Aya Younes
email: aya.younes@inria.fr

Félix Miranda-Villatoro
email: felix.miranda-villatoro@inria.fr

Bernard Brogliato
email: bernard.brogliato@inria.fr

Trajectory Tracking in Linear Complementarity Systems with and without State-Jumps: A Passivity Approach

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Complementarity dynamical systems are a class of nonsmooth nonlinear systems, that has received attention in the Automatic Control and Applied Mathematics literature because of their applications in many fields like electrical circuits with nonsmooth components, mechanical systems with unilateral contact, networks with unilateral interactions, economics with projected dynamical systems, genetics, traffic flow and neural networks, etc, see [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF][START_REF] Brogliato | Dissipative dynamical systems with set-valued feedback loops: Well-posed set-valued Lur'e dynamical systems[END_REF][START_REF] Heemels | The Complementarity Class of Hybrid Dynamical Systems[END_REF]. Trajectory tracking in fully-actuated complementarity Lagrangian systems has been tackled in [START_REF] Menini | Trajectory tracking of a bouncing ball in a triangular billiard by unfolding and folding the billiard table[END_REF][START_REF] Menini | Algebraic Methods for Multiobjective Optimal Design of Control Feedbacks for Linear Systems[END_REF][START_REF] Menini | Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts[END_REF][START_REF] Galeani | Trajectory tracking for a particle in elliptical billiards[END_REF][START_REF] Forni | Follow the Bouncing Ball: Global Results on Tracking and State Estimation With Impacts[END_REF][START_REF] Brogliato | On the control of finite-dimensional mechanical systems with unilateral constraints[END_REF][START_REF] Brogliato | On tracking control of a class of complementary-slackness hybrid mechanical systems[END_REF][START_REF] Bourgeot | Tracking Control of Complementarity Lagrangian Systems[END_REF][START_REF] Morarescu | Trajectory tracking control of multiconstraint complementarity Lagrangian systems[END_REF][START_REF] Morarescu | Passivity-based switching control of flexible-joint complementarity mechanical systems[END_REF][START_REF] Leine | Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints[END_REF][START_REF] Heck | Guaranteeing stable tracking of hybrid position-force trajectories for a robot manipulator interacting with a stiff environment[END_REF][START_REF] Rijnen | Hybrid systems with state-triggered jumps: Sensitivity-based stability analysis with application to trajectory tracking[END_REF][START_REF] Rijnen | Reference spreading: Tracking performance for impact trajectories of a 1DoF setup[END_REF]. The stabilization is studied in [START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Posa | Stability analysis and control of rigid-body systems with impacts and friction[END_REF][START_REF] Aydinoglu | Stabilization of complementarity systems via contact-aware controllers[END_REF], the output regulation is analyzed in [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF]. The Lyapunov stability of linear complementarity systems (LCS) has been tackled in [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF][START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF][START_REF] Brogliato | The Krasovskii-LaSalle invariance principle for a class of unilateral dynamical systems[END_REF][START_REF] Goeleven | Stability and instability matrices for linear evolution variational inequalities[END_REF]. Tracking in complementarity mechanical systems with state jumps is studied in [START_REF] San Felice | [END_REF][START_REF] Biemond | Tracking control for hybrid systems with state-triggered jumps[END_REF]. The trajectory tracking problem is solved in [START_REF] Menini | Trajectory tracking of a bouncing ball in a triangular billiard by unfolding and folding the billiard table[END_REF][START_REF] Menini | Algebraic Methods for Multiobjective Optimal Design of Control Feedbacks for Linear Systems[END_REF][START_REF] Menini | Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts[END_REF][START_REF] Galeani | Trajectory tracking for a particle in elliptical billiards[END_REF][START_REF] Forni | Follow the Bouncing Ball: Global Results on Tracking and State Estimation With Impacts[END_REF][START_REF] Brogliato | On the control of finite-dimensional mechanical systems with unilateral constraints[END_REF][START_REF] Brogliato | On tracking control of a class of complementary-slackness hybrid mechanical systems[END_REF][START_REF] Bourgeot | Tracking Control of Complementarity Lagrangian Systems[END_REF][START_REF] Morarescu | Trajectory tracking control of multiconstraint complementarity Lagrangian systems[END_REF][START_REF] Morarescu | Passivity-based switching control of flexible-joint complementarity mechanical systems[END_REF][START_REF] Leine | Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints[END_REF][START_REF] Heck | Guaranteeing stable tracking of hybrid position-force trajectories for a robot manipulator interacting with a stiff environment[END_REF][START_REF] Rijnen | Hybrid systems with state-triggered jumps: Sensitivity-based stability analysis with application to trajectory tracking[END_REF][START_REF] Rijnen | Reference spreading: Tracking performance for impact trajectories of a 1DoF setup[END_REF] using different strategies. It is noteworthy that in most of these articles, the desired trajectory is suitably modified in the neighborhood of state jumps (impacts), and/or the tracking error dynamics 'stability is tailored to the problem. The tracking of cyclic trajectories which undergo constrained/unconstrained/impacting phases of motion (including multiple constraints and multiple impacts phenomena) is studied in [START_REF] Brogliato | On the control of finite-dimensional mechanical systems with unilateral constraints[END_REF][START_REF] Brogliato | On tracking control of a class of complementary-slackness hybrid mechanical systems[END_REF][START_REF] Bourgeot | Tracking Control of Complementarity Lagrangian Systems[END_REF][START_REF] Morarescu | Trajectory tracking control of multiconstraint complementarity Lagrangian systems[END_REF][START_REF] Morarescu | Passivity-based switching control of flexible-joint complementarity mechanical systems[END_REF] for n-degree-of-freedom (dof) systems. Basically, the controllers are extended passivity-based algorithms which switch between three subcontrollers (persistent contact with constant number of activated constraints, mode obtained from the deactivation of one or several constraints, impacting transition phase between two persistent contact modes when the number of activated constraints increases). The authors introduce the notion of weak stability, which extends Lyapunov stability by disregarding the Lyapunov function variation during the transition phases (nonmonotonic Lyapunov functions are used). Trajectory tracking in planar billiards is studied in [START_REF] Menini | Trajectory tracking of a bouncing ball in a triangular billiard by unfolding and folding the billiard table[END_REF][START_REF] Menini | Algebraic Methods for Multiobjective Optimal Design of Control Feedbacks for Linear Systems[END_REF][START_REF] Menini | Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts[END_REF][START_REF] Galeani | Trajectory tracking for a particle in elliptical billiards[END_REF][START_REF] Forni | Follow the Bouncing Ball: Global Results on Tracking and State Estimation With Impacts[END_REF]. The objective is to control a two-dof particle inside a closed compact domain when trajectories collide with the domain's boundary and never undergo persistent contact with it (a kind of vibro-impact systems [15, section 1.3.2]). This article is dedicated to trajectory tracking in LCS, which make a class of complementarity systems different from Lagrangian complementarity systems studied in the above references. The main differences between Lagrangian systems with unilateral constraints and impacts, and LCS as in this paper, are that impacts in mechanical systems are state-dependent, see (134), and passivity does not hold when the complementarity variables (i.e., the contact force multiplier and the gap function which usually is a nonlinear function of the position) are used as the input and output variables to define the supply rate (see Remark 14.1). In passive LCS, state jumps are essentially exogenous (i.e., they are triggered by the signal F u(t) in the variable w in (2) below), and passivity holds between the complementarity variables. Moreover the complementarity variables are not of the same form in mechanical systems and in LCS. In the latter they take the form of a linear function of the state, the multiplier and an exogenous signal (see Definition 4 below and w in (2)), while unilateral constraints in Lagrangian systems always have D = 0 and F = 0 (hence excluding the strong passivity of the closed-loop system), and usually are a nonlinear function of one part of the state only, i.e., positions. However even linear unilaterally constrained Lagrangian systems do not fit within the class of LCS studied in this article, because of the different nature of state jumps as explained above. One major consequence is that state jumps can be triggered instantaneously by the exogenous signal, which is not possible in mechanical systems where exogenous signals (external forces and torques) do not act directly in the complementarity constraints. The way the desired trajectories are designed in the above works and in this article, are quite different as well. When the unilateral constraints are replaced by unilateral springs, things differ significantly as illustrated in section 14. Certainly, the frameworks that are the closest to what is presented in the sequel, are in [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF] and [START_REF] Leine | Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints[END_REF] (though the class of studied systems is not the same, and the robustness analysis is not led in [START_REF] Leine | Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints[END_REF]). This is detailed below in Remark 3.7.

The main contributions of this article are to propose a solution for the trajectory tracking in LCS, in the case where all parameters are known, and when parameter uncertainties are taken into account. The case of absolutely continuous solutions is treated, as well as the case of solutions with state jumps (when parameters are known). Passivity and maximal monotonicity of suitable operators are the tools which allow to design stable closed-loop systems. Matrix inequalities stemming from passivity are central in the design process. The theoretical developments are illustrated by numerical examples on nonsmooth circuits, networks with unilateral interactions, and mechanics with unilateral springs. This article is organised as follows: useful notations and definitions are given in the next paragraph; section 2 is dedicated to recall the main results about the well-posedness (existence and uniqueness of solutions) of LCS, as well as state jump mappings; section 3 presents in details the control design and the error dynamics analysis, in the known parameters case, with and without state jumps; section 4 is devoted to extend the results to the case when parametric uncertainties are considered; the particular case of FOSwP is analysed in section 5; a simple scalar example is studied in section 6, and many circuit examples are analysed in sections 7 through 13, a simple mechanical system with a unilateral spring is presented in section 14, and a network with unilateral interactions is presented in section 7; conclusions end the article in section 16, and several useful tools, results, code design details, are provided in Appendices A.1 through A.8 and in Appendix B.

Notation and Definitions

The elements of a matrix M ∈ IR n×m are denoted M ij . M •,j is its jth column, and M i,• is its ith row. A matrix M ∈ IR n×n , possibly nonsymmetric, is said to be positive definite, M ≻ 0 (resp. positive semi-definite, M ≽ 0) if x ⊤ M x > 0 for all x ̸ = 0 ∈ IR n (resp. x ⊤ M x ≥ 0). M is a P-matrix if all its principal minors are positive; a positive definite matrix is a P-matrix. The minimum and the maximum eigenvalues of M are denoted λ min (M ) and λ max (M ). The maximum singular value of M ∈ IR n×m is denoted σ max (M ), Im(M )is its image, Ker(M ) is its null space. The singular values are ordered as σ max (M ) = σ 1 (M ) ≥ σ 2 (M ) > . . . > σ r (M ) > 0, where r = rank(M ). Let M = M ⊤ , its eigenvalues are real and ordered as λ max (M ) = = esssup t∈I ||f (t)|| (when I is obvious from the context it may be avoided). The range of the function f (•) is Im(f ), its domain is dom(f ) = {x | f (x) < +∞}. Right and left limits of the function f (•) at t are denoted f (t + ) and f (t -). Let S be a nonempty set, rint(S) is its relative interior [START_REF] Rockafellar | Convex Analysis[END_REF]. Let S and S ′ be closed sets, d H (S, S ′ ) denotes the Hausdorff distance between S and S ′ . AC is for absolute continuity, BV is for bounded variation, LBV is locally BV, and RCLBV is for right-continuous LBV. Let S ⊆ IR m be a closed convex set containing 0. Its polar cone S • = {x ∈ IR m | x ⊤ v ≤ 0, ∀v ∈ S} = -S ⋆ where S ⋆ is the dual cone. Let us recall some useful definitions [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF][START_REF] Rockafellar | Convex Analysis[END_REF][START_REF] Rockafellar | Variational Analysis[END_REF][START_REF] Cottle | The Linear Complementarity Problem[END_REF][START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF]. Definition 1. Let S ⊂ IR n be a closed, non-empty and convex set. The normal cone to S at a point x ∈ S is given by

N S (x) = g ∈ IR n | g ⊤ (z -x) ≤ 0, ∀z ∈ S .
(1)

For x ̸ ∈ S one usually sets N S (x) = ∅. The tangent cone T S (•) is the polar to the normal cone.

The domain of a set-valued mapping T :

IR n ⇒ IR n is Dom(T ) = {x ∈ IR n | T (x) ̸ = ∅}.
Definition 2. A multivalued mapping T : Dom(T ) ⊆ IR n ⇒ IR n is monotone if for all u 1 , u 2 ∈ Dom(T ), for all v 1 ∈ T (u 1 ), v 2 ∈ T (u 2 ), then

⟨u 1 -u 2 , v 1 -v 2 ⟩ ≥ 0.
It is maximal if its graph cannot be enlarged without destroying monotonicity. Definition 3. A multivalued mapping T : Dom(T ) ⊆ IR n ⇒ IR n is hypomonotone if there exists a real k > 0 such that for all u 1 , u 2 ∈ Dom(T ), for all v 1 ∈ T (u 1 ), v 2 ∈ T (u 2 ), we have:

⟨u 1 -u 2 , v 1 -v 2 ⟩ ≥ -k||u 1 -u 2 || 2
The normal cone to a convex closed nonempty set, defines a maximal monotone set-valued mapping. More generally, let φ : IR n → IR ∪ {+∞} be a proper, convex and lower semicontinuous function. Its subdifferential at x is defined as the set ∂φ(x)

= {g ∈ IR n | φ(v) -φ(x) ≥ g ⊤ (v -x), ∀v ∈ IR n },
and it defines a maximal monotone mapping. The normal cone in [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] is the subdifferential of the indicator function defined as Ψ S (x) = 0 if x ∈ S, Ψ S (x) = +∞ if x ̸ ∈ S. Another function plays an important role: the support function of the set S, defined as σ S (y) = sup v∈S v ⊤ y. It is related to the indicator function as: η ∈ N S (v) = ∂Ψ S (v) ⇔ v ∈ ∂σ S (η) = N -1 S (η) (the indicator and the support functions are conjugate functions, and their subdifferentials are inverse mappings. Definition 4. A linear complementarity system (LCS) with inputs is a nonsmooth dynamical system defined as:

(a) ẋ(t) = Ax(t) + Bλ(t) + Eu(t) (almost everywhere) (b) 0 ≤ λ(t) ⊥ w(t) = Cx(t) + Dλ(t) + F u(t) ≥ 0 (2)
where x ∈ IR n , λ ∈ IR m and w ∈ IR m , A, B, C, D, E and F are constant matrices of appropriate dimensions, u : IR + → IR p is an exogenous signal or a control input. The constraint in (2) (b) is a linear complementarity problem (LCP) in λ, denoted LCP(D, Cx + F u).

A mode of the LCP (2) (b) corresponds to a pair (λ, w) with λ i = 0 and w i > 0 (noncontact indices), λ j > 0 and w j = 0 (contact indices), λ k = w k = 0 (degenerate indices), i ∈ I ⊆ {1, . . . , m}, j ∈ J ⊆ {1, . . . , m}, k ∈ K ⊆ {1, . . . , m}, I ∪ J ∪ K = {1, . . . , m}, I ∩ J ∩ K = ∅, I ∩ J = ∅, I ∩ K = ∅, and J ∩ K = ∅. The switching times between modes i and j correspond to junction times, or detachment times. It is noteworthy that most of the results in this article apply to linear cone complementarity systems (LCCS), with (b ′ ) K ∋ λ(t) ⊥ w(t) = Cx(t) + Dλ(t) + F u(t) ∈ K ⋆ , where K ⊆ IR m is a nonempty closed convex cone, K ⋆ = {z ∈ IR m | z ⊤ w ≥ 0, ∀ w ∈ K} is its dual cone. Using tools from Convex Analysis it is possible to rewrite equivalently (2) as the differential inclusion (DI):

ẋ(t) ∈ Ax(t) + Eu(t) -B(D + N -1 I R m + ) -1 (Cx(t) + F u(t)) ∆ = -H(t, x(t)), (3)
where λ(t) ∈ -(D + N -1

I R m +
) -1 (Cx(t) + F u(t)). We shall see later that the DI may sometimes be written in a slightly different way when the input is given a specific form, see [START_REF] Brogliato | On tracking control of a class of complementary-slackness hybrid mechanical systems[END_REF]. A state x * is an equilibrium point of (2) if and only if there exist λ * and w * ∈ R n such that the mixed LCP (MLCP) 0 = Ax * + Bλ * 0 ≤ λ * ⊥ w * = Cx * + Dλ * ≥ 0 (4) holds. Passivity plays an important role in the sequel of this work. Let us define Moreau's first order sweeping process: Definition 5. Moreau's first order sweeping process (FOSwP) is a differential inclusion of the form: ẋ(t) ∈ -N S(t) (x(t)) + f (t, x) [START_REF] Mosek Aps | MOSEK Optimizer API for Python[END_REF] where N S(t) (x(t)) is the normal cone to the closed convex non-empty set S(t) ⊆ IR n at x(t), defined in [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF], and f (t, x) is a single-valued map such that f : IR × IR n → IR n . Definition 6. A quadruple (A, B, C, D) is said to be passive, or dissipative with respect to the supply rate u ⊤ y, if there exists a non-negative function V : IR n → IR + , called a storage function, such that for all t 0 ≤ t 1 and all time functions (u, x, y) ∈ L 2 ([t 0 , t 1 ]; IR m × IR n × IR m ) such that ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t), the following inequality holds:

V (x(t 0 )) + t1 t0 u ⊤ (t)y(t) dt ≥ V (x(t 1 )) ( 6)

The inequality [START_REF] Ch | A Pontryagin maximum principle for the controlled sweeping process[END_REF] is called the dissipation inequality. Equivalently, the following LMI

A ⊤ P + P A P B -C ⊤ B ⊤ P -C -D -D ⊤ ≼ 0 ( 7
)
has a solution P = P ⊤ ≽ 0. Then V (x) = 1 2 x ⊤ P x. The quadruple (A, B, C, D) is said strictly state passive if the LMI [START_REF] Aydinoglu | Stabilization of complementarity systems via contact-aware controllers[END_REF] holds with P = P ⊤ ≻ 0 and A ⊤ P + P A + ϵP ≼ 0 for some ϵ > 0. It is said strongly passive if the LMI [START_REF] Aydinoglu | Stabilization of complementarity systems via contact-aware controllers[END_REF] holds with strict inequality and P = P ⊤ ≻ 0.

When LCS are considered, then the multiplier λ is chosen as the input u while w is chosen as the output y for the passivity property. Passivity is then to be seen as a structural property related to internal variables of the LCS, and motivated by the fact that it corresponds to a physical property in some applications like circuits. Definition 7. [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF]Definitions 2.34,2.58,2.78] The transfer function H(s) = C(sI -A) -1 B +D ∈ C mxm is positive real (PR) if: H(s) has no pole in Re[s] > 0, H(s) is real for all positive real s, H(s) + H ⋆ (s) ≽ 0 for all Re[s] > 0. The transfer function H(s) ∈ C mxm is strictly positive real (SPR) if H(s -ϵ) is PR for some ϵ > 0 [25, Definition 2.58]

and it is strong SPR (SSPR) if H(s) is analytic in Re[s] ≥ 0 and Re[H(jω)]

≥ δ > 0, for all ω ∈ [-∞, ∞] and some δ ∈ IR [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF]Definition 2.78].

Strict state passivity is related to SPR tranfer functions, while strong passivity is related to SSPR transfer functions [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF][START_REF] Madeira | On the equivalence between strict positive realness and strict passivity of linear systems[END_REF]. Using the Schur complement theorem (see e.g., [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF]Theorem A.65]), strong passivity implies that D ≻ 0 and -A ⊤ P -P A ≻ 0.

Well-posedness of Time-Varying LCS

LCS with External Inputs

LCS as in [START_REF] Acary | An introduction to Siconos[END_REF] are nonlinear nonsmooth dynamical systems with inputs, whose well-posedness (existence, uniqueness, and continuous dependence of solutions) has been investigated. Let us provide a short summary of well-posedness results in [START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF][START_REF] Camlibel | Passivity and Complementarity[END_REF][START_REF] Camlibel | Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators[END_REF][START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF][START_REF] Brogliato | On the equivalence between complementarity systems, projected systems and differential inclusions[END_REF][START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF][START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF]. Most of them apply to larger classes of nonsmooth systems than LCS. Let us remind that a right-continuous function f : I ⊆ IR → IR of locally bounded variation (RCLBV), has a countable set of discontinuities on I (allowing for left accumulations of jump instants), and possesses a differential measure df , see [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems[END_REF]. The conditions associated with the results recalled below, are sufficient. However they are well fitted with the conditions imposed in this article for stability purposes, i.e., passivity.

(Well-posedness based on reduction to an ODE)

• Assume that D is a P-matrix. Then from the fundamental theorem of Complementarity Theory [START_REF] Cottle | The Linear Complementarity Problem[END_REF], it follows that λ is a piecewise continuous, Lipschitz single-valued function of x and u. Hence the LCS in [START_REF] Acary | An introduction to Siconos[END_REF] is the ODE: ẋ(t) = Ax(t) + Bλ(x(t), u(t)) + Eu(t).

Provided u(•) satisfies Lebesgue integrability conditions, an AC solution exists on IR + for any bounded initial condition, with uniqueness and continuous dependence on initial data [START_REF] Coddington | Theory of Ordinary Differential Equations[END_REF][START_REF] Cartan | Cours de Calcul Différentiel. 3rd[END_REF].

• Assume that D ≻ 0 (not necessarily symmetric), then using the formalism in [START_REF] Adly | Well-posedness of nonsmooth Lurie dynamical systems involving maximal monotone operators[END_REF], the result can be extended to systems with operators (D+M t ) -1 (•) where M t (•) is a maximal monotone operator for each fixed t. Using [19, Proposition 1] or [START_REF] Adly | Well-posedness of nonsmooth Lurie dynamical systems involving maximal monotone operators[END_REF], the mapping (D + M t ) -1 (•) is then single-valued, defined everywhere, and Lipschitz continuous with constant 1 λmin(D+D ⊤ ) for each fixed t. Therefore the mapping H(t, x) in ( 3) is single-valued Lipschitz continuous in x and classical results on ODE's well-posedness apply.

2. (Well-posedness based on passivity and relaxations of passivity) Let us summarize the results presented in [START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF][START_REF] Camlibel | Passivity and Complementarity[END_REF][START_REF] Camlibel | Linear passive systems and maximal monotone mappings[END_REF][START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF][START_REF] Camlibel | Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators[END_REF], which deal with the case with external inputs.

(a) [START_REF] Camlibel | On linear passive complementarity systems[END_REF]Theorem 7.5] Assume that:

• i) (B ⊤ , D + D ⊤ ) ⊤ has full column rank,

• ii) (A, B, C, D) is passive with P = P ⊤ ≻ 0, and minimal,

• iii) u(•) is piecewise continuous with rational Laplace transform. Then for any x(0) = x 0 , there exists a unique global solution to [START_REF] Acary | An introduction to Siconos[END_REF] such that (λ, x, w) ∈ L 2 δ (IR + ; IR m+n+m ), the space of Schwartz' distributions, with regular parts in L 2 loc ⊆ L 1 loc , and atomic part with isolated atoms (instants of state jumps) that is represented by sums of Dirac measures (higher degree distributions do not occur). (b) [START_REF] Camlibel | Passivity and Complementarity[END_REF]Theorems 7,[START_REF] Bernstein | Matrix Mathematics. Theory, Facts and Formulas[END_REF][START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF] Assume that:

• i) (A, B, C, D) is passive with P = P ⊤ ≻ 0,

• ii) F u(t) ∈ Q ⋆ D + Im(C) for all t ≥ [0, T ), T > 0 (see section 2.1 for the definition of Q D and of its dual cone),

• iii) u(•) is a Bohl function (i.e., u(t) = M exp(N t)R for some constant matrices M , N , R). Then x(0 + ) is calculated as in [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF], there exists a unique AC solution x : (0, T ) → IR n , and λ(•) is locally Lebesgue integrable. The solution is a forward solution, i.e., it is the concatenation of Bohl functions defined between state jumps. Left accumulations of state jumps are allowed. (c) [82, Corollary 2] [23, Corollary 5.9] Assume that:

• i) D ≽ 0, Ker(D + D ⊤ ) ⊆ Ker(P B -C ⊤ ) for some

P = P ⊤ ≻ 0, • ii) rint(IR m + -F u(t)) ⊆ rint(Im(∂σ I R m + -F u(t) + D)), • iii) DIR m + ⊆ Im(C), • iv) Im(C) -IR m + = IR m , • v) for each x ∈ IR n and each t ≥ 0, if the set Λ = {λ ∈ IR m + | v = Cx+Dλ+F u(t) ≥ 0, λ ⊤ v = 0} has a nonzero element, and Λ ∩ Im(D + D ⊤ ) ̸ = ∅.
Then if F u(•) is AC (resp. RCLBV), there exists a unique AC (resp. RCLBV) solution to (2) for any x(0) such that Cx(0) ∈ Im(∂σ I R m + -F u(0) + D). (d) [START_REF] Camlibel | Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators[END_REF]Theorems 11,[START_REF] Brogliato | On the equivalence between complementarity systems, projected systems and differential inclusions[END_REF] Let:

• i) (A, B, C, D) be passive, • ii) Im(C) ∩ rint(Im(D + N I R m + ) -F u(t)) ̸ = ∅, • iii) F u(•) be AC.
Then there exists a unique AC solution for any x(0) = x 0 ∈ Dom(H(0, •)). (e) [19, Theorem 1, Corollaries 2, 3] Assume that:

• i) D = 0 0 0 D 2 ∈ IR m2×m2 , m 2 ≤ m • ii) F = 0, • iii) u(•) is continuous with u ∈ L 1 loc (IR + ; IR p ), • iv) there exists a full-rank R = R ⊤ ∈ IR n×n such that R 2 B 1 = C ⊤ 1 , B = (B 1 B 2 ), C = (C ⊤ 1 C ⊤ 2 ) ⊤ , • v) there exists w 0 ∈ IR m1 at which the operator w → ∂Ψ I R m 1 + (C 1 R -1 w) is continu- ous, • vi) the operator w → RB 2 (D 2 +∂Ψ I R m 2 + ) -1 (-C 2 R -1 w
) is well-defined, single-valued and Lipschitz continuous. Then there exists a unique continuous right-differentiable solution for any x(0 [START_REF] Le | Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions[END_REF]Theorems 5.3,5.4] and [START_REF] Le | On a class of Lur'e dynamical systems with state-dependent set-valued feedback[END_REF]Theorems 1,[START_REF] Acary | An introduction to Siconos[END_REF] deal with DIs of the form:

) ∈ Dom(∂Ψ I R m 1 + (C 1 R -1 •)) = {w ∈ IR n | C 1 R -1 w ≥ 0}, with ẋ ∈ L ∞ (IR + ; IR n ). The result holds if m 2 = 0 and i-v) are verified, or if m 2 = m and vi) is verified (then item 1 applies). Condition vi) holds if D 2 is a P-matrix [19, Proposition 1, Corollary 1]. (f)
ẋ(t) ∈ g(t, x(t)) -B(D + F -1 t,x ) -1 (Cx(t))
, where F t,x : IR m ⇒ IR m is maximal monotone for each (t, x). For briefness we focus on [START_REF] Le | On a class of Lur'e dynamical systems with state-dependent set-valued feedback[END_REF]. Assume that:

F t,x (•) = N K(t,x) (•)
• i) D ≽ 0, Im(D) ⊂ Im(C), Ker(D + D ⊤ ) ⊆ Ker(P B -C ⊤ ) for some P = P ⊤ ≻ 0, • ii) K(t, x) has closed convex values, K(t, x)∩Im(C) ̸ = ∅, d H (K(t, x)∩Im(C), K(s, y)∩ Im(C)) ≤ k 1 |t -s| + l 2 ||x -y|| for all t, s, x, y, l 1 ≥ 0, 0 ≤ l 2 ≤ λ2 ||C|| , λ 2 the smallest positive eigenvalue of CC ⊤ , • iii) (N -1 K(t,x) + D) -1 (Cx) ̸ = ∅ ⇒ Im(D + D ⊤ ) ∩ (N -1 K(t,x) + D) -1 (Cx) ̸ = ∅, • iv) Im(C) ∩ rint(Im(N -1 K(t,x) + D)) ̸ = ∅, • v) g(•, •) is continuous in t and Lipschitz continuous in x.
Then existence and uniqueness of Lipschitz continuous solutions is guaranteed for each initial condition such that (N -1 K(0,x0) + D) -1 (Cx 0 ) ̸ = ∅. Moreover, ∥ ẋ(t)∥ ≤ α + β∥x(0)∥ for some α > 0 and β > 0.

3. (Well-posedness by transformation to a sweeping process [START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF][START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF][START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF]) Assume that D = 0, and that: 2) can be rewritten equivalently as a FOSwP (see Definition 5,[START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF][START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF] for details on the transformation, and [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems[END_REF][START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF] for details on FOSwP). The basic condition i) is implied by, but does not imply, the passivity of (A, B, C, 0). Depending on the signal u(•) being AC (resp. RCLBV), the solution is shown to be AC (resp. RCLBV), defined on IR + , and uniqueness holds. For AC to hold it is needed that x(0) ∈ S(0) (no initial jump).

• i) P B = C ⊤ for some P = P ⊤ ≻ 0, • ii) u ∈ L 1 loc (IR + ; IR p ), • iii) Im(C) -IR m + = IR m . Then the LCS in (
The various assumptions made in items 2 and 3 (which may be thought as constraint qualifications in many instances, a concept which is familiar in Optimization) have different meanings: This assumption thus secures that there exists a bounded multiplier λ(t) which solves the LCP (notice that at the end time T , there may be a state jump if this assumption does not hold). This is an assumption quite similar to item 2 (c) ii), and item 2 (d) ii). Roughly speaking, they all guarantee that the various operators have nonempty domains so that the problem makes sense. They stem from [START_REF] Rockafellar | Variational Analysis[END_REF]Theorem 12.43] to guarantee maximal monotonicity (see [19, section 3.2.1]). Such CQ may not be easy to check and may require some developments [START_REF] Brogliato | Existence, uniqueness of solutions and stability of nonsmooth multivalued Lur'e dynamical systems[END_REF]. Condition in item 2 (c) iii) and (f) i) (Im(D) ⊂ Im(C)) are used to compute Vladimirov's pseudo distance (see [START_REF] Vladimirov | Nonstationary dissipative evolution equations in a Hilbert space[END_REF] for the definition, which is beyond the scope of this article) between time-varying and/or state-dependent sets.

• property of the plant's model: item 2 (a) ii), (b) i), (c) i), (d) i), (e) i) ii) iv), item 3 i), • constraints qualifications: item 2 (a) i), (b) ii), (c) ii), iii), iv), v), (e) v), (f) ii), iii), iv), item 3 iii)

Multipliers boundedness (AC solutions)

Well-posedness analyses usually focus solely on the state. However, the boundedness of the multiplier is an important feature, since it may be used for feedback in this article. The condition in item 2 (b) ii) guarantees the existence of a solution to the LCP in (2 The uniqueness of λ can be deduced from the conditions in item 2 (a) and (b), where passivity is the main property, plus some constraints qualification.

State jumps

Let the solution x(•) of (2) be RCLBV. Then x(•) may undergo discontinuities and the LCS has to be interpreted in the Measure Differential Inclusion (MDI) formalism [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems[END_REF][START_REF] Fenel | BV solutions of nonconvex sweeping process differential inclusion with perturbation[END_REF][START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF]. In other words, the DI (3) has to be embedded into an MDI: Let the set of time instants at which the state undergoes a jump, be denoted as J x , and the set of time instants at which F u(•) is discontinuous be denoted as J F u . Then if conditions of item 2 (a) hold, J x ⊆ {0} ∪ J F u [START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF]. In other words, the state can jump only initially, or at times of discontinuity in F u(•) (or in the set S(t) in the FOSwP formalism). This means that state jumps can be chosen and dwell times can be imposed (contrarily to complementarity mechanical systems), by suitably choosing F u(t).

dx ∈ (Ax(t) + Eu(t))dt -B(D + N -1 I R m + ) -1 (Cx(t + ) + F u(t))
In the case of BV solutions, and under the conditions stated above, the term Bλ is a Dirac measure at times of state jumps, and we may denote λ = λ imp + λ reg . The input in (15) below, may be defined with the total λ ("impulsive" input), or just with its function part λ reg . In both cases, the stability analysis has to incorporate the jumps. At an instant of jump t, using (3) and [82, section 3.4], we have [82, section 3.4]:

x(t + ) -x(t -) ∈ -B(D + ∂σ S(t + ) ) -1 (Cx(t + )) (8)
with S(t + ) = IR m + -F u(t + ). State jump rules formulations are given in [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF]Lemma 2.3], compiling results in [START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF][START_REF] Camlibel | Passivity and Complementarity[END_REF][START_REF] Greenhalgh | On preserving dissipativity properties of linear complementarity dynamical systems with the theta-method[END_REF]. They are based on a maximum dissipation principle (a kind of plastic impact). Let us introduce such rules briefly. If (A, B, C, D) is passive with P = P ⊤ ≻ 0 and Lemma 3], and the state jump rule is given by:

if 0 ∈ K ∆ = {z ∈ IR n | Cz + F u(t + ) ∈ Q ⋆ D } (notice that the set K ̸ = ∅ by the condition in item 2 (b) ii)), Q D = {z ∈ IR m | 0 ≤ z ⊥ Dz ≥ 0}, then the jumps dissipate "energy", i.e., V (x(t + )) ≤ V (x(t -)) [50,
x(t + ) = argmin x∈K 1 2 (x -x(t -)) ⊤ P (x -x(t -)). (9)
Notice that Q D is a polyhedral set, and so is its dual cone [77, Lemma 6.45, Theorem 3

.52]. If D ≻ 0 then Q D = {0}, Q ⋆ D = IR m , K = IR n and using (9), x(t + ) = x(t -). If D = 0 then Q D = IR m + = Q ⋆ D , K = {z ∈ IR n | Cz + F u(t + ) ≥ 0}, hence 0 ∈ K if and only if F u(t) ≥ 0.
Thus strongly passive systems have continuous-time solutions, but strictly state passive systems may have state jumps. Notice that : x(t + ) -x(t -) = Bσ, with λ imp = σδ t , and 0 ≤ σ ⊥ Dσ ≥ 0. As shown in [START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF][START_REF] Greenhalgh | On preserving dissipativity properties of linear complementarity dynamical systems with the theta-method[END_REF] the state jump does not depend on a particular choice of the storage function matrix P . The meaning of the complementarity conditions at discontinuity times is explained in [START_REF] Camlibel | On linear passive complementarity systems[END_REF]Theorems 6.1,9.1], see also [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF]Lemma 2.3].

Zeno behavior

LCS undergo two classes of events, i.e., switching between modes of the LCP (2) (b) (AC solutions), and state discontinuities (BV solutions). Zenoness with AC solutions is tackled in [START_REF] Shen | Linear complementarity systems: Zeno states[END_REF][START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF][START_REF] Shen | Robust non-Zenoness of piecewise affine systems with applications to linear complementarity systems[END_REF] when F u(•) = 0. It states that if BSOL(D, Cx) is a singleton, then the LCS (2) is Zeno-free. This holds if D is a P-matrix. Thus junction and detachment times are separated (dwell time). The conditions stated in [START_REF] Camlibel | Passivity and Complementarity[END_REF]Theorem 7] also prevent Zeno behaviour in LCS with external inputs and state jumps, by imposing condition ii) in item 2 (b) between jumps.

Time-Varying LCS (TVLCS)

We may also encounter TVLCS of the form:

(a) ẋ(t) = A(t)x(t) + B(t)λ(t) + E(t)u(t) (almost everywhere) (b) 0 ≤ λ(t) ⊥ w(t) = C(t)x(t) + D(t)λ(t) + F (t)u(t) ≥ 0 (10)
when we deal with uncertainties which may be time-varying. The TVLCS is rewritten equivalently as:

ẋ(t) ∈ A(t)x(t) + E(t)u(t) -B(t)(D(t) + N -1 I R m + ) -1 (C(t)x(t) + F (t)u(t)) = A(t)x(t) + E(t)u(t) -B(t)(D(t) + N -1 I R m + -F (t)u(t) ) -1 (C(t)x(t)) (11)
Clearly, some of the results described above can be used to analyse [START_REF] Bourgeot | Tracking Control of Complementarity Lagrangian Systems[END_REF], in particular cases. One such case is as follows. Assume that A = A(t), E = E(t), F = F (t), D = 0, C and B are constant. Assume that there exists P = P ⊤ ≻ 0 such that P B = C ⊤ . Then [START_REF] Bourgeot | Tracking Control of Complementarity Lagrangian Systems[END_REF] is rewritten equivalently as:

ẋ(t) ∈ A(t)x(t) + E(t)u(t) -BN I R m + -F (t)u(t) (Cx(t)) (12)
After the classical state change 12) is rewritten equivalently as:

z = Rx, R 2 = P , R = R ⊤ ≻ 0 [23], (
ż(t) ∈ RA(t)R -1 z(t) + RE(t)u(t) -N I R m + -F (t)u(t) (z(t)), (13)
which is a FOSwP with affine perturbation. Then the material in item 3 section 2.1 can be used to analyse the well-posedness with both AC and RCLBV solutions. The general problem of well-posedness of ( 11) is not tackled in this work.

Trajectory Tracking in the Nominal Case

In this section, it is assumed that the plant model and parameters have no uncertainties. That is, the matrices A, B, C, D, E, F in (2) are known.

Controller Design

Let us consider the LCS with external input in [START_REF] Acary | An introduction to Siconos[END_REF]. Let us define the system which generates the desired trajectory to be tracked, as follows:

ẋd (t) = Ax d (t) + Bλ d (t) + Eu d (t) 0 ≤ λ d (t) ⊥ w d (t) = Cx d (t) + Dλ d (t) + F u d (t) ≥ 0 ( 14
)
where u d (•) is an input. Following the LCS of the desired system in [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF], the designer has to perform a preliminary analysis of the desired dynamics to determine a suitable trajectory for tracking. This analysis may involve numerical methods or the approaches presented in [START_REF] Sessa | A complementarity approach for the computation of periodic oscillations in piecewise linear systems[END_REF][START_REF] Heemels | Time-stepping methods for constructing periodic solutions in maximally monotone set-valued dynamical systems[END_REF]. The aim is to design a feedback controller such that the error dynamics with state vector e ∆ = x -x d possesses some stability property, to be defined later. Let us state the following that will be used later: Assumption 3.1. The solution x d : IR + → IR n of the LCS ( 14) is AC and uniformly bounded, and the multiplier vector

λ d (u d , x d ) is a bounded function of time.
Conditions such that this holds can be obtained from the results stated in section 2.

Assumption 3.2. The state x(•) and the multiplier λ(•) are available for measurement.

In practice, the multiplier may be a physical quantity (like voltages and currents in circuits, contact force in mechanics), and Assumption 3.2 is reasonable (and studying state observers in the loop is outside the scope of this article). Thus the feedback controller in the plant (2) is chosen generically as:

u(x, λ, t) = K[x -x d (t)] + G[λ -λ d (t)] + u d (t). ( 15
)
where the example in Appendix A.3 plays the role of a motivating example for introducing feedback from λ in the controller [START_REF] Brogliato | Nonsmooth Mechanics. Models, Dynamics, and Control. 3rd[END_REF]. Inserting ( 15) into (2) for some feedback gains K ∈ IR p×n and G ∈ IR p×m gives rise to the closed-loop LCS:

ẋ(t) = (A + EK)x(t) + (B + EG)λ(t) -EKx d (t) -EGλ d (t) + Eu d (t) (a.e.) 0 ≤ λ(t) ⊥ w(t) = (C + F K)x(t) + (D + F G)λ(t) -F Kx d (t) -F Gλ d (t) + F u d (t) ≥ 0 ( 16
)
It is important to note that in general both λ d (•) and u d (•) in ( 14) may be discontinuous timefunctions, hence ( 16) is in general an LCS as (2) with a potentially discontinuous term -F Kx d (t)-F Gλ d (t) + F u d (t) inside the complementarity constraints. As a consequence jumps in x(•) may occur unless conditions hold (see the end of section 3.2). We shall come back on state jumps in section 3.3. Rewriting ( 14) equivalently as

ẋd (t) = (A + EK)x d (t) + (B + EG)λ d (t) + Eu d (t) -EKx d (t) -EGλ d (t) 0 ≤ λ d (t) ⊥ w d (t) = (C + F K)x d (t) + Dλ d (t) + F u d (t) -F Kx d (t) ≥ 0 (17)
gives rise to the error dynamics:

ė(t) = (A + EK)e(t) + (B + EG)(λ(t) -λ d (t)) 0 ≤ w(t) w d (t) = C + F K -C -F K e(t) + D + F G -F G 0 D λ(t) λ d (t) + 0 C C + F K -F K x(t) x d (t) + F u d (t) F u d (t) ⊥ λ(t) λ d (t) ≥ 0 (18)
Clearly, the LCS in ( 18) is well-posed if both ( 14) and ( 16) are. However, the LCS in ( 18) cannot be used for well-posedness directly, because x(•) acts as an exogenous signal in complementarity constraints, whose properties have to be proved. The closed-loop' well-posedness has to be tackled with ( 16) and ( 17) (or ( 14)). In view of the structure of the closed-loop LCS in [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF], let us state the following assumptions, which will be used in the sequel for both well-posedness and stability purposes.

Assumption 3.3. There exist matrices K and G such that the plant's closed-loop quadruple

(A + EK, B + EG, C + F K, D + F G) is strictly state passive.
From Assumption 3.3, there exist gain matrices K and G such that the following nonlinear matrix inequality with unknowns P , K and G:

M ∆ = (A + EK) ⊤ P + P (A + EK) P (B + EG) -(C + F K) ⊤ (B + EG) ⊤ P -(C + F K) -(D + F G) -(D + F G) ⊤ ≼ -ϵP 0 0 0 , ( 19
)
has a solution P = P ⊤ ≻ 0. This may be replaced by the more stringent assumptions: Assumption 3.4. There exist matrices K and G such that the plant's closed-loop quadruple

(A + EK, B + EG, C + F K, D + F G) is strongly passive.
From Assumption 3.4, there exist gain matrices K and G such that the nonlinear matrix inequality:

M ≺ 0 ( 20
)
has a solution P = P ⊤ ≻ 0. For numerical purposes, it is important to note that the BMIs in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] and [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF] are transformed into LMIs, as detailed in A.2. Using Convex Analysis, the closed-loop system ( 16) is rewritten equivalently as the DI:

ẋ(t) ∈ (A+EK)x(t)-(B+EG)((D+F G)+∂σ S(t) ) -1 ((C+F K)x(t))-EKx d (t)-EGλ d (t)+Eu d (t). ( 21
) where S(t) = {v ∈ IR m + | v + F Kx d (t) + F Gλ d (t) -F u d (t) ≥ 0} is closed nonempty convex for each t, and σ S(t) (•) is the support function of S(t). The DI in (21) has the form ẋ(t) ∈ -M(t, x(t)) + f (t), (22)
with

f (t) = -EKx d (t)-EGλ d (t)+Eu d (t) and M(•, •
) is a set-valued operator. In view of ( 14) and Assumption 3.1, depending on the system's parameters, the multiplier λ d (•) may be discontinuous at some instants, hence the set S(t) may also be discontinuous at those instants (D ≻ 0 prevents such jumps). The LCS in ( 14) is also equivalently rewritten as the DI:

ẋd (t) ∈ Ax d (t) -B(D + ∂σ S d (t) ) -1 (Cx d (t)), (23)
with

S d (t) = {v ∈ IR m | v -F u d (t) ≥ 0}
, and the set S d (t) is continuous as long as u d (t) is. Therefore the DI (23) also fits with [START_REF] Brogliato | On the control of finite-dimensional mechanical systems with unilateral constraints[END_REF]. Let us now examine [START_REF] Brogliato | The Krasovskii-LaSalle invariance principle for a class of unilateral dynamical systems[END_REF]. Using that x(t) = e(t) + x d (t), the error dynamics may be rewritten as:

ė(t) ∈ Ãe(t) -B( D + N -1 S(t,e(t)) ) -1 ( Ce(t)) (24)
with S(t, e) = {z ∈

IR 2m | z + Ẽe + F (t) ≥ 0}, Ẽ = 0 C + F K , F (t) = Cx d (t) + F u d (t) Cx d (t) + F u d (t) , D = D + F G -F G 0 D , C = C + F K -C -F K , B = (B + EG -B -EG), Ã = A + EK.
The DI in [START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF] has a state and time-dependent polyhedral set S(•, •) which renders its study more complex.

For each t and e the normal cone N S(t,e) defines a maximal monotone mapping, which may allow to recast [START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF] in the framework of item 2 (f) in section 2, and deduce conditions such that (24) has a Lipschitz solution. A second equivalent way to write the LCS ( 18) is:

ė(t) ∈ Āe(t) -B( D + N -1 S(t) ) -1 ( Ce(t)) ⇔ ė(t) = Āe(t) + Bλ (t) 0 ≤ λ(t) ⊥ Ce(t) + Dλ (t) + F (t) ≥ 0 , ( 25
) with Ā = Ã, B = B, C = C + F K 0 , D = D, S(t) = {z ∈ IR 2m | z + F (t) ∈ IR 2m + }, F (t) = Cx d (t) + F u d (t) Cx d (t) + F u d (t)
. It is noteworthy that some conditions have to be imposed so that the construction of the DIs in ( 24) and ( 25) is possible, see section 3.3 for more details. Now we have at our disposal several (equivalent) formalisms for the closed-loop plant dynamics ( 16) and ( 21), the desired trajectory generator ( 14) and [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF], and the error dynamics ( 18) and ( 24) and ( 25). This is useful for the well-posedness analyses relying on the various results recalled in section 2. A difficulty is to determine under which conditions the passivity in Assumptions 3.3 or 3.4, implies the passivity of the quadruples ( Ã, B, C, D), or ( Ā, B, C, D), see [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF].

Remark 3.5 (Closed-loop system's well-posedness). In the framework of this article, the wellposedness of the plant dynamics is not a fundamental issue, since it is only the closed-loop system which is used in the analysis. In a similar way as feedback can be used to make an unstable system, stable in closed-loop, it may be used to render an ill-posed plant's model, well-posed in closed loop. The well-posedness of the closed-loop plant LCS [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF] (equivalently the DI in [START_REF] Brogliato | On tracking control of a class of complementary-slackness hybrid mechanical systems[END_REF]) can be inferred from [START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF][START_REF] Camlibel | Convergence of proximal solutions for evolution inclusions with time-dependent maximal monotone operators[END_REF][START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF][START_REF] Le | Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions[END_REF][START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF], assuming that the desired signals

x d (•), λ d (•) and u d (•) are AC or L 2 loc (⊆ L 1 loc )
, and that some basic constraint qualification conditions hold. Clearly in this setting Assumption 3.1 is important. When D + F G ≻ 0 (this is the case if Assumption 3.4 holds) then the results reported in item 1 in section 2.1 apply. If D + F G = 0 (which may be the case if Assumption 3.3 holds) then the results in item 3 in section 2.1 apply. When D + F G ≽ 0 but not null, then the various results in item 2 in section 2.1 can be used. This requires to check the constraint qualifications listed in item 2 (a) through (f). This is not tackled in this article whose primary goal is tracking control. Remark 3.6. The trajectory tracking problem as tackled in this work, can be interpreted as a synchronization problem between the master system [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF], and the plant [START_REF] Acary | An introduction to Siconos[END_REF].

Remark 3.7. The output regulation problem tackled in [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF] differs from the problem tackled in this work: First the desired systems are different: The desired system to be tracked is defined by the dynamics of the quadruple (A r , G r , H r , J r ) as mentioned in [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF] which is different from the real system defined. It is noticeable that the controller u in [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF] is introduced only in the ordinary differential equation (ODE) of the real system, but not in the variational inequality (VI). Besides, there is no presence of a desired controller u d in the desired system, which is autonomous. Second the objectives are different: In the presence of uncertainties, the real and the desired systems have different dynamics. Hence, in this case, the aim of the regulation problem in [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF] is to design a controller using internal model principle that achieves a zero steady-state regulation error: e ∆ = x -Πx r for some matrix Π. But, the aim of the tracking problem tackled in this work, in the presence of uncertainties, is to ultimately bound the tracking error: e ∆ = x -x d , where x d (•) is generated by desired dynamics. Third the controller in [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF] is state feedback, while state and multiplier feedback are considered in this article. The authors in [START_REF] Leine | Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints[END_REF] study tracking for Measure Differential Inclusions (MDIs), which differ from LCS investigated in this paper. They give sufficient conditions for the uniform convergence of MDIs with maximal monotonicity properties. In the tracking control of MDIs, the authors considered any desired trajectory x d of locally bounded variation, and control inputs with an impulsive part (which we avoid, see section 3.3 and condition (C1)). In our approach, the desired trajectory x d is designed independently by a LCS as shown in [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF]. Another difference with [START_REF] Leine | Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints[END_REF] is that the authors are interested in designing a controller such that the MDI of the closed-loop system is uniformly convergent with zero tracking error, while this paper is primarily concerned with studying the stability of the error dynamics without explicitly considering the convergence of the closed-loop system.

Error Dynamics Stability Analysis (no state jumps)

This section is dedicated to the stability analysis of the error dynamics [START_REF] Brogliato | The Krasovskii-LaSalle invariance principle for a class of unilateral dynamical systems[END_REF], and it is assumed that all trajectories are at least absolutely continuous. The arguments used for the proof of Proposition 3.8 are similar to those employed in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF][START_REF] Adly | Nonsmooth Lur'e dynamical systems in Hilbert space[END_REF][START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF], and are given for completeness. Proposition 3.8. Suppose that Assumptions 3.1, 3.2 and 3.3 hold, and that the solution x(•) of the closed-loop LCS ( 16) is AC on IR + . Then, the error dynamics in [START_REF] Brogliato | The Krasovskii-LaSalle invariance principle for a class of unilateral dynamical systems[END_REF] has a globally exponentially stable equilibrium point e * = 0.

Proof. Let ∆λ(t) ∆ = λ(t) -λ d (t) and ∆w(t) ∆ = w(t) -w d (t) = (C + F K)e(t) + (D + F G)∆λ(t).
Consider the Lyapunov function candidate V (e) = e ⊤ P e. From the assumptions e(•) is absolutely continuous and thus it has a derivative almost everywhere. Along the error dynamics trajectories [START_REF] Brogliato | The Krasovskii-LaSalle invariance principle for a class of unilateral dynamical systems[END_REF] it holds that:

V (t) = e ⊤ [(A + EK) ⊤ P + P (A + EK)]e + 2e ⊤ P (B + EG)∆λ(t)
From the complementarity conditions in ( 16) and ( 17) we obtain equivalently:

λ(t) ∈ -N S(t) ((C + F K)x(t) + (D + F G)λ(t)) and λ d (t) ∈ -N S(t) ((C + F K)x d (t) + (D + F G)λ d (t))
with S(t) defined after [START_REF] Brogliato | On tracking control of a class of complementary-slackness hybrid mechanical systems[END_REF]. It follows from the monotonicity of the normal cone mapping that:

[w(t) -w d (t)] ⊤ [λ(t) -λ d (t)] = ∆w(t) ⊤ ∆λ(t) ≤ 0. (26)
In matrix form (the time argument is dropped on the right-hand side):

V (t) = e ∆λ ⊤ M e ∆λ + e ∆λ ⊤ 0 (C + F K) ⊤ C + F K D + F G + (D + F G) ⊤ e ∆λ = e ∆λ ⊤ M e ∆λ + 2∆λ ⊤ ∆w ≤ e ∆λ ⊤ M e ∆λ ≤ 0, (27)
where (C + F K)e = ∆w -(D + F G)∆λ. So, for all e and ∆λ and using the strict state passivity, it is inferred: V (t) ≤ -ϵ e(t) ⊤ P e(t) = -ϵ V (t).

Classical arguments yield V (t) ≤ V (0) exp(-ϵt). Using the inequality: λ min (P )∥e∥ 2 ≤ e ⊤ P e, the following is obtained

∥e(t)∥ 2 ≤ V (t) λ min (P ) ≤ V (0) λ min (P ) exp(-ϵt)
Therefore, the equilibrium point of the error dynamics is globally exponentially stable.

Notice that using [START_REF] Brogliato | The Krasovskii-LaSalle invariance principle for a class of unilateral dynamical systems[END_REF] it follows that λ(t) -λ d (t) converges exponentially fast to Ker(B -EG).

Comments on passivity Following on from the proof of Proposition 3.8, the supply rate in [START_REF] Brogliato | On the equivalence between complementarity systems, projected systems and differential inclusions[END_REF] indicates that the passivity, in this case, is studied between the input λ -λ d and the output w -w d . The variation of the storage function in [START_REF] Camlibel | On linear passive complementarity systems[END_REF] is written as V (t) ≤ 2∆λ ⊤ ∆w. This shows the incremental passivity [START_REF] Pavlov | Incremental passivity and output regulation[END_REF][START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF] of the LCS defined in [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF].

Continuity of the solutions

1. Im(G ⊤ F ⊤ ) ⊆ Im(D + D ⊤ ), 2. D + F G + (D + F G) ⊤ ≽ F G(D + D ⊤ ) † G ⊤ F ⊤ . A sufficient condition for this to hold, is D + F G ≻ 0, and σ max (F G(D + D ⊤ ) † G ⊤ F ⊤ ) < σ max (D + F G + (D + F G) ⊤ ) = λ max (D+F G+(D+F G) ⊤ ).
From [8, Corollary 9.6.5] we have: [START_REF] Bernstein | Matrix Mathematics. Theory, Facts and Formulas[END_REF]Fact 6.3.28], and

σ max (F G(D+D ⊤ ) † G ⊤ F ⊤ ) ≤ σ 2 max (F G)σ max ((D + D ⊤ ) † ). If D + D ⊤ = 0 the matrix inequality is satisfied. If D + D ⊤ ̸ = 0, rank(D + D ⊤ ) = d, then σ max ((D + D ⊤ ) † ) = σ -1 d (D + D ⊤ )
σ d (D + D ⊤ ) = λ d (D + D ⊤ ) since D + D ⊤ ≽ 0 is symmetric. Thus a sufficient condi- tion is σ 2 max (F G) < λ d (D + D ⊤ )λ max (D + F G + (D + F G) ⊤ ). Another sufficient condition, following similar steps, is D +D ⊤ ≻ 0, σ max (F G) < λ d (D +F G+(D +F G) ⊤ )λ max (D +D ⊤ ).
We see that in both cases we allow for D + D ⊤ of D + F G + (D + F G) ⊤ to be low rank, which hampers D ≻ 0. Finally, we may use Lemmas A.3, A.5, as well as the matrix decompositions described after Lemma A.5, to get necessary and sufficient conditions.

(b) Im( D) ⊆ Im( C): ⇔ Im(D + F G) + Im(F G) ⊆ Im(C + F K) and Im(D) ⊆ Im(C + F K). (c) Ker( D + D⊤ ) ⊆ Ker(P B -C⊤ ) for some IR n×n ∋ P = P ⊤ ≻ 0: ⇔ {(x, y) ∈ IR m×m | ((D + F G) + (D + F G) ⊤ )x -F Gy = 0, -G ⊤ F ⊤ x + (D + D ⊤ )y = 0} ⊆ {(x, y) ∈ IR m×m | x -y ∈ Ker(P (B + EG) -(C + F K) ⊤ )}. This is verified if F G = 0
F G(D+D ⊤ ) -1 G ⊤ F ⊤ ̸ = D+F G+(D+F G) ⊤ , then Ker( D+ D⊤ ) = {0}, hence the inclusion holds. If F G(D +D ⊤ ) -1 G ⊤ F ⊤ = D +F G+(D +F G) ⊤ , then Ker( D + D⊤ ) = IR m ×IR m , hence the inclusion holds only if P (B + EG) = (C + F K) ⊤ .
The various conditions stated in section 2 are sufficient only, so it may be that some of them are unnecessary in some cases (e.g., condition b) above is needed for continuity arguments using the Vladimirov's pseudo-distance defined in [START_REF] Vladimirov | Nonstationary dissipative evolution equations in a Hilbert space[END_REF] as done in [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF][START_REF] Le | Well-posedness and nonsmooth Lyapunov pairs for state-dependent maximal monotone differential inclusions[END_REF][START_REF] Le | On a class of Lur'e dynamical systems with state-dependent set-valued feedback[END_REF]], but it is not necessary at all [23, section 5.3] [28]). Condition b) is not necessary for the well-posedness of (18) when both D and D + F G are P-matrices. The point is also that the error system [START_REF] Brogliato | The Krasovskii-LaSalle invariance principle for a class of unilateral dynamical systems[END_REF] or [START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF], is a specific interconnection of both subsystems, which does not necessarily inherits good properties of the subsystems (e.g., D ≽ 0 and D + F G ≽ 0 may not imply D ≽ 0). Conditions in a) and c) are necessary for the passivity of ( Ã, B, C, D) [START_REF] Camlibel | Passivity and Complementarity[END_REF] [25, section 3.8]. The stability proof shows that under the proposition's assumptions (like the continuity of the solutions) the generalized equation 0 ∈ Āe * -B( D + N -1 S(t) ) -1 ( Ce * ) has the unique solution e * = 0 for all t ≥ 0.

Error Dynamics Stability Analysis including State Jumps

An important question is whether or not the tracking control framework developed above is suitable for trajectories with discontinuities. Let us study how to relax Assumption 3.1 and the continuity of the closed-loop state x(•). State jumps are known to add difficulty to the trajectory tracking problem, especially when the jump times are unknown. The so-called peaking phenomenon, due to non-synchronized jumps in the plant and the desired trajectories, has long been known to be one of the obstacles. The jumps in x(•) and x d (•) may arise from different reasons:

1. Let the conditions in section 2.1 item 2 (a) or item [START_REF] Adly | Well-posedness of nonsmooth Lurie dynamical systems involving maximal monotone operators[END_REF]

T x ⊆ {t 0 } ∪ T F u d ∪ T F Gλ d . Proof. (a) is obvious. (b): T x ⊆ {t 0 } ∪ T -F Kx d -F Gλ d +F u d ⊆ {t 0 } ∪ T F Kx d ∪ T F Gλ d ∪ T F u d ⊆ {t 0 } ∪ T F Gλ d ∪ T F u d since T F Kx d ⊆ T F u d .
Lemma 3.9 indicates that when F Gλ d is continuous, then either x d (•) jumps while x(•) is continuous, or the inverse, or they jump simultaneously. This will be illustrated on examples. Notice that when x d (•) jumps, then λ d is a Dirac measure, and the meaning of the feedback in [START_REF] Brogliato | Nonsmooth Mechanics. Models, Dynamics, and Control. 3rd[END_REF] has to be carefully studied, as well as the mere meaning of the complementarity constraints in [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF]

+ F u d ) i = 0 for some 1 ≤ i ≤ m).
Conditions in item 2 are not straightforward, because we want that F G ̸ = 0, but F G also multiplies λ, see [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF], and passivity implies D + F G ≽ 0. The scalar case m = 1 is analysed in section 6. The first step is to characterize the jumps in [START_REF] Brogliato | The Krasovskii-LaSalle invariance principle for a class of unilateral dynamical systems[END_REF], then to study the variation ∆V (e(t)) = V (e(t + )) -V (e(t -)) = (e(t + ) -e(t -)) ⊤ P (e(t + ) -e(t -)).

Case G = 0 and D = 0

In this case λ d does not enter [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF], and D = 0. Let Assumptions 3.1 and 3.3 hold, and let us study the case when x(•) jumps while x d (•) is continuous. Then (A + EK, B, C + F K, 0) is strictly state passive, and P B = (C + F K) ⊤ for some P = P ⊤ ≻ 0. The closed-loop system in (21) becomes:

ẋ(t) ∈ (A + EK)x(t) -BN S(t) ((C + F K)x(t)) -EKx d (t) + Eu d (t). ( 28
)
where

S(t) = {v ∈ IR m + | v -F Kx d (t) + F u d (t) ≥ 0}
. This DI can be rewritten equivalently as a FOSwP, see item 3 in section 2 and [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF][START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF][START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF], as:

ζ(t) ∈ R(A + EK)R -1 ζ(t) + RE(-Kx d (t) + u d (t)) -N Φ(t) (ζ(t))
where

R 2 = P , R = R ⊤ ≻ 0, ζ = Rx, Φ(t) = {Rx | (C +F K)x-F Kx d (t)+F u d (t) ∈ S(t)}. We see that Φ(t) ̸ = ∅ for each t if and only if there exists x such that (C + F K)x -F Kx d (t) + F u d (t) ≥ 0.
This is guaranteed if a condition as in item 3 iii) holds, i.e., Im(C + F K) -IR m + = IR m (which is a constraint qualification). Assume in addition that u d (•) is continuous on IR + . Suppose that Cx d (0) + F u d (0) ≥ 0, hence x d (•) is continuous at t = 0. Then a jump in x(•) (thus in e(•)) can occur only at the initial time and V (0 + ) -V (0 -) = e(0 + ) ⊤ P e(0 + ) -e(0 -) ⊤ P e(0 -) = (e(0 + ) + e(0 -)) ⊤ P (e(0 + ) -e(0 -)) = (x(0 + ) + x(0 -) -2x d (0)) ⊤ P (x(0 + ) -x(0 -)) = x(0 + ) ⊤ P x(0 + )x(0 -) ⊤ P x(0 -)-2x d (0)) ⊤ P (x(0 + )-x(0 -)). Now using the passivity of (A+EK, B, C +F K, 0), we have (see the paragraph State Jumps at the end of section 2): x(0

+ ) ⊤ P x(0 + )-x(0 -) ⊤ P x(0 -) ≤ 0, provided that 0 ∈ {z ∈ IR n | (C + F K)z -F Kx d (0) + F u d (0) ≥ 0}, equivalently 0 ∈ Φ(0), equivalently -F Kx d (0) + F u d (0) ≥ 0. In this case, V (0 + ) -V (0 -) ≤ -2x d (0)) ⊤ P (x(0 + ) -x(0 -)). We have P (x(0 + ) -x(0 -)) ∈ -N K (x(0 + )) [23, Lemma 2.3]. Thus it is necessary and sufficient that x d (0) ∈ (N K (x(0 + ))) • = T K (x(0 + ))
to guarantee that the right-hand side is nonpositive. Therefore, the following has been proved:

Lemma 3.10. Assume that u d (•) is time-continuous, G = 0, D = 0, -F Kx d (0)+F u d (0) ≥ 0, and x d (0) ∈ T K (proj P [K; x(t - 0 )]).
Then at an initial state jump, we have V (0 + ) -V (0 -) ≤ 0, where P = P ⊤ ≻ 0 is a solution of the passivity LMI associated with the triple

(A + EK, B, C + F K) and V (t) = V (e(t)).
The condition on x d (0) is certainly not easy to check in general (but x d (0) = 0 is always suitable). The interest of Lemma 3.10 is that it allows for a jump in x(0) while x d (0) does not jump. But applying it at any time of state jump t k > 0 implies to impose a suitable desired state, which may not be possible in our framework where x d is generated by [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF]. Under the same conditions, assume that F u d (•) has a discontinuity at time t c . From ( 14) and ( 16), both x(•) and x d (•) may jump at t c , so λ and λ d are Dirac measures at t c . From ( 16) the discontinuity may act in both terms F Kx d and F u d . Using ( 21) and [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF], this implies that the jumps' magnitudes in the sets S(t) and S d (t), may not be equal. The post-jump states are computed using [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF], where

K = {z ∈ IR n | (C + F K)z -F Kx d (t) + F u d (t) ≥ 0}
for the closed-loop plant, and

K d = {z ∈ IR n | Cz + F u d (t) ≥ 0} for the desired system.

Extension for further jumps at t > 0

Let us now place ourselves in another perspective. Until now we have assumed that x d (•) is timecontinuous. First notice that condition (C1) can be relaxed, if we admit that the state x d (•) of the dynamics in ( 14) can be arbitrarily reset to some value at arbitrary times, without considering λ d as a Dirac measure. This has important consequences, because as we shall see this means that the desired dynamics is no longer autonomous (the desired state has to be modified online, a common feature in trajectory tracking for systems undergoing state jumps [START_REF] Morarescu | Trajectory tracking control of multiconstraint complementarity Lagrangian systems[END_REF][START_REF] Morarescu | Passivity-based switching control of flexible-joint complementarity mechanical systems[END_REF][START_REF] Rijnen | Reference spreading: Tracking performance for impact trajectories of a 1DoF setup[END_REF]). Consider [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF], with Assumption 3.3. Proposition 3.11. Let V (e) = e ⊤ P e, with P a solution of the closed-loop passivity LMI, and

let {0} ∈ K = {z ∈ IR n | (C + F K)z -F Kx d -F Gλ d (t + ) + F u d (t) ∈ Q * D+F G }. Assume that x d (t + ) ∈ T K (proj P [K; x(t -)]) and x d (t + ) ⊤ P x(t -) ≥ x d (t -) ⊤ P x(t -), then ∆V (e(t)) ≤ (x d (t + ) - x d (t -)) ⊤ P (x d (t + ) + x d (t -)).
Proof. At a state jump time:

∆V (e(t)) = (x(t + ) -x(t -)) ⊤ P (x(t + ) + x(t -)) -(x(t + ) -x(t -)) ⊤ P (x d (t + ) + x d (t -)) -(x d (t + ) -x d (t -)) ⊤ P (x(t + ) + x(t -)) + (x d (t + ) -x d (t -)) ⊤ P (x d (t + ) + x d (t -)) ≤ -(x(t + ) -x(t -)) ⊤ P (x d (t + ) + x d (t -)) -(x d (t + ) -x d (t -)) ⊤ P (x(t + ) + x(t -)) +(x d (t + ) -x d (t -)) ⊤ P (x d (t + ) + x d (t -)) = -(x d (t + ) + x d (t -)) ⊤ P (x(t + ) -x(t -)) -(x d (t + ) -x d (t -)) ⊤ P (x(t + ) -x(t -)) -2(x d (t + ) -x d (t -)) ⊤ P x(t -) + (x d (t + ) -x d (t -)) ⊤ P (x d (t + ) + x d (t -)) = -2x d (t + ) ⊤ P (x(t + ) -x(t -)) -2(x d (t + ) -x d (t -)) ⊤ P x(t -) +(x d (t + ) -x d (t -)) ⊤ P (x d (t + ) + x d (t -)) (29)
The first term in the first equality in ( 29) is nonpositive from the passivity. It is noteworthy that we cannot infer the same conclusions about the last term in ( 29) from ( 17) because these dynamics are equivalent to that in [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF]. Let us consider the last equality in [START_REF] Camlibel | Passivity and Complementarity[END_REF]. We know that [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF] the first condition follows. The second condition is obvious.

P (x(t + ) -x(t -)) ∈ -N K (x(t + )) [23, Lemma 2.3]. Thus the nonpositivity of the first term is equivalent to x d (t + ) ∈ (N K (x(t + ))) • = T K (x(t + )). Using
Thus under the conditions of Proposition 3.11, ∆V (e(t)) ≤ 0 if and only if x d (t + ) ⊤ P x d (t + ) ≤ x d (t -) ⊤ P x d (t -), which means that the desired state jump is dissipative with respect to the closedloop storage function (in general there is no reason that it should satisfy this property). This makes a set of constraints that the reset desired state x d (t + ) has to satisfy. It is noteworthy that the conditions of Proposition 3.11 are sufficient only. Examples show that they may not be satisfied, while ∆V (e(t)) ≤ 0, see section 6.4. Also, we note that the reset mechanism is not needed when G = 0 (then condition C1) holds true), while the characterization of ∆V (e(t)) as in (29) remains valid.

Using the DI (25)

The formalism in [START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF] is not convenient for the state jumps analysis, because it involves a statedependent set. Let us recall the error dynamics in [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF] to analyze the state discontinuities.

ė(t) ∈ Āe(t) -B( D + N -1 S(t) ) -1 ( Ce(t)) ⇔ ė(t) = Āe(t) + Bλ(t) 0 ≤ λ(t) ⊥ Ce(t) + Dλ(t) + F (t) ≥ 0 , with Ā = Ã, B = B, C = C + F K 0 , D = D, S(t) = {z ∈ IR 2m | z + F (t) ∈ IR 2m + }, F (t) = Cx d (t) + F u d (t) Cx d (t) + F u d (t)
. At a state jump time t, the DI in (25) may be written as:

e(t + ) -e(t -) ∈ -B( D + N -1 S(t + ) ) -1 ( Ce(t + )).
which has a solution (possibly with jumps) if the quadruple ( Ā, B, C, D) is passive with positive definite storage function. Then ( 9) holds. Thus it is inferred that ∆V (e(t)) ≤ 0 provided that 0

∈ K = {e ∈ IR n | Ce + F (t) ∈ Q ⋆ D }, and:   -P (A + EK) -(A + EK) ⊤ P -P (B + EG) + (C + F K) ⊤ P (B + EG) -(B + EG) ⊤ P + C + F K D + F G + (D + F G) ⊤ F G (B + EG) ⊤ P (F G) ⊤ D + D ⊤   ≽ 0, (30)
where P = P ⊤ ≻ 0 is a solution to -M ≽ 0 with M in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF]. Notice that [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF] holds only if M ⪯ 0, hence only if P is also a solution of the plant's closed-loop LMI. Thus, the solution P of ( 30) must be the solution of NLMI in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF]. From Assumption 3.3 we have -M ≽ 0. Using Lemma A.3, we infer that (30) holds if and only if:

1. D + D⊤ ≽ 0 (see item a) at the end of section 3.2 for sufficient conditions),

Im -(B + EG)

⊤ P + C + F K (B + EG) ⊤ P ⊆ Im( D + D⊤ ),
3.

-M ≽ -P (B + EG)

+ (C + F K) ⊤ P (B + EG) ( D + D⊤ ) † -(B + EG) ⊤ P + C + F K (B + EG) ⊤ P (31)
Some comments arise:

• A sufficient condition for (31) to hold is D = -D⊤ ⇐⇒ D + F G = -(D + F G) ⊤ , D = -D ⊤ , F G = 0
. This is allowed by strict state passivity in Assumption 3.3. However item 2 then implies that -(B + EG) ⊤ P + C + F K = 0 and (B + EG) ⊤ P = 0, hence C + F K = 0 and B + EG = 0 since P is full-rank. Thus in ( 16) the ODE part and the complementarity part are decoupled.

• Assume that D = D 1 0 0 0 , D 1 ≻ 0. Item 1 implies that F G = (F G) 1 0 0 0 . Then D + D⊤ = D1 + D⊤ 1 0 0 0 , and ( D + D⊤ ) † = ( D1 + D⊤ 1 ) † 0 0 0 , with D1 + D⊤ 1 =   D 1 + (F G) 1 + D 1 + (F G) ⊤ 1 0 -(F G) 1 0 0 0 -(F G) ⊤ 1 0 D 1 + D ⊤ 1 
 . Similar calculations can be done for M in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF], and also for the ranges inclusion in item 2. Pursuing the calculations and matrices partitions allows to simplify [START_REF] Camlibel | Linear passive systems and maximal monotone mappings[END_REF]. 16) are meaningless at t c . This means that applying an impulsive feedback control to an LCS (2) with F Gu ̸ = 0, has to be avoided as pointed out in condition (C1). In fact, the DIs in ( 24), ( 25), [START_REF] Brogliato | On tracking control of a class of complementary-slackness hybrid mechanical systems[END_REF], and ( 22) are constructed with the underlying assumption that the complementarity conditions can be rewritten equivalently as inclusion into a normal cone (see, e.g., [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF]Equ. (B.1)] [76, Corollary 23.5.4]). For instance, f (t) in ( 22) is a Dirac measure if EGλ d is. This might let one think that in this case, x(•) jumps at t c . We set that this is true if F = 0 only, in which case x d (•) can jump only initially, see [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF]. Obviously, if λ d is a Dirac measure, both formalisms (complementarity and inclusion) require further analysis for their understanding if F G ̸ = 0.

• The conditions in this paragraph are different from those of the foregoing paragraph, because now we consider both the desired system and the closed-loop plant simultaneously, instead of looking at (28) only. However, jumps in both x(•) and x d (•) are permitted under some conditions as stated above. A quick examination of the LMI in [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF] shows that the occurrence of state jumps is quite restricted in this context. Indeed, if D + D ⊤ = 0, then [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF] implies that B + EG = 0 and F G = 0. In fact, the LMI in [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF] shows that studying state jumps from the passivity error dynamics in [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF] is almost impossible.

• The controller may be impulsive if G ̸ = 0 and a jump occurs in x(•). Condition (C1) still applies.

Analysis of jump sets K and K d

These sets are crucial in the state jump characterization and computation, see [START_REF] Bernstein | Matrix Mathematics. Theory, Facts and Formulas[END_REF]. We have:

K = {z ∈ IR n | Cz + F u d (t + ) + F K(z -x d (t + )) ∈ Q ⋆ D+F G } (32)
and

K d = {z ∈ IR n | Cz + F u d (t + ) ∈ Q ⋆ D } (33)
In general both sets are different. However if F G = 0 then they differ only by the term F K(zx d (t + )). Since x(t + ) belongs to K, both sets are almost equal if the tracking error x(t -) -x d (t -) and F K are very small. So if in addition both storage functions matrices P (calculated from the closed-loop system LMI) and P d (computed from the desired dynamics LMI when this dynamics is passive) are closed one to each other, both state jumps are almost the same as well. On the other hand, the state jump as computed in ( 8) is independent on P (or P d , respectively), provided it is a solution of the passivity LMI. Thus, if EG = 0, which implies that the "input" matrix of the closed-loop system is equal to the input matrix of the desired system, the forms of K and K d in ( 32) and (33) will imply that both jumps in x and x d are close one to each other.

The jump-mismatch (peaking) phenomenon

This is a well-known phenomenon in trajectory tracking when state jumps are present, as recalled in the introduction. The problem that is faced in the following is twofold: 1) is the mismatch issue present? If it is, how can it be coped with? It will appear that if the plant's parameters are known, the peaking is not present in our systems, because passivity constraints prevent it from occurring: did we prove this really? passivity forces the jumps (in either x or x d or both) to occur in such a way that the Lyapunov function of the error system decreases. Things are different when parameter uncertainty is considered. In the scalar case treated in section 6, it will be shown that if both states jump then they jump at the same time. However, it is also possible that x jumps while x d does not, and vice versa.

Robustness Analysis: Parametric Uncertainties

It is of interest to analyse the tracking problem when the plant's dynamics have uncertainties.

In this case, the desired dynamics' matrices in [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF], and the plant's model matrices, differ. The desired dynamics [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF] has to be designed using a nominal plant model.

Controller Design

The controller

u(t) = K 0 [x(t)-x d (t)]+G 0 [λ(t)-λ d (t)]+u d (t)
is designed from the plant's nominal quadruple (A 0 , B 0 , C 0 , D 0 ), along the same procedure as in the foregoing section. Therefore the desired system ( 14) is represented by the following LCS:

ẋd (t) = A 0 x d (t) + B 0 λ d (t) + E 0 u d (t) 0 ≤ λ d (t) ⊥ w d (t) = C 0 x d (t) + D 0 λ d (t) + F 0 u d (t) ≥ 0. ( 34
)
If strong passivity is used (Assumption 3.4) instead of strict state passivity as in the foregoing section (Assumption 3.3), the controller gains are computed assuming there exist matrices K 0 and G 0 such that the inequality ( 20) is satisfied for the nominal plant, i.e.,

M 0 ∆ = (A 0 + E 0 K 0 ) ⊤ P 0 + P 0 (A 0 + E 0 K 0 ) P 0 (B 0 + E 0 G 0 ) -(C 0 + F 0 K 0 ) ⊤ (B 0 + E 0 G 0 ) ⊤ P 0 -(C 0 + F 0 K 0 ) -D 0 -F 0 G 0 -(D 0 + F 0 G 0 ) ⊤ ≺ 0 ( 35
)
has a solution

P 0 = P ⊤ 0 ≻ 0. The plant dynamics is represented as follows ẋ(t) = (A 0 + ∆A)x(t) + (B 0 + ∆B)λ(t) + (E 0 + ∆E)u(t) 0 ≤ λ(t) ⊥ w(t) = (C 0 + ∆C)x(t) + (D 0 + ∆D)λ(t) + (F 0 + ∆F )u(t) ≥ 0 ( 36
)
where ∆A = A -A 0 , ∆B, ∆C, ∆D, ∆E and ∆F represent additive uncertainties. Sufficient conditions on the uncertainties upperbounds are calculated in the next section so that some stability is guaranteed. The closed-loop system is given by:

         ẋ(t) = (A 0 + ∆A + (E 0 + ∆E)K 0 )x(t) + (B 0 + ∆B + (E 0 + ∆E)G 0 )λ(t) -(E 0 + ∆E)(K 0 x d (t) + G 0 λ d (t) -u d (t)) 0 ≤ λ(t) ⊥ w(t) = (C 0 + ∆C + (F 0 + ∆F )K 0 )x(t) + (D 0 + ∆D + (F 0 + ∆F )G 0 )λ(t) -(F 0 + ∆F )(K 0 x d (t) + G 0 λ d (t) -u d (t)) ≥ 0 (37)
It is noteworthy that the well-posedness of (37) may not be guaranteed for any uncertainties. Some of the results in section 2 can be used. The following assumption is supposed to hold in this section:

Assumption 4.1. The closed-loop system ( 37) is well-posed, i.e., it has unique AC solutions for any initial condition x(t 0 ) = x 0 satisfying w(t 0 , x 0 ) ≥ 0, equivalently: [START_REF] Colombo | Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction[END_REF]. As pointed out in section 2.2, it is only in few particular cases that the well-posedness of time-varying LCS has been studied. In this section stability relies on the fact that the closed-loop nominal system is strongly passive, and that the closed-loop plant feedthrough matrix (D 0 +∆D+(F 0 +∆F )G 0 ) ≻ 0. Thus item 1 in 2 applies if this matrix is constant.

x 0 ∈ Dom (D 0 + ∆D + (F 0 + ∆F )G 0 + ∂σ S(t) ) -1 = Im (D 0 + ∆D + (F 0 + ∆F )G 0 + ∂σ S(t) ) , with S(t) = {ν ∈ IR m | ν + (F 0 + ∆F )(K 0 x d + G 0 λ d -u d ) ≥ 0}.

Remark 4.2. It is clear that time-varying uncertainties give rise to a time-varying LCS in

Error Dynamics Stability Analysis

The error dynamics is given by:

           ė(t) = (A 0 + E 0 K 0 )e(t) + (B 0 + E 0 G 0 )∆λ(t) + ∆Ax(t) + ∆Bλ(t) + ∆EK 0 e(t) + ∆EG 0 ∆λ(t) +∆Eu d (t) ∆w(t) = w(t) -w d (t) = (C 0 + F 0 K 0 )e(t) + (D 0 + F 0 G 0 )∆λ(t) + ∆Cx(t) + ∆Dλ(t) + ∆F K 0 e(t) +∆F G 0 ∆λ(t) + ∆F u d (t) 0 ≤ w(t) ⊥ λ(t) ≥ 0 and 0 ≤ w d (t) ⊥ λ d (t) ≥ 0 (38) with ∆λ(t) = λ(t) -λ d (t). Let: p(x, t, λ) ∆ = ∆Ax(t) + ∆Bλ(t) + ∆EK 0 e(t) + ∆EG 0 ∆λ(t) + ∆Eu d (t) (39)
and q(x, t, λ)

∆ = ∆Cx(t) + ∆Dλ(t) + ∆F K 0 e(t) + ∆F G 0 ∆λ(t) + ∆F u d (t). (40)
Let us now state the stability result.

Proposition 4.3. Let Assumptions 3.1, 3.2, 3.4 hold for the nominal system. Let

∆A ⊤ Λ -1 A ∆A ≼ I n , ∆B ⊤ Λ -1 B ∆B ≼ I m , ∆C ⊤ Λ -1 C ∆C ≼ I n ∆D ⊤ Λ -1 D ∆D ≼ I m , ∆E ⊤ Λ -1 E ∆E ≼ I p , ∆F Λ -1 F ∆F ⊤ ≼ I m ( 41
)
hold for any

Λ k = Λ ⊤ k ≻ 0, k ∈ {A, B, C, D, E, F },
and assume there exist P 0 , K 0 and G 0 such that the matrix inequality

              -(M 0 ) 11 -(M 0 ) 12 P 0 P 0 K ⊤ 0 I n 0 0 0 -(M 0 ) 12 -(M 0 ) 22 0 0 0 0 G ⊤ 0 I m I m P 0 0 Λ -1 1 0 0 0 0 0 0 P 0 0 0 Λ-1 1 0 0 0 0 0 K 0 0 0 0 Λ-1 F 0 0 0 0 I n 0 0 0 0 1 2 I n 0 0 0 0 G 0 0 0 0 0 Λ-1 F 0 0 0 I m 0 0 0 0 0 Λ -1 2 0 0 I m 0 0 0 0 0 0 Λ-1 2               ≻ 0 ( 42
)
and

P 0 = P ⊤ 0 ≻ 0, with ΛF ∆ = I p + Λ F , Λ1 ∆ = Λ A + 2Λ E + Λ B , Λ2 ∆ = Λ C + Λ D + 4I m .
Then, the solution of the error dynamics in [START_REF] Colombo | Optimal control of sweeping processes in robotics and traffic flow models[END_REF] is globally uniformly ultimately bounded. Note that (M 0 ) ij denotes the element of the matrix M 0 in (35) located at the ith row and jth column where i, j ∈ {1, 2}.

Proof. The derivative of the Lyapunov function candidate V (t) = e ⊤ P 0 e along the closed-loop trajectories is calculated as follows:

V = e ⊤ (A 0 + E 0 K 0 ) ⊤ P 0 + P 0 (A 0 + E 0 K 0 ) e + 2e ⊤ P 0 (B 0 + E 0 G 0 )∆λ + 2e ⊤ P 0 p(x, t, λ) In matrix form V (t) = e ∆λ ⊤ (A 0 + E 0 K 0 ) ⊤ P 0 + P 0 (A 0 + E 0 K 0 ) P 0 (B 0 + E 0 G 0 ) (B 0 + E 0 G 0 ) ⊤ P 0 0 e ∆λ +2e ⊤ P 0 p(x, t, λ) ± e ∆λ ⊤ 0 (C 0 + F 0 K 0 ) ⊤ C 0 + F 0 K 0 D 0 + F 0 G 0 + (D 0 + F 0 G 0 ) ⊤ e ∆λ
The purpose of the ± term is to obtain the matrix in [START_REF] Chen | Perturbation bounds of P-matrix linear complementarity problems[END_REF] and

e ∆λ ⊤ 0 (C 0 + F 0 K 0 ) ⊤ C 0 + F 0 K 0 D 0 + F 0 G 0 + (D 0 + F 0 G 0 ) ⊤ e ∆λ = 2∆λ ⊤ [(C 0 + F 0 K 0 )e + (D 0 + F 0 G 0 )∆λ] = 2∆λ ⊤ (∆w -q(x, t, λ))
Then,

V (t) = e ∆λ ⊤ M 0 e ∆λ + 2e ⊤ P 0 p(x, t, λ) + 2∆λ ⊤ (∆w -q(x, λ)).
Let us substitute the values of p and q and write explicitly the following

2e ⊤ P 0 p(x, t, λ) -2∆λ T q(x, λ) = 2e ⊤ P 0 [∆Ax + ∆Bλ + ∆EK 0 e + ∆EG 0 ∆λ + ∆Eu d ±∆Ax d ± ∆Bλ d )] -2∆λ T [∆Cx + ∆Dλ + ∆F K 0 e + ∆F G 0 ∆λ +∆F u d ± ∆Cx d ± ∆Dλ d ].
Hence,

2e ⊤ P 0 p(x, t, λ) -2∆λ ⊤ q(x, λ) = e ∆λ ⊤ (∆A + ∆EK 0 ) ⊤ P 0 + P 0 (∆A + ∆EK 0 ) P 0 (∆B + ∆EG 0 ) -(∆C + ∆F K 0 ) ⊤ (∆B + ∆EG 0 ) ⊤ P 0 -(∆C + ∆F K 0 ) -∆D -∆F G 0 -(∆D + ∆F G 0 ) ⊤ e ∆λ +2e ⊤ P 0 (∆Ax d + ∆Bλ d + ∆Eu d ) -2∆λ ⊤ (∆Cx d + ∆Dλ d + ∆F u d ) Let ∆M 0 ∆ = -(∆A + ∆EK 0 ) ⊤ P 0 -P 0 (∆A + ∆EK 0 ) -P 0 (∆B + ∆EG 0 ) + (∆C + ∆F K 0 ) ⊤ -(∆B + ∆EG 0 ) ⊤ P 0 + (∆C + ∆F K 0 ) ∆D + ∆F G 0 + (∆D + ∆F G 0 ) ⊤ Thus, V ≤ - e ∆λ ⊤ (-M 0 +∆M 0 ) e ∆λ +2e ⊤ P 0 (∆Ax d +∆Bλ d +∆Eu d )-2∆λ ⊤ (∆Cx d +∆Dλ d +∆F u d ) Let a(t) ∆ = ∆Ax d + ∆Bλ d + ∆Eu d and b(t) ∆ = ∆Cx d + ∆Dλ d + ∆F u d . For any Λ ⊤ 1 = Λ 1 ≻ 0 and Λ ⊤ 2 = Λ 2 ≻ 0, it holds that: 2e ⊤ P 0 a(t) ≤ e ⊤ P 0 Λ 1 P 0 e + a ⊤ (t)Λ -1 1 a(t) 2∆λ ⊤ b(t) ≤ ∆λ ⊤ Λ 2 ∆λ + b ⊤ (t)Λ -1 2 b(t) So, V ≤ -z ⊤ -M 0 + ∆M 0 - P 0 Λ 1 P 0 0 0 Λ 2 z + a ⊤ (t)Λ -1 1 a(t) + b ⊤ (t)Λ -1 2 b(t) ≤ -λ min -M 0 + ∆M 0 - P 0 Λ 1 P 0 0 0 Λ 2 ∥z∥ 2 + λ -1 min (Λ 1 )∥a(t)∥ 2 + λ -1 min (Λ 2 )∥b(t)∥ 2
where z(t)

∆ = e ∆λ .
Let us prove that if the conditions, ( 41) and ( 42) hold, then the matrix

-M 0 + ∆M 0 - P 0 Λ 1 P 0 0 0 Λ 2
is positive definite. For this purpose, the upper-bounding of ∆M 0 can be done term by term as:

e ∆λ ⊤ ∆M 0 e ∆λ = -2e ⊤ P 0 (∆A + ∆EK 0 )e -2e ⊤ P 0 (∆B + ∆EG 0 )∆λ + 2e ⊤ (∆C + ∆F K 0 ) ⊤ ∆λ + 2∆λ ⊤ (∆D + ∆F G 0 )∆λ Thus, for any Λ k = Λ ⊤ k ≻ 0, k ∈ {A, B, C, D, E, F } 2e ⊤ P 0 (∆A + ∆EK 0 )e ≥ -e ⊤ P 0 (Λ A + Λ E )P 0 + ∆A ⊤ Λ -1 A ∆A + K ⊤ 0 ∆E ⊤ Λ -1 E ∆EK 0 e 2∆λ ⊤ (∆B+∆EG 0 ) ⊤ P 0 e ≥ -e ⊤ [P 0 (Λ B + Λ E )P 0 ] e-∆λ ⊤ ∆B ⊤ Λ -1 B ∆B + G ⊤ 0 ∆E ⊤ Λ -1 E ∆EG 0 ∆λ 2∆λ ⊤ (∆C + ∆F K 0 )e ≥ -∆λ ⊤ Λ C + ∆F Λ -1 F ∆F ⊤ ∆λ -e ⊤ ∆C ⊤ Λ -1 C ∆C + K ⊤ 0 Λ F K 0 e 2∆λ ⊤ (∆D + ∆F G 0 )∆λ ≥ -∆λ ⊤ Λ D + ∆D ⊤ Λ -1 D ∆D + G ⊤ 0 Λ F G 0 + ∆F Λ -1
F ∆F ⊤ ∆λ Assume that the constraints on uncertainties in [START_REF] Colombo | Optimal control of the sweeping process[END_REF]

Λ1 P 0 + K ⊤ 0 ΛF K 0 + 2I n 0 0 Λ2 + G ⊤ 0 ΛF G 0 e ∆λ
Thus:

z ⊤ -M 0 + ∆M 0 - P 0 Λ 1 P 0 0 0 Λ 2 z ≥ z ⊤ (-M 0 ) 11 -P 0 Λ 1 P 0 -P 0 Λ1 P 0 -K ⊤ 0 ΛF K 0 -2I n -(M 0 ) 12 -(M 0 ) 12 -(M 0 ) 22 -Λ2 -Λ 2 -G ⊤ 0 ΛF G 0 z (43)
Applying the Schur complement Lemma A.3 for positive definiteness, it is inferred that the matrix obtained in ( 43) is positive definite if and only if the matrix inequality [START_REF] Colombo | Optimal control of the sweeping process: the polyhedral case[END_REF] in Proposition 4.3 holds. The inequality obtained in ( 42) is a nonlinear matrix inequality due to the upper-left sub-matrix M 0 . Then, in order to solve this inequality, it must be transformed into LMI, according to Appendix A.2, by multiplying the left and right-hand sides with the matrix

      Q 0 0 ..... 0 0 I . . . . . . . . . 0 0 • • • 0 I      
where

Q 0 = P -1 0
and by defining a new variable

N 0 = K 0 Q 0 . It follows that M 0 lin ∆ = - -Q 0 A ⊤ 0 -Q 0 A 0 -N ⊤ 0 E ⊤ 0 -E 0 N 0 -B 0 -E 0 G 0 + Q 0 C ⊤ 0 + N ⊤ 0 F ⊤ 0 -B ⊤ 0 -G ⊤ 0 E ⊤ 0 + C 0 Q 0 + F 0 N 0 D 0 + F 0 G 0 + (D 0 + F 0 G 0 ) ⊤ (44)
So, the inequality in (42) is written as;

              -(M 0 lin ) 11 -(M 0 lin ) 12 I n I n N ⊤ 0 Q 0 0 0 0 -(M 0 lin ) 21 -(M 0 lin ) 22 0 0 0 0 G ⊤ 0 I m I m I n 0 Λ -1 1 0 0 0 0 0 0 I n 0 0 Λ-1 1 0 0 0 0 0 N 0 0 0 0 Λ-1 F 0 0 0 0 Q 0 0 0 0 0 1 2 I n 0 0 0 0 G 0 0 0 0 0 Λ-1 F 0 0 0 I m 0 0 0 0 0 Λ -1 2 0 0 I m 0 0 0 0 0 0 Λ-1 2               ≻ 0 (45)
where (M 0 lin ) ij denotes the element of the matrix M 0 lin in (44) located at the ith row and jth column with i, j ∈ {1, 2}. The LMI in (45) can be solved, under some conditions, in the new variables Q 0 = Q ⊤ 0 ≻ 0, N 0 and G 0 . Thus, it is possible to say that the inequality:

z ⊤ -M 0 + ∆M 0 - P 0 Λ 1 P 0 0 0 Λ 2 z ≻ 0 holds.
As a consequence of the result obtained about positive definiteness of the matrix obtained in [START_REF] Corless | Continuous State Feedback Guaranteeing Uniform Ultimate Boundedness for Uncertain Dynamic Systems[END_REF] and due to symmetry, there exists µ > 0 such that -λ min M 0 + ∆M 0 -

P 0 Λ 1 P 0 0 0 Λ 2 = -µ < 0.
V ≤ -µ∥z∥ 2 + λ -1 min (Λ 1 )β 1 + λ -1 min (Λ 2 )β 2 ≤ -µ∥e∥ 2 -µ∥∆λ∥ 2 + λ -1 min (Λ 1 )β 1 + λ -1 min (Λ 2 )β 2 ≤ -µ∥e∥ 2 + λ -1 min (Λ 1 )β 1 + λ -1 min (Λ 2 )β 2 It shows that V < 0 outside the ball B r (0) ⊂ IR n , with r ∆ = λ -1 min (Λ1)β1+λ -1 min (Λ2)β2 µ
. Then, the solution of ( 38) is GUUB [START_REF] Khalil | Nonlinear Systems. 3rd[END_REF][START_REF] Corless | Continuous State Feedback Guaranteeing Uniform Ultimate Boundedness for Uncertain Dynamic Systems[END_REF]. According to Theorem A.2, the ultimate bound with α 1 (∥e∥) = λ min (P 0 )∥e∥ 2 and α 2 (∥e∥) = λ max (P 0 )∥e∥ 2 is given by

∥e∥ ≤ α -1 1 (α 2 (r)) = λ max (P 0 )r 2 λ min (P 0 ) = λ max (P 0 ) λ -1 min (Λ 1 )β 1 + λ -1 min (Λ 2 )β 2 λ min (P 0 )µ
Notice that under the conditions of the proposition, then M 0 + ∆M 0 ≻ 0. This implies that the closed-loop system's quadruple in [START_REF] Colombo | Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction[END_REF] is strongly passive (in other words the uncertainties do not destroy the strong passivity of the nominal closed-loop system). In particular this implies that D 0 + ∆D + (F 0 + ∆F )G 0 ≻ 0. Consequently, Assumption 3.1 and item 1 in section 2 guarantee that the solutions of ( 37) are AC as long as uncertainties are constant matrices. Time-varying uncertainties yield a closed-loop system which can be analyzed along the lines of section 2.2.

Relaxing Strong Passivity to Strict State Passivity

It is of interest to relax the strong passivity condition (Assumption 3.4) of Proposition 4.3, with strict state passivity (Assumption 3.3). The strict state passivity BMI in [START_REF] Chen | Perturbation bounds of P-matrix linear complementarity problems[END_REF] modified to M 0 ≼ -ϵP 0 0 0 0 . This means that the nominal closed-loop system may have D 0 + F 0 G 0 ≽ 0, hence M 0 ≼ 0. Therefore we have to find conditions which guarantee

-M 0 + ∆M 0 - ϵ ′ P 0 0 0 0 - P 0 Λ 1 P 0 0 0 Λ 2 ≽ 0 (46)
for some 0 < ϵ ′ < ϵ, where M 0 corresponds to the strict state passivity LMI. To this end, we may rely on Lemma A.3 in Appendix A.5.

1. R ∆ = (D 0 + F 0 G 0 ) + (D 0 + F 0 G 0 ) ⊤ + (∆D + ∆F G 0 ) + (∆D + ∆F G 0 ) ⊤ -Λ 2 ≽ 0, 2. Q ∆ = -(A 0 + E 0 K 0 + ϵ ′ 2 I n ) ⊤ P 0 -P 0 (A 0 + E 0 K 0 + ϵ ′ 2 I n ) -(∆A + ∆EK 0 ) ⊤ P 0 -P 0 (∆A + ∆EK 0 ) -P 0 Λ 1 P 0 ≽ 0, with -(A 0 + E 0 K 0 ) ⊤ P 0 -P 0 (A 0 + E 0 K 0 ) ≽ ϵP 0 , ϵ > ϵ ′ > 0, 3. Im(S ⊤ ) ⊆ Im(R), with S ∆ = -P 0 (B 0 + E 0 G 0 ) + (C 0 + F 0 K 0 ) ⊤ -P 0 (∆B + ∆EG 0 ) + (∆C + ∆F K 0 ) ⊤ , 4. Q ≽ SR † S ⊤ .
Proof. Notice that Q and R are symmetric by construction. The proof follows from Lemma A.3.

The tracking error ultimate boundedness can be shown as follows.

Proposition 4.5. Assume that Proposition 4.4 holds, then the tracking error e is globally ultimately bounded as:

||e|| ≤ λ max (P 0 )η r λ 2 min (P 0 )ϵ ′ (47)
Proof. Under the proposition's assumptions, the rate of change of the Lyapunov function V (e) = e ⊤ P 0 e where P 0 = Q -1 0 is the solution of the strict state passivity LMI M 0 lin ≼ -ϵQ 0 0 0 0 where

M 0 lin is defined in (44) is: V ≤ - e ∆λ ⊤ -M 0 + ∆M 0 - P 0 Λ 1 P 0 0 0 Λ 2 e ∆λ + a ⊤ (t)Λ -1 1 a(t) + b ⊤ (t)Λ -1 2 b(t) r(t) ≤ -ϵ ′ e ⊤ P 0 e + r(t)
where ϵ ′ > 0 and r(t) is a positive bounded term determined by the parametric uncertainties ∆A, ∆B, ∆C, ∆D, ∆E, and ∆F such that

||r(t)|| ≤ η r . So, V ≤ -ϵ ′ λ min (P 0 )||e|| 2 + η r
Hence, the rate of change of the storage function V < 0 outside the ball B r (0) ⊂ IR n , with r ∆ = ηr ϵ ′ λmin(P0) . Therefore, the tracking error is GUUB [START_REF] Khalil | Nonlinear Systems. 3rd[END_REF][START_REF] Corless | Continuous State Feedback Guaranteeing Uniform Ultimate Boundedness for Uncertain Dynamic Systems[END_REF] and the expression of the ultimate bound is in [START_REF] Forni | Follow the Bouncing Ball: Global Results on Tracking and State Estimation With Impacts[END_REF].

The difference between the case with strong passivity and the case with strict state passivity, is mainly that there is no -||∆λ|| 2 that helps to accelerate the convergence in the second case (see (4.2)). Moreover the strong passivity allows us to dispense with strict conditions on the structure of the uncertainties. In a sense, strict state passivity allows us to obtain a more fragile robustness.

It is of interest to analyze under which conditions on the plant's nominal model and on the uncertainties, the conditions in items 1-4 in Proposition 4.4 hold. For item 1, assume that ∆D = 0, ∆F = 0, ∆C = 0 (the complementarity constraint has no uncertainty) then we may take Λ 2 = 0 (since b(t) = 0), so that R ≽ 0 by closed-loop strict-state passivity. For item 2, using the fact that Q 0 ≻ 0, Corollary A.6 can be used to derive sufficient condition such that Q ≻ 0:

σ max (Q 0 ) > ϵλ max (P 0 ) > 2σ max (P 0 )(σ max (∆A) + σ max (∆EK 0 )) + σ 2 max (P 0 )σ max (Λ 1 )
, where we used [8, Fact 9.14.15, Corollary 9.6.5]. Let us focus on items 3 and 4. We have

-M 0 = Q 0 S 0 S ⊤ 0 R 0 ,
and

-M 0 +∆M 0 - P 0 Λ 1 P 0 0 0 Λ 2 = Q 0 + ∆Q S 0 + ∆S S ⊤ 0 + ∆S ⊤ R 0 + ∆R
, where the matrices stem from [START_REF] Chen | Perturbation bounds of P-matrix linear complementarity problems[END_REF] and items 1, 2 and 3 in Proposition 4.4. Using Lemmata A.3 and A.5, the strict-state passivity LMI for the nominal system implies that

Q 0 ≻ Q 0 -ϵP 0 ≽ S 0 R † 0 S ⊤ 0 , and Im(S 0 R † 0 S ⊤ 0 ) ⊆ Im(Q 0 -ϵP 0 ) (we also have Im(S 0 R † 0 S ⊤ 0 ) ⊆ Im(Q 0 ) = IR n since strict state passivity implies that Q 0 ≻ 0), and Im(S ⊤ 0 ) ⊆ Im(R 0 ) = Im(R † 0 ) [9, Proposition 8.1.7]. Assume that Im(∆S ⊤ ) ⊆ Im(S ⊤ 0 ). Then Im(S ⊤ ) = Im(S ⊤ 0 + ∆S ⊤ ) ⊆ Im(S ⊤ 0 ). Thus item 3 is satisfied if Im(R) = Im(R 0 ) (
which holds if ∆D = 0, ∆F = 0, ∆C = 0, since in this case ∆R = 0). We can therefore state the following: Lemma 4.6. Let Assumption 3.3 hold for the nominal system. Assume that Im(∆S ⊤ ) ⊆ Im(S ⊤ 0 ), and that Im(R) = Im(R 0 ), then Im(S ⊤ ) ⊆ Im(R).

Notice that item 4 is Q 0 + ∆Q ≽ (S 0 + ∆S)(R 0 + ∆R) † (S ⊤ 0 + ∆S ⊤ ). In the next proposition we derive conditions on the uncertainty matrices, that guarantee that the inequality in item 4 holds true when the nominal system is strictly state passive.

Proposition 4.7. Let Assumption 3.3 hold for the quadruple

(A 0 , B 0 , C 0 , D 0 ). Assume that: (i) R ≽ 0, (ii) Im(∆R) ⊆ Im(R 0 ), (iii) rank(R 0 ) = r, R 0 ̸ = 0, σ max (∆R) < σ r (R 0 ), (iv) σ max (OT )+σ max (O((R † 0 ∆R) 2 ))+σ max (∆Q) < ϵλ max (P 0 )
, where OT is in [START_REF] Galeani | Trajectory tracking for a particle in elliptical billiards[END_REF]. Then Q ≽ SR † S ⊤ .

Proof. Since ∆R = ∆R ⊤ , using [9, Fact 8.4.3] and [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF]Fact 8.4.38] and (ii), we have (R

0 + ∆R) † = (I m + R † 0 ∆R) † (R † 0 + R † 0 ∆RR † 0 )(I m + ∆RR † 0 ) † .
I m + R † 0 ∆R ≻ 0. Thus (I m + R † 0 ∆R) † = (I m + R † 0 ∆R) -1 = ∞ k=0 (-R † 0 ∆R) k = I m -R † 0 ∆R + O((R † 0 ∆R) 2 ) [9, Proposition 11.3.10]. Therefore (R 0 + ∆R) † = (I m -R † 0 ∆R)(R † 0 + R † 0 ∆RR † 0 )(I m -∆RR † 0 ) + O((R † 0 ∆R) 2 ) = R † 0 -R † 0 ∆RR † 0 + O((R † 0 ∆R) 2 ). Consequently, (S 0 +∆S)(R 0 +∆R) † (S ⊤ 0 +∆S ⊤ ) = (S 0 +∆S)(R † 0 -R † 0 ∆RR † 0 +O((R † 0 ∆R) 2 ))(S ⊤ 0 + ∆S ⊤ ) = S 0 R † 0 S ⊤ 0 + OT + O((R † 0 ∆R) 2 )
, where OT stands for other terms defined as:

OT = S 0 R † 0 ∆S ⊤ -S 0 R † 0 ∆RR † 0 S ⊤ 0 -S 0 R † 0 ∆RR † 0 ∆S ⊤ + ∆SR † 0 S ⊤ 0 + ∆SR † 0 ∆S ⊤ -∆SR † 0 ∆RR † 0 S ⊤ 0 -∆SR † 0 ∆RR † 0 ∆S ⊤ . ( 48
)
The result follows from

Q 0 -S 0 R † 0 S ⊤ 0 ≽ ϵP 0 , (
≻ OT -∆Q + O((R † 0 ∆R) 2 ), so that Q 0 + ∆Q ≻ (S 0 + ∆S)(R 0 + ∆R) † (S ⊤ 0 + ∆S ⊤ ).
Item (iv) in Proposition 4.7 means that a class of nonzero, sufficiently small uncertainties are allowed. The matrix Λ 1 which appears in ∆Q can be chosen small if ∆A, ∆B, ∆E are small. As said above, ∆R can be made small if the uncertainties inside the complementarity constraints are small. In some cases there are no uncertainties in the complementarity constraints.

Tracking Control for First-order Sweeping Process (FOSwP)

Under some conditions (see, e.g., section A.8), LCS can be equivalently rewritten as a FOSwP. Let us first rewrite both the closed-loop plant and the desired dynamics under the perturbed FOSwP format, when D = 0 in [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF] and D + F G = 0 in [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF]. Here we are interested to see how the material in sections 3.2, 3.3 and 4.1, 4.2 and 4.3, adapt to trajectory tracking applied to FOSwP. Indeed the control of FOSwP has recently received much attention, especially its optimal control, stabilization, and the existence of periodic orbits [START_REF] Tan | Optimization of fully controlled sweeping processes[END_REF][START_REF] Colombo | Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction[END_REF][START_REF] Tan | Optimization and discrete approximation of sweeping processes with controlled moving sets and perturbations[END_REF][START_REF] Colombo | Optimization of a perturbed sweeping process by con-strained discontinuous controls[END_REF][START_REF] Ch | A Pontryagin maximum principle for the controlled sweeping process[END_REF][START_REF] Colombo | Optimal control of sweeping processes in robotics and traffic flow models[END_REF][START_REF] Colombo | The minimum time function for the controlled Moreau's sweeping process[END_REF][START_REF] Colombo | Optimal control of the sweeping process: the polyhedral case[END_REF][START_REF] Colombo | Optimal control of the sweeping process[END_REF][START_REF] Cao | Applications of controlled sweeping processes to nonlinear crowd motion models with obstacles[END_REF][START_REF] Tan | Optimization of controlled free-time sweeping processes with applications to marine surface vehicle modeling[END_REF][START_REF] Mordukhovich | Discrete approximations and optimal control of nonsmooth perturbed sweeping processes[END_REF][START_REF] Niwanthi Wadippuli | Global asymptotic stability of nonconvex sweeping processes[END_REF][START_REF] Kamenskii | A continuation principle for periodic BV-continuous state-dependent sweeping processes[END_REF]. Thus it is of interest to investigate the trajectory tracking issue for such DIs.

The LCS of the desired system in ( 14) can be represented equivalently as a FOSwP when D = 0 if the conditions in section 2, item 3 are satisfied. These conditions require the system represented by the quadruple (A, B, C, 0) to be passive which implies the condition P B = C ⊤ where P = P ⊤ ≻ 0 is the solution of the matrix inequality of strict passivity in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF], the controller u ∈ L 1 loc (IR + : IR p ) and the constraint qualification Im(C)

-IR m + = IR m holds. The FOSwP is represented as follows (see section A.8) ζd (t) ∈ R d AR -1 d ζ d (t) -N ϕ d (t) (ζ d (t)) ( 49
)
where

R 2 d = P d , R d = R ⊤ d , ζ d = R d x d and ϕ d (t) = {R d x d | Cx d ∈ S d (t)} with S d (t) = {v ∈ IR m | v + F u d (t) ≥ 0} and x d (t) = R -1 d ζ d (t). Thus, ϕ d (t) = {ζ d ∈ IR n | CR -1 d ζ d + F u d (t) ≥ 0}. Recall that the closed-loop system in (3.3) when D + F G = 0 is the following FOSwP: ζ(t) ∈ R(A + EK)R -1 ζ(t) + RE(-KR -1 d ζ d (t) + u d (t)) -N Φ(t) (ζ(t)) (50)
where

ϕ(t) = {Rx | (C + F K)x ∈ S(t)} with S(t) = {v ∈ IR m | v -F Kx d (t + ) + F u d (t) ≥ 0} and ζ(t) = R -1 x(t).
It is noteworthy that the set ϕ(t) depends on the post-jump desired state x + d . This means that it can be characterized only once the desired state jump has been calculated. Thus,

ϕ(t) = {ζ ∈ IR n | (C + F K)R -1 ζ -F KR -1 d ζ d (t + ) + F u d (t) ≥ 0}. Recall that f (t + ) = lims→t s>t f (s) and f (t -) = lims→t s<t f (s). It is noteworthy that x ∈ K if and only if ζ ∈ ϕ(t) and x d ∈ K d if and only if ζ d ∈ ϕ d (t).
Taking advantage of the FOSwP formalism, the aim in the following is to show under which conditions x + d = x + holds. As a first step, let us check if x + d ∈ ∂K, it is required to show that x + d ∈ K (the set K is defined for ( 16)). Knowing that x + d ∈ K d , then

Cx + d + F u d ≥ 0 ⇔ (C + F K)x + d -F Kx + d + F u d ≥ 0 ( 51
)
which means that x + d satisfies the condition of the set K. Hence x + d ∈ K. In addition, given that

x + d ∈ ∂K d , it follows that Cx + d + F u + d k = 0 ⇔ (C + F K)x + d -F Kx + d + F u + d k = 0 for some k ∈ {1, . . . , m} (52)
Thus, the following is proved:

Lemma 5.1. The post-jump desired state satisfies:

x + d ∈ ∂K ∩ ∂K d . So a necessary condition to have x + = x + d is that x + ∈ ∂K, which is equivalent to x -̸ ∈ Int(K). Assume that P (x --x + d ) ∈ N K (x + d ), equivalently x + d = Proj P [K;
x -] = x + (the last equality holds from ( 9)). The question that arises is: what are the conditions such that the first inclusion holds? Let us present the given information in the following. The optimization problem in (9) for the desired system leads to the following:

x + d = Proj P d [K d ; x - d ] ⇔ P d (x - d -x + d ) ∈ N K d (x + d ) (53)
Similarly, according to the optimization problem in [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF] for the closed-loop system, the following is derived:

P (x --x + ) ∈ N K (x + ) ( 54
)
Using the definition of the normal cone in (1), the equations in ( 53) and ( 54) are written as follows, for the desired and closed-loop systems, respectively:

(x - d -x + d ) ⊤ P d (x d -x + d ) ≤ 0 ∀ x d ∈ K d (55) (x --x + ) ⊤ P (x -x + ) ≤ 0 ∀ x ∈ K (56)
And the third inclusion for x + d = x + is equivalent to the VI:

(x --x + d ) ⊤ P (x -x + d ) ≤ 0 ∀ x ∈ K (57) Let ζ d = R d x d and ζ = Rx. Note that x + d ∈ ∂K if and only if Rx + d ∈ ∂ϕ(t) ⇔ RR -1 d ζ + d ∈ ∂ϕ(t), then the state RR -1 d ζ d becomes relevant. Hence, the set RR -1 d ϕ d (t) is considered in the following since RR -1 d ζ d ∈ RR -1 d ϕ d (t). Let ζ ′ d = RR -1 d ζ d = Rx d and ϕ ′ d (t) = RR -1 d ϕ d (t) = RK d which is defined as: ϕ ′ d (t) = {ζ ′ d ∈ IR 2 | CR -1 ζ ′ d + F u d ≥ 0} (58)
The desired system in ( 49) is written, in the terms of the new variable

ζ ′ d = RR -1 d ζ d , as follows: ζ′ d ∈ RAR -1 ζ ′ d -RP -1 d RN ϕ ′ d (t) (ζ ′ d ) (59)
Similarly, x + ∈ ∂K if and only if ζ + ∈ ∂ϕ((t). Hence, equivalently the three VIs are obtained:

(ζ ′ - d -ζ ′ + d ) ⊤ (ζ ′ d -ζ ′ + d ) ≤ 0 ∀ ζ ′ d ∈ ϕ ′ d (t) (60)
(ζ --ζ + ) ⊤ (ζ -ζ + ) ≤ 0 ∀ ζ ∈ ϕ(t) (61) (ζ --ζ ′ + d ) ⊤ (ζ -ζ ′ + d ) ≤ 0 ∀ ζ ∈ ϕ(t) (62)
The optimization problem in [START_REF] Heemels | Time-stepping methods for constructing periodic solutions in maximally monotone set-valued dynamical systems[END_REF] for the closed-loop FOSwP in ( 50) is written as follows:

ζ + = Proj[ϕ(t), ζ -] ⇔ ζ --ζ + ∈ N ϕ(t) (ζ + ) (63)
Then, it is required to prove under which conditions

ζ --ζ ′ + d ∈ N ϕ(t) (ζ ′ + d ) (which is the VI (62)) holds, equivalently ζ ′ + d = ζ + (i.e., R -1 ζ + = R -1 d ζ + ⇔ x + d = x + ).
Let us study this on a particular example. [START_REF] Goeleven | Stability and instability matrices for linear evolution variational inequalities[END_REF]. Take A = -I, B = I, and C = I where I is the 2 × 2 identity matrix. Let

Example 1. Consider the desired FOSwP in

ζ + ′ 1d ζ + ′ 2d + 1.07 0.034 u d -N ϕ(t) (ζ) ( 65
)
where

N ϕ(t) (ζ) = {v ∈ IR 2 | v ⊤ (ζ -η) ≤ 0 ∀ η ∈ ϕ(t)
} and ϕ(t) is given by:

ϕ(t) = {ζ ∈ IR 2 | (C + F K)R -1 ζ -F KR -1 ζ ′ d (t + ) + F u d (t) ≥ 0} = ζ ∈ IR 2 1.
ζ + ′ 1d ζ + ′ 2d + 2 1 u d ≥ 0 0 ( 66
)
The desired FOSwP in (59) is written as follows:

ζ′ 1d ζ′ 2d ∈ ζ ′ 1d ζ ′ 2d - 1.14 0.07 0.07 1.035 N ϕ ′ d (t) (ζ ′ d ) ( 67
)
where

N ϕ ′ d (t) (ζ ′ d ) = {v ∈ IR 2 | v ⊤ (ζ ′ d -η d ) ≤ 0 ∀ η d ∈ ϕ ′ d (t)} and ϕ ′ d (t
) is given by:

ϕ ′ d (t) = {ζ d ∈ IR 2 | CR -1 ζ ′ d +F u d ≥ 0} = ζ ′ d ∈ IR 2 0.936 -0.031 -0.031 0.984 ζ ′ 1d ζ ′ 2d + 2u d u d ≥ 0 0 (68)
Let us define the set of active constraints for the desired system in (67) as:

A d (ϕ ′ d (t)) = {i ∈ {1, 2} | (CR -1 ) i• ζ ′ + d + F i• u d (t) = 0 and ζ ′ + d ∈ ϕ ′ d (t)} ( 69
)
and we define [START_REF] Morarescu | Passivity-based switching control of flexible-joint complementarity mechanical systems[END_REF] as shown below in Fig. 1:

Ω id ∆ = ζ ′+ d ∈ IR 2 0.936 -0.031 -0.031 0.984 i• ζ ′ + d + 2u d u d i• = 0 and ζ ′ + d ∈ ϕ ′ d (t) , for i ∈ {1, 2}. Let us draw the set ϕ ′ d (t) in
2u d u d ϕ ′ d (t) Ω 1d Ω 2d ζ 1 ζ 2 Figure 1
It is noticeable that the set ϕ(t) in [START_REF] Menini | Trajectory tracking of a bouncing ball in a triangular billiard by unfolding and folding the billiard table[END_REF]

ϕ(t) = {ζ ∈ IR 2 | ((C + F K)R -1 )ζ + (F KC -1 F )u d (t) + F u d (t) ≥ 0} (70)
Figure 2 below shows both sets ϕ(t) and ϕ d (t).

2u d u d ζ ′ + d ϕ(t) N ϕ(t) (ζ ′ + d ) I II III IV ζ 1 ζ 2 Figure 2
By observing Figure 2

A(ϕ(t)) = {i ∈ {1, 2} | ((C + F K)R -1 ) i• ζ + + (F KC -1 F ) i• u d (t) + F i• u d (t) = 0 for ζ + ∈ ϕ(t)} (71)
and we define

Ω i ∆ = ζ + ∈ IR 2 1.065 0.034 0.034 1.016 i• ζ + + 2.35u d 1.75u d i• = 0 for ζ + ∈ ∂ϕ(t) , for i ∈ {1, 2}. • If ζ -∈ Region I (i.e., ζ -/ ∈ ϕ(t))
and the active constraint is Ω 1 defined in [START_REF] Padula | MIMO tracking control of LTI systems: A geometric approach[END_REF], then the post-jump state ζ + ∈ Ω 1 and is defined as

ζ + = Proj[Ω 1 ; ζ -]. • If ζ -∈ Region II (i.e., ζ -∈ int ϕ(t)), then ζ + = ζ -∈ int ϕ(t). • If ζ -∈ Region III (i.e., ζ -/ ∈ ϕ(t))
and the active constraint is Ω 2 as defined in [START_REF] Padula | MIMO tracking control of LTI systems: A geometric approach[END_REF], then

ζ + ∈ Ω 2 . • If ζ -∈ Region IV (i.e., ζ --ζ ′ + d ∈ N ϕ(t) (ζ ′ + d )
) and both constraints Ω 1 and Ω 2 are active as defined in [START_REF] Padula | MIMO tracking control of LTI systems: A geometric approach[END_REF],

then ζ + = ζ ′ + d Thus, regardless of the active constraint, ζ + = ζ ′ + d if and only if ζ --ζ ′ + d ∈ N ϕ(t) (ζ ′ + d ).

Case 2: the state ζ

′ + d ∈ Ω 1d
In this section, only one of the constraints for the desired system is considered active which is Ω 1d as defined in [START_REF] Morarescu | Trajectory tracking control of multiconstraint complementarity Lagrangian systems[END_REF].

Let ζ ′ + d = (-2.104u d , u d ) ⊤ ∈ Ω 1d
and let us substitute the value of ζ ′ + d in the set ϕ(t) in [START_REF] Menini | Trajectory tracking of a bouncing ball in a triangular billiard by unfolding and folding the billiard table[END_REF], then:

ϕ(t) = ζ ∈ IR 2
1.065 0.034 0.034 1.016

ζ 1 ζ 2 - 0.129 0.065 0.064 0.032 -2.1036u d u d + 2u d u d ≥ 0 ( 72
)
and it is represented graphically in Figure 3 below showing the different regions to be studied.

2u d u d ϕ ′ d (t) ϕ(t) ζ ′ + d N ϕ(t) (ζ ′ + d ) ζ 1 ζ 2 Figure 3
By observing Figure 3, the state ζ ′ + d is shared between the active constraint of the desired system Ω 1d and the set ϕ(t). It can be noticed that the set ϕ(t) in Figure 3 is different from that in Figure 1 due to the new selection of ζ ′ + d . Let us define the active constraints for the closed-loop system in (65) as follows:

A(ϕ(t)) = {i ∈ {1, 2} | ((C+F K)R -1 ) i• ζ + +(F KR -1 ) i• ζ ′ + d (t)+F i• u d (t) = 0 for ζ + ∈ ϕ(t)} (73)
and we define

Ω i = ζ + ∈ IR 2 Ω i ∆ = 1.
= ζ ′ + d holds if and only if ζ -is selected such that ζ --ζ ′ + d ∈ N ϕ(t) (ζ ′ + d ). Otherwise, the resulting feasible solution ζ + ̸ = ζ ′ +
d , but it belongs to the boundary of the set ϕ(t), that is, ζ + ∈ ∂ϕ(t), which can be any of the three possible domains: Ω 1 , Ω 2 , or Ω 1 ∩ Ω 2 as defined in [START_REF] Posa | Stability analysis and control of rigid-body systems with impacts and friction[END_REF].

Case 3: the state ζ

′ + d ∈ Ω 2d
In this section, only one of the constraints for the desired system is considered active which is Ω 2d defined in [START_REF] Morarescu | Trajectory tracking control of multiconstraint complementarity Lagrangian systems[END_REF].

2u d u d Ω 2 Ω 1 ϕ(t) ζ ′ + d N ϕ(t) (ζ ′ + d ) ζ 1 ζ 2 Figure 4
The state ζ ′ + d is shared by both sets ϕ(t) and ϕ d (t), as shown in Figure 4. More precisely,

ζ ′ + d ∈ Ω 2d ∩ ϕ(t).
Let us define the active constraints for the closed-loop system in (65) as follows:

A(ϕ(t)) = {i ∈ {1, 2} | ((C+F K)R -1 ) i• ζ + +(F KR -1 ) i• ζ ′ + d (t)+F i• u d (t) = 0 for ζ + ∈ ϕ(t)} (75)
and let

Ω i ∆ = ζ + ∈ IR 2 1.065 0.034 0.034 1.016 i• ζ + + 1.542u d 0.773u d i• = 0 for ζ + ∈ ϕ(t) , for i ∈ {1, 2}.
Let us examine, in the following cases, under which conditions the equality ζ ′ + d = ζ + holds, taking into account the active constraints of the closed-loop system.

Simple Scalar Example with State Jumps

This section and sections 7 through 13 are dedicated to analyse and mainly to present numerical simulations of the above theoretical developments. The numerical simulations are done with the INRIA software package siconos1 [START_REF] Acary | An introduction to Siconos[END_REF], and the LMIs are solved with mosek 9.3.14 solver [START_REF] Mosek Aps | MOSEK Optimizer API for Python[END_REF].

Dynamics and Closed-loop System

Consider the scalar LCS with d = 0:

ẋ(t) = ax(t) + bλ(t) + u(t) 0 ≤ λ(t) ⊥ w(t) = cx(t) + v(t) ≥ 0 ( 76
)
where a, b, and c ∈ IR, u and v are two inputs. The desired system is represented by an LCS as follows:

ẋd (t) = ax d (t) + bλ d (t) + u d (t) 0 ≤ λ d (t) ⊥ w d (t) = cx d (t) + v d (t) ≥ 0 ( 77
)
If a > 0, then the real system in (76) and the desired system (77) are unstable (i.e the quadruple (a, b, c, d) has a positive real pole). But, in the context of trajectory tracking, the stability of the error dynamics is the main concern and not the stability of the real and the desired trajectories.

Let [START_REF] Scherer | Linear Matrix Inequalities in Control[END_REF] so that K = (k 1 , k 2 ) ⊤ and G = (g 1 , g 2 ) ⊤ . By substituting the equations of u and v in [START_REF] Rockafellar | Convex Analysis[END_REF], the closed-loop system is written in the form of ( 16) as follows:

u = k 1 (x -x d ) + g 1 (λ -λ d ) + u d and v = k 2 (x -x d ) + g 2 (λ -λ d ) + v d
ẋ(t) = (a + k 1 )x(t) + (b + g 1 )λ(t) -k 1 x d (t) -g 1 λ d (t) + u d (t) 0 ≤ λ(t) ⊥ w(t) = (c + k 2 )x(t) + g 2 λ(t) -k 2 x d (t) -g 2 λ d (t) + v d (t) ≥ 0 ( 79
0 ≤ λ(t) ⊥ w(t) = cx(t) + v(t) ≥ 0
In the case when d = 0 and at contact (i.e., cx(t) + v(t) = 0), the sign of the variable ẇ is studied. For this purpose, the following LCP is obtained: As shown in Figure 5, the linear complementarity problem is not valid for both desired and closedloop systems since the complementarity variables w < 0 and w d < 0, Thus, the LCS in (77) and ( 79) is ill-posed and it is pointless to study trajectory tracking for an ill-posed system. Therefore, the conditions stated in Remark 6.1 must be respected in the following.

0 ≤ λ(t) ⊥ cax(t) + cbλ(t) + cu(t) + v(t) ≥ 0 (80

State-jumps Analysis

State-jumps in x(•) and x d (•) can occur for different reasons discussed in section 3.3. The purpose of this section is to analyze state jumps in the desired and closed-loop systems in different cases.

If the controller gain G = 0

Notice that the dynamics (79) may not be well-posed due to the possible presence of Dirac measures that stem from both the complementarity conditions in (79) which create an impulsive multiplier λ and from the complementarity in (77) which implies an impulsive multiplier λ d . Therefore we shall consider G = 0 (which means that (C1) in section 3.3 is satisfied). In order to analyze state jumps, the following sets are calculated as follows (See section 2.1). For the desired system in (77):

Q d,D = {λ d ∈ IR | 0 ≤ λ d ⊥ dλ d ≥ 0} = {λ d ∈ IR | 0 ≤ λ d ⊥ 0 ≥ 0} = {λ d ∈ IR | λ d ∈ IR + } Q ⋆ d,D = {w d ∈ IR | ⟨w d , λ d ⟩ ≥ 0} = {w d ∈ IR | w d ∈ IR + } K d = {x d ∈ IR | cx d + v d (t + ) ∈ Q ⋆ d,D } = {x d ∈ IR | cx d ≥ -v d (t + )} (
Q D = {λ ∈ IR | 0 ≤ λ ⊥ dλ ≥ 0} = {λ ∈ IR | λ ∈ IR + } Q ⋆ D = {w ∈ IR | ⟨w, λ⟩ ≥ 0} = {w ∈ IR | w ∈ IR + } K = {x ∈ IR | (c + k 2 )x -k 2 x d (t + ) + v d (t + ) ∈ Q ⋆ d,D } = {x ∈ IR | (c + k 2 )x ≥ k 2 x d (t + ) -v d (t + )} (82)
The state x of the closed-loop system in (79) jumps at any

t = t c , t c ∈ [0, +∞), if and only if cx(t - c ) + k 2 [x(t - c ) -x d (t + c )] < -v d (t + c
). Thus, when G = 0, the jumps at t > 0 in x(•) and x d (•) occur only if v d is discontinuous at t, as expected.

If the controller gain G ̸ = 0 and x d is continuous

In this case, a state jump in x (if any) is caused by the discontinuity in λ d , which can arise when the complementarity problem switches (i.e., cx(t) + v(t) = 0) and v d (t) remains continuous. The desired system's state is considered continuous to avoid the Dirac measure λ d appearing in [START_REF] Sessa | A complementarity approach for the computation of periodic oscillations in piecewise linear systems[END_REF]. For the closed-loop system represented by (a + k 1 , b + g 1 , c + k 2 , g 2 ) to be strictly passive, one must have g 2 > 0 given that d = 0. But, g 2 multiplies λ as shown in [START_REF] Sessa | A complementarity approach for the computation of periodic oscillations in piecewise linear systems[END_REF]. The LCP of the closed-loop system (79) is

0 ≤ λ(t) ⊥ w(t) = (c + k 2 )x(t) + g 2 λ(t) -k 2 x d (t) -g 2 λ d (t) + v d (t) ≥ 0
has a unique piecewise continuous solution λ. Hence, the LCS in ( 79) is an ODE with an AC solution on IR + provided that u(t) and v(t) are continuous (see section 2.1, item 1). Thus, the closed-loop system's state is jump-free in the scalar case with G ̸ = 0 and the desired system's state x d is continuous. Therefore, the error dynamics e(t) is continuous in this case.

The explanation of this result can be further understood by analyzing the following sets of the closed-loop system.

Q D = {λ ∈ IR | 0 ≤ λ ⊥ (d + g 2 )λ ≥ 0} = {0} Q ⋆ D = {w ∈ IR | ⟨w, λ⟩ ≥ 0} = {IR} K = {x ∈ IR | (c + k 2 )x(t) + g 2 λ(t) -k 2 x d (t) -g 2 λ d (t) + v d (t) ∈ Q ⋆ d,D } = {IR} (83)
Using the optimization problem in (9), x(t + ) = x(t -). Thus, the closed-loop system's state x(t) is continuous even if λ d is discontinuous. Therefore, the error dynamics e(t) is continuous.

Stability Analysis of Error Dynamics with State-jumps (G = 0)

This section is dedicated to the stability analysis of the error dynamics in [START_REF] Vladimirov | Nonstationary dissipative evolution equations in a Hilbert space[END_REF] when state jumps occur. The error dynamics e(t) = x(t) -x d (t) in the form of ( 18) is represented by:

ė(t) = (a + k 1 )e(t) + b (λ(t) -λ d (t)) 0 ≤ λ(t) λ d (t) ⊥ w(t) w d (t) = c + k 2 -c -k 2 e(t) + 0 c c + k 2 -k 2 x(t) x d (t) + v d (t) v d (t) ≥ 0 (84)
Let us now try the passivity criterion introduced in (30) to cope with state jumps directly from the error dynamics passivity. Let ā

= a + k 1 , b = b -b and c = c + k 2 0
. The passivity of the error dynamics represented by the quadruple (ā, b, c, 0) is determined by checking analytically if the nonlinear matrix inequality in [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF] has no solution for the system [START_REF] Vladimirov | Nonstationary dissipative evolution equations in a Hilbert space[END_REF]. Consider

M ext ∆ =   -2p(a + k 1 ) -pb + c + k 2 pb -bp + c + k 2 0 0 bp 0 0   ⪰ 0 ( 85
)
has a solution for p > 0. The nonlinear matrix inequality in [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF] has a solution for the system [START_REF] Vladimirov | Nonstationary dissipative evolution equations in a Hilbert space[END_REF] if and only if the nonlinear matrix inequality in (85) has a solution for p > 0. This means that x ⊤ M ext x ⪰ 0 for all x ∈ IR 3 and x ̸ = 0. But

x ⊤ M ext x =   x 1 x 2 x 3   ⊤ M ext   x 1 x 2 x 3   = (-2p(a + k 1 ))x 2 1 + 2bpx 1 x 3
Thus, x ⊤ M ext x ⪰ 0 is not satisfied for all x ∈ IR 3 such that x ̸ = 0. Notice that it can be inferred directly from Lemma A.3 that M ext can be positive semidefinite only if

-pb + c + k 2 = pb = 0.
Therefore it is inferred that this approach (imposing the passivity of the error dynamics with the LMI in [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF]) is not fruitful.

The following Lemma states the stability result.

Lemma 6.2. Consider the dynamical systems in [START_REF] Sessa | A complementarity approach for the computation of periodic oscillations in piecewise linear systems[END_REF] and in [START_REF] Rockafellar | Variational Analysis[END_REF]. Suppose that Assumptions 3. ∆V (e(t)

) = V (e(t + )) -V (e(t -)) = (x(t + ) -x d (t + )) ⊤ p (x(t + ) -x d (t + )) -(x(t -) -x d (t -)) ⊤ (x(t -) -x d (t -)) = p (x(t + ) -x d (t + )) 2 -p (x(t -) -x d (t -)) 2 (86)
There are three cases to consider when analyzing the sign of ∆V (e(t)). To lighten notations we denote

f + = f (t + ) and f -= f (t -).
First case: In this case, both x and x d jump at the same time t. The values of the state jump for the desired and the closed-loop system are given by x

+ d = -1 c v + d and x + = k2x + d -v + d c+k2
respectively by referring to section A.7. If we substitute the value of x + d in x + , then

x + = -1 c v + d .
The variation of the storage function at the jump time is written as follows:

∆V (e(t)) = p (x(t

+ ) -x d (t + )) 2 -p (x(t -) -x d (t -)) 2 = p -1 c v + d + 1 c v + d -p (x(t -) -x d (t -)) 2 = -p (x(t -) -x d (t -)) 2 ≤ 0
Thus, when both x and x d jump, the variation of the storage function of the error dynamics ∆V (e) < 0 for all e ̸ = 0.

Second case:

In this case, the closed-loop system's state x jumps and the desired system's state

x d is continuous such that x + d = x - d = x d .
The value of the closed-loop system's state jump is given by

x + = k2x d -v + d c+k2
(see section A.7). If these values are substituted in ∆V (e(t)) in (86), then

∆V (e(t)) = p

k2x d -v + d c+k2 -x d 2 -p(x --x d ) 2 = p -cx d -v + d c+k2 2 -(x --x d ) 2 = p -cx d -v + d c+k2 -x -+ x d -cx d -v + d c+k2 + x --x d = p -(c+k2)x -+k2x d -v + d c+k2 (c+k2)x --(2c+k2)x d -v + d c+k2 = p (c+k2) 2 -(c + k 2 )x -+ k 2 x d -v + d (c + k 2 )x --(2c + k 2 )x d -v + d Let h(x) ∆ = -(c + k 2 )x -+ k 2 x d -v + d and r(x) ∆ = (c + k 2 )x --(2c + k 2 )x d -v + d .

Let us study the signs of h(x) and r(x).

Knowing that x -/ ∈ K, the following inequality holds:

(c + k 2 )x --k 2 x d + v + d < 0 ⇔ -(c + k 2 )x -+ k 2 x d -v + d > 0 ⇔ h(x) > 0
Now, let us check the sign of r(x). If we add and subtract v + d , then

r(x) ± v + d ∆ = (c + k 2 )x --k 2 x d + v + d -2cx d -2v + d ∆ = -h(x) -2cx d -2v + d Provided that -h(x) < 0 and cx d + v + d ≥ 0 (i.e. x d ∈ K d ) ⇔ -2cx d -2v + d ≤ 0.
So, r(x) < 0. Thus, h(x) > 0 and r(x) < 0. Therefore, ∆V (e(t))

∆ = p (c+k2) 2 (h(x)r(x)) < 0.
Third case: In this case, the desired system's state x d jumps and the closed-loop system's state x is continuous such that x + = x -= x(t). The value of the desired system's state jump is [START_REF] Aydinoglu | Stabilization of complementarity systems via contact-aware controllers[END_REF]. If these values are substituted in ∆V (e(t)) in (86), then

x + d = -1 c v + d (check section A.
∆V (e(t)

) = p x + v + d c 2 -p(x -x - d ) 2 = p x + v + d c + x -x - d v + d c + x - d Let h(x) ∆ = x + v + d c + x -x - d and r(x) ∆ = v + d c + x - d .
Let us study the signs of h(x) and r(x).

Knowing that x - d / ∈ K d , then cx - d + v + d < 0 ⇔ x - d + v + d c < 0 ⇔ r(x) < 0
Now, let us check the sign of h(x) by adding and subtracting v + d . Then,

h(x) ± v + d ∆ = 2(cx + v + d ) -cx - d -v + d ∆ = 2(cx + v + d ) -r(x)
Let us write the following:

x ∈ K ⇔ (c + k 2 )x -k 2 x + d + v + d ≥ 0 ⇔ (c + k 2 )x + k2 c v + d + v + d ≥ 0 ⇔ (c+k2) c (cx + v + d ) ≥ 0 According to Remark 6.1, c+k2 c > 0. So, cx + v + d ≥ 0 ⇔ 2cx + 2v + d ≥ 0 and -r(x) > 0. So, h(x) > 0.
Thus, in this case, h(x) > 0 and r(x) < 0 and ∆V (e(t))

∆ = p (h(x)r(x)) < 0.
Therefore, the storage function is strictly decreasing (i.e., V < 0 when t ̸ = t k and ∆V < 0 when t = t k for all e ̸ = 0 where t k are the time instants of jumps) on IR + .

x d (t - k ) / ∈ K d ⇔ cx d (t - k ) + v d (t + k ) < 0 ⇔ v d (t + k ) < -cx d (t - k )
Also, the state x in (79) performs a jump at t = t k if and only if x(t - k ) / ∈ K in [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF]. Let us write the following:

x(t - k ) / ∈ K ⇔ (c + k 2 )x(t - k ) -k 2 x d (t + k ) + v d (t + k ) < 0 (87) Given that x d jumps at t = t k , let us substitute x d (t + k ) = -1 c v d (t + k ) (see (150) in section A.7) in (87). Then, (c + k 2 )x(t - k ) + k2 c v d (t + k ) + v d (t + k ) < 0 ⇔ c+k2 c cx(t - k ) + v d (t + k )
< 0 Let c and c + k 2 have the same sign due to the well-posedness of LCS as in Remark 6.1. So,

v d (t + k ) < -cx(t - k )
According to the solution of the optimization problem in (9) in section A.7, the closed-loop system's state x jumps at t k such that x(t

+ k ) = k2x d (t + k )-v d (t + k ) c+k2 = -1 c v d (t + k ) where x d (t + k ) = -1 c v d (t + k )
. Therefore, both x and x d jumps at t k to the same value

x d (t + k ) = x(t + k ) = -1 c v d (t + k ) (see section A.7) if and only if v d (t + k ) < µ where µ is given by µ = min -cx d (t - k ), -cx(t - k ) .

Numerical Applications with G = 0

The open-loop system with u = v = 0 is not strictly passive but there exist k 1 , k 2 , and p such that the closed-loop quadruple (a + k 1 , b, c + k 2 , 0) is strictly state passive, equivalently the nonlinear matrix inequality -2(a + k 1 )p -ϵp -pb

+ c + k 2 -bp + c + k 2 0 ⪰ 0
has a solution p > 0 with ϵ > 0. According to A.2, the following LMI is obtained

-2qa -2N 1 -ϵq -b + qc + N 2 -b + cq + N 2 0 ⪰ 0 ( 88
)
where q = p -1 , N 1 = k 1 q and N 2 = k 2 q. The LMI in (88) has a solution such that:

k 1 = -1.
198, k 2 = 0.96 and p = 1.9607 for a = 0.5, b = 1 and c = 1. For the sake of numerical simulation, take a = 0.5 (so that the systems in ( 77) and ( 76) diverge slowly). [START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF]. Thus, In Figure 9, the closed-loop system state x performs a jump at initial time t = 0 and a further jump at t = 1s. At the initial time t = 0, the state x jumps such that x(0 + ) = -2.53 ∈ K which is the solution of (9) and according to Lemma 3.10, at an initial state jump we have V (0 + ) -V (0 -) ≤ 0 which is observed in the simulation shown in Figure 9. At t = 1.2s, the state x performs a jump such that x(1 + ) = 2.29 ∈ K which is the solution of the optimization problem in [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF]. Knowing that the controller is chosen such that -2.59 ≤ v d (1

v d (1 + ) such that v d (1 + ) = min{-x d (1 -) -δ, x(1 -) -δ} < min{-x d (1 -), x(1 -)} with δ = 0.5. Based on the value of δ, v d (1 + ) = min{0.2, -2.78} = -2.78.
x d (1 -) / ∈ K d ⇔ cx d (1 -) + v d (1 + ) < 0 ⇒ v d (1 + ) < 0.7 and x(1) ∈ K ⇔ (c + k 2 )x(1) -k 2 x d (1 + ) + v d (1 + ) ≥ 0 ⇒ v d (1 + ) ≥ -2.
-) = 1.83. It is required to choose v d (1 + ) such that x d (1) ∈ K d in (81) and x(1 -) / ∈ K in
x d (1) ∈ K d ⇔ cx d (1) + v d (1 + ) ≥ 0 ⇒ v d (1 + ) ≥ -2.59 and
x(1 -) / ∈ K ⇔ (c + k 2 )x(1 -) -k 2 x d (1 + ) + v d (1 + ) < 0 ⇒ v d (1 + ) < -

Conditions such that Proposition 3.11's conditions are satisfied at state jump, with G = 0

The above conditions that guarantee a negative Lyapunov function jump, may not match with the sufficient conditions of Proposition 3.

x d is discontinuous at t = 1s, then x d (1 + ) = -v d (1 + ). Assume that at t = 1s: 0 ∈ K ⇔ -k 2 x d (1 + ) + v d (1 + ) ≥ 0 ⇔ (1 + k 2 )v d (1 + ) ≥ 0 ⇒ v d (1 + ) ≥ 0 and x d (1 + )px(1 -) ≥ x d (1 -)px(1 -) ⇔ -v d (1 + )px(1 -) ≥ x d (1 -)px(1 -) ⇒ v d (1 + ) ≤ 4 and x d (1 + ) ∈ T K (proj P [K; x(1 -)]) ⇔ x d (1 + ) ∈ T K (x(1 + )) ⇔ x d (1 + ) ∈ (N K (x(1 + ))) • Knowing that -p(x(1 + ) -x(1 -)) ∈ N K (x(1 +
)), then the last condition is written as 1) (i.e., the state x is continuous at t = 1s).

x d (1 + ) ∈ (N K (x(1 + ))) • ⇒ -x d (1 + )p(x(1 + ) -x(1 -)) ≤ 0 which is satisfied since x(1 + ) = x(1 -) = x(

Let us choose v

d (1 + ) such that 0 ≤ v d (1 + ) ≤ 4
+ ) = k2x + d -v + d c+k2
from Appendix A. [START_REF] Aydinoglu | Stabilization of complementarity systems via contact-aware controllers[END_REF]. Assume that at t = 1s:

0 ∈ K ⇔ -k 2 x d (1 + ) + v d (1 + ) ≥ 0 ⇔ v d (1 + ) ≥ 5.65 and x d (1 + )px(1 -) ≥ x d (1 -)px(1 -) ⇔ x d (1)px(1 -) = x d (1)px(1 -) and x d (t + ) ∈ T K (proj P [K; x(t -)]) ⇔ x d (t + ) ∈ T K (x(t + )) ⇔ x d (t + ) ∈ (N K (x(t + ))) • Knowing that -p(x(1 + ) -x(1 -)) ∈ N K (x(1 +
)), then the last condition is written as

x d (1 + ) ∈ (N K (x(1 + ))) • ⇒ -x d (1 + )p(x(1 + ) -x(1 -)) ≤ 0 ⇒ x d (1 + )p k2x + d -v + d c+k2 -x(1 -) ≤ 0 ⇒ v d (1 + ) ≥ 1.5
Let us choose v d (1 + ) ≥ 5.56 such that the above conditions (i.e.,the conditions of Proposition 3.11) are satisfied and -5 ≤ v d (1 + ) < 9.6 (i.e., x d (1) ∈ K d and x(1 -) / ∈ K at t = 1s), so take v d (1 + ) = 6. The numerical simulation is shown in the following with a time step h = 0.001.

Recapitulation

The occurrence of "peaking phenomenon", which occurs when the jump times of two trajectories do not coincide, is not observed in our case. This is due to the fact that the desired system in ( 77) is derived from the dynamics of the real system in [START_REF] Rockafellar | Convex Analysis[END_REF]

Applications: Circuits with Ideal Diodes

The next sections are dedicated to applications on electrical circuits with ideal diodes (recall that hydraulic circuits share same components where ideal diodes are replaced by check valves [START_REF] Kikuuwe | A nonsmooth quasi-static modeling approach for hydraulic actuators[END_REF]). Let us consider the following examples which differ mainly by the structure of the matrix D as well as the position of the controller in the dynamics (i.e the differential equations and/or complementarity problem). The following examples are provided with their numerical simulation which is done with the INRIA software package siconos2 [START_REF] Acary | An introduction to Siconos[END_REF]. The algorithm of the code is explained as follows: the nominal system is defined at the beginning and it is used in order to design an optimal controller with optimal control gains by solving the LMI in (45) using mosek 9.3.14 solver [START_REF] Mosek Aps | MOSEK Optimizer API for Python[END_REF]. Then, the real system is defined with constrained uncertainties. It is noteworthy that the uncertainties are constrained according to the values of Λ k 's for k ∈ {A, B, C, D, E, F }, as shown in [START_REF] Colombo | Optimal control of the sweeping process[END_REF], which are unknown variables determined from the solutions (Λ 1 , Λ 2 , Λ1 , Λ2 and Λ F ) of the LMI in [START_REF] San Felice | [END_REF].

Electrical Circuit with Parametric Uncertainties

Consider the circuit in Figure 12 with the states x 1 : the charge on the capacitor C and x 2 : the current passing through the inductor L. The dynamics of the electrical circuit in Figure 12 is given by the system in (89):

L u C x 2 i C R + - - + w Figure 12: RLCD circuit with one ideal diode      ẋ1 (t) = -x 2 (t) + λ(t) ẋ2 (t) = x1(t) LC + u(t) L 0 ≤ λ(t) ⊥ w(t) = x1(t) C + Rλ(t) ≥ 0 (89)
with L = 1 H, C = 0.025 F, R = 10 Ω.

Study the passivity of (12) with u = 0

The energy stored in the circuit of Figure 12 with u = 0 is:

V (x) = 1 2 C x1 C 2 + 1 2 Lx 2 2 .
The rate of change in stored energy is given by V

(x) = C x1 C ẋ1 C + Lx 2 ẋ2
. If the first two lines of (89) are substituted in the equation of V , then

V (x) = Lx 2 x 1 LC + x 1 C (-x 2 + λ) = x 1 λ C (90) Knowing that x 1 = CV C = C(-V R -V D )
where V R and V D are the voltages across the resistor and the diode respectively, the rate of change of the storage function is represented as follows:

V (x) = λ C (-Rλ -wλ) = - R C λ 2 (91)
where wλ = 0 due to orthogonality. So, V (x) ≤ 0 (i.e., the rate of change in stored energy is less than the power supplied to the system). Thus, the dynamical system in [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF] with u = 0 is passive.

Let us check the passivity of ( 12) by proving that its open loop transfer function (i.e., TF with u = 0) which is given by H

0 (s) = C(sI -A) -1 B + D is positive real. Then, H 0 (s) = w(s) λ(s) = RLCs 2 + Ls + R LCs 2 + 1
The transfer function H 0 is Hurwitz and Re(H 0 (jω)) = R > 0. Then, the transfer function H 0 is strictly positive real [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF]Theorem 2.45]. Therefore, the open loop system in ( 12) is strictly passive.

Passivity of closed-loop system (12) without uncertainties

Let u = K(x -x d ) + G(λ -λ d ) + u d with K = k 1 k 2 and G = g 1 .
In the case of which the system (89) has no uncertainties, the control gains are calculated by solving the BMI in ( 20) after being transformed into LMI according to A.2; the solution is obtained as follows: 20) is solved with the minimum value of the control gain G, then the solution is given by the following: K = 75.0276 -5.2589 , G = -3.9 × 10 -13 and P = 26.06268 -0.99185 -0.99185 0.25716 (93)

K = 493
It is observed that the control gain G has a negligible value and this indicates that the additional feedback from the complementarity variable λ is useless in this example. So, in the following, the value of G is neglected.

In order to decrease the magnitude of the control gain K, the steps below are followed knowing that K = N Q -1 :

• solve the LMI derived from the LMI in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF] (see Appendix A.2) with G = 0 for the minimum Euclidean norm of the matrix N with fixed Q = P -1 in (93).

• Now, fix the value of N obtained from the previous step and solve the LMI in ( 20) again for the maximum trace of the matrix Q.

• repeat the first two steps until there is no change in the values of N and Q obtained.

This methodology gives the following solution when followed: The plots in Figure 13 show the numerical simulation for the desired and closed loop system with two different forms of controllers. The initial state vectors are x(0) = (1, 0) ⊤ and x d (0) = (-1.5, 1) ⊤ , the time step h = 10 -3 and the desired input is u d (t) = 30 sin 5t. In Figures 13a and13b, L 2 and L ∞ norms of the error between real and desired trajectories are observed at steady state. In Figure 13a, ∥e In Figure 14a, the matrix D is perturbed by ∆D due to disturbance in the resistor R. The perturbation is selected such that ∆D is bounded according to the value of Λ D . For this purpose, the following LMI

K = 38
(a) u = K(x -x d ) + G(λ -λ d ) + u d (b) u = K(x -x d ) + u d
    (M 0 lin ) 11 (M 0 lin ) 12 0 0 (M 0 lin ) 21 (M 0 lin ) 22 I m I m 0 I m Λ -1 2 0 0 I m 0 Λ-1 2     ≻ 0 (94)
which is a special case of the LMI in ( 45) is solved such that Λ-1 In Figures 14c and14d, the uncertainties are added to more matrices of the dynamics in (89). So, the controller is designed by calculating K 0 and G 0 such that the quadruple The bounds of the uncertainties are calculated according to the values of Λ-1 1 and Λ-1 2 . In Figure 14c

(A 0 + E 0 K 0 , B 0 + E 0 G 0 , C 0 , D 0 ) is

Electrical circuit that cannot be made strictly passive

Consider the circuit depicted in Figure 15, with the states x 1 which is the charge on capacitor C and x 2 is the current passing through the inductor L.

C L R u i R i L i C λ w 1 u 2 + + - - Figure 15: RLCD circuit with one ideal diode.      ẋ1 (t) = -x1(t) RC + x 2 (t) + 1 R λ(t) -1 R u 1 (t) ẋ2 (t) = -x1(t) LC + 1 L λ(t) -1 L u 1 (t) -1 L u 2 (t) 0 ≤ λ(t) ⊥ w(t) = -x1(t) RC + x 2 (t) + 1 R λ(t) -1 R u 1 (t) ≥ 0 (
H 0 (s) = w(s) λ(s) = s 2 + R L s Rs 2 + 1 C s + R LC where H(s) = C(sI -A) -1 B + D.
The Nyquist plot of H(s) lies in the closed right-half plane (see Figure 16), which is consistent with positive-real systems theory.

Figure 16: Nyquist Diagram Then, from minimality and the KYP Lemma [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF], the quadruple (A, B, C, D) is passive. The storage function of the circuit of Figure 15 with u = 0 is:

V (x) = 1 2 C x1 C 2 + 1 2 Lx 2 2 . It follows that: V (x) = - x 2 1 RC 2 + λ x 2 + x 1 RC (96)
Due to complementarity conditions between λ and w, we have:

• If λ = 0, then V (x) = - x 2 1 RC 2 ≤ 0. • If λ > 0, then w = 0 and λ = x1 C -Rx 2 .
Hence, V (x) = -Rx 2 2 ≤ 0. Thus, the storage function of the circuit of Figure 15 with u = 0 is nonincreasing. Take u = u 1 and u 2 = 0. The transfer function of the closed-loop system is:

H 1 (s) = [C + F K][sI -(A + EK)] -1 [B + EG] + D + F G. Knowing that u = K[x -x d ] + G[λ -λ d ] + u d with K = k 1 k 2
and G = g, the following is obtained:

H 1 (s) = w(s) λ(s) = (1 -g) LCs 2 + RCs RLCs 2 + (L + k 1 LC + k 2 RC)s + k 1 RC + R (97)
Let us consider u 1 = 0 and u = u 2 . The transfer function of the systems is represented as follows:

H 2 (s) = w(s) λ(s) = LCs 2 + (k 2 C + RC -gRC)s RLCs 2 + (k 2 RC + L)s + k 2 + k 1 RC + R (98) If u = (u 1 , u 2 ) ⊤ with K = k 1 k 2 k 3 k 4 and G = g 1 g 2
, then the transfer function of the system is represented as follows:

H 3 (s) = w(s) λ(s) = (1 -g 1 )(LCs 2 + [(k 2 + k 4 )C -g 2 (RC + k 2 C)]s) RLCs 2 + [RC(k 2 + k 4 ) + L + k 1 LC]s + RC(k 1 + k 3 ) + C(k 1 k 4 + k 2 k 3 ) + R + k 4
(99) We can see from ( 97), ( 98) and (99) that the transfer functions of the closed loop system cannot be made strictly positive real, thus the circuit shown in Figure 15 cannot be made strictly passive with u 1 and u 2 of the general form u

= K[x -x d ] + G[λ -λ d ] + u d .
In terms of zero dynamics of the closed-loop system, when w = 0, there is always one zero at zero (i.e ẋ1 = 0).

Study of passivity of the circuit in Figure 15 with current source

The problem that appeared in section 9, which states that the system in (95) cannot be made strictly state passive even with u = (u 1 , u 2 ) ⊤ motivates us to change the type and the connection of the controller for the circuit of Figure 15. If the controller is designed as a current source connected in parallel with the capacitor, then the following is obtained:

C L R i R i L i C λ w i 1 (t)
     ẋ1 (t) = -x1(t) RC + x 2 (t) + 1 R λ(t) + u(t) ẋ2 (t) = -x1(t) LC + 1 L λ(t) 0 ≤ λ(t) ⊥ w(t) = -x1(t) RC + x 2 (t) + 1 R λ(t) ≥ 0 (100) If u(t) = K[x -x d ] + u d (t)
, then the transfer function of the closed loop system is:

H 4 (s) = C[sI -(A + EK)] -1
B + D and represented as follows:

H 4 (s) = LCs 2 + (RC -k 1 LC)s -k 1 RC RLCs 2 + (L -k 1 RLC)s + R + k 2 R (101)
It is noticeable that the zeros of the transfer function are replaced by the feedback controller designed and implemented in the RLCD circuit as shown in Figure 17. This is observed through the zero dynamics of the system presented in (100). If w = 0, then ẋ1 = u which means that the zero dynamics is set in the desired place using the feedback controller.

Proposition 10.1. The transfer function in (101) is strongly strictly positive real (SSPR) if and only if there exists K = k 1 k 2 such that

• k 1 < 0 • k 2 ∈ -1, k 1 (k 1 LC -L R -2RC) + 2 -k 3 1 RLC 2 + k 2 1 LC -k 1 R 2 C -k 1 R
Proof. In order to check the strong strict positive realness of the transfer function H 4 (s), it is required to present the transfer function in its frequency domain as

H 4 (jω) = Re[H 4 (jω)] + Im[H 4 (jω)] such that: Re[H 4 (jω)] = RL 2 C 2 ω 4 + (-k 2 RLC + k 2 1 RL 2 C 2 -k 1 L 2 C)ω 2 -k 1 R 2 C(1 + k 2 ) (-RLCω 2 + R + k 2 R) 2 + (L -k 1 RLC) 2 ω 2 (102) Im[H 4 (jω)] = j (L 2 C -R 2 LC 2 )ω 3 + (R 2 C + k 2 R 2 C -k 2 1 R 2 LC 2 -k 1 k 2 RLC)ω (-RLCω 2 + R + k 2 R) 2 + (L -k 1 RLC) 2 ω 2 (103)
The transfer function H 4 (s) is strongly strictly positive real if and only if the following conditions are satisfied [25, Definition 2.78]:

• H 4 (s) is Hurwitz,

• Re[H 4 (jω)] ≥ δ > 0 for all ω ∈ [-∞, ∞] and some δ ∈ R.

• 1 + k 2 > 0 • L -k 1 RLC > 0
The function Re[H 4 (jw)] ≥ δ > 0 if and only if the function g(ω)

∆ = RL 2 C 2 ω 4 + (-k 2 RLC + k 2 1 RL 2 C 2 -k 1 L 2 C)ω 2 -k 1 R 2 C(1 + k 2 ) has a global minimum at ω = ω 0 such that g(ω 0 ) > 0.
Knowing that 1 + k 2 > 0, it is necessary to have:

-k 1 R 2 C(1 + k 2 ) > 0 ⇒ k 1 < 0 (104)
so that the function g(0) > 0. The minimum of g(ω) is studied as follows:

∂g ∂ω (ω) = 4RL 2 C 2 ω 3 + 2(-k 2 RLC + k 2 1 RL 2 C 2 -k 1 L 2 C)ω
At this step, there are two cases:

First case: If -k 2 RLC + k 2 1 RL 2 C 2 -k 1 L 2 C > 0 ⇔ k 2 < k 1 L R (k 1 RC -1) (105)
then the function g(ω) is convex and has a global minimum at ω = 0 where ∂g ∂ω (0) = 0. The function g(ω) > 0 ∀ω ∈ [-∞, +∞] if and only if the following holds:

g(0) = -k 1 R 2 C(1 + k 2 ) > 0 ⇔ k 1 < 0 ( 106
)
Second case:

If -k 2 RLC + k 2 1 RL 2 C 2 -k 1 L 2 C < 0 ⇔ k 2 > k 1 L R (k 1 RC -1) (107)
then the roots of ∂g ∂ω (ω) are: ω 1 = 0 and ω 2,3 = ± -

-k2RLC+k 2 1 RL 2 C 2 -k1L 2 C 2RL 2 C 2
. The second derivative of g(ω) is:

∂ 2 g ∂ω 2 (ω) = 12RL 2 C 2 ω 2 + 2(-k 2 RLC + k 2 1 RL 2 C 2 -k 1 L 2 C)
The value of

∂ 2 g ∂ω 2 (ω 2,3 ) = -4(-k 2 RLC + k 2 1 RL 2 C 2 -k 1 L 2 C) > 0. So, the function g(ω) has global minima at ω = ± - -k2RLC+k 2 1 RL 2 C 2 -k1L 2 C 2RL 2 C 2
. The value of g(ω 2,3 ) must be positive. It follows that:

g(ω 2,3 ) = -(-k 2 RLC + k 2 1 RL 2 C 2 -k 1 L 2 C) 2 + 4RL 2 C 2 (-k 1 R 2 C(1 + k 2 )) 4RL 2 C 2 > 0 (108) Then, - R 4 k 2 2 + k 2 k 2 1 RLC 2 -k 1 L 2 -k 1 R 2 C -k 4 1 RL 2 C 2 4 + k 3 1 L 2 C 2 -k 2 1 L 2 4R -k 1 R 2 C > 0 (109)
The function g(ω 2,3 ) in ( 109) is a concave function in k 2 with the discriminant:

∆ = 16R 4 C(-k 3 1 RLC + k 2 1 L + k 2 1 R 2 C -k 1 R) > 0 since k 1 < 0
The roots of the quadratic function h(k 2 ) are:

k 21 = k 2 1 LC -k 1 ( L R + 2RC) + 2 -k 3 1 RLC 2 + k 2 1 LC -k 1 R 2 C -k 1 R or k 22 = k 2 1 LC -k 1 ( L R + 2RC) -2 -k 3 1 RLC 2 + k 2 1 LC -k 1 R 2 C -k 1 R So, the function h(k 2 ) is positive iff k 22 < k 2 < k 21 .
Hence, the function g(ω), in this case, is positive, if and only if the conditions (104),( 107) and (108) are satisfied.

It is noteworthy that if

k 2 = k 1 L R (k 1 RC -1), then RL 2 C 2 ω 4 -k 1 R 2 C 1 + k 2 1 LC -k 1 L R > 0 ∀ω ∈ [-∞, +∞].
If the transfer function in (101) is SSPR for k 1 and k 2 satisfying Proposition 10.1, then the nonlinear matrix inequality in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF] always has a solution. Thus, the system is strongly passive. Let us check strong passivity by solving the NLMI in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF]

                                      ẋ1 (t) ẋ2 (t) ẋ3 (t) ẋ4 (t)      =      0 0 0 0 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC           x 1 (t) x 2 (t) x 3 (t) x 4 (t)      +      1 C 0 0 0 0 1 C 1 C 0      λ 1 (t) λ 2 (t) 0 ≤ λ 1 (t) λ 2 (t) ⊥ w 1 (t) w 2 (t) = 1 0 0 1 0 0 1 0      x 1 (t) x 2 (t) x 3 (t) x 4 (t)      + R 0 0 0 λ 1 (t) λ 2 (t) + -1 0 u ≥ 0
(110) Note that there exists P ⪰ 0 such that the quadruple (A, B, C, D) is passive (i.e the LMI in (7) has a solution). In order to check the strict state passivity of (110). Let us observe the strict positive realness (SPR) of the transfer function matrix given by:

G(s) =     R 2 Cs 4 +s 3 (5R-k1R-k4R)+ s 2 C (-3k1+k2+k3-2k4+5) RCs 4 +3s 3 k3RCs 3 +s 2 (k2+2k3+k4-1) RC 2 s 4 +3Cs 3 -s 2 RC 2 s 4 +3Cs 3 Rs 3 + 2s 2 C RCs 4 +3s 3     (111) where G(s) ∆ = G 11 (s) G 12 (s) G 21 (s) G 22 (s) ∆ = w1 λ1 w1 λ2 w2 λ1 w2 λ2
.

In view of the matrix of the transfer functions in (111), it is noticeable that the transfer function is not Hurwitz, hence not strictly positive real due to the pole at zero of multiplicity 3. Thus, the circuit of Figure 20 is not strictly state passive (SSP) and we cannot analyze the stability of the LCS in (110) in the presence of state jumps.

Build a strict state passive circuit without adding controllers

Knowing that strict state passivity is a sufficient condition for stability analysis in the case of state jumps, the aim is to find an SSP circuit with a matrix D ⪰ 0. One approach to make the circuit of Figure 20 strictly passive would be by adding proper resistors, i.e adding dissipativity in the circuit as shown in Figure 21.

C C u R R R R R C C w 2 - w 1 + - + λ 2 i1 1 2 3 4 iR i 3 (a) R R C C C C u w 2 - w 1 + - + λ 2 i1 1 2 3 4 iR i2 i 3 i4 R R (b)
Figure 21: RLCD circuit with two ideal diodes.

Their dynamical equations are given as follows:

(b)

                                      ẋ1 (t) ẋ2 (t) ẋ3 (t) ẋ4 (t)      =      -1 RC 0 0 0 0 -2 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -2 RC           x 1 (t) x 2 (t) x 3 (t) x 4 (t)      +      1 C 0 0 0 0 1 C 1 C 0      λ 1 (t) λ 2 (t) 0 ≤ λ 1 (t) λ 2 (t) ⊥ w 1 (t) w 2 (t) = 1 0 0 1 0 0 1 0      x 1 (t) x 2 (t) x 3 (t) x 4 (t)      + R 0 0 0 λ 1 (t) λ 2 (t) + -1 0 u ≥ 0 (112) (c)                                       ẋ1 (t) ẋ2 (t) ẋ3 (t) ẋ4 (t)      =      -1 RC 0 0 0 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -2 RC           x 1 (t) x 2 (t) x 3 (t) x 4 (t)      +      1 C 0 0 0 0 1 C 1 C 0      λ 1 (t) λ 2 (t) 0 ≤ λ 1 (t) λ 2 (t) ⊥ w 1 (t) w 2 (t) = 1 0 0 1 0 0 1 0      x 1 (t) x 2 (t) x 3 (t) x 4 (t)      + R 0 0 0 λ 1 (t) λ 2 (t) + -1 0 u ≥ 0
(113) It is noteworthy that connecting a resistor across the capacitor C 3 would cancel the property of positive semi-definiteness (PSD) for the matrix D which is a necessary condition for the existence of state jumps.

The transfer function matrix of the closed-loop system in (112) is represented as follows:

H(s) ∆ = w1 λ1 w1 λ2 w2 λ1 w2 λ2 ∆ = H 11 (s) H 12 (s) H 21 (s) H 22 (s)
where

H 11 (s) = R 4 C 3 (k 1 + k 2 + 2)s 3 + R 3 C 2 (5k 1 -k 2 -k 3 + 4k 4 + 9)s 2 + R 2 C(5k 1 -k 2 -2k 3 + 4k 4 + 9)s + R(k 1 -k 3 + k 4 + 2) R 4 C 4 s 4 + 6R 3 C 3 s 3 + 10R 2 C 2 s 2 + 6RCs + 1 +R H 12 (s) = R 4 C 3 k 3 s 3 + R 3 C 2 (-k 2 + 5k 3 -k 4 -1)s 2 + R 2 C(-2k 2 + 4k 3 -2k 4 -2)s + R(-k 2 + 3k 3 -k 4 -1) R 4 C 4 s 4 + 6R 3 C 3 s 3 + 10R 2 C 2 s 2 + 6RCs + 1 H 21 (s) = -R 3 C 2 s 2 -2R 2 Cs -R R 4 C 4 s 4 + 6R 3 C 3 s 3 + 10R 2 C 2 s 2 + 6RCs + 1 and H 22 (s) = R 4 C 3 s 3 + 5R 3 C 2 s 2 + 4R 2 Cs + 3R R 4 C 4 s 4 + 6R 3 C 3 s 3 + 10R 2 C 2 s 2 + 6RCs + 1
The poles are the roots of the characteristic equation:

R 4 C 4 s 4 + 6R 3 C 3 s 3 + 10R 2 C 2 s 2 + 6RCs + 1
and they are respresented as follows:

• s = -1 RC (double root), • s = -2 RC + 3 R 2 C 2 ,
and

• s = -2 RC - 3 R 2 C 2 .
Hence, the problem of the pole at zero with multiplicity 3 which appears in the system represented in Figure 20 is solved in system represented in Figure 21a since all the poles have negative real part. In order to check if there exist K and P such that the quadruple (A, B, C + F K, D) of the closed loop system (112) is strictly passive, the NLMI in ( 19) is solved and the solution is:

K = -0.304 -0.658 0 -0.962 and P =     0.0032 0 0 0 0 0.005 0 0.002 0 0 0.0025 0 0 0.002 0 0.005     (114)
with ϵ = 0.1.

The variable ϵ is chosen arbitrarily by the user such that the system is strictly passive with a desired degree of strict passivity (as ϵ increases, the degree of strict passitvity increases). The value of ϵ can be increased until it reaches a crtitical value, at which the LMI becomes infeasible again. If F = 0, then the open loop system of (112) is strictly passive as well. So, in our case, the open loop system is strictly passive and the role of ϵ is to increase the degree of strict passivity by pushing the eigenvalues of the matrix A + EK to be more negative. But, E = 0 in this system, thus varying the value of ϵ does not have an interesting effect on the control gain. It is noteworthy that the critical value of ϵ (the last value at which the system is strictly passive) is ϵ = 2 in the closed-loop system represented by (112).

In order to check if it is necessary to connect three resistors across C 1 , C 2 and C 4 , let us remove one of these resistors as shown in Figure 21b. The poles of the transfer function can be viewed as the poles of det|sI -A| -1 which is given by:

R 3 C 3 s(RCs + 1)(R 2 C 2 s 2 + 4RCs + 2)
with one pole at zero. Hence, the transfer function is not strictly positive real and the system represented by (113) is not strictly state passive. Thus, the closed-loop as well as the open-loop of the system in (112) of the circuit in Figure 21a are strictly state passive.

Build a strictly state passive circuit from the circuit in Figure 20 by adding controllers

Let us now add controllers (current sources) to the circuit in Figure 20 in different numbers and places in order to study strict state passivity of the closed-loop system as shown in circuits of Figure 22.

w 2 - w 1 + - + λ 2 i1 1 2 3 4 iR i 3 C C u R R C C i c1 i c2 i c3 0 (d) R R C C C C u w 2 - w 1 + - + λ 2 i1 1 2 3 4 iR i2 i 3 ic1 ic2 (e)
Figure 22: RLCD circuit with two ideal diodes and both voltage and current sources.

Their dynamics are given as follows:

(d)                                       ẋ1 (t) ẋ2 (t) ẋ3 (t) ẋ4 (t)      =      0 0 0 0 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC           x 1 (t) x 2 (t) x 3 (t) x 4 (t)      +      1 C 0 0 0 0 1 C 1 C 0      λ 1 (t) λ 2 (t) +      0 -1 C 0 0 0 0 -1 C 0 0 0 0 0 0 0 0 -1 C           u 0 i c1 i c2 i c3      0 ≤ λ 1 (t) λ 2 (t) ⊥ w 1 (t) w 2 (t) = 1 0 0 1 0 0 1 0      x 1 (t) x 2 (t) x 3 (t) x 4 (t)      + R 0 0 0 λ 1 (t) λ 2 (t) + -1 0 0 0 0 0 0 0      u 0 i c1 i c2 i c3      ≥ 0 (115) (e)
                                      ẋ1 (t) ẋ2 (t) ẋ3 (t) ẋ4 (t)      =      0 0 0 0 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC           x 1 (t) x 2 (t) x 3 (t) x 4 (t)      +      1 C 0 0 0 0 1 C 1 C 0      λ 1 (t) λ 2 (t) +      0 -1 C 0 0 0 -1 C 0 0 0 0 -1 C -1 C         u i c1 i c2    0 ≤ λ 1 (t) λ 2 (t) ⊥ w 1 (t) w 2 (t) = 1 0 0 1 0 0 1 0      x 1 (t) x 2 (t) x 3 (t) x 4 (t)      + R 0 0 0 λ 1 (t) λ 2 (t) + -1 0 0 0 0 0    u i c1 i c2    ≥ 0 (116)
If E = 0 and F = 0, then the NLMI in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] has no solution and the open loop system (A, B, C, D) for circuits in Figure 22 is not strictly passive. In order to avoid Dirac measure in the presence of state jumps, let us take G = 0 in the following study (so that (C1) is satisfied). The NLMI in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] for strict state passivity has a solution when solved for the closed loop system (A + EK, B, C + F K, D) represented by (115). The solution is:

K =     0.52 -0.01 0
with

ϵ = 0.1. Let K =     k 11 k 12 k 13 k 14 k 21 k 22 k 23 k 24 k 31 k 32 k 33 k 34 k 41 k 42 k 43 k 44     . The closed loop system with u = u 0 i c1 i c2 i c3 ⊤ = K[x -x d ] + u d is represented as follows:     ẋ1 (t) ẋ2 (t) ẋ3 (t) ẋ4 (t)     =     -k21 C -k22 C -k23 C -k24 C -k31 C -1 C ( 1 R + k 32 ) -1 C ( 1 R + k 33 ) -1 C ( 1 R + k 34 ) 0 -1 RC -1 RC -1 RC -k41 C -1 C ( 1 R + k 42 ) -1 C ( 1 R + k 43 ) -1 C ( 1 R + k 44 )         x 1 (t) x 2 (t) x 3 (t) x 4 (t)     +     1 C 0 0 0 0 1 C 1 C 0     λ 1 (t) λ 2 (t) -     -k21 C -k22 C -k23 C -k24 C -k31 C -k32 C -k33 C -k34 C 0 0 0 0 -k41 C -k42 C -k43 C -k44 C         x 1d (t) x 2d (t) x 3d (t) x 4d (t)     +     0 -1 C 0 0 0 0 -1 C 0 0 0 0 0 0 0 0 -1 C         u 0d i c1d i c2d i c3d     0 ≤ w 1 (t) w 2 (t) = 1 -k 11 -k 12 -k 13 1 -k 14 0 0 1 0     x 1 (t) x 2 (t) x 3 (t) x 4 (t)     + R 0 0 0 λ 1 (t) λ 2 (t) - -k 11 -k 12 -k 13 -k 14 0 0 0 0     x 1d (t) x 2d (t) x 3d (t) x 4d (t)     + -1 0 0 0 0 0 0 0     u 0d i c1d i c2d i c3d     ⊥ λ 1 (t) λ 2 (t) ≥ 0
(118) The desired dynamics are represented as follows:

    ẋ1d (t) ẋ2d (t) ẋ3d (t) ẋ4d (t)     =     0 0 0 0 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC         x 1d (t) x 2d (t) x 3d (t) x 4d (t)     +     1 C 0 0 0 0 1 C 1 C 0     λ 1d (t) λ 2d (t) +     0 -1 C 0 0 0 0 -1 C 0 0 0 0 0 0 0 0 -1 C         u 0d i c1d i c2d i c3d     0 ≤ λ 1d (t) λ 2d (t) ⊥ w 1d (t) w 2d (t) = 1 0 0 1 0 0 1 0     x 1d (t) x 2d (t) x 3d (t) x 4d (t)     + R 0 0 0 λ 1d (t) λ 2d (t) + -1 0 0 0 0 0 0 0     u 0d i c1d i c2d i c3d     ≥ 0 (119)
The error dynamics of (115) is written in the form of (18) as follows:

    ė1 (t) ė2 (t) ė3 (t) ė4 (t)     =     -k21 C -k22 C -k23 C -k24 C -k31 C -1 C ( 1 R + k 32 ) -1 C ( 1 R + k 33 ) -1 C ( 1 R + k 34 ) 0 -1 RC -1 RC -1 RC -k41 C -1 C ( 1 R + k 42 ) -1 C ( 1 R + k 43 ) -1 C ( 1 R + k 44 )         e 1 (t) e 2 (t) e 3 (t) e 4 (t)     +     1 C 0 0 0 0 1 C 1 C 0     λ 1 -λ 1d λ 2 -λ 2d 0 ≤     w 1 (t) w 2 (t) w 1d (t) w 2d (t)     =     1 -k 11 -k 12 -k 13 1 -k 14 0 0 1 0 k 11 -1 k 12 k 13 k 14 -1 0 0 -1 0         e 1 (t) e 2 (t) e 3 (t) e 4 (t)     +     R 0 0 0 0 0 0 0 0 0 R 0 0 0 0 0         λ 1 (t) λ 2 (t) λ 1d (t) λ 2d (t)     +     0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 1 -k 11 -k 12 -k 13 1 -k 14 k 11 k 12 k 13 k 14 0 0 1 0 0 0 0 0                 x 1 x 2 x 3 x 4 x 1d x 2d x 3d x 4d             +     -u 0d 0 -u 0d 0     ⊥     λ 1 (t) λ 2 (t) λ 1d (t) λ 2d (t)     ≥ 0
(120) Let us study state jumps in the system represented by the circuit in Figure 22d whose dynamics is in (119). For the corresponding desired system, the set:

Q d,D = {λ d ∈ IR 2 | 0 ≤ λ d ⊥ Dλ d ≥ 0} = {λ d ∈ IR 2 | λ 1d = 0 and λ 2d ∈ IR + } The dual cone of Q d,D is Q ⋆ d,D = {w d ∈ IR 2 | ⟨w d , λ d ⟩ ≥ 0} = {w d ∈ IR 2 | w 1d ∈ IR and w 2d ∈ IR + } and K d = {x d ∈ IR 4 | Cx d + F u d (t + ) ∈ Q ⋆ d,D } = {x d ∈ IR 4 | x 1d , x 2d , x 4d ∈ IR and x 3d ≥ 0}
According to the conditions stated in the set K, a state jump exists only at initial time in the desired system trajectory x 3d , and it occurs if and only if x 3d (0 -) < 0. It is notable that , by solving the minimization problem in (9), if x 3d (0 -) < 0, then x 3d (0 + ) = 0. Otherwise, if if x 3d (0 -) ≥ 0, then x 3d (0 + ) = x 3d (0 -).

For the closed-loop system whose dynamics is in (118), the sets are calculated as :

Q D = {λ ∈ IR 2 | 0 ≤ λ ⊥ Dλ ≥ 0} = {λ ∈ IR 2 | λ 1 = 0 and λ 2 ∈ IR + }
with a dual cone:

Q ⋆ D = {w ∈ IR 2 | ⟨w, λ⟩ ≥ 0} = {w ∈ IR 2 | w 1 ∈ IR and w 2 ∈ IR + }
and the set:

K = {x ∈ IR 4 | (C + F K)x -F Kx d + F u d (t) ∈ Q ⋆ D } = {x ∈ IR 4 | x 3 ≥ 0 and x 1 , x 2 and x 4 ∈ IR}
In the closed-loop system, it is observed from the set K and the state jump rule in (9) that a state jump occurs in the state x 3 only at t = 0, and it occurs if and only if x 3 (0 -) < 0 so that x 3 (0 + ) = 0.

For the error dynamics in (120):

Q e,D =        λ ∈ IR 4 | 0 ≤     λ 1 λ 2 λ 1d λ 2d     ⊥     R 0 0 0 0 0 0 0 0 0 R 0 0 0 0 0         λ 1 λ 2 λ 1d λ 2d     ≥ 0        = λ ∈ IR 4 | λ 1 = 0, λ 1d = 0, λ 2 and λ 2d ∈ IR +
with the dual cone given by:

Q ⋆ e,D = {w ∈ IR 4 | ⟨w, λ⟩ ≥ 0} = {w ∈ IR 4 | w 1 ∈ IR, w 2 ∈ IR + , w 1d ∈ IR and w 2d ∈ IR + }
and the set Knowing that the minimization problem in ( 9) is the same at initial time for the states x 3 and x 3d , the jump in e 3 vanishes when x 3 (0 -) = x 3d (0 -). A jump occurs at t = 0, in the error dynamics in e 3 if and only if e 3 (0 -) < -x 3d (0 -) or e 3 (0 -) > x 3 (0 -). If x 3d and x 3 have state jumps at t = 0 and if x 3 (0 -) ̸ = x 3d (0 -) , then e 3 jumps.

K e = e ∈ IR 4 | C + F K -C -F K e + 0 C C + F K -F K x x d + F u d F u d ∈ Q ⋆ e,D = {e ∈ IR 4 | e 1 ,
Consider the following numerical simulation with the initial state x(0 -) = (-1, 1, -2, 2) and

x d (0 -) = (1, 0, 1, -2)
and with a time step h = 0.001. Take u 0d = 5 sin 10t, i c1d = sin 5t, i c2d = 3 sin 5t and i c3d = 2 sin 3t. It is observed from the numerical simulation in Figure 23 that the state x 3 of the closed-loop system performs a state jump from x 3 (0 -) = -2 to x 3 (0 + ) = 0. This numerical result confirms the jump rule given by the minimization problem in [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF].

In Figure 23b, a synchronization is noticed between the complementarity variables λ and w in the desired and the real systems. The reason of this result is the fact that the term λ -λ d in (120) converges to Ker(B + EG) which is zero in this example.

It is observable in Figure 23c that the storage function jumps at initial time such that V (0 + ) -V (0 -) < 0. By applying Lemma 3.10, it is given in our example that u 0d (0) is time continuous, x d is continuous at t = 0 and

F Kx d (0)-F u d (0) ≥ 0 ⇔ 0 ∈ {x ∈ IR 4 | (C +F K)x+F Kx d (0)-F u d (0) ≥
0}, then at an initial state jump we have V (0 + ) -V (0 -) ≤ 0 where P = P ⊤ ≻ 0 given in (117).

Let us decrease the number of controllers (current sources) as shown in Figure 22e. The NLMI in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] has no solution for the closed-loop system in (116) which means that the stability of the LCS cannot be analyzed in the presence of state jumps.

Strictly state passive circuit with state jumps at t = t c > 0

Let us add a voltage source in series with the capacitor C 3 to the strictly passive circuit in Figure 22d so that a jump can occur at t = t c > 0 when

u d is discontinuous at t = t c . R R C C C C u 2 u 1 w 2 - w 1 + - + λ 2 i1 1 2 3 4 iR i 3 ic1 ic2 ic3 (f) R R C C C C u 1 w 2 - w 1 + - + λ 2 i1 1 2 3 4 iR i 3 ic1 ic2 ic3 (g)
Figure 24: RLCD circuit with two ideal diodes and both voltage and current sources.

Their dynamics are given by:

(f )                                                 ẋ1 (t) ẋ2 (t) ẋ3 (t) ẋ4 (t)      =      0 0 0 0 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC           x 1 (t) x 2 (t) x 3 (t) x 4 (t)      +      1 C 0 0 0 0 1 C 1 C 0      λ 1 (t) λ 2 (t) +      0 0 -1 C 0 0 0 -1 RC 0 -1 C 0 0 -1 RC 0 0 0 0 -1 RC 0 0 -1 C              u 1 u 2 i c1 i c2 i c3         0 ≤ λ 1 (t) λ 2 (t) ⊥ w 1 (t) w 2 (t) = 1 0 0 1 0 0 1 0      x 1 (t) x 2 (t) x 3 (t) x 4 (t)      + R 0 0 0 λ 1 (t) λ 2 (t) + -1 0 0 0 0 0 -1 0 0 0         u 1 u 2 i c1 i c2 i c3         ≥ 0 (121) (g)
                                      ẋ1 (t) ẋ2 (t) ẋ3 (t) ẋ4 (t)      =      0 0 0 0 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC           x 1 (t) x 2 (t) x 3 (t) x 4 (t)      +      1 C 0 0 0 0 1 C 1 C 0      λ 1 (t) λ 2 (t) +      0 -1 C 0 0 -1 RC 0 -1 C 0 -1 RC 0 0 0 -1 RC 0 0 -1 C           u 1 i c1 i c2 i c3      0 ≤ λ 1 (t) λ 2 (t) ⊥ w 1 (t) w 2 (t) = 1 0 0 1 0 0 1 0      x 1 (t) x 2 (t) x 3 (t) x 4 (t)      + R 0 0 0 λ 1 (t) λ 2 (t) + 0 0 0 0 1 0 0 0      u 1 i c1 i c2 i c3      ≥ 0 (122)
The open loop quadruple (A, B, C, D) (i.e E = 0 and F = 0) of circuits if Figure 24 is not strictly passive. But, while checking if there exist K and P such that the closed loop quadruple (A + EK, B, C + F K, D) is strictly passive, the NLMI in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] has a solution for circuit in Figure 24f such that: with ϵ = 0.01. Consider the circuit of Figure 24g which is strictly passive with less number of controllers. The desired dynamics of (122) is given as follows:

K =       -1.
                                      ẋ1d (t) ẋ2d (t) ẋ3d (t) ẋ4d (t)      =      0 0 0 0 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC 0 -1 RC -1 RC -1 RC           x 1d (t) x 2d (t) x 3d (t) x 4d (t)      +      1 C 0 0 0 0 1 C 1 C 0      λ 1d (t) λ 2d (t) +      0 -1 C 0 0 -1 RC 0 -1 C 0 -1 RC 0 0 0 -1 RC 0 0 -1 C           u 1d i c1d i c2d i c3d      0 ≤ λ 1d (t) λ 2d (t) ⊥ w 1d (t) w 2d (t) = 1 0 0 1 0 0 1 0      x 1d (t) x 2d (t) x 3d (t) x 4d (t)      + R 0 0 0 λ 1d (t) λ 2d (t) + 0 0 0 0 1 0 0 0      u 1d i c1d i c2d i c3d      ≥ 0
(124) For the desired system, the set:

Q d,D = {λ d ∈ IR 2 | 0 ≤ λ d ⊥ Dλ d ≥ 0} = {λ d ∈ IR 2 | λ 1d = 0 and λ 2d ∈ IR + } The dual cone of Q d,D is Q ⋆ d,D = {w d ∈ IR 2 | ⟨w d , λ d ⟩ ≥ 0} = {w d ∈ IR 2 | w 1d ∈ IR and w 2d ∈ IR + } and K d = {x d ∈ IR 4 | Cx d +F u d (t + ) ∈ Q ⋆ d,D } = {x d ∈ IR 4 | x 1d , x 2d , x 4d ∈ IR and x 3d +u 1d (t + ) ≥ 0} Let u = u 1 i c1 i c2 i c3 ⊤ = K(x-x d )+u d with K =     k 11 k
    =     -k21 C -k22 C -k23 C -k24 C -k11 RC -k31 C -1 RC (1 + k 12 ) -k32 C -1 RC (1 + k 13 ) -k33 C -1 RC (1 + k 14 ) -k34 C -k11 RC -1 RC (1 + k 12 ) -1 RC (1 + k 13 ) -1 RC (1 + k 14 ) -k11 RC -k41 C -1 RC -k12 RC -k42 C -1 RC -k13 RC -k43 C -1 RC -k14 RC -k44 C         x 1 (t) x 2 (t) x 3 (t) x 4 (t)     +     1 C 0 0 0 0 1 C 1 C 0     λ 1 (t) λ 2 (t) -     -k21 C -k22 C -k23 C -k24 C -k11 RC -k31 C -k12 RC -k32 C -k13 RC -k33 C -k14 RC -k34 C -k11 RC -k12 RC -k13 RC -k14 RC -k11 RC -k41 C -k12 RC -k42 C -k13 RC -k43 C -k14 RC -k44 C         x 1d (t) x 2d (t) x 3d (t) x 4d (t)     +     0 -1 C 0 0 -1 RC 0 -1 C 0 -1 RC 0 0 0 -1 RC 0 0 -1 C         u 1d i c1d i c2d i c3d     0 ≤ w 1 (t) w 2 (t) = 1 0 0 1 k 11 k 12 1 + k 13 k 14     x 1 (t) x 2 (t) x 3 (t) x 4 (t)     + R 0 0 0 λ 1 (t) λ 2 (t) - 0 0 0 0 k 11 k 12 k 13 k 14     x 1d (t) x 2d (t) x 3d (t) x 4d (t)     + 0 0 0 0 1 0 0 0     u 1d i c1d i c2d i c3d     ⊥ λ 1 (t) λ 2 (t) ≥ 0
(125) For the closed-loop system whose dynamics is in (125), the sets are calculated as :

Q D = {λ ∈ IR 2 | 0 ≤ λ ⊥ Dλ ≥ 0} = {λ ∈ IR 2 | λ 1 = 0 and λ 2 ∈ IR + }
with a dual cone:

Q ⋆ D = {w ∈ IR 2 | ⟨w, λ⟩ ≥ 0} = {w ∈ IR 2 | w 1 ∈ IR and w 2 ∈ IR + }
and the set:

K = {x ∈ IR 4 | (C + F K)x -F Kx d + F u d (t) ∈ Q ⋆ D } = {x ∈ IR 4 | x 1 + x 4 ∈ IR and k 11 (x 1 -x 1d ) + k 12 (x 2 -x 2d ) + k 13 (x 3 -x 3d ) + k 14 (x 4 -x 4d ) + x 3 + u 1d (t + ) ≥ 0}
The error dynamics is written in the form of the dynamics in [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF] as follows: For the circuit in Figure 26, the states x 1 and x 2 are the charge on the capacitor C and the current passing through the inductor L. The diode is connected in parallel with a voltage w and current λ.

    ė1 (t) ė2 (t) ė3 (t) ė4 (t)     =     -k21 C -k22 C -k23 C -k24 C -k11 RC -k31 C -1 RC (1 + k 12 ) -k32 C -1 RC (1 + k 13 ) -k33 C -1 RC (1 + k 14 ) -k34 C -k11 RC -1 RC (1 + k 12 ) -1 RC (1 + k 13 ) -1 RC (1 + k 14 ) -k11 RC -k41 C -1 RC -k12 RC -k42 C -1 RC -k13 RC -k43 C -1 RC -k14 RC -k44 C         e 1 (t) e 2 (t) e 3 (t) e 4 (t)     +     1 C 0 -1 C 0 0 0 0 0 0 1 C 0 -1 C 1 C 0 -1 C 0         λ 1 (t) λ 2 (t) λ 1d (t) λ 2d (t)     0 ≤     w 1 (t) w 2 (t) w 1d (t) w 2d (t)     =     1 0 0 1 k 11 k 12 1 + k 13 k 14 0 0 0 0 0 0 0 0         e 1 (t) e 2 (t) e 3 (t) e 4 (t)     +     R 0 0 0 0 0 0 0 0 0 R 0 0 0 0 0         λ 1 (t) λ 2 (t) λ 1d (t) λ 2d (t)     +     x 1d (t) + x 4d (t) x 3d (t) + u 1d (t) x 1d (t) + x 4d (t) x 3d (t) + u 1d (t)     ⊥     λ 1 (t) λ 2 (t) λ 1d (t) λ 2d (t)     ≥ 0 (
C L R u i R i L iC λ w Figure 26: RLCD circuit      ẋ1 (t) = -x1(t) RC -x 2 (t) + λ(t) -1 R u(t) ẋ2 (t) = x1(t) LC + 1 L u(t) 0 ≤ λ(t) ⊥ w(t) = x1(t) C + u(t) ≥ 0 (127)
Take L = 1H, R = 10Ω and C = 0.25F . The storage function of the circuit of Figure 15 with u = 0 is:

V (x) = 1 2 C x1 C 2 + 1 2 Lx 2 2 .
It follows that:

V (x) = - x 1 RC 2 + λw
Due to orthogonality between λ and w, the rate of change of the storage function V

(x) = -x1 RC 2 ≤ 0. If u(t) = K[x -x d ] + u d (t),
H(s) = s(RL + k RLC) s 2 -s(L + k 1 LC -k 2 RC) + R + k 1 RC
The transfer function H(s) cannot be made strictly state passive by the feedback u. In terms of zero dynamics of the closed-loop system, when w = 0, there is always one zero at zero (i.e ẋ2 = 0). In this example, D = 0. So, the desired system has the following sets:

Q D = {z ∈ IR | 0 ≤ z ⊥ Dz ≥ 0} = IR + = Q ⋆ D and K d = {x d ∈ IR 2 | Cx d + F u d (t) ≥ 0} = {x d ∈ IR 2 | x 1d ≥ -Cu d (t)}. For the closed loop system, when u = K[x -x d ] + u d , the set K = {x ∈ IR 2 | (C + F K)x -F Kx d + F u d (t) ≥ 0} = {x ∈ IR 2 | ( 1 C + k 1 )x 1 + k 2 x 2 ≥ k 1 x 1d + k 2 x 2d -u d }.

Diode Bridge Example

Consider the diode bridge rectifier circuit in Figure 27. Let x 1 be the current passing through the inductor L, x 2 be the voltage across the capacitor C 1 , and x 3 be the voltage across the capacitor C 2 . Take λ(t)

∆ = i DF 1 , i DR1 , -v DF 2 , i DR2 ⊤ and w(t) ∆ = -v DF 1 , -v DR1 , i DF 2 , -v DR2 ⊤ .
The dynamics are written as a linear complementarity system LCS and are given in (128).

C L R u i R i L iC iDF1 iDF2 iDR1 iDR2 1 C2 + - Figure 27: Diode Bridge                                 ẋ1 (t) ẋ2 (t) ẋ3 (t)    =    0 1 L 0 -1 C 0 0 0 0 -1 RC       x 1 (t) x 2 (t) x 3 (t)    +    0 0 0 0 -1 C 0 0 1 C 1 C 1 C 0 0         λ 1 (t) λ 2 (t) λ 3 (t) λ 4 (t)      0 ≤ λ(t) ⊥ w(t) =      0 -1 1 0 0 1 0 0 0 0 1 0         x 1 (t) x 2 (t) x 3 (t)    +      0 0 -1 0 0 0 -1 0 1 1 0 -1 0 0 1 0           λ 1 (t) λ 2 (t) λ 3 (t) λ 4 (t)      +      -1 0 0 1      u(t) ≥ 0
(128) Clearly P B = C ⊤ with P = CI 3 , and D + D ⊤ = 0 (i.e., D is a skew-symmetric matrix). The closed-loop system is not strictly passive.

Let us add more inputs (i.e., voltage sources) in the circuit of the diode bridge in Figure 27 as shown in the following Figure:

C L R i R i L iDF1 iDF2 iDR1 iDR2 1 C2 + - iC + - + - u 2 u 1 u 3 1 Figure 28: Diode Bridge                                 ẋ1 (t) ẋ2 (t) ẋ3 (t)    =    0 1 L 0 -1 C 0 0 0 0 -1 RC       x 1 (t) x 2 (t) x 3 (t)    +    0 0 0 0 -1 C 0 0 1 C 1 C 1 C 0 0         λ 1 (t) λ 2 (t) λ 3 (t) λ 4 (t)      +    1 L 0 0 0 0 0 0 0 -1 RC       u 1 (t) u 2 (t) u 3 (t)    0 ≤ λ(t) ⊥ w(t) =      0 -1 1 0 0 1 0 0 0 0 1 0         x 1 (t) x 2 (t) x 3 (t)    +      0 0 -1 0 0 0 -1 0 1 1 0 -1 0 0 1 0           λ 1 (t) λ 2 (t) λ 3 (t) λ 4 (t)      +      -1 -1 1 0 0 1 0 0 0 1 1 0         u 1 (t) u 2 (t) u 3 (t)    ≥ 0 (129) The closed-loop system of the LCS with the external input u = u 1 u 2 u 3 ⊤ = K(x -x d ) + u d where K =   k 11 k 12 k 13 k 21 k 22 k 23 k 31 k 32 k 33 
 is given by:

                                                                                                     ẋ =    k11 L 1 L + k12 L k13 L -1 C 0 0 -k31 RC -k32 RC -1 RC -k33 RC       x 1 (t) x 2 (t) x 3 (t)    +    0 0 0 0 -1 C 0 0 1 C 1 C 1 C 0 0         λ 1 (t) λ 2 (t) λ 3 (t) λ 4 (t)      -    k11 L k12 L k13 L 0 0 0 -k31 RC -k32 RC -k33 RC       x 1d (t) x 2d (t) x 3d (t)    +    1 L 0 0 0 0 0 0 0 -1 RC       u 1d (t) u 2d (t) u 3d (t)    0 ≤ λ(t) ⊥ w(t) =      -k 11 -k 21 + k 31 -1 -k 12 -k 22 + k 32 1 -k 13 -k 23 + k 33 k 31 k 32 1 + k 33 0 0 0 k 11 + k 21 1 + k 12 + k 22 k 13 + k 23         x 1 (t) x 2 (t) x 3 (t)    +      0 0 -1 0 0 0 -1 0 1 1 0 -1 0 0 1 0           λ 1 (t) λ 2 (t) λ 3 (t) λ 4 (t)      -      -k 11 -k 21 + k 31 -k 12 -k 22 + k 32 -k 13 -k 23 + k 33 k 31 k 32 k 33 0 0 0 k 11 + k 21 k 12 + k 22 k 13 + k 23         x 1d (t) x 2d (t) x 3d (t)    +      -1 -1 1 0 0 1 0 0 0 1 1 0         u 1d (t) u 2d (t) u 3d (t)    ≥ 0 (130)
is strictly passive since there exist K and P such that the NLMI in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] where P B = (C + F K) ⊤ holds.

Nonsmooth Mechanical Systems with Unilateral Springs

It is known that unilateral spring/dashpot contact/impact models, can be written in a complementarity framework [START_REF] Brogliato | Nonsmooth Mechanics. Models, Dynamics, and Control. 3rd[END_REF][START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF][START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF][START_REF] Aydinoglu | Stabilization of complementarity systems via contact-aware controllers[END_REF]. This class of contact/impact models significantly differs from unilateral constraints which yield complementarity constraints and impact models as in Remark 14.1.

mq(t) = u(t) + λ(t) 0 ≤ λ(t) ⊥ w(t) = λ(t) + k(q(t) -l) ≥ 0 (132)
where q-l is the deformation of the spring, k > 0 is the stiffness of the spring. The complementarity constraint is written between the contact force and the signed distance between the spring and the mass. Clearly the dynamics (132) fits with our general framework. Let x = (x 1 , x 2 ) ⊤ = (q -l, q) ⊤ , (132) is rewritten equivalently as the LCS: [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF] and verifying the positive definiteness conditions. Following the matrix M in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF], the matrix inequality is written as:

   ẋ(t) = 0 1 0 0 x(t) + 0 1 m λ(t) + 0 1 m u(t) 0 ≤ λ(t) ⊥ w(t) = λ(t) + kx 1 (t) ≥ 0 (133) Here A = 0 1 0 0 , B = E = 0 1 m , C = (k, 0), D = 1, F = 0,
  -2p12k1 m -p 11 -p12k2 m -p22k1 m -p12 m + k -p 11 -p12k2 m -p22k1 m -2p 12 -2p22k2 m -p22 m -p12 m + k -p22 m 2   ≻ 0 with P = p 11 p 12 p 12 p 22 ≻ 0.
One of the necessary conditions to be satisfied is:

2 × -2p12k1 m -p 11 -p12k2 m -p22k1 m -p 11 -p12k2 m -p22k1 m -2p 12 -2p22k2 m --2p12 m + k 2 0 0 0 ≻ 0
Given that -p12k1 m > 0 which is a necessary condition for positive definiteness, thus for strong passivity of the quadruple (A+EK, B, C, D), it is required to prove that:

-4p12k1 m --2p12 m + k 2 >
0. This condition implies that as the stiffness of the spring k increases, the control gain k 1 increases to satisfy the condition. It is seen that in a certain sense this class of mechanical systems (with D > 0 and F = 0) lies in-between the unilaterally constrained mechanical systems (with always D = 0 and F = 0), and LCS (with possibly both D and F nonzero).

Remark 14.1 (Linear Complementarity Lagrangian Systems). Linear Lagrangian systems with unilateral constraints have the nonlinear nonsmooth dynamics: [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF]Definition 6.12], S ≽ 0 is a stiffness matrix, and it is assumed that q has left and right limits. For simplicity we also assume that there is a unique unilateral constraint, i.e., C q ∈ IR 1×n , λ ∈ IR and F q ∈ IR is a constant (the system's admissible domain in the configuration space, is a convex polyhedral set, assumed to be nonempty). Denoting x = (q ⊤ , q⊤ ) ⊤ , x 1 = q, x 2 = q, we obtain:

   M q(t) + R q(t) + Sq(t) = u(t) + C ⊤ q λ(t) 0 ≤ λ(t) ⊥ w(t) = C q q(t) + F q ≥ 0 C q q(t + ) = -e n C q q(t) if C q q(t) = 0 and C q q(t -) ≤ 0, (134) where q(t) ∈ IR n , M = M ⊤ ≻ 0, the restitution coefficient e n ∈ [0, 1], R ≽ 0 is a Rayleigh dissipation matrix
       ẋ(t) = 0 I n -M -1 S -M -1 R x(t) + 0 M -1 u(t) + 0 M -1 C ⊤ q λ(t) 0 ≤ λ(t) ⊥ w(t) = Cx(t) + F q = (C q , 0)x(t) + F q ≥ 0 C ẋ(t + ) = -e n C ẋ(t -) if Cx(t) = 0 and C ẋ(t -) ≤ 0. (135) Therefore A = 0 I n -M -1 S -M -1 R , B = 0 M -1 C ⊤ q , E = 0 M -1 , F = 0, C = (C q , 0
), D = 0. Since D = 0 and no control acts in the complementarity constraint, passivity implies that P B = C ⊤ whatever the controller u(x) = Kx. It can be checked that this is not possible with P ≻ 0. Adding a multiplier feedback u(x, λ) = Kx + Gλ does not change the conclusion. Fundamentally, systems as (135) have a relative degree 2 when λ is seen as the input and w is seen as the output [START_REF] Brogliato | Nonsmooth Mechanics. Models, Dynamics, and Control. 3rd[END_REF] (hence hampering passivity [START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF]), while systems as in (133) have a relative degree 0 between the same input/output variables.

Networks with Unilateral Interactions

Let us present an example of a network with unilateral interactions, which is introduced in [START_REF] Manfredi | Necessary and sufficient conditions for consensus in nonlinear monotone networks with unilateral interactions[END_REF]. These systems can be treated as LCS [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF], and by adding exogenous signals, they can be formulated as LCS with external inputs as in [START_REF] Acary | An introduction to Siconos[END_REF]. Network systems with unilateral interactions have applications in various fields such as sensor networks, robotics, and game theory. This example focuses on the study of trajectory tracking for a system of networks with unilateral interactions and inputs with the following dynamics:

     ẋ1 = max(0, x 3 -x 1 -u 1 ) + u 2 ẋ2 = x 1 -x 2 + min(0, x 3 -x 2 + u 3 ) + u 4 ẋ3 = max(0, x 1 -x 3 -u 5 ) + u 6 (136)
which is written equivalently in the form of LCS as:

                       ẋ =    0 0 0 1 -1 0 0 0 0    x +    1 0 0 0 -1 0 0 0 1    λ +    u 2 u 4 u 6    0 ≤ λ ⊥    1 0 -1 0 -1 1 -1 0 1    x +    1 0 0 0 1 0 0 0 1    λ +    u 1 u 3 u 5    ≥ 0 (137)
The desired system is given by the following LCS:

                       ẋd =    0 0 0 1 -1 0 0 0 0    x d +    1 0 0 0 -1 0 0 0 1    λ d +    u 2d u 4d u 6d    0 ≤ λ d ⊥    1 0 -1 0 -1 1 -1 0 1    x d +    1 0 0 0 1 0 0 0 1    λ d +    u 1d u 3d u 5d    ≥ 0 (138)
Remark 15.1. In this analysis, the most general form of inputs is chosen (they appear in both the linear and the nonsmooth parts of the dynamics). It is out of the scope of this analysis to justify the feasibility of such inputs.

First case: Complete controller

Let u = u 1 u 2 u 3 u 4 u 5 u 6 ⊤ and the matrices E and F in ( 16) are defined as

E =   0
                       ẋ =    k 21 k 22 k 23 1 + k 41 -1 + k 42 k 43 k 61 k 62 k 63    x +    1 0 0 0 -1 0 0 0 1    λ -    k 21 k 22 k 23 k 41 k 42 k 43 k 61 k 62 k 63    x d +    u 2d u 4d u 6d    0 ≤ λ ⊥    1 + k 11 k 12 -1 + k 13 k 31 -1 + k 32 1 + k 33 -1 + k 51 k 52 1 + k 53    x +    1 0 0 0 1 0 0 0 1    λ -    k 11 k 12 k 13 k 31 k 32 k 33 k 51 k 51 k 53    x d +    u 1d u 3d u 5d    ≥ 0 (139)
The solution of the NLMI in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] for the closed-loop system in ( 139 It is important to note that, in this case, the closed-loop quadruple (A+EK, B, C, D) is strongly passive. This property allows us to analyse the problem of trajectory tracking in the presence of uncertainties (see section 4). However, whether or not considering such uncertainties in networks with unilateral interactions makes sense, is out of the scope of this work.

Strong passivity of the closed-loop system is a fundamental assumption in robustness analysis in section 4 in order to guarantee the boundedness of the tracking error e. Therefore, there is a need to introduce a feedback from λ in the controller u to enhance the passivity of the closed-loop system in certain applications.

A.4 Uniform and Ultimate Boundedness

The following Lyapunov-like theorem is for showing uniform boundedness and ultimate boundedness.

Theorem A.2. [START_REF] Khalil | Nonlinear Systems. 3rd[END_REF]Theorem 4.18] Let D ⊂ IR n be a domain that contains the origin and V : [0, ∞) × D → IR be a continuously differentiable function such that

α 1 (∥x∥) ≤ V (x) ≤ α 2 (∥x∥) (145) ∂V ∂t + ∂V ∂x ∂x ∂t ≤ -W 3 (x), ∀∥x∥ ≥ µ > 0 ( 146
)
for all t ≥ 0, for all x ∈ D, where α 1 and α 2 are class K functions and W 3 (•) is a continuous positive definite function. Take r > 0 such that B r ⊂ D and suppose that,

µ < α -1 2 (α 1 (r)) (147)
Then, there exists a class KL function β and for every initial state x(t 0 ), satisfying ∥x(t 0 )∥ ≤ α -1 2 (α 1 (r)), there exists T > 0 (dependent on x(t 0 ) and µ) such that the solution satisfies

∥x(t)∥ ≤ α -1 1 (α 2 (r)), ∀t ≥ t 0 + T (148)
Moreover, if D = IR n and α 1 belongs to class K ∞ , then (148) holds for any initial state x(t 0 ), with no restrictions on how large µ is.

A.5 Conditions for Positive (Semi) Definiteness

Lemma A.

= Q S S ⊤ R . Assume that Q = Q ⊤ and R = R ⊤ . Then M ≽ 0 if and only if: 1. R ≽ 0, 2. SR † R = S ⇔ Im(S ⊤ ) ⊆ Im(R), 3. Q ≽ SR † S ⊤ .
The equivalence in item 2 follows from [8, Remark after Fact 6.4.5]. The next lemma provides some necessary conditions for item 3 to hold.

Lemma A.4. Assume that Q = Q ⊤ ≽ 0 and R = R ⊤ ≽ 0. Then Q ≽ SR † S ⊤ : 1. ⇒ Im(Q) ⊇ Im(SR † S ⊤ ) ⇐= Im(Q) ⊇ Im(S). 2. ⇒ λ i (Q) ≥ λ i (SR † S ⊤ ) for all i ∈ {1, . . . , n}, and tr(Q) ≥ tr(SR † S ⊤ ) and det(Q) ≥ det(SR † S ⊤ ) ≥ 0.
Proof.

1. From [9, Fact 7.17

Notice that λ

i (SR † S ⊤ ) = λ i (N N ⊤ ) = σ 2 i (N ) (because R † ≽ 0)
, and similarly λ i (Q) = σ 2 i (L). The matrix inequality in Lemma A.3 item 3 can therefore be transformed into a singular values inequality, provided the matrices N and L are computed (Cholesky decomposition [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF]Fact 10.10.42], or the Gram matrix decomposition [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF]Fact 10.10.41] can be chosen). An interesting result follows from [9, Theorem 10.6.2] and the same reasoning as in the proof of Lemma A.5 item 1: there exists α > 0 such that αQ ≽ SR † S ⊤ ⇐⇒ Im(Q) ⊇ Im(SR † S ⊤ ). A first step may be to check the ranges inclusion, then to calculate an α > 0, following the reasoning in the proof of [9, Theorem 10.6.2]: compute the matrices N and L, compute the matrix T such that N = LT , and compute α = λ max (T T ⊤ ). If α ≤ 1 then Lemma A.3 item 3 holds true. Following the proof of [9, Theorem 10.6.2], T can be calculated. Let us provide an excerpt of [START_REF] Chen | Perturbation bounds of P-matrix linear complementarity problems[END_REF]Theorem 2.11], and a corollary of it. Let us recall that for a given M ∈ IR n×n , ||M || 2,2 is the induced matricial norm such that ||M || 2,2 = σ max (M ) (the largest singular value) [8, Proposition 9.4.9], it is a submultiplicative norm [8, Corollary 9.4.12].

Theorem A.5. [START_REF] Chen | Perturbation bounds of P-matrix linear complementarity problems[END_REF] Let M ∈ IR n×n be a positive definite matrix. Then every matrix

A ∈ {A ∈ IR n×n | M +M ⊤ 2 -1 2,2 ||M -A|| 2,2 < 1}
is positive definite.

Corollary A.6. [START_REF] Brogliato | Kinetic quasi-velocities in unilaterally constrained Lagrangian mechanics with impacts and friction[END_REF] Let D = P + N , where D, P and N are n × n real matrices, and P ≻ 0, not necessarily symmetric. If

||N || 2,2 < 1 ∥ P +P ⊤ 2 -1 ∥ 2,2 (149)
then D ≻ 0.

A.6 Numerical computation of state-jump times in (122)

Let us check numerically that the state jumps and the state-jump times are independent of the matrix solution of the passivity LMI, as predicted by theoretical arguments [START_REF] Camlibel | On linear passive complementarity systems[END_REF][START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF][START_REF] Greenhalgh | On preserving dissipativity properties of linear complementarity dynamical systems with the theta-method[END_REF]. It is also shown in [START_REF] Greenhalgh | On preserving dissipativity properties of linear complementarity dynamical systems with the theta-method[END_REF] that the event-capturing time-stepping Moreau-Jean scheme that is implemented in the siconos software package, does approximate this state-jump rule. Hence the numerical results obtained from siconos can be taken as the correct state-jump times when the chosen time-step is small enough.

For the desired system: Knowing that at the time of discontinuity t = 1, the values of the desired state from simulation in Figure 25 at t = 1 -is given by the numerical solver siconos as follows:

x d (1 -) =     3.4615397 -1.832126 0.238885 2.91881    
In order to calculate the values of the states at t = 1 + , the optimization problem in ( 9) is solved using mosek solver with P = P d which is the solution of the LMI for the passivity of the desired system in [START_REF] Aydinoglu | Stabilization of complementarity systems via contact-aware controllers[END_REF]. We solved the optimization problem for different values of the solution P d (i.e., the solution P d is not unique). The results is obtained below: For the closed-loop system: At the time of discontinuity t = 1, the value of the closed-loop state is given by: According to the results shown above, the value of the jump in the desired state and closed-loop state x d and x respectively given by siconos converges to the value of the jump which is the solution of the optimization problem in (9) as the time step h → 0 [50, Definition 8].

x(1 -) =     3
A.7 Solve the optimization problem in [START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF] for the simple example in section 6

In this section, the optimization problem in ( 9) is solved using KKT conditions since we are dealing with quadratic optimization problems with inequality constraints. In this case, KKT conditions are necessary and sufficient conditions for the optimal value because the following optimization problem

x(t + ) = argmin x∈K 1 2 (x -x(t -)) ⊤ P (x -x(t -)).
is convex knowing that P = P ⊤ ≻ 0, x(t) and x(t -) ∈ IR. The optimization problem is solved for the two systems as shown in the following.

For the desired system in [START_REF] Rockafellar | Variational Analysis[END_REF] It is required to find x d (t + ) which is the optimal value to minimize the following

f d (x d ) = min 1 2 (x d -x - d ) 2 s.t.
L(x d , µ) = 1 2 (x d -x - d ) 2 -µ(-cx d -v + d )
where µ is the lagrangian multiplier. Let us state KKT conditions to be satisfied:

                   ∂L ∂x d = 0 ⇔ x d -x - d + cµ = 0 ⇔ x d = x - d -cµ µ(-cx d -v + d ) = 0 ⇔ µ(c 2 µ -cx - d -v + d ) = 0 µ ≤ 0 -cx d -v + d ≤ 0 ⇔ c 2 µ -cx - d -v + d ≤ 0
KKT conditions are written in the form of a linear complementarity problem LCP as shown below:

0 ≤ -µ ⊥ -c 2 µ + cx - d + v + d ≥ 0
The linear complementarity problem (LCP) is denoted by LCP(c 2 , cx - d + v + d ) and it has a unique solution defined by:

-µ =    0 if cx - d + v + d ≥ 0 -1 c x - d -1 c 2 v + d if cx - d + v + d < 0
Let us substitute the values of the solution in the KKT condition ∂L ∂x d = 0 to calculate x d as follows:

x d =    x - d if cx - d + v + d ≥ 0 -1 c v + d if cx - d + v + d < 0 (150)
For the closed-loop system in [START_REF] Sessa | A complementarity approach for the computation of periodic oscillations in piecewise linear systems[END_REF] It is required to find the optimal value x(t + ) in order to minimize the following f (x) = min The necessary and sufficient conditions to be satisfied are the following:

                   ∂L ∂x = 0 ⇔ x -x -+ µ(c + k 3 ) = 0 ⇔ x = x --µ(c + k 3 ) µ -(c + k 3 )x + k 3 x + d -v + d = 0 ⇔ µ -(c + k 3 )x -+ µ(c + k 3 ) 2 + k 3 x + d -v + d = 0 µ ≤ 0 -(c + k 3 )x + k 3 x + d -v + d ≤ 0 ⇔ -(c + k 3 )x -+ µ(c + k 3 ) 2 + k 3 x + d -v + d = 0
KKT conditions are written in the form of a linear complementarity problem LCP as shown below:

0 ≤ -µ ⊥ -(c + k 3 ) 2 µ + (c + k 3 )x --k 3 x + d + v + d ≥ 0
The linear complementarity problem (LCP) is denoted by LCP (c + k 3 ) 2 , (c

+ k 3 )x --k 3 x + d + v + d
and it has a unique solution defined by:

-µ =    0 if (c + k 3 )x --k 3 x + d + v + d ≥ 0 -1 (c+k3) x -+ k3 (c+k3) 2 x + d - 1 (c+k3) 2 v + d if (c + k 3 )x --k 3 x + d + v + d < 0
Let us substitute the values of the solution in the KKT condition ∂L ∂x = 0 to calculate x as follows:

x =      x - if (c + k 3 )x --k 3 x + d + v + d ≥ 0 - k3x + d -v + d (c+k3) if (c + k 3 )x --k 3 x + d + v + d < 0
A.8 Proof of the DI in (3.3) Let us remind the transformation of an LCP into a differential inclusion of the first-order sweeping process (FOSwP) with perturbation type, as proposed in [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF][START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF][START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF]. it is assumed that the constraint qualification stated after (3.

B Python Code

There are two files used in order to simulate the electrical circuits in section 7. The first Python file is implemented to solve the matrix inequalities in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF], [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF] or [START_REF] Colombo | Optimal control of the sweeping process: the polyhedral case[END_REF] after being transformed into linear matrix inequalities using the solver mosek 9.3.14. The solution of the LMI gives the values of the control gain as well as that of the upper bounds of the uncertainties (in the case of robustness analysis). Then, the real system is defined. #p r i n t s o l u t i o n i f e x i s t s i f prob . s t a t u s == cvx .OPTIMAL: print ( " P 0 =" , np . l i n a l g . i n v ( Q 0 . v a l u e ) ) print ( " N 0 =" , N 0 . v a l u e ) print ( " K 0 =" , K 0 . v a l u e ) print ( " G 0 =" , G 0 . v a l u e ) e l s e :

# This l i n e means t h a t you b u i l d t h e u n d e r l y i n g # F i r s t O r d e r L i n e a r D S w i t h A, b0 and x0 . # A i s empty and w i l l be u p d a t e d a t each time s t e p w i t h a

e f i n e r e l a t i o n ( c o n s t r a i n t s ) # y = Cx + Dlambda + e ( t ) # r = B lambda # F i r s t c r e a t e t h e o p e r a t o r s (C, D, . . . ) o f t h e e x t e n d e d s ys t e m which c o n t a i n s t h e d e s i r e d and

# And t h e n b u i l d t h e u n d e r l y i n g r e l a t i o n w i t h t h e s e o p e r a t o r s

super ( m y r e l a t i o n , s e l f ) .

i n i t (C, B) s e l f . s e tD Pt r (D) s e l f . s e t e P t r ( e ) #i f F ̸ = 0 def computeh ( s e l f , time , x , l l , z , y ) : s e l f . computeC ( time , z , s e l f . C ( ) ) ; np . matmul ( s e l f . C( ) , x , y ) s e l f . computeD ( time , z , s e l f .D ( ) ) ; s e l f . computee ( time , z , s e l f . e ( ) ) ; y [ . . . ] += np . matmul ( s e l f .D( ) , l l ) ; + s e l f . e ( ) ; def computeg ( s e l f , time , l l , z , r ) : s e l f . computeB ( time , z , s e l f . B ( ) ) ; r [ . . . ] += np . matmul ( s e l f . B ( ) , l l )

#d e f i n e t h e m a t r i c e s B and C o f t h e e x t e n d e d sy

Now, the seond file is implemented and the first is imported.

# A f u n c t i o n t o b u i l d t h e n s d s and l a u n c h a s i m u l a t i o n # -b e t w e e n s t a r t t i m e and e n d t i m e w i t h a time s t e p s i z e e q u a l t o t i m e s t e p # -f o r some g i v e n v

I t ' s s i z e must be t h e s i z e o f your a l g e b r a i c # s ys t e m d e f i n i n g t h e c o n s t r a i n t s # i . e t h e l e n g t h o f y and lambda v e c t o r s o f your r e l a t i o n . nslaw = sk . ComplementarityConditionNSL (m) #m i s t h e number o f c o m

p l e m e n t a r i t y v a r i a b l e s i n t e r = sk . I n t e r a c t i o n ( nslaw , r e l a t i o n ) # nonsmooth dyn amica l s ys t e m n s d s = sk . NonSmoothDynamicalSystem ( s t a r t t i m e , e n d t i m e ) n s d s . i n s e r t D y n a m i c a l S y s t e m ( ds ) n s d s . l i n k ( i n t e r , ds ) # s i m u l a t i o n s e t u p t h e t a = 0 . 5 i n t e g r a t o r = sk . EulerMoreauOSI ( t h e t a ) osnspb = sk . LCP( ) t i m e d i s c r = sk . T i m e D i s c r e t i s a t i o n ( s t a r t t i m e , t i m e s t e p ) simu = sk . TimeStepping ( nsds , t i m e d i s c r , i n t e g r a t o r , osnspb ) # e s t i m a t e t h e number o f time s t e p s N = math . c e i l ( ( e n d t i m e -s t a r t t i m e ) / simu . t i m e S t e p ( ) ) + 1 # d a t a P l o t : an a r r a y where each l i n e k i s used t o s a v e # v a l u e s a t time-s t e p k d a t a P l o t = np . z e r o s ( (N, 1 2 ) )

# numpy a r r a y t o s a v e d a t a f o r p l o t t i n g . # s t o r e i n i t i a l v a l u e s k = 0 x = ds . x ( ) # s t a t e w1 = i n t e r . y ( 0 ) [ 0 ] w2 = i n t e r . y ( 0 ) [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] lamb1 = i n t e r . lambda ( 0 ) [ 0 ] lamb2 = i n t e r . lambda ( 0 ) [

C Circuits' Dynamics

Let us recall some useful mathematical relations between the voltage V and the current i of the electrical components R, C and L as follows:

• For the resistor, let us recall Ohm's Law: V R = Ri R where R is the resistance of the resistor.

• For the capacitor, we have the relations: Q C = CV C where C is the capacitance and Q C is the charge on the capacitor and QC = i C

• For the inductor, the relation is V L = L di L dt where L is the inductance of the inductor.

C.1 Circuit in Figure 12

Recall that the state x 1 is the charge on the capacitor and the state x 2 is the current passing through the inductor. By applying KVL (Kirchhoff's Voltage Law) we can write the following equations:

-u + V C -V L = 0 ⇔ -u + x 1 C -L ẋ2 = 0 ⇔ ẋ2 = 1 LC x 1 - u L -V C -V R + w = 0 ⇔ - x 1 C -Rλ + w = 0 ⇔ w = x 1 C + Rλ
Let us apply KCL (Kirchhoff's Current Law) at the point of connection between the capacitor C and the voltage source u, then:

i C + i L = i R ⇔ ẋ1 + x 2 = λ ⇔ ẋ1 = -x 2 + λ
The equations derived, along with the complementarity relation (i.e., 0 ≤ λ ⊥ w ≥ 0), are consistent with the LCS in (89).

C.2 Circuit in Figure 15 C.3 Circuit in Figure 20 C.4 Circuit in Figure 27

λ 1 ( 1 p 1 p

 111 M ) ≥ λ 2 (M ) ≥ . . . ≥ λ n (M ) = λ min (M ). M † denotes the Moore-Penrose pseudo-inverse of M . Let 1 ≤ p < +∞, a Lebesgue integrable function f : I ⊆ IR n → IR belongs to L p loc (I; IR) if I ||f (t)|| p dt < +∞ for any open set I (extended spaces). If I = IR n then f ∈ L p (IR n ; IR). Also f ∈ L ∞ (IR n ; IR) if f (•) isLebesgue integrable and ||f (x)|| < +∞ almost everywhere on IR n . We denote ||f || I,p ∆ = I ||f (t)|| p dt for 1 ≤ p < +∞, and ||f || I,∞ ∆

  , • exogenous signals properties: item 2 (a) iii), (b) iii), conclusion of item 2 (c), item 2 (e) iii), (f) v), conclusion of item 3. The condition in item 2 (b) ii) stems from a fundamental result in Complementarity Theory [44, Theorem 3.8.6, Corollary 3.8.10], which guarantees the existence of solutions to the LCP in (2).

  and Assumption 3.3 holds, since in this case Ker(D + D ⊤ ) ⊆ Ker(P (B + EG) -(C + F K) ⊤ ) [29] [25, section 3.8]. If D and D + F G are Pmatrices, and if

Proposition 4 . 4 .

 44 Let Assumptions 3.1, 3.2, 3.3 hold for the nominal system, and Assumption 4.1 holds for the closed-loop system. Then the matrix inequality in[START_REF] Fenel | BV solutions of nonconvex sweeping process differential inclusion with perturbation[END_REF] holds if and only if:

) Remark 6 . 1 .Example 2 (

 612 Consider G = 0. Recall the conditions of well-posedness given in section 2.1, item 3 for the case when d = 0. Assume that there exists p > 0 such that the quadruple (a, b, c, 0) is passive (i.e., the LMI in[START_REF] Aydinoglu | Stabilization of complementarity systems via contact-aware controllers[END_REF] has a solution). If the desired system in (77) is well-posed, then the condition pb = c holds. In addition, if the closed-loop system in (79) is well-posed, then the condition pb = c + k 2 holds. Therefore, knowing that p > 0, the variables b, c and c + k 2 should have the same sign (seeExample 2). Ill-posed LCS). Let b = 1 > 0 and c = -1 < 0. Consider the LCS in[START_REF] Rockafellar | Convex Analysis[END_REF] with the following linear complementarity system (LCP):

)

  According to Theorem A.1, the LCP in[START_REF] Shen | Robust non-Zenoness of piecewise affine systems with applications to linear complementarity systems[END_REF] is ill-posed because cb = -1 < 0. Therefore, the LCS in[START_REF] Rockafellar | Convex Analysis[END_REF] is ill-posed when cb < 0. Following the same steps as before for the desired system in (77) and the closed-loop system in[START_REF] Sessa | A complementarity approach for the computation of periodic oscillations in piecewise linear systems[END_REF], the coefficients cb and b(c + k 2 ) have to to positive. Otherwise, the result of ill-posedness is observed in the following numerical simulation with a = 0.5, b = 1, c = -1, and k 1 = -1.19, k 2 = 2.96, p = 1.96 being the solution of the LMI of strict passivity in (88). Take x(0) = 5, x d (0) = 0, u d (t) = 5 sin 3t and v d a discontinuous periodic function as shown in Figure 5b with the time step h = 0.001.

Figure 5 :

 5 Figure 5: Numerical simulation of ill-posed LCS

  2 and 3.3 hold. Assume that the states x(•) and x d (•) have jumps for t > 0 since d = 0 in the complementarity constraint. Then, the error dynamics in[START_REF] Vladimirov | Nonstationary dissipative evolution equations in a Hilbert space[END_REF] has a globally asymptotically stable equilibrium point e ⋆ = 0.Proof. Consider the Lyapunov candidate function V (e(t)) = e(t) ⊤ P e(t). Recall from section 3.2, Proposition 3.8, that error dynamics in (84) has a globally exponentially stable equilibrium point e * = 0 when both the states x(•) and x d (•) are continuous. This result is proved by showing the variation of the storage function of the error dynamics V (e(t)) < 0 for all e ̸ = 0.In view of the desired dynamics (77), the closed-loop dynamics[START_REF] Sessa | A complementarity approach for the computation of periodic oscillations in piecewise linear systems[END_REF] and the controllers (78), let t k where k > 0 be the set of time instants at which v d (t) is discontinuous. The states x and x d undergoes a jump at t ∈ {0} ∪ {t k }. The goal is to study the sign of ∆V (e(t)) at the jump time which is written as follows in scalar case:

Example 3 (

 3 Both x and x d jump). Let us take b = 1, and c = 1. The numerical simulation below for the closed-loop system and the desired system is implemented with a discontinuous controller v d (t) at t k with k = {1, 2, 3, 4}. Take x(0) = 3, x d (0) = -3, u d = 5 sin 3t and v d = 4 for t ≤ 1s with the time step h = 0.001. Given that x d (1 -) = -0.7 and x(1 -) = 2.28, and following Corollary 6.3, let us choose

Figure 6 :Example 4 .

 64 Figure 6: Numerical simulation of LCS where both x and x d jump

28 with c = 1 Figure 7 :

 2817 Figure 7: Numerical simulation of LCS when only x d jumps

Figure 8 :

 8 Figure 8: Numerical simulation of LCS where only x d jumps

1 . 1

 11 knowing that c = 1, k 2 = 0.96 and x d (1 + ) = x d (1 -) = x d (1) = 2.59 (i.e., x d is continuous). Let us take v d (1 + ) = -2 and observe the following with v d(t) = 4, if t = n, -2, if t = n + 1where n ∈ {1, 2, 3, 4}.

Figure 9 :

 9 Figure 9: Numerical simulation of LCS where only x jumps

11 .Example 6 .

 116 The goal in this section is to find examples such that the sufficient conditions of Proposition 3.11 are satisfied, as an illustration. Let a = 0.5, b = 1, c = 1, p = 1.96 and k 2 = 0.96. Let x(0) = 10, x d (0) = -5, u d = 5 sin 3t and v d is discontinuous as shown in Figure 10. Let us create a jump at t = 1s such that the conditions in Proposition 3.11 are satisfied. At t = 1 -, the values of the states are x d (1 -) = -4 and x(1 -) = 3.45. If we consider the case when

  which is obtained from the above assumption and -3.45 ≤ v d (1 + ) < 4 which comes from the fact that at t = 1 x d (1 -) / ∈ K d and x(1) ∈ K. Take v d (1 + ) = 2. The numerical simulation is shown in the following with a time step h = 0.001.

Figure 10 :Example 7 .

 107 Figure 10: Numerical simulation of LCS where only x d jumps

Figure 11 :

 11 Figure 11: Numerical simulation of LCS where only x jumps

  .89363 -2.145726 , P = 10.911047 -0.26867 -0.26867 0.12605 and N = 3.31969 -9.94634

Figure 13 :

 13 Figure 13: Numerical simulation showing the closed-loop system's trajectory x, the desired system's trajectory x d and the error dynamics e = x -x d of the LCS in (89), without uncertainties and with two forms of controllers. The simulation also presents the controller u and the complementarity variables λ and w.

1 ∥Figure 14 :

 114 Figure 14: Numerical simulation showing the error dynamics e = x -x d of the LCS in (89), with different values of uncertainties. The simulation also presents the controller u and the complementarity variables λ and w.

  , the matrices A, E and D are perturbed by ∆A, ∆E and ∆D respectively due to the uncertainties on the elements R and L. The values of L 2 norm of the error function are ∥e 1 ∥ [2,3],2 = 0.43 and ∥e 2 ∥ [2,3],2 = 1.08 and that of L ∞ norm are ∥e 1 ∥ [2,3],∞ = 0.09 and ∥e 2 ∥ [2,3],∞ = 0.18. These values increase when a new disturbance ∆C is introduced and the results are shown in Figure 14d. This increase is noticed such that ∥e 1 ∥ [2,3],2 = 0.5, ∥e 2 ∥ [2,3],2 = 1.2, ∥e 1 ∥ [2,3],∞ = 0.12 and ∥e 2 ∥ [2,3],∞ = 0.25.

  95) Take L = 1H, C = 0.025F and R = 10Ω. Let us begin with studying the passivity of the open loop system. So, it is required to check passivity of the quadruple (A, B, C, D) with u 1 = u 2 = 0 by checking the positive realness of the transfer function:

Figure 17 :

 17 Figure 17: RLCD circuit with current source

  Take L = 1 H, C = 0.025 F, and R = 10 Ω. The solution is shown below: K = -11.78022858 0.54878816 and P = 27.65215362 -0.26929383 -0.26929383 0.99594605 Using the given values of L, R, C and K parameters, the plot of Re[H 4 (jω)] in (102) is depicted in the following figure:

Figure 18 :

 18 Figure 18: Graphical representation of (102) with given parameters

Figure 19 :Figure 20 :

 1920 Figure 19: Solution of (100) with a controller represented by the current source

Remark 11 . 1 .

 111 According to[START_REF] Brogliato | Dissipative Systems Analysis and Control. 3rd. Communications and Control Eng[END_REF] Definition 2.70], all the principal sub-matrices of the transfer function matrix G 11 , G 22 and |G 11 * G 22 -G 21 * G 12 | are of index = 1 (i.e all the principal submatrices are proper). So, the transfer matrix G(s) is totally index 1. As well, according to [25, Proposition 2.71], the transfer matrix G(s) (111) is totally of index 1 knowing that (A, B, C, D) is passive, (A, B) is controllable and the matrix B D + D ⊤ has full column rank.

e 2 , e 4 ∈

 24 IR, e 3 + x 3d (0 + ) ≥ 0 and -e 3 + x 3 (0 + ) ≥ 0} = {e ∈ IR4 | e 1 , e 2 , e 4 ∈ IR and e 3 ∈ -x 3d (0 + ), x 3 (0 + ) }

Figure 23 :

 23 Figure 23: Numerical Simulation of desired, closed-loop and error dynamics of LCS in (115)

Figure 25 :

 25 Figure 25: Numerical Simulation

  then the transfer function of the closed loop system is:H 4 (s) = [C + F K][sI -(A + EK)] -1 B + Dand represented as follows:

  has a solution given by:

Figure 29 :

 29 Figure 29: Mechanical system with unilateral spring.

 k 11 k 21 k 31 k 41 k 51 k 61 k 12 k 22 k 32 k 42 k 52 k 62 k 13 k 23 k 33 k 43 k 53 k 63   ⊤

 63 The closed-loop system with u = K(x -x d ) + u d where K =  is written in the form of the LCS in[START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF] as follows:

Figure 30 : 6 ⊤

 306 Figure 30: Network with complete controlller

Figure 31 :

 31 Figure 31: Network with input (u 2 , u 4 , u 6 ) ⊤

  .24], there exists matrices L such that Q = LL ⊤ and N such thatSR † S ⊤ = N N ⊤ , since R † is symmetric [8, Proposition 6.1.6]. From [9, Theorem 10.6.1] it follows that LL ⊤ ≽ N N ⊤ ⇒ Im(N ) ⊆ Im(L).Using [8, Theorem 2.4.3] we have that Im(N ) = Im(N N ⊤ ) and Im(L) = Im(LL ⊤ ). The first implication is proved. Using [8, Lemma 2.4.1] we have Im(SR † S ⊤ ) ⊆ Im(S). The second implication is proved. 2. From [9, Theorem 10.4.9, Corollary 10.4.10].

  x d ∈ K d where P = 1 and K d = {x d ∈ IR | cx d + v d (t + ) ≥ 0} = {x d ∈ IR | -cx d -v + d ≤ 0} .Let us write the Lagrangian function of the minimization problem

1 2 (

 2 x -x -) 2 s.t. x ∈ K where K = {x ∈ IR | (c+k 3 )x-k 3 x d (t + )+v d (t + ) ≥ 0} = {x ∈ IR | -(c+k 3 )x+k 3 x d (t + )-v d (t + ) ≤ 0}. Let us write the Lagrangian function L(x, µ) = 1 2 (x -x -) 2 -µ -(c + k 3 )x + k 3 x + d -v + d

+

  3) holds true. Recall the closed-loop system in[START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF] with D = 0 and G = 0 as follows:ẋ(t) = (A + EK)x(t) + Bλ(t) -EKx d (t) + Eu d (t) 0 ≤ λ(t) ⊥ w(t) = (C + F K)x(t) -F Kx d (t) + F u d (t) ≥ 0 (151)As a result of convex analysis, the linear complementarity problem (LCP) is written in the form of a differential inclusion DI as follows:0 ≤ w ⊥ λ ≥ 0 ⇔ w ∈ -N IR m + (λ) ⇔ λ ∈ -N IR m property to the LCP in (151), then λ(t) ∈ -N IR m + ((C + F K)x(t) -F Kx d (t) + F u d (t)) ⇔ λ(t) ∈ -N S(t) ((C + F K)x(t)) (153)whereS(t) = v ∈ IR m + | v -F Kx d (t) + F u d (t) ∈ IR m + .Given that the quadruple of the closedloop system (A+EK, B, C+F K, 0) is strictly passive, then P B = (C+F K) ⊤ for some P = P ⊤ ≻ 0. The closed-loop system in (151) is written as DI: ẋ ∈ (A + EK)x(t) -BN S(t) ((C + F K)x(t)) -EKx d (t) + Eu d (t) (154)

import s i c o n o s . k e r n e l a s sk import numpy a s np import cvxpy a s cvx import math L v a l u e = 1

 1 # i n d u c t a n c e Cvalue = 0 . 0 2 5 # c a p a c i t a n c e Rvalue = 10 # r e s i s t a n c e def compute K0G0 ( ) : #Nominal System A 0 = np . a r r a y ( n , n ) B 0 = np . a r r a y ( n ,m) C 0 = np . a r r a y (m, n ) D 0 = np . a r r a y (m,m) E 0 = np . a r r a y ( n , p ) F 0 = np . a r r a y (m, p ) e p s = 1 e-2 #S t r u c t u r e s o f Q, N and G: p a r a m e t e r s o f LMI Q 0 = cvx . V a r i a b l e ( ( n , n ) , symmetric=True ) N 0 = cvx . V a r i a b l e ( ( p , n ) ) G 0 = np . z e r o s ( ( p ,m) ) #S t r u c t u r e o f Lambdas i n c a s e o f u n c e r t a i n t i e s Lam1 inv = cvx . V a r i a b l e ( ( n , n ) ) L a m 1 t i l d e i n v = cvx . V a r i a b l e ( ( n , n ) ) Lam2 inv = cvx . V a r i a b l e ( (m,m) ) L a m 2 t i l d e i n v = cvx . V a r i a b l e ( (m,m) ) LamF inv = cvx . V a r i a b l e ( (m, p ) )#S t r u c t u r e o f LMI i n t h e c a s e o f u n c e r t a i n t i e sLMI 1 = cvx . bmat ( [ [ -(Q 0 @ ( A 0 . t r a n s p o s e ( ) ) ) -( A 0 @ Q 0 ) -( ( N 0 . T) @ ( E 0 . T) ) -( E 0 @ N 0 ) , -B 0 -( E 0 @ G 0 ) + ( Q 0 @ ( C 0 . T) ) + ( ( N 0 . T) @ ( F 0 . T) ) ,np . e y e ( n ) , np . e y e ( n ) ,( N 0 . T) , Q 0 , np . z e r o s ( ( n , p ) ) , np . z e r o s ( ( n , m) ) , np . z e r o s ( ( n , m) ) , ] , [ -(B 0 . T) -( ( G 0 . T) @ ( E 0 . T) ) + ( C 0 @ Q 0 ) + ( F 0 @ N 0 ) , D 0 + ( F 0 @ G 0 ) + np . tr a n s p o s e ( D 0 ) + ( ( G 0 . T) @ ( F 0 . T) ) , np . z e r o s ( (m, n ) ) , np . z e r o s ( (m, n ) ) , np . z e r o s ( (m, p ) ) , np . z e r o s ( (m, n ) ) , G 0 . T, np . e y e (m) , np . e y e (m) , ] , [ np . e y e ( n ) , np . z e r o s ( ( n , m) ) , Lam1 inv , np . z e r o s ( ( n , n ) ) , np . z e r o s ( ( n , p ) ) , np . z e r o s ( ( n , n ) ) , np . z e r o s ( ( n , p ) ) , np . z e r o s ( ( n , m) ) , np . z e r o s ( ( n , m) ) , ] , [ np . e y e ( n ) , np . z e r o s ( ( n , m) ) , np . z e r o s ( ( n , n ) ) , L a m 1 t i l d e i n v , np . z e r o s ( ( n , p ) ) , np . z e r o s ( ( n , n ) ) , np . z e r o s ( ( n , p ) ) , np . z e r o s ( ( n , m) ) , np . z e r o s ( ( n , m) ) , ] , [ N 0 , np . z e r o s ( ( p , m) ) , np . z e r o s ( ( p , n ) ) , np . z e r o s ( ( p , n ) ) , LamF inv , np . z e r o s ( ( p , n ) ) , np . z e r o s ( ( p , p ) ) , np . z e r o s ( ( p , m) ) , np . z e r o s ( ( p , m) ) , ] , [ Q 0 , np . z e r o s ( ( n , m) ) , np . z e r o s ( ( n , n ) ) , np . z e r o s ( ( n , n ) ) , np . z e r o s ( ( n , p ) ) , 0 . 5 * np . e y e ( ( n ) ) , np . z e r o s ( ( n , p ) ) , np . z e r o s ( ( n , m) ) , np . z e r o s ( ( n , m) ) , ] , [ np . z e r o s ( ( p , n ) ) , G 0 , np . z e r o s ( ( p , n ) ) , np . z e r o s ( ( p , n ) ) , np . z e r o s ( ( p , p ) ) , np . z e r o s ( ( p , n ) ) , np . z e r o s ( ( p , p ) ) , np . z e r o s ( ( p , m) ) , np . z e r o s ( ( p , m) ) , ] , [ np . z e r o s ( (m, n ) ) , np . e y e ( (m) ) , np . z e r o s ( (m, n ) ) , np . z e r o s ( (m, n ) ) , np . z e r o s ( (m, p ) ) , np . z e r o s ( (m, n ) ) , np . z e r o s ( (m, p ) ) , Lam2 inv , np . z e r o s ( (m, m) ) , ] , [ np . z e r o s ( (m, n ) ) , np . e y e ( (m) ) , np . z e r o s ( (m, n ) ) , np . z e r o s ( (m, n ) ) , np . z e r o s ( (m, p ) ) , np . z e r o s ( (m, n ) ) , np . z e r o s ( (m, p ) ) , np . z e r o s ( (m, m) ) , L a m 2 t i l d e i n v , ] , ] ) # C o n s t r a i n t s c o n s t r a i n t s = [ LMI 1 >> 1 . 0 e -3 , #LMI i s p o s i t i v e d e f i n i t e Q 0 >> 1 . 0 e -2 , #t h e i n v e r s e o f P i s PD ] # o b j = cvx . Minimize ( 0 ) prob = cvx . Problem ( obj , c o n s t r a i n t s ) # s o l v e prob . s o l v e ( s o l v e r=cvx .MOSEK, v e r b o s e = 1 ) K 0 = N 0 @ np . l i n a l g . i n v ( Q 0 . v a l u e ) #K = N Q -1

  print ( "No s o l u t i o n " ) return K 0 , G 0 #D e f i n e r e a l s y s te m dynamics i n c l o s e d l o o p c l a s s my dynamics ( sk . F i r s t O r d e r L i n e a r D S ) : def i n i t ( s e l f , K 0 ) : n d o f = n # number o f s t a t e s x0 = np . z e r o s ( n d o f ) #D e f i n e i n i t i a l v a l u e s x0 [ 0 ] = x0 [ 1 ] = . . . x0 [ n-1] = s e l f . K 0 = K 0 A = np . z e r o s ( ( ndof , n d o f ) ) b0 = np . z e r o s ( n d o f )

  c a l l t o computeA . # Same f o r b w i t h computeb . . . super ( my dynamics , s e l f ) . i n i t ( x0 , A, b0 ) #t o r e c a l l t h e v a l u e s o f K and G, u s e s e l f .K . . . def computeA ( s e l f , time ) : s e l f .A( ) [ 0 , 0 ] = s e l f .A( ) [ 0 , 1 ] = . . . s e l f .A ( ) [ n-1 , n-1] =#The f u n c t i o n b i n s i c o n o s i s e q u i v a l e n t t o Eu i n our LCSdef computeb ( s e l f , time ) : s e l f . b ( ) [ 0 ] = . . . s e l f . b ( ) [ n-1] = c l a s s m y r e l a t i o n ( sk . F i r s t O r d e r L i n e a r R ) : # d e f i n i t ( s e l f , arguments ) def i n i t ( s e l f ) : #s e l f . K 0 = K 0 #D

C

  = np . z e r o s ( ( 2 * m, 2 * n ) ) D = np . z e r o s ( ( 2 * m, 2 * m) ) B = np . z e r o s ( ( 2 * n , 2 * m) ) e = np . z e r o s ( 2 * m)

  s te m def computeC ( s e l f , time , z , C ) : s e l f . C ( ) [ 0 , 0 ] = . . . s e l f . C ( ) [ 2 * m -1 , 2 * n -1 ] = def computeB ( s e l f , time , z , B ) : s e l f . B ( ) [ 0 , 0 ] = . . . s e l f . B ( ) [ 2 * n -1 , 2 * m -1 ] = def computeD ( s e l f , time , z , D ) : s e l f .D( ) [ 0 , 0 ] = ( 1 . 0 / Rvalue ) s e l f .D( ) [ 0 , 1 ] = 0 . 0 s e l f .D( ) [ 1 , 0 ] = 0 . 0 s e l f .D( ) [ 1 , 1 ] = ( 1 . 0 / Rvalue ) #Fu def computee ( s e l f , time , z , e ) : s e l f . e ( ) [ 0 ] = . . . s e l f . e ( ) [ 2 * m -1 ] =

import numpy a s np import s c i p y a s sp from s c

  i p y . i n t e g r a t e import quad import s i c o n o s . k e r n e l a s sk#i m p o r t t h e f i l e where t h e dynamics s y s t e m s a r e d e f i n e d importSystem a s ex import m a t p l o t l i b . p y p l o t a s p l t import math

  a l u e s o f K 0 and G 0 def b u i l d a n d r u n ( t i m e s t e p , s t a r t t i m e , e n d t i m e , K 0 , G 0 ) : # c r e a t e dynamics ds = ex . my dynamics ( K 0 ) # c r e a t e i n t e r a c t i o n s r e l a t i o n = ex . m y r e l a t i o n ( K 0 ) # D e c l a r e a nonsmooth law .

#

  This i s where t h i n g s a r e r e a l l y done s t a r t t i m e = 0 . 0 e n d t i m e = 5 0 . 0 # * 10 e-4# B u i l d t h e n s d s and run t h e simu f o r a g i v e n t i m e s t e pK 0 , G 0 = ex . compute K0G0 ( ) c u r r e n t t i m e s t e p = 1 e-2 d a t a P l o t = b u i l d a n d r u n ( c u r r e n t t i m e s t e p , s t a r t t i m e , e n d t i m e , K 0 , G 0 )#p l o t f i g u r e s

  that the minimum norm element is available and that it satisfies a linear growth condition in ∥x∥. Finally, it is worth recalling that even if x(•) is AC, the multiplier λ(•) may have discontinuities at junction times when w(•) reaches the boundary (i.e., w = 0), when D is not a P-matrix. In the case of item 1, the multipliers are AC when F u(t) is AC since they are a Lipschitz continuous function of AC functions. From a more practical point of view, multipliers in LCS can be calculated by constructing the contact LCP [23, section 2.4.1]. Thus they appear to be a nonsmooth function of both state and input.

). If the conditions in item 3 are verified, then explicit upper-bounds ∥ ẋ(t)∥ ≤ v(t) + α(t)∥x(t)∥ can be obtained for some integrable α(•) and AC v(•)

[START_REF] Thibault | Sweeping process with regular and nonregular sets[END_REF] Theorem 4.

3] [64, section 5.2, Theorem 2.1]. In our case, these upper-bounds depend on the properties of the set-valued map t → S(t) (which in turn depend on F u(•) being AC, or Lipschitz continuous, or LBV

[START_REF] Brogliato | Existence and uniqueness of solutions for non-autonomous complementarity systems[END_REF] Proposition 3.2]

). Following the developments in

[START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF]

(see item 2. (c) in section 2), λ = λ im (t, x) + λ ker , where x → λ im (t, x) ∈ Im(D + D ⊤ ) is Lipschitz continuous, while λ ker ∈ Ker(D + D ⊤ ). Moreover, λ im is the least norm element in the (possibly) set-valued right-hand side of the DI, see

[START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF] Equ. (15)

,

Lemma 3]

. It is inferred that in case the existence of AC solutions is proved, then Bλ(•) is a bounded selection of the (possibly set-valued) right-hand side in

[START_REF] Acary | An introduction to Siconos[END_REF]

. The part of λ inside Ker(B) does not play any role in x(•), but we may need the whole multiplier vector for feedback purpose. It is therefore reasonable to assume

  dν, where dx is the differential measure associated with the RCLBV function x(•)[START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems[END_REF], dt is the Lebesgue measure, and dν is a specific Radon measure, see[START_REF] Tanwani | Well-posedness and output regulation for implicit time-varying evolution variational inequalities[END_REF] Definition 4] for a rigorous introduction. The dissipation inequality extends to MDI, see[15, section 5.4.4.5] [25, section 7.2.4.1].

  Kx d -F Gλ d + F u d is discontinuous at t.

	hold for (A, B, C, D). Jumps at t in
	x d (•) occur only if F u d is discontinuous at t and rank(D) < m.
	2. Let the conditions in section 2.1 item 2 (a) or item 3 hold for (A + EK, B + EG, C +
	F K, D + F G). Jumps at t in the closed-loop state occur only if rank(D + F G) < m and
	-F The following is deduced, where T f denotes the set of discontinuity times of the function f (•):
	Lemma 3.9. Let F Gλ d (•) and u d (•) be bounded functions of time. (a) T x d ⊆ {t 0 } ∪ T F u d and (b)

  which contain the term F Gλ d . Also depending on F G, the sum of two Dirac measures, one stemming from λ ̸ = 0, x d is continuous at t and λ d jumps at t (which can occur if u d is continuous at t at a junction time with the constraint boundary (Cx d + Dλ d

	could occur in the differential part of the closed-loop dynamics (see section 6): we may call this a
	forbidden situation (this is not to be confused with the case of Measure Differential Inclusions, or
	with passive LCS with state jumps [23, 24, 27]). It is inferred that:
	(C1) If x d (•) jumps, we impose F G = 0. A sufficient condition to prevent forbidden situations, is
	that F G = 0 if u d (•) is discontinuous. In other words, only continuous u d (•) is allowed if F G ̸ = 0.
	In view of condition (C1), we can refine item 2: Jumps in the closed-loop state can occur at t if:
	1. F G = 0 and -F Kx d + F u d is discontinuous at t,
	2. or F G

d (due to a jump in u d (t)) and one stemming from λ (due to a jump in -F Kx d -F Gλ d +F u d )

  stated in Proposition 4.3 are satisfied, then

	e ∆λ	⊤	∆M 0	e ∆λ	≥ -	e ∆λ

⊤ P 0

  Using Corollary A.6, [9, Fact 8.3.33], [9, Corollary 11.6.5], it follows from (iii) that

  iv), Corollary A.6 and[START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF] Fact 11.16.18], which guarantee that ϵP 0

  such that the closed-loop system's quadruple (A + EK, B, C + F K, 0) is strictly passive. This means that it is required to check if the matrix inequality in[START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] has a solution. The solution of[START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] is given by

			K = 0.07 0.035 and P =	1.14 0.07 0.07 1.035	(64)
	Given that R 2 = P , then R =	1.07 0.034 0.034 1.017	. Thus the FOSwP associated with the closed-loop
	system in (50) is written as follows:
	ζ1 ζ2	∈	-0.93 0.034 0.034 -1	ζ 1 ζ 2	-	0.14 0.07 0.07 0.034

E = 1 0 ⊤ and F = 2 1 ⊤ . The LMI in (19) for the desired system's quadruple (A, B, C, 0) has a solution P d = P ⊤ d = I ≻ 0. Thus, the desired system is strictly state passive. Given that R 2 d = P d , then R d = I. Let us check if there exist K and P

  depends on the value of the state ζ ′ + d . This implies that there exists a unique set ϕ(t) associated with each ζ ′ + d which is the solution to the optimization problem in (53) with a specific set of active constraints applied. Let us consider the following cases based on the different values of ζ ′ + is invertible in this example, it follows that the state variable ζ ′ + d is expressed as ζ ′ + d = -RC -1 F u d ∈ Ω 1d ∩ Ω 2d . By substituting the value of ζ ′ +

d , resulting in different sets of ϕ(t).

5.1 Case 1: the state ζ

′ + d ∈ Ω 1d ∩ Ω 2d

In this case, both constraints Ω 1 and Ω 2 in (69) for the desired system are considered active.

According to the set of active constraints A d (ϕ ′ d (t) in

[START_REF] Morarescu | Trajectory tracking control of multiconstraint complementarity Lagrangian systems[END_REF]

and given that the matrix CR -1 d in the set ϕ(t) in

[START_REF] Menini | Trajectory tracking of a bouncing ball in a triangular billiard by unfolding and folding the billiard table[END_REF]

, it gives the following expression of ϕ(t):

  , it is noticed that the state ζ ′ + Let us define the active constraints for the closed-loop system in[START_REF] Menini | Algebraic Methods for Multiobjective Optimal Design of Control Feedbacks for Linear Systems[END_REF] as follows:

d is located at the corner of the set ϕ ′ d (t) (i.e., ζ ′ + d ∈ Ω 1d ∩ Ω 2d defined in (69)), and this corner represented by ζ ′ + d is shared by both sets ϕ ′ d (t) and ϕ(t) such that ζ ′ + d ∈ ∂ϕ(t) ∩ ∂ϕ ′ d (t).

  Let ζ ′ + d = (4u d , -0.89u d ) ⊤ ∈ Ω 2d and let us substitute the value of ζ ′ + d in the set ϕ(t) in (66), then:

	ϕ(t) = ζ ∈ IR 2	1.065 0.034 0.034 1.016	ζ 1 ζ 2	-	0.129 0.065 0.064 0.032	4u d -0.89u d	+	2u d u d	≥ 0	(74)
	which is represented in Figure 4

  81) A jump can occur in the desired state x d at initial time if and only if cx d (0 -) < -v d (0 + ). The desired state x d jumps at t = t c if and only if cx d (t - c ) < -v d (t + c ). For the closed-loop system in (79):

  Corollary 6.3. Following on from Lemma 6.2, assume that v d (•) is discontinuous at time instants t k , with k = {1, 2, ..., n} where n ∈ IN ⋆ . For any t k , if there exists v d(t + k ) < min -cx(t - k ), -cx d (t - k ) , then both x and x d jump to the value x d (t + k ) = x(t + k ) = -1 c v d (t + k ).Therefore, there will be perfect tracking for all t ≥ t k (i.e., finite-time tracking).Proof. Let the time instants at which v d is discontinuous be denoted by t k with k = {1, 2, . . . , n}. The goal is to show that if the inputv d (t + k ) < min{-cx(t - k ), -cx d (t - k )},then both x and x d jump to the same value at t = t k . According to the set K d in (81), the state x d performs a jump at t = t k if and only if x d (t -

k ) / ∈ K d which is written as follows:

  Then ∆D is selected such that |∆D| ≤ 1.63 as in Figure14a. It is observed that∥e 1 ∥ [2,3],2 = 0.57 and ∥e 2 ∥ [2,3],2 = 4.89. Also, ∥e 1 ∥ [2,3],∞ = 0.108, ∥e 2 ∥ [2,[START_REF] Adly | Well-posedness of nonsmooth Lurie dynamical systems involving maximal monotone operators[END_REF],∞ = 0.85 and ∥u∥[START_REF] Acary | An introduction to Siconos[END_REF][START_REF] Adly | Well-posedness of nonsmooth Lurie dynamical systems involving maximal monotone operators[END_REF],∞ = 31.98. In order to increase the upper bound of ∆D, the constraint on Λ-1

			2	≤ 0.25 and Λ -1 2	≥ 10; the
	solution is given as follows:
	K 0 = 45.815006 -2.158609 , P 0 =	13.497987 -0.304643 -0.304643 0.142712	, Λ -1 2 = 13.86 , Λ-1 2 = 0.15
	2 is modified such that Λ-1 2 ≤ 0.15. The solutions of the LMI in (94) with a new constraint on Λ-1 2 are given as follows:
	K 0 = 52.60207 -2.25634 , P 0 =	15.09418 -0.32588 -0.3258 0.14667	, Λ -1 2 = 14.03 , Λ-1 2 = 0.105.
	Then, it is possible to choose ∆D such that |∆D| ≤ 2.44. It is noticeable that a higher gain is ob-

tained when the bound of the uncertainty is increased. As the value of ∆D increases, it is noticed in Figure 14b that L 2 and L ∞ norms of the error function increase such that ∥e 1 ∥ [2,3],2 = 0.89, ∥e 2 ∥ [2,3],2 = 8.17, ∥e 1 ∥ [2,3],∞ = 0.16 and ∥e 2 ∥ [2,3],∞ = 1.7 with ∥u∥ [2,3],∞ = 35.4.

  strongly passive and the values of the control gains are the solutions of the LMI in[START_REF] San Felice | [END_REF]; they are given as follows:

	K 0 = 55.155 -26.438 , G 0 = 1.292 , P 0 =	146.481 -57.51 -57.51 27.342	, Λ-1 1 =	21.602 -2.11 -2.11 8.916
	Λ-1 2 = 0.169, Λ -1 1 =	18.761 -1.756 -1.756 8.115	and Λ -1 2 = 14.015

  12 k 13 k 14 k 21 k 22 k 23 k 24 k 31 k 32 k 33 k 34 k 41 k 42 k 43 k 44

		
		   . The closed-loop
	system is represented as follows:
		ẋ1 (t)
	  	ẋ2 (t) ẋ3 (t)
		ẋ4 (t)

  and the system has AC solutions with uniqueness, see section 2.1 item 1. The quadruple (A, B, C, D) can be made strongly passive with state feedback u = Kx = k 1 x 1 + k 2 x 2 . This means that there exists a control gain K = (k 1 , k 2 ) such that the quadruple (A + EK, B, C, D) of the closed-loop system is strongly passive. Equivalently, the BMI in[START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF] has a solution after being transformed into an LMI, as detailed in A.2 and the solution is given by: = 5N/m. As the stiffness of the spring k increases, the control gain gives very large numerical values like k 1 = -1.2 × 10 7 for k = 100N/m. This is explained analytically by explicitly writing the matrix inequality in

	P =	7.863 0.538 0.538 0.337	and K = -21.38 -2.148
	with m = 1kg and k

  It is noteworthy that the different values P d1 , P d2 and P d3 comes from solving the LMI in[START_REF] Aydinoglu | Stabilization of complementarity systems via contact-aware controllers[END_REF] using mosek, cvxopt and scs solvers respectively. The numerical solver siconos computes the jump automatically and gives x d (1 + ) =

									Values of P d	Values of x d (1 + )
				1.2165869		1.1915869	0	-1.1915867 		3.4615397	
	P d1 =		1.1915869 0		1.2165866 0	0 0.02499	-1.191587 0		-1.832121  10 
				-1.1915867 -1.191587	0	1.2165868	2.91881
				0.9388826	0.9138827	0	-0.9138835 		3.4615397	
	P d2 =			0.9138827 0	0.9388826 0	0 0.024999	-0.9138835 0 	-1.832126  10 
				0.9138835 -0.9138835	0	0.9388818	2.91881
						0.016668		-0.008336	0	0.008332 		3.4615397	
	P d3 =	-0.008336  0	0.016673 0	0 0.025	0.008336 0 	-1.832126  10 
						0.008332		0.008336	0	0.016668	2.91881
										3.499868	
									  	-1.798264 10	   when the time step h = 0.001 as well as
									2.932387
			3.466	
	x d (1 + ) =	   -1.834 10	   when the time step h = 0.00001.
					2.92

  The value of the state jump is the solution of the optimization problem in[START_REF] Bernstein | Scalar, Vectors, and Matrix Mathematics. Theory, Facts and Formulas[END_REF]. The table below shows the values of x(1 + ) for different value of the storage function matrix P being the solution of NLMI in[START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] for strict passivity.

										3.3986	
	when the time step h = 0.001 as well as x(1 + ) =	   -2.029 9.93	   when the time step h = 0.00001.
									3.4233
									.391813	
									-2.0267 0.723447	  
									3.42975
								Values of P	Values of x(1 + )
							0.029	0.001 -0.006 -0.004 		3.397125 
			P 1 =	0.001 -0.006 0.016 0.033 	0.016 0.032	0.001 0.01			-2.0267 9.9417	
						-0.004 0.001	0.01	0.0304	3.431
			0.04589	0.006245	-0.005963 -0.006264 		3.3952	
	P 2 =	0.006245 -0.005963 	0.03969 0.015618	0.015618 0.032439	-0.001698 0.009719 		-2.0269 9.9415 
			-0.006264 -0.001698	0.009719	0.034227	3.431
				0.03844	0.00339	-0.00596 -0.00489 		3.3958
	P 3 =	0.00399 -0.005963 0.015618 0.032439 0.03742 0.01562 	0 0.009719			-2.0268 9.9415
					-0.00489	0	0.009719	0.03262	3.431
										3.4299	
									   -1.9915 9.9262   
									3.4462

 

The numerical solver siconos computes the jump automatically and gives x(1 + ) =

https://nonsmooth.gricad-pages.univ-grenoble-alpes.fr/siconos/index.html

https://nonsmooth.gricad-pages.univ-grenoble-alpes.fr/siconos/index.html

Third case

The NLMI in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF] has no solution. Hence, the quadruple of the closed-loop system (A, B, C + F K, D) cannot be made strictly passive with such a set of inputs and the chosen feedback, and the framework developed in the foregoing sections does not apply. It is noteworthy that the conclusions are still valid even if we consider G ̸ = 0.

Conclusions

Open issues: robustness with respect to Shockley's model of diodes, output tracking [START_REF] Padula | MIMO tracking control of LTI systems: A geometric approach[END_REF] (tracking of an output y d = Hx d + Jλ d + Lu d with state feedback), output feedback (same problem as in this paper but with only y = Hx + Jλ available for feedback), state observation, adaptive control when parameters are uncertain (does the fact that uncertain parameters are present only in the ODE part and not in the complementarity part -as for friction oscillators-play a role?).

A Appendix

A.1 LCP and P-matrices Definition 8. A matrix M is said to be P -matrix if and only if all its principal minors are positive.

The complementarity between two vectors x, y ∈ IR n is denoted as 0 ≤ x ⊥ y ≥ 0 where the notation x ⊥ y means that x ⊤ y = 0, and nonnegativeness is understood componentwise. When w = M λ + q, this gives rise to a Linear Complementarity Problem (LCP), denoted as LCP(M, q). Theorem A.1 ([1], Theorem 12.14). A matrix M ∈ IR n×n is a P -matrix if and only if LCP(M, q) has a unique solution for all vectors q ∈ IR n .

A.2 Transformation of NLMI to an LMI

The inequalities in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF] and [START_REF] Chen | Perturbation bounds of P-matrix linear complementarity problems[END_REF] are nonlinear matrix inequalities(NLMI). In order to solve these inequalities, the NLMI must be reformulated to an LMI (following a classical technique, e.g., [

This LMI is a feasible problem and it is solved in the new variables Q = Q T ≻ 0, G, and N where

A.3 Motivation for feedback from λ in the controller in u (15)

Let us consider the LCS in (2) with the following matrices:

Let u = K(x -x d ) + u d , then the closed-loop system's quadruple (A, B, C + F K, D) is strictly state passive with K = (1.12, -1) and P = 1.12 -0.006 -0.006 2.012

given by solving the LMI in [START_REF] Brogliato | Well-posedness, stability and invariance results for a class of multivaled Lur'e dynamical systems[END_REF]. However, the LMI for strong passivity in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF] does not have a solution for the closed-loop quadruple (A, B, C + F K, D).

Let us introduce feedback from the complementarity variable λ, resulting in the extended controller

In order to check if the closed-loop quadruple (A, B, C + F K, D + F G) with the extended controller is strongly passive, let us check if the LMI in (20) has a solution. It appears that the LMI in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: passivity approach[END_REF] for strong passivity has a solution given by: K = 1.13 -0.99 , G = 0.59 and P = 1.17 -0.014 -0.014 2.16 (144)

By using the following property from the convex analysis (the chain rule):

where ϕ(t) = {z | M z ∈ S(t)}. Then,