Thierry Floquet

Laurentiu Hetel

Wilfrid Perruquetti

Stabilization of switched linear systems: a sliding mode approach

In this paper, the existence of stabilizing sliding motions in the context of switched linear systems is investigated. We consider the case of switched systems with two subsystems and we propose conditions for the existence of stable sliding hyper-planes. For particular cases of switched systems (planar systems, integrator chains), we provide constructive analytical methods for deriving a stabilizing switching sequence.

INTRODUCTION

Nowadays, control of hybrid systems is extensively studied because of its relevance in numerous applications such as embedded systems. Switched systems can be viewed as higher-level abstractions of hybrid systems, obtained by neglecting the details of the discrete behavior.

Roughly speaking, a switched system is made of a family of dynamical subsystems (linear or nonlinear), and a rule, called the switching law, that orchestrates the switching between them [START_REF] Liberzon | Swithcing in Systems and Control[END_REF]). A taxonomy of switched systems can be defined based on the switching law. One can distinguish the controlled aspect (when the switching function represents a discrete input) from the autonomous aspect (when, for example, switchings due to state space transitions appear). A survey of different switched system classes and the associated stability problems can be found in [START_REF] Liberzon | Swithcing in Systems and Control[END_REF], [START_REF] Decarlo | Perspectives and results on the stability and stabilizability of hybrid systems[END_REF], [START_REF] Sun | Analysis and synthesis of switched linear control systems[END_REF], [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF]). In this paper, we will focus on the controlled aspect of the switching law, that is on the design of switching law that leads to the stabilization of a switched linear system. In particular, we address the case of switched linear systems described by differential equations of the form: ẋ(t) = A σ(t) x(t).

(

) 1
where x ∈ R n represents the state vector and σ(t), σ : R + → I = {1, 2, . . . , N } is a piecewise constant function that represents the switching law. The stabilization problem can be formalized as follows :

Problem : for any t > 0 design the switching signal σ(t) which makes the system (1) asymptotically stable.

Since the problem is trivial when at least one of the A i matrices is Hurwitz stable, it is considered here that none of the matrices A i , i = 1, . . . , N is Hurwitz.

Several authors addressed the switching law design problem in the literature. A necessary condition for the stabilization by switching law was given in [START_REF] Lin | Switching stabilizability for continuous-time uncertain switched linear systems[END_REF]): for at least one index k ∈ I the matrix A T k + A k has a negative eigenvalue. Using quadratic stability arguments, the results of [START_REF] Wicks | Construction of piecewise Lyapunov functions for stabilizing switched systems[END_REF]) provide a state dependent switching law, by verifying the existence of a global stable convex combination. For a system with two modes described by the matrices A 1 , A 2 , this implies finding an α ∈ [0, 1] such that the equivalent matrix

A eq (α) = αA 1 + (1 -α)A 2
(2) is Hurwitz stable. This implies the existence of a symmetric positive definite matrix P = P T ≻ 0 such that A T eq (α)P + P A eq (α) ≺ 0.

(3) For switched systems with two subsystems, the existence of a stable convex combination is strictly equivalent to the quadratic stabilization of the system [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF]), i.e. there exists a stable convex combination if and only if there exists a switching sequence σ * (t) ∀t ∈ [0, ∞) such that the quadratic Lyapunov functions V (x) = x T P x is strictly decreasing along the solutions of the system with σ(t) = σ * (t), ∀t ∈ [0, ∞). Constructive methods for obtaining a state dependent switching law can be obtained using the min-projection strategy [START_REF] Liberzon | Swithcing in Systems and Control[END_REF]). In this case, the system is switching according to conic partitions of the state space.

However, the research of a stable convex combination of matrices is an NP hard problem [START_REF] Blondel | NP-hardness of some linear control design problems[END_REF]). Moreover, these methods are based on quadratic Lyapunov functions which are conservative: there are cases of asymptotically stable switched linear systems that can be stabilized by switching while no quadratic Lyapunov function exists. In fact, it has been recently shown in [START_REF] Blanchini | Stabilizability of switched linear systems does not imply the existence of convex lyapunov functions[END_REF]) that in general the existence of any smooth Lyapunov function is only sufficient for stabilization, but not necessary.

To reduce the conservatism of quadratic Lyapunov functions, several authors (Peleties andDeCarlo [1991], Geromel and[START_REF] José | Stability and stabilization of continuous-time switched linear systems[END_REF]) proposed stabilization conditions based on multiple Lyapunov functions [START_REF] Branicky | Stability of switched and hybrid systems[END_REF]).

Stabilization criteria can be obtained using the S-procedure [START_REF] Rantzer | Piecewise linear quadratic optimal control[END_REF]). The results are expressed in terms of bilinear matrix inequalities (BMIs) which makes the approach difficult to apply in general. Other interesting approaches have been proposed based on polyhedral Lyapunov Functions [START_REF] Lin | Switching stabilizability for continuous-time uncertain switched linear systems[END_REF]) which are necessary and sufficient for stabilization of uncertain switched linear systems if no sliding motion is allowed. Stabilization conditions based on the explicit analysis of vector fields have been proposed in [START_REF] Xu | Stabilization of second-order LTI switched systems[END_REF]).

Sliding mode control has been extensively studied for several years (see ??? for tutorials). Usually, a discontinuous control action is used so that the system trajectories are constrained to lie, after a finite time, within a neighbourhood of a given manifold of the state space. The result is a variable structure system, which may be regarded as a combination of subsystems where each subsystem has a fixed control structure and is valid for specified regions of system behaviour. The resulting behavior, called sliding mode, has some prescribed dynamics totally imposed by the parameters and the equations that define the sliding manifold. Robustness properties against various kinds of uncertainties such as parameter perturbations, external disturbances and measurement errors can be guaranteed. Sliding mode is still the object of many studies for both observation and control problems (see e.g. ???), with a large number of application fields (???).

However, in the case of switched linear systems, most of the existing switching law design approaches try to avoid their existence since they are difficult to characterize.

In this paper, we intend to investigate the existence of stabilizing sliding motions in the context of switched linear systems. The case of systems that switch between two subsystems is considered conditions for the existence of stable sliding hyper-planes are studied. For special cases of switched systems (second-order systems, integrators), constructive analytical methods for deriving a stabilizing switching sequence are provided.

The paper is organized as follows : Section 2 presents general concepts encountered in the domain of sliding mode control and gives a motivating example. In Section 3, conditions for the existence of a stable sliding hyperplane for switched linear systems with two subsystems are given. Numerical examples are given in Section 4.

PRELIMINARIES

Background on sliding modes

Consider a nonlinear controlled system of the form:

ẋ = f (x, u), (4) where f : O × U → R n , with O an open subset of R n and U a subset of R m .
A basic problem in the field of variable structure control is to design a m-dimensional control variable u (which should belong to the given control set U) such that the state variable x has to satisfy some a given constraint s(x(t)) = 0 for all t ≥ T, T > 0 (5) where s : R n → R m is a fixed mapping which defines the sliding manifold S = {x : s(x) = 0}. Such a problem can be tackled by designing u * (x) as a discontinuous control where, in general, the components of u are switching between two functions. As a result the closed-loop system is usually described using a differential inclusion setting, ẋ ∈ F (x, u * (x)), (6) where the vector field f (x, u) in ( 4) with the discontinuous control u * (x) is replaced by the set valued map F (x, u * (x)). The map F is assumed to have good "properties" such as being "nonempty compact convex valued and upper semi-continuous" (ensuring thus existence of classical (i.e., almost everywhere) solutions to the initial value problem). Several approaches can be used for the construction of the set valued map F . One of them is due to Filippov [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]): 7) where conv{A} denotes the closed convex hull of A, i.e., the intersection of all closed convex sets containing the set A. Definition 1. A Filippov solution y to ( 6) is a locally absolutely continuous function y

F (x) = ∩ ǫ>0 conv {f [B(x, ǫ) \ L] : meas L = 0} (
: [0, T ) → R n such that ẏ(t) ∈ F (y(t)) (8) for almost every t ∈ [0, T ).
Thus, Filippov definition replaces the discontinuous differential equation ( 4) by the differential inclusion (6). In this context, assume that m = 1 and that f in ( 4) is switching according:

f (x, u * (x)) = f + (x), if s(x) > 0, f -(x), if s(x) < 0. (9)
Then, F is defined by

F (x) = f (x, u * (x)) if s(x) = 0, lim x→x * ∈S conv {f + (x), f -(x)} if s(x) = 0.
Using the projection operator Pr normal onto the normal to the manifold S (oriented from {x : s(x) > 0} to {x : s(x) < 0}), define

f + n (x) = Pr normal lim s(x)→0 + f + (x), (10)
f - n (x) = Pr normal lim s(x)→0 -f -(x). ( 11
)
Then, Filippov results on the existence of solutions (see [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]) lead to the existence of a "sliding motion" along the manifold S when f + n < 0 and

f - n > 0 1 : x ∈ S, ẋ = f 0 (x), ( 12
) with f 0 (x) ∈ conv {f + (x), f -(x)} ∩ T x S.
The "equivalent dynamic" or "sliding dynamic" is given by:

dx dt = grad s, f - grad s, f --f + f + - grad s, f + grad s, f --f + f -, (13)
where a, b denotes the scalar product of the two vectors a and b. Indeed, the convex hull is defined by

f 0 (x) = αf + (x) + (1 -α)f -(x), α ∈ [0, 1] and f 0 ∈ T x S ⇐⇒ grad s, f 0 = 0, leads to α = grad s,f -(t,x) grad s,(f -(t,x)-f + (t,x)) .

Motivating example

Here we provide an example that motivates the study of the existence of sliding modes in switched linear systems. Consider a second order switched system with two subsystems:

dx dt = A i x, i = 1, 2 (14)
with

A 1 = 1 1/µ -µ 1 , A 2 = T -1 1 -1/µ µ 1 T, (15)
T = cos(π/4) sin(π/4) sin(π/4) cos(π/4) ( 16)

where µ = 0.1. Both systems have unstable complex eigenvalues. The corresponding vector fields are represented in Figures 1.

- It can be shown that no stable combination exists for ( 14) and therefore there is no constant α that solves the bilinear matrix inequality (3). Yet, it is rather intuitive that the system can be stabilized. In fact, through a qualitative analysis, one can define the switching lines Γ 1 x = 0 and Γ 2 x = 0 with Γ

1 = [1 1] and Γ 2 = [0 1], such that σ(t) = 1, if x T Γ T 1 Γ 2 x < 0; 2, otherwise . (17)
Then, the solution of the system exhibits a stable sliding motion along the surface Γ 1 x = 0, as it can be seen in Figure 2.

This implies, from the point of view of Filippov solutions, that, for all x such that Γ 1 x = 0, there exists locally an equivalent convex combination A eq (α(x)) such that the solution of the system converges towards zero on the given surface in spite of the fact that no Hurwitz stable A eq (α) exists with a constant α for all x. This example illustrates the usefulness of defining stable sliding motions for switched systems when no global stable convex combination exists.

Bilinear formulation

Consider a switched system of the form (1) with N = 2. This system can be formulated as a bilinear system ẋ(t) = Ax(t) + Bx(t)u(t) (18) with

A = A 1 + A 2 2 and B = A 1 -A 2 2 ,
where the control u(t) ∈ {-1, 1} is defined by

u(t) = 1, if σ(t) = 1; -1, if σ(t) = 2. ( 19
)
Using this notation, the control law design Problem 1 can be regarded as a particular sliding mode control synthesis problem for bilinear systems. The idea of this work is to investigate the existence of stable sliding modes on the manifold of the form: s(t) = Γx(t) = 0 (20) where Γ ∈ R 1×n is a design parameter, for system (18). The problem is to design a switching control law u(t) ∈ {-1, 1} as well as the set of vectors Γ such that: (i) s(t) = 0 is an attractive and invariant sliding hyperplane, (ii) the resulting sliding motion on s(t) = 0 is asymptotically stable.

Attractivity Conditions

In order to define the region of attractivity of the sliding manifold, we analyze the domain in R n for which

s(t) ṡ(t) < 0, that is s(t) ṡ(t) = s(t) (ΓAx(t) + ΓBx(t)u(t)) < 0.
Consider the following switching rule:

u(t) = -sign(ΓBx(t).s(t)). ( 21
)
Using the switching rule ( 21), one can notice that

s(t) ṡ(t) ≤ |s(t)| (|ΓAx(t)| -|ΓBx(t)|) .
This implies that the sliding manifold is attractive in the domain characterized by |ΓAx| < |ΓBx| i.e.

|Γ (A 1 + A 2 ) x| < |Γ (A 1 -A 2 ) x| .
(22) This condition defines a conic partition of the state space:

x T (A 1 + A 2 ) T Γ T Γ (A 1 + A 2 ) - (A 1 -A 2 ) T Γ T Γ (A 1 -A 2 ) x < 0 (23)
which gives the attractivity domain:

x : x ∈ R n , x T A T 1 Γ T ΓA 2 x < 0 . ( 24
)

Equivalent sliding motion

Here, Γ has to be chosen such that the equivalent sliding motion on s = 0 is asymptotically stable. Under the Filippov definition, the equivalent motion is given by the convex combination of vector fields. This implies that for all t such that Γx(t) = 0, there exist a scalar α (x(t)) ∈ [0, 1] such that the state of the system evolves according to the differential equation:

ẋ(t) = α (x(t)) A 1 x(t) + (1 -α (x(t))) A 2 x(t).
Using the definition of the sliding surface s = Γx = 0, the function α (x) satisfies the relation:

α(x) = grad s, A 2 x (x) grad s, (A 2 -A 1 ) x = ΓA 2 x Γ (A 2 -A 1 ) x .
This implies that the equivalent dynamic on the sliding hyperplane is given by

ẋ = 1 Γ (A 2 -A 1 ) x ((ΓA 2 x) A 1 x + (ΓA 1 x) A 2 x) (25)
In the general case, the study of the stability of ( 25) is not straightforward. Hereafter two classes of systems are considered for which choices of a Γ leading to stable equivalent motions are fully characterized.

Stabilization of second order switched linear systems

For n = 2, one can choose without loss of generality:

Γ = [γ 1 1], s(x) = γ 1 x 1 + x 2 .
(26) Denoting by a i kl the entries of the matrices A i , the switching law given by ( 21) gives rise to a "sliding motion" as soon as ( 22) holds. Moreover α is independent of x and is given by:

α = -γ 2 1 a 2 12 + γ 1 (a 2 11 -a 2 22 ) + a 2 21 aγ 2 1 + bγ 1 + c (27) a = (a 1 12 -a 2 12 ) b = (a 2 11 -a 1 11 + a 1 22 -a 2 22 ) c = (a 2 21 -a 1 21 ) (28)
which gives the following equivalent dynamic ẋ1 = βx 1 (29)

x 2 = -γ 1 x 1 ( 30
)
where β is given by

β = dγ 2 1 + eγ 1 + f aγ 2 1 + bγ 1 + c (31) d = (a 1 22 a 2 12 -a 2 22 a 1 12 ) e = (a 1 11 a 2 22 -a 2 11 a 1 22 + a 2 21 a 1 12 -a 1 21 a 2 12 ) f = (a 2
11 a 1 21 -a 1 11 a 2 21 ). γ 1 has to be chosen such that 0 < α < 1 and β < 0 (for a stable sliding motion). From equation ( 22), one can characterize the attractivity domain of the obtained sliding mode. For the case presented here, the attractivity domain is the conic region given by x :

x ∈ R n , x T M (γ 1 )x < 0 . (32) with M (γ 1 ) = (γ 1 a 1 11 + a 1 21 )(γ 1 a 2 11 + a 2 21 ) (γ 1 a 1 11 + a 1 21 )(γ 1 a 2 12 + a 2 22 ) (γ 1 a 1 12 + a 1 22 )(γ 1 a 2 11 + a 2 21 ) (γ 1 a 1 12 + a 1 22 )(γ 1 a 2 12 + a 2 22 )
.

Notice that any x for which s(x) = 0 can be expressed on the form x = c × (1 -γ 1 ) T with c ∈ R. Therefore the sliding line is contained in the attractivity domain if the condition (1 -γ 1 )M (γ 1 )(1 -γ 1 ) T < 0 (33) is satisfied.

Stabilization of integrator cascade

Consider the case of two linear switching systems described by the following relation:

y (n) + n-1 i=1 a j i y (i) = 0, j = 1, 2 (34)
In a state space representation, (34) is given by ( 1) where the two matrices A j are

A j =       0 1 0 . . . . . . . . . 1 -a j 1 -a j n-1       (35)
(x being defined obviously). In that case, the resulting bilinear system (18) is defined by the matrices:

A = A 1 + A 2 2 =       0 1 0 . . . . . . . . . 1 - a 1 1 +a 2 1 2 - a 1 n-1 +a 2 n-1 2      
(36) and

B = A 1 -A 2 2 =       0 0 0 . . . . . . . . . 0 - a 1 1 -a 2 1 2 - a 1 n-1 -a 2 n-1 2      
.

(37) Define the sliding variable by:

s = Γx, (38)
where Γ = [γ 1 , ..., γ n ] ∈ R (1×n) (one can set γ n = 1 without loss of generality). According to section 2.1, the equivalent dynamic is obtained using (13) where f 0 is computed as

f 0 (x) = (αA 1 + (1 -α)A 2 )x where αA 1 + (1 -α)A 2 =       0 1 0 . . . . . . . . . 1 -(αa 1 1 + (1 -α)a 2 1 ) -(αa 1 n-1 + (1 -α)a 2 n-1 )      
.

(39) From Γx = 0, the resulting equivalent dynamics is described by : ẋ1 = x 2 ẋ2 = x 3 . . . = . . .

ẋn-1 = - n-1 i=1 γ i x i (40)
which is stable if the vector Γ contains the coefficient of a Hurwitz polynomial. In this case, the domain of attraction obtained in ( 22) is described by the equation

n i=1 γ i-1 - a 1 i + a 2 i 2 x i < n i=1 a 1 i -a 2 i 2 x i . (41)

EXAMPLES

Two examples of planar systems are given here.

Example 1

Consider again the example described by ( 14),( 15), ( 16) with µ = 0.1. Using the sliding line (26) and the obtained computation for α and β, different sliding motions can be described. For this example, the parameter α is constant and is given by:

α = 101γ 2 1 + 198γ 1 + 101 301γ 2 1 + 198γ 1 + 103 . The constraint α ∈ [0, 1] is satisfied for any γ 1 ∈ R.
Stability of the sliding motion implies that

β = - 1 10 6890γ 2 1 + 8019γ 1 -1129 301γ 2 1 + 198γ 1 + 103 < 0,
which holds for any

γ 1 ∈ R \ -8019 - √ 95419601 13780 , -8019 + √ 95419601 13780
.

Simulations have been carried out with γ 1 = 3 and with x 1 (0) = -2 and x 2 (0) = 1 as initial conditions (this implies that the conditions (32), (33) are satisfied). It can be seen in Figure 3 that the states tend to the origin once the sliding surface is reached.

Example 2

Consider the switched system given described by :

dx dt = A i x, i = 1, 2 (42)
(0) = -2, x 2 (0) = 1 with A 1 = 1 0 0 -1 , A 2 = µ -1 1 µ (43)
where µ = 1. In [START_REF] Blanchini | Stabilizability of switched linear systems does not imply the existence of convex lyapunov functions[END_REF]), it was shown by means of this counterexample that a system which can be stabilized via switching control, does not necessarily admit a convex Lyapunov function, thus making difficult to apply usual methods given in the literature.

However, the constructive approach given here allows for the stabilization of this system. Indeed, the conditions

α = γ 2 1 + 1 (γ 1 -1) 2 ∈ [0, 1], β = - γ 2
1 + 2γ 1 -1 (γ 1 -1) 2 < 0, can be satisfied by choosing γ 1 such that γ 1 < 0 and

γ 1 ∈ R \ [-1 - √ 2, -1 + √ 2].
In the simulation reported in Figure 4, γ 1 = -10 (satisfying (33)) and x 1 (0) = 1 and x 2 (0) = -0.3. This particular set of initial conditions has been chosen because the attractivity condition ( 22) is not globally fulfilled in this case. Nevertheless, since the matrix A 2 has complex eigenvalues, the trajectories of the system meet the attractivity domain after some finite time. Three specific behaviors can be seen in Figure 4(a): 0 ≤ t ≤ 1.7 unstability, 1.7 ≤ t ≤ 2.2 attractivity, t ≥ 2.2 sliding motion. This shows that global convergence can be obtained, even if the attractivity condition is not initially satisfied. This fact will be discussed in further work.

CONCLUSION

In this paper, conditions for the existence of stabilizing sliding motions in the context of switched linear systems have been given. The case of switched systems with two subsystems has been discussed and conditions for the existence of stable sliding hyper-planes have been provided. For particular cases of switched systems (planar systems, integrator chains), constructive analytical methods for deriving a stabilizing switching sequence have been proposed.

In future work, we intend to investigate the use of more complex switching strategies, that ensure the global attractiveness of sliding hyper-planes.

Fig. 1 .

 1 Fig. 1. Vector fields.

  Fig. 2. Stabilization using (17) 3. A SLIDING MODE BASED STABILIZING SWITCHING LAW

Fig. 3 .

 3 Fig.3. Evolution of the states x(t) and of the sliding variable s(x) for the system in Example 1 with γ 1 = 3 and x 1 (0) = -2, x 2 (0) = 1

Fig. 4 .

 4 Fig. 4. Evolution of the states x(t) and of the sliding variable s(x) for the system in Example 2.

The two conditions for the existence of a sliding motion along the manifold S : f + n (t, x) < 0 and f - n (t, x) > 0 are equivalent to the well-known condition s ṡ < 0 (condition for the local attractivity of S).


Yang Tian
email: yang.tian@inria.fr

Thierry Floquet
email: thierry.floquet@ec-lille.fr

Lotfi Belkoura
email: lotfi.belkoura@univ-lille1.fr

Wilfrid Perruquetti
email: wilfrid.perruquetti@ec-lille.fr

Algebraic switching time identification for a class of linear hybrid systems

Keywords: Linear systems, switched systems, switching time estimation, distribution theory

In this paper, a method for the finite time estimation of the switching times in linear switched systems is proposed. The approach is based on algebraic tools (differential algebra, module theory and operational calculus) and distribution theory. Switching time estimates are given by explicit algebraic formulae that can be implemented in a straightforward manner using standard tools from computational mathematics. Simulations illustrate the proposed techniques.

INTRODUCTION

Many systems encountered in practice exhibit switchings between several subsystems, inherently by nature, such as when a physical plant has the capability of undergoing several operational modes, or as a result of the controller design, such as in switching supervisory control.

Switched systems can be seen as higher-level abstractions of hybrid systems, obtained by neglecting the details of the discrete behavior. A switched system is composed of a family of dynamical (linear or nonlinear) subsystems and a rule, called the switching law, that orchestrates the switching between them (see [START_REF] Liberzon | Switching in systems and control[END_REF]; [START_REF] Sun | Switched linear systems: Control and design[END_REF] for surveys). In the recent years, there has been an increasing interest in the control problems of switched systems due to their significance from both a theoretical and practical point of view. Important results for switched systems have been achieved for problems including stability [START_REF] Agrachev | Lie-algebraic stability criteria for switched systems[END_REF]; [START_REF] Boscain | Stability of planar switched systems: The linear single input case[END_REF]; [START_REF] Branicky | Multiple Lyapunov functions and other analysis tools for switched and hybrid systems[END_REF]; [START_REF] Mancilla-Aguilar | A converse Lyapunov theorem for nonlinear switched systems[END_REF]; [START_REF] Vu | Common Lyapunov functions for families of commuting nonlinear systems[END_REF], stabilization [START_REF] Moulay | Stabilization of nonlinear switched systems using control Lyapunov functions[END_REF]; [START_REF] Persis | Supervisory control with state-dependent dwell-time logic and constraints[END_REF]; [START_REF] Wicks | Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems[END_REF]; [START_REF] Zhai | Quadratic stabilizability of switched linear systems with polytopic uncertainties[END_REF], tracking [START_REF] Bourdais | Towards a model-free output tracking of switched nonlinear systems[END_REF] or controllability [START_REF] Sun | Controllability and reachability criteria for switched linear control[END_REF]; [START_REF] Xie | Controllability of switched linear systems[END_REF].

Observability and state estimation is a key problem for such systems because both the active mode and the continuous state have to be estimated and this during a finite time interval. The notion of state estimation for switched systems was introduced in [START_REF] Ackerson | On state estimation in switching environnements[END_REF]. Observability notions for some classes of hybrid systems such as switched linear systems has been discussed and characterized in recent works such as [START_REF] Babaali | Observability of switched linear systems in continuous time[END_REF], [START_REF] Santis | On observability and detectability of continuous-time linear switching systems[END_REF], [START_REF] Vidal | Observability of linear hybrid systems[END_REF]. The problem is to recover from available measurements the state of the system and/or the switching signal, and eventually the switching time. Different observation and identification methods have been performed during the last years (Alessan-dri and Coletta (2001); [START_REF] Balluchi | Design of observers for hybrid systems[END_REF]; [START_REF] Domlan | Switching systems: Active mode recognition, identification of the switching law[END_REF]; [START_REF] Fantuzzi | Identification of piecewise affine models in noisy environment[END_REF]; [START_REF] Hashambhoy | Recursive identification of switched arx models with unknown number of models and unknown orders[END_REF]; [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF]; [START_REF] Huang | Identification of hybrid linear time-invariant systems via subspace embedding and segmentation[END_REF]; [START_REF] Paoletti | Identification of hybrid systems: a tutorial[END_REF]; [START_REF] Saadaoui | Exact differentiation and sliding mode observers for switched lagrangian systems[END_REF]). Usually, the hybrid observer consists of two parts: an index estimator of the current active sub-model and a continuous observer that estimates, asymptotically in most cases, the continuous state of the hybrid system.

The aim of this paper is to estimate in "real-time" the switching time sequence of some class of switched linear systems with the knowledge of the continuous state only. The possibility to have finite time estimate for that kind of systems is clearly important, not to say crucial. The approach considered here takes root in recent works developed by [START_REF] Fliess | An algebraic framework for linear identification[END_REF] for parameter identification of linear time-invariant systems. This method is based on algebraic tools (differential algebra, module theory and operational calculus) and results in finite estimates given by explicit algebraic formulae that can be implemented in a straightforward manner using standard tools from computational mathematics. Those results have been extended to the problems of closed-loop parametric estimation in [START_REF] Fliess | Closed-loop parametric identification for continuous-time linear systems via new algebraic techniques[END_REF], state estimation of linear systems with time-varying parameters in [START_REF] Tian | Fast state estimation in linear time-variant systems: an algebraic approach[END_REF] or with delays in [START_REF] Belkoura | On-line identification of systems with delayed inputs[END_REF], fault diagnosis in [START_REF] Fliess | Robust residual generationfor linear fault diagnosis: an algebraic setting with examples[END_REF], nonlinear systems with unknown inputs in [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF] or some class of linear infinite dimensional systems governed by partial differential equations in [START_REF] Rudolph | An algebraic approach to parameter identification in linear infinite dimensional systems[END_REF]. This approach was also applied in Fliess et al. (2008) for the estimation of the index corresponding to the current active subsystem, and the state variable of this subsystem. In this paper, the algebraic approach is extended to the problem of the finite time identification of the switching occurrences. Using some properties of the distribution theory, the switching time estimation is given by an explicit formulae, as a function of the integral of the output, in order to attenuate the influence of measurement noises.

PROBLEM STATEMENT

In this work, we study a class of linear hybrid systems called switched linear systems (definition see [START_REF] Vidal | Observability of linear hybrid systems[END_REF]), i.e. systems whose evolution is determined by a collection of linear models (Q subsystems) with continuous state x ∈ R n connected by switches among a number of discrete states q ∈ I Q {1, . . . , Q}, modeled by linear ordinary differential equations of the following form:

ẋ = A q x, (1)
A q ∈ R n×n are constant matrices. For the sake of convenience and without loss of generality, it is assumed that at each time t ∈ R only one discrete event can act on the system. It is also assumed that there is no jump, i.e. that if t k is a switching instant lim

t→t - k x(t) = lim t→t + k x(t).
The objective of this paper is to estimate on line the switching times of the system (1) using the dynamics of the continuous state only. It will be assumed that all the state is available for measurement. The more general case with partial state measurements will be studied in further work. The method that will be developed will involve high order time derivatives of x.

Because of the presence of non smooth dynamics, derivation has to be understood in the distribution sense.

Distribution Framework

We recall here some standard definitions and results from distribution theory developed in [START_REF] Schwartz | Théorie des distributions[END_REF], and fix the notations to be used in the sequel. The space of C ∞ -functions having compact support in an open subset Ω of R is denoted by D(Ω), and D (Ω) is the space of distributions on Ω, i.e., the space of continuous linear functionals on D(Ω).

When concentrated at a point {τ}, the Dirac distribution δ (tτ) is written δ τ .

Functions are considered through the distributions they define and are therefore indefinitely differentiable. Hence, if y is a continuous function except at a point a with a finite jump σ a , its derivative writes

ẏ = dy/dt + σ a δ a , ( 2
)
where dy/dt is the distribution stemming from the usual derivative of y.

A distribution is said to be of order r if it acts continuously on C r -functions but not on C r-1 -functions. Measures and functions are of order 0. By virtue of Schwartz Theorem (see [START_REF] Schwartz | Théorie des distributions[END_REF]), α × T = 0 for any smooth function α whose derivatives of appropriate order vanish on the support of a given distribution T ∈ D (Ω). In particular, one has for the Dirac distribution 1 :

f (t) • δ t i = 0, ∀ f ∈ C ∞ (R) if f (t i ) = 0, f (t) • δ t i = f (t i ) • δ t i , ∀ f ∈ C ∞ (R) if f (t i ) = 0.
1 The existence of the product f (t) • δ t i is ensured as soon as f is a C ∞ -function.

SWITCHING TIME IDENTIFICATION

The case of two subsystems and one commutation

In this case, q ∈ {1, 2}. Assume that the system (1) switches from the subsystem 1 to the subsystem 2 at the time t c . Then, the dynamical behavior of the system can be rewritten as follows:

ẋ = Γ(t)x (3) Γ(t) = A 1 H(t) + A 2 (1 -H(t))
with the function H given by:

H(t) = 1, 0 ≤ t < t c 0, t c ≤ t < T.
The aim is to get a finite time estimation of the switching time t c when the system commutes from q = 1 to q = 2.

Assume that A 1 and A 2 are two square commuting matrices and consider the following change of variable:

z = e Gt x (4) with G = -A 1 +A 2 2 . One has: ż = Ge Gt x + e Gt ẋ = Gz + e Gt Γ(t)x = (G + e Gt Γ(t)e -Gt )z This leads to ż = M(t)z (5) with M(t) = (G + e Gt A 1 e -Gt )H(t) + (G + e Gt A 2 e -Gt )(1 -H(t)).
Since A 1 and A 2 are two square commuting matrices, one gets:

M(t) = (G + A 1 )H(t) + (G + A 2 )(1 -H(t)) = A 1 -A 2 2 H(t) + A 2 -A 1 2 (1 -H(t)) = -AH(t) + A(1 -H(t))
with A = A 2 -A 1 2 . The choice of the change of coordinates (4) is motivated by the fact that the subsequent computations involve M 2 = A 2 which is a known quantity independent of the switching time.

Differentiating (5) over the interval t ∈ [0, T [, one has:

z -A 2 z = -2Aδ 0 z(0), 0 ≤ t < t c z -A 2 z = -2Aδ 0 z(0) + 2Aδ t c z(t c ), t c ≤ t < T which can be rewritten in the following compact form: z -A 2 z = -2Aδ 0 z(0) + 2A(1 -H(t))δ t c z(t c ), 0 ≤ t < T (6)
Note that if the initial subsystem is q = 2, a similar expression can be obtained as follows:

z

-A 2 z = 2Aδ 0 z(0) -2A(1 -H(t))δ t c z(t c
), 0 ≤ t < T the terms in the right hand have the opposite signum. As it will be seen hereafter, this slight difference has no influence on the result because the right hand side of (6) will be canceled by suitable algebraic manipulations. Thus the proposed method does not need the knowledge of the current mode.

Explicit computation of the switching instant

Take any function with the following properties:

(i) f (t,t c )δ t c = 0, (ii) f (0,t c ) = ḟ (0,t c ) = 0.
Multiplying (6) by f (t,t c ), one obtains:

f (t,t c )(z -A 2 z) = 0. (7) Integrating (7) from 0 to t c < t ≤ T leads to t 0 f (τ,t c )(z(τ) -A 2 z(τ))dτ = 0 (8)
Integration by parts gives:

t 0 f (τ,t c )z(τ)dτ = [ f (τ,t c )ż(τ)] τ=t τ=0 - t 0 ḟ (τ,t c )ż(τ)dτ = f (t,t c )ż(t) - t 0 ḟ (τ,t c )ż(τ)dτ and t 0 ḟ (τ,t c )ż(τ)dτ = ḟ (τ,t c )z(τ) τ=t τ=0 - t 0 f (τ,t c )z(τ)dτ = ḟ (t,t c )z(t) - t 0 f (τ,t c )z(τ)dτ. Hence t 0 f (τ,t c )z(τ)dτ = f (t,t c )ż(t) -ḟ (t,t c )z(t) + t 0 f (τ,t c )z(τ)dτ (9)
Integrating one more time (8) from 0 to t c < t ≤ T and using (9) leads to the following relation:

t 0 [ f (τ,t c )ż(τ) -ḟ (τ,t c )z(τ)]dτ + t 0 τ 1 0 ( f (τ,t c ) -A 2 f (τ,t c ))z(τ)dτdτ 1 = 0
Thus, using the property

t 0 t ν-1 0 • • • t 1 0 x(τ)dt ν-1 • • • dt 1 dτ = t 0 (t -τ) ν-1 (ν -1)! x(τ)dτ
one has:

f (t,t c )z(t) -2 t 0 ḟ (τ,t c )z(τ)dτ + t 0 (t -τ)( f (τ,t c ) -A 2 f (τ,t c ))z(τ)dτ = 0. ( 10
)
The following function

f (t,t c ) = t 2 (t -t c ) satisfies the two properties (i)-(ii) and ḟ (t,t c ) = t(3t -2t c ) f (t,t c ) = 6t -2t c
Using the result of ( 10), the estimate of t c is given by the following formula:

D(t, z)t c = N(t, z) with N(t, z, A) = t 3 z(t) - t 0 6τ 2 z(τ)dτ + t 0 (t -τ)(6τI -A 2 τ 3 )z(τ)dτ D(t, z, A) = t 2 z(t) - t 0 4τz(τ)dτ + t 0 (t -τ)(2I -A 2 τ 2 )z(τ)dτ (11)
where I ∈ R n×n is the unit matrix. Note that N(t, z, A) and D(t, z, A) are column vectors of dimension n. So

t c = N g (t, z, A) D g (t, z, A) , ( 12
)
for any 1 ≤ g ≤ n, where N g (t, z, A) and D g (t, z, A) are the components of N(t, z, A) and D(t, z, A), respectively.

Thus, one obtains the exact formula for the identification of the switching time t c . Hereafter, the method is extended to the problem of the identification of the switching times between an arbitrary number of subsystems.

The case of Q subsystems and S commutations

Assume that every pair of matrices A i and A j (i, j ∈ I Q and i = j) in system (1) are commuting. It has been seen that an estimator E i, j that computes the quantities D(t, z, A i, j ) and N(t, z, A i, j ), with A i, j = A j -A i 2 can be used to determine a switching time t i, j that occurs between the two subsystems i and j (either from mode i to mode j or from mode j to mode i).

Hence, in order to identify all the switches among Q modes, one can use C 2 Q estimators in parallel. Then, the output signals of each estimator can be analyzed as follows to determine the occurrence of a switch and its associated mode. Indeed, assume that the system (1) is in the mode i for t ∈ [t 0 ,t i, j [ and in mode j for t ∈ [t i, j , T [, where times t 0 and T stands for other switch occurrences. Then, one has:

(1) D g (t, z, A i, j ) = 0 and N g (t, z, A i, j ) = 0 for t ∈ [t 0 ,t i, j [: in practice several successive points have to be tested before the switching instant; (2) D g (t, z, A i, j ) and N g (t, z, A i, j ) are straight lines for t ∈ [t i, j , T [: in practice several points have to be used to detect the slope of the line is constant.

(3) for t ∈ [t i, j , T [ the ratio of N g (t,z,A i, j ) D g (t,z,A i, j
) is constant and equal to t i, j : to detect when this ratio is constant, successive values within a boundary of a given thickness are needed.

Properties (1) and ( 2) can be seen from the fact that Dirac distributions in equation ( 6) are integrated twice. Thus, the estimator that fulfills those three conditions provides the switching instant as well as the index of the subsystems between which the switch occurs. Furthermore, with the knowledge of the first active mode, the sequence of the all active models can be estimated. The implementation of the numerical algorithm which allows to identify all the switching instants is done on a sliding window as follows: The numerical implementation of the above obtained algorithm has also to address the following points:

• integration window: when none of the C 2 Q estimator detect the commutation, algorithm will calculate the value of N(t, z, A i, j )(k), D(t, z, A i, j )(k) in a new integration window (k ←-k + 1) whose width is one sample time larger than the old one (lower bound of the integration no change while upper bound increase one sampling), until a switching instant is detected with one of the estimator.

• update: when t c is detected, one has to update the scheme by resetting the integral computation and replacing the lower bound of the integration with t c in order to identify the next switching time (for this in the integral approximation).

EXAMPLE

Consider a linear switched system consisting of three linear time invariant scalar systems given by:

A 1 = -8 A 2 = -1 A 3 = 0.4
The switching instants of this switched system are detected using the algorithm proposed in Section 3 with T e = 0.001s and 3 estimators (C 2 3 ). The first active model is subsystem 1 and the switching sequences among the subsystems are A

1 -→ A 2 -→ A 3 -→ A 1 -→ A 2 -→ A 3 -→ A 2 -→ A 1 .
Since estimator E 1,2 detects the commutation between A 1 ←→ A 2 , estimator E 2,3 corresponds to the commutation between A 2 ←→ A 3 and estimator E 1,3 detects the switch between A 1 ←→ A 3 , it is known that estimators E 1,2 -→ E 2,3 -→ E 1,3 -→ E 1,2 -→ E 2,3 -→ E 2,3 -→ E 1,2 work respectively. The switching times are assumed to occur at: t 1 = 0.2s, t 2 = 1.2s, t 3 = 2.4s, t 4 = 2.6s, t 5 = 3.6s, t 6 = 4s and t 7 = 4.9s. In the simulation, we choose 30 samplings to identify the switching time, so the dwell time has to be at least larger than 0.03s. Fig. 1 shows that the estimator performs well. Note that t 4 occurs shortly (0.2s) after t 3 but that the algorithm can detect it accurately. Fig. 2 gives the estimation of the sequence of active model. It can be seen that the estimated switched system S e tracks exactly the real switched system S r .

Note that identification of the switching time does not need the knowledge of the first active mode, but is just required in order to obtain the sequence of the active model.

CONCLUSION

In this paper, an algebraic approach for switching time estimation of a class of hybrid systems has been introduced. An explicit algorithm which computes on-line the switching time instants in a fast way has been derived. In future works, this approach will be extended to more general cases with partial state measurements.

  function detection of the switching time t c read x(kT e ) T e : sample time; h: number of points chosen to verify the constraints with h.T e less than the dwell time of the system; t s = 0: left bound of the integration window; m = length(x): length of the sampled state x; z i, j (kT e ) = e - A i +A j 2 x(kT e ): new state for k = h : mReset all the components of N(t, z, A i, j )(k) and D(t, z, A i, j )(k) to zero for i = t s : k Approximate numerically the integrals as sums using the trapezoidal rule end Compute the value of N(t, z, A i, j )(k) and D(t, z, A i, j )(k) in an integration window with(11) if an estimator satisfies the three detection criterions update the integration window: t s ←-k -h; print this estimator and the active model.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Identification of the switching times t c .