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Nonlinear Analysis 63 (2005) 423 – 438 www.elsevier.com/locate/na Observability of the discrete state for dynamical piecewise hybrid systems Salim Chaib a , , Driss Boutat a , Abderraouf Benali a , Jean Pierre Barbot b a Laboratoire deVision et Robotique, Ecole Nationale Supérieure d’Ingénieurs de Bourges, 10 Boulevard de Lahitolle, 18020 Bourges, France b Equipe Commande des Systèmes, Ecole Nationale Supérieure de l’Electronique et de ses Applications, 6 Avenue du Ponceau, 95014 Cergy-Pontoise, France Received 10 May 2005; accepted 10 May 2005 Abstract In this paper, we deal with the observability of piecewise-affine hybrid systems. Our aim is to give sufficient conditions to observe the discrete and continuous states, in terms of algebraic and geometrical conditions. Firstly, we will give the algebraic conditions to observe the discrete state based on the switch function reconstruction for linear hybrid systems. Secondly, we will give a geometrical condition based on the transversality concept for nonlinear hybrid systems. Throughout this paper, we illustrate our propositions with examples and simulations. 2005 Elsevier Ltd. All rights reserved. Keywords: Observability; Hybrid system; Piecewise-affine systems; Switching function; Transversality 1. Introduction Hybrid systems are systems containing mixtures of logic or discrete and continuous dynamics. The hybrid models have become very important in the last few years as tools for modelling systems. Hybrid behavior is generally described as intervals of piecewise This work was supported by “la Région Centre de France”. Corresponding author. E-mail addresses: [email protected] (S. Chaib), [email protected] (D. Boutat), [email protected] (A. Benali), [email protected] (J.P. Barbot). 0362-546X/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.05.028

Observability of the discrete state for dynamical piecewise hybrid systems

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Page 1: Observability of the discrete state for dynamical piecewise hybrid systems

Nonlinear Analysis 63 (2005) 423–438www.elsevier.com/locate/na

Observability of the discrete state for dynamicalpiecewise hybrid systems�

Salim Chaiba,∗, Driss Boutata, Abderraouf Benalia,Jean Pierre Barbotb

aLaboratoire de Vision et Robotique, Ecole Nationale Supérieure d’Ingénieurs de Bourges, 10 Boulevard deLahitolle, 18020 Bourges, France

bEquipe Commande des Systèmes, Ecole Nationale Supérieure de l’Electronique et de ses Applications, 6 Avenuedu Ponceau, 95014 Cergy-Pontoise, France

Received 10 May 2005; accepted 10 May 2005

Abstract

In this paper, we deal with the observability of piecewise-affine hybrid systems. Our aim is togive sufficient conditions to observe the discrete and continuous states, in terms of algebraic andgeometrical conditions. Firstly, we will give the algebraic conditions to observe the discrete statebased on the switch function reconstruction for linear hybrid systems. Secondly, we will give ageometrical condition based on the transversality concept for nonlinear hybrid systems. Throughoutthis paper, we illustrate our propositions with examples and simulations.� 2005 Elsevier Ltd. All rights reserved.

Keywords:Observability; Hybrid system; Piecewise-affine systems; Switching function; Transversality

1. Introduction

Hybrid systems are systems containing mixtures of logic or discrete and continuousdynamics. The hybrid models have become very important in the last few years as toolsfor modelling systems. Hybrid behavior is generally described as intervals of piecewise

� This work was supported by “la Région Centre de France”.∗ Corresponding author.E-mail addresses:[email protected](S. Chaib), [email protected](D. Boutat),

[email protected](A. Benali),[email protected](J.P. Barbot).

0362-546X/$ - see front matter� 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2005.05.028

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424 S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438

continuous behaviors (modes) interspersed with discrete transitions that occur at pointsin time. Examples of hybrid systems include Networks, multi-agent systems, mechanicaldevices, air traffic management where different types of dynamics coexist and interact innonobvious ways, interactive distributed simulation, robot design and path planning.Thecombinationof continuousand thediscretedynamicsmakes thesesystems interesting

and complex such that new mathematical tools are needed for their analysis, observationand control.Several works have been recently carried out on the algebraic properties of hybrid sys-

tems as well as their stability and hyperstability properties[9–12,15,16,20,21]. In the lastdecade many researchers were interested in the study of the observability of such systems,but the definitions and the testing criteria proposed in the literature varied depending onthe class of hybrid systems considered and the type of knowledge on the output. In[2,3]we found sufficient conditions for final state observability and a methodology to designa dynamical observer of a class of hybrid systems that reconstructs the discrete state andthe continuous state from the knowledge of the continuous and discrete outputs. It wasintroduced in[4,5] for the class of piecewise (PWA) systems. It implies that different initialstates always give different outputs independent of the applied input, which is a strongerversion of observability requiring a minimum amount of distinguishability between thestates. Observability in the case of multiple discrete states was discussed in[1,7]. Theauthors in[13,14,24]proposed a definition of observability based on the concept of in-distinguishability of continuous initial states and discrete states from the outputs in freeevolution and gave the necessary and sufficient conditions for the observability of a class ofhybrid systems called jump linear systems. Mixed logical dynamical (MLD) formulationof hybrid systems is given in[4,5,15]. Since the MLD formulation naturally lends itself tobeing analyzed throughmixed-integer linear programming (MILP), and since very efficientalgorithms have recently been developed for MILP, several computational approaches haverecently been proposed to analyze various observability concepts for hybrid systems. In[15], an algorithm based on multi-parametric programming was proposed for computingthe maximal observability region of a discrete-time hybrid system, i.e., the set of initialstates that can only be determined by the output.Based on these works and on the results presented in[6] on the observability analysis

for piecewise-affine hybrid systems without input, in this paper we give the algebraic andgeometrical conditions to observe the discrete state. Our attention will be restricted topiecewise hybrid systems.The outline of this paper is as follows: in the next section, we give some definitions,

notations and problem statement. In Section 3, we give the first results on the observabilityof the discrete state for linear hybrid systems, using the switch function reconstruction.In Section 4, we analyze the observability for nonlinear hybrid systems by means of atransversality concept. We give a simulation in Section 5.

2. Notations and problem formulation

The class of systems considered in this section have the following form:

x(t)= Aqx(t)+ Bqu(t),

y(t)= Cqx(t), (1)

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S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438 425

wherex(t) ∈ Rn is the continuous state,q ∈ Q = {1,2} is the discrete state,u(t) ∈ R isthe input, andy(t) ∈ R is the output.Aq ∈ Rn×n, Bq ∈ Rn×1, Cq ∈ R1×n. In this paperwe deal with a subclass of systems(1) where the discrete state is given by

q = 1 if Hx(t)�0,

q = 2 if Hx(t)>0 (2)

andH is a 1× n matrix. The observability matrixOq of each subsystemq is given by

Oq�

CqCqAq...

CqAn−1q

. (3)

Assumption 1. Weassume that thesystemsunderconsiderationhavenoZenophenomenon.Therefore, a solution to the hybrid system does not exhibit infinitely many discontinuitiesin a finite time interval.

For each integerr�1, we define the following matrix:

Gr�

C1 C2C1A1 C2A2...

...

C1Ar−11 C2A

r−12

. (4)

As we work in the finite-dimensional vector spaceRn, there exists��2n, such that:

rank(Gr)={r ∀ r <�,� ∀ r��.

(5)

More precisely, we have

� =maxr�1

[rank(Gr)]. (6)

We will call � the global joint observability index, andG� the joint observability matrixassociated with system (1).For�>0, we set

h��(H �H). (7)

Finally, we denote

Y(t)�

y

y...

y(n−1)

, U(t)�

u

u...

u(n−1)

,

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426 S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438

�q(k)�

0 0 . . . 0 0CqBq 0 . . . 0 0CqAqBq CqBq . . . 0 0

... · · · . . ....

...

CqAn−1q Bq CqA

n−2q Bq . . . CqBq 0

.

It is easy to show the following observation equation:

Y(t)= Oqx(t)+ �qU(t). (8)

In this paper, we can say that system (1) is observable if we can recover the whole state: thecontinuous and the discrete states.Our problem is to deal with the observability of the discrete state. The idea to solve this

problem is to give the sufficient conditions to know which subsystem of (1) evolves.

3. Algebraic conditions for discrete-state observability

In this section, we give the sufficient conditions for the observability of the discrete stateusing the switch function reconstruction. For this we give the following theorem.

Theorem 1. Let us assume the following conditions:

(i) ∃�>0 and∃k, 0�k�� such that: hT� ∈ Im[GTk ]

(ii) C1Ai1B1 = C2A

i2B2 , ∀i = {0, . . . , k − 1}.

Then, we haveH1x(t) = �H2x(t) and then from(2) we know which subsystem evolves.Therefore, the discrete state is observable for all admissible control inputs.

Proof. If hT� ∈ Im[GTk ] then rank[GT

k hT� ] = k. As rank[Gk] = k this means that vector

h� is a linear combination of the rows ofGk. Then, there exists an� = (�0 . . . �k−1 )T

solution to the following algebraic equation:

GTk

�0

...

�k−1

= hT� . (9)

Then

h� =k−1∑i=0

�i ((C1Ai1)T, (C2A

i2)T) (10)

which gives two values ofHx(t)

H1x(t)= �TO(C1, A1)x(t)

= �T[Y(t)− �1U(t)], (11)

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S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438 427

�H2x(t)= �TO(C2, A2)x(t)

= �T[Y(t)− �2U(t)]. (12)

Both expressions can be written in the following form:

H1x(t)= �0y +k∑

m=1

�m

(y(m) −

m−1∑i=0

C1Am−1−i1 B1u

(i)

), (13)

H2x(t)= 1

[�0y +

k∑m=1

�m

(y(m) −

m−1∑i=0

C2Am−1−i2 B2u

(i)

)]. (14)

If �>0 and thanks to condition (ii),H1x(t) andH2x(t) have the same sign. Thus, we knowwhich subsystem of (1) evolves.�

Remarks. (i) If each subsystem is observable,rank(Oq)=n for q=1and2and if conditions(i) and (ii) of Theorem 1 are fulfilled, then we can recover the whole hybrid state: thecontinuous and the discrete one.(ii) If � = 1; thenH1x(t)=H2x(t); thus, we measure the same switch function.(iii) When � = 2n, we always have� = 1.(iv) For k >1, condition (ii) becomes

(B1,−B2) ∈ ker(G(k−1)), k�1. (15)

Let us assume thatC1=C2=C; then fork�1, condition (15) of the theorem is equivalentfor 0� i�k − 1 to

Ai1B1 − Ai2B2 ∈ ker(C) (16)

(v) If B1=B2=0, then we obtain the same result given in[6] relative to piecewise-affinehybrid systems without control input.

Example 1. Let us consider the hybrid system formed by the following two subsystems:

A1 =(

0 1−1 0

), B1 =

(10

), C1 = (1 0) .

A2 =(

0 1−1 −1

), B2 =

(10

), C2 = (1 0) . (17)

Let us consider the following switch function:

Hx(t)= x1(t)+ x2(t). (18)

We havek = � = 4 and

[GT4 h

T� ] =

1 0 −1 0 10 1 0 −1 11 0 −1 1 �0 1 −1 0 �

. (19)

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428 S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438

Here� = 1 and�T = (1 1 0 0). Condition (ii) of Theorem 1 is fulfilled, which givesus a single value of the switch function given by

H1x(t)= y(t)+ y(t)+ u(t). (20)

Then, we can know which subsystem evolves. Therefore, the discrete state is observable.Now, let us consider the system given by the parametersA1, A2, B1, B2 as in (17);C1

andC2 are given by

C1 = (1 0) , C2 = (0 1) . (21)

We consider the same switch function as in (18),k = � = 4, rankG4 = 4 and for� = 1 wehaverank[GT

4hT� ] = 4. It is easy to see that:

C1Ai1B1 �= C2A

i2B2, ∀i = 0,1,2. (22)

This means that condition (ii) of Theorem 1 is not fulfilled. Thus, we cannot determine thesign of the switch functionHx(t). Therefore, the discrete state is not observable with thealgebraic equality of Theorem 1.

For the linear dynamical systemswhichdonot satisfy thealgebraic conditionsofTheorem1 and for the nonlinear systems, in the next section we will propose another method toobserve the discrete state, based on a geometrical condition.

4. Geometrical conditions

In this section, we give a geometrically sufficient condition to observe the discrete state.We consider nonlinear hybrid systems consisting of the two nonlinear subsystems:

x(t)= fq(x(t))+ gq(x(t))u(t),

y(t)= cq(x(t)),

q = 1 if h(x(t))�0,

q = 2 if h(x(t))>0, (23)

wherefq(x) for q = 1,2 is the drift smooth vector field,gq(x) is the input smooth vectorfields,cq(x) is the smooth output andh(x) is a smooth switch function onR

n.As mentioned in Section 3, if we know at each moment which subsystem evolves then

the discrete state is observable. In the following we will use dynamical equations satisfiedby the output to give another method to observe the discrete state.In this section, we assume that each subsystem of (23) is observable i.e., forq = 1,2 we

have:

rank

dcqdLfq+gqucq

...

dL(n−1)fq+gq ucq

= n, (24)

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S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438 429

whereL is the Lie–Bäcklund derivative given by

Lf+gu� = d�dt

+ Lf� + Lg�u (25)

andLf� is the Lie derivative of a given function� in the direction of the vector fieldf .

Remark. As the following Lie–Bäcklund isomorphisms[18]:

�q1 = cq ,

�q2 = Lfq+gqucq ,�q3 = L2

fq+gqucq...

�qn = Ln−1fq+gqucq (26)

are different for each subsystem(q = 1 or 2) we have a jump on the canonical states whenthe system switches even if the original coordinates do not jump.For each subsystem we have

y(n) = L(n)fq+gqucq . (27)

Next, we set

y(n) = �q(y, y, . . . , y(n−1), u, u, . . . , u(n−1)) (28)

as defined in (27). For a given inputu, if the outputy satisfies one and only one of the twodynamical equations of (28), then, we know which subsystem evolves, and we can thusobserve the discrete state.

Let us now define the following submanifold:

Mu = {v ∈ Rn/�1(v, u, u, . . . , u(n−1))= �2(v, u, u, . . . , u

(n−1))}. (29)

The following assumption is one of the keys to the observability of the discrete state.

Assumption 2. We setLu to be the submanifold of the common singularities of the twosubsystems of (23). IfLu contains more than one element, then we assume thatLu isincluded in only one of the two spaces shared by the switching functionHx(t) i.e., for allx inLu we have

Hx�0 or elseHx >0. (30)

It is clear thatLu ⊂ Mu. The following result gives a sufficient condition to observe thediscrete state.

Theorem 2. Under Assumptions1 and2 and if the two subsystems of(23)are transversetoMu except on a discrete subset, then, there exists a finite time wheny(t) satisfies oneand only one equation of(28).

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430 S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438

Fig. 1. The behavior of the vector fields onMu atv between two instants.

Proof. Thanks to the assumption about the observability of each subsystem, the followingchange of coordinates:

�q1 = y, �q2 = y, . . . , �qn = y(n−1) (31)

transform (23) into the following forms:for subsystem 1:{

�1i = �1i+1 for i = 1 : n− 1,

�1n = �1(�, u, u, . . . , u

(n−1)).(32)

for subsystem 2:{�2i = �2i+1 for i = 1 : n− 1,

�2n = �2(�, u, u, . . . , u

(n−1)),(33)

where�q is given by (28).Let v /∈Mu, asMu is a closed submanifold; then, there exists a neighborhoodV of v

such thatV ∩ Mu = ∅. InV, we have�1(v, u, u, . . . , u

(n−1))|v∈V �= �2(v, u, u, . . . , u(n−1))|v∈V . (34)

This inequality implies that the output satisfies oneandonly oneequation of (28) forq=1,2.If v ∈ Mu ∩ Lu then, by Assumption 2 we can conclude on the observability of the

discrete state. Now, ifv ∈ Mu and v /∈Lu at t = t∗ then,�1(v, u, u, . . . , u(n−1)) =

�2(v, u, u, . . . , u(n−1)) at t∗, but�1(v, u, u, . . . , u

(n−1)) and�2(v, u, u, . . . , u(n−1)) are

transverse toMu except on a discrete subset and thanks to Assumption 1 we have

�1(v, u, u, . . . , u(n−1)) �= �2(v, u, u, . . . , u

(n−1)) (35)

for a certain momentt = t∗ + � (�>0) (seeFig. 1), which leads us to the previouscase. �

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S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438 431

Mu

P1

p2

x1

x2

Vector fieldsof subsysems

1 and 2

Fig. 2. The behavior of the vector fields onM0 for u= 0.

Example 2. Let us consider the system described by the following subsystems:

{x1 = x2,

x2 = −x32 + u= �1(x1, x2, u),

y = x1,

(36)

{x1 = x2,

x2 = x1 + u= �2(x1, x2, u),

y = x1.

(37)

We have

Mu = {(x1, x2) ∈ R2/x1 = −x32} (38)

The two systems (36) and (37) are transverse toMu except on the common singularity(0,0). At the points solution to the following equations:

{x1 = −x32,x2(1− 3x42 + 3x2u)= 0.

(39)

The dynamics of the two subsystems are tangent toMu, the vector fields of the two subsys-tems at these points have two nonnull components along the axisx1 andx2 and therefore, the

discretestate isobservable.Forexample foru=0 thesepointsaregivenbyp1=(− 1

33/4, 131/4

)andp2 =

(1

33/4,− 1

31/4

); the vector fields of subsystems 1 and 2 are tangent toM0 (see

Fig. 2). Therefore, the discrete state is observable for any switch function.

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432 S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438

Fig. 3. A double cart with elastic coupling.

5. Simulation: double cart with elastic coupling

In this section,wegivea linear example in order to illustrate our algebraic andgeometricalmethods. Let us consider the mechanical system described inFig. 3.The linear hybrid model of the plant under investigation is as follows: forz2(t)�0:

m1z1 = −1z1 + 2(z2 − z1)+ 1(z2 − z1)+ u and

m2z2 = −2(z2 − z1)− 1(z2 − z1) (40)

for z2(t)>0:

m1z1 = −1z1 + 2(z2 − z1)+ 1(z2 − z1)+ u and

m2z2 = −2(z2 − z1)− 1(z2 − z1)− 2z2. (41)

Let xT = (x1, x2, x3, x4)= (z1, z1, z2, z2); then, the state representation of (40) and (41) is

x(t)=

0 1 0 0

−1 + 2m1

− 1m1

2m1

1m1

0 0 0 12m2

1m2

− 2m2

− 1m2

x(t)+

0100

u(t), (42)

y(t)= (1 0 0 0) x(t) if x3�0, (43)

x(t)=

0 1 0 0

−1 + 2m1

− 1m1

2m1

1m1

0 0 0 1

2m2

1m2

− 2m2

−1 + 2m2

x(t)+

0100

u(t), (44)

y(t)= (1 0 0 0) x(t) if x3>0. (45)

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S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438 433

0 5 10 15 20 25

Time[s]

Hx(

t) e

stim

ed

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Fig. 4. Real switching function.

We will consider the following two configurations:

(1) The system without damper 1, i.e.(1 = 0).(2) The system with damper 1, i.e.(1 �= 0).

We will study the first one by using Theorem 1. In the second one, if the conditions ofTheorem 1 are not fulfilled, we will then apply the geometrical method. Letm1 = m2 =1Kg, 1 = 2 = 1Nm−1, 2 = 1N sm−1.Case1: (1 = 0):The matricesA1 andA2 are given by:

A1 =

0 1 0 0−2 0 1 00 0 0 11 0 −1 0

, A2 =

0 1 0 0−2 0 1 00 0 0 11 0 −1 −1

. (46)

It is easy to show that,� = 8, butk = 3 and� = (2 0 1)T. We haveC1B1 = C2B2 andC1A1B1 = C2A2B2; then, we estimate a single expression of the switch function given by

Hx(t)= 2y(t)+ y(t)− C1A1B1u(t)− C1B1u(t). (47)

Thus, we can observe the discrete state. Asrank(O1) = rank(O2) = 4 we can observe thewhole state.The control input used for the simulation is sinusoidal.Fig. 4shows the time evolution

of the real switching functionHx(t). The estimated switching function is represented byFig. 5. Fig. 6 shows the evolution of the discrete state of the hybrid system. We observethat we have a series of fast switching atx3= 0, which is explained by the vibrations at thepercussion time.Fig. 7shows the evolution of trajectory in space(x3, x4).

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434 S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438

0 5 10 15 20 25

Time[s]

Hx(

t) e

stim

ed

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Fig. 5. Estimated switching function.

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

Time[s]

q

Fig. 6. Discrete state.

Case2: (1 �= 0):Let 1 = 1N sm−1; in this case the two matricesA1 andA2 are

A1 =

0 1 0 0−2 −1 1 10 0 0 11 1 −1 −1

, A2 =

0 1 0 0−2 −1 1 10 0 0 11 1 −1 −2

. (48)

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S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438 435

x3[m]

x4[m

/s]

Hx(t)=0

subsystem 1 subsystem 2

-1-0.4

-0.4

-0.2

-0.1

-0.2

-0.3

0

0

0.2

0.2

0.1

0.4

0.4

0.3

0.6

0.6

0.5

-0.8 -0.6

Fig. 7. State trajectory(x3, x4).

Using Eq. (28) we have

y(4)(t)= −y(t)− y(t)− 3y(t)− 2y(3)(t)+ u(t)+ u(t)+ u(t). (49)

For the second subsystem

y(4)(t)= −y(t)− 5y(t)− 5y(t)− 4y(3)(t)+ u(t)+ 3u(t)+ u(t). (50)

Mu is a subspace of dimension 3, given by

Mu = {(�2, �3, �4) ∈ R3/2�2 + �3 + �4 = u}. (51)

For piecewise continuous inputs, the transversality of (40) and (41) toMu is verified excepton the subset ofMu defined by{

2�2 + �3 + �4 = u,

�1 − �2 = 0.(52)

In this simulation, we used the same control input as in the first case.Figs. 8and9 show,respectively, the time evolution of the output of the first and of the second subsystem. Thereal output is represented byFig. 10. Fig. 11shows the time evolution of the discrete stateassociated with each subsystem.

6. Conclusion

The main contribution of this paper is the extension of the result presented in[6] topiecewise hybrid systems with control input. We have proposed a method to study theobservability of a class of hybrid systems; a linear and a nonlinear case are considered. We

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436 S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438

0 5 10 15 20 25Time[s]

y[m

]

-0.3

0.3

-0.4

-0.2

0

0.2

0.4

-0.1

0.1

Fig. 8. Output of subsystem 1.

0 5 10 15 20 25Time[s]

y[m

]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Fig. 9. Output of subsystem 2.

gave some sufficient conditions to observe the discrete state of such hybrid systems. Westudied two cases: the first was related to the possibility of the switch function reconstructionfrom the output, input and their derivatives. In the second case, we studied the observabilitybased on a transversality concept.The latter approach is promising for the hybrid observer design, where the discrete-

state observability will be necessary. The hybrid observer design will then be based on

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S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438 437

0 5 10 15 20 25-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time[s]

y[m

]

Fig. 10. Output of the hybrid system.

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

Time[s]

q

Fig. 11. Discrete state.

the observation of both the continuous and the discrete state. The continuous state willbe observed using a family of continuous linear or nonlinear observers; each continuousobserver will be designed using the subsystem forming the hybrid system. The design of thediscrete part of the hybrid observer will be based on a discrete-state observability methodwhich can be inspired by the concepts given in this paper.

Page 16: Observability of the discrete state for dynamical piecewise hybrid systems

438 S. Chaib et al. / Nonlinear Analysis 63 (2005) 423–438

Acknowledgements

The authors are grateful to professor Vincent Maki for his reading, corrections andremarks.

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