Observability of Internal Variables in Interconnected Switching Systems

Elena De Santis, Maria D. Di Benedetto and Giordano Pola

Abstract— Given a pair of linear switching systems, we studyconditions that ensure the observability of internal variablesarising in the input–output interconnection of the switchingsystems on the basis of the knowledge of input and hybrid out-put of the interconnected system. Some sufficient and necessaryconditions are provided for the reconstruction of hybrid statesand of (hybrid) latent variables associated to the interconnectedsystem.

keywords: observability, interconnection, internal vari-ables, latent variables, switching systems, hybrid systems.

I. INTRODUCTION

Hybrid systems have been the subject of intense researchover the past few years because of their expressive powerthat is able to capture various non–smooth phenomena indiverse application areas (e.g. [6], [8] and [7]). Howeverin many situations the resulting hybrid systems are verycomplex and therefore a rigorous analysis is very difficultto approach. Researchers focused on several subclasses ofhybrid systems with the attempt of characterizing theirdynamical properties and to synthesize controllers ensuringsome prescribed performance. In particular, the observabilityproperty has been studied for different subclasses of hybridsystems. The definitions and testing criteria vary dependingon the class of systems under consideration (see [5] fora review on observability notions). For example, in [17]observability of autonomous switching systems is studied. In[5] and [1] observability of (controlled) switching systems isaddressed. Observability of piecewise affine systems is fo-cused in [4] and in [3]. In [2], the notion of generic final–statedeterminability proposed in [15] is extended to linear hybridsystems. In [9] observability of autonomous hybrid systemsis studied by using abstraction techniques. The commondenominator of those works is to focus on a global notion ofobservability where the aim is to reconstruct the (full) hybridstate associated with the hybrid system under consideration.The aim of this paper is to give a first contribution towards acharacterization of a local notion of observability where thefocus is on the reconstruction of internal variables arisingfrom the interconnection of the systems involved. This callsfor formal notions of interconnection for hybrid systems.Notions of interconnection in discrete domains (e.g. [19])and in continuous domains (e.g. [10]) have been extensivelystudied before. By combining notions of [19] and [10], wepropose here a notion of input–output interconnection for

This work has been partially supported by the HYCON Network of Ex-cellence, contract number FP6-IST-511368 and by Ministero dell’Istruzionedell’Universita’ e della Ricerca under Projects MACSI and SCEF (PRIN05).

The authors are with the Department of Electrical Engineering andComputer Science, Center of Excellence DEWS, University of L’Aquila,Poggio di Roio, 67040 L’Aquila, Italy, desantis,dibenede,pola@ing.univaq.it

(a subclass of) hybrid systems. We consider the subclassof linear switching systems [5], where discrete transitionsare caused by a discrete input that acts as an external andunknown disturbance and where continuous dynamics aregiven by linear dynamical control systems. The motivationfor considering this particular subclass of hybrid systems liesin:

• switching systems are an appropriate abstraction formodelling important complex systems such as ATMsystems (e.g. [6]) or automotive engines (e.g. [7]);

• the semantics of switching systems allows the deriva-tion of necessary and sufficient checkable observabilityconditions that become sufficient for the general classof hybrid systems, where the transitions may depend onthe continuous component of the hybrid state.

Given a pair of switching systems S1 and S2 and theinput–output interconnection S 1||S2 of S1 and S2, internalvariables of S1||S2 are: hybrid state of S1, hybrid state ofS2 and hybrid latent variable [18]. A notion of observabilityof internal variables of S1||S2 is proposed, which is basedon the reconstruction of the internal variable evolution intime knowing the input and (hybrid) output of S 1||S2. Theconnections between the observability of various internalvariables is discussed and sufficient and necessary condi-tions are provided for the reconstruction of (hybrid) internalvariables of S1||S2.

The paper is organized as follows: in Section II, weintroduce the class of linear switching systems and thenotion of observability. In Section III relationships betweenobservability and interconnection of switching systems areinvestigated. Section IV studies conditions for the observabil-ity of internal variables in interconnected switching systems.Conclusions are offered in Section V.

II. PRELIMINARIES AND BASIC DEFINITIONS

In this paper, we consider the class of linear switchingsystems, that generalize the class defined in [5], followingthe general model of hybrid automata (see e.g. [11], [16]).

The inputs of a linear switching system are a discreteand unknown disturbance σ and a continuous control inputu. The hybrid state ξ is composed of two components: thediscrete state qi belonging to a finite set Q and the continuousstate x belonging to the linear space Rni , whose dimensionni depends on qi. The evolution of the discrete state isgoverned by a Discrete Event System (DES); a transitione = (qi, σ, qh) may occur at time t from the discrete state qi

to the discrete state qh, if the discrete input σ occurs at timet. The evolution of the continuous state is described by a setof linear dynamical systems, controlled by the continuous

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input u, and whose matrices depend on the current discretestate qi. Whenever a transition e occurs, the continuous statex is instantly reset to a new value R(e)x, where R(e) is amatrix depending on the transition e. The hybrid output hasa discrete and a continuous component as well, the formerassociated to the transitions between discrete states, the latterassociated to the continuous state. More formally:

Definition 1: (Linear switching system) A linear switchingsystem S is a tuple (Ξ, Θ,Υ,S, E, γ, R) , where:

• Ξ =⋃

qi∈Q {qi} × X(qi) is the hybrid state space,where:◦ Q = {qi, i ∈ J} is the set of discrete states, J ={1, 2, . . . , N};◦ X(qi) = Rni is the continuous state space associatedwith the discrete state qi ∈ Q;

• Θ = Σ × U is the hybrid input space, where:◦ Σ = {σh, h ∈ J1} is the set of discrete disturbances,J1 = {1, 2, . . . , N1};◦ U = Rm is the continuous input space;

• Υ = P × Y is the hybrid output space, where:◦ P = {pi, i ∈ J2} ∪ {ε} is the set of discrete outputs,J2 = {1, 2, . . . , N2}, with ε the null output;◦ Y = Rl is the continuous output space;

• S is a map associating to each discrete state qi ∈ Q thelinear dynamical control system:{

x(t) = A(qi)x(t) + Bi(qi)u(t),y(t) = C(qi)x(t),

where x ∈ X(qi) is the continuous state, u ∈ U isthe continuous input, y ∈ Y is the continuous output,A(qi) ∈ Rni×ni, B(qi) ∈ Rni×m and C(qi) ∈ Rl×ni ;

• E ⊂ Q × Σ × Q is a collection of transitions;• γ : E → P associates an output symbol to each

transition;• R is the reset function that associates to every e =

(qi, σ, qh) ∈ E the reset matrix R(e) ∈ Rnh×ni .Given a switching system S, the tuple DS =

(Q, Σ, P, E, γ) can be viewed as a Discrete Event System(DES) [14] having state space Q, input set Σ, output set P ,transition relation E and output function γ. A transition efor which γ(e) = ε will be called silent transition [13].

We now define the semantics of linear switching systems.We assume that the discrete disturbance is not available formeasurement, and that the class of admissible continuousinputs is the set U of piecewise continuous functions u :R → Rm. As defined in [11], a hybrid time basis τ is aninfinite or finite sequence of sets Ij = {t ∈ R : tj ≤ t ≤ t′j},with t′j = tj+1; let be card(τ ) = L + 1. If L < ∞, then t′Lcan be finite or infinite. A hybrid time basis τ is said to befinite if L < ∞ and t′L < ∞ and infinite, otherwise. Givena hybrid time basis τ , time instants t′j are called switchingtimes. Denote by T the set of all hybrid time bases. Theswitching system temporal evolution is defined, as follows.

Definition 2: (Switching system execution) An execu-tion χ of a linear switching system S is a collection(ξ0, τ, σ, u, ξ, η) , with ξ0 ∈ Ξ, τ ∈ T , σ : N → Σ, u ∈ U ,ξ : R × N → Ξ, η : R × N → Υ, where ξ and η are defined

for any j = 0, 1, ..., L− 1, as follows:

ξ (t0, 0) = ξ0,ξ (t, j) = (q (t, j) , x(t, j)) , t ∈ Ij,ξ (tj+1, j + 1) = (q (tj+1, j + 1) , R(ej)x(t′j, j)),η (t, j) = (p (t, j) , y(t, j)) , t ∈ Ij,

where:

• q(t, j) = q(tj , j), ∀t ∈ Ij and q(tj+1, j + 1) is suchthat ej = (q(t′j , j), σ (j) , q (tj+1, j + 1)) ∈ E;

• x (t, j) and y(t, j) are respectively the state andoutput evolutions at time t of the dynamical sys-tem S (q (tj, j)), with initial time tj , initial conditionx (tj , j) and control law u;

• p (t, j) = γ(ej−1), if t = tj and p (t, j) = ε, otherwise,where γ (e−1) = ε.

In hybrid system control theory [12], some assumptionsare often made for avoiding undesired phenomena on theexecutions, such as Zeno behaviours. We assume here thatthe system is non Zeno and that multiple simultaneoustransitions are not allowed:

Assumption 1: For any execution χ, the hybrid time basisτ is such that t′j − tj > 0 and for any t > 0, the cardinalityof the set {tj : tj ≤ t} is finite.

Moreover the semantics of linear switching systems as inDefinition 2, implies that this subclass of hybrid systems isnon–blocking (see [12]).

Let be ηo : R →Υ, with ηo (t) = η (t, j), t ∈ [tj, t′j),j = 0, 1, ..., L. The restriction of ηo to the interval [t0, t], issaid to be the observed output at time t of the execution χand the symbol Y0 denotes the set of all observed outputs atsome t.

In this paper we consider the structural property of ob-servability that deals with the exact reconstruction of thehybrid state of a linear switching system S, on the basis ofthe knowledge of the continuous input and of the observedhybrid (discrete and continuous) output. The following def-inition is based on the existence of at least an input–outputexperiment such that, after some transitions, the hybrid stateis reconstructed:

Definition 3: (Observability of switching systems) A lin-ear switching system S is said to be observable if there exista function u ∈ U , a function ξ : Y0 × U → Ξ and a timet > 0 such that:

ξ(

yo|[t0,t] , u|[t0,t)

)= ξ (t, j) ,

∀t ∈ (tj , t′j] ∩ [t,∞), ∀j = j, j + 1, ..., L,

j = min{j : t ∈ Ij},(1)

for any execution χ with control input u. A switching systemis said to be unobservable if it is not observable.

If one wants to reconstruct the only discrete componentof the hybrid state the following notion of observability canbe considered.

Definition 4: (Location observability of switching sys-tems) A linear switching system S is said to be locationobservable if there exist a function u ∈ U , a function

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q : Y0 × U → Q and a time t > 0 such that:

q(

yo|[t0,t] , u|[t0,t)

)= q (t, j) ,

∀t ∈ (tj , t′j] ∩ [t,∞), ∀j = j, j + 1, ..., L,

j = min{j : t ∈ Ij}.for any execution χ with control input u.

Remark 1: Definitions 3 and 4 generalize the notionsproposed in [5] for the class of switching systems withminimum dwell time and also hold for the general classof hybrid systems since they involve only the notion ofexecution.

Remark 2: Definition 3 requires the reconstruction of thehybrid state for every time t ≥ t and t ∈ (tj , t′j]. We rule outthe time instants tj , j = j, j+1, ..., L since it is well knownthat, for observable linear systems (which are a particularcase of linear switching systems), the current state may bereconstructed only for every time strictly greater than theinitial time.

Denote by T (S) the transfer matrix associated with alinear system:

S :{

x(t) = Ax(t) + Bu(t),y(t) = Cx(t), t ≥ 0,

i.e. T (S)(s) = C(sI−A)−1B, s ∈ C, where I is the identitymatrix. By a straightforward generalization of results in [5],one obtains:

Proposition 1: A linear switching system S is observableif and only if:

• qi �= qh ⇒ T (S(qi)) �= T (S(qh)), a.e., ∀qi, qh ∈ Q;• S(qi) is observable ∀qi ∈ Q.Remark 3: The first condition of the result above, is

necessary and sufficient for the location observability ofS, while the second condition ensures the continuous stateevolution reconstruction. Notice that a necessary conditionfor reconstructing the continuous component of the hybridstate is that the switching system is location observable.

III. INTERCONNECTION AND OBSERVABILITY

In this section we introduce a notion of interconnectionfor switching systems. Moreover we investigate relationshipsbetween observability and interconnection of switching sys-tems.

Given a pair of switching systems S1 and S2, the input–output interconnection between S1 and S2 is a switchingsystem S1||S2, whose hybrid inputs are the ones of S1,hybrid outputs are the ones of S2 and where hybrid outputsof S1 coincide with hybrid inputs of S2. More formally:

Definition 5: (Input–Output Interconnection) Given a pairof linear switching systems:

S1 = (Ξ1, Θ1,Υ1,S1, E1, γ1, R1) ,

S2 = (Ξ2, Θ2,Υ2,S2, E2, γ2, R2) ,

where Υ1 ⊂ Θ2, the interconnection S1||S2 between S1 andS2 is a linear switching system S = (Ξ, Θ,Υ,S, E, γ, R),where:

• Ξ =⋃

q∈Q {q}×X(q) is the hybrid state space, where:

◦ Q = Q1 × Q2 is the discrete state space;◦ X(q) = X1(qi)×X2(qh) is the continuous state spaceassociated with the discrete state q = (qi, qh) ∈ Q;

• Θ = Θ1 is the hybrid input space;• Υ = Υ2 is the hybrid output space;• S is a map associating to each discrete state (qi, qh) ∈ Q

the input–output interconnected linear system [10]:

S1(qi)||S2(qh) :{

x(t) = Aihx(t) + Bihu(t),y(t) = Cihx(t),

where:

Aih =[

A1(qi) 0B2(qh)C1(qi) A2(qh)

], Bih =

[B1(qi)

0

],

Cih =[

0 C2(qh)],

and Ak(q), Bk(q), Ck(q) are dynamical matrices asso-ciated with linear system Sk(q), k = 1, 2;

• E is the set of transitions of the form e =((q1, q2), σ1, (q′1, q′2)) where e1 = (q1, σ1, q

′1) ∈ E1 and

one of the following conditions hold:◦ (q2, γ1(e1), q′2) ∈ E2;◦ (q2, γ1(e1), q′2) /∈ E2 and q2 = q′2;given e = ((q1, q2), σ1, (q′1, q′2)) ∈ E, set e1 =(q1, σ1, q

′1) and e2 = (q2, γ1(e1), q′2);

• γ : E → P2 associates to every e ∈ E the followingdiscrete output:

γ(e) ={

γ2(e2), if e2 ∈ E2,ε, otherwise;

• R is the reset function that associates to every e ∈ Ethe following matrix:

R(e) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

[R1(e1) 0

0 R2(e2)

], if e2 ∈ E2,

[R1(e1) 0

0 I

], otherwise,

where I is the identity matrix with appropriate dimen-sions.

Remark 4: Definition above of switching systems inter-connection has been obtained by combining classical notionsof interconnection of DESs (see for example [19]) withinput–output interconnection of linear systems, e.g. [10]. Inparticular while [19] assumes that, given a pair of DESs D1

and D2, transitions of D1 causes transitions of D2 in D1||D2

and vice versa, we assume in Definition 5 that transitions ofS1 causes transition of S2 in S1||S2 while the converse isnot allowed; this being motivated by the input–output notionof interconnection that we are focusing on here.

Remark 5: By definition above and given the semanticsof linear switching systems, input–output interconnectionof linear switching systems is non–blocking, i.e. for anypair of switching systems S1 and S2, the switching systemS1||S2 is non–blocking. It is worth to point out that classicalinterconnection of DESs may lead to blocking behaviours.However since in our framework the semantics of switching

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systems (as formalized in Definition 2) allow the discretestate of S1||S2 to remain in a given location for any time,(while in DESs classical formulation this is not allowed), theDESs blocking behaviour cannot occur.

As in the classical interconnection of DESs (see e.g. [19]),interconnection of a pair of switching systems S1 and S2

with no silent transitions may give rise to a switching systemS1||S2 with silent transitions, as the following exampleshows.

Example 1: Consider two switching systems:

S1 = (Ξ1, Θ1,Υ1,S1, E1, γ1, R1) ,

S2 = (Ξ2, Θ2,Υ2,S2, E2, γ2, R2) ,

where:

• Ξ1 = {q1, q2, q3} × R2, Θ1 = {σ} × R, Υ1 ={p1, p2} × R, E1 = {e1, e2, e3, e4}, e1 = (q1, σ, q2),e2 = (q2, σ, q3), e3 = (q3, σ, q2), e4 = (q2, σ, q1),γ1(e1) = γ1(e3) = p1, γ1(e2) = γ1(e4) = p2,R1(ei) = I, i = 1, ..., 4;

• Ξ2 = {q4, q5}×R2, Θ2 = {p1, p2}×R, Υ2 = {p3}×R, E2 = {e5, e6}, e5 = (q4, p1, q5), e6 = (q5, p2, q4),γ2(e5) = γ2(e6) = p3, R2(e5) = R2(e6) = I,

and transfer matrices associated with linear systems in-volved are such that:

T (S1(q1)) = T (S1(q3)) �= T (S1(q2)), a.e. (2)

T (S2(q4)) = T (S2(q5)).

DESs associated with switching systems S1 and S2 aredepicted in Figure 1 and DES associated with S1||S2 isdepicted in Figure 2. No silent transitions are in S1 and S2,while Figure 2 shows that transitions outgoing from discretestates (q1, q5), (q2, q4) and (q3, q5) are silent.

As in the case of linear systems, interconnection ofobservable switching systems may lead to an unobservableswitching system. In fact, suppose to consider a pair ofobservable switching systems S1 and S2; Proposition 1implies that S1||S2 is unobservable if and only if one ofthe following conditions hold:

• There exist q1 ∈ Q1 and q2 ∈ Q2 such thatS1(q1)||S2(q2) is unobservable;

• There exist q1, q3 ∈ Q1 and q2, q4 ∈ Q2 such that:

T (S1(q1)||S2(q2)) = T (S1(q1)) · T (S2(q2)) == T (S1(q3)) · T (S2(q4))= T (S1(q3)||S2(q4)).

While the first condition above is a direct consequence ofstandard linear system theory, the second condition is a par-ticular phenomenon arising in the context of interconnectedhybrid systems. The following example aims at clarifyingthis phenomenon.

Example 2: Consider two switching systems:

S1 = (Ξ1, Θ1,Υ1,S1, E1, γ1, R1) ,

S2 = (Ξ2, Θ2,Υ2,S2, E2, γ2, R2) ,

where:

Fig. 1. DESs associated with switching system S1 (in the left) and withswitching system S2 (in the right) of Example 1.

Fig. 2. DES associated with interconnected switching system S1||S2 ofExample 1.

• Ξ1 = {q1, q2} × R2, Θ1 = Υ1 = {σ} × R, E1 ={e1, e2}, e1 = (q1, σ, q2), e2 = (q2, σ, q1), γ1(e1) =γ1(e2) = σ, R1(e1) = R1(e2) = I;

• Ξ2 = {q3, q4} × R2, Θ2 = Υ2 = {σ} × R, E2 ={e3, e4}, e3 = (q3, σ, q4), e4 = (q4, σ, q3), γ2(e3) =γ2(e4) = σ, R2(e3) = R2(e4) = I;

and for any i, h, linear systems Si(qh) are observable andcharacterized by the following transfer functions:

T (S1(q1))(s) = s+1s(s+2) ; T (S1(q2))(s) = s+1

s(s+4) ;

T (S2(q3))(s) = s+3s(s+4) ; T (S2(q4))(s) = s+3

s(s+2) ; s ∈ C.

DESs associated with switching systems S1 and S2 aredepicted in Figure 3. Switching systems S1 and S2 areobservable since they satisfy conditions of Proposition 1.Moreover linear systems S1(q1)||S2(q3), S1(q1)||S2(q4),S1(q2)||S2(q3) and S1(q2)||S2(q4) are observable. However

T (S1(q1)||S2(q3)) = T (S1(q2)||S2(q4))

Fig. 3. DESs associated with switching system S1 (in the left) and withswitching system S2 (in the right) of Example 2.

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and hence by Proposition 1, switching system S1||S2 is notobservable.

IV. OBSERVABILITY OF INTERNAL VARIABLES

Given a pair of switching systems S1 and S2, variablesarising in the interconnected switching system S1||S2 maybe classified as external and internal variables:

• External variables of S1||S2 are hybrid inputs (σ1, u1)and hybrid outputs (p2, y2);

• Internal variables of S1||S2 are hybrid state (q1, x1) ofS1, hybrid state (q2, x2) of S2 and hybrid interconnec-tion variable (p1, y1).

Hybrid interconnection variable (p1, y1) is known in theliterature as (hybrid) latent variable (see e.g. [18]). Variablep1 will be called discrete latent variable.

In this section, we derive some conditions that ensureobservability of internal variables of S1||S2 on the basis ofthe knowledge of the external variables of S1||S2.

More formally, denote by v any internal variable takingvalue in V , in the interconnected switching system S1||S2.The following table reports internal variables of S1||S2 thatwe are interested to observe:

v : q1 q2 (q1, x1) (q2, x2) p1 (p1, y1)V : Q1 Q2 Ξ1 Ξ2 P1 Υ1

The following observability definition of internal variablev is considered.

Definition 6: (Observability of internal variables) Given apair of switching systems S1 and S2, consider the intercon-nected switching system S1||S2. Internal variable v is said tobe observable from S1||S2, if there exist a function u ∈ U ,a function v : Y0 ×U → V and a time t > 0 such that:

v(

yo|[t0,t] , u|[t0,t)

)= v(t, j),

∀t ∈ (tj , t′j] ∩ [t,∞), ∀j = j, j + 1, ..., L,

j = min{j : t ∈ Ij},

for any execution χ with control input u.In the following picture we summarize relationships

among observability of switching systems S1, S2 and S1||S2

and of internal variables from S1||S2. We do not include re-lationships between location observability and observabilityof S1, S2 and S1||S2 for the sake of simplicity.

Loc Obsof S1

← Loc Obsof S1||S2

→ Loc Obsof S2

↑ ↙ ↘ ↑Obs of q1

fromS1||S2

→ Obs of p1

from S1||S2

Obs of q2

fromS1||S2

↑ ↑ ↑Obs of(q1, x1)fromS1||S2

−→Obs of(p1, y1)

from S1||S2

Obs of(q2, x2)

fromS1||S2

↓ ↖ ↗ ↓Obs of S1 ← Obs of S1||S2 → Obs of S2

By table above, observability of switching system S1||S2

(as in Definition 3) implies observability of all internalvariables of S1||S2 (as in Definition 6). However, by focusingon each internal variable to be observed, some weakerconditions can be assessed on the switching system S1||S2,as the following results highlight.

The first result gives a checkable necessary and sufficientcondition for characterizing observability of discrete state q i

from S1||S2, i = 1, 2.Theorem 1: Internal discrete state q1 of switching system

S1 (resp. q2 of switching system S2) is observable fromS1||S2 if and only if for any (q1, q2), (q3, q4) ∈ Q such thatq1 �= q3 (resp. q2 �= q4):

T (S1(q1)||S2(q2)) �= T (S1(q3)||S2(q4)), a.e.

We now address observability of internal hybrid states(q1, x1) and (q2, x2) from S1||S2. For doing so, we needthe following result that gives conditions for reconstructingthe projection of the initial state of a given linear system ona given subspace, on the basis of information coming fromthe continuous output. Given a subspace F ⊂ Rn denote byPF (x) the projection of a vector x ∈ Rn on F .

Lemma 1: Given a linear system:

S :{

x(t) = Ax(t), x ∈ Rn

y(t) = Cx(t), t ≥ 0,

and a subspace F ⊂ Rn, PF (x(0)) is reconstructable fromthe output y of S if and only if ker(O) ⊂ F ⊥, where O isthe observability matrix associated with system S.

The following result shows, under the assumption oflocation observability of S1||S2, a checkable necessary andsufficient condition for the observability of hybrid state(qi, xi) from S1||S2, i = 1, 2.

Theorem 2: Assume that S1||S2 is location observable.Then internal hybrid state (q1, x1) of switching system S1

(resp. (q2, x2) of switching system S2) is observable fromS1||S2 if and only if for any (q1, q2) ∈ Q,

ker(O) ⊂ {0} × X2(q2) (resp. ker(O) ⊂ X1(q1) × {0}),

where O is the observability matrix associated withS1(q1)||S2(q2).

The proof is a direct consequence of Lemma 1 andtherefore is omitted.

We can now characterize observability of internal (hybrid)latent variables from S1||S2. The following result gives asufficient condition for assessing observability of discretelatent variable p1 from S1||S2. Denote by γ−1 the inverseoperator of the function γ, i.e. γ−1(p) = {q : p = γ(q)}.

Theorem 3: Discrete latent variable p1 is observable fromS1||S2, if for any e = ((q1, q2), σ1, (q3, q4)) ∈ γ−1(ε) ⊂ E,the following condition holds

T (S1(q1)||S2(q2)) �= T (S1(q3)||S2(q4)), a.e., (3)

and for any p2 ∈ P2, and any e1 = ((q1, q2), σ1, (q3, q4)),e2 = ((q5, q6), σ2, (q7, q8)) ∈ γ−1(p2) ⊂ E, such that

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e1 �= e2 and γ1((q1, σ1, q3)) �= γ1((q5, σ2, q7)), one of thefollowing conditions hold:

• T (S1(q1)||S2(q2)) �= T (S1(q5)||S2(q6)), a.e.• T (S1(q3)||S2(q4)) �= T (S1(q7)||S2(q8)), a.e.

Remark 6: It is worth to point out that condition (3)ensures the reconstruction of all switching times associatedto any execution of S1||S2, while the latter ones ensureto distinguish between discrete outputs γ1((q1, σ1, q3)) andγ1((q5, σ2, q7)).

Observability of hybrid state of switching system S 1 fromS1||S2 implies observability of hybrid latent variable (p 1, y1)from S1||S2, while the converse implication is not true ingeneral. The following example illustrates a case wherediscrete state of switching system S1 is not observable fromS1||S2, while discrete latent variable is observable fromS1||S2.

Example 3: Consider the switching systems of Example1. By Proposition 1 switching systems S1 and S2 areunobservable and hence in particular discrete state q1 ofswitching system S1 is not observable from S1||S2. However(2) implies:

T (S1(q1)||S2(q4)) �= T (S1(q2)||S2(q5)), a.e.T (S1(q3)||S2(q4)) �= T (S1(q2)||S2(q5)), a.e.T (S1(q1)||S2(q5)) �= T (S1(q2)||S2(q4)), a.e.T (S1(q3)||S2(q5)) �= T (S1(q2)||S2(q4)), a.e.

thus, conditions of Theorem 3 are satisfied and hence discretelatent variable p1 is observable from S1||S2.

In the following we give, under the assumption of locationobservability of S1||S2, a checkable necessary and sufficientcondition for characterizing the observability of internal(hybrid) latent variables.

Theorem 4: Assume that S1||S2 is location observable.Then hybrid latent variable (p1, y1) is observable fromS1||S2, if and only if for any (q1, q2) ∈ Q,

ker(O) ⊂ Im([

C ′1(q1)0

])⊥,

where O is the observability matrix associated withS1(q1)||S2(q2) and C ′

1(q1) is the transpose of C1(q1).The proof is a direct consequence of Lemma 1 and

therefore is omitted.

V. CONCLUSIONS

In this paper, we studied the observability of internalvariables arising in the input–output interconnection of apair of linear switching systems. Sufficient as well as nec-essary conditions for reconstructing the discrete and hybridstate variable of both switching systems involved and thediscrete and hybrid latent variable arising in the intercon-nected switching system are provided. Our current workis focused on the generalization of the results obtained inthis framework to the context of more general notions ofinterconnection as well as to more general classes of hybridsystems.

Acknowledgements: The authors wish to thank Stefano DiGennaro and Alessandro D’Innocenzo for useful discussionson the topic of this paper.

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