Stabilisation and control synthesis of switching systems subject to actuator saturation

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Stabilisation and control synthesis of switching systemssubject to actuator saturationAbdellah Benzaouia a , Ouassima Akhrif b & Lahcen Saydy ca Faculty of Science Semlalia , EACPI, University Cadi Ayyad , BP 2390, Marrakech, Moroccob Département de Génie Electrique , Ecole de Technologie Supérieure , 1100, Rue Notre-Dame Ouest, Montréal, Québec, Canada H3C 1K3c Electrical Engineering Department , Ecole Polytechnique de Montréal , P.O. Box 6079,Montréal, Québec, Canada H3C 3A7Published online: 16 Mar 2010.

To cite this article: Abdellah Benzaouia , Ouassima Akhrif & Lahcen Saydy (2010) Stabilisation and control synthesisof switching systems subject to actuator saturation, International Journal of Systems Science, 41:4, 397-409, DOI:10.1080/00207720903045791

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International Journal of Systems ScienceVol. 41, No. 4, April 2010, 397–409

Stabilisation and control synthesis of switching systems subject to actuator saturation

Abdellah Benzaouiaa*, Ouassima Akhrif b and Lahcen Saydyc

aFaculty of Science Semlalia, EACPI, University Cadi Ayyad, BP 2390, Marrakech, Morocco; bDepartement de GenieElectrique, Ecole de Technologie Superieure, 1100, Rue Notre-Dame Ouest, Montreal, Quebec,Canada H3C 1K3; cElectrical Engineering Department, Ecole Polytechnique de Montreal,

P.O. Box 6079, Montreal, Quebec, Canada H3C 3A7

(Received 18 December 2007; final version received 28 November 2008)

This article presents sufficient conditions for the stabilisation of switching discrete-time linear systems subject toactuator saturations. These conditions are obtained by using successively state and output feedback control laws.The obtained results are formulated in terms of linear matrix inequalities (LMIs). The saturating and non-saturating controllers are synthesised for both cases in this work. Three sets of LMIs are presented for outputfeedback case. Numerical examples are used to illustrate these techniques by using a linear optimisation problemsubject to LMI constraints.

Keywords: switching systems; actuator saturations; state feedback control; output feedback control; multipleLyapunov functions; LMIs

1. Introduction

Switched systems are a class of hybrid systems

encountered in many practical situations, which involve

switching between several subsystems depending on

various factors. Generally, a switching system consists

of a family of continuous-time subsystems and a rule

that supervises the switching between them. This class

of systems has numerous applications in the control of

mechanical systems, the automotive industry, aircraft

and air traffic control (Zhu, Akhrif and Hentabli 2000),

switching power converters and many other fields.

Although this class of systems can be seen as a

particular case of linear parameter varying (LPV)

systems, it has specific characterisations: the first is

that the switch occurs between a finite number of

subsystems; the second is that the switching sequence,

which plays an important rule in the problem, has to be

taken into account as is the case in practical situations.

For instance, one can only act on this sequence to

stabilise the switched system. For these reasons, this

subject constitutes an important area of research

independently of LPV systems. Hence, two main

problems are widely studied in the literature according

to the classification given in Blanchini and Savorgnan

(2006): the first one deals with the obtention of testable

conditions that guarantee the asymptotic stability of a

switching system under arbitrary switching rules, while

the second is to determine a switching sequence that

renders the switched system asymptotically stable

(Liberzon and Morse 1999 and the reference therein).Also, for hybrid systems described by switched lineardifference equations, Bemporad (2004) proposes algo-rithms for transforming such systems described inmixed logical dynamical form into an equivalentpiecewise affine form.

A main problem that is always inherent to alldynamical systems is the presence of actuators satura-tions. Even for linear systems this problem has been anactive area of research for many years. Two mainapproaches have been developed in the literature: thefirst is the so-called positive invariance approach thatis based on the design of controllers, which work insidea region of linear behaviour where saturations do notoccur (Benzaouia and Burgat 1988; Benzaouia andHmamed 1993; Blanchini 1999 and the referencestherein). This approach was applied to a class of hybridsystems involving jumping parameters (Boukas andBenzaouia 2002). It was also used to design controllersfor switching systems with constrained control givenunder complete modelling as reset functions anddifferent system’s dimension (Benzaouia, DeSantis,Caravani and Daraoui 2007). The second approach,however, allows saturations to take effect whileguaranteeing asymptotic stability (Nguyen andJabbari 1999, 2000; Tarbouriech, Prieur and Gomesda Silva 2006; Hu, Lin and Chen 2002a; Hu andLin 2002; Benzaouia, Tadeo and Mesquine 2006; Caoand Lin 2003 and the references therein). The mainchallenge in these two approaches is to obtain a large

*Corresponding author. Email: [email protected]

ISSN 0020–7721 print/ISSN 1464–5319 online

� 2010 Taylor & Francis

DOI: 10.1080/00207720903045791

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enough domain of initial states, which ensures asymp-

totic stability for the system despite the presence of

saturations (Gilbert and Tan 1991; Benzaouia and

Baddou 1999; Benzaouia, Baddou and Elfaiz 2002; Hu,

Lin and Chen 2002b).The objective of this article is to study a switching

system subject to actuator saturations by following two

methods: the first concerns the synthesis of non-

saturating controllers (controllers working inside alarge region of linear behaviour where the saturations

do not occur), while the second extends the resultsobtained for unsaturated switching systems by

Mignone, Ferrari-Trecate and Morari (2000), Ferrari-Trecate, Cuzzola, Mignone and Morari (2001) and

Daafouz, Riedinger and Iung (2001, 2002) leading tosaturating controllers (controllers tolerating saturations

to take effect). The second method was firstly used in

Benzaouia, Saydy and Akhrif (2004) with the use of acomposite Lyapunov function Vðt, xÞ ¼ xTðtÞ

ðPN

i¼1 �iðtÞPiÞxðtÞ. However, only the intersection of allthe corresponding level sets of the functions Viðt, xÞ ¼

xTðtÞ � PixðtÞ, i ¼ 1, . . . ,N was considered as a regionof asymptotic stability of the switching system. This

drawback was improved in Benzaouia, Akhrif andSaydy (2006) by considering a large set of asymptotic

stability composed by the union of all the level sets. Inthis work, the main contribution is to prove that the

same conditions also allow that the union of all the

corresponding level sets of functions Viðt, xÞ ¼xTðtÞPixðtÞ, i ¼ 1, . . . ,N constitutes a large region of

asymptotic stability of the switching system for bothstate feedback control and output feedback control.

In this context, two main different sufficient

conditions of asymptotic stability are obtained forswitching systems subject to actuator saturations.

Furthermore, these conditions are presented in theform of three sets of linear matrix inequalities (LMIs)

for the state feedback control case.A particular attention is given to the output

feedback case, which has an additive complexity due

to the output equation. For this class of switchingsystems, the problem has already been studied in

Daafouz et al. (2001, 2002) and extended to positiveswitching systems by Benzaouia and Tadeo (2008).

Recently, Bara and Boutayeb (2006) has presented a

promising technique based on a certain decompositionof the input-to-state matrix B. In this sense, three

different LMIs are presented for this case. It is alsoshown that the LMIs obtained for computing con-

trollers working inside a large region of linear behav-iour are less conservative.

The article is organised as follows: the second

section deals with the problem formulation while thethird section presents some preliminary results.

The main results of this work together with numerical

examples are presented in the fourth section.

2. Problem formulation

In this section, we give a more precise problemstatement for the class of systems under consideration,namely, discrete-time switching linear systems withinput saturation and state or output feedback. Anequivalent description of such systems, based on theindicator function is also used in this work.

Thus, we consider systems described by:

xðtþ 1Þ ¼ A�xðtÞ þ B� satðuðtÞÞ

yðtÞ ¼ C�xðtÞ ð1Þ

where x2Rn is the state, u2R

m is the control, y2Rp is

the output, sat(�) is the standard saturation functionand � a switching rule, which takes its values in thefinite set I :¼ f1, . . . ,Ng, t2Z+. The saturation func-

tion is assumed here to be normalised, i.e.(jsat(u)j ¼min{1, juj}).

Definition 2.1: (Lygeros, Tombin and Sastry 1999) Ahybrid time basis � is an infinite or finite sequence ofsets Is ¼ t 2 N : ts � t � �ts

� �, with �ts ¼ tsþ1 for

s 2 L ¼ 0, . . . ,Lf g, and if cardð�Þ ¼ Lþ 151 then�tL can be finite or infinite.

It is assumed that:

. the switching system is stabilisable;

. matrices C� are of full rank;

. tsþ1 � ts þ 1, 8s 2 L;

. the switching rule is not known a priori but�(t) is available at each t.

The second assumption ensures that at each time only

one subsystem is active. The fourth assumptioncorresponds to practical implementations where theswitched system is supervised by a discrete-eventsystem or operator allowing for �(t) to be known inreal time.

In this work, we are interested by the synthesis ofstabilising controllers for this class of hybrid systemssubject to actuator saturation. We use a feedbackcontrol law as:

uðtÞ ¼ F�xðtÞ ¼ K�yðtÞ, ð2Þ

and write the closed-loop system as

xðtþ 1Þ ¼ A�xðtÞ þ B� satðF�xðtÞÞ:

¼ A�xðtÞ þ B� satðK�yðtÞÞ ð3Þ

Upon introducing the indicator function:

�ðtÞ ¼ ½�1ðtÞ, . . . , �NðtÞ�T ð4Þ

398 A. Benzaouia et al.

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where �i(t)¼ 1 if the switching system is in mode i and�i(t)¼ 0 if it is in a different mode, one can write theclosed-loop system (3) as follows:

xðtþ 1Þ ¼XNi¼1

�iðtÞ½AixðtÞ þ Bi satðFixðtÞÞ�:

¼XNi¼1

�iðtÞ½AixðtÞ þ Bi satðKiCixðtÞÞ� ð5Þ

3. Preliminary results

In this section, we recall two results on which our workis based. Let � be fixed. Then system (3) becomes alinear time-invariant system with input saturationgiven by:

xðtþ 1Þ ¼ AxðtÞ þ B satðFxðtÞÞ ð6Þ

Define the following subsets of Rn:

"ðP, �Þ ¼ fx 2 RnjxTPx � �g, ð7Þ

LðF Þ ¼ fx 2 Rnjj flxj � 1, l ¼ 1, . . . ,mg, ð8Þ

with P a positive definite matrix, �4 0 and fl the l-throw of the matrix F2R

m�n. Thus "(P, �) is an ellipsoidwhile LðF Þ is a polyhedral consisting of states forwhich the saturation does not occur.

Lemma 3.1: (Hu et al. 2002b) For all u2Rm and

�2Rm such that j�lj5 1, l2 [1, m]

satðuÞ 2 cofDsuþD�s v, s 2 ½1, ��g ; � ¼ 2m ð9Þ

where co denotes the convex hull.

Consequently, there exists �1� 0, . . . , �� 0 withP�s¼1 �s ¼ 1 such that,

satðuÞ ¼X�s¼1

�s½DsuþD�s v� ð10Þ

Here, Ds is an m�m diagonal matrix with elementseither 1 or 0 and D�s ¼ Im �Ds: There are 2m possiblesuch matrices. One can also consult the work ofBenzaouia et al. (2006) for more details and otherextensions to linear systems with both constraints onthe control and the increment or rate of the control.

Consider the following autonomous switchingsystem:

xðtþ 1Þ ¼XNi¼1

�iðtÞAixðtÞ ð11Þ

The use of the Lyapunov functions revealed twoways:

. The existence of a common Lyapunov func-tion to the various subsystems guarantees the

asymptotic stability of the switching system.

In general, the search for such a function is

not always obvious (Shorten and Narenda,

1997, 1998).. The Lyapunov-like functions or multiple

Lyapunov functions were introduced in

Branicky (1998). They are considered as a

strong tool in the analysis of the stability of

the hybrid systems and in particular the

switching systems.

Proposition 3.1: (Branicky 1998) If there exists a

Lyapunov-like function V(t, x),2 Is, with V :N�Rn!

R+, such that the following hold:

. DV(t, x(t))5 0, for any time t2 Is

. Vðtk, xðtkÞÞ5Vðtk�1, xðtk�1Þ, 8k 2 L

then, the switching autonomous system (11) is asympto-

tically stable.

We now recall a useful stability result for switching

systems with no input saturations presented by many

authors (see Mignone et al. 2000 and Daafouz et al.

2001, 2002) firstly used for linear time-varying systems

(Daafouz and Bernussou 2001).

Theorem 3.1: The closed-loop switching system (11) is

asymptotically stable at the origin if there exist N

symmetric and definite positive matrices P1, . . . ,PN

satisfying,

ATi PjAi � Pi 5 0, 8 ði, j Þ 2 I � I ð12Þ

A corresponding Lyapunov function for the system is

then given by:

Vðt, xÞ ¼ xTðtÞ

�XNi¼1

�iðtÞPi

�xðtÞ ð13Þ

It is worth noting that function V(t, x), which is a

Lyapunov-like function candidate involving matrices

Pi, can be seen as a standard Lyapunov function

candidate. It was the way followed by Daafouz et al.

(2001, 2002). Further, condition (12) is equivalent, by

using Schur complement to,

Pi ATi Pj

� Pj

" #4 0, 8 ði, j Þ 2 I � I ð14Þ

where * denotes the transpose of the off diagonal

element of the LMI. Subsequently, we will need the

following equivalent LMI representation of (14):

Xi XiATi

� Xj

" #4 0, 8 ði, j Þ 2 I � I ð15Þ

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where Xi ¼ P�1i . The LMI (15) is also in turn equiv-

alent to the following LMI, generally used to relax the

previous one (Daafouz et al. 2001):

Gi þ GTi � Xi GT

i ATi

� Xj

" #4 0, 8 ði, j Þ 2 I � I ð16Þ

where matrices Gi called slack variables are

non-singular matrices and Xi the positive definite

matrices.It is worth noting that condition (12) has to be

satisfied 8 ði, j Þ 2 I � I , in particular for i¼ j. This

means that each mode is necessarily asymptotically

stable and Vi(t,x)¼ xT(t)Pix(t) is the associated

Lyapunov function.Recall that a level set of the Lyapunov function

V(t,x) given by (13) and associated to the switching

system (11) is given by the set "(P, �) defined by (7)

with P ¼PN

i¼1 �iðtÞPi. This region of asymptotic sta-

bility is very difficult to construct. Nevertheless, an

estimate can be obtained as large as possible as

presented in the next section based on the union of

the sub-level ellipsoid sets "(Pi, 1). Other type of level

sets obtained with different Lyapunov functions for

switched systems can be found in Hu, Ma and

Lin (2006).

4. Main results

In this section, the region of local asymptotic stability

associated to the saturated switching system is firstly

studied. The design of a stabilising controller for the

class of switching system with actuator saturation is

then presented in the following two ways: the first

concerns controllers working inside the region of linear

behaviour where the saturations do not occur while the

second is based on Lemma 3.1 tolerating saturations to

occur.

4.1. Region of asymptotic stability

The link between the result of Theorem 3.1 and the

level set (7) is not mentioned in Mignone et al. (2000)

nor in Daafouz et al. (2001, 2002). The absence of

saturation on the control of the switching systems in

these works does not necessitate to take care with the

level sets since the asymptotic stability is global. We

present here after an interesting result on this subject

inspired from the general class studied in Benzaouia

et al. (2007).

Theorem 4.1: If there exists N symmetric matrices

P1, . . . ,PN such that the inequalities (12) are satisfied

then the set given by,

� ¼[Ni¼1

"ðPi, 1Þ ð17Þ

is a level set of the Lyapunov-like function V(t, x) given

by (13) with respect to the trajectories of the autonomous

switching system (11).

Proof: Note that condition (12) for i¼ j ensures that

DV(t,x)5 0, for t 2 Is, s 2 L (Daafouz et al. 2001,

2002). Now let xð�tk�1Þ 2 "ðPi, 1Þ. What happens at the

switching time �tk�1?For this, compute x(tk). With an identity reset

function we have,

xðtkÞ ¼ xð �tk�1Þ ð18Þ

The switch to the next region "(Pj, 1) will be seen an

unit time after,

xðtk þ 1Þ ¼ xð �tk�1 þ 1Þ

¼ Aixð�tk�1Þð19Þ

We have,

xTðtk þ 1ÞPjxðtk þ 1Þ ¼ xTð�tk�1Þ ATi PjAi

� �xð�tk�1Þ ð20Þ

According to condition (12), one can obtain,

xTðtk þ 1ÞPjxðtk þ 1Þ5 xTð�tk�1ÞPixð�tk�1Þ ð21Þ

Since xð �tk�1Þ 2 "ðPi, 1Þ, then, xðtk þ 1Þ 2 "ðPj, 1Þ,

8i, j 2 I � I . This proves that V(tk, x(tk))5V(tk�1,

x(tk�1)), 8k 2 L. Consequently, function V(t, x) is a

Lyapunov-like function of the switching system

according to Proposition 3.1 and the set � is a level

set associated to this function. œ

Note that an important remark is to be made at this

level: since the studied system is a switching one

composed of N subsystems (modes), the switching

system can be initialised inside any level set "(Pi, 1);

however, the initial mode to be selected is the

corresponding mode i.

4.2. State feedback control

We assume that the state is available, so a state

feedback control can be performed. The first result we

present is a simple extension of the known result given

by Gutman and Hagandar (1985) for linear systems

and concerns the synthesis of non-saturating control-

lers ensuring that a large region of linear behaviour is a

region of asymptotic stability.


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X1, . . . ,XN and N matrices Y1, . . . ,YN solutions of the

following LMIs:

Xi ðAiXi þ BiYiÞT

� Xj

" #4 0, ð22Þ

1 yli

� Xi

� 4 0,

8 ði, j Þ 2 I � I , 8l 2 ½1,m�

ð23Þ

where yli is the l-th row of matrix Yi ; then the switching

system with saturations in closed loop (3), with,

Fi ¼ YiX�1i , Pi ¼ X�1i ð24Þ

is asymptotically stable at the origin 8x02� and for any

sequences of switching �(t).

Proof: In order to guarantee that the control by state

feedback is always admissible, each sub-level set of

the Lyapunov function has to be contained inside the

polyhedral set where the saturations do not occur

"ðPi, 1Þ LðFiÞ, 8i 2 I (Gutman and Hagandar 1985).

Using Boyd, El Ghaoui, Feron and Balakrishnan

(1994), this inclusion condition can also be trans-

formed to the equivalent LMI (23) by letting Xi ¼ P�1i

and FiXi¼Yi. Let x(t)2�, 8t2 Is. According to

condition (23), the system in closed loop (5) can be

equivalently written as,

xðtþ 1Þ ¼XNi¼1

�iðtÞAcixðtÞ: ð25Þ

with, Aci¼AiþBiFi. The proof follows from the

asymptotic stability conditions of the switching

system given by Theorem 3.2, which is expressed

equivalently by (15), with matrix Aci instead of Ai.

Further, Theorem 4.1 ensures that the set � is a set

of asymptotic stability of the switching system with

saturations in closed loop (3). œ

To achieve a domain of attraction as large as

possible, we can solve the following optimisation

problem:

ðPb:1Þ :

supðXi,YiÞTraceðXiÞ

s:t: ð22Þ, ð23Þ,

i ¼ 1, . . . ,N

8><>:

When this optimisation problem is feasible, the

obtained ellipsoid volumes are maximum with respect

to the data of the system.The obtained LMIs (22) of Theorem 4.2 are similar

to those obtained in Mignone et al. (2000) and Ferrari-

Trecate et al. (2001) for non-saturated switching

systems. The presence of saturation on the control in

our problem leads to the additional LMIs (23), which

will obviously restrict the set of solutions.

Nevertheless, the associated large region of asymptotic

stability � for the saturated switching system enables

one to conclude that these LMIs are not conservative.The second result we present concerns the synthesis

of saturating controllers tolerating saturations to take

effect inside a large region of asymptotic stability.

Theorem 4.3: If there exist symmetric matrices

P1, . . . ,PN2Rn�n and matrices H1, . . . ,HN2R

m�n

such that,

Pi ½Ai þ BiðDisFi þD�isHiÞ�TPj

� Pj

" #4 0,

8 ði, j Þ 2 I � I , 8s 2 ½1, �� ð26Þ

and,

"ðPi, 1Þ LðHiÞ, 8i 2 I ð27Þ

then, the closed-loop switching system (3) is asymptoti-

cally stable at the origin 8x02� and for all sequences of

switching �(t).

Proof: Assume that there existsNmatricesH1, . . . ,HN

and N symmetric matrices P1, . . . ,PN such that condi-

tion (26) and (27) are satisfied. Using the expression in

(10) and rewriting System (3) as in (5) yields that:

satðFixðtÞÞ ¼X�s¼1

�siðtÞ½DisFi þD�isHi�xðtÞ; ð28Þ

�siðtÞ � 0,X�s¼1

�siðtÞ ¼ 1 ð29Þ

and, subsequently:

xðtþ 1Þ ¼X�s¼1

XNi¼1

�iðtÞ�siðtÞAcisxðtÞ;

Acis :¼ Ai þ BiðDisFi þD�isHiÞ, s 2 ½1, �� ð30Þ

The rate of change of the Lyapunov function candidate

(13) along the trajectories of (30) is given by:

DVðt,xðtÞÞ ¼ xTðtþ 1Þ

�XNj¼1

�j ðtþ 1ÞPj

�xðtþ 1Þ

� xTðtÞ

�XNi¼1

�iðtÞPi

�xðtÞ

¼ xTðtÞX�s¼1

XNi¼1

�iðtÞ�siðtÞAcis

!T24

�XNj¼1

�j ðtþ 1ÞPj

! X�s¼1

XNi¼1


!

�XNi¼1

�iðtÞPi

#xðtÞ ð31Þ


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Let condition (26) be satisfied. At this level, for each i,

multiply the j¼ 1, . . . ,N inequalities (26) by �j(tþ 1)

and sum. Multiply the resulting i¼ 1, . . . ,N inequal-

ities by �i(t) and sum. Multiply again the resulting

s¼ 1, . . . , � inequalities by �s(t) and sum. AsPNi¼1 �iðtÞ ¼

PNj¼1 �j ðtþ 1Þ ¼

P�s¼1 �siðtÞ ¼ 1, one gets:

PNi¼1 �iðtÞPi �

PNj¼1 �j ðtþ 1ÞPj

�PN

j¼1 �j ðtþ 1ÞPj

" #4 0, ð32Þ

with � ¼ ½P�

s¼1

PNi¼1 �iðtÞ�siðtÞAcis�

T. The use of Schur

complement allows us to write condition (32) under the

equivalent form,

X�s¼1

XNi¼1


!T XNj¼1

�j ðtþ 1ÞPj

!

�X�s¼1

XNi¼1


!�XNi¼1

�iðtÞPi 5 0,

8 ði, j Þ 2 I � I , 8s 2 ½1, �� ð33Þ

Letting be the largest eigenvalue among all the above

matrices, we obtain that

DVðt, xðtÞÞ � xTðtÞxðtÞ5 0: ð34Þ

which ensures the desired result. Noting that condition

(26) is also satisfied for i¼ j, this implies that each set

"(Pi, 1) is a contractively invariant set for the corre-

sponding subsystem. Further, by taking account of

condition (27), one can guarantee that for every

xo 2 "ðPi, 1Þ LðHiÞ, each subsystem is asymptotically

stable at the origin. Besides, following Theorem 4.1,

conditions (26) and (27) also allow for a state

belonging, before the switch, to a set "ðPi, 1Þ LðHiÞ,

if a switch occurs at any time tk, that the switch will

handle the state to the desired set "ðPj, 1Þ LðHj Þ.

That means that the set � is a set of asymptotic

stability of the switching system. œ

This stability result can be used for control

synthesis as follows.


X1, . . . ,XN and 2N matrices Y1, . . . ,YN and Z1, . . . ,ZN

solutions of the following LMIs:

Xi ðAiXi þ BiDisYi þ BiD�isZiÞ

T

� Xj

" #4 0, ð35Þ

1 zli

� Xi

� 4 0,

8 ði, j Þ 2 I � I , 8s 2 ½1, ��, 8l 2 ½1,m�

ð36Þ

then the switching system with saturations in closed loop

(3), with,

Fi ¼ YiX�1i , Hi ¼ ZiX

�1i , Pi ¼ X�1i ð37Þ

is asymptotically stable at the origin 8x02� and for any

sequences of switching �(t).

Proof: The Inequalities (26) can be transformed

equivalently by Schur complement to the following:

Pj Ai þ BiðDisFi þD�isHiÞ �� P�1i Ai þ BiðDisFi þD�isHiÞ

�TPj � Pj 5 0 ð38Þ

By pre and post-multiplying the latter by P�1j , it

follows:

Ai þ BiðDisFi þD�isHiÞ �� P�1i Ai þ BiðDisFi þD�isHiÞ

�T�P�1j 5 0 ð39Þ

Pose Xi ¼ P�1i , inequality (39) can then be rewritten as,

Ai þ BiðDisFi þD�isHiÞ �� Xi Ai þ BiðDisFi þD�isHiÞ

�T�Xj 5 0,

that is

AiXi þ BiðDisFiXi þD�isHiXiÞ �� X�1i AiXi þ BiðDisFiXi þD�isHiXiÞ

�T�Xj 5 0:

The use of the Schur complement a second time

leads to:

Xi ðAiXi þ BiDisFiXi þ BiD�isHiXiÞ

T

� Xj

" #4 0 ð40Þ

By letting FiXi¼Yi and HiXi¼Zi, the LMI (35)

follows readily. Using Boyd et al. (1994), the inclusion

condition (27) can also be transformed to the equiv-

alent LMI (36). œ

To obtain larger ellipsoid domains "(Pi, 1), we can

apply the following optimisation problem:

ðPb:2Þ :

supðXi,Yi,ZiÞTraceðXiÞ

s:t: ð35Þ, ð36Þ,

i ¼ 1, . . . ,N

8><>:

When this optimisation problem is feasible, the

obtained ellipsoid volumes are maximum with respect

to the data of the system.

Remark 4.1: The results presented in this section can

be extended easily to level sets "(Pi, �i) where the

scalars �i, i¼ 1, . . . ,N can be priori fixed, by using

the variables Xi¼Pi/�i in the LMIs (22)–(23) and

(35)–(36).


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Comment 4.1: The resolution of the LMIs (35) for

s2 [1, �] may be very restrictive. One can relax this

resolution by accepting controllers working in a region

of linear behaviour where the saturations do not occur.The obtained results in this work, where the switching

system is with saturated control, are a set of

(2m�N2þNm) LMIs for computing stabilising con-

trollers working inside the saturated regions LðHiÞ or

only a set of (N2þNm) LMIs for computing stabilising

controllers working inside regions of linear behaviour

LðFiÞ. Note also that the LMIs of Theorem 4.2 can beobtained from LMIs of Theorem 4.4 by letting Dis ¼ I

and D�is ¼ 0.

Example 4.1: Consider a numerical switching dis-

crete-time system with saturation specified by the two

subsystems:

A1 ¼�0:7 1

�0:5 �1:5

� ; B1 ¼

1

0

� ;

A2 ¼0:9 �1

1:7 �1:5

� ; B2 ¼

0

�1

� :

We have to solve 5 LMIs with 4 variables to computecontrollers working inside a region of linear behavior.

The use of the Matlab LMI Toolbox to check our

conditions leads to the following results without usingthe optimisation problem (Pb.1).

P1 ¼7:3328 18:7125

18:7125 55:9711

" #; F1 ¼ 1:2244 0:7535

�;

Ac1 ¼0:5244 1:7535

�0:5000 �1:5000

" #;

P2 ¼4:8237 �5:1001

�5:1001 5:6715

� ; F2 ¼ 1:7559 �1:5699

�;

Ac2 ¼0:9000 �1:0000

�0:0559 0:0699

� ;

Figure 1 presents the two ellipsoid sets of invariance

and contractivity "(Pi, 1) together with the sets oflinear behaviour LðFiÞ computed with LMIs (22)–(23).

For the same example, the use of the Matlab LMI

Toolbox to check conditions (35)–(36) leads to the

following results by using the optimisation problem(Pb.2) with 10 LMIs and 6 variables.

P1 ¼7:4032 18:1841

18:1841 56:6345

" #; H1 ¼ 1:2534 0:8569

�;

F1 ¼ 1:2641 0:8526 �

; Ac1 ¼0:5641 1:8526

�0:5000 �1:5000

" #;

P2¼4:5202 �4:3908

�4:3908 6:5909

� ; H2¼ 1:7693 �1:5845

�;

F2¼ 1:7697 �1:5836 �

; Ac2¼0:9000 �1:0000

�0:0697 0:0836

� ;

Note that the optimal values of the optimisation

problem (Pb.2) are given by TraceðP�11 Þ ¼ 0:7227 and

TraceðP�12 Þ ¼ 1:0568.Figure 2 presents the two ellipsoid sets of invar-

iance and contractivity "(Pi,1) together with the set of

saturations LðHiÞ computed with (Pb.2).Comparing the results obtained by LMIs (35)–(36)

and LMIs (22)–(23), one can note that the matrices Hi

–3 –2 –1 –0 1 2 3–4

–3

–2

–1

0

1

2

3

4

x1

x 2

Figure 1. The ellipsoids sets of invariance and contractivityfor the switching discrete-time linear system computed withLMIs (22)–(23).

–1.5 –1 –0.5 0 0.5 1 1.5–4

–3

–2

–1

0

1

2

3

4

Figure 2. The ellipsoids sets of invariance and contractivityfor the switching discrete-time linear system computed withLIMI’s (35)–(36) and (Pb.2).


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are very closed up to matrices Fi obtained with LMIs(35)–(36), that is, LðHiÞ ffi LðFiÞ. This means that in thecase of this example, the saturations allowed by thistechnique are not very important. Recall that insidethe sets LðFiÞ no saturations occur. Furthermore, theresults obtained with LMIs (22)–(23) are less con-servative. This is due to the fact that among the numberof LMIs to be solved in (35)–(36), one find the LMIs(22)–(23). In conclusion of the study of this example,a compromise has to be taken between a less conser-vative synthesis of non-saturating controllers and amore conservative synthesis of saturating controllers.

4.3. Output feedback control

Static output-feedback control plays a very importantrole in control applications: the purpose is to designcontrollers such that the resulting closed-loop system isasymptotically stable without using any reconstructionmethod of the unavailable states. In this section, webegin by the synthesis of the saturating controllerssince the non-saturating ones can be derived as aparticular case.

The synthesis of the stabilising controller by outputfeedback control for the class of switching system withactuator saturation is presented by applying the resultsof Theorem 4.3.

Theorem 4.5: If there exist symmetric matrices Xi,matrices Vi, Yi and Zi solutions of the following LMIs:

Xi ðAiXi þ BiDisYiCi þ BiD�isZiÞ

T

� Xj

" #4 0, ð41Þ

1 Zil

� Xi

� 4 0, ð42Þ

ViCi ¼ CiXi

8 ði, j Þ 2 I2, 8s 2 ½1, ��, 8l 2 ½1,m�ð43Þ

then the closed-loop saturated switching system (3) with

Ki ¼ YiV�1i , Hi ¼ ZiX

�1i , Pi ¼ X�1i ð44Þ

is asymptotically stable 8x02� and for all switchingsequences �(k).

Proof: The same development as (38)–(40) whilereplacing Fi by KiCi, can be followed to obtain,

Xi ðAiXi þ BiDisKiCiXi þ BiD�isHiXiÞ

T

� Xj

" #4 0:

According to Equation (43), we have KiCiXi¼KiViCi.By letting KiVi¼Yi and HiXi¼Zi, the LMI (41)follows together with relations (44). Similarly, the

inclusion condition (27) can also be transformed to the

equivalent LMI (42). Finally, the inequality (41)

ensures that the obtained solutions Xi are positive

definite while equalities (43) guarantee that matrices Vi

are non-singular.It is worth noting that the state feedback follows

readily from Theorem 4.5 by letting Ci ¼ Im. In this

case, Vi¼Xi.The LMI (41), relating matrices Ci to matrix Xi by

means of Equation (43), can be relaxed by using the

LMI (16), where the new variables Gi are related to

matrices Ci instead of the matrix Xi. œ

Corollary 4.1: If there exist symmetric matrices Xi,

matrices Gi,Vi,Yi and Zi solutions of the following

LMIs:

Gi þ GTi � Xi �T

is

� Xj

" #4 0, ð45Þ

1, Zil

� Gi þ GTi � Xi

� 4 0, ð46Þ

ViCi ¼ CiGi ð47Þ

8 ði, j Þ 2 I2, 8s 2 ½1, ��, 8l 2 ½1,m�

where �is ¼ ðAiGi þ BiDisYiCi þ BiD�isZiÞ, then the

closed-loop saturated switching system (3) with

Ki ¼ YiV�1i , Hi ¼ ZiG

�1i , Pi ¼ X�1i ð48Þ

is asymptotically stable 8x02� and all switching

sequences �(k).

Proof: The proof uses the equivalent LMI (16) and

is similar to that of Theorem 4.5. However, the

inclusion condition "ðPi, �Þ LðHiÞ 8i 2 I holds if

1�HilXiHTil 4 0, 8l 2 ½1,m�, which is equivalent to,

1� ðHiGiÞl ðGTi X�1i GiÞ

�1ðHiGiÞ

Tl 4 0. That is, by virtue

of (48) 1� ðZilÞðGTi X�1i GiÞ

�1ðZilÞ

T 4 0. Since,

ðGi � XiÞT X�1i ðGi � XiÞ � 0, then GiX

�1i GT

i � Giþ

GTi � Xi. It follows that 1� ZilðGi þ GT

i � XiÞ �

ZTil 4 0. By Schur complement, the LMI (46) is

obtained. œ

To achieve a domain of attraction as large as

possible, we can solve a similar optimisation problem

as for state feedback control.Another way to deal with the problem by relaxing

the more constraining relations (43) of Theorem 4.5,

especially in MIMO case, is presented by the following

result which uses the same idea as in El Ghaoui, Oustry

and AitRami (1997) and Chadli, Maquin and Ragot

(2002).


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Theorem 4.6: If there exist symmetric matrices Xi and

Si, matrices Yi and Zi solutions of the following LMIs:

Xi ðAi þ BiDisYiCi þ BiD�isZiÞ

T

� Sj

" #4 0, ð49Þ

1 Zil

� Xi

� 4 0, ð50Þ

Xi, I

� Si

� � 0 ð51Þ

8 ði, j Þ 2 I2, 8s 2 ½1, ��, 8l 2 ½1,m�

such that Trace (XiSi)¼ n, then the closed-loop saturatedswitching system (3) with

Ki ¼ Yi, Hi ¼ Zi, Pi ¼ Xi ð52Þ

is asymptotically stable 8x02� and all switchingsequences �(t).

Proof: The inequality (26) can be equivalently trans-formed via the Schur Complement to the following:

Ai þ BiðDisKiCi þD�isHiÞ �T� Pj Ai þ BiðDisKiCi þD�isHiÞ

�� Pi 5 0 ð53Þ

The use of the Schur complement a second time leadsto

Pi ðAi þ BiDisKiCi þ BiD�isHiÞ

T

� P�1j

" #4 0 ð54Þ

By letting Xi¼Pi, Sj ¼ P�1j , Yi¼Ki and Zi¼Hi, theLMI (49) follows together with relations (52). The

inclusion condition (27) can also be transformed to theequivalent LMI (50) (Boyd et al. 1994): HilP

�1i HT

il � 1can be rewritten by Schur complement, while usingZi¼Hi as (50). Note that the LMI (49) ensures that the

obtained solutions Xi and Si are positive definite.Finally, the LMI (51) is equivalent to SiXi � I. œ

This result presents the advantage of computingdirectly the matrices Ki and Hi. Nevertheless, an

optimisation problem must be solved to achieveXiSi ’ I by minimizing iteratively the trace of matrixXiSi using an algorithm presented in El Ghaoui et al.(1997). This heuristic is based on a linear approxima-tion of Trace(XiSi) by Trace(X0Si+S0Xi) where X0

and S0 are particular solutions of the LMI constraints(49), (50) and (51).

ðPb:3Þ :

minðSj,Xi,Yi,ZiÞ TraceðXiSiÞ

s:t: ð49Þ, ð50Þ, ð51Þ

i, j ¼ 1, . . . ,N

8><>:

An output feedback non-saturating controller canbe obtained, as noted in Comment 4.1, as a particularcase of the saturating controller results by lettingDis ¼ I and Dþis ¼ 0. It is also obvious that the

inclusion condition becomes "ðPi, 1Þ LðKiCiÞ. Thefollowing results are then directly obtained.

Corollary 4.2: If there exist symmetric matrices Xi,matrices Vi and Yi solutions of the following LMIs:

Xi ðAiXi þ BiYiCiÞT

� Xj

" #4 0, ð55Þ

1 ðYiCiÞl

� Xi

� 4 0, ð56Þ

ViCi ¼ CiXi ð57Þ

8 ði, j Þ 2 I 2, 8l 2 ½1,m�

then the closed-loop saturated switching system (3) with

Ki ¼ YiV�1i , Pi ¼ X�1i ð58Þ

is asymptotically stable 8x02� and for all switchingsequences �(t).

Proof: This proof can be obtained easily by lettingDis ¼ I and D�is ¼ 0 in Theorem 4.5. However, theinclusion condition "ðPi, 1Þ LðKiCiÞ leads toðKiCiÞlP

�1i ðKiCiÞ

Tl � 1 which is equivalent to,

ðKiCiP�1i ÞlPiðKiCiP

�1i Þ

Tl � 1,

by using (57), ðKiViCiÞlX�1i ðKiViCiÞ

Tl � 1, can be

rewritten by Schur complement, while using Yi¼KiVi

as (56). œ

In a similar way, the result of Theorem 4.6 can beapplied for designing non-saturating controls by lettingDis ¼ I and D�is ¼ 0. Finally, one can cite a recent work(Bara and Boutayeb 2006) where the authors claimthat they present a less conservative sufficient condi-tion of stabilizability than the one of Daafouz et al.(2002) on which our work is based, but only appliedfor a class of switching linear systems. This newdirection, using the congruence transformation, hasalso to be explored for saturated switching systems.

Example 4.2: In order to illustrate our results, con-sider a single-input single-output (SISO) saturatedswitching discrete-time system with two modes givenby the following matrices:

A1 ¼1 1

0 1

� ; B1 ¼

10

5

� ; C1 ¼ 1 1

�

A2 ¼0 �1

0:0001 1

� ; B2 ¼

0:5

�2

� ; C2 ¼ 2 3

�


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For this example with n¼ 2, m¼ p¼ 1 and N¼ 2, we

have to solve 13 LMIs with 9 variables by using the

LMIs of Corollary 4.1. Let the scalar �¼ 1. The use of

the Matlab LMI Toolbox yields the following results:

P1 ¼0:0016 0:0006

0:0006 0:0025

� ; H1 ¼ �0:0196 �0:0479

�;

K1 ¼�0:0743; Ac1 ¼0:2574 0:2574

�0:3713 0:6287

� ;

P2 ¼0:0010 0:0004

0:0004 0:0027

� ; H2 ¼ 0:000007 0:0483

�;

K2 ¼ 0:0347; Ac2 ¼0:0615 �0:9077

�0:2460 0:6309

�

The main conditions (12) are also satisfied:

AcT1P2Ac1 � P1 ¼ 10�3�0:0012 �0:0011

�0:0011 �0:0013

" #;

AcT2P1Ac2 � P2 ¼ 10�3�0:0008 �0:0007

�0:0007 �0:0011

" #

Figure 3 shows the level set as the union of two

ellipsoid sets of invariance and contractivity for the

switching discrete-time linear system obtained by

Corollary 4.1. Each ellipsoid set is contained inside

the set of admissible saturations LðHiÞ. Inside this set,

the asymptotic stability of the saturated switching

system is guaranteed for any arbitrary switch

from any linear subsystem to another. This SISO

example is studied only in order to illustrate the results

of this work by plotting the level set in the plane. The

use of Theorem 4.6 leads to the following results

for �¼ 1:

P1 ¼1:0961 0:0848

0:0848 1:0290

" #; P2 ¼

1:5366 0:1255

0:1255 1:4821

" #;

S1¼0:6553 �0:0555

�0:0555 0:6794

" #; S2¼

0:9182 �0:0756

�0:0756 0:9781

" #;

H1 ¼ �0:0786 �0:1208 �

; K1 ¼¼ �0:1007;

H2 ¼ 0:00004 0:5473 �

; K2 ¼ 0:1812

The level set obtained with this approach is

plotted in Figure 5 together with few trajectories.

The use of Corollary 4.2 leads to the following results

2 4 6 8 10 12 14 16 18 20

0

0

5 10 15 20–1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.8

1

–1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

2

2.5

3

Figure 4. The level set as the union of two ellipsoid sets of invariance and contractivity for the switching discrete-time linearsystem obtained by Corollary 4.2 and the corresponding switching sequences.

–40 –30 –20 –10 0 10 20 30 40–40

–30

–20

–10

0

10

20

30

40

x1

x 2

Figure 3. The level set as the union of two ellipsoid sets ofinvariance and contractivity for the switching discrete-timelinear system obtained by Corollary 4.1.


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for �¼ 1:

P1 ¼1:0000 �0:0000012

�0:0000012 1:0000

� ;

P2 ¼2:2750 �0:0373

�0:0373 1:1188

� ;

K1 ¼ �0:0998; K2 ¼ 0:2234

The level set obtained with this approach is plotted inFigure 4 together with few trajectories.

One can notice that the obtained level sets withCorollary 4.1 are the most larger while the onesobtained with Corollary 4.2 are similar to thoseobtained with Theorem 4.6 with an important reductionof the number of LMIs. One can conclude, via thisexample, that the result of Corollary 4.1 is the lessconservative among all the other results due to theintroduced slack variables Gi. Combining the compar-isons made for state feedback control and outputfeedback control, one can expect that the introductionof slack variables in the Corollary 4.2 can lead to themore less conservative results.

5. Conclusion

In this work, two main different sufficient conditions ofasymptotic stability are obtained for switching discrete-time linear systems subject to actuator saturations foreach case: state feedback and output feedback control.The first allows the synthesis of stabilising controllersinside a large region of linear behaviour while thesecond applies the idea of Lemma 3.1 which rewrites thesaturation function under a combination of 2m elementsto obtain stabilising controllers tolerating saturations totake effect. Particular attention is given to the output

feedback case, which has additive complexity due to theoutput equation. In this sense, three different LMIs arepresented for this case. Also, the last one recentlypresented in Bara and Boutayeb (2006) is cited to beexplored in a future work. The main results of thisarticle are given under LMIs formulation leading to thedesign of the stabilising state feedback and outputfeedback controllers for the system. Even the dynamicalsystem is a switching system; it is shown that the set� formed by the union of all the ellipsoid level setsassociated to each subsystem constitutes a set ofasymptotic stability. It is the first time that thisimportant result is established for saturated switchingsystems. Two illustrative examples are studied by usingthe solution of the proposed LMIs. A comparison of theobtained solutions is also given.

Acknowledgement

The authors would like to thank the anonymous reviewersfor their valuable comments and suggestions, which helpedimproving the quality of the article.

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Documents

Stabilisation and control synthesis of switching systems subject to actuator saturation