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Systems & Control Letters 53 (2004) 157–169www.elsevier.com/locate/sysconle

Stability analysis of piecewise-a%ne systems using controlledinvariant setsLuis Rodrigues∗

Department of Mechanical and Industrial Engineering, Concordia University, 2160B Bishop Street, B-304, Montr$eal,Canada QC H3G 2E9

Received 16 July 2003; received in revised form 9 February 2004; accepted 1 April 2004

Abstract

This paper presents new stability conditions for closed-loop piecewise-a%ne (PWA) systems. The result is based oncontrolled invariant sets for PWA systems, which are de3ned by extending the notion of semi-ellipsoidal invariant setsfor constrained linear systems reported in previous research. The paper shows that by proper use of the control input,concatenations of semi-ellipsoidal sets can be made invariant for the trajectories of PWA systems. Furthermore, basedon these controlled invariant sets, the paper presents a result for stability of a closed-loop PWA system which is lessconservative than existing approaches in the literature. In this result, it is shown that a PWA system is stable inside theintersection of any level set for a local Lyapunov function and the design set where the function is de3ned, provided the7ow points inwards at the boundaries of the intersection. This result is less conservative than previous approaches and itenables the designer to have an estimation of a much larger region of exponential stability then it would be possible usingprevious results. A numerical example is presented, in which it is made clear by comparison with previous approachesthat the estimated region of stability can be made signi3cantly larger using the new stability conditions developed in thispaper.c© 2004 Elsevier B.V. All rights reserved.

Keywords: Piecewise-a%ne systems; Invariant sets; Tangent cone; Ellipsoids; Lyapunov function

1. Introduction

Invariant sets in control of dynamical systems are an important notion because they enable a designer toguarantee that state trajectories starting inside them will remain inside a given desirable set. Examples ofdesirable sets are: a stability region, a safety set or a set corresponding to control and state constraints. Thestudy of invariant sets in control has been a very active area of research. Blanchini [2] o>ers a detailedsurvey of this area up until 1999. A related area is viability theory, which uses many concepts pertinent toinvariant set theory (see [1] and references therein). Earlier contributions with connections to piecewise-a%nesystems were devoted to constrained linear systems [5,17]. Gutman and Cwikel [5] have shown that fordiscrete-time linear systems with the control and state constrained to convex polyhedra, some control laws

∗ Tel.: +1-514-8482424x3135; fax: +1-514-8484524.E-mail address: luisrod@me.concordia.ca (L. Rodrigues).

0167-6911/$ - see front matter c© 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2004.04.001

158 L. Rodrigues / Systems & Control Letters 53 (2004) 157–169

verifying the constraints were piecewise linear. For constrained linear systems, O’Dell and Misawa [17]have de3ned the notion of semi-ellipsoidal set as the intersection of a controlled invariant ellipsoid and stateconstraints. They have investigated semi-ellipsoidal invariant sets as possible solutions to the maximal viabilityset for continuous-time linear systems with a bounded input and the state constrained to convex polyhedra.Invariant sets have also been used to solve the reachability problem for veri3cation of hybrid systems [7,19]where the continuous dynamics of the hybrid systems are either of very simple form (constant derivative)or are a simple abstraction (rectangular di>erential inclusions) of more complicated dynamics. A similarconcept to invariant set, maximal safety set, has also been studied in the literature by Sastry et al. [27] fora class of discrete-time linear hybrid systems and for a class of continuous-time linear hybrid systems [26].Robust-controlled invariant sets for a class of uncertain hybrid systems have been investigated by Lin andAntsaklis [14] and the maximal-controlled invariant set of switched linear systems has been studied by Juliusand van der Schaft [11].

Although an active area of research, very limited work has been concentrated on invariant sets for eithercontinuous-time PWA systems [8,25] or discrete-time PWA systems [12,15]. The 3rst attempt to de3ne invari-ant sets for PWA systems appears to have been under the framework of linear hybrid systems [8]. Jirstrand [8]has de3ned classes of invariant sets for PWA systems with each local dynamics restricted to be nonsingularand to have a real eigenvalue. The invariant sets were formed as concatenations of ellipsoids, paraboloids andquadratic cones. An iterative update algorithm for computation of these sets is presented but it is not shownthat it has a 3xed point. Therefore, although an important 3rst step, convergence of the algorithm and decid-ability of the formulated problem are not guaranteed. Roll [25] presented an approach to investigate invarianceof approximating automata for continuous-time PWA systems to changes in the dynamics and translation ofthe switching hyperplanes. The 7ow at the boundaries of the switching hyperplanes has been used to developconditions for invariance in the three possible cases: the 7ow points always outwards, the 7ow points alwaysinwards, and there is at least one point where the 7ow is parallel to the boundaries. When only the dynamicsor the switching boundaries vary, the solution set is convex. Unfortunately, when both the dynamics andswitching surfaces change, the conditions lead to a nonconvex solution set that is hard to e%ciently represent.A 3nite union of polytopes was shown to be a robustly controllable set for constrained discrete-time PWAsystems by Kerrigan and Mayne [12]. Although the authors present an interesting formulation of the problem,as pointed out in [12], algorithms based directly on the results presented there might be too ine%cient to berealizable for large or complex systems, which motivates further research. A technique for receding horizoncontrol for discrete-time piecewise-linear systems based on invariant sets has been reported by Mukai et al.[15]. The proposed control strategy is implemented by solving a min–max type optimization problem at eachtime step.

As seen in the last paragraph, previous research on invariant sets for PWA systems has been very limitedand, except for Refs. [8,25], it has been geared towards discrete-time systems. Furthermore, it has not beenconcerned with the important problem of estimation of regions of exponential stability for systems underpiecewise-a%ne control. The powerful connection between invariant sets and Lyapunov functions has also notbeen explored to estimate larger regions of attraction for the closed-loop trajectories. The current work triesto 3ll this gap based on previous work of the author, which addressed synthesis methods for PWA systems[23], for a PWA approximation of a class of nonlinear systems [22] and for classes of nonlinear [24] andhybrid systems [20]. The current paper builds on this research and presents a less conservative result forthe stability of a closed-loop continuous-time PWA system. The approach given in the paper is based on aconcatenation of semi-ellipsoidal sets [17] that are shown to be controlled invariant sets for PWA systems.The result presented here enables the designer to have an estimation of a larger region of exponential stabilitythan it would be possible using previous results for PWA systems [9,23]. Since the paper will derive stabilityconditions for PWA systems based on Lyapunov theory, we start by reviewing Lyapunov stability results forclosed-loop PWA systems under state feedback. Next, for the same closed-loop PWA systems, a new andless conservative stability result is presented based on controlled invariant sets. The important improvements

L. Rodrigues / Systems & Control Letters 53 (2004) 157–169 159

enabled by this new result relative to the estimation of the size of the region of guaranteed exponential stabilityare then illustrated in a numerical example. It should be mentioned that although state feedback closed-loopsystems were selected for illustrative purposes, the new result can be used for any controlled PWA system.Finally, the paper 3nishes by presenting the conclusions.

2. Lyapunov stability results for piecewise-a�ne systems

It is assumed that a PWA system and a corresponding partition of the state space with polytopic cellsRi, i∈I = {1; : : : ; M} are given (see [22] for generating such a partition). Following [6,10,18], each cell isconstructed as the intersection of a 3nite number (pi) of half-spaces

Ri = {x |HTi x − gi ¡ 0}; (1)

where Hi = [hi1 hi2 : : : hipi ], gi = [gi1 gi2 : : : gipi ]T. Moreover the sets Ri partition a subset of the state space

X ⊂ Rn such that⋃M

i=1Ri =X, Ri ∩Rj = ∅, i = j, where Ri denotes the closure of Ri. Within each cell thedynamics are a%ne of the form

x(t) = Aix(t) + bi + Biu(t); (2)

where x(t)∈Rn and u(t)∈Rm. Any two cells sharing a common facet will be called level-1 neighboring cells.Let Ni = {level-1 neighboring cells of Ri}. It is assumed that vectors cij ∈Rn and scalars dij exist such thatthe facet boundary between cells Ri and Rj is contained in the hyperplane described by {x∈Rn | cTijx−dij=0},for i = 1; : : : ; M , j∈Ni. A parametric description of the boundaries can then be obtained as [6]

Ri ∩Rj ⊆ {lij + Fijs | s∈Rn−1} (3)

for i=1; : : : ; M , j∈Ni, where Fij ∈Rn×(n−1) (full rank) is the matrix whose columns span the null space ofcij, and lij ∈Rn is given by lij = cij(cTijcij)

−1dij. For a de3nition of trajectories or solutions for system (2)see [23].

Following prior analysis of PWA systems [6,9,18], consider the piecewise-quadratic Lyapunov functioncontinuous at the boundaries and de3ned in

⋃Mi=1 Ri by the expression

V (x) =M∑i=1

�i(x)Vi(x); V (x)¿ 0;

Vi(x) = (xTPix + 2qTi x + ri); (4)

where Pi = PTi ∈R(n×n), qi ∈Rn, ri ∈R and

�i(x) =

{1; x∈Ri ;

0; x∈Rj; j = i(5)

for i = 1; : : : ; M . The stability method described in this paper is restricted to systems for which there is apartition of the state space for which a piecewise-quadratic Lyapunov function parameterized by (4) can befound for the closed-loop system. If this is the case, there will be a local quadratic Lyapunov function sector,strictly decreasing in each region Ri and, consequently, there will be a closed-loop equilibrium point xicl foreach region Ri. This restriction of the method, namely restricting the search to a piecewise-quadratic Lyapunovfunction, also exists in previous results for the stability of piecewise-a%ne systems [6,10,18,23] against whichthe new stability method of this paper will be compared. The expression for the candidate Lyapunov functionin region Ri can be recast as [9]

Vi(x) =

[x

1

]T [Pi qi

qTi ri

] [x

1

]= PxTPi Px: (6)

160 L. Rodrigues / Systems & Control Letters 53 (2004) 157–169

Let i be the desired decay rate for this Lyapunov function in each region Ri. Then, we de3ne the performancecriterion

J= mini=1:::M

i; (7)

Using a PWA control signal u= Kix + mi, the closed-loop state equations in each region Ri are

x = (Ai + BiKi)x + (bi + Bimi) ≡ PAix + Pbi: (8)

Setting xicl to be the closed-loop equilibrium point for region Ri yields

(Ai + BiKi)xicl + (bi + Bimi) = 0: (9)

Using the boundary description (3), continuity of the candidate Lyapunov function across the boundaries isenforced for each region Ri and for j∈Ni by [6]

FTij(Pi − Pj)Fij = 0;

FTij(Pi − Pj)lij + FT

ij(qi − qj) = 0;

lTij(Pi − Pj)lij + 2(qi − qj)Tlij + (ri − rj) = 0: (10)

Remark 1. Note that because V (x)¿ 0 (de3ned in Rn) is continuous, the fact that V is piecewise-quadraticalso implies that V is radially unbounded, i.e., V (x) → +∞ as ‖x‖ → ∞, provided Pi ¿ 0, i = 1; : : : ; M .

The function V (x) in (4) will be a Lyapunov function with a decay rate of i for region Ri if, for 3xed"¿ 0,

x∈Ri ⇒

Vi(x)¿"‖x − xcl‖2;

ddt

Vi(x)¡ − iVi(x);(11)

where xcl is the desired closed-loop equilibrium point of the system. Using the polytopic description of thecells (1) and the S-procedure [3], it can be shown [20] that conditions (11) are implied by the existence ofPi = PT

i ¿ 0, qi, ri, and matrices Zi and $i with nonnegative entries satisfying[(Pi − "In − PHT

i Zi PHi) (qi + "xcl + PHTi Zi Pgi)

(·)T ri − "xTclxcl − PgTi Zi Pgi

]¿ 0; (12)

[( PAT

i Pi + Pi PAi + PHTi $i PHi + iPi) (Pi Pbi + PAT

i qi − PHTi $i Pgi + iqi)

(·)T 2 PbTi qi + PgTi $i Pgi + iri

]¡ 0; (13)

where PHi=[0 hi1 hi2 : : : hipi ]T, Pgi=[1 gi1 gi2 : : : gipi ]

T, and In is the identity matrix. It must also be ensured thatthe trajectories do not stay at a switching boundary in a sliding mode. Ref. [23] shows that su%cient constraintsfor avoidance of sliding modes are obtained by making the component of the vector 3eld perpendicular tothe boundaries be continuous, i.e.,

cTij(Ai + BiKi − Aj − BjKj)Fij = 0;

cTij[(Ai + BiKi − Aj − BjKj)lij + bi + Bimi − bj − Bjmj] = 0 (14)

for i = 1; : : : ; M and j∈Ni.

L. Rodrigues / Systems & Control Letters 53 (2004) 157–169 161

De�nition 2.1. The state feedback synthesis optimization problem is

max mini

i

s:t: (9); (10); (12); (13); (14);

Zi � 0; $i � 0; i ¿ l0¿ 0;

−l1 ≺ Ki ≺ l1; −l2 ≺ mi ≺ l2;

i=1; : : : ; M , where �, ≺ mean component-wise inequalities, l0 is a scalar bound and l1, l2 are vector bounds.Note that the optimization variables are xicl, Ki, mi, i, Pi, qi, ri, Zi and $i.

To simplify the optimization problem, the desired closed-loop equilibrium points for each polytopic region,xicl, are selected a priori using the optimization algorithm described in [22]. Then, expression (13) is called abilinear matrix inequality (BMI) [4]. Optimization problems with BMI constraints are known to be NP-hardand while branch and bound algorithms are available for 3nding the optimal solution [4], these algorithms runin nonpolynomial time because they consist of global searches. Therefore, they are typically not computation-ally tractable for problems that use a medium/large number of polytopic regions. Although suboptimal, localalgorithms consist of an iterative scheme that, at each iteration, executes e%cient polynomial-time algorithmsand thus can provide (suboptimal) solutions in a reasonable amount of computational time. Suboptimal solu-tions to the optimization problem can be obtained with the algorithms presented in [23]. Alternative approachesfor solving this problem can also be found in [21].

Theorem 2.1. Assume the Lyapunov function (4) is de:ned in X ⊆ Rn. If there is a solution to the designproblem from De:nition 2.1, the closed-loop system is locally asymptotically stable inside any subset ofthe largest level set of the control Lyapunov function (4) that is fully contained in X. If "¿ 0 then theconvergence is exponential. If, furthermore, X = Rn then the exponential stability is global.

Proof. See [23].

3. Stability of the closed-loop PWA system

In this section, a new stability result that is less conservative than Theorem 2.1 and than other stabilityanalysis results for PWA systems [9,18,6] will be presented. The result is based on the notion of controlledinvariant set.

De�nition 3.1 (Blanchini [2]): A set C is (positively) invariant for a dynamical system if once the state x(t)enters that set at any given time t0, it remains in it for all future time, i.e., x(t0)∈C ⇒ x(t)∈C, ∀t¿t0 . Acontrolled invariant set is a set that can be made positively invariant by a proper choice of a control action.

Stability will be de3ned within a set called the design set [23]. Design sets are sets where a candidateLyapunov function will be de3ned and convex design sets will be of particular importance in this paper toanalyze the stability of the system trajectories. Design sets and a particular class of convex design sets areformally de3ned next.

De�nition 3.2 (Rodrigues and How [23]): A design set D is a nonempty closed subset of the domain of thestate variable x.

162 L. Rodrigues / Systems & Control Letters 53 (2004) 157–169

De�nition 3.3. A design set D is called a polytopic design set if

1. D ⊆ X where X =⋃M

i=1 Ri,2. D is the intersection of a 3nite number of half-spaces Dj = {x | cTj x6dj} in the form

D= {x | cTj x6dj; j = 1; : : : ; p}:

The main result in this section states formally that a closed-loop PWA system is stable inside the intersectionof any level set for a local Lyapunov function and the design set where the function is de3ned, provided the7ow points inwards at the boundaries of the intersection. To prove this result we need the following de3nitionand lemma.

De�nition 3.4 (Khalil [13]): Given a real number &¿ 0, a &-level set of a Lyapunov function V is the set{x∈Rn|V (x)6 &}.

Remark 2. Note that the &-level set of the kth sector (corresponding to region Rk) of the Lyapunov function(4) is invariant under the 7ow x(t) = PAkx(t) + Pbk . This invariant set is described by the ellipsoid

"k& = {x∈Rn | (x − xkcl)TPk(x − xkcl) + rk6 &}: (15)

Lemma 3.1 (Nagumo [16] and Blanchini [2]). Consider the system x(t)=f(x(t)) and assume that, for eachinitial condition in a set D, it admits a globally unique solution. Let L ⊆ D be a closed and convex set andlet @L be its boundary. Then the set L is positively invariant for the system if and only if, f(x)∈CL(x)for all x∈ @L, where CL(x) is the tangent cone for the set L at the point x.

Remark 3. Roughly speaking, Lemma 3.1 just states that for a set to be positively invariant, the vector 3eldhas to point inwards at the boundaries of the set.

Remark 4. For the boundary of smooth sets L the tangent cone CL(x) at the point x is just the half-spacetangent to L at x shifted to the origin. Thus, the tangent cones for the boundaries of the ellipsoid "k& anddesign set constraints Dj are C"k& = {y | (x − xkcl)

TPky6 0} and CDj = {y | cTj y6 0}, respectively. Note that,as expected, the orientation of the boundary of CDj (x) is 3xed and depends only on cj. It does not dependon the point x.

The two sets de3ned in (16) will be needed in part 2 of the proof of Theorem 3.1. That part of the prooffollows closely the one presented in [17] for a related theorem relative to constrained linear systems:

@Dj≡ {x | cTj x = 0};

D+j ≡ {x | cTj x ¿ 0}: (16)

Theorem 3.1. Let the design set D be polytopic as in De:nition 3.3 and let @Dj be the boundary of Dj and@D be the boundary of D. Assume there exists a parametric description for @Dj , j = 1; : : : ; p of the form

x∈ @Dj ⊆ Hj = {vj + Tjs | s∈Rn−1}; (17)

where all involved matrices have appropriate dimensions. Given &¿ 0 de:ne the closed set "& =⋃M

k=1("k& ∩

Rk) and the closed subset "& = "& ∩ D. If there is a solution to the controller synthesis problem from

L. Rodrigues / Systems & Control Letters 53 (2004) 157–169 163

De:nition 2.1 then the following are veri:ed:

1. The set "& is a &-level set of the Lyapunov function (4). As such, "& is an invariant set for the PWAsystem for all &¿ 0 such that "& ⊆ D and the closed-loop system is exponentially stable inside allsuch "&;

2. The set "& is an invariant set of the system, if and only if "& ∩ @Dj ∩ D+j = ∅, j = 1; : : : ; p,

3. The closed-loop system is exponentially stable inside "& for any &¿ 0 for which there are "kj ¿ 0 suchthat ∀(j; k) |Rk ∩Hj = ∅, expression (18) is veri:ed.

[TTj PkTj TT

j Pk(vj − xkcl) − "kjTTj A

Tk cj

(TTj Pk(vj − xkcl) − "kjTT

j ATk cj)

T (vj − xkcl)TPk(vj − xkcl) + rk − & − 2"kj(vTj A

Tk + bTk )cj

]¿ 0: (18)

Proof. (1) The Lyapunov function (4) is radially unbounded (see Remark 1). Therefore its level sets are closedcurves [13]. Within each region Rk the level curves of the piecewise-quadratic function need to coincide withthe ellipsoidal level curves "k& of the corresponding quadratic sector. This ensures that in

⋃Mi=1 Ri, the &-level

sets are represented by⋃M

k=1("k& ∩ Rk). To see that in fact "& =

⋃Mk=1("

k& ∩ Rk) are the level &-level sets

of the Lyapunov function (4) we need now only analyze what happens at the boundaries between regions.Observe that the ellipsoidal level curves for each region meet at the boundaries and form closed concatenatedellipsoidal level curves for the piecewise-quadratic function because the Lyapunov function is continuous atthe boundary. Because of this fact, for all &¿ 0 such that "& ⊆ D, "& is a &-level set of the Lyapunov function(4) which is a local Lyapunov function de3ned in D. Therefore, as a &-level set of a Lyapunov function, "&is an invariant set of the system. Moreover, Theorem 2.1 applies which leads to the conclusion and 3nishesthe proof of this part.

(2) (⇒) Assume that "& is invariant but "& ∩@Dj ∩ D+j = ∅ for some j=1; : : : ; p. This implies that cTj x0 ¿ 0

for at least one point x0 ∈ "& ∩ @Dj . Thus, from Remark 4, f(x0) = x0 ∈ CDj ⇒ f(x0) ∈ C"& , since at x0 thetwo cones coincide. By Theorem 3.1, "& is not an invariant set, violating the assumption.

(2) (⇐) We want to prove now that given the assumption "&∩@Dj ∩D+j =∅, j=1; : : : ; p we have f(x)∈C"& ,

∀x∈ @"& , which guarantees invariance from Theorem 3.1. Note that D ⊆ X ≡ ⋃Mi=1 Ri and (as observed in

1) "& is a &-level set of the Lyapunov function (4) inside D. Because of these facts, it su%ces to check theinvariance for the portions of the boundary @"& corresponding to x∈ @"k& , x∈ @Dj and any region of @"& formed

by the intersection of two or more sets of the form "k& or Dj. For x∈ @"& such that x∈ @"k& de3ned in (15),f(x)∈C"k& is guaranteed by making the Lyapunov function negative de3nite since this condition implies that

(x− xkcl)TPkx¡ 0. For x∈ @"& such that x∈ @Dj , the assumption "& ∩ @Dj ∩ D+

j = ∅, j=1; : : : p implies cTj x6 0,which in turn implies that f(x)∈CDj . If x∈ @"& is such that x∈ [@"k& ∩ @Dj ] then from the above we havef(x)∈C"k& and f(x)∈CDj and therefore f(x)∈ (C"k& ∩CDj), which implies that f(x)∈C"& . More generally thisholds for any region of @"& formed by the intersection of two or more sets (ellipsoids or sets Dj) by repeatedlyapplying the above reasoning. This argument 3nishes this part of the proof.

(3) In the previous parts of the proof we have shown that for "& to be an invariant set, a necessary andsu%cient condition is

cTj x6 0; ∀x∈ "& ∩ @Dj ; j = 1; : : : ; p: (19)

However, this necessary and su%cient condition is not easy to formulate as an LMI problem. A su%cientcondition for "& being an invariant set is

x∈Hj ∩ "& ⇒ cTj x ¡ 0; j = 1; : : : ; p: (20)

164 L. Rodrigues / Systems & Control Letters 53 (2004) 157–169

D

εΚ

RK

__D

U UεΚ

γ

γ

Fig. 1. Invariant regions of Lyapunov function.

Indeed, (20) implies (19) because "& ⊆ "& by de3nition and @Dj ⊆ Hj from (17). Given the de3nition of "&this su%cient condition will be relaxed to (Fig. 1)

x∈Hj ∩ "k& ⇒ cTj x ¡ 0; ∀(j; k) |Rk ∩Hj = ∅: (21)

Again, (21) implies (20) because (Hj ∩ "k& ∩Rk) ⊆ (Hj ∩ "k&). Using (17) and (15) yields

x∈Hj ∩ "k& ⇔[s

1

]T [TTj PkTj TT

j Pk(vj − xkcl)

(·)T (vj − xkcl)TPk(vj − xkcl) + rk − &

] [s

1

]6 0: (22)

Using (17) yields

x∈Hj ∩ {x | cTj x ¡ 0} ⇔[s

1

]T [0 TT

j ATk cj

(TTj A

Tk cj)

T 2(vTj ATk + bTk )cj

] [s

1

]¡ 0: (23)

Using (22), (23) and the S-procedure the condition of the theorem results, which 3nishes the proof.

Remark 5. Note that the necessary and su%cient condition for positive invariance from Theorem 3.1 wasreplaced by a su%cient condition, which is less sharp and more conservative. However, the problem wasformulated as an LMI, which can be solved e%ciently by convex optimization software packages.

Remark 6. Part 2 of the theorem can be seen as a generalization to PWA systems of the results presented in[17] for state-feedback of constrained linear systems. The geometrical ideas in [17] were also generalized torelax the necessary and su%cient condition to a su%cient condition for the case of PWA systems. Unfortu-nately, the procedure for computing appropriate subspaces adopted in [17] to formulate the relaxation as anLMI is not straightforward to generalize to PWA systems. Therefore, the formulation of the relaxation intoan LMI di>ers substantially from the procedure adopted in [17]. Instead of computing a subspace, the currentwork uses a parametric description of the boundaries of D, as described in (17). Given that these boundariesare hyperplanes, assuming the existence of a parametric description is appropriate and it enables a simpli3edformulation of the LMI for the more complex case of PWA systems.

L. Rodrigues / Systems & Control Letters 53 (2004) 157–169 165

Fig. 2. Circuit with nonlinear resistor.

Remark 7. For a given 3xed Lyapunov function, one can search for the maximum value of & that veri3es(18) by solving the semide3nite program

max &

s:t: "jk ¿ 0; &¿ 0; (18)

∀(j; k) |Rk ∩Hj = ∅:

Remark 8. Another interesting problem would be to 3nd the Lyapunov function that maximizes the volumeof the invariant set verifying the conditions of either Theorem 2.1 or Theorem 3.1. However, it does not seemthat such problem can be formulated as a convex program.

4. Example

This example considers a circuit with a nonlinear resistor shown in Fig. 2 [6]. This nonlinear resistor modelis sometimes used to approximate the behavior of a tunnel diode. With time expressed in 10−10 seconds, theinductor current in milliAmps and the capacitor voltage in volts, the dynamics are written as[

x1

x2

]=

[ −30 −20

0:05 0

] [x1

x2

]+

[24

−50g(x2)

]+

[20

0

]u:

Following [6], the characteristic of the nonlinear resistor g(x2) is de3ned to be the piecewise-a%ne functionshown in Fig. 3 which generates the polytopic regions

R1 = {x∈R2 | 0¡x2 ¡ 0:2}; R2 = {x∈R2 | 0:2¡x2 ¡ 0:6}; R3 = {x∈R2 | 0:6¡x2 ¡ 0:8}:Using the method explained in Section 2, a piecewise-a%ne state feedback controller has been designed tostabilize the open-loop equilibrium point of region R3

xcl = x3ol =

[0:3714

0:6429

]:

For the design set

D= {x∈R2 | − 36 x16 3; 06 x26 0:78}

166 L. Rodrigues / Systems & Control Letters 53 (2004) 157–169

Fig. 3. Nonlinear resistor characteristic.

the resulting controller parameters are

K1 = [0:3523 + 1:8371]; m1 = 0:7094;

K2 = [0:3523 − 3:3611]; m2 = 1:7605;

K3 = [0:3523 + 3:0258]; m3 = −2:076

and the Lyapunov function parameters are

P1 =

[1:5587 −4:3011

−4:3011 688:18

]; P2 =

[1:5587 8:8930

8:8930 249:18

]; P3 =

[1:5587 19:587

19:587 687:64

];

q1 =

[ −4:1153

−785:52

]; q2 =

[ −6:7541

−195:6238

]; q3 =

[ −13:171

−449:33

];

r1 = 365:54; r2 = 147:14; r3 = 293:74:

Using Theorem 2.1 (or any related theorem from [9]), exponential stability can only be proven inside thelargest level set of the Lyapunov function fully contained in the design set (see closed curve in Fig. 4).For the same 3xed Lyapunov function, using Theorem 3.1 and the optimization from Remark 7, exponentialstability can be proven inside the non3lled region in Fig. 4. This region corresponds to the intersection of thedesign set (whose boundary is composed of the dashed lines in Fig. 4) and the sublevel set of the Lyapunovfunction for the obtained maximum value of & = 239:73. From Fig. 4, it can be clearly seen that Theorem3.1 yielded a much larger region of guaranteed exponential stability. Fig. 5 represents the level sets of the

L. Rodrigues / Systems & Control Letters 53 (2004) 157–169 167

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-3

-2

-1

0

1

2

3

x2

x 1

boundary of stabilityregion using previous methods

boundary of stabilityregion using method from this paper

equilibriumpoint

Fig. 4. Regions of stability.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-3

-2

-1

0

1

2

3

x2

x 1

Fig. 5. Level curves and system trajectories.

168 L. Rodrigues / Systems & Control Letters 53 (2004) 157–169

obtained Lyapunov function over-plotted with some system trajectories. It can be seen that, as expected, forall the trajectories in the 3gure, the 7ow points inwards at the boundaries of the region of stability. Theseresults show one example where the conservatism of Theorem 2.1 can be greatly reduced by using the newresult presented in Theorem 3.1.

5. Conclusions

This paper presented a new stability theorem for closed-loop PWA systems. If the conditions of the theoremare veri3ed then a larger region of exponential stability can be guaranteed than it would be possible usingprevious results. The theorem is based on controlled invariant sets for PWA systems. Invariant sets forpiecewise-a%ne systems were de3ned by extending the notion of semi-ellipsoidal invariant sets for constrainedlinear systems reported in previous research. A numerical example has shown a particular case for which amuch larger region of guaranteed exponential stability can be obtained using the result presented in this paperinstead of previous approaches.

Acknowledgements

The author would like to thank AntSonio Pascoal, Arash Hassibi, Claire Tomlin, Jonathan How, ManfredMorari, Mikael Johansson and Steve Boyd for interesting discussions about piecewise-a%ne systems. Theauthor would also like to thank the anonymous reviewers for their careful reading and for their suggestions.

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