if there exist positive definitematrices PðjÞ which satisfy conditions (18). Such an existence problem is a classical LMI convexfeasibility problem which can be solved by using one of the algorithms described in Reference[20].

Remark 2Note that, when applying Theorem 1, the parameter to be chosen is the number of sub-intervalsni in which we partition the interval ½

%pi; %pi� i ¼ 1; . . . ; r; which gives place to the following

ip ……… ……...)(kpi 1, +iip v

Figure 3. Possible transitions of parameter pi in the interval ½k; kþ 1� in the case mi ¼ 2:



partition of }:

} ¼[n1j¼1

J1; j �[n2j¼1

J2; j � � � � �[nrj¼1

Jr; j ð20Þ

It is evident that, for a given partitioning of the original hyper-rectangle }; the faster theparameters are allowed to vary in time, the larger is the set (16) to which they can jump in onetime step and hence the greater are the integers mi; i ¼ 1; . . . ; r; which satisfy (13).

Since the integers ni and mi ði ¼ 1; . . . ; rÞ increase the dimension of the LMI problem to besolved, one should try to keep the numbers ni as low as possible, compatibly with the rate ofvariation of the parameters.

Remark 3As said in the introduction, interesting connections can be established between the approachfollowed in our paper and the works [21, 22]. In these papers a condition similar to Theorem 1 isderived for the stability of discrete-time piecewise affine systems; this condition is obtained bypartitioning the state space into the union of cells and analysing the switches of the statevariables between the cells. The link between our approach and the approach of References[21, 22] follows from the fact that a discrete-time parameter dependent system can be viewed as aswitched system. The bounds on the rate of variation of parameters we consider in our paper, isequivalent, in the view of References [21, 22], to restrict the set of admissible jumps between onecell and another.

The next theorem is a sufficient condition for stability with guaranteed L2 performances. Notethat this result recovers, as particular case, Theorem 1 when B1 ¼ C1 ¼ D11 ¼ 0:

Theorem 2If there exist n1n2 . . . nr positive definite matrices PðjÞ ¼ Pðj1; j2; . . . ; jrÞ; such that the following setof LMIs are satisfied for ji ¼ 1; 2; . . . ; ni and li ¼ �mi; . . . ;�1; 0; 1; . . . ;mi; i ¼ 1; 2; . . . ; r:

ATðpÞPðj_�lÞAðpÞ � Pð jÞ ATðpÞPðj

_�lÞB1ðpÞ CT

1 ðpÞ

BT1 ðpÞPðj

_�lÞAðpÞ BT

1 ðpÞPðj_�lÞB1ðpÞ � I DT

11ðpÞ

C1ðpÞ D11ðpÞ �I

0BB@

1CCA50 8p : pi 2 fpi; ji ; pi; ji_�1g ð21Þ

or equivalently

�Pðj_�lÞ Pðj

_�lÞAðpÞ Pðj

_�lÞB1ðpÞ 0

ATðpÞPðj_�lÞ �PðjÞ 0 CT

1 ðpÞ

BT1 ðpÞPðj

_�lÞ 0 �I DT

11ðpÞ

0 C1ðpÞ D11ðpÞ �I

0BBBBBB@

1CCCCCCA50 8p : pi 2 fpi; ji ; pi; ji_�1g ð22Þ

then system (8) is uniformly asymptotically stable for all time-varying realizations of theparameter vector pð�Þ satisfying Assumptions 1 and 2 and has an L2 norm bound on the w-z



channel less than one, that is

supp2}

supjjwjj2=0w2L2

jjzjj2jjwjj2

41

ProofCombining standard LMI results [20, Chapter 9] with Theorem 1 in Reference [26], followingthe guidelines of Theorem 1 and the fact that the matrix

AðpÞ B1ðpÞ

C1ðpÞ D11ðpÞ

!

is multi-affine, it readily follows that, if there exist n1n2 . . . nr positive definite matricesPðjÞ ¼ Pðj1; j2; . . . ; jrÞ such that the following set of LMIs are satisfied for ji ¼ 1; 2; . . . ; ni andli ¼ �mi; . . . ;�1; 0; 1; . . . ;mi; i ¼ 1; 2; . . . ; r:

AðpÞ B1ðpÞ

C1ðpÞ D11ðpÞ

!TPðj

_�lÞ 0

0 I

!AðpÞ B1ðpÞ

C1ðpÞ D11ðpÞ

!�

PðjÞ 0

0 I

!50 8p : pi 2 fpi;ji ; pi;ji_�1g

ð23Þ

system (8) is uniformly asymptotically stable for all time-varying realizations of the parametervector pð�Þ satisfying Assumptions 1 and 2 and has an L2 norm bound on the w-z channel lessthan one.

By applying Schur complements the proof follows. &

3.2. Comparison with the QS approach

Consider the autonomous, uncertain linear system (1). According to the classical definition, it issaid to be quadratically stable (see Reference [5]) if there exists a time-invariant quadraticLyapunov function VðxÞ ¼ xTPx whose first difference along the solutions of (1) is negativedefinite for all p 2}: Using properties of quadratic forms we can state the following equivalentdefinition.

Definition 1 (Quadratic stability)System (1) is said to be quadratically stable if there exists a symmetric positive definite matrix P

such that

ATðpÞPAðpÞ � P50 8p 2} ð24Þ

It is readily seen from the definition that QS guarantees uniform asymptotic stability of system(1) for all time-varying realization (that is regardless the rate of variation) of parameters pð�Þwith pðkÞ 2} for all k50:

As said in the introduction, the approach proposed in this paper can be applied to obtainsufficient conditions that are less conservative than the ones proposed in the standard QSapproach even in the case of parameters with unknown bounds on the rate of variation.

The key point is that, in the discrete-time context, the rate of variation of the ith parameter piis in any case bounded if such parameter is assumed to belong to a finite interval. Indeed pi



cannot vary in the unit time step more than the length j%pi � %pij of the interval. Hence, we have

that inequality (3) is always satisfied with %hi ¼ j%pi � %pij:

Therefore, if there is no explicit bound on the parameters rate of variation we can establishthe following corollary of Theorem 1 which guarantees uniform asymptotic stability of system(1) for any possible time realization of the unknown parameter pð�Þ satisfying Assumption 1.Such a corollary is obtained by letting mi ¼ ni in Theorem 1; indeed if no bound on the rate ofvariation of the parameters is available, one has to consider that in one step the ith parameter pican reach every subinterval of the interval ½

%pi; %pi�:

Corollary 1If there exist n1n2 . . . nr symmetric positive definite matrices PðjÞ ¼ Pð j1; j2; . . . ; jrÞ; such that thefollowing LMIs are satisfied for ji ¼ 1; 2; . . . ; ni and li ¼ �ni; . . . ;�1; 0; 1; . . . ; ni; i ¼ 1; 2; . . . ; r:

ATðpÞPðj_�lÞAðpÞ � Pð jÞ50 8p : pi 2 fpi; ji ; pi; ji_�1g ð25Þ

then system (1) is uniformly asymptotically stable for all time-varying realization of theparameter vector pð�Þ satisfying Assumption 1.

Corollary 1 converts once again a stability analysis problem into an LMI [20] standardfeasibility problem. Note that, by assuming ni ¼ 1; i ¼ 1; . . . ; r; Corollary 1 recovers the classicalQS condition; indeed in this case the Lyapunov function reduces to a fixed quadratic function inthe form xTPx; where P is a constant positive definite matrix. Conversely, by assuming, forsome i; ni > 1; we can obtain a less conservative condition for asymptotic stability, as it will beshown in Example 2 in the next subsection.

We end the section by noting that a condition equivalent to Theorem 1 can be stated asfollows.

Theorem 3If there exist n1n2 . . . nr positive definite matrices XðjÞ ¼ Xð j1; j2; . . . ; jrÞ; such that the followingLMIs are satisfied for ji ¼ 1; 2; . . . ; ni and li ¼ �mi; . . . ;�1; 0; 1; . . . ;mi; i ¼ 1; 2; . . . ; r:

AðpÞXðjÞATðpÞ � Xðj_�lÞ50 8p : pi 2 fpi; ji ; pi; ji_�1g ð26Þ

or equivalently

�Xðj_�lÞ AðpÞXðjÞ

XðjÞATðpÞ �XðjÞ

" #50 8p : pi 2 fpi; ji ; pi; ji_�1g

then system (1) is uniformly asymptotically stable for all time-varying realizations of theparameter vector pð�Þ satisfying Assumptions 1 and 2.

ProofThe condition

ATðpÞPðj_�lÞAðpÞ � PðjÞ50



is equivalent to

�Pðj_�lÞ Pðj

_�lÞAðpÞ

ATðpÞPðj_�lÞ �PðjÞ

24

3550

Pre- and post-multiplying the last inequality by

P�1ðj_�lÞ 0

0 P�1ðjÞ

" #

we obtain

�P�1ðj_�lÞ AðpÞP�1ðjÞ

P�1ðjÞATðpÞ �P�1ðjÞ

" #50

The proof follows by letting XðjÞ ¼ P�1ðjÞ: &

The condition in Theorem 3, will be useful in the design context. In the same way a conditionequivalent to Theorem 2 can be stated as follows.

Theorem 4If there exist n1n2 . . . nr positive definite matrices XðjÞ ¼ Xðj1; j2; . . . ; jrÞ; such that the followingset of LMIs are satisfied for ji ¼ 1; 2; . . . ; ni and li ¼ �mi; . . . ;�1; 0; 1; . . . ;mi; i ¼ 1; 2; . . . ; r:

�Xðj_�lÞ AðpÞXðjÞ B1ðpÞ 0

XðjÞATðpÞ �XðjÞ 0 XðjÞCT1 ðpÞ

BT1 ðpÞ 0 �I DT

11ðpÞ

0 C1ðpÞXðjÞ D11ðpÞ �I

0BBBBBB@

1CCCCCCA50

8p : pi 2 fpi; ji ; pi; ji_�1g ð27Þ

then system (8) is uniformly asymptotically stable for all time-varying realizations of theparameter vector pð�Þ satisfying Assumptions 1 and 2 and has an L2 norm bound on the w-zchannel less than one.

4. THE ROBUST STABILIZATION PROBLEM

We now turn our attention to the synthesis Problems 2 (SF) and 3 (OF) in order to find somesufficient conditions for the stabilizability of a discrete-time system subject to time-varying andbounded rate parameters. In the next sections, passing through the SF problem solution, we willgradually approach to the synthesis of a robustly stabilizing output feedback linear dynamiccompensator which can be obtained by means of the solution of an LMI standard feasibilityproblem.



4.1. The SF problem

Consider the discrete-time uncertain system described by the state space equation (5a); the nextresult converts the state feedback problem into a standard LMIs feasibility problem.

Theorem 5The SF problem is solvable if there exist positive definite matrices XðjÞ ¼ Xðj1; j2; . . . ; jrÞ andmatrices Wð jÞ ¼Wð j1; j2; . . . ; jrÞ 2 RðnþmÞ�ðnþmÞ such that the following LMIs are satisfied forji ¼ 1; . . . ; ni; li ¼ �mi; . . . ;�1; 0;þ1; . . . ;þmi; i ¼ 1; 2; . . . ; r:

�Xðj_�lÞ AðpÞXðjÞ þ BðpÞWðjÞ

XðjÞATðpÞ þWTðjÞBTðpÞ �XðjÞ

" #50 8p : pi 2 fpi; ji ; pi; ji_�1g ð28Þ

In this case, a state feedback gain scheduled controller guaranteeing uniform asymptoticstability of system (5a) is the following:

*K : p 2 J1; j1 � J2; j2 � � � � � Jr; jr ! KðjÞ ¼WðjÞX�1ðjÞ ð29Þ

ProofFrom Theorem 3 the SF problem is solvable if there exist positive definite matricesXð jÞ ¼ Xð j1; j2; . . . ; jrÞ and a state feedback memoryless gain scheduled controller

*Kð�Þ : p 2 J1; j1 � J2; j2 � � � � � Jr; jr ! KðjÞ ¼ Kð j1; j2; . . . ; jrÞ

such that for all ji ¼ 1; . . . ; ni; li ¼ �mi; . . . ;�1; 0;þ1; . . . ;þmi; i ¼ 1; 2; . . . ; r

�Xðj_�lÞ ðAðpÞ þ BðpÞKðjÞÞXðjÞ

XðjÞðAðpÞ þ BðpÞKðjÞÞT �XðjÞ

" #50 8p 2 J1; j1 � J2; j2 � � � � � Jr; jr

The proof follows by letting KðjÞXðjÞ ¼WðjÞ and observing that the LHS of the resulting LMI ismultiaffine (see Reference [27]). &

Remark 3Note that the control gain matrix

*Kð�Þ : p 2 J1;j1 � J2;j2 � � � � � Jr;jr ! Kðj1; j2; . . . ; jrÞ; ji ¼ 1; . . . ; ni; i ¼ 1; 2; . . . ; r ð30Þ

coming from Theorem 5, turns out to be gain scheduled with the parameters. In particular thehyper-rectangle to which the parameter vector is assumed to belong is partitioned into a certainnumber of sub-hyper-rectangles and to each sub-hyper-rectangle the corresponding statefeedback gain matrix is associated.

The following result problem also accounts for performance requirements.

Theorem 6Consider system (8). If there exist n1n2 . . . nr positive definite matrices XðjÞ 2 Rn�n; and matricesWðjÞ 2 Rm�n such that the following LMIs are satisfied for ji ¼ 1; 2; . . . ; ni and li ¼ �mi; . . . ;



�1; 0; 1; . . . ;mi; i ¼ 1; 2; . . . ; r :

�Xðj_�lÞ AðpÞXðjÞ þ B2ðpÞWðjÞ B1ðpÞ 0

XðjÞATðpÞ þWðjÞBT2 ðpÞ �XðjÞ 0 XðjÞCT

1 ðpÞ þWðjÞDT12ðpÞ

BT1 ðpÞ 0 �I DT

11ðpÞ

0 C1ðpÞXðjÞ þD12ðpÞWðjÞ D11ðpÞ �I

0BBBBBB@

1CCCCCCA50


then the SF problem is solvable with an L2 norm bound on the w-z channel less than one. A gainscheduled compensator is given by KðjÞ ¼WðjÞX�1ðjÞ:

ProofThe connection between system (9) and a gain scheduled state feedback control law in the formu ¼ *KðpÞx with %KðpÞ given by (30) reads

xðkþ 1Þ ¼ ðAðpÞ þ B2ðpÞKðjÞÞxðkÞ þ B1ðpÞwðkÞ

zðkÞ ¼ ðC1ðpÞ þD12ðpÞKðjÞÞxðkÞ þD11ðpÞwðkÞ

Applying Theorem 4 we have that the SF problem is solvable if there exist n1n2 . . . nr positivedefinite matrices XðjÞ 2 Rn�n; and matrices KðjÞ 2 Rm�n such that

�Xðj_�lÞ ðAðpÞ þ B2ðpÞKðjÞÞXðjÞ B1ðpÞ 0

XðjÞðAðpÞ þ B2ðpÞKðjÞÞT �XðjÞ 0 XðjÞðC1ðpÞ þD12ðpÞKðjÞÞ

T

BT1 ðpÞ 0 �I DT

11ðpÞ

0 ðC1ðpÞ þD12ðpÞKðjÞÞXðjÞ D11ðpÞ �I

0BBBBBB@

1CCCCCCA50


As usual, the proof follows by letting KðjÞXðjÞ ¼WðjÞ: &

4.2. The OF problem

Let us consider the uncertain discrete-time linear systems in the form (5). To prove the mainresult of the section we need the following technical lemma.

Lemma 3Given matrices F; T 2 Rn�n; there exist positive definite matrices P and Q such that

FPFT �Q50 ð33Þ

if and only if there exist positive definite matrices %P and %Q such that

FPFT� %Q50 ð34Þ

where %F ¼ T�1FT:



ProofAssume there exists positive definite matrices P and Q such that (33) holds. Pre-multiplying andpost-multiplying both members of (33) by T�1 and T�T respectively we obtain

T�1FPFTT�T � T�1QT�T50

which can be rewritten as

T�1FTT�1PT�TTTFTT�T � T�1QT�T50

from which (34) follows with %P ¼ T�1PT�T and %Q ¼ T�1QT�T:The only if part of the theorem can be proven in a similar way. &

Now we are ready to state the main result of the section.

Theorem 7The OF Problem is solvable if the following two conditions are satisfied:

(a) There exist positive definite matrices Xð jÞ ¼ Xð j1; j2; . . . ; jrÞ and matrices WðjÞ ¼Wðj1; j2; . . . ; jrÞ such that the following LMIs are satisfied for ji ¼ 1; . . . ; ni and li ¼ �mi; . . . ;�1; 0;þ1; . . . ;þmi; i ¼ 1; 2; . . . ; r:

�Xðj_�lÞ AðpÞXðjÞ þ BðpÞWðjÞ

XðjÞATðpÞ þWTðjÞBTðpÞ �XðjÞ

" #50 8p : pi 2 fpi; ji ; pi; ji_�1g ð35Þ

(b) There exist positive definite matrices PðjÞ ¼ Pðj1; j2; . . . ; jrÞ and matrices ZðjÞ ¼ Zðj1; j2;. . . ; jrÞ such that the following LMIs are satisfied for ji ¼ 1; . . . ; ni and li ¼ �mi; . . . ;�1;0;þ1; . . . ;þmi; i ¼ 1; 2; . . . ; r:

�PðjÞ ATðpÞPðj_�lÞ þ CTðpÞZTðj

_�lÞ

Pðj_�lÞAðpÞ þ Zðj

_�lÞCðpÞ �Pðj

_�lÞ

264

37550 8p:pi 2 fpi; ji ; pi; ji_�1g

ð36Þ

In this case an output feedback gain scheduled stabilizing compensator in controller-observerform is

xcðkþ 1Þ ¼AðpÞxcðkÞ þ BðpÞuðkÞ � *LðpÞðyðkÞ � CðpÞxcðkÞÞ

uðkÞ ¼ *KðpÞxcðkÞ ð37Þ

where

*Kð�Þ : p 2 J1; j1 � J2; j2 � � � � � Jr; jr ! KðjÞ ¼WðjÞX�1ðjÞ

*Lð�Þ : p 2 J1; j1 � J2; j2 � � � � � Jr; jr ! LðjÞ ¼ P�1ðjÞZðjÞ



ProofIf conditions (35) and (36) are satisfied, it follows that there exist matrices KðjÞ 2 Rn�m;LðjÞ 2 Rp�n such that the autonomous systems

xðkþ 1Þ ¼ ðAðpðkÞÞ þ BðpðkÞÞKðjðpðkÞÞÞÞx

xoðkþ 1Þ ¼ ðAðpðkÞÞ þ LðjðpðkÞÞÞCðpðkÞÞÞx0

are uniformly asymptotically stable for all the admissible realizations of the parameters.Now, by virtue of Theorem 4.3 in Reference [28], for the given realization of pðkÞ there exist

two bounded non-negative definite matrix sequences X1ðkÞ and X2ðkÞ satisfying the followinginequalities (the dependence of p on k is omitted for brevity):

ðAðpÞ þ BðpÞKðjðpÞÞÞX1ðkÞðAðpÞ þ BðpÞKðjðpÞÞÞT � X1ðkþ 1Þ50

ðAðpÞ þ LðjðpÞÞCðpÞÞX2ðkÞðAðpÞ þ LðjðpÞÞCðpÞÞT � X2ðkþ 1Þ50

Since the set J1; j1 � J2; j2 � � � � � Jr; jr to which the parameters are assumed to belong is compactand the matrix sequences X1ðkÞ and X2ðkÞ are bounded, there exists a sufficiently small positivescalar e such that the last pair of inequalities can be rewritten

AðpÞ þ BðpÞKðjðpÞÞ �eBðpÞKðjðpÞÞ

0 AðpÞ þ LðjðpÞÞCðpÞ

" #X1ðkÞ 0

0 X2ðkÞ

" #

�AðpÞ þ BðpÞKðjðpÞÞ �eBðpÞKðjðpÞÞ

0 AðpÞ þ LðjðpÞÞCðpÞ

" #T�

X1ðkþ 1Þ 0

0 X2ðkþ 1Þ

" #50

Now we apply Lemma 3 with

T ¼I 0

1=eI �1=eI

" #F ¼

AðpÞ þ BðpÞKðjðpÞÞ �eBðpÞKðjðpÞÞ

0 ðAðpÞ þ LðjðpÞÞCðpÞÞ

" #

in this way we obtain that there exist two bounded non-negative definite matrix sequences %X1ðkÞ;%X2ðkÞ 2 Rn�n such that for ji ¼ 1; . . . ; ni; and li ¼ �mi; . . . ;�1; 0;þ1; . . .þmi; i ¼ 1; 2; . . . ; r

AðpÞ BðpÞKðjðpÞÞ

�LðjðpÞÞCðpÞ AðpÞ þ LðjðpÞÞCðpÞ þ BðpÞKðjðpÞÞ

" #%X1ðkÞ 0

0 %X2ðkÞ

" #

�AðpÞ BðpÞKðjðpÞÞ

�LðjðpÞÞCðpÞ AðpÞ þ LðjðpÞÞCðpÞ þ BðpÞKðjðpÞÞ

" #T�

%X1ðkþ 1Þ 0

0 %X2ðkþ 1Þ

" #50

for p 2 J1; j1 � J2; j2 � � � � � Jr; jr : This last inequality implies uniform asymptotic stability of thefollowing system:

xðkþ 1Þ

xcðkþ 1Þ

" #¼

AðpÞ BðpÞ*KðpÞ

�*LðpÞCðpÞ AðpÞ þ *LðpÞCðpÞ þ BðpÞ*KðpÞ

" #xðkÞ

xcðkÞ

" #ð38Þ

which is exactly the closed loop connection between controller (37) and system (5). From thisfact the proof follows. &



Next, we consider the output feedback robust stabilization with L2 performance. The closedloop connection between system (9) and controller (7) exhibits the L2 performance bound g if,according to Theorem 2, there exist n1n2 � � � vr positive definite matrices PðjÞ such that

�Pðj_�lÞ Pðj

_�lÞAclðpÞ Pðj

_�lÞB1cl ðpÞ 0

ATclðpÞPðj

_�lÞ �PðjÞ 0 CT

1clðpÞ

BT1clðpÞPðj

_�lÞ 0 �I DT

11clðpÞ

0 C1cl ðpÞ D11cl ðpÞ �I

0BBBBBB@

1CCCCCCA50 8p : pi 2 fpi; ji ; pi; ji_�1g ð39Þ

where

AclðpÞ ¼AðpÞ þ B2ðpÞ *DcðpÞC2ðpÞ B2

*CcðpÞ

*BcðpÞC2ðpÞ *AcðpÞ

" #

B1clðpÞ ¼B1ðpÞ þ B2ðpÞ *DcðpÞD21ðpÞ

*BcðpÞD21ðpÞ

" #

C1clðpÞ ¼ ½C1ðpÞ þD12ðpÞ *DcðpÞC2ðpÞ D12ðpÞ*CcðpÞ �

D11clðpÞ ¼D11ðpÞ þD12ðpÞ *DcðpÞD21ðpÞ

and

½*Acð�Þ; *Bcð�Þ; *Ccð�Þ; *Dcð�Þ� : p 2 J1; j1 � J2; j2 � � � � � Jr; jr

! ½Acðp; jÞ;Bcðp; jÞ;Ccðp; jÞ;Dcðp; jÞ�

note that in each sub-hyper-rectangle J1; j1 � J2; j2 � � � � � Jr; jr the controller matrices areallowed to depend on p:

Now we follow the approach of Reference [29] to reduce the last inequality to an LMI.Partition PðjÞ and its inverse as follows:

PðjÞ ¼SðjÞ MðjÞ

MTðjÞ ?

" #P�1ðjÞ ¼

QðjÞ NðjÞ

NTðjÞ ?

" #

where ? means ‘don’t care’, and define

P1ðjÞ ¼QðjÞ I

NTðjÞ 0

" #

note that

PðjÞP1ðjÞ ¼ P2ðjÞ ¼I SðjÞ

0 MTðjÞ

" #

Pre- and post-multiplying (39) by blockdiagðPT1 ðj_�lÞPT

1 ðjÞ I IÞ and its transpose, respectively,we obtain for p 2 J1; j1 � J2; j2 � � � � � Jr; jr that (39) is equivalent to the satisfaction of the



following inequalities

�Qðj_�lÞ �I AðpÞQðjÞ þ B2ðpÞ #Ccðp; jÞ AðpÞ þ B2ðpÞDcðp; jÞC2ðpÞ

* �Sðj_�lÞ #Aðp; j; j

_�lÞ Sðj

_�lÞAðpÞ þ #Bcðj

_�lÞC2ðpÞ

* * �QðjÞ �I

* * * �SðjÞ

* * * *

* * * *

26666666666664

B1ðpÞ þ B2ðpÞDcðp; jÞD21ðpÞ 0

Sðj_�lÞB1ðpÞ þ #Bcðj

_�lÞD21ðpÞ 0

0 QðjÞCT1 ðpÞ þ #CT

c ðp; jÞDT12ðpÞ

0 CT1 ðpÞ þ CT

2 ðpÞDTc ðp; jÞD

T12ðpÞ

�I DT11ðpÞ þDT

21ðpÞDTc ðp; jÞD

T12ðpÞ

* �I

3777777777777550

QðjÞ I

I SðjÞ

" #> 0 ð40Þ

where the last inequality guarantees the reconstruction of PðjÞ starting from QðjÞ and SðjÞ (seeReference [29]), the symbol * at the place ij means the transpose of the element at the place ji;and

#Ccðp; jÞ ¼Dcðp; jÞC2ðpÞQðjÞ þ Ccðp; jÞNTðjÞ

#Bcðp; jÞ ¼SðjÞB2ðpÞDcðp; jÞ þMðjÞBcðp; jÞ

#Acðp; j; j_�lÞ ¼Sðj

_�lÞðAðpÞ þ B2ðpÞDcðp; jÞC2ðpÞÞQðjÞ þMðj

_�lÞBcðp; jÞC2ðpÞQðjÞ

þ Sðj_�lÞB2ðpÞCcðp; jÞNTðjÞ þMðj

_�lÞAcðp; jÞNTðjÞ ð41Þ

Theorem 8Consider system (9). If there exist positive definite matrices QðjÞ and SðjÞ and continuous matrix-valued functions #Acð . ; j; j

_�lÞ; #Bcð . ; jÞ; #Ccð . ; jÞ such that inequalities (40) are satisfied for ji ¼

1; 2; . . . ; ni; pi 2 ½pi; j1 ; pi; ji_�1� and li ¼ �mi; . . . ;�1; 0; 1; . . . ;mi; i ¼ 1; 2; . . . ; r; then the OFproblem is solvable with an L2 norm bound on the w-z channel less than one.

Theorem 8 is an existence result. To exploit the theorem in order to design a controller, we have

(a) To invert (41); to make this possible we have to allow Ac to depend also on j_�l:



(b) To look for parameter independent matrices #Ac; #Bc; #Cc; #Dc (or, at least, we should fix agiven structure for these matrices, but, in that case, the problem becomes computationallyintractable).

Moreover, in order to obtain an extreme points result:

(c) Either the pair ðB2ðpÞ;D12ðpÞÞ or the pair ðC2ðpÞ;D21ðpÞÞ has to be parameter independent.

Note that, also if case (b) holds, when (41) are inverted to obtain Ac; Bc; Cc; these matrices willdepend on both p and j:

Now, let us consider the case in which in (40) #Ac; #Bc; #Cc; Dc are parameter independent. If weextract from (40) the block composed by the first four rows and column we obtain a sufficientcondition for robust stability (without performance):

�Qðj_�lÞ �I AðpÞQðjÞ þ B2ðpÞ #CcðjÞ AðpÞ þ B2ðpÞDcðjÞC2ðpÞ

* �Sðj_�lÞ #Aðj; j

_�lÞ Sðj


_�lÞC2ðpÞ

* * �QðjÞ �I

* * * �SðjÞ

2666664

377777550

or equivalently,

�Qðj_�lÞ AðpÞQðjÞ þ B2ðpÞ #CcðjÞ �I AðpÞ þ B2ðpÞDcðjÞC2ðpÞ

* �QðjÞ #ATc ðj; j

_�lÞ �I

* * �Sðj_�lÞ Sðj


_�lÞC2ðpÞ

* * * �SðjÞ

26666664

3777777550

It is simple to recognize that the feasibility of the last condition is equivalent to the feasibility ofthe condition in Theorem 7. Indeed the existence of matrices QðjÞ; SðjÞ; #BcðjÞ; #CcðjÞ whichguarantee negative definiteness of the 1,1 and 2,2 blocks in the last LMI implies the existence ofmatrices QðjÞ; SðjÞ #Acðj; j

_�lÞ; #BcðjÞ; #CcðjÞ; DcðjÞ which satisfy the whole LMI. Therefore we can

conclude that, for what concerns the simple robust stabilization problem, there is no loss ofgenerality in considering the controller–observer structure for the output feedback controller.

4.3. Numerical examples

Example 3Consider the discrete-time uncertain system depending on two parameters:

xðkþ 1Þ ¼0 1

1 1

!þ

0 0

0 1

!p1 þ

0 0

0:05 0

!p2

" #xðkÞ

þ0

1

!þ

0

0:1

!p1 þ

0

0:05

!p2

" #uðkÞ ð42aÞ

yðkÞ ¼ ½ð1 0Þ þ ð0:1 0Þp1 þ ð0:05 0Þp2�xðkÞ ð42bÞ

with jp1ðkÞj41:5; jp2ðkÞj41:5; jDp1ðkÞj40:75 and jDp2ðkÞj41:5:



We partition the hyper-rectangle ½�1:5; 1:5�2 to which the parameters are assumed to belonginto 8 sub-hyper-rectangles of equal area ðn1 ¼ 4; n2 ¼ 2Þ and we let m1 ¼ m2 ¼ 1: With thesevalues of ni and mi; i ¼ 1; 2; the LMI problem (35), (36) admits a feasible solution and thefollowing are the gain scheduled controller and observer gains obtained:

Kð1; 1Þ ¼ ½�1:0603 � 0:3916�T; Lð1; 1Þ ¼ ½�1:0726 � 0:3787�

Kð2; 1Þ ¼ ½�1:0094 � 0:8262�T; Lð2; 1Þ ¼ ½�1:0366 � 0:8338�

Kð3; 1Þ ¼ ½�0:9573 � 1:0750�T; Lð3; 1Þ ¼ ½�0:9818 � 1:0904�

Kð4; 1Þ ¼ ½�0:9082 � 1:4363�T; Lð4; 1Þ ¼ ½�0:9263 � 1:4584�

Kð1; 2Þ ¼ ½�1:0590 � 0:3749�T; Lð1; 2Þ ¼ ½�1:0326 � 0:3869�

Kð2; 2Þ ¼ ½�1:0086 � 0:7953�T; Lð2; 2Þ ¼ ½�0:9766 � 0:8027�

Kð3; 2Þ ¼ ½�0:9600 � 1:0408�T; Lð3; 2Þ ¼ ½�0:9299 � 1:0509�

Kð4; 2Þ ¼ ½�0:9138 � 1:3606�T; Lð4; 2Þ ¼ ½�0:8981 � 1:3621�

We finally note that, by using the classical QS approach with the procedure described inReference [18], which does not take into account the rate of variation of the parameters, thestabilization of system (42) would have resulted impossible.

Example 4 (Analysis of the computational burden)To make a rough evaluation of the computational burden required to apply the proposedtechnique we consider the following realistic fourth-order discrete-time system depending onfour uncertain parameters

xðkþ 1Þ ¼

0 1 0 0

0 0 1 0

0 0 0 1

�1þ p4 p3 2þ p2 p1

0BBBBB@

1CCCCCAxðkÞ

þ

0

0

0

1þ 0:1p1 þ 0:05p2 þ 0:05p3 þ 0:05p4

0BBBBB@

1CCCCCAuðkÞ ð43Þ

We have that:

1. If we try to solve the classical state feedback quadratic stabilization problem by using oneconstant matrix P; we can stabilize the system for p 2 ½�0:25 0:25�4;

2. By our technique we find that, in the absence of bounds on the rate of variation of theparameters, we can stabilize the system for p 2 ½�0:35 0:35�4 by choosing ðn1 ¼ n2 ¼ n3 ¼



n4 ¼ 2Þ; in this way an improvement of about 30% w. r. t. the classical approach of point 1 isobtained;

3. If we assume that a bound on the parameter rate of variation exists ðn1 ¼ n2 ¼ n3 ¼ n4 ¼ 3Þwe can stabilize the system for p 2 ½�0:45 0:45�4:

We report some data concerning the computational power required to solve the above threeproblems [30]:

1. Number of inequalities ¼ 17; Flops ¼ 3875; elapsed time on a Pentium III 600 MHzprocessor ð128 Mb RAMÞ ¼ 0:54 s;

2. Number of inequalities ¼ 1296; Flops� 270 000; elapsed time on a Pentium III 600 MHzprocessor ð128 Mb RAMÞ ¼ 449 s;

3. Number of inequalities ¼ 8290; Flops� 1 700 000; elapsed time on a Pentium III 600MHz processor ð128 Mb RAMÞ � 3790 s:

Generally speaking, the computational burden depends on three main factors: the order of thesystem, the number of uncertain parameters and the number of parts in which each parameterinterval is split. To fix ideas we assume that each interval is divided into at most three sub-intervals, like in problem 3 of the above example; in this case some further investigation seems toshow that the proposed approach is suitable (computation time limited to a few hours) formedium size systems from fourth (four or five parameters) to tenth (three or four parameters)order. Obviously the use of more powerful computers than the one we have used above woulddrastically reduce the computation time required and enlarge the class of systems to which theapproach can be applied.

5. CONCLUSIONS

In this paper linear discrete-time systems depending on time-varying, bounded rate parametershave been considered. In the analysis context a sufficient condition for uniform asymptoticstability has been found with the aid of piecewise constant quadratic parameter dependentLyapunov functions. It has been shown that the proposed approach is less conservative than theclassical QS approach both when the parameters are actually bounded rate and when no boundson the rate of variation of parameters is available. Then the synthesis problem has beenconsidered; it has been shown how the state and output feedback problems can be reduced tofeasibility problems involving LMIs. Both for the analysis and the synthesis problems L2

performance requirements have been taken into account. Some examples, which illustrate theproposed technique and show its effectiveness have been provided. In particular, an analysis ofthe computational burden required to apply the proposed methodology has been carried out.This analysis has shown that the technique is suitable for medium size systems depending on atmost five parameters. Obviously, with a reasonable optimism on the growth of the computercomputational power and on the numerical algorithm developments, it is reasonable to expectthat the class of tractable problems will increase.

ACKNOWLEDGEMENTS

The authors wish to thank reviewer 1 who has brought to their attention the theory of PWA systems andsuggested investigating the conditions for performance requirements. Moreover, he or she has providedmany suggestions to improve the quality of this paper.



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Gain scheduled control for discrete-time systems depending on bounded rate parameters