792 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 3, MARCH 2013

On Reliable Stabilization via Rectangular DilatedLMIs and Dissipativity-Based Certifications

Getachew K. Befekadu, Member, IEEE, Vijay Gupta, Member, IEEE,and Panos J. Antsaklis, Fellow, IEEE

Abstract—In this short note, we consider the problem of reliable stabi-lization for multi-channel systems, where our main objective is to main-tain the robust stability of the perturbed closed-loop system when thereis a single controller failure in any of the control channels. Specifically,we present a less-conservative result than results in the literature in termsof a set of dilated LMIs for the reliable state-feedback stabilization of thenominal system, while a dissipativity-based certification is used to extendthe stability condition when there is an additive model perturbation in thesystem. Finally, a numerical example is presented to illustrate the applica-bility of the proposed technique.

Index Terms—Dilated LMI, dissipativity, multi-channel system, reliablecontrol, stabilization.

I. INTRODUCTION

Reliable control using multi-controllers is used for enhancing ro-bustness against possible component failures that may occur in actu-ators, sensors or controllers. In a multi-channel control configuration(e.g., see [1], [2]), the objective is to guarantee certain stability andperformance criteria for the closed-loop system when all of the con-trollers work together and when some controllers turn faulty or de-viate from nominal conditions. In the past, several major theoreticalresults have been achieved in the context of reliable stabilization viathe so-called factorization approach [3]–[6] and in the context of per-formance via Riccati and/or Lyapunov equations [7], [8]. For example,in the case of a single input-output channel, a complete characteriza-tion of plants that can be reliably stabilized using two controllers, whereeither of them may fail, was considered in [3]. Alternative character-izations have also been derived for a more general situation where aplant with two input-output channels is stabilized by two controllers in[4]–[6]. It should be noted that the problem of reliable stabilization fora multi-channel system is, in general, a very difficult problem. This isbecause reliable stabilization is essentially equivalent to a strong sta-bilization problem [3] which is known to be intractable [9], [10]. We,therefore, need a tractable and less conservative design technique forreliable stabilization of multi-channel systems.Recently, the problem of reliable stabilization for a multi-channel

continuous-time system with a single failure in any of the control chan-nels has been addressed in [11] via dilated LMIs and unknown distur-bance observers. In this note, we present an extension to the problem ofreliable stabilization for a perturbed multi-channel system, where ourmain objective is to maintain the robust/simultaneous stability for allperturbed instances of closed-loop systems even in the presence of asingle controller failure. We introduce a new dilated LMI framework

Manuscript received July 08, 2011; revised November 14, 2011, May 11,2012, and July 16, 2012; accepted July 23, 2012. Date of publication August02, 2012; date of current version February 18, 2013. This work was supportedin part by the National Science Foundation under Grant CNS-1035655 and theMoreau Fellowship of the University of Notre Dame. Recommended by Asso-ciate Editor H. Trentelman.The authors are with the Department of Electrical Engineering, University

of Notre Dame, Notre Dame, IN 46556 USA (e-mail: gbefekadu1@nd.edu;vgupta2@nd.edu; antsaklis.1@nd.edu).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.2012.2211443

(i.e., a rectangular dilated LMI framework) which provides a less con-servative solution for the reliable stabilization of multi-channel system.In addition, a dissipativity-based certificate is used to extend the sta-bility condition for all instances of perturbation in the system. Specif-ically, using a panel of storage functions and a common supply rate,we verify the stability property possessed by the perturbed closed-loopsystems (see [12]–[14] and references therein for a review of systemswith dissipativity properties).This short note is organized as follows. In Section II, we present

some preliminary results on the stability condition for a continuous-time linear system in terms of dilated LMIs. Section III presents ourmain result. A verifiable necessary and sufficient condition in terms ofa set of dilated LMIs is given for the reliable state-feedback stabiliza-tion of the nominal multi-channel system, with a dissipativity-basedcertification being used to extend the stability condition when there isan additive model perturbation in the system. In Section IV, we presenta simple numerical example. Finally, Section V provides some con-cluding remarks. A preliminary version of this note, which deals witha special case of square dilated LMIs, was presented at the 19th IEEEMediterranean Conference onControl andAutomation (MED’11) [15].Throughout this note, we use the following notations. For a matrix

, denotes a hermitian matrix defined by, where is the transpose of . For a matrix

with , denotes an orthogonal complementof , which is a matrix satisfies and .denotes the set of non-negative real numbers, which is

. denotes the set of strictly positive definite realmatrices. denotes a compact uncertainty set in .

II. PRELIMINARIES

Consider a continuous-time linear system

(1)

with and .As is well known, the system in (1) is stable (or equivalently is a

Hurwitz matrix) if and only if there exits satisfying

(2)

(3)

However, this coupling of and leads to several difficulties espe-cially in the context of robust stability analysis (e.g., see references[10], [16]–[18]). To address some of the concerns, a relaxed stabilitycondition has recently been derived using a set of dilated LMIs. Thefollowing gives an example of dilated LMIs, which is a version of theresult given in [11], [19], [20].Lemma 1: (Type I—Square Dilated LMIs): The system (1) is stable

if and only if there exist , and such that

(4)

The above result can be verified by using Finsler’s lemma [21], whichis a specialized version of the elimination lemma, with

Note that the condition in (4) is an LMI with respect to and if wefix . This scalar parameter can be chosen with a line-search method.

0018-9286/$31.00 © 2012 IEEE

HUI: WU et al.: BEFEKADU et al.: ABERKANE: BOUNDED REAL LEMMA FOR NONHOMOGENEOUS MARKOVIAN JUMP LINEAR SYSTEMS 793

Next consider the following matrix representation that will be usefulin the sequel (i.e., in the context of reliable control using rectangulardilated LMIs framework)

(5)

where for ; andand are with dimensions of .We then obtain the following result which is more flexible dilation,

i.e., a rectangular dilated LMI framework, of (2) and (3).Lemma 2: (Type II—Rectangular Dilated LMIs): The system (1)

with (5) is stable if and only if there exist , forand such that

(6)

where is a block-diagonal matrix given by.

Proof: Sufficiency: Note that

(7)

Then, eliminating from (6) by using these matrices, we have twoinequalities

Thus we see that (2) and (3) actually hold.Necessity: Define

which has the same size as of the matrix . Then, we note that thefollowing relation holds:

(8)

Suppose (2) and (3) hold, then there exists a sufficiently smallwhich satisfies

(9)

Since , employing the Schur complement transformation for(9), we have

This means that (6) holds with .Note that Lemma 2 contains Lemma 1 as a special case. To see that,

if we multiply (6) from the left-side by the following matrix:

(10)

and from the right-side by the transpose matrix . Finally, makinguse of the following relation and settingfor all (this latter condition further gives

, c.f., (8) above), then the condition in (6) reduces to

(11)

If we further let , we see that (11) is equivalent to (4).

III. MAIN RESULTS

Consider a continuous-time -channel system

(12)

with , , is the state of the system,and is the control input to the -th channel.For this system, consider the following state-feedback controllers:

(13)

where for .Moreover, we model component failures that occur in an actuator,

sensor or controller by extracting the corresponding controller and set-ting the control input of the corresponding channel to zero.1 That is, ifthe -th control channel fails, we remove the -th controller and set

(14)

To describe the closed-loop systems with/without failures in compactforms, introduce the following notations:

and

where .Then, we can write the closed-loop systems for all

as

(15)

Here, we remark that the closed-loop system under normal operationis obtained if , while the closed-loop system under the -th con-troller failure is obtained if .We now formally state the state-feedback stabilization problem as

follows.Problem 1: (Reliable State-Feedback Stabilization Problem): Find

for such that all the closed-loop systemsfor all are stable.

Remark 1: Note that Problem 1 is solvable only if the pairsfor all are stabilizable which is assumed

in this note.

1In this note, we do not discuss transient situations in failures. This is justifiedin the context of stability since stability is definedwith behaviors over the infinitetime interval.

794 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 3, MARCH 2013

If we simply apply standard stability analysis based on the aforemen-tioned equations of (2) and (3) for all instances of closed-loop systemsin (15), we have to introduce a common quadratic Lyapunov stabilitycertificate for all which will give us a suffi-cient condition for solving the stabilizing gains for .However, such coupling of and usually leads eitherto conservative or infeasible solution set.Here, we employ instead a dilated LMI technique to check

whether all instances in (15) share a common solution set forthat maintains stability.

To this end, introduce the following matrix notations:

where , and are with dimensions of ,and , respectively

where , , and for ; and, and are with dimensions of , and

, respectively.In light of Lemma 2 and the previous discussion, we have the fol-

lowing theorem.Theorem 1: (Reliable State-Feedback Stabilizing Controllers):

Problem 1 is solvable if there exist , andfor and , for

that satisfy

(16)

for all .Once this condition is fulfilled, the state-feedback controllers in (13)

that achieve reliable stabilization are recovered by

(17)

with nonsingular solutions for and for.

Proof: Suppose that the condition in (16) is satisfied for allclosed-loop systems indexed by . Note that we canalways obtain nonsingular solutions for for and

for by introducing small perturbation if necessary,since the condition in (16) is described with a strict inequality. Thus,the solutions for the reliable state-feedback gains are well-defined asin (17). Then, we have

(18)

and the rest of the proof follows the same lines as that of Lemma 2. Infact, replacing the following:

in Lemma 2 immediately gives the condition in (16) of Theorem 1.We remark that (16) describes a set of dilated LMI conditions in

terms of , , , and ,

, . Note also that a common set of andis used for all failure modes, i.e., for all . This is

because we need a set of reliable state feedback gains thatworks well for all possible closed-loop systems. However, it should benoted that, since we use a rectangular dilated LMI framework, this doesnot require to employ a common as in the case of quadraticLyapunov approach or a common (i.e., for

and for ) as in the case of squaredilated LMIs approach (where the latter is based on (4) or equivalentlybased on (11)) for all possible failure modes. In this sense, the proposedmethod, which is based on Theorem 1, is not as conservative as thequadratic Lyapunov approach or the square dilated LMI approach. Thisfact will be further clarified in Section IV.Consider next a multi-channel system with a perturbation term, i.e.

(19)

where , is the uncertainty level andis the basic perturbation term in the system. Here we assume that theperturbed matrix lies in a compact uncertainty set .2

Consider now the following problem where we are interested in es-timating the effect of perturbation on the stability of system.Problem 2: (Robust/Reliable State-Feedback Stabilization

Problem): For a given uncertainty set , find the controller gainsfor and an upper bound on the level of

perturbation for which the perturbed closed-loop systems,i.e., for all , are stable.Solving this problem is not easy in general since it is a non-convex

optimization problem. In what follows, we assume there exists a set ofstate-feedback gains for that maintains the stabilitycondition for all instances of Problem 1.We will then estimate an upperbound on the uncertainty level for which the state-feedbackgains preserve robust/reliable stability property of the perturbed multi-channel system.We, therefore, provide a precise statement based on the existence of

“a panel of dissipativity certificates” that implies reliable stability forall instances of perturbed multi-channel system.Theorem 2: (A Panel of Dissipativity Certificates): Let ,

, for and ,for satisfy Theorem 1. Suppose ,

and , then there exist an upper bound andfor that satisfy

(20)

(21)

Moreover, the system in (19) is reliably stable for all instances of per-turbation .

Proof: To prove the above theorem, we require the following sys-tems:

(22)

to satisfy certain dissipativity property for all andinstances of perturbation in the system.

2Note that the existence of a solution for state trajectories is well-defined andit is always upper semicontinuous with compact values (e.g., see [22]).

HUI: WU et al.: BEFEKADU et al.: ABERKANE: BOUNDED REAL LEMMA FOR NONHOMOGENEOUS MARKOVIAN JUMP LINEAR SYSTEMS 795

Define the following supply rate:

(23)

with and .We clearly see that if the systems in (22) are stable for all instances

of perturbation. Then, the following dissipation inequalities will hold:

(24)

for all with non-negative quadratic storage functions, for all that satisfy

.Condition (24) with (23) further implies the following:

(25)

Therefore, there exists an upper bound for which the dissipa-tivity conditions in (25) will hold true for all instances of perturbationin the system.Using (18), then we have the following result:

(26)

where .On the other hand, let us define the following matrix interval set in:

(27)

with and which is assumed to be known a priori.Suppose that satisfies the conditions in (20) and (21), then the

trajectories of the perturbed closed-loop system

for all satisfy

(28)

Note that the condition in (28) further implies the following conditions:

(29)

and

(30)

for all .Hence, the conditions (28), (29) and (30) stating, equivalently, that

the set with consists of a panel of dissi-

pativity certificates, with a common supply rate of (23), for all instancesof perturbation in (22).3

Note that if there exists a solution set for the Problem1 that gives a minimum distance between and the set

for all ,i.e., for all ,then we have essentially a weak optimal solution for the problem weposed in Problem 2. This solution set is also unique since is anon-empty compact and convex set (e.g., see [23]).Therefore, the precise statement behind the result in Theorem

2 comes from the fact that such a panel of certificates, i.e., theset with , ensures all the perturbedmulti-channel systems to possess a dissipativity property. A similaridea has been explored by Barb et al. [25] in the context of a commondissipativity certificate for uncertain systems.Remark 2: Here, we remark that finding an upper bound on

and a set of solutions for all from a non-emptycompact and convex set is equivalent to solving the verificationproblem that we posed in Theorem 2 (e.g., see [26]).

IV. NUMERICAL EXAMPLE

Consider the following simple example where the system matricesfor the nominal system are given by:

with the following base perturbation in the system:

Note that for this system we cannot design reliable stabilizing state-feedback controllers based on a common solution of Lyapunov in-equalities, i.e., we cannot find a set which satisfiesthe conditions in (2) and (3). However, if we employ Theorem 1, thestate-feedback gains, for , 2, 3, that achievereliable stabilization are ,

andfor with .Furthermore, if we define a matrix interval

using the following positive definite matrix:

For and , Theorem 2 guarantees reliable stabilityfor all instances of perturbation in the system. Here, theupper bound on the perturbation level, which is computed togetherwith a panel of dissipativity certificates from the set , is given by

.The eigenvalues of the perturbed closed-loop systems with the con-

trollers in the system, when the perturbations are uniformly sam-pled from the interval , are shown in Fig. 1. As can be seen

3Note that the parameter determines the long-term behavior of thesystem, whereas the parameter bounds its short-term or transient be-havior. In general, these parameters can be chosen so as to guarantee the reli-ably stability of the system with acceptable decay and transient behavior [24].

796 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 58, NO. 3, MARCH 2013

TABLE IEIGENVALUES FOR THE NOMINAL SYSTEM

Fig. 1. Eigenvalues of the perturbed closed-loop system.

from this figure, all of the eigenvalues reside in the left half -plane.Moreover, the eigenvalues for the nominal closed-loop system withthe controllers in the system are given in Table I. For comparison, weprovide here the results obtained using the square dilated LMIs ap-proach that are actually given by [15] in the context robust/reliablestabilization of multichannel systems. In fact, the computed reliablestate feedback gains, for , 2, 3, are

,and ; and the corresponding upperbound on the perturbation level is , which is a conservativeestimate compared to the result of . We further remark thatwe require, for the case of the square dilated LMI approach, a common(i.e., for , 2, 3 and for , 1, 2, 3) for

all possible failure modes.

V. CONCLUSION

In this short note, we considered the problem of reliable stabiliza-tion for a perturbed multi-channel system. We provided a less conser-vative result in terms of a set of rectangular dilated LMIs for the reli-able state-feedback controllers of the nominal system, while a dissipa-tivity-based certification is employed for extending the stability con-dition for an additive model perturbation in the system. Moreover, theframework in which we have defined the problem provides a compu-tationally tractable treatment for handling the issue of robust/reliablestabilization and model uncertainty.Although we have considered reliable state feedback stabilization

problem in this technical note, the problem of reliable stabilization viamulti-controller configuration that was actually explored by Fujisakiand Befekadu [11] can be treated in the same way.

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