Prof Bernard

First and foremost, I wish to express my gratitude to my supervisors,

 2.8 LCS (2.19) with d = 0, e = 1, f = 0 . . . . . . . . . . . . . . . . . . . 2.9 A circuit with two ideal diodes and a voltage source . . . . . . . . . . . 2.10 Desired trajectory simulated by SICONOS with h = 10 -4 , θ = 0.5 . . . 2.11 Simulation results by using SICONOS with h = 10 -4 and θ = 0.5. . . . 2.12 A semi-log plot of the error between x and x d in time t (c =

Introduction

With a large range of applications in various fields like mechanics, electrical circuits, control systems, etc., complementarity systems have recently been the object of strong attention. In mathematical programming, complementarity conditions are well-known and they arise, for instance, in the Karush-Kuhn-Tucker conditions for optimality. In terms of electrical circuits, imposing complementarity conditions simply means that some ports are terminated by ideal diodes (see [START_REF] Camlibel | On linear passive complementarity systems[END_REF]).

Because of its applications, like other dynamical systems, dynamic systems with complementarity conditions, called complementarity dynamical systems, have been carefully studied as witnessed by many articles, see for instance [2,3,[START_REF] Brogliato | Existence and uniqueness of solutions for nonautonomous complementarity dynamical systems[END_REF][START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Goeleven | On the stability of stationary solutions of first order evolution variational inequalities[END_REF][START_REF] Heemels | The complementarity class of hybrid dynamical systems[END_REF].

The well-posedness has been treated and another interesting problem has arisen. That is controlling such dynamical systems and it is also our object in this report. Due to some limitations, in this paper, we restricted our attention to linear underlying dynamics with consistent initialization, such that no state jump occurs (solutions are continuous). Our goals are:

• To survey the well-posedness of Linear Complementarity System,

• To find a controller such that the state x and desired trajectory x d satisfy x(t) -

x d (t) → 0, as t → +∞.

This work was developed in my internship funded by INRIA (The French National Institute for Research in Computer Science and Control) from February to July 2019. The report consists of two main chapters, Introduction, Conclusion and Appendix.

Chapter 1 is devoted to the study of the existence and uniqueness of solution of Linear Complementarity Systems with feedthrough matrix satisfying D = 0, D 0 and D 0, which arise from physical examples, for instance, electrical circuits with a network of ideal diodes. The well-posedness results of the multivalued Lur'e system have been considered as a generalization of so-called Linear Complementarity Systems.

That may be important in practice in case D is a nonzero feedthrough matrix, possibly positive semidefinite and non-symmetric.

In Chapter 2, our main results are stated and proved to reach our target, that is designing controller u for a Linear Complementarity Systems satisfying reference inputs.

A brief introduction to the SICONOS Platform and modeling of a nonsmooth dynamic system via examples, in the scalar case as well as the higher dimensions, aims at obtaining numerical simulations of these problems. Furthermore, we also deal with extended controller form and a way to choose the "best" gain K.

In Appendix we recall some basic facts and notations which will be used throughout this report. It includes some definitions and properties in Convex analysis concerning convex sets and convex functions. The solution concept of a dynamic system such as equilibrium point, stability was introduced as well. Appendix also provides detailed scripts written in C++ and MATLAB that we have used in Chapter 2. We can find there the codes to find feedback gain K and script to execute in SICONOS.

Chapter 1 Well-posedness of Linear Complementarity Systems

In this chapter, we will deal with Linear Complementarity Systems (LCS) and focus on the existence and uniqueness of their solutions, which are absolutely continuous (AC). Three sub-classes of LCS are treated, including D = 0, D 0 and D 0 that come from physical examples, specifically electrical circuits with resistors, inductors, capacitors and ideal diodes (RLCD). In order to summarize briefly, the following results are presented without their proofs. The references here are [3,[START_REF] Brogliato | Existence and uniqueness of solutions for nonautonomous complementarity dynamical systems[END_REF][START_REF] Camlibel | On linear passive complementarity systems[END_REF].

Linear Complementarity Systems

Let us now consider the following complementarity dynamical system

   ẋ(t) = a(x(t)) + b(x(t))λ(t) + e(x(t), u(t)) 0 ≤ λ(t) ⊥ y(t) = c(x(t)) + g(u(t)) ≥ 0,
where u(t) ∈ R m , y(t) ∈ R p . This class is very general and needs to be restricted to obtain well-posedness results. Thus, in this paper, we will restrict our attention to the linear complementarity systems of the form

               x(0) = x 0 , ẋ(t) = Ax(t) + Bλ(t) + Eu(t), y(t) = Cx(t) + Dλ(t) + F u(t), 0 ≤ λ(t) ⊥ y(t) ≥ 0, (1.1)
where A, B, C, D, E, and F are real constant matrices with appropriate sizes,

x(t) ∈ R n , λ(t) ∈ R p , u(t) ∈ R m . The last line is a complementarity relation between λ(t)
and y(t), which are forced to remain non-negative and always orthogonal one to each other. At each time t, both λ i (t) and y i (t) must be non-negative, and at least one of them should be zero. Using the equivalence

0 ≤ λ(t) ⊥ y(t) ≥ 0 ⇔ -λ(t) ∈ ∂ψ R p + (y(t)),
which holds for vectors λ(t), w(t) ∈ R p , the LCS above is equivalently rewritten as

         x(0) = x 0 , ẋ(t) = Ax(t) + Bλ(t) + Eu(t), λ(t) ∈ -∂ψ R p + (Cx(t) + Dλ(t) + F u(t)).
(1.2)

We define the Linear Complementarity Problem (LCP) with data q ∈ R m and M ∈ R m×m by the problem of finding z ∈ R m such that 0 ≤ z ⊥ q + M z ≥ 0. We know that this LCP has a unique solution for any q if and only if M is an m × m P-matrix.

A square matrix is said to be a P-matrix if all its principal minors are positive (see [START_REF] Murty | Linear Complementarity, Linear and Nonlinear Programming[END_REF] for more details).

Remark 1.1. Complementarity relations appear in many fileds. For instance, voltagecurrent characteristic of an ideal diode is a complementarity relation. Before starting the study of linear complementarity systems with external inputs and outputs, we first need to establish some results on existence and uniqueness of solutions. The following results are taken from [3,[START_REF] Brogliato | Existence and uniqueness of solutions for nonautonomous complementarity dynamical systems[END_REF][START_REF] Camlibel | On linear passive complementarity systems[END_REF].

v(t) i(t) (a) Ideal diode v(t) i(t) O (b) Ideal diode characteristic

Case D = 0

Let us recall some facts and developments from [START_REF] Brogliato | Existence and uniqueness of solutions for nonautonomous complementarity dynamical systems[END_REF].

Assumption 1.1. There exists a constant matrix P = P 0 such that

P B = C .

INTERNSHIP REPORT

Denoting by R, the symmetric positive definite square root of P , and letting z = Rx, one gets from (1.1)

   ż(t) = R ẋ(t) = RAR -1 z(t) + REu(t) + RBλ(t) 0 ≤ λ(t) ⊥ y(t) = CR -1 z(t) + F u(t) ≥ 0. (1.3) Since the equivalence 0 ≤ λ(t) ⊥ CR -1 z(t) + F u(t) ≥ 0 ⇔ -λ(t) ∈ ∂ψ R p + (CR -1 z(t) + F u(t)),
consequently, one can rewrite (1.3) as

-ż(t) ∈ -RAR -1 z(t) -REu(t) + RB∂ψ R p + (CR -1 z(t) + F u(t)). Now using R 2 B = C , it follows that -ż(t) ∈ -RAR -1 z(t) -REu(t) + R -1 C ∂ψ R p + (CR -1 z(t) + F u(t)). (1.4) For each t ∈ [0, +∞[ the closed set K(t) := {x ∈ R n : Cx + F u(t) ≥ 0}
and R m are convex polyhedral and

ψ K(t) (x) = (ψ R m + -F u(t) • C)(x). Therefore, C ∂ψ R p + (CR -1 z(t) + F u(t)) = ∂ψ K(t) (x)
for any x ∈ R n . So the inclusion in (1.4) can be rewritten as

-ż(t) + RAR -1 z(t) + REu(t) ∈ R -1 ∂ψ K(t) (R -1 z(t)).
Consider the closed convex polyhedral set

S(t) := R(K(t)) = {Rx : x ∈ K(t)}, we can see that ψ S(t) (x) = (ψ K(t) • R -1 )(x) for all x ∈ R n .
Since R is invertible and symmetric, again we have

∂ψ S(t) (x) = R -1 (∂ψ K(t) )(R -1 x) for all x ∈ R n
and hence, since N S(t) (x) = ∂ψ S(t) (x), the differential inclusion may be written in the form -ż(t) + RAR -1 z(t) + REu(t) ∈ N S(t) (z(t)).

(1.5)

Remark 1.2. By the definition of the normal cone, the inclusion (1.5) is equivalent to the evolution variational inequality

ż(t) -RAR -1 z(t) -REu(t), v -z(t) ≥ 0, ∀v ∈ S(t), z(t) ∈ S(t).
If F = 0 then K does not vary with the time and Theorem 2.2 in [START_REF] Goeleven | On the stability of stationary solutions of first order evolution variational inequalities[END_REF] may applied with u(•) a continuous mapping with locally L 1 derivative. Consequently, we conclude the well-posedness of the LCS. However, here we let K hence S be time-varying, which complicates the analysis.

Theorem 1.1 ([5], Theorem 3.5). Assume that u(•) ∈ L 1 loc (R + ; R p ) and the set-valued mapping S(•) = R(K(•)
) is locally absolutely continuous with nonempty values. Then,

-ż(t) + RAR -1 z(t) + REu(t) ∈ N S(t) (z(t)) with initial condition z(0) = z 0 ∈ R(K(0))
         ẋ1 (t) = x 2 (t) ẋ2 (t) = -1 LC x 1 (t) -R L x 2 (t) -1 L λ(t) + 1 L u(t) 0 ≤ λ(t) ⊥ -x 2 (t) ≥ 0
where u denotes the voltage applied to the system, for instance u(t) = sin(t), t ≥ 0, x 1 is the charge of the capacitor and x 2 is current though inductor. The matrices A, B, C, D, E, F can be easily identified. In this situation, we have D = 0. Following the same lines, we can rewrite the dynamics as

-ż(t) + 0 1 -1 LC -R L z(t) + 0 -1 L √ L u(t) ∈ N S (z(t)).

INTERNSHIP REPORT

Here Proposition 3.2,[START_REF] Brogliato | Existence and uniqueness of solutions for nonautonomous complementarity dynamical systems[END_REF]). Now, applying Theorem 1.1, we claim that for any z 0 ∈ S, the dynamics with initial condition z(0) = z 0 has a unique locally absolutely continuous solution z(•) on [0, +∞[.

z = Rx with R = 1 √ L I and S = {(x 1 , x 2 ) ∈ R 2 : x 2 ≤ 0}. Since F = 0, S(•) = S is locally absolutely continuous (see

Case D 0

In this section our goal is just to point out that extensions of the above developments led for D = 0 are possible when D = 0. We first consider the positive semidefinite matrix D of the form

D = D 1 0 0 0 ,
where D 1 0 is square of dimension q < p, and not necessarily symmetric. We partition the multiplier as

λ = λ 1 λ 2 , such that Dλ = D 1 λ 1 0
. The vector y is partitioned similarly. Then the complementarity conditions in (1.1) can be written as the following

   0 ≤ λ 1 (t) ⊥ y 1 (t) = C 1 x(t) + D 1 λ 1 (t) + F 1 u(t) ≥ 0, 0 ≤ λ 2 (t) ⊥ y 2 (t) = C 2 x(t) + F 2 u(t) ≥ 0.
The first set of conditions is a LCP with unknown λ 1 and matrix D 1 . It possesses a unique solution that is continuous piecewise linear in x(t) and u(t), and we denote as λ 1 (x, u). The LCS in (1.1) is rewritten as

   ẋ(t) = Ax(t) + B 1 λ 1 (x, u) + B 2 λ 2 (t) + Eu(t) 0 ≤ λ 2 (t) ⊥ y 2 (t) = C 2 x(t) + F 2 u(t) ≥ 0.
If we assume that P 2 B 2 = C 2 for some symmetric positive definite matrix P 2 , and R 2 the symmetric positive definite square root of P 2 , then following the same steps as in case D = 0, one gets

-ż(t) + R 2 (AR -1 2 z(t) + B 1 λ 1 (R -1 2 z(t), u(t))) + R 2 Eu(t) ∈ N S(t) (z(t)), here S(t) = R 2 (K(t)) = {R 2 x : x ∈ K(t)} and K(t) = {x ∈ R n : C 2 x+F 2 u(t) ≥ 0}.
Then Theorem 1.1 may be applied since the steps of its proof can be redone for a Lipschitz continuous field.

Example 1.2. Let us consider the electrical system (Figure 1.3) taken from [3] that is composed of two resistors R and four capacitors C and two ideal diodes with char- The dynamics is given by

acteristics 0 ≤ v 1 (t) ⊥ i 1 (t) ≥ 0 and 0 ≤ v 2 (t) ⊥ i 3 (t) ≥ 0, respectively. The state variables are x 1 (t) = t 0 i 1 (t)dt, x 2 (t) = t 0 i 2 (t)dt, x 3 (t) = v 2 (t), and λ 1 (t) = i 3 (t), λ 2 (t) = v 1 (t). v 1 i 1 R R i 2 C C C C v 2 i 3
   ẋ1 (t) ẋ2 (t) ẋ3 (t)    =    -2 RC 1 RC 0 1 RC -2 RC 1 R -1 RC 2 2 RC 2 -1 RC       x 1 (t) x 2 (t) x 3 (t)    +    0 1 R 0 0 1 C 0    λ 1 (t) λ 2 (t) , 0 ≤ λ(t) ⊥ y(t) = 0 0 1 -2 RC 1 RC 0    x 1 (t) x 2 (t) x 3 (t)    + 0 0 0 1 R λ(t) ≥ 0.
The matrices A, B, C and D are easily identified. Set B 1 = (0, 0, 1 C ) , B 2 = ( 1 R , 0, 0) , C 1 = (0, 0, 1), and C 2 = ( -2 RC , 1 RC , 0). The linear complementarity conditions can be rewritten as follows

   0 ≤ λ 1 (t) ⊥ C 1 x(t) ≥ 0, 0 ≤ λ 2 (t) ⊥ C 2 x(t) + 1 R λ 2 (t) ≥ 0. (1.6)
From the second line, we have λ 2 (t) = -R min{0, C 2 x(t)}. The matrix D is positive semidefinite and relation

P 1 B 1 = C 1 holds with P 1 = CI 3 . Let z = R 1 x with R 1 = √ CI 3 , one obtains the differential inclusion -ż(t) + Az(t) - R √ C B 2 min{0, C 2 z(t)} ∈ N S(t) (z(t)),
where

K(t) = {x ∈ R 3 : C 1 x ≥ 0} = {(x 1 , x 2 , x 3 ) ∈ R 3 : x 3 ≥ 0} and S(t) = {R 1 x : x ∈ K(t)} = { √ C(x 1 , x 2 , x 3 ) ∈ R 3 : x 3 ≥ 0}.
Since the steps of the proof of Theorem 1.1 can be redone for a Lipschitz continuous field, we can conclude that the dynamics has a unique solution with an initial condition x 0 = (x 1 , x 2 , x 3 ) satisfying x 3 ≥ 0. Now we will deal with the matrix D 0 but not in the form D = D 1 0 0 0 .

However, the following lemma gives us a way to feedthrough matrix D. Lemma 1.1 ([5], Lemma 3.8). Let D ∈ R p×p have rank q < p, and suppose that there exist full-rank matrices V and W with W = V , such that D = V DW with

D = D 1 0 0 0
, where D 1 is a q × q full-rank matrix. Then the LCS in (1.1) can be equivalently rewritten as the cone complementarity system (CCS)

   ẋ(t) = Ax(t) + Eu(t) + BW -1 λ(t) C W -1 λ(t) ⊥ ȳ(t) = V -1 Cx(t) + V -1 F u(t) + Dλ (t) ∈ C V (1.7)
where

C V = {z ∈ R p : V z ∈ R p + }, and C W -1 = {z ∈ R p : W -1 z ∈ R p + } are two dual polyhedral cones.
The second line in (1.7) is so-called a cone complementarity problem (CCP).

Remark 1.3. Following the same lines as in the case D = 0 and in the case matrix D of the form D 1 0 0 0 and since a CPP with two dual cones can be equivalently written as an inclusion into a normal cone, the CCS in (1.7) can be transformed into a sweeping process. Further conditions on D 1 are necessary to ensure that these manipulations are doable.

Case D 0

The case D is positive definite is worth stating separately.

u(•) ∈ L 1 loc (R + ; R p ).
Remark 1.4. The continuity of the solution is also stated by Corollary 10.6 in [6].

Example 1.3. Let us consider the circuit in Figure 1.4 with a voltage source u. The dynamics is given by

         ẋ1 (t) = -1 RC x 1 (t) + x 2 (t) -λ(t) R + u(t) R ẋ2 (t) = -1 LC x 1 (t) -λ(t) L + u(t) L 0 ≤ λ(t) ⊥ λ(t) R + x 1 (t) RC -x 2 (t) -u(t) R ≥ 0.
(1.8)

Here, we considered the current through the inductor for the variable x 2 , and for the variable x 1 the charge on the capacitor as state variables. It can be seen that (1.8) is an LCS with matrix D = 1 R > 0. According to Proposition 1.1, we claim that this problem has a unique continuous solution for any initial state x 0 and input u. In our framework, the assumption on existence a matrix P that is symmetric, positive definite satisfying P B = C is significant to perform the steps after; however, in a certain situation, finding a such matrix is not easy even difficult. The following proposition provides a sufficient condition for existence of matrix P . It is considered as a corollary of Lemma 10.4 in [6].

Proposition 1.2. Let B ∈ R n×p , C ∈ R p×n ∈ and let B be full column-rank. Then, there exists a symmetric positive definite matrix P such that P B = C if and only if CB is symmetric positive definite.

The multivalued Lur'e system

Suppose that

A ∈ R n×n , B ∈ R n×p , C ∈ R p×n , D ∈ R p×p are given matrices, f ∈ C 0 (R + ; R) such that ḟ ∈ L 1 loc (R + ; R n ) and ϕ i : R → R ∪ {+∞}(1 ≤ i ≤ p)
given proper convex and lower semicontinuous functions. Let x 0 ∈ R n be some initial condition, we consider the problem: Find

x ∈ C 0 (R + ; R n ) such that ẋ ∈ L ∞ loc (R + ; R n ) and x right-differentiable on R + , λ ∈ C 0 (R + ; R p ) and y ∈ C 0 (R + ; R p ) satisfying the nonsmooth dynamical system                                  x(0) = x 0 ẋ(t) = Ax(t) + Bλ(t) + f (t) a.e. t ≥ 0 y(t) = Cx(t) + Dλ(t) λ 1 (t) ∈ -∂ϕ 1 (y 1 (t)) λ 2 (t) ∈ -∂ϕ 2 (y 2 (t))
. . . λ p (t) ∈ -∂ϕ p (y p (t))

(1.9)

INTERNSHIP REPORT

The case D = 0 was considered in [START_REF] Brogliato | Existence and uniqueness of solutions for nonautonomous complementarity dynamical systems[END_REF], so we shall study the case D = 0. The following results can be found in [3]. Let us now denote by p I (and set p II = p -p I ) the largest integer such that the matrix D can be written as follows:

D = 0 p I ×p I 0 p I ×p II 0 p II ×p I D II
with D II = 0 p II ×p II . In using this notation, it is convenient to suppose that p I = 0 (resp. p II = 0) means that the terms indexed by I (resp. II) are useless and not considered.

For z ∈ R p I , we set z = (z I z II ) with z I ∈ R p I and z II ∈ R p II , B = B I B II , C = C I C II , here B I ∈ R n×p I , B II ∈ R n×p II , C I ∈ R p I ×n and C II ∈ R p II ×n . Finally, we set ∀y ∈ R p I : Φ I (y) := ϕ 1 (y 1 ) + ϕ 2 (y 2 ) + • • • + ϕ p I (y p I ) and ∀y ∈ R p II : Φ II (y) := ϕ p I +1 (y 1 ) + ϕ 2 (y p I +2 ) + • • • + ϕ p (y p II ).
We also set for all y ∈ R p , and z ∈ R

Φ(y) = Φ I (y I ) + Φ II (y II ) and Φ * ,-(z) = Φ * (-z).
By using these notations, we can see that the nonsmooth dynamical system reduces to the system

                           x(0) = x 0 ẋ(t) = Ax(t) + B I λ I (t) + B II λ II (t) + f (t) for all t ≥ 0 y I (t) = C I x(t) λ I (t) ∈ -∂Φ I (y I (t)) y II (t) = C II x(t) + D II λ II (t) y II (t) ∈ -∂Φ * ,- II (λ II (t)).
(1.10)

Assumption 1.2. For each 1 ≤ i ≤ p, there exists z 0,i ∈ R at which ϕ * ,- i (•) is continu- ous, with ϕ * ,- i (z) = ϕ * i (-z), ∀z ∈ R.
Assumption 1.2 is required to guarantee that for all z ∈ R,

∂ϕ * ,- i (z) = -∂ϕ * i (-z).
Then

λ i ∈ -∂ϕ i (y i ) ⇔ y i ∈ -∂ϕ * ,- i (λ i ). Assumption 1.3 (If p I ≥ 1)
. There exists a symmetric and invertible matrix

W ∈ R n×n such that W 2 B I = C I .
Then, we set

V =    W if p I ≥ 1 I if p I = 0 and ∀w ∈ R p I , Ξ I (w) =    Φ I (C I V -1 w) if p I ≥ 1 0 if p I = 0. Assumption 1.4 (If p I ≥ 1). There exists a point w 0 in R p I at which Ξ I (•) is continuous.
Assumptions 1.3 and 1.4 ensure that in case p I ≥ 1, one has for all w ∈ R p I ,

∂Ξ I (w) = V -T C I ∂Φ I (C I V -1 w) = V -1 C I ∂Φ I (C I V -1 w).
The multivalued mapping Ξ I (•) is maximal monotone, being the subdifferential of a convex, proper, lower semicontinuous function. Let us denote

∀x ∈ R n , Λ II (x) =    V B II (D II + ∂Φ * ,- II ) -1 (-C II V -1 x) if p II ≥ 1 0 if p II = 0. Assumption 1.5 (If p II ≥ 1). The operator Λ II : R n → R n , x → Λ II (x) is well-
defined, single-valued and Lipschitz continuous.

Proposition 1.3 ([3], Proposition 1). Suppose that D II is positive definite and Φ II (•)
is proper convex and lower semicontinuous. Then the operator (D II + ∂Φ * ,- II ) -1 is well-defined, single-valued and Lipschitz continuous. Proposition 1.4 ([3], Proposition 2). Suppose that D II is a P-matrix and

Φ II (•) = ψ R p II + (•).
Then the operator (D II + ∂Φ * ,- II ) -1 is well-defined, single-valued and Lipschitz continuous.

INTERNSHIP REPORT

We recall that for a nonempty closed and convex set C ⊂ R n , the recession cone of C is defined by

C ∞ = λ>0 1 λ (C -x 0 ),
where x 0 is an element in C. We also denote by ker(M ) the kernel of M and by R(M ) the range of M .

Proposition 1.5 ([3], Proposition 3). Suppose that D II is positive semidefinite, dom(Φ * ,- II ) = dom(Φ * ,- II ), (dom(Φ * ,- II )) ∞ ∩ ker(D II + D II ) ∩ {z ∈ R p II : D II z ∈ ((dom(Φ * ,- II )) ∞ ) * } = {0}. Moreover, R(C II ) ⊂ R(D II + D II ) ⊂ ker(B II ).
Then the operator

x → V B II (D II + ∂Φ * ,- II ) -1 (-C II V -1 x
) is well-defined, singlevalued and Lipschitz continuous.

The problem in the nonsmooth dynamical system (1.10) can be reduced, by setting X(t) = V x(t), ∀t ≥ 0, to the following dynamical variational inequality problem:

Find X ∈ C 0 (R + ; R n ) such that Ẋ ∈ L ∞ loc (R + ; R n
) and X right-differentiable on R + with X(0) = X 0 and satisfying for almost everywhere t ≥ 0 the variational inequality

Ẋ(t) -V AV -1 X(t) -Λ II (X(t) -V f (t), v -X(t) + Ξ I (v) -Ξ I (X(t)) ≥ 0, holds for all v ∈ R n .
Indeed, in case p I ≥ 1 and p II ≥ 1, from (1.10), it is clear that

ẋ(t) -B II (D II + ∂Φ * ,- II ) -1 (-C II x(t)) -Ax(t) -f (t) ∈ -B I ∂Φ I (C I x(t)).
Notice that

D II z -q ∈ -∂Φ * ,- II (z) ⇔ z ∈ (D II + ∂Φ * ,- II ) -1 (q),
for all q ∈ R p I . Multiply by invertible matrix V , then we obtain

Ẋ(t)-V AV -1 X(t)-V B II (D II +∂Φ * ,- II ) -1 (-C II V -1 X(t))-V f (t) ∈ -∂Ξ I (X(t))
from which one deduces the variational inequality. The case p I = 0 (resp. p II = 0) can be deduced from the previous relations by removing the terms indexed by I (resp. II).

Well-posedness by Kato's Theorem

The following result relies on Kato's theorem.

Theorem 1.2 ([3], Theorem 1). Suppose that Assumptions 1.2-1.5 hold. Then for any

X 0 ∈ Dom(∂Ξ I ) there exists a unique X ∈ C 0 (R + ; R n ) such that Ẋ ∈ L ∞ loc (R + ; R n ), X right-differentiable on R + and for all v ∈ R n , X(t) ∈ Dom(∂Ξ I ), ∀t ≥ 0, Ẋ(t) -V AV -1 X(t) -Λ II (X(t) -V f (t), v -X(t) + Ξ I (v) -Ξ I (X(t)) ≥ 0, X(0) = X 0 .
We can check that Assumptions 1.2-1.5 are satisfied for the system given in Example 2.2. Then, using Theorem 1.2, one concludes that the dynamical system has a unique solution.

Corollary 1.1. Let p I = p. Suppose that Assumptions 1.2-1.4 hold. Then for any

X 0 ∈ dom(Ξ) there exists a unique X ∈ C 0 (R + ; R n ) such that Ẋ ∈ L ∞ loc (R + ; R n ), X right-differentiable on R + satisfying          X(t) ∈ Dom(∂Ξ), ∀t ≥ 0, Ẋ(t) -V AV -1 X(t) -V f (t), v -X(t) + Ξ(v) -Ξ(X(t)) ≥ 0, ∀v ∈ R n , a.e. t ≥ 0, X(0) = X 0 . It follows that for x 0 ∈ R n such that Cx 0 ∈ Dom(∂Φ), the function x(•) = V -1 X(•) is the unique solution of the problem                x(0) = x 0 ẋ(t) = Ax(t) + Bλ(t) + f (t) a.e. t ≥ 0 y(t) = Cx(t), ∀t ≥ 0 λ(t) ∈ -∂Φ(y(t)), ∀t ≥ 0 Corollary 1.2. Let p II = p. Suppose that Assumptions 1.2 and 1.5 hold. Then for any X 0 ∈ R n there exists a unique X ∈ C 0 (R + ; R n ) such that Ẋ ∈ L ∞ loc (R + ; R n ), X right-differentiable on R + satisfying    Ẋ(t) -AX(t) -B(D + ∂Φ * ,-) -1 (-CX(t)) -f (t) = 0, a.e. t ≥ 0, X(0) = X 0 . INTERNSHIP REPORT Then for x 0 ∈ R the function X(•) is the unique solution of Problem 1.9, that is                x(0) = x 0 ẋ(t) = Ax(t) + Bλ(t) + f (t) a.e. t ≥ 0 y(t) = Cx(t) + Dλ(t), ∀t ≥ 0 λ(t) ∈ -∂Φ(y(t)), ∀t ≥ 0 1.2.

Well-posedness by maximal monotonicity

It is noteworthy that the condition R(C II ) ⊂ R(D II + D II ) ⊂ ker(B II ) in Propostion 1.5 implies that the symmetric part of D has a large enough range, but in some cases of D, that is not satisfied. This motivates us to look for another assumptions to show the well-posedness of Problem (1.9). Assumption 1.6. There exists a symmetric and positive definite matrix P such that

P B = C and D is positive semidefinite.
It is clear that the Lur'e system in (1.9) may be rewritten by

ẋ(t) ∈ Ax(t) + B(D + ∂Φ * ,-) -1 (-Cx(t)) + f (t).
(1.11)

Defining R as R 2 = P , the symmetric positive definite square root of P , and letting z = -Rx, one gets from (1.11)

ż(t) ∈ RAR -1 z(t) -R -1 C (D + ∂Φ * ,-) -1 (-CR -1 z(t)) -Rf (t). Since R is symmetric, the operator z → R -1 C (D + ∂Φ * ,-) -1 (CR -1 z) is maximal monotone, provided R(CR -1 ) ∩ rint(Dom((D + ∂Φ * ,-) -1 )) = ∅.
Theorem 1.3 ([3], Theorem 2). Let Assumptions 1.2 and 1.6 hold and suppose that

R(CR -1 ) ∩ rint(Dom((D + ∂Φ * ,-) -1 )) = ∅. Let x 0 ∈ R n such that -Cx 0 ∈ Dom((D + ∂Φ * ,-) -1
). Then the Lur'e system in (1.9) possesses a unique solution that is Lipschitz continuous.

Before ending this section, we consider the example as follows.

Example 1.4. Let us consider the Filtered full wave rectifier in Figure 1.5 which is taken from [3] with the dynamics given by

E(t) i DF 1 i DR1 i DF 2 i DR2 R v R i R C Figure 1.5: Filtered full wave rectifier                      ẋ(t) = -1 RC x(t) + 1 C 0 1 C 0 λ(t) 0 ≤ λ(t) ⊥ y(t) =         0 -1 0 0 1 0 1 -1 0 -1 0 0 0 1 0 0         λ(t) +         1 0 1 0         x(t) ≥ 0, (1.12)
where the state x is the voltage across the capacitor,

y 1 = v DR1 , y 2 = i DF 2 , y 3 = v DF 1 , y 4 = v DR2 , and λ = (i DR1 , v DF 2 , i DF 1 , i DR2 ) . The matrix D is positive semidefinite since it is skew symmetric. The relation P B = C holds with P = C > 0. We choose R = √ C. Then, R(CR -1 ) = {α(1, 0, 1, 0) , α ∈ R}. It is noteworthy that Dom((D + ∂Φ * ,-) -1 ) = R(D + ∂Φ * ,-
) and the condition of Theorem 1.3 is satisfied and the system (1.12) has a unique Lipschitz solution (see [3]).

Remark 1.5. [3] In the case y = Cx + Dλ + F u and f (t) = Eu(t), where u(•)

is some m-dimensional control input, then the results above may be used to analyse feedback controllers of the form u = Sx + Gλ. It suffices to replace (A, B, C, D) by

(A + ES, B + EG, C + F S, D + F G) in the analysis.

Dissipativity and stability results

Suppose that the Lur'e system is well-posed, we are now at the position to study stability properties of the system. Denote

X 0 =    {x ∈ R n : C I x ∈ Dom(∂Φ I )} if p I ≥ 1 R n if p I = 0.
Then, for x 0 ∈ X 0 , the following problem

               ẋ(t) = Ax(t) + Bλ(t) a.e. t ≥ 0 y(t) = Cx(t) + Dλ(t), ∀t ≥ 0 y(t) ∈ -∂Φ * ,-(λ(t)), ∀t ≥ 0 x(0) = x 0 ,
shortly denoted as the system (A, B, C, D, Φ * ,-), has a unique solution

x(•; x 0 ) : R + → R n t → R n .
In order to make sure that x(t; 0) = 0 for all t ≥ 0, we need one more assumption.

Assumption 1.7. Suppose that the initial data satisfies 0 ∈ X 0 and λ II (0) ∈ ∂Ξ I (0).

The notion of passivity (dissipativity) has played an important role in various contexts such as stability issues, adaptive control, identification, etc. First, let us present some definitions of dissipative systems.

Definition 1.1. The system (A, B, C, D) is called passive or dissipative, if there exists a symmetric and positive definite matrix P ∈ R n×n such that the matrix

A P + P A P B -C B P -C -(D + D ) 0.
Definition 1.2. The system (A, B, C, D) is said to be strictly passive, if there exists a symmetric and positive definite matrix P ∈ R n×n such that the matrix

A P + P A + εP P B -C B P -C -(D + D ) 0,
for some ε > 0.

Definition 1.3. One says that the system (A, B, C, D, Φ * ,-) is passive provided that there exists a symmetric and positive definite matrix P ∈ R n×n such that

P Ax, x + (P B -C )z, x -Dz, z + Φ * ,-(0) -Φ * ,-(z) ≤ 0, holds for all x ∈ R n , z ∈ R p .
Definition 1.4. One says that the system (A, B, C, D, Φ * ,-) is strictly passive provided that there exists a symmetric, positive definite matrix P ∈ R n×n and a real ε > 0 such that

P Ax + 1 2 εP x, x + (P B -C )z, x -Dz, z + Φ * ,-(0) -Φ * ,-(z) ≤ 0, holds for all x ∈ R n , z ∈ R p . Proposition 1.6 ([3], Propositions 4-5). If the system (A, B, C, D) is passive (strictly passive) and ∀z ∈ R p , Φ * ,-(z) ≥ Φ * ,- (0)
then the system (A, B, C, D, Φ * ,-) is passive (strictly passive), respectively.

Theorem 1.4 ([3], Theorem 3). Assume that the system (A, B, C, D, Φ * ,-) is passive.

Then there exists a constant c > 0 such that for each x 0 ∈ X 0 ,

x(t; x 0 ) ≤ c x 0 , ∀t ≥ 0.
Furthermore, if the system (A, B, C, D, Φ * ,-) is strictly passive then the system is globally exponentially stable, i.e.,

x(t; x 0 ) ≤ c x 0 e -αt , ∀t ≥ 0, for some c > 0, α > 0.

Remark 1.6. The system (A, B, C, D, Φ * ,-) is passive that ensures the stability of the trivial solution. However, if the system is strictly passive, then the trivial solution is stable and global attractive as well.

Chapter 2

Controller design for LCS

In this chapter, we will continue to study the LCS given by

         ẋ(t) = Ax(t) + Bλ(t) + Eu(t), 0 ≤ λ(t) ⊥ Cx(t) + Dλ(t) + F u(t) ≥ 0, x(0) = x 0 , (2.1)
where , where D 1 0 is square of dimension q < p.

x(t) ∈ R n , u(t) ∈ R m , λ(t) ∈ R p with D = 0,
Our goal is to design the controller u such that x(t) -x d (t) → 0 as t → ∞, where x d is a desired function. The main idea comes from the conclusions in Chapter 1. Generally, to find a controller satisfying a given reference is a hard problem. Thus our ambitious in this paper will only deal with some sub-classes of problems which we can handle.

The main references here are [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics[END_REF]3,[START_REF] Brogliato | Dissipative Systems Analysis and Control, Theory and Applications[END_REF][START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Heemels | The complementarity class of hybrid dynamical systems[END_REF][START_REF] Murty | Linear Complementarity, Linear and Nonlinear Programming[END_REF].

Sufficient conditions for stability

Recall that the quadruple (A, B, C, D) is called passive if

A P + P A P B -C B P -C -(D + D ) 0
for some matrix P = P 0. It is called strictly passive provided that

A P + P A + εP P B -C B P -C -(D + D ) 0 (2.2)
for some ε > 0 and matrix P = P 0.

The results in this section are extended from the passivity theorems that are formulated for the absolutely stability problems for Lur'e systems (see [3]).

Consider an input-free LCS given by

   ẋ(t) = Ax(t) + Bλ(t), t ≥ 0 0 ≤ λ(t) ⊥ Cx(t) + Dλ(t) ≥ 0 (2.
* ∈ R p satisfying    Ax * + Bλ * = 0 0 ≤ λ * ⊥ Cx * + Dλ * ≥ 0. (2.4)
Moreover, if A is invertible, then all equilibrium points of (2.3) are characterized by

x * = -A -1 Bλ * and 0 ≤ (-CA -1 B + D)λ * ⊥ λ * ≥ 0. (2.5)
From Lemma 2.1 it follows that x * = 0 is an equilibrium point. Next, we will deal with sufficient conditions for nature of equilibrium points.

State feedback controller

As we know, Dissipativity theory gives a framework for the design and analysis of control systems using an input-output description based on energy-related considerations. The main idea behind this is that many important physical systems have certain input-output properties related to the conservation, dissipation and transport of energy [START_REF] Brogliato | Dissipative Systems Analysis and Control, Theory and Applications[END_REF]. Our problems in this topic are derived from linear electrical networks consisting of resistors, inductors, capacitors, and ideal diodes, thus it allows us for an approach to control systems design and analysis. Now we focus on the linear complementarity system give by (2.1). First, we need to make some assumptions for our problem.

Assumption 2.1. There exists a multiplier λ d such that desired trajectory x d satisfies

   ẋd (t) = Ax d (t) + Bλ d (t) + Eu d (t) 0 ≤ λ d (t) ⊥ Cx d (t) + Dλ d (t) + F u d (t) ≥ 0, for a given input u d ∈ L 1 loc (R + ; R m ). Remark 2.1. For asymptotic tracking, x(t) -x d (t) → 0, as t → ∞, Assumption 2.1 is necessary. Let us consider the following problem    ẋ(t) = -x(t) + λ(t) 0 ≤ λ(t) ⊥ x(t) + λ(t) + u(t) ≥ 0. (2.6)
From Proposition 1.1, the system (2.6) has a unique solution for any initial condition x(0) = x 0 ∈ R due to d = 1 > 0. Our aim is to find u such that the solution x(t) → -a, as t → ∞ with a > 0. It is easy to check that Assumption 2.1 is violated. We will prove that there is no controller u satisfying this problem. One can rewrite (2.6) as

ẋ(t) =    -x(t) if x(t) + u(t) ≥ 0 -2x(t) -u(t) if x(t) + u(t) ≤ 0.
It yields

x(t) =    C 1 e -t if x(t) + u(t) ≥ 0 C 2 e -2t - t t 0 e -2(t-s) u(s)ds if x(t) + u(t) ≤ 0,
for some C 1 and C 2 . Since the trajectory x(t) → -a < 0, t → ∞, the controller u must satisfy x(t) + u(t) ≤ 0 for large t. That means the second mode has to be activated. It follows that lim

t→∞ u(t) ≤ -lim t→∞ x(t) = a. Since x(t) has to converge to -a when t → ∞, lim t→∞ t t 0 e -2(t-s) u(s)ds = a.
Using L'Hospital rule, one gets

lim t→∞ u(t) 2 = a.
We got a contradiction. Therefore, there is no controller u for this problem.

Note that we can solve the regulation problem (x(t) → 0). In fact, if we choose u(t) = kx(t), then from (2.6) we obtain

   ẋ(t) = -x(t) + λ(t) 0 ≤ λ(t) ⊥ (1 + k)x(t) + λ(t) ≥ 0.
(2.7) Now, we choose k such that (-1, 1, 1 + k, 1) is strictly passive, for instance k = 0. Using Theorem 2.1, we claim that 0 is the only equilibrium point of (2.7), which is asymptotically stable. Assumption 2.2. There exists a matrix K such that the quadruple

(A + EK, B, C + F K, D)
is strictly passive. Lemma 2.2. Suppose that (A, B, C, D) is (strictly) passive. Then, the system

         ẋ(t) = Ax(t) + Bλ(t) + Eu(t), 0 ≤ λ(t) ⊥ Cx(t) + Dλ(t) + F u(t) ≥ 0, x(0) = x 0 .
(2.8) has a unique global solution for any input u ∈ L 1 loc (R + ; R m ). Moreover, the solution is continuous on [0, +∞[. Proof. We will only prove the lemma in case (A, B, C, D) is strictly passive. In case (A, B, C, D) is passive, the proof of the lemma is similar but ε = 0 in (2.2).

Let P and ε > 0 be a solution to P = P 0 and

Q := A P + P A + εP P B -C B P -C -(D + D ) 0.
Since Q 0, D is necessarily non-negative definite.

Case 1: D = 0. From Lemma 2.3 below, we have P B -C = 0. Following the same lines as in Section 1.1.1, the system (2.8) may be written as a differential inclusion. Then the well-posedness of the solution is claimed by Theorem 1.1.

Case 2: D = D 1 0 0 0 , where D 1 0 is square of dimension q < p. We can see that the Moore-Penrose pseudo inverse of the matrix D + D is

(D + D ) + := (D 1 + D 1 ) -1 0 0 0 .
Again, from Lemma 2.3, one obtains

(P B -C )[I p -(D + D ) + (D + D )] = 0.
(2.9)

By writing B = B 1 B 2 , C = C 1 C 2 where B 1 , B 2 , C 1 , C 2 have appropriate sizes,
then (2.9) is equivalent to

P B 1 -C 1 P B 2 -C 2 0 0 0 I p-q = 0.
That implies P B 2 -C 2 = 0. Following the lines in Section 1.1.2, we can conclude that the system (2.8) has a unique solution.

Case 3: D 0. Using Proposition 1.1, the system (2.8) has a unique solution.

The continuity of the solution in such cases is also obtained since the input u ∈ L 1 loc (R + ; R m ).

We are at the position to state the main result in this section. The following proposition gives us a form of controller u to solve our problems. Proposition 2.1. Suppose that Assumptions 2.1 and 2.2 hold. Then the closed-loop system (2.1) with the state feedback controller

u(t) = K[x(t) -x d (t)] + u d (t)
has a unique global solution x(•), and x(t) -x d (t) → 0 as t → +∞.

Proof. With the state feedback controller u = K(x -x d ) + u d , the closed-loop system (2.1) can be rewritten as

         ẋ(t) = (A + EK)x(t) + Bλ(t) + E(-Kx d (t) + u d (t)), 0 ≤ λ(t) ⊥ (C + F K)x(t) + Dλ(t) + F (-Kx d (t) + u d (t)) ≥ 0, x(0) = x 0 .
(2.10)

The well-posedness can be claimed by using Lemma 2.2 because the quadruple (A + EK, B, C + F K, D) is strictly passive. We just need to show that x(t; x 0 ) -x d (t) → 0, as t → +∞.

Set z x -x d . From Assumptions 2.1 and 2.2, we obtain

ż(t) = Az(t) + B[λ(t) -λ d (t)] + E[u(t) -u d (t)].
(2.11)

Using the state feedback controller

u(t) = K[(x(t) -x d (t)] + u d (t), we obtain ż(t) = (A + EK)z(t) + B(λ(t) -λ d (t)).
From Assumption 2.2, let P and ε > 0 be a solution to P = P 0 and

Q := (A + EK) P + P (A + EK) + εP P B -(C + F K) B P -(C + F K) -(D + D ) 0.
Since P = P 0, it follows that P x, x ≥ λ 1 (P ) x 2 for all x ∈ R n , where λ 1 (P ) > 0 is the smallest eigenvalue of P .

Consider the Lyapunov function V : R → R + defined as V (z) = z P z, then V (•) is continuously differentiable and its derivative along the trajectories of (2.11) satisfies the following for almost all t ∈ [0, ∞):

V (z(t)) = z(t) [(A + EK) P + P (A + EK)]z(t) + 2z(t) P B[λ(t) -λ d (t)].
To shorten notation, we denote Ā := A + EK, C := C + F K and z(t) := z(t; z 0 ). Then, we have

V (z(t)) = z(t) ( Ā P + P Ā)z(t) + 2z(t) P B[λ(t) -λ d (t)], = z(t) ( Ā P + P Ā)z(t) + 2z(t) (P B -C )[λ(t) -λ d (t)] + 2[λ(t) -λ d (t)] Cz(t).
holds for any t ≥ 0. From the complementarity conditions in Assumption 2.2, we have

0 ≤ λ(t) ⊥ Cx(t) + Dλ(t) + F u(t) ≥ 0, 0 ≤ λ d (t) ⊥ Cx d (t) + Dλ d (t) + F u d (t) ≥ 0. Using the fact that (u 1 -u 2 ) (v 1 -v 2 ) ≤ 0 holds for u 1 , u 2 , v 1 , v 2 ∈ R p satisfying 0 ≤ u 1 ⊥ v 1 ≥ 0 and 0 ≤ u 2 ⊥ v 2 ≥ 0, one obtains [λ(t) -λ d (t)] [C(x(t) -x d (t)) + D (λ(t) -λ d (t)) + F (u(t) -u d (t))] ≤ 0. or equivalently, [λ(t) -λ d (t)] [ Cz(t) + D (λ(t) -λ d (t))] ≤ 0. It follows that V (z(t)) ≤ z(t) ( Ā P + P Ā)z(t) + 2z(t) (P B -C )[λ(t) -λ d (t)] -[λ(t) -λ d (t)] (D + D )[λ(t) -λ d (t)] = (z(t) λ(t) -λ d (t)) Q z(t) λ(t) -λ d (t)
-εz(t) P z(t).

Since Q 0, V (z(t)) ≤ -ε P z(t), z(t) ≤ -ελ 1 (P ) z(t) 2 . Thus P z(t), z(t) ≤ P z 0 , z 0 -ελ 1 (P ) t 0 z(s) 2 ds.
Hence

z(t) 2 ≤ P z 0 , z 0 λ 1 (P ) -ε t 0 z(s) 2 ds.
From Gronwall's lemma, one deduces that

z(t) 2 ≤ P z 0 , z 0 λ 1 (P ) e -εt .
That implies z(t) → 0 or equivalently x(t) -x d (t) → 0 as t → ∞.

In summary, controller u stated in Proposition 2.1 depends on tracking error (x-x d ) and input u d and feedback gain K, while K depends on the matrices A, B, C, D, E and F , the coefficient matrices of the problem. This is depicted in Figure 2.1.

+ u d ẋ = Ax + Bλ + Eu 0 ≤ λ ⊥ Cx + Dλ + F u ≥ 0 u - x d + x K K Figure 2
.1: LCS with state feedback.

Determine gain K 2.3.1 Linear Matrix Inequality (LMI) Problems

To complete the design of the controller u, it is necessary to find K that ensures that the matrix inequality

(A + EK) P + P (A + EK) + εP P B -(C + F K) B P -(C + F K) -(D + D ) 0 (2.12)
has a solution P = P 0 for some ε > 0. For problems with small dimensions, Lemma 2.3 can be used to find a consistent matrix K by solving a system of equations and inequations. However, for higher dimension, we must have a useful tool to treat the problems. To do this, (2.12) needs to be reformulated to an LMI problem which is solved by a powerful software, MATLAB.

Let us multiply the left-hand side of (2.12) by Q 0 0 I , where Q is the inverse matrix of P . Then, we have

Q(A + EK) + (A + EK)Q + εQ B -Q(C + F K) B -(C + F K)Q -(D + D ) 0. (2.13)
Let us define a second new variable L = KQ. It follows that .14) This gives an LMI feasibility problem in the new variables Q = Q T 0 and L.

QA + AQ + L E + EL + εQ B -QC -L F B -CQ -F L -(D + D ) 0. ( 2
After solving this LMI, the feedback gain K can be recovered from K = LQ -1 .

Solving LMI using MATLAB Toolboxes

LMI Toolbox of MATLAB is a powerful toolbox providing a set of useful functions to solve LMIs. To solve an LMI problem in MATLAB, there are 3 steps.

Step 1. Initializing the LMI description.

Step 2. Defining the Decision Variables.

Step 3. Defining the LMIs one by one.

In this section, we will show the code to get feedback gain K by solving the LMI given in (2.14). The code is given in Section B.1 in Appendix. For more details about the script, we refer the reader to [START_REF] Gahinet | LMI Control Toolbox For Use with MATLAB[END_REF] or visit https://fr.mathworks.com/help/robust/lmis.html.

Another toolbox that may be helpful to solve LMI problems is YALMIP. It is available freely to be used by researchers. Like LMI toolbox, to solve an LMI problem using YALMIP requires 3 steps:

Step 1. Defining Decision Variables.

Step 2. Defining Constraints.

Step 3. Solving Optimization Problems.

For more information please log on to the following: https://yalmip.github.io.

Numerical simulation 2.4.1 Numerical scheme

The numerical simulation for our problems is based on the Moreau-Jean's time-stepping scheme which is reported in [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics[END_REF].

We denote by

0 = t 0 < t 1 < • • • < t k < • • • < t N =
T a finite partition of the time interval [0, T ] (T > 0). The integer N stands for the number of time intervals in the partition. The length of a time step is denoted by h k = t k+1 -t k . For simplicity sake, we only consider a constant time length h = h k (0 ≤ k ≤ N -1).

Let us consider the dynamic system (2.1) with state feedback controller u = K(x -

x d ) + u d . Then,    ẋ(t) = (A + EK)x(t) + Bλ(t) -EKx d (t) + Eu d (t), 0 ≤ λ(t) ⊥ (C + F K)x(t) + Dλ(t) + F (-Kx d (t) + u d (t)) ≥ 0.
(2.15) Integrating the above ODE over a time step [t k , t k+1 ] of length h, it follows

t k+1 t k ẋ(t)dt = t k+1 t k (A+EK)x(t)dt+ t k+1 t k Bλ(t)dt+ t k+1 t k [Eu d (t)-EKx d (t)]dt.
The left-hand side is equal to x(t k+1 ) -x(t k ). The right-hand terms are approximated by using the θ-method and an implicit Euler integration. Specifically,

t k+1 t k (A + EK)x(t)dt ≈ h(A + EK)[θx(t k+1 ) + (1 -θ)x(t k )], t k+1 t k Eu d (t)dt ≈ hE[θu d (t k+1 ) + (1 -θ)u d (t k )], t k+1 t k -EKx d (t)dt ≈ -hEK[θx d (t k+1 ) + (1 -θ)x d (t k )], t k+1 t k Bλ(t)dt ≈ hBλ(t k+1 ),
for some θ ∈ [0, 1]. By replacing the accurate solution x(t k ) by the approximated value x k , we obtain

[I -h(A + EK)θ]x k+1 =[I + h(A + EK)(1 -θ)]x k + hE[θu d (t k+1 ) + (1 -θ)u d (t k )] -hEK[θx d (t k+1 ) + (1 -θ)x d (t k )] + hBλ k+1 .
(2.16)

Then, the time discretization of the non-smooth law is given by

0 ≤ λ k+1 ⊥ (C + F K)x k+1 + Dλ k+1 + F (-Kx d (t k+1 ) + u d (t k+1 )) ≥ 0.
Suppose that [I -h(A + EK)θ], denoted by W , is invertible. By substituting x k+1 from (2.16) to time discretization of the non-smooth law, one gets the LCP:

0 ≤ λ k+1 ⊥ M λ k+1 + q k+1 ≥ 0, in which M = h(C + F K)W -1 B + D, q k+1 = (C + F K)W -1 [ W x k + hE(u d,k+θ -Kx d,k+θ )] + F (-Kx d,k+1 + u d,k+1 ), W = I + h(A + EK)(1 -θ), u d,k+θ = θu d (t k+1 ) + (1 -θ)u d (t k ), x d,k+θ = θx d (t k+1 ) + (1 -θ)x d (t k ).
Thus, h, θ can be chosen such that M is a positive definite matrix or a P-matrix in order to ensure that the time discretization of the non-smooth law possesses a solution λ k+1 .

SICONOS Platform

The SICONOS Platform is a scientific computing software dedicated to modeling, simulation, control, and analysis of nonsmooth dynamical systems. It is developed in the Tripop team-project at INRIA (The French National Institute for Research in Computer Science and Control) in Grenoble and distributed under GPL GNU license [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics[END_REF]. For more information, the reader can visit https://nonsmooth.gricad-pages.univ-grenoble-alpes. fr/siconos/index.html.

Nonsmooth Dynamical System Nonsmooth Dynamical System

Dynamical System

ẋ = f (x, t) + r Dynamical System ẋ = f (x, t) + r Nonsmooth Law 0 ≤ λ ⊥ y ≥ 0 Nonsmooth Law 0 ≤ λ ⊥ y ≥ 0 y = h(x) r = g(λ) y = h(x) r = g(λ)
Input/Output Relation Input/Output Relation Nonsmooth Interaction Nonsmooth Interaction The SICONOS software is mostly written in C++ and it aims at providing a general and common tool for nonsmooth problems in various scientific fields such as applied mathematics, mechanics, robotics, electrical circuits, and so on. The general writing process for a problem treated with SICONOS includes two steps.

Step 1. Building a Nonsmooth Dynamical System

Step 2. Defining Simulation Strategy All steps are built relying on its general principles of modeling and simulation. SICONOS Nonsmooth Dynamical System Modeling Principle is shown in Figure 2.2 (see also [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics[END_REF]).

Examples in scalar case

This part is dedicated to a short presentation of the general writing process for some examples, with x(t) ∈ R, i.e., n = 1, m = 1, p = 1.

Let us consider a dynamical system given by

   ẋ(t) = -x(t) + λ(t) + u(t), 0 ≤ λ(t) ⊥ x(t) + λ(t) + u(t) ≥ 0.
(2.17)

with initial state x(0) = 1. Our goal is to find a controller u such that x(t) → x d (t) as t → +∞, where

x d satisfies    ẋd (t) = -x d (t) + λ d (t) + u d (t), 0 ≤ λ d (t) ⊥ x d (t) + λ d (t) + u d (t) ≥ 0.
(2.18)

for given input u d (t) = sin(t), t ≥ 0 and x d (0) = 0.

According to Proposition 1.1, we claim that x d is unique. We try to find a matrix (scalar) k such that (-1 + k, 1, 1 + k, 1) is strictly passive. Then we have

2(k -1)p + εp p -k -1 p -k -1 -2 0
for some p > 0 and ε > 0. From Lemma 2.3, we get 2(k -1)p + εp ≤ 0. This gives 2(k -1) + ε ≤ 0, or equivalently k ≤ 1 -ε/2. For instance, we can choose u = 0.25(x -x d ) + u d as a solution for our problem.

To get a simulation of this problem on SICONOS platform, we use the script as in Section B.2 in Appendix. In the following paragraphs, we turn our attention to Nonsmooth Dynamical System related and Simulation-related components in the platform which are used in our problem.

• FirstOrderLinearDS class, which describes the linear dynamical systems of first order in form (coefficients may be time invariant or not)

   ẋ(t) = A(t, z)x(t) + b(t, z) + r, x(t 0 ) = x 0 .
• A subclass of FirstOrderLinearDS class, FirstOrderLinearTIDS, which describes the linear dynamical systems of the form

   ẋ(t) = Ax(t) + b + r, x(t 0 ) = x 0 .
• FirstOrderLinearTIR class, which describes the first-order linear and time invariant relations:

   y = Cx + F z + Dλ + e, r = Bλ.
• ComplementarityConditionNSL class models a complementarity condition as 0 ≤ y ⊥ λ ≥ 0.

• LCP class represents the linear complementarity problem

   w = M z + q, 0 ≤ w ⊥ z ≥ 0.
Numerical simulation of Problem 2.17 with time-step h = 5.10 -4 and θ = 0.5 is shown in Figure 2.3. Now, we consider more dynamical systems by changing the constant coefficients in (2.17). That is, Notice that in the first two problems, the initial states can be chosen arbitrary due to d > 0, but three last problems (shown in , the initial states ensure that x(0) ≥ 0. As we can see, in all problems, the state converges to desired trajectory perfectly. Moreover, there are two modes of the complementarity conditions activated in each problems, indicated by the value of the multiplier λ. Especially, in Figure 2.8, the complementarity condition for this problem is 0 ≤ λ ⊥ x ≥ 0. From the plot in Figure 2.8, relation between x and λ is a complementarity one.

   ẋ(t) = -x(t) + λ(t) + eu(t), 0 ≤ λ(t) ⊥ x(t) + dλ(t) + f u(t) ≥ 0,

Example in higher dimension

Let us consider the circuit with two ideal diodes in Figure 2.9, with R 1 , R 2 , R 3 > 0, L 1 , L 2 > 0, which is obtained from the circuit of Figure 2.3 in [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics[END_REF] by dropping the capacitor. The state variables are x 1 , x 2 , where x 1 is the current across the inductor L 1 and x 1 is the current across the inductor L 2 . λ 1 , λ 2 are the voltage of the diodes D 1 and D 2 , respectively. The dynamics is given by For instance, we choose

               ẋ1 (t) = -R 1 +R 3 L 1 x 1 (t) + R 3 L 1 x 2 (t) + λ 1 (t) L 1 + λ 2 (t) L 1 ẋ2 (t) = R 3 L 2 x 1 (t) -R 2 +R 3 L 2 x 2 (t) -λ 2 (t) L 2 + u(t) L 2 , 0 ≤   λ 1 (t) λ 2 (t)   ⊥   x 1 (t) x 1 (t) -x 2 (t)   ≥ 0.
R 1 = R 2 = R 3 = 1Ω and L 1 = L 2 = 1H
. Then, the system can be written compactly as

   ẋ(t) = Ax(t) + Bλ(t) + Eu(t), 0 ≤ λ(t) ⊥ Cx(t) ≥ 0, (2.21)
with

A = -2 1 1 -2 , B = 1 1 0 -1 , C = 1 0 1 -1
, and E = 0 1 .

We can see that the relation P B = C holds with P = I. From Section 1.1.1, (2.21) has a unique solution for any initial state x(0) = x 0 such that Cx 0 ≥ 0. Now, let us find controller u such that x(t) -x d (t) → 0, as t → +∞, where x d is the desired trajectory, given as the solution of the LCS:

   ẋd (t) = Ax d (t) + Bλ d (t) + Eu d (t), 0 ≤ λ d (t) ⊥ Cx d (t) ≥ 0,
for given input u d (t) = sin(t), t ≥ 0. The numerical simulation of the desired trajectory is shown in Figure 2.10 with time step h = 10 -4 and θ = 0.5. From Proposition 2.1, Then we can choose the matrix K = (-2.0, -1.0) as a feedback gain for our problem.

LCS (2.19) with d = 0, e = 1, f = 0 R 3 x 1 -x 2 λ 2 R 2 L 2 x 2 u(t) λ 1 R 1 L 1 x 1
Proceeding with simulation of this problem in SICONOS, the results we obtained are depicted in Figure 2.11. Notice that (A + EK, B, C, 0) is strictly passive, it means

(A + EK) P + P (A + EK) + εP P B -C B P -C 0 0 (2.22)
holds for some P = P 0, ε > 0. As the solution obtained by MATLAB, we have

P = Q -1 = I, ε = 4.
From the proof of Proposition 2.1, for all t ≥ 0, we get

x(t) -x d (t) 2 ≤ P (x(0) -x d (0)), x(0) -x d (0 λ 1 (P ) e -εt or x(t) -x d (t) ≤ √ 2 2 e -2t .
Hence the curve y = √ 2 2 e -2t is considered as an upper-bound of the tracking error x(t) -x d (t) . From Figure 2.12, it is obvious that the inequality x(t) -x d (t) ≤ √ 2 2 e -2t holds for all t ≥ 0 and x(t) -x d (t) tends to 0 as expected. Finally, the simulation of the controller u is shown in Figure 2.13. is valid for some P = P 0 and K. In fact, we suppose that K = (k 1 , k 2 ). Since D = 0, it requires P B = C . This relation holds if and only if P = I. Furthermore, we must ensure that (A + EK) P + P (A + EK) + εP 0.

This gives us

-4 + ε k 1 + 2 k 1 + 2 2(k 2 -2) + ε 0.
It follows that -4 + ε ≤ 0, or equivalently ε ≤ 4.

Extended controller form 2.5.1 Motivation

Let us consider the circuit that investigated in Example 1.1. The dynamics can be written compactly as

   ẋ(t) = Ax(t) + Bλ(t) + Eu(t), 0 ≤ λ(t) ⊥ Cx(t) ≥ 0 (2.23) with A = 0 1 -1 LC -R L , B = 0 -1 L , C = 0 -1 , E = 0 1 L .
Suppose that the desired trajectory given as the solution of the LCS:

   ẋd (t) = Ax d (t) + Bλ d (t) + Eu d (t), 0 ≤ λ d (t) ⊥ Cx d (t) ≥ 0,
for given input u d (t) = 1, t ≥ 0 and our aim is finding a controller u such that x(t) -

x d (t) → 0, as t → +∞.
It is clearly that the relation P B = C holds with P = p 1 0 0 L , for p 1 > 0.

From results in Section 1.1.1, the desired trajectory x d is unique. According to Proposition 2.1, if we try to find a controller u in the form u = K(x -x d ) + u d , then the matrix K is chosen such that (A + EK, B, C, 0) is strictly passive, i.e., (A + EK) P + P (A + EK) + εP P B -C B P -C 0 0 holds for some P = P 0 and ε > 0. Since D = 0, we must have P B = C .

That implies P = p 1 0 0 L , p 1 > 0. Suppose that K = (k 1 , k 2 ). Then, the matrix inequality (A + EK) P + P (A + EK) + εP 0 is written as

εp 1 p 1 + k 1 -1 C p 1 + k 1 -1 C 2k 2 -2R 0.
It follows that εp 1 ≤ 0. That is impossible because both ε and p 1 are positive real numbers.

One question has arisen. That is whether we can improve the form of the controller u to solve this problem. Fortunately, the answer is yes.

Form of the extended controller

Assumption 2.3. There exist matrices K, G such that the quadruple

(A + EK, B + EG, C + F K, D + F G)
is strictly passive and D + F G is either a zero matrix, or a positive definite matrix, or a matrix in the form D 1 0 0 0 with D 1 0.

Proposition 2.2. Suppose that Assumptions 2.1 and 2.3 hold. Then the closed-loop system (2.1) with the controller

u(t) = K[x(t) -x d (t)] + G[λ(t) -λ d (t)] + u d (t)
has a unique global solution x(•) and x(t) -x d (t) → 0 as t → +∞.

Proof. The proof is similar to the proof of Proposition 2. As pointed out in Section 2.3, to complete the design of the controller u, we must determine matrices K, G such that

(A + EK) P + P (A + EK) + εP P (B + EG) -(C + F K) (B + EG) P -(C + F K) -D -F G -(D + F G) 0 (2.24)
has a solution P = P 0 for some ε > 0. Let us multiply the left-hand side of (2.24) by Q 0 0 I , where Q = P -1 and define a second new variable L = KQ. Then, we have

QA + AQ + L E + EL + εQ B + EG -QC -L F B + G E -CQ -F L -D -F G -(D + F G) 0.
This is an LMI feasibility problem in the new variables Q = Q 0, G and L. The matrix K can be recovered from K = LQ -1 . In summary, the closed-loop system structure with extended controller is depicted in Figure 2.14. It may be consider as an generalize of Figure 2.1. Example 2.2. Let us investigate the circuit in Figure 2.15. The dynamics is given as Assume that P = p 1 p 2 p 2 p 3 0. If G is chosen such that D + F G = 0, then

ẋ = Ax + Bλ + Eu 0 ≤ λ ⊥ Cx + Dλ + F u ≥ 0 ẋd = Ax d + Bλ d + Eu d 0 ≤ λ d ⊥ Cx d + Dλ d + F u d ≥ 0 G K - - + + z → 0 + + + u d λ d x d λ x u
         ẋ1 (t) = x 2 (t) ẋ2 (t) = -1 LC x 1 (t) + λ(t) L 0 ≤ λ(t) ⊥ λ(t) R + x 2 (t) -u(t) R ≥ 0,
A P + P A + εP 0. It follows that -2p 2 LC p 1 -p 3 LC p 1 -p 3 LC 2p 2 + ε p 1 p 2 p 2 p 3 0.
This implies

-2p 2 LC + εp 1 ≤ 0, 2p 2 + εp 3 ≤ 0.
Thus, we get εp 1 + εp 3 LC ≤ 0, which is impossible since p 1 , p 3 > 0. If G satisfies D + F G > 0, then we must have

A P + P A + εP + [P B -(C + F K) ](D + F G) + [B P -(C + F K)] 0. INTERNSHIP REPORT Note that (D + F G) + = c > 0. One deduces that -2p 2 LC p 1 -p 3 LC p 1 -p 3 LC 2p 2 +ε p 1 p 2 p 2 p 3 +c p 2 L + k 1 R p 3 L + k 2 R -1 p 2 L + k 1 R p 3 L + k 2 R -1 0.
It follows that

-2p 2 LC + εp 1 + c( p 2 L + k 1 R ) 2 ≤ 0, (2.25
)

2p 2 + εp 3 + c( p 3 L + k 2 R -1) 2 ≤ 0. (2.26)
From (2.25), (2.26), we obtain

0 < εp 1 + ε LC p 3 + c( p 2 L + k 1 R ) 2 + c LC ( p 3 L + k 2 R -1) 2 ≤ 0.
This contradicts our assumption since p 1 , p 3 , ε, c are positive numbers. Therefore, there are not K and G such that (A, B, C + F K, D + F G) is strictly passive. If we add a voltage source u 2 to the circuit in will have a new dynamics given by

         ẋ1 (t) = x 2 (t) ẋ2 (t) = -1 LC x 1 (t) + λ(t) L + u 2 (t) L 0 ≤ λ(t) ⊥ λ(t) R + x 2 (t) -u 1 (t) R ≥ 0.

Now, we have the LCS with

A = 0 1 -1 LC 0 , B = 0 1 L , C = 0 1 , D = 1 R , E = 0 0 0 1 L , F = -1 R 0 . For instance, let us consider R = C = L = 1,
given inputs u d = (u 1,d , u 2,d ) = (1, 1) and initial state x 0 = (0.5, 0.5) . From The matrices are obtained by using YALMIP with 6-digit accuracy. Using SICONOS to get the numerical results of this problem, the results are depicted in Figure 2.17 and 2.16. In both cases, we use explicit method (θ = 0) with time step h = 5.10 -4 . The multiplier λ 1 and λ 2 when using controller u 1 and u 2 .

Decay rate of tracking error

As pointed out, for each trajectory tracking problem presented in the report, the feedback gain K is chosen satisfying Assumption 2.2 or Assumption 2.3 in the extended controller form. However, in general, such K is not unique. For instance, in Problem 2.17, Section 2.4.3, the matrix (scalar) k can be chosen such that k ≤ 1 -ε/2 for ε > 0.

It is clear that the set of feasible k is not unique even infinity. In some situations, we have more than one choice for K. Thus, a criterion has to be proposed with the aim at finding the "best" K in a set of feasible solutions.

From the proof of Proposition 1.1, for all t ≥ 0, we have

x(t) -x d (t) 2 ≤ P (x(0) -x d (0)), x(0) -x d (0) λ 1 (P ) e -εt .
In which, P and ε > 0 are a solution to P = P 0 and

P ε := (A + EK) P + P (A + EK) + εP P B -(C + F K) B P -C -F K -(D + D ) 0.
The first idea in our mind is finding K such that ε as large as possible in order to increase decay rate of x(t) -x d (t) . With the relation k ≤ 1 -ε/2, for each ε > 0, there exists k that solves our problem. And k → -∞ provided that ε → +∞. Then, if we apply the controller u = k(x -x d ) + u d , it may be unbounded. That may be unsuitable in the case the controller has been bounded, for example, a voltage source with a limited voltage. Hence, we continue to find a consistent K ensures that it satisfies assumptions and LMI holds with the largest ε but K is bounded by some given constant. Here, the norm of K is defined as the square root of the maximum eigenvalue of K K. Suppose that K ≤ κ, for a given κ > 0. Our goal can be formulated under the form of an optimal problem (P κ ). max

(ε,Q,K) ε subject to P = P 0, ε > 0, P ε 0, K ≤ κ.
Let us change variable Q = P -1 and define a new varibale L = KQ. The constraint K ≤ κ can be replaced by sufficient conditions

Q I, κI L L κI 0,
where I stands for identity matrices with appropriate sizes.

In fact, from κI L L κI 0, it follows that κ 2 I LL . That implies L ≤ κ.

Using Q I, we obtain

K = LQ -1 ≤ L Q -1 ≤ κ. Now, the problem (P κ ) is completely rewritten as (Q κ ): max (ε,Q,L) ε subject to Q = Q I, ε > 0, Q ε 0, L κ 0, in which
Q ε := QA + AQ + L E + EL + εQ B -QC -L F B -CQ -F L -(D + D ) , and
L κ := κI L L κI .
Due to the product between ε and Q, the problem (Q κ ) cannot be solved directly using linear semidefinite programming (YALMIP).

Let us define a feasible set for each ε > 0 given by

F ε := {(Q, L) : Q = Q I, Q ε 0, L κ 0}.
Then, Problem (Q κ ) is considered as finding the largest ε such that F ε is non-empty.

Proposition 2.3. Let ε > ε > 0. Suppose that F ε = ∅. Then F ε = ∅.
Proof. Suppose that F ε is non-empty and (Q, L) ∈ F ε . Then, we have

Q ε = Q ε + (ε -ε ) Q 0 0 0 0.
That inequality holds because Q ε 0 and Q 0. This implies (Q, L) ∈ F ε , which contradicts our assumption.

This property gives us a method to estimate the solution of the problem (Q κ ). The method called Bisection Algorithm can be described in a simple way as below. For more details, we refer the reader to https://yalmip.github.io/example/decayrate/.

1. Find a lower bound on optimal ε (any feasible ε you can compute).

2. Find an upper bound on optimal ε (increase the lower bound until a feasibility problem for fixed ε is infeasible). The dynamics is written as

   ẋ(t) = Ax(t) + Bλ(t) + Eu(t), 0 ≤ λ(t) ⊥ Cx(t) ≥ 0, with A = -2 1 1 -2 , B = 1 1 0 -1 , C = 1 0 1 -1
, and E = 0 1 .

If the feedback gain K do not require being bounded, then K = (-2, -1) is a solution for our problem. But it is not a solution when we add a constraint in K, for instance K ≤ κ with κ = 2. Among consistent matrices K, we can choose K = (-2, 0)

which makes ε to obtain the largest value (ε = 4) by solving Problem (Q 2 ).

It is easy to see that the feasible set of (Q κ ) is larger and larger when we increase the κ and vice versa. Thus, the optimal value of (Q κ ) is non-decreased in κ. But it may be bounded (see Figure 2.19) or not, see for instance Problem 2.17. For the problem (Q 4 ), we have at least two solutions. They are K = k 1 = (-2, 0) and K = k 2 = (-2, -1) with k 1 < k 2 ≤ 4. The question is among feasible K of (Q 4 ), which one does have a minimal norm? According to Figure 2.19, we can see that the minimal norm is approximate to 2, for example K = (-2, 0). This leads us to study an interesting problem (K):

min (κ,Q,L) κ subject to Q = Q I, κ ≥ 0, Q εmax 0, L κ 0.
Here, we denote by ε max the optimal value of (Q κ ). Using YALMIP, we obtain the most suitable matrix K that solves our problem. The code can be found in Section B.3.

Running that code, we get the best gain K for problem in Example 2.3 is (-2, 0). Then, ε max = 4 and K = 2.

Remark 2.3. In the case we would like to use extended controller form, the results above may be used to analyse feed back gain K by replacing Q ε by

Qε := QA + AQ + L E + EL + εQ B + EG -QC -L F B + G E -CQ -F L -D -F G -(D + F G) ,
and we consider a new problem ( Qκ ):

max (ε,Q,L) ε subject to Q = Q I, ε > 0, Qε 0, L κ 0.
Then, all previous steps can be redone and we have the most suitable K for our problem.

Conclusion

Besides reviewing several concepts of Convex analysis and the nature of equilibrium points of a dynamical system, during my internship, I have achieved several goals. When D = 0 and D 0, with some suitable conditions (see Assumption 1.1, Theorem 1.1), we claim its well-posedness. For D 0, according to Proposition 1.1, the LCS has a unique solution for any initial state x 0 and input u.

Secondly, I have investigated the multivalued Lur'e system. Some results about its well-posedness relying on Kato's Theorem and maximal monotonicity are stated as basis facts, they deal with a class of systems in which D is a nonzero feedthrough matrix, possibly positive semidefinite and non-symmetric. Dissipativity and stability results are background notions to construct a controller for a closed-loop, Proposition 1.6, Theorem 1.4.

For our main goal, we make some suitable assumptions and propose a form of state feedback controller u = K(x -x d ) + u d (see Proposition 2.1) based on Dissipativity theory to solve our problem. I suggest that we can use Dissipativity theory to establish a form of controller when the desired trajectory have state jumps.

Next, we consider an example that assumptions in Proposition 2.1 are not satisfied. From this, we give an extended form of the controller u to handle that example, see Proposition 2.2. We also deal with the problems that the controllers u are bounded. From that we propose a way to choose gain K that makes decay rate of tracking error to reach maximal value.

Finally, I used SICONOS Platform to model and simulate nonsmooth dynamical systems via examples in the scalar cases and higher dimension case. SICONOS is a useful tool to treat such problems. For the future works, we are going to work on:

• Shockley law versus complementarity conditions,

• robustness of controller with Shockley law with respect to Shockley law,

• robustness property with respect to uncertainties as A, B, C, D.

• advantage of an LCS approach versus an exponential one for control synthesis.

(i) α 1 f 1 + α 2 f 2 , (ii) sup{f 1 , f 2 }, (iii) f 1 • A and f 2 • A.
Definition A.9. Let f : R n → R ∪ {+∞} be a proper function and let x 0 ∈ R n . We say that f is lower semicontinuous at x 0 if, for every sequence (x n ) n∈N in R n such that x n → x 0 as n → +∞, we have lim inf

A.2 Equilibrium point of a dynamic system

For an interval I ⊂ R, and a function f : I → R n , the variation of f (•) on the interval I is the supremum of k i=1 f (x i ) -f (x i-1 ) over the set of of all finite sets of points x 0 < x 1 < • • • < x k of I. When this supremum is finite, the function f (•) is said to be of bounded variation on I. An absolutely continuous (AC) function f : I → R n is a function that can be written as f (t) = f (t 0 ) + t t 0 ḟ (s)ds for any t 0 , t ∈ I, t 0 ≤ t, and some ḟ ∈ L 1 (I, R n ).

Let us consider a nonlinear system represented as

   ẋ(t) = f (x(t), t)
x(t 0 ) = x 0 ,

(A.2)
where f (•) is a nonlinear vector function, and x ∈ R n is the state vector. We suppose that the system is well-posed, i.e., a unique solution exists globally.

Definition A.12 (Equilibrium). A state x is an equilibrium point of (A.2) if f (x , t) = 0 for all t ≥ t 0 .

Definition A.13 (Stable). The equilibrium point x = 0 is stable at t = t 0 if for any ρ > 0 there exists an r(ρ, t 0 ) > 0 such that x(t) < ρ for all t ≥ t 0 whenever x(t 0 ) < r. Otherwise the equilibrium point x = 0 is unstable.

Definition A.14. The equilibrium point x = 0 is said to be:

• asymptotically stable at t = t 0 if it is stable and if it exists r(t 0 ) > 0 such that x(t 0 ) < r(t 0 ) ⇒ x(t) → 0 as t → ∞.

• exponentially stable if there exist two positive numbers α and λ such that x(t) ≤ α x(t 0 ) e -λ(t-t 0 ) , ∀t ≥ t 0 , for x(t 0 ) sufficiently small.

• globally asymptotically stable at t = t 0 if it is stable and x(t) → 0 as t → ∞ for all x(t 0 ) ∈ R n .

B.3 YALMIP codes for solving (Q κ ) and (K)

Assuming that the matrices A, B, C, D, E, and F already exist in the environment.

Introduction 1 1

 11 Well-posedness of Linear Complementarity Systems 1.1 Linear Complementarity Systems . . . . . . . . . . . . . . . . . . . . 1.1.1 Case D = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Case D 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Case D 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The multivalued Lur'e system . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Well-posedness by Kato's Theorem . . . . . . . . . . . . . . . 1.2.2 Well-posedness by maximal monotonicity . . . . . . . . . . . . 1.2.3 Dissipativity and stability results . . . . . . . . . . . . . . . . . Subdifferential and normal cones . . . . . . . . . . . . . . . . . . . . . A.1.1 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Equilibrium point of a dynamic system . . . . . . . . . . . . . . . . . . B Scripts B.1 MATLAB Code solving LMI problems . . . . . . . . . . . . . . . . . B.2 SICONOS Script for LCS . . . . . . . . . . . . . . . . . . . . . . . . . B.3 YALMIP codes for solving (Q κ ) and (K) . . . . . . . . . . . . . . . . Bibliography iii List of Figures 1.1 A complementarity relation in physics . . . . . . . . . . . . . . . . . . 1.2 RLC circuit with an ideal diode . . . . . . . . . . . . . . . . . . . . . . 1.3 Electrical circuit with capacitors, resistors and ideal diodes. . . . . . . . 1.4 A circuit with an ideal diode in case D 0 . . . . . . . . . . . . . . . 1.5 Filtered full wave rectifier . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 LCS with state feedback. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nonsmooth Dynamical System Modeling Principle [1]. . . . . . . . . . 2.3 SICONOS simulation results for Problem 2.17 . . . . . . . . . . . . . . 2.4 LCS (2.19)with d = 1, e = 0, f = 1 . . . . . . . . . . . . . . . . . . . 2.5 LCS (2.19) with d = 1, e = 1, f = 0 . . . . . . . . . . . . . . . . . . . 2.6 LCS (2.19) with d = 0, e = 1, f = 1 . . . . . . . . . . . . . . . . . . . 2.7 LCS (2.19) with d = 0, e = 0, f = 1 . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: A complementarity relation in physics

2 Figure 1 . 2 :

 212 Figure 1.2: RLC circuit with an ideal diode

Figure 1 . 3 :

 13 Figure 1.3: Electrical circuit with capacitors, resistors and ideal diodes.

Figure 1 . 4 :

 14 Figure 1.4: A circuit with an ideal diode in case D 0

  D 0, and D 0. As pointed out in Chapter 1, it is worth considering such cases because many systems like electrical circuits leading us to study LCS. In case D 0, we restrict to positive semidefinite matrices D of the form D 1 0 0 0

Theorem 2 . 1 .

 21 ([9], Theorem 7.15) Consider an LCS given by(2.3) such that (A, B, C, D) is passive, matrices [B AB . . . A n-1 B] and [C A C . . . (A ) n-1 C] have full rank, and B D + Dhas full column rank. Then this LCS has only Lyapunov stable equilibrium points x * given by (2.5). Moreover, if (A, B, C, D) is strictly passive, then x * = 0 is the only equilibrium point, which is asymptotically stable.

Lemma 2 . 3 (

 23 [START_REF] Brogliato | Dissipative Systems Analysis and Control, Theory and Applications[END_REF], Lemma A.65). Let us assume that Q and R are symmetric. Then Q S S R 0 if and only if R 0, Q -SR + S 0, and S(I -R + R) = 0. Here R + is the Moore-Penrose pseudo inverse of the matrix R.

Figure 2 . 2 :

 22 Figure 2.2: Nonsmooth Dynamical System Modeling Principle [1].

Figure 2 . 3 :

 23 Figure 2.3: SICONOS simulation results for Problem 2.17

  e, f are real constants. By studying the same inputs u d (t) = sin(t), t ≥ 0 and only changing d, e, f , we consider 5 different problems whose numerical results are shown in the following figures.

Figure 2 . 4 :Figure 2 . 5 :

 2425 Figure 2.4: LCS (2.19) with d = 1, e = 0, f = 1

Figure 2 . 6 :Figure 2 . 7 :

 2627 Figure 2.6: LCS (2.19) with d = 0, e = 1, f = 1

  Figure 2.8: LCS (2.19) with d = 0, e = 1, f = 0

Figure 2 . 9 :

 29 Figure 2.9: A circuit with two ideal diodes and a voltage source

Figure 2 . 10 :Figure 2 . 11 :Figure 2 .

 2102112 Figure 2.10: Desired trajectory simulated by SICONOS with h = 10 -4 , θ = 0.5

Figure 2 . 13 :

 213 Figure 2.13: Simulation result of controller u

1 .

 1 It suffices to replace B and D by B + EG and D + F G in our argument.

Example 2 . 1 .

 21 Again, let us consider the circuit in Example 1.1. We have shown that there is no controller u in the form u = K(x -x d ) + u d that satisfies Proposition 2.1. For R = 10 2 Ω, L = 10 -6 H, C = 10 -1 F, Assumption 2.3 holds with ε = 0.0001, G = 0.999999, and K = (9.945811, -324.213572). From Proposition 2.2, the controller u = K(x -x d ) + G(λ -λ d ) + u d solves our problem.

Figure 2 .Figure 2 .

 22 Figure 2.14: Closed-loop system structure with extended controller

where x 1

 1 is the charge on the capacitor C, x 2 is the current through the inductor L. The matrices A, B, C, D, E, F can be easily identified.Suppose that there existsK = (k 1 , k 2 ) and G such that (A + EK, B + EG, C + F K, D +F G), or equivalently (A, B, C +F K, D +F G) is strictly passive. This means that A P + P A + εP P B -(C + F K) B P -(C + F K) -(D + F G) -(D + F G)0holds for some P = P 0 and ε > 0.

Figure 2 1 RFigure 2 .

 212 Figure 2.16: RLCD circuit with two voltage sources.

Figure 2 .

 2 Figure2.17: The desired trajectory and the state when using controller u 1 and u 2 .

2 Figure 2 .

 22 Figure 2.18: The multiplier λ 1 and λ 2 when using controller u 1 and u 2 .

Figure 2 . 19 :

 219 Figure 2.19: Relation between κ and optimal value of (Q κ )

  First of all, I have surveyed the existence and uniqueness of LCS in cases D = 0, D 0 and D 0 and several examples in each case (see Examples 1.1, 1.2, 1.3).

  n→+∞ f (x n ) ≥ f (x 0 ).Proposition A.3. Let f : R n → R ∪ {+∞} be a proper function. Then the following assertions are equivalent:(i) f is lower semicontinuous, (ii) epi(f ) is closed in R n × R, (iii) lev σ (f ) is closed in R n for all σ ∈ R. Example A.1. Let C ⊂ R n . The indicator function ψ C is lower semicontinuous if and only if C is closed in R n . Definition A.10. Let f : R n → R ∪ {+∞} be a proper and convex function. The subdifferential ∂f (x) of f at x ∈ R n is defined by ∂f (x) = { p, y -x ≤ f (y) -f (x) for all y ∈ R n }.We associate to f a set-valued operator ∂f : R n ⇒ R n , x → ∂f (x). The domain of the operator ∂f , denoted by Dom(∂f ), is defined byDom(∂f ) = {x ∈ R n : ∂f (x) = ∅.} Definition A.11. Let f : R n → [-∞, +∞] be a function. The conjugate function f * : R n → [-∞, +∞] associated with f is defined by f * (p) = sup x∈R n { p, x -f (x)}.This is known in the literature as the Legendre-Fenchel transformation or the Fenchel conjugate.Proposition A.4. Let f : R n → R ∪ {+∞} be a proper convex function. Then p ∈ ∂f (x) if and only if f (x)+f * (p) = p, x . Moreover, if f is lower semicontinuous, then each of the above properties is equivalent to x ∈ ∂f * (p), which means that for every proper, lower semicontinuous and convex function f on R n we get p ∈ ∂f (x) ⇔ x ∈ ∂f * (p), i.e., (∂f ) -1 = ∂f * .

1 : 2 :

 12 Find a lower bound cons=[Q>=eye(n),Q * A'+A * Q+L' * E'+E * L+eps_0 * Q<=0, B-Q * C'-L' * F'==0,[k * eye(m), L;L', k * eye(n)]>=0]; ops = sdpsettings ( ' solver ' , 'sedumi', 'verbose' ,0, 'warning' ,0) ; sol = optimize (cons ,[], ops) ; while sol .problem==1 %F_eps_0 is empty eps_0 = eps_0 /2; cons=[Q>=eye(n),Q * A'+A * Q+L' * E'+E * L+eps_0 * Q<=0, B-Q * C'-L' * F'==0,[k * eye(m), L;L', k * eye(n)]Find an upper bound cons=[Q>=eye(n),Q * A'+A * Q+L' * E'+E * L+eps_0 * Q<=0, B-Q * C'-L' * F'==0,[k * eye(m), L;L', k * eye(n)]>=0]; sol = optimize (cons ,[], ops) ; while ~( sol .problem==1) %F_eps_0 is nonempty eps_0 = eps_0 * 2; cons=[Q>=eye(n),Q * A'+A * Q+L' * E'+E * L+eps_0 * Q<=0, B-Q * C'-L' * F'==0,[k * eye(m), L;L', k * eye(n)]>=0]; sol = optimize (cons ,[], ops) ; end eps_upper=eps_0;

  3. Check value between lower and upper bound. If feasible, update lower bound, if infeasible update upper bound. Repeat until bounds are sufficiently close. Now, by using YALMIP we definitely compute the optimal value of the problem (Q κ ). The code is given in Section B.3. Example 2.3. Let us review the dynamical system which is surveyed in Section 2.4.4.

Acknowledgements

Appendix A Introductory Material

This introductory chapter aims at providing some basic facts and notations concerning convex analysis, convex functions, and equilibrium point of a dynamic system, which will often be useful for us throughout this report. The material which follows is taken from [2].

A.1 Subdifferential and normal cones

Definition A.5 (Tangent cone). Let C ∈ R n be a nonempty subset. A vector d ∈ H is a direction tangent to C at a point x ∈ C if there exists a sequence (x n ) n∈N ⊂ C and a real sequence (τ n ) n∈N such that

The set of all such directions d ∈ R n is called the tangent cone to C at x ∈ C, denoted by T C (x).

A.1.2 Convex functions

An extended real-valued function is a function f : R n → R ∪ {+∞}.

Definition A.6. Given a subset C of R n , we denote by ψ C (•) the indicator function of C given by

Definition A.7. Let f : R n → R ∪ {+∞} be an extended real-valued function.

(i) The effective domain of f is defined by dom

The function f is said to be proper if its effective domain is nonempty.

(ii) The epigraph of f is defined by epi

(iii) Given σ ∈ R, the lower σ-level set of f is defined by

for every λ ∈ [0, 1] and every x, y ∈ dom(f ).

If (A.1) holds with strict inequality for λ ∈ (0, 1) and x = y, then f is called strictly convex.

Proposition A.1. Let f : R n → R ∪ {+∞} be a proper function. Then % First LMI: inequation (3.9)


A Lionel Burgaud

B Gerard Ladier

"Certification Together" : Unique and innovative approach to leverage the Certification-related efforts

Keywords: Certification, DO178B, DO254, ARP4754, ARP4761, User Groups

Generally the aeronautical certification context is considered as being very demanding, but necessary because of Safety objectives we can easily understand when we are a simple passenger. However in this context we can consider several improvements:

-When a new standard is discussed, joint committees gather all the stakeholders and allow discussions between them. But once the new standard is released, the committee no longer exists and it is difficult to share experiences about application of the new standard, -Discussions with Certification Authorities are usually possible only in the context of a real project, and generally they are based on specific cases. There is no real way for the Industry in general to express a point of view. In addition, it is important to have a commercial offer for training, expertise and support, managed in a way that we can iterate with customers and their subcontractors in order the share their objectives and consider work packages on which experts are fully committed. The "Certification Together" initiative is an innovative approach addressing those challenges, by mixing open approaches and a commercial offer. .

Introduction

When standards, or new versions of existing standards, are being written, Committees (RTCA, Eurocae,...) allow to exchange about the acceptable way to address challenges and meet certification objectives for a given domain.

But once the documents are released, the Committee no longer exists and companies are faced with the challenges of applying a generic document on specific and concrete projects. Challenges are even bigger for companies which were not part of the Committee. Until now, there has been no formal way to exchange opinions about the additional documents and guidance released by the Certification Authorities about application of the new or updated standard.

In addition, the only opportunities to have exchanges between the industry and the certification authorities are in the context of project audits, and this context makes these discussions quite difficult. It is necessary to create a context allowing generic discussions about management of emerging techniques or technologies, or special topics, without replacing project or program specific discussions of course. It also allows Industry to be proactive and propose acceptable approaches to the certification authorities.

Last but not least, such exchanges also allow to build best-of-art expert support and training, permanently updated thanks to these discussions, in order to collaborate with solution providers, R&D labs and schools. This set of objectives is what the "Certification Together" initiative addresses, as detailed in this paper.

Background

The ED80/DO254 [1] context is a good example: these guidelines have been released in April 2000 in their first version. This document has been difficult to handle, and can be source of a lot of discussions, misunderstanding or misinterpretation.

That's why we decided in 2004 to create the "DO254 User Group", involving industry players in the field of airborne projects.

• Aerospace companies,

• Certification Experts and Trainers,

• Tool vendors,

• Component Manufacturers,

• Subcontractors... We wanted to improve the applicants knowledge level and support the challenges of both EASA / FAA certification. This new standard introduced a lot of confusion, uncertainty and misunderstanding. We wanted to facilitate and understand the value and share lessons learned.

The current status is a unique collaboration between companies acting in 2 subgroups : the original one in Europe and one in the US [4].

Each subgroup is composed of about 20 major companies, and we have around 4 meetings per year with an average of 30 attendees per meeting. Each group defines its roadmap, and because the challenges in the US and FAA context may differ from challenges in the European and EASA context, roadmaps are shared and activities are synchronized when necessary between EU and US groups. This DO254 User Group is a great success, and has been recognized internationally as a model to use when new standards are released. There is no fee to join, and each member benefits come from the quality of the exchanges among a large number of stakeholders. All this initiative is also called "DO254 Together".

In the DO254 context, we also had the opportunity to leverage these exchanges to develop a training and expertise offer and partnerships between Expert companies, including FAA DERs, and solution providers.

The Toulouse area, mainly because of the Airbus related eco-system, concentrates a lot of synergetic skills in the scope of systems, software, hardware certification, but also for environmental (DO160), CNS/ATM and Security aspects.

This was a unique opportunity to extend the good experience we had in the DO254 context, in order to address all the domains: The first needs we can expect are related to the short term release of updated standards for Aircraft and System (ARP4754 [3]) and Software (DO178B [2]) aspects of Certification.

In addition, the international "Aerospace Valley" competitiveness cluster was representing a natural host of such an international, aeronautical-oriented initiative.

Certification Together

Certification Together is an initiative to create exchanges between all kinds of companies involved in airborne projects, facing certification challenges, and to provide expert solutions.

One main aspect is to leverage the expertise and synergies of several internationally recognized expert companies in the certification of critical systems. This international network and experts can be extended to additional companies providing a synergetic added value, as described later in this paper.

This initiative also involves graduate schools like ISAE, ENSEEIHT, the ONERA R&D lab, as well as tool and component providers.

The Aerospace Valley competitiveness cluster, with headquarters located in Toulouse (France), has selected Certification Together as their international Center of Expertise for Certification resource. It confirms that our joint initiative is providing benefits for the entire Industry community.

Certification Together provides a structured offer to help companies achieve their certification objectives. This offer includes:

• Training for all the certification domains: ARP4754, ARP4761, DO178B (C), DO254, DO160, DO278,… Certification Together also integrates animation and management of collaborative initiatives:

• Creation and management of User Groups, to exchange experiences and difficulties about new standards or updates of standards. We are currently preparing creation of equivalent User Groups related to the DO-178C, ARP 4754A, etc.

• Exchanges with schools to provide them with good input for their training programs, to share training resources, etc. It is also a resource for R&D labs who need to position their technologies and solutions in the avionics domain.

• Exchanges with tool vendors, component providers, and other stakeholders who need to understand their customers' certification needs, in order to provide new features, dedicated reports, specific documentation packages, and so on.

As an example of the animation of exchanges in this community, the first international conference on certification of systems, software and airborne electronics hardware in Europe will be organized in 2010, with participation of the certification authorities.

The Certification Authorities like FAA, EASA or others are not part of "Certification Together", which is an Industry initiative. The positioning is somehow an industry counterpart to the CAST (Certification Authority Software Team), which exists for years. It is of course not built "against" CAST: it is anticipated to discuss and exchange on a regular basis with CAST, which will benefit from consolidated industry positions.

How can this work ?

This innovative approach can work because we satisfy several key parameters:

The strategy of Certification Together is compliant with the individual strategies of all the actors and contributors:

• Experts companies want to keep their expertise at a top level, and find good partners they already know if they need to collaborate to take in charge larger work packages,

• Industrial companies want to exchange, learn from others and find proper support in order to reduce risks and save time for the certification of their projects,

• Solution providers want to find an opportunity to exchange with a community of industrial companies, users and non-users, about their roadmaps and plans. They also want to find recognized experts they can partner with, in order to concentrate on their core business and services and rely on expert partners if a customer needs a certification support.

• R&D labs and projects need a certificationoriented support in order to evaluate usability of the solution, technology or technique they are working on,

• Schools need to be aware of short term and long term challenges, find experts from the industry and from expert company to train students and show internationally their involvement in understanding the certification context.

The commercial partnership in place is managed and limited to partners between which the synergy is analyzed and validated. Of course this point is a sensitive and challenging one, that's why we defined a formal process with a set of conditions to accept new candidates.

Another key parameter is our experience on creation of User Groups creating added value for all the members, recognized as an acceptable contributor by Certification Authorities (i.e. the CAST group).

To have this working, we manage and monitor the initiative with the objective to have all the investments reasonable and generating benefits for all the actors.

The key aspect is also to have all the actors and contributors working and collaborating with a proactive and open spirit. We progressed a lot on that point, as it is not so easy to have companies which are sometimes, at least partially, competitors working together.

The public entities (regional or national entities) were also very interested by this initiative and wanted to be part of it: Subcontractors at the end of the supply chain are often small or medium companies (SME), and for these companies the effort to meet certification objectives, or become autonomous as required by their customers, is not always affordable.

Thanks to the recognition of Certification Together by the Aerospace Valley competitiveness cluster, public entities can analyze the needs of these companies and decide to financially support them on a case by case basis for training, audits and definition of action plans for process improvements. Then they will have a chance to be part of the competition.

Conditions related to the network of experts

Even if experts are involved in open and collaborative initiatives, the network of experts represents a commercial aspect of Certification

Together. This commercial partnership can be extended to companies or individual experts if the synergy can be established.

In practice it means any company can propose its participation to our network, according to the following steps and conditions:

A preliminary exchange is necessary (phone call, discussion during an event,…) to validate understanding of the Certification Together objectives, preferably with a partner already part of the network of experts, The company sends a first presentation to the Certification Together leader, with at least the following inputs:

• Short introduction of the company, including a focus on the activities and domain for which the contribution is proposed,

• Description of the proposed contribution: perimeter, domain and related standards, activities (training, support,…)

• What is expected from Certification Together,

• Proposed positioning compared to the expert partners already in place.

If the preliminary analysis of those inputs does not highlight any showstopper, the company is invited to a live presentation during a steering committee meeting.

The decision to integrate the company or not is based on the consensus between the partners in place and attending the meeting.

More details about this process can be obtained by contacting the Certification Together leader.

Access to the expertise and support offer

As "Certification Together" aggregates different concepts and contexts, from commercial aspects to open and non-commercial initiatives, we did not create a specific legal entity. A given legal status can be a good answer to one part of the initiative, but not appropriate to others.

"Certification Together" is an entry point, then customers who need training, support and expertise will contract classically with the expert company which will provide these services.

About User Group membership

Meetings are mainly technical, and not marketing or sales events.

The highest priority is given to industrial companies (equipment suppliers or A/C manufacturers). Other contributors such as tool vendors and IP providers,… can be concerned but it is usually more effective for them and for the Group to invite them for specific topics, after preliminary discussion with these contributors,

Companies who want to be part of the group send some preliminary information about:

• The company and its representative, especially concerning their experience in the domain addressed by the User Group.

• How they can contribute,

• What they expect from the group,

The group needs to be of a representative but reasonable size, allowing discussions to quickly reach an optimum level of maturity. Thus, highest priority is given to participants who have:

• Experience on several DAL A projects, and perspectives of future projects at the same level

• Experience of certification audits, and of several certification contexts like EASA/FAA,…

• Technical capabilities for the topics and contexts addressed by the group

In addition the company needs:

• To allocate time for 3-4 meetings per year, in Europe and/or in the US

• To allocate time for subgroup discussions, through meetings and phone calls,

• To allocate time for the review of papers, presentations, etc.

Companies who wish to simply obtain answers to the problems they may encounter, without the level of contribution required by User Groups, should consider training or consulting activities.

Important point : currently there is no fee to be part of the user group. Each company invests time and budget for attending the meetings and gets benefits because these exchanges allow them to save time and reduce risks on projects.

So we recommend being very careful about other initiatives looking similar but with a very small number of participants and a participation fee…

About collaboration with solution providers

Because a key aspect of "Certification Together" is certification expertise and credibility, it is necessary to keep independence and neutrality in mind regarding commercial solutions and their promotion.

The objective is clearly not to sell or distribute solutions.

As opposed to promoting the solutions themselves, through the website and seminar presentations, for example, Certification Together aims to promote the efforts of solution providers to help their users meet certification objectives.

Even a market-leader solution will not be promoted by "Certification Together" unless technical or marketing materials are available (white papers, specific presentations, dedicated packages,…) to explain how the solution helps meet certification objectives.

The experts of the "Certification Together" network will evaluate the added value of the certificationoriented marketing material that would be presented by the solution providers, before promoting this material.

A similar evaluation of the added value will be applied to a "success story" paper that would be written by users and that we could publish with the users' authorization.

Some partnerships are already in place, and others can be discussed, but usually the starting point is based on the following respective contributions:

Contribution from Solution Providers

• Integrate the Certification Together expertise offer during seminars presenting the solution.

The certification experts can participate in such events to introduce/explain the certification context and highlight how the solution provider's efforts can help in this context.

• Share information with their customers concerning Certification Together events, such as training sessions or seminars focusing on their domain.

• Consider preferential pricing to the experts of the Certification Together network for product training sessions.

• Solicit the Certification Together experts for customer needs that go beyond what the solution provider can address directly through its own consulting services.

Contribution from the Certification Together network of experts:

• Help the solution provider evaluate their internal processes regarding certification objectives, building certification-oriented packages, etc. (These activities are usually included in a specific commercial agreement.)

• Review technical presentations and marketing material (whitepapers, etc.): provide an external point of view and contribute to the added value of the solution in a certification context,

• Introduce the certification context and challenges to the solution provider's sales team and application engineers, to give them better understanding of the customer expectations, where the certification-oriented packages can help,…

• Offer preferential pricing for training of solution provider employees, for sessions organized by the Certification Together network of experts,

• On the Certification Together website, publish the white papers and added value material released by the solution provider, which will highlight the credibility of this material.

• On the "Solutions" page of the Certification Together website, include a summary of the certification-oriented efforts performed by the solution provider, with a link to their webpage.

• Publish information about certification-oriented events organized by the solution provider.

• Integrate screenshots of or references to the pertinent solution and/or certification-oriented material during Training sessions and Seminars.

• Provide a first introduction to the added value of the solution to customers who may have relevant needs, then transfer the opportunity to the solution provider.

• Keep the solution provider informed of all opportunities coming from the Certification Together website, discussions during training sessions or seminars, etc.

Next steps

Based on the success of the DO254 group, we are considering the future DO178C and ARP4754A User Groups.

We are exchanging with their respective chairmen in order to define the schedule and conditions.

Another key step is the next Certification Together International Conference (CTIC). This conference will take place in Toulouse in October 2010, with about 250 participants representing Certification Authorities, Aircraft manufacturer, System and equipment suppliers, subcontractors, certification experts, solution providers, schools and R&D labs.

The participation should be 70% Europe, 15% US and 15% RoW (China, Russia, Canada,…) All the news and events are published and available from the Certification Together website.

In 2010, we will also extend the Steering Committee to include not only Partners and representatives from the Aerospace Valley competitiveness cluster, but also large Aerospace companies relying on this innovative approach:

• To ensure that initial objectives are met • To ensure that emerging needs are identified, discussed and addressed

Acknowledgements

The authors acknowledge the contribution of other Certification Together partners, and especially representatives from expert companies EADS-Apsys, ACG Solutions, Cap Gemini, who contributed to the definition of the input data summarized in this paper.

Conclusion

Certification Together is an initiative launched by and for the Industry to create and animate exchanges about aeronautical certification: on one hand through collaborative approaches like User Groups, on the other hand through a training and consulting offer supported by international expert partners.

In addition to its current activity and its recognition by major companies, Certification Together has been selected as the "Center of Expertise for Certification" of the Aerospace Valley international cluster.

Through all these collaborations and exchanges, the objective is to create a set of solutions in order to make the certification objectives easier to meet, by addressing Safety and Security issues in relation with industrial and productivity aspects.

[1] DO254-ED80 "Design assurance guidance for airborne electronic hardware", April 2000

[2] DO178B-ED12B "Software Considerations in Airborne Systems and Equipment Certification", December 1992.

[3] ARP4754-ED79: "Certification considerations for highly integrated or complex aircraft systems", April 1996.

[4] DO254 User website : www.do254.com

[5] Certification Together website: www.certification-together.com