Automatica 43 (2007) 1249–1256www.elsevier.com/locate/automatica

Brief paper

Stability of sampled-data piecewise-affine systems under state feedback�

Luis Rodrigues∗

Department of Mechanical and Industrial Engineering, Concordia University, 1515 St. Catherine Street, EV4.243 Montréal, Canada QC H3G 2W1

Received 10 November 2004; received in revised form 11 October 2006; accepted 19 December 2006Available online 10 May 2007

Abstract

This paper addresses stability of sampled-data piecewise-affine (PWA) systems consisting of a continuous-time plant and a discrete-timeemulation of a continuous-time state feedback controller. The paper presents conditions under which the trajectories of the sampled-data closed-loop system will exponentially converge to a neighborhood of the origin. Moreover, the size of this neighborhood will be related to bounds onperturbation parameters related to the sampling procedure, in particular, related to the sampling period. Finally, it will be shown that when thesampling period converges to zero the performance of the stabilizing continuous-time PWA state feedback controller can be recovered by theemulated controller.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Piecewise-affine systems; Switched systems; Sampled-data systems; Stability analysis; Linear matrix inequalities

1. Introduction

Piecewise-affine (PWA) systems are multi-model systemsthat offer a good modeling framework for complex dynamicalsystems involving nonlinear phenomena. State and output feed-back control of continuous-time PWA systems have receivedincreasing interest over the last years. The research work hasconcentrated on Lyapunov-based controller synthesis methodsfor continuous-time PWA systems (Hassibi & Boyd, 1998; Jo-hansson, 2003; Johansson & Rantzer, 2000; Rodrigues & Boyd,2005; Rodrigues & How, 2003). However, none of these ap-proaches would be applicable directly to controller synthesis forcomputer-controlled or sampled-data PWA systems. This is thescenario mostly encountered in applications given the flexibil-ity of control implementation in a microprocessor. Hassibi andBoyd (1998), Johansson (2003), Johansson and Rantzer (2000),Rodrigues and How (2003) and Rodrigues and Boyd (2005)consider continuous-time processes controlled by continuous-time controllers while the implementation in a microproces-sor requires emulation of a continuous-time controller as a

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor DraganNesic under the direction of Editor Hassan Khalil.

∗ Tel.: +1 514 8482424; fax: +1 514 8483175.E-mail address: luisrod@encs.concordia.ca.

0005-1098/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2006.12.016

discrete-time controller. Although linear sampled-data systemsare a well-studied matter (Chen & Francis, 1995), controlleremulation for systems with possible discontinuities at theswitching, such as sampled-data PWA systems, has not hadmany research contributions. In fact, only recently these sys-tems have started to be addressed in the literature in referencessuch as Imura (2003a,b), Azuma and Imura (2004), Sun andGe (2002), Sun (2004) and Zhai, Lin, and Yasuda (2004). Theapproach by Sun and Ge (2002) established that, under certainconditions, the controllable subspaces of a continuous-timeswitched linear system and its discrete-time counterpart arethe same. Canonical forms of switched linear systems basedon controllability are presented in the more recent work ofSun (2004). The approach by Zhai et al. (2004) considers sta-bility analysis of switched systems that can switch betweena set of continuous-time plants and a set of discrete-timeplants but does not handle sampled-data systems involving acascade of a discrete-time system between a sample-and-holdand a continuous-time system. Furthermore, it does not ad-dress controller design. The approach by Imura (2003a,b) andAzuma and Imura (2004) was probably the first where the term“sampled-data PWA systems” is used, although the systemsdescribed in this work do not possess the typical structure of acontinuous-time plant being controlled by a discrete-time con-troller. The problem addressed in Imura (2003a,b) and Azuma

1250 L. Rodrigues / Automatica 43 (2007) 1249–1256

and Imura (2004) is the one where the controller is continuous-time and the switching events are the ones controlled by thesystem logic inside a computer. In other words, in these systemsit is assumed that the designer has command over the switchingtimes of the system. The preliminary study of Imura (2003a,b)is interesting as it highlights important limitations of currentdiscrete-time PWA control methodologies when applied to thecontrol of a physical continuous-time system. As mentioned inImura (2003a) unexpected phenomena such as chattering canoccur, depending on the switching times. This increases theinterest in studying computer implementations of controllersdesigned in continuous-time.

This paper addresses the classical structure of a sampled-datasystem whereby the system is continuous-time and the con-troller is being implemented (emulated) in discrete-time inside acomputer. Previous approaches to this classical structure can beclassified into two categories: (i) discrete-time controller designto a discrete-time approximation of the continuous-time plantand (ii) continuous-time controller design to a continuous-timeplant followed by discrete-time emulation of the controller. Tothe best of the author’s knowledge the only previous work insampled-data PWA systems is the work of Imura (2003a,b) andAzuma and Imura (2004) which, as already stated, does notaddress the classical structure of interest in this paper. For pa-pers in sampled-data control for nonlinear systems that fall intocategory (i) we refer the reader to Nesic and Laila (2002) andreferences therein. For papers in sampled-data control for non-linear systems that fall under category (ii) we refer the readerto Khalil (2004) and references therein. Note that these pa-pers always assume the plant dynamics to be locally Lipschitz.Therefore they do not include the possibility of having PWAdynamics that are switched with possible discontinuities in theplant dynamics at the switching. The interesting paper by Nesicand Teel (2004) also falls under category (i) described above butoffers the advantage of treating the plant model as a differentialinclusion, thus possibly enabling discontinuous vector fields.In fact, one of the examples described in Nesic and Teel (2004)deals with a hysteresis switched controller. Although potentiallyapplicable to PWA systems, the work of Nesic and Teel (2004)does not address the problem of interest here, namely stabilityand performance recovery by emulation of a continuous-timePWA controller. Furthermore, in the framework of Nesic andTeel (2004), the plant dynamics must be embedded in a dif-ferential inclusion, which can potentially lead to conservativeresults instead of handling the PWA dynamics directly.

The paper starts by stating the problem assumptions. Then,the stability of the sampled-data system when a continuous-timecontroller is emulated in discrete-time is analyzed. A numericalexample is included to show an application of the main stabilityresult. Finally, the paper closes by stating the conclusions.

2. Problem assumptions

It is assumed that a PWA system and a corresponding par-tition of the state space with polytopic cells Ri , i ∈ I ={1, . . . , M} are given (see Rodrigues & How, 2001 for generat-ing such a partition). Following Johansson (2003) and Hassibi

and Boyd (1998), each cell is constructed as the intersection ofa finite number (pi) of half spaces

Ri = {z ∈ Rn |HTi z − gi < 0}, (1)

where Hi = [hi1 . . . hipi] ∈ Rn×pi , gi = [gi1 . . . gipi

]T ∈ Rpi .Moreover the sets Ri partition a subset of the state space X ⊂Rn such that

⋃Mi=1 Ri = X,Ri ∩ Rj = ∅, i �= j , where Ri

denotes the closure of Ri . Within each cell the dynamics areaffine of the form

z(t) = Aiz(t) + bi + Biu(t), (2)

where z(t) ∈ Rn, u(t) ∈ Rm and bi ∈ Rn. For system (2), weadopt the following definition of solutions.

Definition 1 (Johansson, 2003). Let z(t) ∈ X be an absolutelycontinuous function. Then z(t) is a trajectory of the system (2)on [t0, tf ] if, for almost all t ∈ [t0, tf ] and Lebesgue measurableu(t), the equation z(t) = Aiz(t) + bi + Biu(t) holds for all isuch that z(t) ∈ Ri .

Any two cells sharing a common facet will be called level-1 neighboring cells. Let Ni = {level-1 neighboring cells ofRi}. Vectors cij ∈ Rn and scalars dij will then exist suchthat the facet boundary between cells Ri and Rj is containedin the hyperplane described by {z ∈ Rn | cT

ij z − dij = 0},for i = 1, . . . , M , j ∈ Ni . A parametric description of theboundaries is

Ri ∩ Rj ⊆ {z = lij + Fij s | 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 of cT

ij and

lij ∈ Rn is given by lij = cij (cTij cij )

−1dij . It is further assumedthat matrices Ei and fi exist such that Ri ⊆ εi ,

εi = {z| ‖Eiz + fi‖�1}. (4)

This ellipsoidal covering is especially useful in the case whereRi is a slab because in this case the matrices Ei and fi areguaranteed to exist and the covering (having one degenerate el-lipsoid εi) is exact, i.e., εi ⊆ Ri and Ri ⊆ εi . More precisely,if Ri = {z | d1 < cT

i z < d2}, then the degenerate ellipsoid is de-scribed by Ei = 2cT

i /(d2 − d1) and fi = −(d2 + d1)/(d2 − d1).Finally it is assumed, without loss of generality, that the controlobjective is to stabilize the system to the origin.

3. Stability of sampled-data PWA systems

In this section a stability result is presented for the closed-loop sampled-data system that is obtained when a continuous-time state feedback controller is implemented on a digitalcomputer. It is assumed that a continuous-time state feedbackcontroller parameterized by Ki ∈ Rm×n and mi ∈ Rm in theform

u = Kiz + mi, z ∈ Ri (5)

L. Rodrigues / Automatica 43 (2007) 1249–1256 1251

has already been designed such that the continuous-time closed-loop system is exponentially stable.1 It is also assumed thatthe state z of the system is measured at a sampling rate fs =T −1, T > 0, and that the controller in the feedback loop ap-pears between a sampler and a zero-order-hold. At the sam-pling instants, the plant state and the sampled state overlap andtherefore the sampled-data system is described by

z = Ajz + bj + BjKjz(kT ) + Bjmj (6)

for z(t) ∈ Rj , z(kT ) ∈ Rj . However, for a given time t thatis not a sampling instant, the general situation that should beconsidered is the one in which the state of the plant is in regionRi and the most recently sampled state is in region Rj withpossibly i �= j . The system is then described by the differentialequation

z = Aiz + bi + BiKjz(kT ) + Bimj (7)

for z(t) ∈ Ri , z(kT ) ∈ Rj . This equation can be rewritten inthe perturbed form

z = Aiz + bi + Bi�ij (8)

for z(t) ∈ Ri , z(kT ) ∈ Rj , where Ai =Ai +BiKi, bi = bi +Bimi and

�ij = Kj(z(kT ) − z(t)) + (Kj − Ki)z(t) + (mj − mi). (9)

Note that the first term in (9) represents the perturbation due tothe error between the last available sample of the state and itscurrent value. The second and third terms are associated withthe perturbation due to the state and its most recent sample beingpossibly in different regions. The second term represents theperturbation due to a difference in the gain matrices in regionsRi and Rj and the third term represents the perturbation dueto a difference in the affine control terms. Given a continuous-time controller of the form (5), the first step in the procedureoutlined in this paper is to search for a quadratic Lyapunovfunction of the form

V (z) = zTPz (10)

that proves stability of the continuous-time closed-loop system.This can be done by solving for fixed ��0 the following setof LMIs (see for example Rodrigues & Boyd, 2005 for detailson the derivation of these conditions):

P = P T > 0, �i < 0, i = 1, . . . , M , (11)[AT

i P + P Ai + �P + �iETi Ei P bi + �iE

Ti fi

(P bi + �iETi fi)

T −�i (1 − f Ti fi)

]< 0.

The results that follow assume that such a Lyapunov functioncan be found. Note, however, that not all continuous-time PWAsystems that are stable admit a globally quadratic Lyapunovfunction (see Johansson, 2003 for counter-examples).

1 For optimization programs whose solution (when it exists) yieldsexponentially stabilizing PWA controllers see Rodrigues and How (2003) andRodrigues and Boyd (2005).

3.1. Conditions independent of the sampling period

We now present the first result of this section. It gives condi-tions under which the trajectories of the sampled-data system(8) converge to a region around the closed-loop equilibriumpoint. Furthermore, it relates the size of this region to a mea-sure of the perturbation term in the closed-loop system. In whatfollows, unless otherwise indicated, the time dependence of theindices i, j will be omitted for simplicity.

Theorem 2. Assume a Lyapunov function of the form (10)is found and is defined in X ⊆ Rn. Let the condition num-ber of P be �P = �max(P )/�min(P ). Assume there are fi-nite constants Nij > 0, �Kij

�0, such that ‖�ij‖�Nij +�Kij

‖z‖, i, j = 1, . . . , M . Let N = maxi,j=1,...,M(Nij ), �K =maxi,j=1,...,M(�Kij

), B = maxi=1,...,M‖Bi‖. Define

� = 2�P B

� − 2�P B�K

N

and the region

S = {z ∈ X|‖z‖��}for any positive constant < 1 that verifies

�K <�

2�P B. (12)

Then, if (12) is verified, the trajectories of the closed-loopsampled-data system (8) converge exponentially to the set

= {z ∈ X|V (z)��max(P )�2}.

Proof. For z(t) ∈ Ri , z(kT ) ∈ Rj , using the dynamics (8),the derivative of the candidate Lyapunov function (10) alongthe trajectories of the system is

V (z) =[

z

1

]T [AT

i P + P Ai P bi

(P bi)T 0

] [z

1

]+ 2zTPBi�ij .

However, note that if a quadratic Lyapunov function is found bysolving (11), using the S-procedure (see Rodrigues & Boyd,2005 for details) it can be shown that for z ∈ Ri ,[

z

1

]T [AT

i P + P Ai P bi

(P bi)T 0

] [z

1

]< − �zTPz.

Therefore, for z ∈ Ri , z(kT ) ∈ Rj it follows that

V (z) < − �zTPz + 2zTPBi�ij .

Taking norms and using the bounds

‖�ij‖�Nij + �Kij‖z‖�N + �K‖z‖

and −zTPz� − �min(P )‖z‖2 yields

V (z) < − ��min(P )‖z‖2 + 2‖z‖�max(P )B(N + �K‖z‖)

1252 L. Rodrigues / Automatica 43 (2007) 1249–1256

or, for any positive constant < 1,

V (z) < − (1 − )��min(P )‖z‖2

− ��min(P )‖z‖2 + 2‖z‖�max(P )B(N + �K‖z‖).Therefore, for 0 < < 1, we have

V (z) < − (1 − )��min(P )‖z‖2 � − (1 − )�−1P �V (z) (13)

for

‖z‖ >2�P B

� − 2�P B�K

N

provided

�K <�

2�P B. (14)

Note that although condition (13) is only valid inside eachregion of the partition of the state space, it also guarantees thatno unstable sliding modes can be generated at the boundariesbecause the Lyapunov function is of class C1 (see Johansson,2003, p. 84 for more details). As a result of (13), for z ∈ Rn\S,

V (z(t)) < V (z(t0))e−(1−)�−1

P �(t−t0).

Using the relation �min(P )‖z‖2 �V (z)��max(P )‖z‖2 we canconclude that for z ∈ Rn\S,

‖z(t)‖�‖z(t0)‖�1/2P e−0.5(1−)�−1

P �(t−t0).

Thus, there will be a positive and finite time t1 such that z(t1 ) ∈S for any positive constant < 1 that verifies (14). Note thatS ⊆ . This can be proved by contradiction. Assume thatit is not true that S ⊆ . Then, there exists at least onez0 ∈ S for which zT

0 Pz0 > �max(P )�2, a contradiction. Since

V �0 at the boundary of , is an invariant set for system (8).Consequently, since z(t1 ) ∈ S ⊆ , z(t) ∈ for all t � t1and for all 0 < < 1 that verifies (14). �

Remark 3. This result relates the size of the region to whichthe trajectories converge to the size of the perturbations. Thesize of the region decreases with the size of the perturbations,as expected. Note that for the case where Ki = Kj , �K = 0and (12) is automatically verified.

Remark 4. Bounds on �ij can be easily obtained in the casewhere all polytopic regions are bounded, by noticing that‖z(kT ) − z(t)‖�maxx∈Ri ,y∈Rj

‖x − y‖. These bounds are,however, potentially conservative and better ways of obtainingthem should be investigated. In particular, the bound shoulddepend on the sampling period T.

The next section relates the bound on ‖z(kT ) − z(t)‖ to thesampling period T and offers a less conservative result thatenables us to prove that if the sampling period converges tozero then the system is practically exponentially stable to theorigin and the continuous-time behavior is recovered.

3.2. Conditions dependent of the sampling period

Integrating Eq. (7) for t ∈ [kT , (k + 1)T ] yields

z(t) − z(kT ) =∫ t

kT

Ai(�)z(�) d� +∫ t

kT

bi(�) d�

+∫ t

kT

Bi(�) d�(Kjz(kT ) + mj). (15)

Thus, letting A=maxi=1,...,M ‖Ai‖, b=maxi=1,...,M ‖bi‖, B=maxi=1,...,M ‖Bi‖ yields

‖z(t) − z(kT )‖�A

∫ t

kT

‖z(�)‖ d�

+ (t − kT )(b + B‖Kjz(kT ) + mj‖). (16)

Since all possible dynamics in a PWA system with coefficientsindependent of the partition are affine, finite escape times can-not occur and therefore there will be a finite constant Z(k, T )=supkT � t �kT +T ‖z(t)‖ such that

‖z(t)‖kT � t �kT +T �Z(k, T ). (17)

Using the bound (17) in expression (16) leads to

‖z(t) − z(kT )‖�(t − kT )(AZ(k, T ) + b

+ B‖Kjz(kT ) + mj‖). (18)

Remark 5. Note that the bound Z(k, T ) gets smaller as thesampling time decreases and when T → 0, Z(k, T ) →‖z(kT )‖. Note further that the Euler approximation for integra-tion would lead to Z(k, T )=‖z(kT )‖ because

∫ t

kT‖z(�)‖ d� �

‖z(kT )‖(t − kT ).

Letting K = maxi=1,...,M ‖Ki‖, m = maxi=1,...,M ‖mi‖,

‖z(t) − z(kT )‖�(t − kT )(AZ(k, T ) + b

+ BK‖z(kT )‖ + Bm). (19)

The worst possible (highest) bound is the one corresponding tot = (k + 1)T , which leads to

‖z(t) − z(kT )‖�T (AZ(k, T )

+ b + BK‖z(kT )‖ + Bm). (20)

Recall that the expression for the perturbations developed in(9) was

�ij = Kj(z(kT ) − z(t)) + (Kj − Ki)z(t) + (mj − mi). (21)

Let now �Kij= ‖Kj − Ki‖, �mij

= ‖mj − mi‖. Then we canwrite

‖�ij‖�K‖z(t) − z(kT )‖ + �Kij‖z(t)‖ + �mij

(22)

and therefore using (17) and (20) this finally yields

‖�ij‖�Nij (k, T ) + �Kij‖z‖, i, j = 1, . . . , M , (23)

where

Nij (k, T ) = �mij+ KT (AZ(k, T ) + b) (24)

L. Rodrigues / Automatica 43 (2007) 1249–1256 1253

and A=A+BK, b=b+Bm. Using this bound and Theorem2 the following result can now be stated.

Corollary 6. Assume a Lyapunov function of the form (10) isfound and is defined in X ⊆ Rn. Let the condition number of Pbe �P = �max(P )/�min(P ). Let Nij (k, T ) be defined as in (24)where Z(k, T ) = supkT � t �kT +T ‖z(t)‖. Define

Nij (T ) = �mij+ KT (AZ(T ) + b) (25)

where

Z(T ) = limL→∞ max

k∈{0,...,L} Z(k, T ) (26)

and

A = maxi=1,...,M

‖Ai‖, b = maxi=1,...,M

‖bi‖,

B = maxi=1,...,M

‖Bi‖, �K = maxi,j=1,...,M

‖Kj − Ki‖.

Furthermore, let N(T ) = maxi,j=1,...,M(Nij ). Define

�(T ) = 2�P B

� − 2�P B�K

N(T )

and the region

S(T ) = {z ∈ X | ‖z‖��(T )}for any positive constant < 1 that verifies

�K <�

2�P B. (27)

Then, in the absence of sliding modes, if (27) is verified itfollows that:

(1) The trajectories of the closed-loop sampled-data system(8) converge exponentially to the set

(T ) = {z ∈ X |V (z)��max(P )�2(T )}.

(ii) When T → 0, the trajectories of the closed-loop sampled-data system (8) are practically exponentially stable to theorigin. By this it is meant that z(t) → 0 exponentially a.e.when T → 0.

Proof. Result (1) follows directly from the proof of Theorem 2.Result (2) follows from the facts that:

In the absence of sliding modes, chattering phenomena isruled out in closed-loop. Therefore, for any finite t�, there willbe a finite number N�(t�, T ) of switchings in the time period[0, t�].

At t = kT the dynamics of the system will be governed by(6) where j is the index of the region where the state lies ator immediately after kT. Notice that (6) is an affine differen-tial equation with constant coefficients and finite escape timesare not possible, so z(t) is bounded. Notice also that until aswitch occurs, the solution of (6) is continuous and given by

the variation of constants

z(t) = �j (t − kT )z(kT ) + j (t − kT )uj ,

uj = bj + Bj (Kjz(kT ) + mj),

where �j (t −kT )=eAj (t−kT ) and j (t −kT )=∫ t−kT

0 eAj � d�.Note also that

‖z(t) − z(kT )‖�‖�j + jBjKj − I‖Z(T ) + ‖ j‖‖bj‖and thus ‖z(t) − z(kT )‖ is also bounded since Z(T ) muststay bounded due to the impossibility of finite escape times forPWA systems with constant coefficients not dependent on thepartition. Furthermore, the first switching outside of region jwill only occur when wji(z)=hT

jiz−gji =0 for some i ∈ Nj .Since z(t) and wji(z) are continuous and wji(z) �= 0 rightafter the update of the state at t = kT , wji(z) �= 0 for at leasta time interval with some positive measure �ij �T until theoccurrence of the next switch. Because of this fact, N�(t�, T )

cannot grow unbounded as T → 0. Until the occurrence of aswitch, �mij

= 0, �Kij= 0.

From the previous point, we conclude that for anyT > 0, k�0 the Lebesgue measure of the set

Si,jkT = {t ∈ [kT , (k + 1)T ] |�Kij

(t), �mij(t) > 0}

verifies

�(Si,jkT )�T − �ij (T ) < T .

Thus, for any finite t� the set S= ⋃k:kT <t�

Si,jkT will have bounded

Lebesgue measure �(S)�N�T . As T → 0, the Lebesguemeasure of this set will also converge to zero because N�(t�, T )

cannot grow unbounded as T → 0.Therefore �Kij

→ 0, �mij→ 0 as T → 0 except possibly

on a set of time instants that has Lebesgue measure convergingto zero. Thus, the set of times t for which i(t), j (t) are differentconverges to zero almost everywhere as T → 0 so i = j a.e.when T → 0.

Z(k, T ) → ‖z(kT )‖ when T → 0 and, as seen before,�mij

→ 0 a.e. as T → 0. This together with the fact that‖z(kT )‖ is bounded for any k�0 implies by (25) and (26) thatNij (T ) → 0 a.e. when T → 0. Thus �(T ) → 0 a.e. whenT → 0.

Following the rationale in the proof of Theorem 2, the pre-vious points show that the closed-loop system trajectories con-verge exponentially to the set (T ) whose size converges tozero as �(T ) → 0 a.e. when T → 0. In fact, following the ar-guments of the proof of Theorem 2, when T → 0 the Lyapunovfunction V (z) decreases exponentially except possibly for a setof times whose Lebesgue measure converges to zero as T →0. However, for this set of times, as discussed, z(t) remainsbounded and, by expression (20), z(t)−z(kT ) remains boundedand is small for small T. Since the Lyapunov function is of classC2 on z, the Lyapunov function also remains bounded. Giventhat the Lyapunov function remains bounded for these sets oftime and these time instants form a set whose Lebesgue measure

converges to zero as T → 0, V (t) < V (t0)e−(1−)�−1P �(t−t0),

1254 L. Rodrigues / Automatica 43 (2007) 1249–1256

a.e. and the result of the theorem is established by the fact that�min(P )‖z(t)‖2 �V (z)��max‖z(t)‖2. �

Remark 7. This result formally establishes the very importantand desired property that a sampled-data PWA system con-verges to a closed-loop continuous-time PWA system when thesampling period converges to zero. As desired, all the stabilityguarantees for the closed-loop continuous-time system can berecovered.

Remark 8. The result assumes the absence of sliding modes.Sliding modes can indeed be ruled out in feedback for PWA sys-tems with hyperplane boundaries if the component of the vectorfields perpendicular to the boundaries is continuous across theboundaries. This idea was first suggested for PWA systems inRodrigues and How (2003) to avoid the generation of slidingmodes in closed-loop. If the feedback construction suggested inRodrigues and How (2003) is used, it can be shown followingthe reasoning explained in Rodrigues and How (2003) that slid-ing modes are still ruled out in feedback for sampled-data PWAsystems if the additional constraints Bi = Bj = B, cT

ijB(Kj −Ki +mj −mi)=0, ∀i =1, . . . , M,∀j ∈ Ni are verified. No-tice that these constraints are linear in the controller parametersand can easily be included in the optimization procedure sug-gested in Rodrigues and How (2003) for systems with a con-stant input matrix B (such as the one presented in the exampleof the next section).

Remark 9. Note that for the case of continuous PWA systems,the continuous vector field from the state equation (2) given byf (z, u)=Aiz+bi+Biu and f (z(kT ), u(kT ))=Ajz(kT )+bj +Bju(kT ) is locally Lipschitz in z with Lipschitz constant L =maxi=1,...,M ‖Ai‖. In this case, following the ideas presentedin Khalil (2004), the Gronwall–Bellman inequality applied tothe integral of the dynamical equation (2) between kT andt �kT + T ,

z(t) = z(kT ) + (t − kT )f (z(kT ), u(kT ))

+∫ t

kT

[f (z(�), u(kT )) − f (z(kT ), u(kT )] d�

enables us to show that

‖z(t)−z(kT )‖� 1

L[e(t−kT )L−1] [‖Ajz(kT )+bj+Bju(kT )‖],

kT � t �kT + T .

When the control input is replaced by its value u(kT ) =Kjz(kT ) + mj , it finally yields the bound

‖z(t) − z(kT )‖� 1

L[eT L − 1][‖Aj‖‖z(kT )‖ + ‖bj‖], (28)

where we have used the fact that t − kT �T for kT � t �kT +T and Aj , bj are defined as before. Following the reasoningleading to (24) a new value for Nij (k, T ) can be found as

Nij (k, T ) = �mij+ K

L[eT L − 1][‖Aj‖‖z(kT )‖ + ‖bj‖]. (29)

Note that (24) and (29) become very similar (A, b are replacedby ‖Aj‖, ‖bj‖) for very small T if the Euler approximation isused in (24) to approximate Z(k, T ) by ‖z(kT )‖. Note, how-ever, that (24) is more general and less restrictive than (29) be-cause it is valid even for discontinuous PWA systems that aretherefore not locally Lipschitz. The important point to makefrom expression (29) is that when T → 0, Nij (T ) → 0 a.e.since ‖z(kT )‖ is bounded and z(kT ) → z(t) so that i, j be-come the same, except on a set of measure zero. This leads tothe same result obtained in Corollary 6 when one replaces (24)by (29) for the special case of continuous PWA systems.

4. Example

The objective of this example is to design a controller thatforces a cart on the x–y plane to follow the straight line y =0 with a constant velocity U0 = 1 m/s. It is assumed that acontroller has already been designed to maintain a constantforward velocity. The cart’s path is then controlled by the torqueu about the z-axis according to the following dynamics:⎡⎢⎣

�

r

y

⎤⎥⎦ =

⎡⎢⎣

0 1 0

0 − kI

0

0 0 0

⎤⎥⎦⎡⎢⎣

�

r

y

⎤⎥⎦ +

⎡⎢⎣

0

0

U0 sin(�)

⎤⎥⎦ +

⎡⎢⎣

0

1I

0

⎤⎥⎦ u, (30)

where � is the heading angle with time derivative r, I =1 kg m2

is the moment of inertia of the cart with respect to the centerof mass and k = 0.01 N ms is the damping coefficient. Notethat for this example Bi = Bj = B, cT

ij = [1 0 0], cTijB = 0.

The state of the system is (z1, z2, z3) = (�, r, y). Assume thetrajectories can start from any possible initial angle in the range�0 ∈ [−3�/5, 3�/5] and any initial distance from the line.The function sin(�) is approximated by a PWA function (seeRodrigues & How, 2001) yielding

R1 ={z ∈ R3 | z1 ∈

(−3�

5, −�

5

)},

R2 ={z ∈ R3 | z1 ∈

(−�

5, − �

15

)},

R3 ={z ∈ R3 | z1 ∈

(− �

15,

�

15

)}and R4 is symmetric to R2 and R5 is symmetric to R1, all withrespect to the origin. A controller was designed to stabilize theorigin (inside region R3) yielding

K1 = [−49.908 − 9.467 − 13.926], m1 = 2.70 × 10−6,

K2 = [−48.316 − 9.330 − 13.812], m2 = 3.75 × 10−7,

K3 = [−50.148 − 9.468 − 13.742], m3 = 0.00 × 100,

K4 = [−48.316 − 9.330 − 13.812], m4 = −m2 × 100,

K5 = [−49.908 − 9.468 − 13.926], m5 = −m1 × 100.

The trajectory in the x–y plane using this controller is shown inFig. 1 where it is clear that the controller makes the cart trajec-tory converge to the desired straight line. For a sampling period

L. Rodrigues / Automatica 43 (2007) 1249–1256 1255

-10 -8 -6 -4 -2 0 2 4 6 8 10

0

2

4

6

8

10

12

14

16

18

y(m)

x(m)

Fig. 1. x–y trajectory with continuous-time controller, �0 =�/2, r0 =0 rad/s,y0 = 1 m.

-10 -8 -6 -4 -2 0 2 4 6 8 10

0

2

4

6

8

10

12

14

16

18

y(m)

x(m)

Fig. 2. x–y trajectory for a sampling period of T = 0.05 s.

-0.005 0 0.005 0.01 0.015 0.02 0.025

12

14

16

18

20

22

24

26

28

Fig. 3. x–y trajectory zoomed for a sampling period of T = 0.2 s.

of T =0.05 s the same controller was emulated in discrete-timebetween a sampler and a zero-order-hold and the results of thecorresponding x–y trajectory are shown in Fig. 2. It can be seen

that the trajectory still follows approximately the one obtainedwith the continuous-time controller. When the sampling periodis further increased to T = 0.2 s the simulation of the x–y tra-jectory close to the line is zoomed in Fig. 3. It is clear thatthe trajectory converges to a region around the desired straightline, as predicted by the results of this paper.

5. Conclusions

This paper has presented stability results for closed-loopsampled-data PWA systems under state feedback. It was shownthat the emulation of a state feedback controller designed incontinuous-time to exponentially stabilize the system to a targetpoint would still exponentially stabilize the system to a regionaround the target point. The size of this region was related tothe sampling period. It was shown that when the sampling pe-riod converges to zero the exponential stability results for theclosed-loop continuous-time system are recovered.

Acknowledgments

The author would like to thank the Associate Editor andthe anonymous reviewers for the detailed and interesting sug-gestions, especially regarding the proof of the corollary, thatgreatly improved the quality of the paper. The author wouldlike to also thank Daniel Liberzon and João Hespanha for in-teresting discussions on this topic. Finally, the author wouldlike to acknowledge the Natural Sciences and Engineering Re-search Council of Canada (NSERC) for partially funding thisresearch.

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Luis Rodrigues earned his “licenciatura” andhis M.Sc. degrees in Electrical Engineering andComputers from IST, Technical University ofLisbon. He obtained his Ph.D. in Aeronauticsand Astronautics from Stanford University in2002. He worked as a consultant in speech mod-eling and recognition for Eliza Corporation inUSA and as a project manager for Ydreamsin Portugal before joining Concordia as an As-sistant Professor in 2003. His research inter-ests lie in the areas of switched, hybrid andoptimal control with applications to aerospace,

automotive, manufacturing and biological systems (such as the vocal tract).