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Guaranteed Cost Certification for Discrete-TimeLinear Switched Systems With a Dwell Time
Marc Jungers and Jamal Daafouz, Member, IEEE
Abstract—This technical note studies the guaranteed cost of a quadraticcriterion associated with a linear discrete-time switched system for all theset of admissible switching laws. The admissible switching laws are here theones exhibiting a dwell time. The approach provided here is to design anupper bound and a parametrized family of lower bounds of the guaranteedcost as close as possible in order to obtain a certification of the guaranteedcost. The upper bound is determined via a switched Lyapunov function andthe lower bounds are obtained via the numerical computation of the costinduced by particular periodic switching laws. The features of the proposedapproach are illustrated by a numerical example.
Index Terms—Performance certification, periodic Lyapunov equations,switched systems.
I. INTRODUCTION
A linear switched system is an association of a set of a finite numberof LTI dynamic systems and a switching law [1]. In practice, there aremany applications where switched systems modelling is appropriate,like embedded systems in automotive industry, aerospace and energymanagement. The last decades have witnessed an increasing interestfrom the scientific community in the study of the stability property forthis class of hybrid systems [1]–[4], by emphasizing tools among themLyapunov functions depending on the switching parameter [5] or mul-tiple Lyapunov functions [6], [7]. A situation which is often encoun-tered in real applications is the one where a mode must remain activeat least during a constant time interval called a dwell time.Beyond questions of stability, the performance aspects of switched
systems have been investigated and still remain an open issue [8], likethe notion of the guaranteed cost of a performance index. Determiningsuch an extremum over all the admissible switching laws is a diffi-cult task, even if theoretical answers have been provided via hybridversions of the Pontryagin Maximum Principle [9], [10], or dynamicprogramming [11], [12]. Another way to study the guaranteed cost of aperformance index is to consider a set of its upper bounds and to selectin this set the smallest value. Obtaining an upper bound of a guaranteedcost as small as possible does not provide information on its gap. Thisis the issue of the certification by exhibiting upper and lower boundsas close as possible.
Manuscript received May 25, 2011; revised November 09, 2011; acceptedJuly 24, 2012. Date of publication August 02, 2012; date of current versionFebruary 18, 2013. This work was supported in part by ANR project ArHyCo,Programme ”Systèmes Embarqués et Grandes Infrastructures” – ARPEGE,contract ANR-2008 SEGI 004 01-30011459 and by the European Commu-nity’s Seventh Framework Programme (FP7/2007-2013) under grant 257462:HYCON2 Network of Excellence ”Highly-Complex and Networked ControlSystems.” Recommended by Associate Editor P. Shi.M. Jungers is with the Université de Lorraine, CRAN, UMR 7039,
Vandœuvre-lès-Nancy Cedex 54516, France and also with CNRS, CRAN,UMR 7039, Vandœuvre-lès-Nancy Cedex 54516, France (e-mail: [email protected]).J. Daafouz is with the Université de Lorraine, CRAN, UMR 7039,
Vandœuvre-lès-Nancy Cedex 54516, France, with CNRS, CRAN, UMR7039, Vandœuvre-lès-Nancy Cedex 54516, France, and also with the InstitutUniversitaire de France, Vandœuvre-lès-Nancy Cedex 54516, France (e-mail:[email protected]).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.2012.2211441
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 3, MARCH 2013 769
This manuscript deals with the problem of the performance certi-fication for a discrete-time linear switched system with a switchinglaw under a dwell time assumption. The performance is chosen as the2-norm of the output. Firstly the upper bound of the guaranteed costwill be designed by the help of the initial value of a switched Lya-punov function. Secondly the lower bounds are determined from thefollowing observation: the exact value of the criterion associated witha particular admissible switching law is less than the guaranteed costand is one of its lower bounds. Such a certification was firstly illustratedin [13], [14] by considering for the lower bounds the simple case ofconstant switching laws. Here we propose a less restrictive case usingswitching laws exhibiting the following specifications: being periodicand satisfying a dwell time. The exact value of the criterion, in thisframework, is obtained via periodic Lyapunov equations.The periodic Lyapunov equations have been studied [15], for both
the theoretic (see for instance [16]–[19]) and numerical aspects [20],[21]. Here an algorithm based on the shift-invariant representation andadapted to include the dwell time as a parameter is proposed.The technical note is organized as follows. Section II gives the
problem formulation. Section III provides the main results to designupper and lower bounds of the guaranteed cost by taking into accountthe dwell time. Section IV presents the certification of the guaranteedcost and offers computational methods for its obtaining. Section Vemphasizes the efficiency of our certification on an academic example.
Notations
, , are respectively the sets of real, natural and complex num-bers and . If is a set, is the cartesian product
, times. For two symmetric matrices, and ,means that is positive definite. denotes the transpose of .The operator is a block diagonal matrix of and .
II. PROBLEM DEFINITION
Let us consider the autonomous linear switched system in discrete-time
(1)
(2)
where is the state of the system, the performanceoutput and , where isthe switching rule which indicates the active mode at each time :
(respectively ) belonging to the finite set of matrices(respectively ). We assume that the
switching law verifies a dwell time, defined as follows.Definition 1: For an integer , the set of the switching laws
satisfying a dwell time at least equal to is defined by
The subsequence , induced in consists in the switchingtimes related to the switching law . Due to the definition of wehave the following properties .In addition is the set of arbitrary switching laws and
is the set of the constant switching laws. The performance of thesystem (1) is chosen as the norm of the performance output
(3)
The aim of this note is to obtain an evaluation of the guaranteed cost
(4)
which is a quadratic form with respect to the initial state .Obtaining analytically the value of is not possible in general
via the PontryaginMaximum Principle or dynamic programming, evenin the case of switched linear systems, due to the time-dependency andthe infinite time horizon of the criterion . Moreover, one may applydynamic programming over a truncated horizon but at the expense of anumerical explosion when increasing the size of the truncated horizon.The exact numerical computation of an upper bound and a parametrizedfamily of lower bounds for particular switching laws is considered inthis note.
III. UPPER AND LOWER BOUNDS OF THE GUARANTEED COST
This section is devoted to designing upper bounds and a familyof lower bounds of . The associated main results are given inTheorem 1.
A. Preliminaries and Notations
Preliminaries about tools for periodic systems [15], [17] are recalledhere.Definition 2: For , the set of the -periodic switching laws
is defined by .Definition 3: For , let us define
(5)
A switching law belonging to has a dwell-time equal to andis -periodic. is then a subset of and and one gets
.The set contains exactly switching laws, which
are characterized by the -uplet defined by. We note the natural bijection
allowing the link between and the cartesian product by
Definition 4: The state-transition matrix related tothe system (1) is defined by , and
, .Definition 5: For and , the monodromy matrix at
time is defined as . The eigenvalues of are called thecharacteristic multipliers of the system (1).The characteristic multipliers of the system (1) are independent on
the time (see [15, Chapter 3] for more details and a proof). The system(1) is globally exponentially stable if its characteristic multipliers be-long strictly to the unit circle. The inclusion will be assumedhere stable in order to have all the possible monodromy matrices stableand ensure the finiteness of .Definition 6: For , a characteristic multiplier of is
said to be -observable at time if
770 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 3, MARCH 2013
implies or , , where
....
The pair is called observable if it is observable at anytime , .
B. Main Results
Theorem 1: For , let us consider the system(1)-(2) and the performance index defined by (3). Assume that for
, the pair is observable. If there exists sym-metric positive definite matrices , ( ), symmetric pos-itive definite matrices , ( ) and a scalar
satisfying in one hand the LMIs, ,
(6)
(7)
(8)
where , and in the other handthe equalities, for a -uplet ,
(9)
(10)
then , and for any , the origin isglobally exponentially stable and in addition the guaranteed costis finite and verifies
(11)
(12)
where is an upper bound when is known, and whenis unknown.Proof: The proof is decomposed into two parts dedicated to the
bounds (12) and (11).The conditions (7) and (8) imply and
, . These conditions arethe same as those of [22, Theorem 1] leading to the global asymptoticstability of the origin. The relation (12) is proven in the same way of[22, Theorem 2], by noticing that is constant betweentwo consecutive switching times: ,
By induction, we infer that ,and for any
(13)
By reordering the set via the switching times:, we have
By the definition of , one gets
because is positive semidefinite. Withthe help of inequality (13), the inequality
holds.Furthermore, it has been shown that the function
is a switched Lyapunov function for this system. Due toand to the structure of a telescopic series,
we have
(14)
By multiplying the inequality (6) from the left by and from theright by , we have . By taking the supremumover all of the inequality (14), we obtain the inequality (12).The second part of the proof is based on the design of a -periodic
Lyapunov function associated with the -periodic switching law, by the help of the symmetric positive definite matrices ,
.Set . Define by the (19), that is
(15)
For , we set
(16)
It is noteworthy that for , we have
(17)
In the same way, it induces that, which proves that such a sequence
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 3, MARCH 2013 771
is positive definite, symmetric and solves the peri-odic Lyapunov equation associated with ;
(18)
(19)
Due to inequalities (7) and (8), the system (1) is stable for any, that is the characteristic multipliers belong strictly to the unit
circle. The infinite series
(20)
are well defined, symmetric and solve the periodic Lyapunov (18)-(19),by arguments proposed in [16], [17], [23]. By invoking [17, Theorem1], the solution of the periodic Lyapunov equation is unique. This im-plies that the infinite series, defined by (20) are equal to . Because thepair is assumed observable, classical arguments allow toprove the positive-definitness of matrix .Let us prove now that the value of the criterion is given
by , because
(21)
The switching law belonging to which is a subset of ,we obtain the inequality (11), which ends the proof.The main results proposed in Theorem 1 deserve some comments.• The stability of and the observability of the pairs ,
are necessary conditions for the existence of symmetricpositive definite matrices satisfying inequalities (7).
• The upper bound of is less thanbut depends on the direction of instead of .
is the maximal eigenvalue of matrices .• For , we recover the method and the conditions proposedoriginally in [13], [14].
IV. CERTIFICATION AND NUMERICAL METHODS
Theorem 1 offers tools to obtain classes of upper bounds (12) andlower bounds (11) of the guaranteed cost . The smallest upperbound is given by the following optimization problemOptimization Problem 1: Under the assumptions of Theorem 1, the
smallest upper bound (12) is solution of
subject to the inequalities (6); (7) and (8).The Theorem 1 provides a family of lower bounds, based on the
solution of the -periodic Lyapunov (18)-(19) parametrized by a-uplet . The numerical computation of the lower bounds maybe conveniently achieved by using a shift-invariant representation[20], [21], with a block-diagonal structure of the extended Lyapunovmatrix. An optimization problem under linear matrix inequalities isthen proposed to solve the periodic Lyapunov equations [24], whenthe symmetric positive definite solution exists and is unique. Thisallows to obtain numerically the lower bound defined by (11)and satisfying also , due to the factthat depends on the -uplet . The numerical result is gatheredin the optimization problem 2. For ,
we denote ;and
. . .
. . .. . .
Optimization Problem 2: For , consider the op-timization variables as symmetric positive definite matrices ,
. If the system (1) is stable and if the pair isobservable, then the optimization problem
(22)
leads to the unique symmetric and positive solution of (9)-(10).Moreover, it is possible to quantify more precisely the smallest
gap between and . Let introduce the scalar. By definition of
and , we have.
The certification of the guaranteed cost is especially good foran particular initial condition as is close to . The decreasinginclusion of implies that is a decreasing function of . Because
, we have , where is the solution ofthe optimization problem 1, without inequality (8) as a constraint.In addition, when , the periodic Lyapunov solutions are re-
stricted to be constant and they depend only on the initial mode, wehave thus , for any dwell time . One gets
. Furthermore, if is amultiple of , thenwhich implies that . Nevertheless, is not an in-creasing function of .
V. ILLLUSTRATION
Let us consider the next example, with , ,
, , and . We
can check in Fig. 1 that is a decreasing function of the dwell time,converging, as expected, to .For a dwell time , we define , with
, because is quadratic. We consider here . Thevalue of the upper bound and the lower bounds( ) are depicted on Fig. 2 in function of . It can be seenthat the periodic behavior helps to take into account the evolution ofthe switched system and improves the lower bound. The values( ) are compared with in Table I. We couldverify that , that is is not an increasing function of ,but we have and .The gap between the bounds is characterized by ,which drastically improves the rate .
VI. CONCLUSION
The performance certification of a quadratic cost associated withan autonomous linear discrete-time switched system for the class ofswitching laws with a dwell time has been studied. The certificationproposed here consists in designing, for a dwell time, upper and lowerbounds as close as possible. The presented upper bound is obtained viathe initial value of a switched Lyapunov function taking into accountthe dwell time. A family of lower bounds is determined by computingthe cost value for a specific periodic switching law. The algorithm used
772 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 3, MARCH 2013
Fig. 1. Upper bound in function of the dwell time .
Fig. 2. For , bounds (depicted by ’+-’) and (, depicted by ’-’) in function of .
TABLE IVALUES IN FUNCTION OF THE PARAMETER .
for this computation is based on solving a periodic Lyapunov equationwith a period multiple of the dwell time. An academic example illus-trates the features of our approach.
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