Mod´elisation, Identifiabilit´e et Observabilit´e pour le diagnostic et la commande d’une classe de syst`emes Complexes `a Structure Variable. N.K. M’sirdi, L. H. Rajaoarisoa and A. Naamane Laboratoire des Sciences de l’Information et des Syst`emes ; LSIS, CNRS UMR 6168. Domaine Univ. St J´erˆome, Av. Escadrille Normandie - Niemen 13397. Marseille Cedex 20. France. e-mail : [email protected] 24 juin 2007

R´ esum´ e Les progr`es technologiques font que la complexit´e des syst`emes industriels augmente. Ils op`erent dans diff´erents environments avec des conditions et caract´eristiques qui changent (rapidement ou brutalement). L’identification de ces syst`eme avec un mod`ele global unique ne donne pas de bons r´esultats. Il en est de mˆeme pour l’observation car la dynamique pr´esente des changements brusques. L’observabilit´e, l’identifiabilit´e et la controlabilit´e sont analys´ees pour une classe de syst`emes hybrides pouvant ˆetre repr´esent´es par multi-mod`ele avec commutation. La mod´elisation et la structure de commutation doivent ˆetre choisie de mani`ere appropri´ee pour l’identification, l’observation et le diagnostic. Quelques exemples sont consid´er´es pour illustrer les particularit´es des syst`emes consid´er´es.

Table des mati` eres 1 INTRODUCTION

2

2 Formulation du Probl` eme 2.1 Description de la Classe de Syst`emes `a commutation . . . . . . . . . . . . . . . . . . . . . 2.2 Syst`emes Continus par Morceaux (SCM) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 3

3 Stabilit´ e pour syst` emes ` a Commutation 3.1 Outils d’analyse de Stabilit´e . . . . . . . . . . . . . . . . . . 3.2 Exemple 1 : Syst`eme Masse Ressort `a 2 DDL . . . . . . . . 3.3 Cycles Limites Contrˆol´es (CLC) . . . . . . . . . . . . . . . . 3.3.1 Convergence et stabilit´e de mouvements p´eriodiques 3.3.2 Mod`ele Hybride : Supervision et Commutations . . .

. . . . .

4 4 4 6 6 7

4 R´ esultats d’applications 4.1 Syst`emes m´ecaniques `a commutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Identification et zone de fonctionnement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Mod´elisation nominale d’une serre agricole . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 8 9 10

5 Perspectives et conclusions

10

6 R´ ef´ erences Keywords : Structure Variable, Syst`emes `a commutation, Syst`eme Hybrides, multi-mod`eles.

10

1

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

1

INTRODUCTION

Several complex systems operate in different environments with changing features, or the input output characteristics (smoothly or even discontinuously). The description model must change also. For identification or diagnosis if only one standard model is used, it will have to be adapted each time a change appear somewhere before that an appropriate control can be taken. In linear systems, where different environments are described by different functions, a single model may not be adequate to identify the changes, a model may not exists to match the environment in the assumed framework. Hence, multiple models are required both to identify the different dynamics as well as to control them rapidly [2][1]. Dynamics of several physical systems can be well described by use of ordinary differential equations despite the fact that some times nonlinearities are introduced and can be taken into account. In mechanical systems we can find also systems where these descriptions may be multiple and conditioned by some events like contact between two mechanical bodies or activation of some part of the system like braking, for example, in mobile robots and vehicles. If in identification of a such system we use a single model, then it will have to quickly adapt itself to the new environment after each change. An appropriate control can be computed using several models combined adequately in function of the operating region. In literature different approach have proposed use of switches and variable structures with supervision, for complex system. Approaches are based either on a classic linear modeling and control using statistical method [1] or on a hierarchical fuzzy logic used in the same time for models identification and commutation supervision [2], or non linear extension of PCA [3] or driven by a prediction error generated [4], or using a bank of Kalman filters [5] or Markov chains [6]. We suggest here a modeling approach based on behavioral models. Different models are supervised by discrete events which affect system operating zone, and switched via different selectors associated to events. In several applications involving this kind of complex systems, learning approaches are applied such as neural networks, fuzzy logic based methods or neuro fuzzy techniques [2]. These approaches uses black box representations or combined black box models with learning and data processing methods, like eg Principal Component Analysis to extract information which will be used for the system supervision [3]. In several cases these models are local representations, operating points dependant and do not allow physical interpretations Another way may be to combine local grey box representations based on partial knowledge on the process and the involved phenomena. Several possible subsystems, local models are combined in a hybrid representation. The interconnected subsystem and structure are driven by some discrete events which are present and depend on involved phenomena. Some knowledge on this kind of events is necessary to built up detection procedures in order to be able to follow the structure commutations and then estimate the models combination and their parameters. We can propose some structures for the sub models and nonlinear functions combining commutations and switching between structures. Then we have to define some methods for supervision and Fig. 1 – Varying Structure Systems with non lineacontrol of the main partial models. In general, the rities and commutations. global process needs first to be stabilized in its global behavior around some operational point. This corresponds to some operating conditions for the main components or subsystems in coordination of the discrete events. Switching and commutations have also to be managed.

2 2.1

Formulation du Probl` eme Description de la Classe de Syst` emes ` a commutation

Definition of a class of systems having variable structures, commutations in their dynamic behavior, non linearities hard or smooth, non stationarity, varying parameters and other non standard features, is difficult to be done in general. So we restrict ourself, here in a first step, to some simple situations

2

with known involved physical phenomena. Figure 1 can be obviously used to depict the features we are interested by. In figure 1 we can remark that several models (Mp )are involved. We can consider multi-model approaches [7]. Use of these models and even their combinations is orchestrated by some occurring events noted discrete event in figure 1. Switching between models or ODEs (Ordinary Differential Equations) or combinations can appear. It is driven by a Commutation Logic. The same holds for the pertinent variables selection at each time interval or period of operating. It is obvious that importance of output variables and input commands may change in function of the selected model(s). Some models combinations may also be used in function of : time periods leading to selection of some kind or category of model equations, some running logic in the energetic behavior or command of the system. some connection and or interaction with other dynamics or environment Discrete event systems can be considered for supervision of the switchings and commutations which drive the model selection and the the multi-model representation. The switchings and commutations often appear abruptly but changes from one representation to another one may be very smooth or not. In another hand we must note also that such systems representation is not unique and differences can appear between the behavioral representation and physical system description or modeling for diagnosis and control. This can be referred to as a switched multi-model representation driven by some Discrete Event Systems.

2.2

Syst` emes Continus par Morceaux (SCM)

Discrete Events System formalism is developed by Bernard Zeigler in the mid-seventies [16]. It allows representing all the systems whose input-output behavior can be described by sequence of events, called ”trace” [9]. The discrete events taken into account and the choice of their effect may be driven by some higher level or simply selected according of optimization of some criterion or performance index [8] [9] [16]. The system is defined by the following contents : - A set of sub-models, representing process in different regions Dp ⊂ Rn of dynamic state space vector x(t) ∈ Rn - Switching rules or conditions for each operating zone which drives the switchings between models. Suppose that we have several simple sub- models : x˙ = fp (x, u, t), t ∈ R for p = 1, 2, ...q and that each one of these models is valid in some state subspace Dp ⊂ Rn . u is the set of input of the system. We can distinguish two events kinds : ”exogenous events”, setting by the external environment and ”endogenous events” which are state switches result (Bel et al., 2001). The moment where events appear are called ”occurrence date” and action during the time delimited by two events called ”Activity”. So, when one event appears, the system switches from one behavioral state, noted S, to another and it’s done instantaneously. Formally, the model (Figure 1) is defined by the following 7-tuple : Hed = { S, I, O, δ int , δ ext , λ, tαi }

(1)

Such that : S is the discrete state of the system (i.e. depending on the region of operation and valid sub-model). The discrete system is in Sp when operation in the region Dp . I is the set of input event values (i.e. all the values that an input event can take). O is the set of output event values λ isthe output function which warrants the activity execution. δ int is the internal transition function. It assures state switches when no exogenous events come out before courses time ti and liveness of state [14] tαi : Sj = δ int (Si , ti + tαi ) δ ext is the external transit function. This function is used when exogenous events come out [14] : Sj = δ ext (Si , ti , Ii ) To describe the global system’s dynamic behavior, we have to gather all the locally valid model

3

equations and then : .

x = fp (x, u, t) t ∈ R+ and x(t) ∈ Dp ⊂ Rn Sed = {S, I, O, δint , δext , λ, ta }

(2) (3)

The events may be considered as defined by state x of the system and any internal x-dependant variable, and then, events depend on input values of the Sed system noted I(t) [10],. As consequence the validity domain Dp (I(t)) depends also on this input.

3 3.1

Stabilit´ e pour syst` emes ` a Commutation Outils d’analyse de Stabilit´ e

The need of tools for stability analysis and control design, when switched systems are involved, is important. In this section we simply recall useful recent result presented for Hybrid systems and using multiple Lyapunov functions as a generalization of the Lyapunov’s second method. Stability proof depends on the existence and/or construction of an appropriate Lyapunov candidate function V and is rather not obvious for hybrid systems. The inherent discontinuous nature of hybrid system suggests use of multiple Lyapunov functions concatenated together in function of sub models commutations and transitions. This may produce a non-traditional multiple Lyapunov functions useful to prove stability [13], [14][15] of the hybrid system. Let us recall a useful theorem based on the second Lyapunov method for stability analysis. A Lyapunov function for the system (2), at an equilibrium point xep in the domain Dp is real valued function Vp (x) defined in the domain Dp satisfying the conditions : (C1) : Positive definiteness : Vp (0) = 0 and Vp (x) > 0 for x 6= 0 ∂Vp (x) (C2) : Negative derivative : for any x ∈ Dp : V˙ p (x) = ∂x fp (x, u, t) ≤ 0 Theorem 1 Given an P -switched non-linear system, suppose that each vector field fp (x, u, t) has an associated Lyapunov function Vp (x) in the domain Dp , each one defined for the equilibrium point xe = 0. Let Sk+1 be a switching sequence of the discrete state such that Sk+1 can take values p only if xk+1 ∈ Dp , and in addition : (C3) : Vp (x(tp , k + 1)) ≤ Vp (x(tp , k)) for all tpk the switching times Beginning with different assumptions, this more general result assumes a so-called weak Lyapunov function for Vp , in which condition (C3) is replaced by : (C4) : Vp (x(t)) ≤ h.Vp (x(tp ) with t ∈ (tp , tp+1 ) thus, the set V contains a number of candidate Lyapunov functions that are used as a measure of the hybrid system energy, V = {V1 , . . . , Vp }. Since the energy changes according to : ∂Vq (x) V˙ q (x) = ∂x fp (x, u, t) ≤ 0 for an arbitrary Vq ∈ V , this means that the change of energy depends on the vector field fp (x, u, t) and thus on the discrete state Sp . To express where in the continuous state space the energy decreases when there is switching from the Lyapunov function Vq to Vr , the following sets are defined [18] : Dpq = ∂Vq (x) {x ∈ Rn | ∂x fp (x, u, t) ≤ 0} q DR = {x ∈ Rn | Vq (x) ≥ Vr (x)} So, assume that the candidate Lyapunov functions Vq and Vr are used as a measure of the energy for different discrete behavioral states Si and Sj , and consider i 6= j and q 6= r. If the discrete state is Si and the threshold point Rij is reached, then the discrete state becomes Sj , implying that the vector is changed.

3.2

Exemple 1 : Syst` eme Masse Ressort ` a 2 DDL

Several examples may be considered like vehicles, mobile robots, legged robots, greenhouses and flying robots. A one DoF mass-spring system has been considered in our previous work [11][12]. The system

4

switches between two ODEs. ξ(z) = 12 (1−sign(z −zo )) is equal to unity when the spring is in contact with . . ground (we have the ODE 1 x = f1 (x, u, t)) and zero otherwise (we then have the ODE 0 x = f0 (x, u, t)). 2 DOF Mechanical system (see figure (2)). The system is made of two springs (whit stiffness kl and kr we assume kr = kl = k) connected to a mass M,. The spring are attached at distance l from the gravity center whit (respectively) two displacements ul and ur assumed controllable (see figure (2)). The system is in translation along the vertical axis z, and the masse otation is noted θ. z is the height of the gravity center. zr and zl are the positions of the springs and I is the mass inertia. ˙ [11]. ˙ θ, θ) Fig. 2 – Mass - Spring 2DOF System The state vector is (z, z, – Contact Phases. In these phases the controls are active when the springs are in contact. - The two springs are in contact : x˙ = f1 (x, u, t) t ∈ R+ and x(t) ∈ D1 ⊂ R4  x˙ 1    x˙ 2 x ˙    3 x˙ 4

= x3 = x4 kl kr = z¨ = − M (zl − zl0 − ul ) − M (zr − zr0 − ur ) − g lkl lkr ¨ = θ = I (zl − zl0 − ul ) − I (zr − zr0 − ur )

- The right spring is in contact : x˙ = f2 (x, u, t) t ∈ R+ and x(t) ∈ D2 ⊂ R4  kr x˙ 3 = z¨ = − M (zr − zr0 − ur ) − g lkr ¨ x˙ 4 = θ = − I (zr − zr0 − ur )

(4)

(5)

- The left spring is in contact : x˙ = f3 (x, u, t) t ∈ R+ and x(t) ∈ D3 ⊂ R4 

kl x˙ 3 = z¨ = − M (zl − zl0 − ul ) − g lkl ¨ x˙ 4 = θ = I (zl − zl0 − ul )

– Flying Phase : x˙ = f4 (x, u, t) t ∈ R+ and x(t) ∈ D4 ⊂ R4  x˙ 3 = z¨ = −g x˙ 4 = θ¨ = 0

(6)

(7)

The system has a ballistic trajectory determined by knowledge of the mass lift off velocity vd = z˙d (t = 0). Let us note tαf the flight phase duration. During the flight motion the system has a purely ballistic trajectory with as initial velocity z˙d and position zd . It is worthwhile to note that the system is uncontrolable in the flying phase. Control inputs do not appear in the system equations. In the succeeding phases changes appear only in the last two equations. let us consider the functions ξr (zr , ur ) and ξl (zl , ul ) equating 0 in the flying phase and 1 when there is contact whit the corresponding spring. ξr (zr , ur ) = 12 (1 − sign(zr − z0 − ur )) and ξl (zl , ul ) = 12 (1 − sign(zl − z0 − ul )) The robot model can be gathered as follows :  k k x˙ 3 = z¨ = −ξl (zl , ul ) M (zl − z0 − ul ) − ξr (zr , ur ) M (zr − z0 − ur ) − g lk lk ¨ x˙ 4 = θ = ξl (zl , ul ) I (zl − z0 − ul ) − ξr (zr , ur ) I (zr − z0 − ur ) zr et zl can be expressed in function of z if we assume θ small : zl = z − l sin θ ≈ z − lθ and zr = z + l sin θ ≈ z + lθ

5

(8)

3.3

Cycles Limites Contrˆ ol´ es (CLC)

For control the main objective is to damp any rotational motion and maintain hopping along z axis. We want to stabilize a periodic motion. Let us consider the following Lyapunov function (energy of the system) : VLC = 12 z˙ 2 + gz I ˙2 (9) V = 12 z˙ 2 + gz + 2M θ V = VLC + VT whit VT = I θ˙2 2M

Energy is split in two parts : VLC the energy corresponding to the desired periodic hopping motion and VT the transverse motion energy. It is clear that one of this energy has to be regulated to some level and the other must be damped. The right an left control inputs are ξr ur =

1 2

(uLC + uT )

ξl ul =

1 2

(uLC − uT )

(10)

uT is the control which has to damp transverse energy VT and then rotational motions. This has as consequence to keep the system state in the plane (z, z) ˙ with θ = θ˙ = 0. The uLC has to stabilize periodic cycle (cyclic motion). The two control inputs uT and uLC have to be applied in the time period where the corresponding spring is in contact with ground. This is made by displacement of the springs attach points ur and ul . 3.3.1

Convergence et stabilit´ e de mouvements p´ eriodiques

The transverse motion and its energy VT have to be damped. Let us use as Lyapunov candidate function V1 ; its time derivative is : V1 = 21 VT2

V˙ 1 = VT V˙ T

V˙ T =

I ˙¨ M θθ

(11)

using expression (8), leads : lk V˙ T = (ξl (zl − zl0 ) − ξr (zr − zr0 ) + (ξr ur − ξl ul )) θ˙ M

(12)

Substituting controls ξr ur , ξl ul by equation (10) in V˙ T , we have : lk V˙ T = (ξl (zl − zl0 ) − ξr (zr − zr0 ) + uT ) θ˙ M

(13)

We propose a transverse control input uT as follows : uT = ξr ur − ξl ul = −Γ1 ψ(VT )θ˙ − ξl (zl − zl0 ) + ξr (zr − zr0 )

(14)

ψ is a positive function and can be sign or saturation function ; we then obtain : lk V˙ T = − M Γ1 ψ(VT )θ˙2

V˙ 1 = VT V˙ T = −Γ1 VT ψ(VT )θ˙2 ≤ 0

(15)

VT V˙ T is negative then the transverse energy VT converges to zero. We can conclude that ∀1 > 0, ∃t1 ≥ 0, such as | VT |< 1 , ∀t > t1 2 Consequently θ˙ is bonded and goes to zero. The constant λ is chosen such as : Γ1 θ˙max ≤   θ˙max = max θ˙ .

V˙ 1 ≤ − √λ2 |VT |

V1 = 12 VT2 ⇒ VT =

√

√ 2V1 ⇒ V˙ 1 ≤ −λ V1

which implies finite time convergence to the attractive sliding surface VT = 0 : p V1 (t1 ) V1 (t) = 0 for t ≥ t1 + 2 and VT = 0 λ

6

√λ , 2

avec

∗ . Let us choose another Lyapunov candidate Convergence of VLC to desired reference VLC function 1 1 ∗ 2 V2 = VT2 + (VLC − VLC ) (16) 2 2 ∗ the constant reference energy is defined at the lift off point z˙ or at the maximal desired height VLC d zmax : 1 ∗ ∗ (zmax , 0) = gzmax VLC (0, zd ) = z˙d2 or VLC 2 when t > t1 we have (VT , V˙ T ) = (0, 0), then ∗ V˙ 2 = (VLC − VLC ) V˙ LC

with V˙ LC = (¨ z + g) z˙ =



∀t > t1

k k k −ξl (zl − zl0 ) − ξr (zr − zr0 ) + uLC M M M

 z˙

∀t > t1

(17)

we propose as control input : ∗ uLC = ξr ur + ξl ul = −Γ2 ψ(V − VLC )z˙ + ξl (zl − zl0 ) + ξr (zr − zr0 )

(18)

k ∗ )z˙ 2 . Then such as the derivative V˙ 2 will be negative : V˙ LC = − M Γ2 ψ(V − VLC

k ∗ ∗ ∗ V˙ 2 = (VLC − VLC ) V˙ LC = − Γ2 (VLC − VLC )ψ(VLC − VLC )z˙ 2 ≤ 0 M The second Lyapunov function ensures global asymptotic convergence of the system trajectories z to ˙ converge to zero with the following control : the orbit Ω θ(t), θ(t)  uT = −Γ1 ψ(VT )θ˙ − ξl (zl − zl0 ) + ξr (zr − zr0 ) (19) ∗ )z˙ + ξ (z − z ) + ξ (z − z ) uLC = −Γ2 ψ(V − VLC r r r0 l l l0 3.3.2

Mod` ele Hybride : Supervision et Commutations

Supervision device is composed by the following contents which represented in the figure 4 : - Input variables are given by ξ(z) . It can get the following values ξ(z) which is 0 during flight and equal to 1 in contact phase. So, we can considerate the following table which give us the input values each moment of transition. - We have four states such that : * Two springs are in contact * Left spring is in contact Fig. 3 – State Behavioral Model * Right spring is in contact and finally * Two springs are in flight

State Contact RightContact LeftContact Flight

IrC 1 1 0 0

Input IlC IrF 1 0 0 0 1 1 0 1

IlF 0 1 0 1

Tab. 1 – Transition table by Input values

7

State S1 S2 S3 S4

Name Contact RightContact LeftContact Flight

Input Variable IrC , IlC IrC , IlF IrF , IlC IrF , IlF

Tab. 2 – State designation of the supervision device

- The set of the output variables are given by the selector element device (Select1 and Select2). These selectors are associated respectively to Contact model and Flight model. Thus, we can write : −Oc = Select1 = 2contact Contact model = 1 −OrC = Select2 = Right Contact model = 2 For Contact model = 3 Fig. 4 – (a) Flight State to Right Contact b) −OlC = Select3 = Lef t model = 4 Right Contact State to Left Contact (c) Left −Of = Select4 = F light beginning the simulation, we consider as initial state Contact to Flight and (d) Execution of Flight the Flight phase. Thus one has : Initial state : S = S4 = (F light, IrF , IlF ) ( S2 if I = IrC ∗ ∗ δext = δext (F light, ti , Ii ) = S with S = S3 if I = ( IlC Or δint = δint (S ∗ , tαf = ∞) = S ∗ λ(S ∗ ) = O∗ with O∗ = Ol = (RightContact, I , I ) Set S = S 2 rC lF   S if I = I 1 lC  δext = δext (RightContact, ti , Ii ) = S ∗ with S ∗ = S3 if I1 = IrF and I2 = IlC   S4 if I = IrF  ∗  OrF if S = S4 δint = δint (S ∗ , tαf = ∞) = S ∗ λ(S ∗ ) = O∗ with O∗ = OlC if S ∗ = S3   OC if S ∗ = S1 These two phases are illustrated in Figure(4ab ). Now, let S = S3 = (Lef tContact, IrF , IlC ) as illustrated in Figure(4cd )   S1 if I = IrC ∗ ∗ δext = δext (Lef tContact, ti , Ii ) = S with S = S2 if I1 = IrC and I2 = IlF   S4 if I = IlF  ∗  OrF if S = S4 δint = δint (S ∗ , tαf = ∞) = S ∗ λ(S ∗ ) = O∗ with O∗ = OrC if S ∗ = S2   OC if S ∗ = S1 In short, the effectiveness of this approach depends especially on exact estimation of events to observe, so that the output values on the level of the selectors is a value which selects, as well as possible, the best model to use for calculating the continuous state of the system.

4

R´ esultats d’applications

4.1

Syst` emes m´ ecaniques ` a commutations

– First test : we consider that θ0 = 1.5rad and Control inputs uT = −Γ1 ψ(VT )θ˙ and uLC = ∗ )z, −Γ2 ψ(VLC − VLC ˙ whit zl0 and zr0 null ; gain values are : Γ1 = 0.2 et Γ2 = 0.008. the desired height zmax = 3.5m. Let us consider as initial condition θ0 = 1.5 rad , a value near to π2 .which is the limit angle for the mass M . The cycle is retrieved after few seconds (see figure (5 and 6)). 8

We remark in this figure that the angle θ converge quickly to zero. With the same conditions for simulation Figure(6) shows the good results with the desired height zm = 3.5m obtain after only 12s of simulation. This corresponds to the necessary time interval to damp the transverse energy VT and stabilize the cycle in vertical direction (VLc goes to its imposed reference (Figure(6)). Figure(7) shows convergence of the angle θ and the Figure (8) illustrated the switching model result.

Fig. 5 – Energy of the system with θ0 = 1.5rad – Second test : : For the sequel, the goal is to show the effectiveness of our supervision and commutation device. In this effect, we change the corrector gain to attempt to retrace all commutation that can exist in our system. Let θ0 = 1.5rad and Γ2 = 0.0002. The simulations show good results with the desired height and in the Figure (8) we can see one more variation of the system Energy. These commutations are illustrated in the Figure(9).

Fig. 6 – Energy of the system with θ0 = 1.5rad and Γ2 = 0.0002

4.2

Identification et zone de fonctionnement

Let us associate the observations via external environment with events notion. The observations will be described [17] to specify a both the events trace and occurrence date where they appear. These events are defined by input value of the system. The output value is calculated Fig. 7 – Stable swing angle of the mass where by comparison with the state where the system is at each instant [10]. As we have said previously, each sub model θ0 = 1.5rad is defined for specific operating point (or region). So, we can consider that this operating point as the center of a specific domain, or sub model validity domain, for description of behavior. Let Dq (I(k)) be the validity domain when we are in the state Sq . Sq can switch to Sr , if the threshold Rq,r (I(k)) is true and the input value I(k) defines exactly this validity domain. Thus, we have the switching of q → r if I(k)Dr (I(k)). So, the behavioral state r is got if the following switching rules are respected : Dr (I(k)) > Dq (I(k)) + Rq,r (I(k)) which gives us at the same moment the estimation of the operating point of the system at each appearance of an event and the state or the specific behavior of the system at this instant. Thus, the state j is chosen when fulfilling the following relation : r = arg min Dq (I(k)) + Rq,r (I(k)) and the transition limited area SL can be written : SLq, r = I(k), Dr (I(k)) = Dq (I(k)) + Rq,r (I(k)) The switches from a model to another are instantaneous. This is due to the fact that the events emerge compared to a well defined zone of the system operation. In the goal to build the best prediction of system outputs, we have to get the best switching and supervision device depending on operating point, the behavior and environment. We can remark, as shown in figures, transitions occur such as the system is the more often in either the flying phase or in the double contact phase. The single contact phases do not appear often and if 9

Fig. 8 – a) Commutation result for the system when θ0 = 1.5rad and Γ2 = 0.0002. b) Representation of the switching sequence of model they occur, they are very short. This means that identifiability is very difficult to obtain in such part of operating. The control is difficult to adjust such as the system gives this phase during a long time. Another important remark is that the system is not controllable every where in the space phase. So this simple examples emphasizes several limitations of the hybrid system modeling using multi model approach. For observability and identifiability, we have to accept that some times short phases will not be observable, despite that a good placement of sensors will allow to detect them and determine the commutation event time. Contact forces are not defined in flying phase or single contact for opposite spring. So they will not be observable. Owing to the fact that some commutations are very short, a nominal representation can be set up. Then it can be used to define an approach to identify, control and supervise such systems represented by switched models. The nominal description will be composed parsimoniously by some selected sub-models. Each sub model switches to another quasi instantaneously depending on some operating points or zone. For example double contact is reached after flying phase in a bounded time interval (maximum depends on the control gains) with a bounded perturbation.

4.3

5

Mod´ elisation nominale d’une serre agricole

Perspectives et conclusions

In this paper we use supervision of discrete events commutations, setting an approach of multi model, for modeling complex systems. The setting device used give possibility of optimizing estimation of the desired output of the system, according to its nominal behavior. The representation is not unique and identifiability is not always fulfilled. The extension of the technique towards the monitoring and the diagnosis of the complex systems (perspective of this work) must be adapted regard to observability and identifiability features we have emphasized. The defaults in system operation or measurement have to be distinguished from commutations or other events to which the system is subjected. The fact that some commutations are very short, a nominal representation can be defined to identify the behavior. A nominal description can be proposed. The presented results emphasize that efficiency of modeling have to be considered building a nominal model to describe the behavior and for prediction. We have shown, in our first results using this approach, an important difference of performance of the prediction regard to the case when using fuzzy logic for estimation and prediction. This approach will be used for diagnosis, fault detection and monitoring. A diagnostic framework will be considered to detect defaults and control the system.

6

R´ ef´ erences

R´ ef´ erences [1] R. Murray-Smith, T.A. Johansen, R. Shorten. Multiple model approaches to modeling and Control. Taylor and Francis London, 1997. [2] J. Duplaix, J.F. Balmat, F. Lafont, N. Pessel, Data Analysis For Neuro-Fuzzy Model Approach, STIC-LSIS, Universit´e du Sud Toulon-Var, Sofa 2005. [3] N. Pessel, J-F. Balmat, N. K. M’sirdi. Analyse discriminante pour le diagnostic – Application ` a une serre agricole exp´erimentale, in : IEEE Conf´erence Internationale Francophone d’Automatique, CIFA 2006, Bordeaux, France, 30-31 Mai et 1er Juin 2006. 10

[4] K. S. Narendra, J. Balakrishnan, M. K. Ciliz, Adaptation and Learning Using Multiple Models, Switching, and Tuning,, IEEE Transactions on Systems, June 1995. [5] D. Theillol, J. C. Ponsart, H. Noura, L. G. Vel Valdes, A Multiple Model Based Approach for Sensor Fault Tolerant Control of Nonlinear Systems, Universit´e Henri Poincar´e, Nancy, 2001. [6] R. Murray-Smith, Modelling Human Control Behaviour with Context-Dependent Markov-Switching Multiple Models, Dept of Mathematical Modelling, Technical University of Danemark, IFAC Man Machine System, 1998. [7] R. Murray-Smith, T.A. Johansen, R. Shorten, On Transient Dynamics, Off Equilibrium Behaviour and Identification in Blended Multiple Model Structures,1997. ´ enements Discrets, LSIS [8] Giambiasi N., Doboz R., Ramat E., Introduction `a la Mod´elisation `a Ev` UMR CNRS 6168, LIL Calais, Rapport interne N LIL-2002-02. [9] Allur R., Dill D. L., A Theory of Timed Automata, Theoretical Computer Science, ACM., vol. 126, pp. 116-146, N◦ 2, 1994. [10] Baruch I., F. Thomas, R. Garrido, E. Gorthcheva, A Hybrid Multimodel Neural Network for Nonlinear Systems Identification, Cinvestavipn, Mexico, grant 611/1996 [11] N.K.M’Sirdi, N.Manamani and N.Nadjar- Gauthier “Methodology based on CLC for control of fast legged robots, Intl. Conference on intelligent robots and systems, proceedings of IEEE/RSJ, October 1998. [12] N.K. M’sirdi, L. H. Rajaoarisoa, J. Duplaix, J. F. Balmat Modeling for Control and diagnosis for a class of Non Linear complex switched systems. AVCS2007, Buenos Aires, Argentina. [13] M. S. Branicky, Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems, IEEE Transactions on Automatic Control, Vol. 43, N◦ 4, April 1998 [14] V. Donde, Development of Multiple Lyapunov Functions for a Hybrid Power System with a Tap Changer, ECE 497DL, Spring 2001 [15] R. A. Decarlo, M. S. Branicky, S. Pettersson, B. Lennartson, Perspectives and Results on the Stability and Stabilizability of Hybrid Systems, Proceedings of the IEEE, Vol. 88, N◦ 7, July 2000 [16] Zeigler, B. Theory of Modelling and Simulation. John Wiley & Sons, New York, 1976. [17] J. Lunze, Discrete Event modelling and Diagnosis of Quantised Systems, Rurh-University Bochum, Germany July 19-22, 2005 [18] S. Pettersson, B. Lennartson, Stability and Robustness for Hybrid Systems, in Proc. 35th IEEE conf. Decision and Control, Kobe, Japan, 1996, pp. 1202-1207.

11