2922 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013

Robust Sampled – Data Control of Switched Affine Systems

Laurentiu Hetel and Emilia Fridman, Senior Member, IEEE

Abstract—This technical note considers the stabilization problem forswitched affine systems with a sampled-data switching law. The switchinglaw is assumed to be a function of the system state at sampling instantsand the sampling interval may be subject to variations or uncertainty. Weprovide a robust switching law design that takes into account the sam-pled-data implementation and uncertainties. The problem is addressedfrom the continuous-time point of view. The method is illustrated bynumerical examples.

Index Terms—Linear matrix inequalities (LMIs), sampled-data control,switched affine systems, switching control.

I. INTRODUCTION

In the last decade, the design of switching controllers has representedan important problem in the hybrid system community [3], [4], [15],[16], [21], [23], [25]. This problem is very challenging for the case ofswitched affine systems where, generally, the different subsystems donot share a common equilibrium point. The study is motivated by thewide range of applications to power electronics (see e.g., [5]). Differentstabilization solutions exist in the literature based on the existence ofstable convex combinations [2], [5], on optimal control methods [12],[20], or on the use of sliding modes [6], [24]. A characterization of theset of attainable equilibrium points using quadratic Lyapunov functionsand conic switching laws has been provided in [2], [5].The next phase toward practical application of switching control is

to study its sampled-data implementation. For results on sampled-datacontrol we point to the discrete-time methods in [1], [14], [19] the inputdelay-approach [8], [17], [22] and the impulsive system method [18].Recently, increasing attention has been given to the sampled-data con-trol of switched systems [9], [12]. This aspect is crucial in the switchedaffine system context since, due to sampling, one can no longer drivethe state exponentially towards the equilibrium point, but only towardsa limit cycle or to some attractive compact set containing the equilib-rium. Moreover, in many practical applications timing imperfectionsdue to sampling jitters or delays in switching law may affect the con-trol performances. Recently, a discrete-time analysis has been providedin [12] based on control Lyapunov functions and the use of a nonlinearoptimization solvers for quadratically constrained quadratic programs.However, it is still a problem to choose the control Lyapunov functionso as to optimize the robustness with respect to sampling or the sizeof the attractive set around the desired equilibrium. Moreover, for anaccurate study, it is of interest to exactly describe the continuous-time

Manuscript received March 19, 2012; revised October 14, 2012; acceptedApril 11, 2013. Date of publication April 18, 2013; date of current version Oc-tober 21, 2013. This paper was presented in part at the IFAC ROCOND 2012Symposium, Aalborg, Denmark, June20–222012. This work was supported inpart by the European Community’s 7th Framework Programme (under grant257462) HYCON2Network of Excellence, the Interreg IV A 2Mers Seas Zeeenprogram SYSIASS and Israel Science Foundation (under grant 754/10). Recom-mended by Associate Editor F. Blanchini.L. Hetel is with the University Lille Nord de France, LAGIS, UMR CNRS

8219, Ecole Centrale de Lille, Cite Scientifique, BP 48, 59651 Villeneuved’Ascq cedex, France (e-mail: laurentiu.hetel@ec-lille.fr).E. Fridman is with the School of Electrical Engineering, Tel-Aviv University,

Tel-Aviv 69978, Israel (e-mail: emilia@eng.tau.ac.il).Color versions of one or more of the figures in this technical note are available

online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAC.2013.2258786

behavior. For systems with sampled-data switching control, this studyis very challenging. The switching control is often described by a dis-crete-event system with transitions ruled by a partition of the statespace. Then the sampling usually induces a delay in the discrete-eventsystem variable. This may imply a mismatch in the control: one systemmode may be active in other state zones then the one for which it hasbeen designed. If not appropriately taken into account, the samplingmay be a source of poor performance and even may lead to unboundedsolutions.The goal of the technical note is to present a continuous-time ap-

proach to sampled-data switching control design that ensures robust-ness with respect to sampling and to potential implementations imper-fections (jitters, uncertainty etc.). Simple criteria are given to optimizethe choice of Lyapunov functions.The technical note is organized as follows: in Section II we formalize

the problem under study and we provide simple conditions for practicalstabilization based on the existence of stable convex combinations. Thecase of systems with parametric uncertainties is also presented. In Sec-tion III it is shown how the presented methodology may be improvedby using switching Lyapunov functions. Numerical examples are givenin Section IV. Preliminary results on the quadratic stabilization of un-certainty-free systems were presented in [13].

Notations

We denote the transpose of a matrix by . The symbol de-notes a block that can be inferred by symmetry. By (or ) we de-note the identity (or the null) matrix with the appropriate dimension.

denotes the Euclidean vector norm. For a square symmetric ma-trix, ( ) indicates that is positive (negative) defi-nite. By we denote the minimum eigenvalue of a squaresymmetric matrix . For a given set , the symbol denotes theconvex hull of the set. Given a finite set of index and a set of scalarsindexed by the elements in , we denote

.

II. PRACTICAL STABILIZATION BASED ON

STABLE CONVEX COMBINATIONS

A. Problem Formulation

Given positive integers and , consider matricesand vectors . We

are interested in the class of switched systems described by

(1)

where represents the system state anda switching control. Consider the simplex

the

convex combinations of matrices ,, and a subset of associated to Hurwitz matrices:

It hasbeen shown in [2], [5] that associated to each there existsan equilibrium point to which (1) may beexponentially stabilized using a continuous-time switching function.Without loss of generality, we may consider system’s (1) stabiliza-

tion with respect to the equilibrium point , i.e., we may considerthat there exists s.t. . The stabilization with respectto an equilibrium point , may always bereformulated as a null equilibrium point problem by considering theerror state vector and the model representing theerror dynamics , with and

0018-9286 © 2013 IEEE

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013 2923

, for which . Under thehypothesis that there exists s.t. , system (1) is sta-bilized to the origin using a switching law of the form

(2)

where is a symmetric positive definite matrix satisfying.

In a sampled-data implementation, the values of system state areavailable at sample times , with

. The sampling interval may beunknown and time-varying, with where is aknown bound. In this technical note we consider that the implementedswitching law is a function of , although an implementation based on

may also be of interest in practical applications. With a sampled-data implementation, the switched system becomes

(3)

where is constant for all and represents the dis-crete-time implementation of the switching law (2), i.e.

(4)

Note that with the sampled-data control sliding modes may not occursince and only one system mode is actif for .However, due to the sampling, for and wegenerally have , i.e., the equilib-rium point is no longer invariant. Moreover, the system state cannotbe driven to the equilibrium point of the continuous-time switchedsystem, but only to a neighborhood of the equilibrium, whose size maygrow with the sampling interval. Our goal is to provide methods forthe design of sampled-data switching laws that are practicallystabilizing the system (3) to a ball, i.e., to find switching laws of theform (4) that guarantee that is exponentially converging to the ball

when , where is positive andsatisfies as . The latter recovers the ex-ponential stability of the system under the continuous-time switching,where .

B. Design Conditions

In this subsection, simple design conditions for switching law (4)are provided by requiring the decrease of the function

with respect to its value at sampling instants. The idea isto use a continuous function , differentiable over

, with and , ,satisfying the following condition:

(5)

for some , . By the comparison principle,(5) yields

i.e.

(6)

which means that exponentially converges to the attractive ellip-soid given by

(7)

By -procedure, (5) leads to the following implication:

(8)

Hence, (5) guarantees thatwhenever . Then the function

may be related to a Lyapunov function for the discrete-time system representation. For the particular case where

(9)

with , , the functionbecomes a Lyapunov–Krasovskii functional as in [8]. Sta-

bilization conditions are presented in the following theorem, as shownin (10) at the bottom of the page.Theorem 1: Consider system (3), (4) with and a given

scalar tuning parameter . Assume that , and that thereexists a s.t. .

(10)

2924 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013

(i) Let there exist matrices in , a scalarsuch that

(11)

(12)

, where

Then (3), (4) is practically stabilizable to the set , that isis exponentially attracted to the ellipsoid for .

(ii) Under conditions of (i), if for some , thenfor all .

Proof:(i) We will show that (11), (12) imply (5) with andgiven by (9).Assume that and denote ,with , for all . Using the Jenseninequality it is seen that

Furthermore,. Therefore, (5) holds with and

if

(13)

for all , where . Notethat for all if satisfies the relation

.Multiplying the latter inequality by , and summing foryields i.e.

(14)

where the relation has been used. Sincewhenever (14) holds, using the – procedure by adding the leftside of (14) to the left side of (13), we arrive to

where

Since

, then

(15)

(16)

are sufficient to guarantee that the function sat-isfies the condition (5) along the solutions of system (3) withthe switching law (4). Note that implies

. Therefore (11), (12) yield (15), (16).(ii) If (11), (12) are feasible, then (5) and, thus, (8) are satis-

fied. The inequality impliessince . Let . Assume

that for some we haveand that

for all . Then for, which after integration yields

. The latter implies that, since . Hence, the system solution

cannot exit between the sampling instants.Remark 1: The parameter from Theorem 1 corresponds to the

system decay rate. For fixed , conditions (11), (12) represent LMIs.The optimization of the decay rate may be addressed by combiningLMI-based methods for (11), (12) with a line search on .Remark 2: For the case of , the conditions from

the previous theorem are reduced to the existence of insuch that

(17)

, are feasible. In this case the zero solution of (3), (4) is expo-nentially stable.Remark 3: Note that solutions of (11), (12) depend on. Given , the feasibility of (11), (11) with some

guarantees that for the trajecto-ries of the resulting system approach to the ball ,where

LMIs (11), (12) for are reduced to

(18)

which guarantees the exponential stability of the continuous-timesystem. For small enough , the feasibility of (18) implies thefeasibility of (11), (12):Corollary 1: Given and , let be the solution

of (18). Then for any there exists a sufficiently smallsuch that (11), (12) are feasible withand for all . Thus, for allthe solutions of (3), (4) are exponentially attracted to the

ball as , where (the – independent)constant is given by .

Proof: Given , there exists s.t. and. When (18) holds, there exists a suffi-

ciently small such that

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013 2925

and

and . By Schur complements and classicalmanipulation, and lead to (11) and (12), respectively(with , ).Remark 4: Theorem 1 guarantees that is decreasing out-

side with respect to its values at sampling times only. Forthe case when is outside there may exist time instants

such that while. However, once is in the attractive ellipsoid, the

state will stay there as shown by (ii) of Theorem 1.

C. LMI Conditions for Systems With Parametric Uncertainties

The conditions (11), (12) presented in the previous subsection are notaffine in the systems parameters. This makes difficult their use for time-varying parametric uncertainties. The conditions may be modified tocope with this important case using the descriptor method [7]. Considerthe system

(19)

where for all the system matrices have the form, withand . represent the nominal

parameters while represent the vertices of perturbations withrespect to the nominal values. Note that for the uncertain case, even inthe continuous-time case, we only have practical stabilization to a ballaround the origin and not exponential stabilization. Robust practicalstabilization conditions for the sampled-data case are given as shownin (20) at the bottom of the page.Corollary 2: Consider (19), (4) with . Assume that

and that there exists a such that .Given the scalar tuning parameter , let there exist matrices

, and scalars such thatconditions (10) are feasible. Then for solutions of (19),(4) are exponentially attracted to the ellipsoid

(21)

Proof: Following the arguments in the proof of Theorem 1 withand given by (9), a sufficient condition for (5) is

(22)

whenever . Using the descriptor method

(23)

Summing (14), (22), (23) and using convexity arguments, the condi-tions (10) are obtained.Remark 5: The set of matrix inequalities (10) for are

reduced to the conditions

(24)

, which ensures that under the continuous-time switching law, the (uncertain) system state is exponentiallyattracted to the ellipsoid as(and not to the equilibrium point).

III. PRACTICAL STABILIZATION BASED ON

SWITCHED LYAPUNOV FUNCTIONS

In this section we show how the presented methodology may be gen-eralized for systems that do not admit a stable convex combinationby using switched Lyapunov functions. We extend the methodologyin [15], where the switched linear systems were studied, to the case of

(20)

2926 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013

switched affine systems under the sampled-data measurements. Con-sider the sets

Similarly to the case of quadratic stabilization, it may be shown thatthe stabilization to an equilibrium point may always be re-formulated as a stabilization problem with a null equilibrium point byconsidering the dynamics of the error vector ,

withand , for which . For the casewhen the barycentric coordinates are not unique, i.e., there existvectors such that ,we may use switched Lyapunov functions. To each we asso-ciate a quadratic form , where . For thequadratic forms with the minimum value are indicated by the index set

(25)

Consider a pair

(26)

We define as a switching function. The function asso-ciated to is used for indicating the active index in the switchedLyapunov function

(27)

The following theorem provides practical stabilization conditions:Theorem 2: Consider system (3), (26) with and scalar

tuning parameters , . Assume thatand that there exists vectors , such that , .Let there exist matrices in and ascalar such that the set of matrix inequalities (20) (given at thebottom of the page) hold. Then for solutions of (3), (26)are exponentially attracted to the ball

(28)

Proof: The definition of switching and Lyapunov functions in(26) and (27) respectively guarantees, for the fixed , a uniquedefinition for the Lyapunov function as the function with theindex associated to . is piecewise differentiablefor any and does not grow in the jumps, i.e.

(29)

Moreover, it is positive definite, radially unbounded and zero at .As in the quadratic case, we use a continuous function given by (9)and show that (20) imply (5), where is changed by .From (5), (29) it follows that for

i.e., that is exponentially attracted to as .The inequality (5) is satisfied for

if

for all whenever

(30)

(31)

Following arguments of Theorem 1, a sufficient condition for (5) tohold with and is:

(32)

for all , with , .Multiplying (31) by and (30) by , summing and expressing

leads to

(33)

To end the proof, add (33) to (32) and use convexity arguments.Remark 6: Theorem 2 allows to reduce the conservatism of the

stabilization conditions by using switching Lyapunov functions (27).It may be used to stabilize a switched affine system to equilibriumpoints in the set for which is not Hurwitz. However,the resulting stabilization conditions represent essentially more com-plicated matrix inequalities with many additional tuning parameters

comparatively to the conditions of Theorem 1 andCorollary 2. They represent a non-convex optimization problem dueto the products of variables and . It is similar to the problemof choosing the Metzler matrix in the design conditions related toswitching strategies [10] for switched linear systems with continuous-time switching laws. Such Bilinear Matrix Inequalities (BMIs) maybe addressed via LMI-based numerical approaches by combining thepath-following method in [11] and the direct iteration with as vari-ables [15]. However, for the moment, there is no guarantee to find theglobal optimal solution.Remark 7: There are two main sources of conservatism for the

results in the technical note. The first one stems from the choice of(quadratic or switching) Lyapunov functions. Another is related tothe form of . Using additional terms in may reduce theconservatism of the design conditions, but on the account of thecomputational complexity.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013 2927

Fig. 1. Example 1: evolution of system states under the control law based onTheorem 1 with a fixed sampling interval .

IV. NUMERICAL EXAMPLES

Example 1: Quadratic Stabilization: Consider a switched affinesystem (1) consisting of four affine subsystems with andthe following matrices [2]:

Each individual subsystem is unstable. For , ,and , the is Hurwitz and . Using

Theorem 1 we find that the system is practically stabilizable under vari-able sampling with . Conditions (11), (12) arefound to be feasible with

(34)

, and . An illustration of system evo-lution with an arbitrary initial condition is shown in Fig. 1. Numericalsimulations under uniform sampling show that the system is praticallystable for bigger sampling intervals with , whichillustrates the conservatism of the proposed method.Example 2: Uncertain System: We illustrate the applicability of our

results on an example from power electronics. Consider the DC-DCconverter from [12], where the model has the form

with

(35)

, with , ,and . For , the matrix

is Hurwitz and the system may be stabilized to the equilibriumpoint using a continuous-timeswitching law. Consider the error dynamics

For the numerical tests, the time scale change with isused to cope with large numerical values in the system matrices and toavoid ill conditioned matrix inequalities. The system of the form (3) isobtained with , , ,

. Note that the trajectories are invariant with respect to timescaling. Furthermore, the switching laws are equivalent, since

Concerning the robust switching law design, conditions (11), (12) ofTheorem 1 are feasible for any (time-varying) sampling intervals with

.In order to compare with [12] the estimates of the attraction domains,

consider the particular case studied in [12], with .The conditions in Theorem 1 allow to design a switching law that guar-antees that for , , which is not very far from thevalue for obtained in [12] by usingthe exact integration over a sampling interval. The obtained numericalvalues (solutions of (11), (12)) are ,

To illustrate the use of our method for uncertain systems, chooseand assume that the resistor is subject to unknown

time-varying uncertainties . Then each of thematrices is varying in a polytope corresponding to the two vertices

. Considering the equivalent error dynamics and using thesame time scale change, a model of the form (19) is obtained with twovertices for each subsystem. The conditions in Corollary 2 are feasiblewith

, , which implies that as. The error system evolution with the initial condition

is shown in Fig. 2. The figure presents the attractive ellipsoids for boththe sampled-data case ( , obtained based on Corollary 2) and forthe continuous-time switching implementation ( , representing thelimiting set when ). Due to sampling and to parametricuncertainties the system state (in black) does not converge to the equi-librium point (the center of the ellipsoid) but only to a bounded regionwhich is not far from the border of . Numerical simulations underan uniform sampling show that the same attractiveellipsoid is achieved for bigger , to be comparedwith proved in theory under the variable sampling.The latter may illustrate the conservatism of the method.Example 3: Switched Lyapunov Functions: Consider the example

adapted from [15] with

(36)

, All of thematrices , ,2, 3, are neutrally stable and there exists no Hurwitz convex combina-

2928 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 11, NOVEMBER 2013

Fig. 2. Example 2: trajectory in the error state space with a control law based onCorollary 2 under variations in the resistor value from 35 to 65 with a fixedsampling interval (solid black line), the attractive setsobtained for the continuous-time case (dashed line) and for(solid line).

tion . Therefore the quadratic stabilizationmethodology cannot beapplied. We achieve practical stabilization by using a switching Lya-punov function (Theorem 2) of the form (27), switching amongquadratic forms. BMIs (20) are feasible for with

, and

, . We useda dichotomy search for , at each step alternating and asdecision variables. Then, for fixed or conditions (20) are LMIs.The matrices were initialized with the values from [15].

V. CONCLUSION

This technical note presented a sampled-data switching designmethod for practical stabilization of switched affine systems. Theresults are robust with respect to sampling and to potential im-plementation imperfections such as sampling jitters or parametricuncertainties. The method uses LMI-based methods for the optimiza-tion of Lyapunov functions and it is illustrated by numerical examples.Improvement of the presented method by e.g., using sum-of-squaresor by modifying the Lyapunov–Krasovskii functional may be topicsfor the future research.

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