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A complexity reduction approach to detectability ofswitching systemsElena De Santis a , Maria Domenica Di Benedetto a & Giordano Pola aa Department of Electrical and Information Engineering, University of L'Aquila , Center ofExcellence DEWS , Poggio di Roio, 67040 L'Aquila, ItalyPublished online: 18 Aug 2010.

To cite this article: Elena De Santis , Maria Domenica Di Benedetto & Giordano Pola (2010) A complexity reduction approachto detectability of switching systems, International Journal of Control, 83:9, 1930-1938, DOI: 10.1080/00207179.2010.501387

To link to this article: http://dx.doi.org/10.1080/00207179.2010.501387

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International Journal of ControlVol. 83, No. 9, September 2010, 1930–1938

A complexity reduction approach to detectability of switching systems

Elena De Santis, Maria Domenica Di Benedetto and Giordano Pola*

Department of Electrical and Information Engineering, University of L’Aquila,Center of Excellence DEWS, Poggio di Roio, 67040 L’Aquila, Italy

(Received 22 July 2009; final version received 11 June 2010)

In this article, we address detectability for the class of linear switching systems. We focus on some hybrid state-space decompositions of the original switching system based on hybrid invariant subspaces, which yield acomplexity reduction in checking detectability. We show that the reduced system extracted from the originalsystem is the minimal bisimilar switching system associated with it. An example is also included which shows theapplicability and benefits of the proposed approach.

Keywords: hybrid systems; switching systems; detectability

1. Introduction

In the past few years, there has been a growing interestfrom the control systems community in the study ofhybrid systems because of their expressive power whichis general enough to provide an accurate modelling ofmany processes of interest, ranging from engineering tosystems biology, finance, etc. Due to their richexpressive power the analysis and control of generalhybrid systems is quite hard, in general. This is thereason why many researchers focused on someimportant subclasses of hybrid systems as for exampleswitching systems, piecewise affine systems amongmany others, with the aim of providing more solidresults in the analysis and control. In this article weconsider the class of linear switching systems(LSw-systems). LSw-systems are an important subclassof hybrid systems arising in many application domains,such as, among many others, mechanical systems,power train control, aircraft and air traffic control,switching power converters, see e.g. Liberzon (2003),De Santis, Di Benedetto, and Pola (2006b) and thereferences therein.

The analysis of structural properties of switchingsystems is essential towards the design of controlalgorithms. In many application domains, hybridcontroller synthesis problems are addressed by assum-ing full hybrid state information, although in manyrealistic situations state measurements are not avail-able. Hence, to make hybrid controller synthesisrelevant, the design of hybrid state observers is offundamental importance. A step towards a procedurefor the synthesis of these observers is the analysis ofobservability and detectability of hybrid systems.

Results on observability and detectability of switchingsystems have been established in the works ofDe Santis, Di Benedetto, and Pola (2003), Vidal,Chiuso, Soatto, and Sastry (2003), Babaali and Pappas(2005), Elhamifar, Petreczky, and Vidal (2008) and DeSantis, Di Benedetto, and Pola (2009), among manyothers. In particular, the work in De Santis et al. (2009)provides sufficient and necessary conditions for asystem to be detectable, and shows that checkingdetectability can be decoupled into two sub-problems:

(i) the problem of checking location observabil-ity, i.e. the one of reconstructing the discretecomponent of the hybrid state from theknowledge of the output for a suitable choiceof the control input;

(ii) the problem of checking detectability for anautonomous switching system (i.e. a switchingsystem with autonomous continuousdynamics) associated with the given system.

Checkable necessary and sufficient conditions for aswitching system to be location observable have beenestablished in the works of Babaali and Pappas (2005)and De Santis et al. (2009). For this reason in thisarticle we focus on autonomous switching systems. Inthe work of De Santis et al. (2009), it has been shownthat problem (ii) is equivalent to checking asymptoticstability of a suitable autonomous switching sub-system with guard conditions enabling the transitions.However, the current literature on stability analysis ofswitching systems with guard conditions is ratherscant, at the present. On the other hand, there is awealth of results on stability of switching systems

*Corresponding author. Email: giordano.pola@univaq.it

ISSN 0020–7179 print/ISSN 1366–5820 online

� 2010 Taylor & Francis

DOI: 10.1080/00207179.2010.501387

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(with no guard conditions), see e.g. Liberzon (2003),Sun and Ge (2005) and the references therein. This isthe reason why in this article we exploit hybrid state-space decompositions that are alternative to the onesproposed in De Santis et al. (2009) and which allow theoriginal detectability problem to be split into simplersub-problems. In particular, we reduce the problem ofassessing whether a switching system S is detectable todetectability of a switching system So and asymptoticstability of a switching system Sh with no guards,where both systems So and Sh are suitable sub-systemsextracted from S. The approach that we propose in thisarticle makes analysis of detectability easier than theone based on the original system since the size ofswitching systems So and Sh is smaller than the one ofS. We further show that the switching system So is theminimal switching system, in terms of size of the state-space, which is bisimulation equivalent (Park 1981;Milner 1989) to the original system S. While theapproach presented in this article differs from the onepursued in De Santis et al. (2009) the resultsestablished in both the works can be efficientlycombined in order to check detectability of switchingsystems. Such a discussion is reported at the end ofSection 3.

Dual results on stabilisability-based state-spacereductions have been established in the companionpaper (De Santis, Di Benedetto, and Pola 2008). Apreliminary investigation on detectability-based reduc-tion of switching systems appeared in the conferencepaper (De Santis, Di Benedetto, and Pola 2006a). Inthis article, we present a detailed and mature descrip-tion of the results announced in De Santis et al.(2006a), which includes proofs and examples.

The organisation of this article is as follows. Wefirst introduce the class of switching systems and thenotion of detectability in Section 2. Section 3 illustratesthe techniques for state-space reduction of a switchingsystem in order to check detectability. In Section 4, weshow that the reduced system obtained in Section 3coincides with the minimal bisimilar system of theoriginal switching system. An academic example ispresented in Section 5. Some concluding remarks areoffered in Section 6.

Notations: The symbols 0 and I denote the origin andthe identity matrix, in the appropriate linear vectorspaces.

2. Switching systems

In this article, we consider the class of autonomousLSw-systems, obtained as a specialisation of themodels studied in De Santis et al. (2003) andDe Santis et al. (2009).

The hybrid state � of a LSw-system is composed of

two components: the discrete state i belonging to the

finite set Q¼ {1, 2, . . . ,N} with N2N and the con-

tinuous state x belonging to the linear space Rni, whose

dimension ni� 0 depends on i. A map S associates to

any i2Q the dynamical system S(i) defined by

_x tð Þ ¼ fi x tð Þð Þ,

yðtÞ ¼ hi xðtÞð Þ,

�ð1Þ

where t2R, x(t)2Rni, fi is linear on R

ni and hi is linear

on Rni. When ni¼ 0 system S(i) is characterised by a

state-space of zero dimension and hence, it is not

dynamic. When ni4 0 there exist matrices Ai2Rni�ni

and Ci2Rl�ni such that

fi xðtÞð Þ ¼ AixðtÞ, hi xðtÞð Þ ¼ CixðtÞ:

The hybrid state-space of a LSw-system is given by the

set � ¼S

i2Q if g �Rni and a hybrid state is (i, x). The

hybrid output has a discrete and a continuous

component as well: the former coincides with the

discrete state, while the latter is a vector associated

with the continuous state. Therefore, the hybrid output

space of a LSw-system is the set �¼Q�Rl with l4 0.

The evolution of the discrete state is governed by a

finite state machine (FSM) and formalised by a

collection of transitions E�Q�Q. When a transition

e¼ (i, j)2E occurs, the continuous component x�2Rni

of the hybrid state (i, x�) is instantly reset to a new

value xþ¼R(e, (i, x�))2Rnj, where R :E��!� is

the reset function, which is assumed to be a linear

function on Rni. When njni4 0 there exists a matrix

Me2Rnj�ni such that:

Rðe, ði, x�ÞÞ ¼Mex�:

A LSw-system is then specified by means of the tuple:

�,�,S,E,Rð Þ,

where all entities have been defined before. We now

formally define the semantics of LSw-systems. We

recall from Lygeros, Tomlin, and Sastry (1999) that a

hybrid time basis � is an infinite or finite sequence of sets

Ij ¼ ft 2 R : tj � t � t0jg, with t0j ¼ tjþ1. If card(�)51,

then t0card ð�Þ�1 can be finite or infinite. A hybrid time

basis � is said to be finite if card(�)51 and

t0card ð�Þ�1 51 and, infinite otherwise. Given a hybrid

time basis �, a time instant t0j is called a switching time.

Denote by T the set of all hybrid time bases. The

switching system temporal evolution is formalised by

the notion of execution (Lygeros et al. 1999) as follows:

Definition 1 (Switching system execution): An execu-

tion � of a LSw-system S is a collection:

�0, �, �, �ð Þ,

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where �02�, � 2T , � :R�N!� and � :R�N!�.The hybrid state evolution � is defined as follows:

� t0,0ð Þ ¼ �0,

� t, jð Þ ¼ q jð Þ,xðt, jÞð Þ, t2 Ij, j¼ 0,1, . . . ,card ð�Þ�1,

� tjþ1, jþ1� �

¼ðq jþ1ð Þ, Rðej,xðt0j, jÞÞ, j¼ 0,1, . . . ,card ð�Þ,

where q :N!Q and for any j¼ 0, 1, . . . , card(�)�1,ej¼ (q( j), q(jþ 1))2E and x(t, j) is the (unique) solutionat time t of the dynamical system S(q(j)), with initialtime tj and initial condition x(tj, j). The outputevolution is given by the function � :�!�, defined by:

�ðt, j Þ ¼ ðqð j Þ, hqð j Þðxðt, j ÞÞÞ, t 2 Ij,

j ¼ 0, 1, . . . , card ð�Þ � 1:

For any hybrid state (i, �0), there always exists anexecution with initial condition (i, �0) which is acceptedby the switching system S. Indeed, starting from (i, �0),the evolution of S follows the linear system S(i), whosesolution exists and it is unique. If a transition occursfrom the discrete state i to the discrete state j, thecontinuous state of S is reset to a new value and theevolution of S follows the linear system S( j). If furthertransitions occur, the evolution of S follows the linearsystems S(k), where k is the discrete state visited in thetransitions. If a transition does not occur, the evolutionof S follows the linear system S(i). While as discussedabove, existence of executions of switching systems isguaranteed, uniqueness of executions cannot beguaranteed in general. This is because the discretestates transition mechanism of switching systems isnondeterministic.

Given an execution �, the pair (�, q) will be calleddiscrete-state evolution. Checkable sufficient and neces-sary conditions for the reconstruction of the discrete-state evolution from the hybrid output can be found inthe work of Babaali and Pappas (2005) and De Santiset al. (2009). We therefore assume that the discrete-state evolution of switching systems is known.

In hybrid system control theory some assumptionsare often made to avoid undesired phenomena on theexecutions, as Zeno behaviours (Lygeros, Johansson,Simic, Zhang, and Sastry 2003). Since in LSw-systemsmultiple switchings can occur at the origin LSw-systems may exhibit Zeno behaviour. Hence, in orderto ensure non-Zenoness, conditions on uncontrolleddiscrete transitions have to be introduced. In thisarticle we suppose that LSw-systems are non-Zeno, orequivalently we have the following assumption.

Assumption 2: Executions �¼ (�0, �, �, �) ofLSw-systems satisfy:

Xcard ð�Þ�1

j¼0

tjþ1 � tj ¼ 1:

Given t2R, the restriction of � to the sett0, t½ � � 0, 1, . . . , j

� �, where j ¼ max j 2 N : t 2 Ij

� �, is

said to be the observed output at time t of the execution� and the symbol Yo denotes the collection of allobserved outputs at some t. We now introduce thenotion of detectability. For doing so, we need to equipthe hybrid state-space of switching systems with ametric, as follows. Given a LSw-system S, define thefollowing function:

� : ���! R [ f1g,

such that, for any (i, x), ( j, z)2�,

� i, xð Þ, j, zð Þð Þ ¼1, if i 6¼ j,

x� zk kni if i ¼ j,

�where kxkni is the Euclidean norm of x in R

ni. It is easyto see that (�, �) is a metric space (Kelley 1955). LetB ¼

Si2Q if g � Bi, with Bi¼ {x2R

ni : kxkni� 1}, and forsome scalar �2R let �B ¼

Si2Q if g � �Bi. We can now

introduce the following definition:

Definition 3: A LSw-system S is (exponentially)detectable if there exists a function b� : Yo! � andtwo positive reals M and � such that for any execution� there exists a time t4 0 for which:

�ðb�ð�j t0, t½ �Þ, � t, jð ÞÞ �Me�� t,

8t � t, t 6¼ tj, j ¼ 0, 1, . . . , card ð�Þ � 1:ð2Þ

The above definition differs from the one intro-duced in De Santis et al. (2009) since it requiresexponential convergence of the error in the estimation,while the one in De Santis et al. (2009) only requiresasymptotic convergence of such error to zero. In thisarticle we will show that working with this strongernotion of detectability allows us to derive some resultswhich reduce the computational effort required inchecking such structural properties. For furtherpurposes we recall that an autonomous LSw-systemS is asymptotically stable if for any "4 0 and for any4 0 there exists t4 t0 so that �(t, j)2 "B, for anyt � t, j¼ 0, 1, . . . , card(�), for any execution � withhybrid initial state �02 B. Sufficient and necessaryconditions for checking asymptotic stability of switch-ing systems have been extensively studied in theliterature, see e.g. Liberzon (2003) and the referencestherein.

3. State-space reductions based on detectability

In the case of linear dynamical systems detectabilityholds if and only if a suitable sub-system extractedfrom it is asymptotically stable. In the context ofgeneral switching systems, the conditions become

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more involved. In this section, we show how to extract

a pair of switching systems from a given LSw-system S

in such a way that the detectability of one of them and

the asymptotic stability of the other one imply the

detectability of S. This reduces the computational

effort required to check the detectability property. Our

procedure is based on a reduction of the state-space of

the autonomous LSw-system S by means of invariant

hybrid subspaces. We recall that a set ��� is said to

be a hybrid linear subspace if it is of the form

� ¼S

i2Q if g ��i,

where �i is a linear subspace of Rni. Moreover we have

the following definition.

Definition 4: Given a LSw-system S¼ (�,�,S,E,R),

a set

� ¼S

i2Q if g ��i � �,

is S-invariant if for any hybrid initial state �02� and

for any execution �¼ (�0, �, �, �),

�ðt, j Þ 2 � 8t 2 Ij, 8j ¼ 0, 1, . . . , card ð�Þ:

The following result gives necessary and sufficient

conditions for a hybrid linear subspace to be invariant.

Proposition 5 (De Santis et al. 2008): Given a LSw-

system S, a hybrid linear subspace

� ¼S

i2Q if g ��i

is S-invariant if and only if the following conditions hold:

. Ai�i��i, for any i2Q with ni4 0;

. R(e, (i, x))��j, for any i2Q, for any

e¼ (i, j)2E and for any x2�i.

A result in De Santis (2008) shows that the maximal

S-invariant set contained in a hybrid linear subspace

is a hybrid linear subspace. In the further develop-

ments we will be interested in the maximal S-invariant

hybrid subspace contained inS

i2Q if g � Oi, where:

Oi ¼ker

Ci

CiAi

..

.

CiAni�1i

0BBBB@1CCCCA, if ni 4 0,

R0, if ni ¼ 0:

8>>>>>>><>>>>>>>:We denote by

I ¼S

i2Q if g � I i,

such maximal hybrid linear subspace. The following

result shows that the set I exists and it can be

computed in a finite number of steps.

Theorem 6: Define the sequences of subspaces

�ki � R

ni , k2N, i2Q, as

�0i ¼ Oi,

�ki ¼ x 2 Oi : Rðe, ði, xÞÞ 2 �k�1

j , 8e ¼ ði, j Þ 2 En o

,

k ¼ 1, 2 . . . : ð3Þ

This sequence converges in at most n �PN

i¼1 ni steps

to the subspaces I i, i2Q, i.e.

I ¼S

i2Q if g ��ni :

Proof: The sequence of sets in (3) is a special case of

the sequence defined in De Santis, Di Benedetto, and

Berardi (2004) to solve general safety problems. As a

consequence, such a sequence converges to the set I .

By definition �kþ1i � �k

i , k¼ 0, 1, 2, . . . , and the

condition �sþ1i ¼ �s

i , for any i¼ 1, 2, . . . ,N, implies

that I ¼S

i2Q if g ��si . Therefore, given k� 0 either

�kþ1i is strictly contained in �k

i for some i¼ 1, 2, . . . ,N

or the convergence has been obtained in k steps. Since

by definition �ki is a linear subspace of R

ni, then the

sequence converges after at mostPN

i¼1 ni steps. œ

The S-invariant hybrid subspace I induces a

decomposition of S into two switching systems Soand Sh so that the hybrid state-space �h of Sh coincides

with I while the hybrid state-space �o of So is so that

�o��h¼�. Since by construction, subspace I i�Ri

is invariant for system S(i), there exists a linear

transformation x ¼ Tix so that if x ¼ x1 x2� �

with

x2 2 Ri then any trajectory of the system (in the new

coordinates) starting from any state of the form

0 x2� �

remains in the subspace Ti I i, for any time

t� 0. Define a hybrid state-space transformation on S,

that associates to a hybrid state (i, x) a new hybrid state

(i,Tix). By performing such hybrid state-space trans-

formation a new switching system S is obtained

which is algebraically equivalent (Pola, van der

Schaft, and Di Benedetto 2006) to the original one.

For any ni4 0, the dynamical matrices of system S

take the form

bAi ¼ TiAiT�1i ¼

Að11Þi 0

Að21Þi A

ð22Þi

!,

bCi ¼ CiTi ¼ Cð1Þi 0

� �,

where Að22Þi 2 R

i�i . For any e¼ (i, j)2E with ninj4 0,

the reset matrix of S takes the form

bMe ¼ T�1j MeTi ¼Mð11Þe 0

Mð21Þe Mð22Þe

!,

where Mð22Þe 2 Rj�i . Note that in general, the

pair ðAð11Þi ,C

ð1Þi Þ is not observable. The above hybrid

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state-space transformation induces the definition of a

pair of switching systems, as follows. Define the

LSw-system:

So ¼ ð�o,�,So,E,RoÞ,

where �o ¼S

i2Q if g �Rni�i , system So(i) is given by

the following dynamics:

_zðtÞ ¼ f oi zðtÞð Þ,

yðtÞ ¼ hoi zðtÞð Þ,

�ð4Þ

where f oi zð Þ ¼ Að11Þi z and hoi zð Þ ¼ C

ð1Þi z, if i5 ni, and

Ro e, i, zð Þð Þ ¼Mð11Þe z, if i5 ni and j5 nj. Moreover,

define the LSw-system:

Sh ¼ ð�h,�,Sh,E,RhÞ,

where �h ¼S

i2Q if g �Ri , system Sh(i) is characterised

by the following dynamics:

_zðtÞ ¼ f hi ðzðtÞÞ, ð5Þ

where f hi zð Þ ¼ Að22Þi z if i4 0 and Rhðe, ði, zÞÞ ¼Mð22Þe z

if i4 0 and j4 0. The above switching system

captures dynamics of S which is unobservable. In the

next result (Theorem 8), we will show that detectability

of the original switching system S can be assessed by

studying detectability of the (sub-)switching system Soand asymptotic stability of the (sub-)switching system

Sh. For the sake of simplicity we suppose in the further

developments existence of a minimum dwell time in the

evolution of the switching system under consideration.

More formally we have the following assumption.

Assumption 7 (Minimum dwell time): There exists a

real �m4 0, called minimum dwell time, such that

t0j � tj � �m for all j¼ 0, 1, . . . , card(�)� 1 and any

hybrid time basis �.

Existence of minimum dwell time is a widely used

assumption in the analysis and control of switching

systems, see e.g. Liberzon (2003) and the references

therein.We now have all the ingredients to give the main

result of this article:

Theorem 8: An LSw-system S satisfying Assumption 7

is detectable if and only if So is detectable and Sh is

asymptotically stable.

Proof: (Necessity) Obvious. (Sufficiency). Let

�o(t, j)¼ (q( j), xo(t, j)) be the hybrid state of system Soat time (t, j). Since So is detectable there exists a

function b�o : Yo! �o with b�o ¼ ðq,bxoÞ and two

positive reals � and M such that:

�ðb�oðyo�� t0, t½ �Þ, �o t, jð ÞÞ 2Me��t, 8t � 0, t 6¼ tj,

8j ¼ 0, . . . , card ð�Þ � 1,

whereb�oðyoj t0, t0½ �Þ can be set to (q(0), 0), without loss ofgenerality. Then xo t, jð Þ ¼ bxo t, jð Þ � d ðt, j Þ, with d(t, j)exponentially converging to the origin, i.e.

ð j,d ðt, j ÞÞ 2 e�� tB, 8t 2 Ij, 8j¼ 0, 1, . . . , card ð�Þ � 1:

In what follows all the block matrices are assumed ofappropriate dimension, which can be trivially deducedfrom the context. Let us consider the switching system

eSh ¼ �h,�,eSh,E, eRh

� ,

where eShðqiÞ is characterised by the followingdynamics:

_zðtÞ ¼ Að22Þi zðtÞ þ A

ð21Þi 0

� �wðtÞ

where w(t)2Rp, p¼maxi ni and the hybrid state after

the transition e2E is defined by

e�h tjþ1, jþ 1� �

¼ Rð22Þee�h t0j, j�

þ Rð21Þe 0� �

wðt0jÞ,

where e�h denotes the hybrid state of eSh and wðt0jÞ ¼limt!ðt0

jÞ� wðtÞ. If we partition the continuous compo-

nent of the state of the original system S in appropriatecoordinates, as x(t, j)¼ (x(1)(t, j), x(2)(t, j)), according tothe block dimensions of the matrices bAi, then x(2)(t, j)equals the state of the system eSh at time (t, j) withinitial state x(2)(t0, 0) and w(t)¼ (x(1)(t, j), 0), t 2 ½tj, t

0jÞ,

j¼ 0, 1, . . . , card(�)� 1. Let us consider the functionb� : Yo! � defined byb�ðyo�� t0, t½ �Þ ¼ ðqð j Þ, ðbxoðt, j Þ,bzðt, j ÞÞÞ, ð6Þ

where j is such that t 2 ðtj, t0j� and bzðt, j Þ is the

continuous state evolution of system eSh with initialstate (q(0), 0) and wðtÞ ¼ ðbxoðt, j Þ, 0Þ. We now show thatfunctionb� defined in (6) satisfies properties required inDefinition 3. We first note that

�ðb�ðyo�� t0, t½ �Þ, �ðt, j ÞÞ ¼ ðqð j Þ, ðd ðt, j Þ,ezðt, j ÞÞÞ,

whereezðt, j Þ is the continuous state evolution of systemeSh with initial state x(2)(t0, 0) and w(t)¼ (d(t, j), 0),t 2 ½tj, t

0jÞ, j¼ 0, 1, . . . , card(�)� 1. Since Sh is asympto-

tically stable by applying Theorem 14 in De Santiset al. (2008) the component ezðt, j Þ converges to theorigin. Hence, functionb� satisfies the requirements ofDefinition 3 and the result follows. œ

The result obtained above allows reducing theproblem of assessing whether a switching system S isdetectable to the one of assessing whether S0 isdetectable and Sh is asymptotically stable, a signifi-cantly lower complexity task. Asymptotic stability ofSh can be checked by leveraging a wealth of results onstability analysis of switching systems (see e.g.Liberzon 2003), while detectability of So can be

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analysed by using the results of De Santis et al. (2009).In particular by Theorem 3.5 in De Santis et al. (2009),switching system So is detectable if and only if asuitable switching system with guards Sg extractedfrom it, is asymptotically stable. Therefore, bycombining Theorem 8 and Theorem 3.5 in De Santiset al. (2009) detectability of a switching system Sreduces to asymptotic stability of switching systems Shand Sg. An alternative approach to check detectabilityof S could be the direct application of Theorem 3.5 inDe Santis et al. (2009), resulting in checking asympto-tic stability of a suitable switching system Sgg withguards extracted from S. However, in the currentliterature (see e.g. Liberzon 2003) stability analysis ofswitching systems is often based on the search forLyapunov functions which is in general a nontrivialtask when the dimension of the state-space of theswitching systems involved is large. Since dimensionsof the state-spaces of switching systems Sh and Sg aresmaller than the one of Sgg, checking asymptoticstability of Sh and Sg is in general an easier task thanchecking asymptotic stability of Sgg.

4. Relations with bisimulation equivalence

In this section, we establish some relationships betweenthe switching system S and the reduced one So. Inparticular we show that So is the switching system withminimal size in the hybrid state-space which isequivalent by bisimulation to S. Bisimulation is astandard mechanism developed in theoretical compu-ter science (Park 1981; Milner 1989) to reduce spacecomplexity of models of computations, while preser-ving properties of interest. This theory has beenrecently applied to continuous and hybrid systemswith the aim of reducing complexity of such systems,see e.g. van der Schaft (2004), Haghverdi, Tabuada,and Pappas (2005), Pola et al. (2006) and the referencestherein.

In this section we suppose, for simplicity, that thedimension ni of state-spaces associated with systemsS(qi) is greater than zero. The generalisation to the casewith ni� 0 is straightforward but is cumbersome fromthe notational point of view. We start by recalling fromPola et al. (2006) the notion of bisimilar switchingsystems.

Definition 9 (Bisimulation Equivalence): Considertwo LSw-systems:

Si ¼ ð�i,�i,Si,Ei,RiÞ, i ¼ 1, 2:

A hybrid bisimulation between S1 and S2 is a relation:

R � �1 ��2,

where for any (�10, �20)2R the following property

holds: for any execution �1¼ (�10, �, �1, �1) of S1, thereexists an execution �2¼ (�20, �, �2, �2) of S2 (and

conversely, for any execution �2 of S2 there exists an

execution �1 of S1) such that for any time t2 Ij and any

j¼ 0, 1, 2, . . . , card(�):

. (�1(t, j), �2(t, j))2R,

. �1(t, j)¼ �2(t, j).

Two LSw-systems S1 and S2 are bisimilar if there

exists a hybrid bisimulation R��1��2 such that the

projection of R on each hybrid state-space equals this

hybrid state-space, i.e. �j�i(R)¼�i, i¼ 1, 2.

Sufficient and necessary conditions for a pair of

switching systems to be bisimilar can be found in

Theorem 2 of Pola et al. (2006). It is readily seen that:

Proposition 10: The switching systems S and So are

bisimilar.

Proof (Sketch.): The relation R���o defined by

R ¼ fðði, xÞ, ð j, yÞÞ 2 ���o : i ¼ j

and 9z s:t: x ¼ T�1i ð y z Þg

is a hybrid bisimulation between S1 and S2 satisfying

�j�(R)¼� and �j�o(R)¼�o. œ

Given any R��1��2, let QR be the projection of

R on the space Q1�Q2 and

Rði, j Þ ¼ ðx, yÞ 2 Rni � R

nj : ði, xÞ, ð j, yÞð Þ 2 R� �

:

Then R can be written w.l.o.g as:

R ¼[ði,j Þ2QR

fði, j Þg � Rði, j Þ,

where R(i, j) is a suitable subspace of Rni�R

nj. From

Pola et al. (2006), given a pair of LSw-systems S1 and

S2, there exists a unique maximal hybrid bisimulation

R* between S1 and S2, i.e. a hybrid bisimulation R*

such that R�R* for any hybrid bisimulation R

between S1 and S2.It was shown in Pola et al. (2006) that the minimal1

bisimilar LSw-system of a given LSw-system S is

obtained by reducing the state-space of S by means of

the maximal hybrid bisimulation R* between S and

itself. Hence, we first characterise

R ¼[

ði,j Þ2QR

fði, j Þg � Rði, j Þ,

where R*(i, j) is a suitable subspace of Rni�R

nj. An

algorithm converging in a finite number of steps to the

maximal bisimulation can be found in Pola et al.

(2006). A direct consequence of the definition of

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bisimulation and of the semantics of LSw-systems

is that:

QR ¼ fði, j Þ 2 Q�Q : i ¼ jg:

Given the LSw-system S, define the aggregated

LSw-system

Sa¼ ð�a,�a,Sa,E, �,RaÞ,

where:

. �a ¼S

i2Q i, if g � R2ni ;

. �a¼Q�Q�R

l�R

l;. Sa associates to the discrete state i2Q the

following linear dynamical system:

_xðtÞ ¼AðiÞ 0

0 AðiÞ

!xðtÞ,

yðtÞ ¼ CðiÞ �CðiÞ� �

xðtÞ;

8>>>><>>>>:. for any e2E, Ra(e, x)¼Ma(e)x, where:

MaðeÞ ¼MðeÞ 0

0 MðeÞ

!:

Lemma 11: The maximal hybrid bisimulation R*

between S and itself is the maximal invariant hybrid

linear subspace contained in

Si2Q ði, iÞ� �

� ker Ci �Ci

� �for the aggregated LSw-system Sa.

The proof of the above result is a direct

consequence of Proposition 5 and of the results in

van der Schaft (2004) and Pola et al. (2006) and is

therefore omitted. We now show how to construct

the minimal bisimilar switching system associated

with S starting from the maximal bisimulation R*

between S and itself. The set R* is an equivalence

relation on the hybrid state-space of S (Pola et al.

2006). For performing the hybrid state-space reduc-

tion of S, we use this equivalence relation on the

hybrid state-space � in such a way that all hybrid

states belonging to the same equivalence class are

reduced to the same hybrid state. Given R* for any

(i, i)2QR* define:

�RðiÞ ¼ x1 � x2 j ðx1, x2Þ 2 Rði, iÞ

� �,

�R ¼S

i2Qfig ��RðiÞ:

The hybrid state-space � of the LSw-system S under

consideration may be factored by �R. We write �= �R to

denote the reduced hybrid state-space of S naturally

induced by �R, i.e.

�= �R ¼S

i2Q if g �Rni= �RðiÞ:

The following result highlights the connections

between the maximal hybrid bisimulation R* and the

hybrid subspace I .

Proposition 12: �R ¼ I .

Remark 13: It was shown in Pappas (2003) that given

a linear system

S :_x ¼ Axþ Bu,

y ¼ Cx,

(ð7Þ

and a subspace ker(H )�ker(C), for some matrix H, it

is possible to define an abstraction �S of S, by reducing

the state-space of S by means of ker(H ). Moreover, it

was shown that the relation

R ¼ fðx, zÞ : z ¼ Hxg

is a bisimulation between S and �S if and only if ker(H )

is an invariant set for system (7). Proposition 12 can be

viewed as a generalisation of those results to switching

systems.

Let SR* be the reduced system of S, under the

equivalence relation R*. For a formal definition of

such reduced system the interested reader can refer to

Pola et al. (2006). We can now give the following result

which formalises the relationships between S and So in

terms of bisimulation theory.

Theorem 14: The LSw-system So is the minimal

bisimilar switching system of S, i.e.

So ¼ SR :

Proof: Direct consequence of Proposition 12 and of

Corollary 4 is given in Pola et al. (2006). œ

5. An illustrative example

In this section, we present an example that shows the

interest and applicability of our results. Consider the

LSw-system

S ¼ �,�,S,E,Rð Þ,

where:

. �¼ ({1}�R2)[ ({2}�R

2)[ ({3}�R3)[

({4}�R);. �¼Q�R, where Q¼ {1, 2, 3, 4};

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. S associates to any i2Q the linear control

system S(i) of (1), where:

A1 ¼1 0

2 �1

�, C1 ¼ 2 0

� �,

A2 ¼�2 2

0 �1

�, C2 ¼ 0 0

� �,

A3 ¼

1 0 0

�1 �2 0

2 1 �3

0B@1CA, C3 ¼ 1 0 0

� �,

A4 ¼ 1, C4 ¼ 1;

. E¼ {(1, 2), (1, 3), (2, 1), (3, 2), (3, 4), (4, 3)};

. R is defined by:

Rð1, 2Þ ¼2 0

1 0

�, Rð1, 3Þ ¼

1 0

1 0

2 1

0B@1CA,

Rð2, 1Þ ¼0 0

0 1

�, Rð3, 2Þ ¼

1 2 0

2 1 0

�,

Rð3, 4Þ ¼ ð 1 2 0 Þ, Rð4, 3Þ ¼ 1 1 1� �0

:

The FSM associated with system S is depicted in

Figure 1(a).Let us analyse detectability properties of S by

applying Theorem 8. We first need to define switching

systems So and Sh. For doing so, we need to compute

the maximal invariant hybrid subspace I . A straight-

forward inspection of dynamical matrices and reset

maps of S reveals that:

I ¼ ðf1g � I 1Þ [ ðf2g � I2Þ [ ðf3g � I3Þ [ ðf4g � I 4Þ,

where:

I 1 ¼ fðx1, x2Þ 2 R2 : x1 ¼ 0g,

I 2 ¼ R2,

I 3 ¼ fðx1, x2, x3Þ 2 R3 : x1 ¼ x2 ¼ 0g,

I 4 ¼ R0:

On the basis of the hybrid subspace I , the

switching systems So and Sh can be defined. The

resulting switching system:

So ¼ ð�o,�,So,E,RoÞ,

is characterised by the FSM in Figure 1(b) and

described by:

. �o¼ ({1}�R)[ ({3}�R

2)[ ({4}�R);. System So(i) is given by the dynamics in (4),

where:

A111 ¼ 1, C1

1 ¼ 2,

A113 ¼

1 0

�1 �2

�, C1

3 ¼ 1 0� �

,

A114 ¼ 1, C1

4 ¼ 1;

. Ro e, i, zð Þð Þ ¼Mð11Þe z, where:

Mð11Þð1,3Þ ¼ 1 1

� �0, M11

ð3,4Þ ¼ 1 2� �

, M11ð4,3Þ ¼ 1 1

� �0:

The resulting switching system:

Sh ¼ ð�h,�,Sh,E,RhÞ,

is characterised by the FSM in Figure 1(c) and

described by:

. �h¼ ({1}�R)[ ({2}�R

2)[ ({3}�R);. System Sh(i) is given by the dynamics in (5),

where:

A221 ¼ �1, A22

2 ¼�2 2

0 �1

�, A3 ¼ �3;

. Rh e, i, zð Þð Þ ¼Mð22Þe z, where:

Mð22Þð1,2Þ ¼ 0 0

� �0, M22

ð2,1Þ ¼ 0 1� �

,

M22ð1,3Þ ¼ 1, Mð3,2Þ ¼ 0 0

� �0:

By Theorem 8, system S is detectable if and only if

the following conditions are satisfied:

(C1) Switching system So is detectable;(C2) Switching system Sh is asymptotically stable.

We start by checking condition (C1). Since

dynamics in locations 1 and 4 of S0 are observable

by using the results from De Santis, Di Benedetto, and

Pola (2004) So is detectable if and only if dynamics in

location 3 is detectable, which is the case. We proceed

with a further step by checking condition (C2). Since

M22ð1,2Þ ¼M22

ð3,2Þ ¼ 0 0� �0

, Sh is asymptotically stable if

and only if systems in (5) with i¼ 1, 2, 3 are asympto-

tically stable, which is the case. Hence, Sh is

asymptotically stable. Thus, by Theorem 8 switching

system S is detectable.

1 2

3 4

1

3 4

1 2

3

Figure 1. (Left panel) Finite State Machine associated withthe linear switching system S. (Central panel) Finite StateMachine associated with the linear switching system So.(Right panel) Finite State Machine associated with the linearswitching system Sh.

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6. Conclusions

In this article, we considered autonomous switchingsystems and we proposed some state-space decomposi-tions, based on hybrid invariant subspaces, which yielda complexity reduction in checking detectability. Wealso showed that the reduced system, extracted fromthe original system, is the minimal bisimilar switchingsystem associated with the original one. Future workwill investigate the implications of the results presentedin this article in the construction of hybrid observers.

Acknowledgements

This work was partially supported by European Commissionunder STREP project n.TREN/07/FP6AE/S07. 71574/037180 IFLY.

Note

1. i.e. with dimension of the continuous state-spaceassociated with discrete state i minimal, in the class ofall systems bisimilar with S, for all i2Q.

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