Ryan P C De Souza

Zohra Kader
email: kader@laplace.univ-tlse.fr

Stéphane Caux
email: caux@laplace.univ-tlse.fr

LMI-based Control of Singularly Perturbed Switched Affine Systems

Keywords: Control of switched systems, Switched affine systems, Stabilization of hybrid systems, Singularly perturbed systems, Linear Matrix Inequalities. Proof. See

In this paper, we tackle the problem of stabilizing singularly perturbed switched affine systems. Even though efficient control strategies based on the solution of Linear Matrix Inequalities (LMIs) have been presented in the switched affine systems literature, the presence of the small singular perturbation parameter may introduce numerical difficulties in the solution of these LMIs due to ill conditioning. Moreover, this parameter may be uncertain or its use in the control law may not be practical. Here, we propose an LMI-based control design strategy that avoids the conditioning issues and that also provides an estimation of the domain of attraction. Furthermore, the resulting controller does not depend on the singular perturbation parameter. The proposed method is illustrated on a numerical example.

INTRODUCTION

Switched Affine Systems (SAS) have been an interesting topic of research in the past few decades [START_REF] Liberzon | Switching in systems and control[END_REF]. Their dynamics can be described as follows: there is a finite set of subsystems (or modes) and a switching law that dictates at each time instant which one is the active mode. Moreover, the dynamics of each mode are affine on the system states. Their study is interesting from a practical viewpoint since they are suitable for modeling systems in different applications, including power converters.

Stabilization of SAS is challenging due to the fact that the goal is usually to stabilize the system at an equilibrium point which is different from the equilibrium of any individual mode. In the SAS literature, some state-dependent switching control laws have been proposed for stabilization [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF][START_REF] Hetel | Local stabilization of switched affine systems[END_REF][START_REF] Kader | Stabilization of switched affine systems with disturbed state-dependent switching laws[END_REF][START_REF] Egidio | Stabilization of rank-deficient continuous-time switched affine systems[END_REF]. One important feature of these methods is that control design is carried out through the solution of Linear Matrix Inequalities (LMIs), in a process that is numerically efficient when the problem is well conditioned.

Many physical systems exhibit dynamics evolving in two time scales. The difference in the time scales is usually represented by the singular perturbation parameter ε in the system model [START_REF] Kokotović | Singular perturbation methods in control: analysis and design[END_REF], which is consequently ill-conditioned. For switched systems, taking into account the time-scale separation between the slow and fast dynamics is a very challenging problem. Moreover, designing LMI-based control laws in this case is not trivial due to the ill conditioning of the system matrices. Singular perturbation theory is a mature tool for stability analysis and control design for both linear and nonlinear systems [START_REF] Kokotović | Singular perturbation methods in control: analysis and design[END_REF][START_REF] Lizarraga | Control of singularly perturbed systems under actuator saturation[END_REF][START_REF] Tognetti | LMI-based output feedback control of singularly perturbed systems with guaranteed cost[END_REF], including practical applications in power electronics [START_REF] Kimball | Singular perturbation theory for DC-DC converters and application to PFC converters[END_REF][START_REF] Ghanes | Fuel cell system control under converter losses with experimental results[END_REF]. These techniques have been extended to the case of stabilization of singularly perturbed switched linear systems in [START_REF] Deaecto | H 2 and H ∞ performance optimization of singularly perturbed switched systems[END_REF]. Stability criteria have equally been proposed for singularly perturbed switched systems with arbitrary switching laws in [START_REF] Rejeb | Stability analysis of a general class of singularly perturbed linear hybrid systems[END_REF]; [START_REF] Yang | Exponential stability of singularly perturbed switched systems with all modes being unstable[END_REF], and for a slow switched affine system interconnected with a fast linear time invariant system in [START_REF] Tang | About switched affine system interconnected with fast LTI dynamics[END_REF]. Recently, in [START_REF] De Souza | Stabilization of a class of singularly perturbed switched systems[END_REF], state-dependent control design for a subclass of singularly perturbed SAS has been addressed. To the best of the authors' knowledge, however, the design of a statedependent controller for general singularly perturbed SAS remains an open problem.

In this paper, we propose an LMI-based control design strategy for the stabilization of singularly perturbed SAS. Moreover, an estimation of the domain of attraction is provided when only local exponential stability is guaranteed. Since, in practice, the parameter ε may not be wellknown or its use in the controller may not be desirable [START_REF] Tognetti | LMI-based output feedback control of singularly perturbed systems with guaranteed cost[END_REF], the proposed controller and domain of attraction do not depend explicitly on ε. Moreover, we provide an estimation of the upper bound of ε under which the stability of the closed-loop singularly SAS is ensured. Global stabilization conditions are also proposed.

The paper is structured as follows. In Section 2, we describe the system and state the problem addressed here. In Section 3, we give a generic stabilization result valid for singularly perturbed SAS. Then, in Section 4, we present the main result of the paper using the results obtained in Section 3. The proposed technique is illustrated in Section 5 on a numerical example and then conclusions are drawn in Section 6.

Notation: If v is a vector, then v (j) denotes its j-th component. If M is a matrix, then M T denotes its transpose and sym {M } := M + M T . If M is symmetric, then M 0 (resp. 0) means that M is positive definite (resp. semidefinite). The notation for negative definiteness (or semidefiniteness) is analogous. Moreover, the minimum (resp. maximum) eigenvalue of M is denoted as λ min (M ) (resp. λ max (M )). The notation E(M ) := {η ∈ R n : η T M η ≤ 1} is used to denote an ellipsoid in R n , and B(r 2 ) := {η ∈ R n : η T η ≤ r 2 } denotes a ball of radius r centered at the origin. The identity matrix in R n×n is denoted as I n . The superscript i in M i is used as an index. The symbol 0 can be used as the scalar zero or a null matrix of appropriate dimensions, depending on the context. The symbol in a symmetric matrix means that the corresponding sub-matrix block can be inferred from symmetry. The convex hull of a finite set V is denoted as conv {V}. Given a set of real values {f (v

) : v ∈ V}, let arg min v∈V f (v) := {v ∈ V : f (v) ≤ f (v ), ∀v ∈ V}.
The interior of a region R is denoted as Int {R}. Given an integer N , the unit simplex in R N is denoted as ∆

N := {α ∈ [0, 1] N : N i=1 α (i) = 1}. Set I N is defined as I N := {1, 2, . . . , N }. Given a domain D ⊆ R n and a function V : D → R, ∂V /∂η := [∂V /∂η (1) ∂V /∂η (2) • • • ∂V /∂η (n) ].

PROBLEM STATEMENT

Consider a singularly perturbed switched affine system written as follows:

ẋ = A 11 (u)x + A 12 (u)z + B 1 u ε ż = A 21 (u)x + A 22 (u)z + B 2 u, (1a) (1b
) where x ∈ R nx is the slow state, z ∈ R nz is the fast state, ε > 0 is a small parameter, and u ∈ R m is a control variable that can only belong to a finite set V := {v 1 , v 2 , . . . , v N }. Matrices A 11 (u), A 12 (u), A 21 (u) and A 22 (u) are functions of u. The following assumption regarding V is made. Assumption 1. conv {V} has nonempty interior and 0 ∈ Int {conv {V}}.

2 Remark 1. System (1) can be rewritten in the more familiar form: ẋ = Ãσ 11 x + Ãσ 12 z + bσ

1 ε ż = Ãσ 21 x + Ãσ 22 z + bσ 2 , (2a) (2b)
where σ ∈ I N denotes the switching function, Ãi

11 = A 11 (v i ), Ãi 12 = A 12 (v i ), Ãi 21 = A 21 (v i ), Ãi 22 = A 22 (v i ), bi 1 = B 1 v i and bi 2 = B 2 v i , ∀i ∈ I N .
Moreover, under Assumption 1, it can be shown using (Hetel and Bernuau, 2015, Prop. 1) that one can always pass from the representation in (2) to the one in (1).

2

Define the full state η := [x T z T ] T ∈ R n , where n = n x + n z . Let us also define A ε (u) and B ε as:

A ε (u) := A 11 (u) A 12 (u) ε -1 A 21 (u) ε -1 A 22 (u) , (3)
B ε := B 1 ε -1 B 2 . ( 4
)
We address in this paper the use of state-dependent switching laws of the form:

u(η) ∈ arg min v∈V η T Hv, (5)
with H a parameter to be determined. The closed-loop system (1), ( 5), can be equivalently described as:

η = A ε (u(η))η + B ε u(η). (6)
The closed-loop system ( 6) is a discontinuous system [START_REF] Cortes | Discontinuous dynamical systems[END_REF] and its solutions are considered here in the Filippov sense [START_REF] Filippov | Differential equations with discontinuous righthand sides[END_REF]. Thus, to the system (1), ( 5) we associate the differential inclusion η(t) ∈ F(η(t)), (7) where F(η) is the set-valued map defined by

F(η) = conv A ε (ṽ)η + B ε ṽ : ṽ ∈ arg min v∈V η T Hv . (8)
Definition 1. The trajectory η(t) is a solution of (6) if it is absolutely continuous in the interval [0, t 1 ] ⊂ R for any t 1 > 0 and satisfies the associated differential inclusion ( 7), ( 8) for almost all t ∈ [0, t 1 ].

2

Note that in the case of piecewise constant control inputs, the existence of at least one solution of ( 7), ( 8) is guaranteed. This is because the obtained set-valued map ( 8) is Lebesgue measurable, upper-semicontinuous, compact, non-empty and convex (see [START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF]). Definition 2. The origin is an equilibrium point of ( 1), ( 5), if 0 ∈ F(0). 2

The statement of Definition 2 implies that η = 0 is an equilibrium of the closed-loop system (6

) if 0 ∈ conv {B ε v : v ∈ V}.
Note that this is implied by Assumption 1. Definition 3. The origin of the closed-loop system (1), ( 5) (or equivalently the associated differential inclusion ( 7), ( 8)) is said to be locally exponentially stable if there exist D ⊆ R n with 0 ∈ Int {D}, κ 1 > 0 and κ 2 > 0, such that for all Filippov solutions η(t) of ( 7), ( 8), with η(0) ∈ D, η(t) ≤ κ 1 e -κ2t η(0) , ∀t ≥ 0. If D = R n , then the origin is said to be globally exponentially stable.

2

Sufficient conditions for the local exponential stability of systems modeled by a differential inclusion ( 7), ( 8), are given by the existence of a strict Lyapunov function

V : D → R, with 0 ∈ Int {D}, V (0) = 0, V (η) > 0, ∀η ∈ D \ {0}, such that sup ς∈F (η) ∂V ∂η ς < -2χV (η), ∀η ∈ D \ {0}, (9)
where χ > 0 is a desired decay rate.A sufficient condition for global stability is obtained if (9) holds with D = R n and V (•) radially unbounded.

The difficulty in designing a controller for the stabilization of (1) lies in the numerical issues induced by the small parameter ε. This may pose a challenge in the LMI-based techniques which are usually adopted in the SAS literature (such as those used, e.g., in [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF]). Additionally, ε may not be well-known, or it may not be practical to use it in the control law. In this case, it is desirable that the controller does not depend on this parameter, and as a result the control design strategy must be carried out accordingly. The problem addressed in this paper is summarized as follows.

Problem. Design an LMI-based ε-independent switching law of the form (5) that ensures exponential stabilization of the origin of the singularly perturbed switched affine system (1) (or, equivalently, (2)), while alleviating the numerical issues due to the presence of parameter ε.

SYSTEM STABILIZATION

In this section, we present some generic results concerning the stabilization of the closed-loop system (1), ( 5), as well as the estimation of the domain of attraction.

Firstly, note that, thanks to Assumption 1, we can write conv {V} as a region in R m delimited by n h hyperplanes in the following manner:

conv {V} = {v ∈ R m : h T q v ≤ 1, q = 1, . . . , n h }, (10)
where h q ∈ R m , ∀q ∈ I n h . This representation of conv {V} will be useful throughout this paper for estimating the domain of attraction.

Consider the next lemma, which provides an important result that will be useful later on. It gives ε-dependent conditions for the stabilization of the origin of (1). This result is similar to others reported in the literature on switched systems, such as Hetel et al. (2015) and [START_REF] Delpoux | Parameter-dependent relay control: Application to PMSM[END_REF]. However, these works do not consider general switched affine systems, differently from the current paper.

Lemma 1. Given a parameter χ > 0, assume that there exist ρ > 0 and matrices

Q = Q T 0 and Y ∈ R m×n such that: sym {A ε (v i )Q + B ε Y } + 2χQ ≺ 0, ∀i ∈ I N , (11)
1 h T q Y Q 0, ∀q ∈ I n h , (12)
ρI n I n Q 0. ( 13
)
Then, the origin of the closed-loop system (1), (5), is locally exponentially stable by choosing

H = Q -1 B ε in (5). In addition, Ω = E(Q -1
) is an estimation of the domain of attraction that contains the ball B(0, 1/ρ). 2

Proof. Using (11) and considering P = Q -1 , then:

2η T P A ε (v i )η + 2η T P B ε Kη + 2χη T P η < 0, ∀i ∈ I N , (14) for all η ∈ R n \ {0}, with K = Y P . Define the set C V (K) := {η ∈ R n : Kη ∈ conv {V}}.
Using Assumption 1 and continuity arguments, it can be seen that C V (K) has nonempty interior. On the other hand, using the Schur complement, ( 12) is equivalent to 1

- h T q Y Q -1 Y T h q > 0, ∀q ∈ I n h .
It can be shown (Boyd et al., 1994, Sec. 5.2.2) that this implies E(P ) ⊂ C V (K). Thus, there exists α(η) ∈ ∆ N such that Kη = N j=1 α (j) (η)v j , for all η ∈ E(P ). Substituting in ( 14) and taking into account the fact that

N j=1 α (j) (η) = 1: N j=1 α (j) (η) η T P Aε(v i )η + η T P Bεv j + χη T P η < 0, ( 15
)
for every i ∈ I N and for all η ∈ E(P )\{0}. Since α (j) (η) ∈ [0, 1], ∀j ∈ I N , then for each η there must be at least one minimizer v * j for which η T P A ε (v i )η + η T P B ε v j * + χη T P η is negative. Choosing j = j * such that v j * ∈ arg min v∈V η T P B ε v then guarantees that:

η T P A ε (v i )η + η T P B ε v j * + χη T P η < 0, ( 16
)
for every i ∈ I N and for all η ∈ E(P )\{0}. In particular, ( 16) is valid for i = j * .

Consider the Lyapunov function V (η) = η T P η. Since ∂V /∂η = 2η T P and taking i = j * , we obtain from ( 16):

∂V ∂η (A ε (v j * ) η + B ε v j * ) < -2χη T P η, (17)
for all η ∈ E(P )\{0}. By considering H = P B ε and noting that F(η) is compact and convex:

sup

ς∈F (η) ∂V ∂η ς < -2χV (η), ∀η ∈ E(P )\{0}, (18)
and therefore the origin is locally exponentially stable with decay rate χ. Since Ω = E(P ) is a level set of V (η), then it can be taken as an estimation of the domain of attraction.

Finally, LMI (13) implies that ρη T η -

η T Q -1 η > 0, ∀η ∈ R n . Thus, η ∈ B(1/ρ) ⇐⇒ ρη T η ≤ 1 =⇒ η T Q -1 η < 1 ⇐⇒ η ∈ Ω. Therefore, B(1/ρ) ⊆ Ω.
Remark 2. From ( 11), one can see that stabilizability of the pairs (A ε (v i ), B ε ), i = 1, . . . , N , is a necessary condition for applying the results of Lemma 1. 2

Variable ρ has been included in order to offer a way to maximize the size of the domain of attraction. Indeed, this can be accomplished by minimizing ρ subject to the constraints ( 11)-( 13), ρ > 0, and Q = Q T 0, which is a standard convex optimization problem. Alternatively, one could also maximize the size of Ω by minimizing log det Q -1 , as in (Boyd et al., 1994, Section 5.

2.2).

There are two drawbacks in directly using Lemma 1 for the design of the controller: (i) since ε can be very small and cause ill conditioning in the system matrices, the solution of (11) may suffer from numerical problems; and (ii) if ε is not well-known, Lemma 1 cannot be applied, since both LMI (11) and the controller require the knowledge of this parameter. In the next section, an ε-independent stabilizing state-dependent switching law that circumvents both of the aforementioned drawbacks is proposed.

MAIN RESULT

Consider the following lemma, which will be useful later in proving the main result of the paper. Lemma 2. Consider symmetric matrices Ψ i , Θ i , Σ i ∈ R n×n , i = 1, . . . , N , of same dimensions, having the following form 1 :

Ψ i = Ψ i 1 Ψ i 12 Ψ i 2 , Θ i = 0 Θ i 12 Θ i 2 , Σ i = 0 0 Σ i 2 , ( 19
)
where Σ i 2 ≺ 0, ∀i ∈ I N . There exists µ > 0 such that

Ψ i + ε -1 Θ i + ε -2 Σ i ≺ 0, ∀i ∈ I N , (20)
holds for all ε ∈ (0, µ) if and only if Ψ i 1 ≺ 0 and Θ i 12 = 0, ∀i ∈ I N . In addition, assuming

Ψ i 2 -Ψ i 12 T Ψ i 1 -1 Ψ i 12
0, ∀i ∈ I N , and defining:

µ i = +∞, if Θ i 2 0, λ -1 max (-Σ i 2 ) -1/2 Θ i 2 (-Σ i 2 ) -1/2 , otherwise. ( 21
)
for every i ∈ I N , then the bound µ can be estimated as µ = min i∈I N µ i . 2

The next theorem presents the main result of the paper.

It provides an LMI-based approach for the design of stabilizing ε-independent switching laws for the singularly perturbed switched affine system (1). The singular perturbation parameter ε does not appear in the LMIs and local exponential stabilization is shown for all ε ∈ (0, ε * ) while both the upper bound ε * and an estimation of the domain of attraction are provided. Theorem 1. Consider system (1). For a given parameter χ > 0, assume that there exist ρ > 0,

Q 1 = Q T 1 0, Q 2 = Q T 2 0, Y 1 ∈ R m×nx , Π i 1 = Π i 1 T 0 and Π i 2 = Π i 2
T , i = 1, . . . , N , such that the following LMI conditions are satisfied:

1 h T q Y 1 Q 1 0, ∀q ∈ I n h , ( 22
)
ρI nx 0 I nx ρI nz 0 Q 1 0, ( 23
)
and sym {A 11 (v i )Q 1 + B 1 Y 1 } + 2χQ 1 + Π i 1 ≺ 0, ( 24
) sym {A 22 (v i )Q 2 } + Π i 2 ≺ 0, ( 25
) Π i 1 A 12 (v i )Q 2 + Q 1 A T 21 (v i ) + Y T 1 B T 2 Π i 2 0, ( 26
)
for every i ∈ I N . Then, there exists ε * > 0 such that, for all ε ∈ (0, ε * ), the origin of the closed-loop system (1), ( 5), is locally exponentially stable with decay rate χ by choosing H as:

H = Q -1 1 B 1 Q -1 2 B 2 . ( 27
)
In addition, Ω * = E(Q -1 ε * ) is an estimation of the domain of attraction, where:

Q -1 ε * = Q -1 1 0 0 ε * Q -1 2 , ( 28
)
and B(1/ρ) ⊆ Ω * . 2

Proof. Consider the following inequality:

sym {A ε (v i )Q ε + B ε Y } + 2χQ ε + Π i ε ≺ 0, ∀i ∈ I N , (29) with Q ε = Q T ε and Π i ε = Π i ε T given by Q ε = Q 1 0 0 ε -1 Q 2 , Π i ε = Π i 1 ε -1 Π i 12 ε -2 Π i 2 , ( 30
)
for every i ∈ I N , and Y = [Y 1 0]. Moreover, note that A ε (v i ) and B ε can be written as:

A ε (v i ) = A 11 (v i ) A 12 (v i ) 0 0 +ε -1 0 0 A 21 (v i ) A 22 (v i ) , (31)
B ε = B 1 0 + ε -1 0 B 2 . ( 32
)
Using ( 31)-( 32) and developing, (29) can be written in the form of expression (20) with Ψ i , Θ i and Σ i having the structure shown in ( 19), where:

                 Ψ i 1 = sym {A 11 (v i )Q 1 + B 1 Y 1 } + 2χQ 1 + Π i 1 ≺ 0; Ψ i 12 = 0; Ψ i 2 = 0; Θ i 12 = A 12 (v i )Q 2 + Q 1 A T 21 (v i ) + Y T 1 B T 2 + Π i 12 ; Θ i 2 = 2χQ 2 ; Σ i 2 = sym {A 22 (v i )Q 2 } + Π i 2 . (33a) (33b) (33c) (33d) (33e) (33f)
The feasibility of (25) implies directly that Σ i 2 ≺ 0 for all i ∈ I N . Thus, we can apply Lemma 2 to show that there exists µ > 0 such that (29) holds for all ε ∈ (0, µ) iff Ψ i 1 ≺ 0 and Θ i 12 = 0, ∀i ∈ I N . It can be readily seen that the feasibility of (24) implies Ψ i 1 ≺ 0. The condition Θ i 12 = 0 is enforced by making

Π i 12 = -(A 12 (v i )Q 2 + Q 1 A T 21 (v i ) + Y T 1 B T 2 )
. Therefore, LMI (26) implies that:

Π i 1 -Π i 12 Π i 2 0, ∀i ∈ I N . (34)
Applying the Schur complement on the first row and column of (34), we have (since

Π i 1 0) that (34) ⇐⇒ Π i 2 -Π i 12 T Π i 1 -1 Π i 12 0 ⇐⇒ ε -2 Π i 2 -ε -2 Π i 12 T Π i 1 -1 Π i 12
0, which, by using once again the Schur complement, is equivalent to Π i ε 0 for any i ∈ I N .

Therefore, we have shown that ( 24)-( 26) are sufficient conditions for (29) and for Π i ε 0, for all i ∈ I N and for all ε ∈ (0, µ), for some µ > 0. Since Π i ε 0, (29) implies that (11) holds with

Q = Q -1 ε and Y = [Y 1 0].
Because of this, using the result from Lemma 1, one can show that the origin of the closed-loop system (1), ( 5) is locally exponentially stable with decay rate χ by selecting

H = Q -1 ε B ε , whose expression is given in (27). More specifically, (18) holds with V (η) = V ε (η) := η T Q -1 ε η and K = Y Q -1 ε . Moreover, exactly as in the proof of Lemma 1, Ω ε := E(Q -1 ε ) is an estimation of the domain of attraction if 1 -h T q Y Q -1 ε Y T h q > 0, ∀q ∈ I n h . Since Y = [Y 1 0], then this can be rewritten as 1 -h T q Y 1 Q -1 1 Y T 1 h q > 0, ∀q ∈ I n h .
Note that, using the Schur complement, this is implied by (22).

We can follow the same arguments as in the proof of Lemma 1 to show that, taking Q = Q ε , then (13) implies B(1/ρ) ⊂ Ω ε . Note that, here, LMI (13) can be written as:

   ρI nx 0 I nx 0 ρI nz 0 I nz Q 1 0 ε -1 Q 2    0. ( 35
)
Since ( 23) holds, then using the Schur complement, it can be shown that (35

) ⇐⇒ ε -1 Q 2 -(1/ρ)I nz 0.
This is valid for all ε ∈ (0, ν), with ν = ρλ min (Q 2 ). Thus, for all ε ∈ (0, ν), we have that B(1/ρ) ⊂ Ω ε .

Let ε * = min{µ, ν}, so that all results proven so far are valid for all ε ∈ (0, ε * ). Note that η ∈ Ω

* ⇐⇒ x T Q -1 1 x + ε * z T Q -1 2 z ≤ 1 =⇒ x T Q -1 1 x + εz T Q -1 2 z ≤ 1 ⇐⇒ η ∈ Ω ε , ∀ε ∈ (0, ε * ), since Q -1 2 0.
This implies that Ω * ⊆ ∩ ε∈(0,ε * ) Ω ε , and therefore Ω * is contained in the domain of attraction. Since ε * ≤ ν and Ω * = Ω ε * , then B(1/ρ) ⊆ Ω * . This concludes the proof.

The bound ε * for which Theorem 1 is valid can be estimated by simply applying Lemma 2 with the expressions for the sub-block matrices given in (33). Remark 3. From ( 26) and the fact that Π i 1 0, we have that Π i 2 0. Then, from (25), we conclude that A 22 (v i ) must be Hurwitz, for all i ∈ I N . Otherwise, Theorem 1 cannot be applied since (25) would not be feasible. 2

Remark 4. If a bound ε is known such that ε ≤ ε < ε * , then a less conservative estimation of the domain of attraction can be obtained as Ω = E(Q -1 ε ). 2

The next result presents conditions under which global stabilization can be ensured. Corollary 1. Assume that there exist matrices

Q 1 = Q T 1 0, Q 2 = Q T 2 0, Π i 1 = Π i 1 T 0 and Π i 2 = Π i 2
T , i = 1, . . . , N , such that LMI conditions ( 24)-( 26) are satisfied with Y 1 = 0. Then, there exists ε * > 0 such that, for all ε ∈ (0, ε * ), the origin of the closed-loop system (1), ( 5), ( 27), is globally exponentially stable with decay rate χ.

2

Proof. It has been shown in the proof of Theorem 1 that LMIs ( 24)-( 26) imply that (18) holds with K = [Y 1 0]Q -1 ε . If these LMIs are satisfied with Y 1 = 0, then (18) holds with K = 0. As a result of Assumption 1, C V (K = 0) = {η ∈ R n : 0 ∈ conv {V}} = R n . Therefore, ( 18) holds for all η ∈ R n \{0}, meaning that the stability result obtained in Theorem 1 holds globally.

Note that Corollary 1 presents stricter conditions with respect to Theorem 1. Indeed, it can be seen that now both A 11 (v i ) and A 22 (v i ) must be Hurwitz for every i ∈ I N in order for the corresponding LMIs to be feasible. However, global stability of the closed-loop system is ensured.

The results presented in Theorem 1 and Corollary 1 allow us to design a controller that does not depend on the value of ε (though it must be bounded by ε * ). It can be seen in the proofs that this is accomplished by considering a Lyapunov function V ε (η) = η T Q -1 ε η that is in fact parameterized by ε. This means that, for any ε ∈ (0, ε * ), there is a Lyapunov function that can be used to prove the stability results. Even if the value of ε is unknown, we know that a corresponding Lyapunov function exists as long as ε < ε * .

NUMERICAL EXAMPLE

In this section, we illustrate our approach by means of a numerical example. Consider system (6) with:

A ε 1 ) = -1 0 0 -4000 , (36)
A ε (v 2 ) = -1 1 -20000 -4000 , (37)
B ε = -2.5 30000 , (38)
v 1 = 0.61, v 2 = -0.39, (39) and u(η) given by (5).

By taking ε = 5 × 10 -5 , the system dynamics can be reexpressed as (1), with:

A 11 (v 1 ) = A 11 (v 2 ) = A 21 (v 2 ) = -1, A 12 (v 1 ) = A 21 (v 1 ) = 0, A 12 (v 2 ) = 1, A 22 (v i ) = -0.2, i = 1, 2, B 1 = -2.5 and B 2 = 1.5.
We can now apply the results of Theorem 1. By solving LMIs ( 23)-( 26) choosing χ = 0.3, we obtain: It is worth noting in Fig. 1 that, at first, the trajectory behaves as a singularly perturbed system: the variation of the fast variable z is large in comparison with that of the slow variable x. However, the trajectory eventually reaches the surface s(η) = 0. Then, the trajectory slides on this surface until converging to the origin. While switching occurs on this surface, the separation between the slow and fast dynamics is not observed. However, before attaining the switching surface, the difference between the two time scales is evident. It is important to remark that the same controller (and hence the same switching surface) can be used to stabilize the system for any ε ∈ (0, ε * ), which ensures some robustness with respect to this parameter.

Q 1 = 25.8, Q 2 = 28.2, Y 1 = 0.76, Π 1 1 = Π 2 1 = 20.7,

CONCLUSION

In this paper, we have proposed an ε-independent switching controller which ensures the stabilization of a singularly perturbed SAS with a given imposed decay rate. Moreover, an estimation of the domain of attraction is also provided and conditions are presented for global stabilization.

This work can be viewed as a preliminary step in the stabilization of switched affine systems from a singular perturbation perspective. The end goal is to apply to the systems addressed here the very useful tools in classical singular perturbations theory, most notably model reduction, in which the control signal is based only on the slow states. Conditions for achieving this are the object of further investigation.

We also hope in future works to relax some of the LMI conditions stated here in order to enlarge the possible applications and also obtain less conservative results. Furthermore, the maximization of the upper bound ε * may also be another interesting result to pursue. A way of dealing with the growth in the number of LMIs when many switching modes are present is also an important point worth looking into.

  and Π 1 2 = Π 2 2 = 5.63. The bound ε * has been estimated as ε * = 0.3. Since ε < ε * , we can use the controller proposed in Theorem 1.

Fig. 1 .

 1 Fig.1. Trajectory of the system in the example.Fig.1shows the simulation results. In Fig.1, the border of the estimation of the domain of attraction (∂Ω * ) is also shown. It can be seen in Fig.1that the trajectory indeed stabilizes at the desired equilibrium point. The switching surface s(η) := η T H(v 1 -v 2 ) = 0 is also shown in Fig.1. The expression for s(η) is obtained from (5), with H determined using (27).

Appendix A. PROOF OF LEMMA 2 From (20), we have that, for each i ∈ I N : (Deaecto et al., 2012, Lem. 1), there exists

the latter being valid for all ε ∈ (0, µ i,1 ). Therefore, (A.1) holds for all ε ∈ (0, µ i ) with µ i = min{µ i,1 , µ i,2 }.

The estimation of µ i,2 is carried out as follows. First, rewrite (A.1) as:

which, using the Schur complement (given that

and thus µ i can be written as in ( 21). The bound µ can be taken as the most conservative one among the bounds for each i ∈ I N . Thus, µ = min i∈I N µ i .


A Blanchard

A Pitti

Rationalizing Humanoïd Platforms for Studies on Navigation, Interaction and Facial Expression

Introduction

In many robotics labs, we have to maintain a fleet of robots that are costly and time consuming to build, and also require a lot of energy for maintenance.

In our lab, we have robots for three main kinds of experiments: navigation (indoor and outpdoor), interaction (human-robot and object-robot) and facial expressions. We present a modular platform which aims to demonstrate all these possibilities in a unique humanoïd robot while keeping it easy to work on the different parts separately.

Needs

For bio-inspired navigation we use robuROC4 robots for outdor and robulabs robots for indoor from Robosoft[5] with some specific equipements. For interaction and manipulation, we use robotic arms like Kinova or Katana [START_REF]Jaco arm from kinova[END_REF]. For facial expressions, we used an home made robotic head [START_REF] Sofiane Boucenna | Imitation as a Communication Tool for Online Facial Expression Learning and Recognition[END_REF] we have then installed this head on a moving platform Robulab which allows the robot to navigate in museum while displaying different facial expressions [START_REF] Moualla | Is it useful for a robot to visit a museum? Paladyn[END_REF] (see Fig. 1).

Figure 1: Berenson navigating in a museum expressing its critical

The difficulty with this set of robots is that it is requiring a lot of maintenance and a various setups. Another difficulty is that the companies building the robots may be abroad (Kinova, Neuronics) or may have stopped to produce the robots (Robosoft).

That why we try to rationalize the platforms. First we wanted to use elements that can be used independently ( a head, a torso, a robotic base) in order to let students or researchers to work in parrallel on specific parts without interfering with the main robot. The parts should be easy to maintain independently of the building companies. We try to minimise the number of platforms for outdoor and indoor.

The proposed platform

For all the precedent reasons, we use on open source humanïd torso Reachy from Pollen-robotics in Bordeaux [START_REF]Pollen robotics[END_REF]. It is 3D printed so if needed we can reprint some parts and customize it. It exactly what we have done for the head in order to make facial expressions. We ask a compagny Animatronix at Ivry-sur-Seine near Paris [START_REF]Animatronix[END_REF] to design and build a compatible head with facial expressions. For the mobile base we wanted the same platform to be able to go indoor and outdoor on uneven terrain and to be able to rotate the torso. We ask a compagny in Cergy-Pontoise to adapt their autonomous mobile robot for logistics and agriculture in order to support and motorize the humanoïd torso.

The resulting robot is presented Fig. 2. We call it Ferdinand in reference to Ferdinand Cheval, a famous french postman which was navigating a lot through the mountain, but who has also very good artistic and building qualities as alone, he did build a magnificent palace [START_REF] Cheval | The palace of Ferdinand[END_REF].

It is smaller than an electric wheelchair so it can 1 Figure 2: Ferdinand in a task of navigation guided by a human navigate indoor but it is well suited to go outdoor as it is studied for agriculture. Everything is ROS compatible and we have enough power to add a powerfull computer (intel i9), powering the torso and the head. We have two cameras and microphones in the head. The torso is easy to remove ( 4 screws and 2 cables) to test the mobile base and the torso separatly. We have added a camera on the mobile base ( not on the picture ) identical to the one we use on the autonomous car which allow us to use the same algorithm of navigation ( with some adaptations due to the size, speed, ...).

Discussion

We will propose some feedbacks on the experience of this platform and we will test different applications. A thing that could be interesting is to compare this platform with the holonomic one proposed by Pollen-robotics for indoor ( it did not exist when we started the project ).