A Tanwani
email: aneel.tanwani@inria.fr

B Brogliato
email: bernard.brogliato@inria.fr

C Prieur

Passivity-Based Observer Design for a Class of Lagrangian Systems with

Perfect Unilateral Constraints Aneel Tanwani, Bernard Brogliato and Christophe Prieur

Abstract-This paper addresses the problem of estimating the velocity variables, using the position measurement as output, in nonlinear Lagrangian dynamical systems with perfect unilateral constraints. The dynamics of such systems are formulated as a measure differential inclusion (MDI) at velocity level which naturally encodes the relations for prescribing the post-impact velocity. Under the assumption that the velocity of the system is uniformly bounded, an observer is designed which is also a measure differential inclusion. It is proved that there exists a unique solution to the proposed observer and that solution converges asymptotically to the actual velocity.

I. INTRODUCTION

In this paper, we consider mechanical systems with unilateral constraints (without friction) on the position of a moving point, whose position and velocity we choose to denote by q and q, respectively. The unconstrained motion of the particle satisfies the equation q = F (t, q, q), (

and the position q is constrained by:

h i (q) ≥ 0, i ∈ {1, . . . , m} (1b)
where F : R × R n × R n → R n denotes a vector field, and h i : R n → R represent the constraints imposed on a moving particle. Mechanical systems with impacts, such as robots and colliding rigid bodies could be seen as systems with unilateral constraints. It is seen that the nonsmooth behavior, or discontinuity in state trajectory, appears in such systems when any of the constraint is active, that is, h i (q) = 0. This is because the velocity must change its direction instantaneously to keep the moving point inside the admissible set. In general, it is seen that the trajectories of such systems are algebraically constrained, they exhibit continuous (due to flow within the interior of the set) as well as discrete dynamics (due to impacts with the boundary of the set), and hence form an important class of nonsmooth systems. There are several modeling frameworks for such nonsmooth systems; one such modeling framework, which is used to model the motion of state-constrained trajectories is the so-called sweeping process [START_REF] Kunze | An introduction to Moreau's sweeping process[END_REF]. The term so-coined because they represent the motion of a point inside a closed set. As the the set moves, the point is swept across by the moving set. If for such processes, the constraint set is parameterized by time only, then we call it the first-order sweeping process. However, for system (1), we first define an admissible set for velocity q(•) which is parameterized by the state q(•), and this formulation leads to a second-order sweeping process. This paper is concerned with the design of observers for estimating the velocity q(•) using the position q(•) as the output, while using the sweeping process formulation to describe the dynamics of the system and the observer. The construction of observers, or state estimators, is a classical problem in the design of control systems and has found many useful applications such as output feedback control, and fault diagnosis. For smooth systems, there are several standard techniques such as Kalman filter, Luenberger observer, or high-gain approach. Such techniques have also been applied to design observers for certain classes of smooth and unconstrained Lagrangian systems with application to output feedback control, see for example, [START_REF] Berghuis | A passivity approach to controllerobserver design for robots[END_REF], [START_REF] Besancon | Global output feedback tracking control for a class of Lagrangian systems[END_REF], [START_REF] Nicosia | Robot control by using only joint position measurements[END_REF], [START_REF] Xian | A discontinuous output feedback controller and velocity observer for nonlinear mechanical systems[END_REF]. A common element of these designs is to assume that the velocity q(•) is uniformly bounded (in time) which is primarily because F (t, q, •) is quadratic in general.

Lately, the researchers have started looking at the stateestimation problem in nonsmooth systems. In this regard, we mention the recent work on observer design of switched systems with ordinary differential equations [START_REF] Shim | Hybrid-type observer design based on a sufficient condition for observability in switched nonlinear systems[END_REF], [START_REF] Tanwani | Invertibility and Observability of Switched Systems with Inputs and Outputs[END_REF], [START_REF] Tanwani | Observability for switched linear systems: characterization and observer design[END_REF], switched differential-algebraic equations [START_REF] Tanwani | Observability of switched differentialalgebraic equations for general switching signals[END_REF], [START_REF] Tanwani | An observer for switched differentialalgebraic equations based on geometric characterization of observability[END_REF], certain classes of differential inclusions [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: A passivity approach[END_REF], [START_REF] Osorio | Dissipative design of observers for multivalued nonlinear systems[END_REF], [START_REF] Tanwani | Stability and observer design for a class of nonsmooth dynamical systems with nonconvex state constraints[END_REF], complementarity systems [START_REF] Heemels | Observer-based control of linear complementarity systems[END_REF], and the references therein for more details. Classical approaches for observer design are based on constructing an auxiliary dynamical system driven by the error between the measured output and the estimated output, where it is shown that the resulting dynamics of the state estimation error converge to the origin. However, for nonsmooth systems subjected to impacts, such schemes are not easily implementable since the impacts, or discrete dynamics, are not influenced by error injection and hence destroy the integration effect.

For nonsmooth Lagrangian systems with impacts, the problem of state estimation has been considered in [START_REF] Menini | Velocity observers for non-linear mechanical systems subject to non-smooth impacts[END_REF] under certain restrictive assumptions, and state estimation with tracking control in [START_REF] Forni | Follow the bouncing ball: Global results on tracking and state estimation with impacts[END_REF] for motions restricted within a convex polyhedral domain. This article, however, deals with a more general class of nonsmooth Lagrangian dynamics, that allow more general domains, using the formalism of differential inclusions. The approach adopted is closely related to the observer design presented in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: A passivity approach[END_REF]. The authors in [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: A passivity approach[END_REF] work with differential inclusions that represent the first-order sweeping process and the resulting trajectories are absolutely continuous. Since the constraint set is a function of time, this information is passed to the state estimator. The state estimator replicates the system dynamics, and hence is a sweeping process of the first-order. The convexity of the constraint set then generates the passivity relation between the error dynamics and the output estimation error, which is the key component in proving the error convergence.

In this paper, however, the system under consideration is a second-order sweeping process. This way, the statetrajectories of the system are allowed to be of locally bounded variation (BV), and hence discontinuous, which introduces the major difference. Since locally BV functions may admit an infinite number of discontinuities in finite time, the Zeno phenomenon is not excluded in our setup. The proposed observer only describes the dynamics for the velocity estimate (and not the position) in the form of a differential inclusion driven by the measured output (position). It is proved that, for each output, there exists a unique solution to such differential inclusions, and that this solution converges to the actual velocity of the system asymptotically.

II. PRELIMINARIES

In this section, we collect some basic definitions and notations that will be used later on. The total variation of a function f :

[a, b] → R n is defined as var f (τ ) = sup k i=1 |f (t i ) -f (t i-1 )|, τ ∈ [a, b],
where the supremum is taken over all integers k ≥ 1, and all possible choices of the sequence

{t i } such that a = t 0 < t 1 < • • • < t k = τ . The function f (•) is said to be of bounded variation (BV) on [a, b] if var f (b) < ∞.
If it is right continuous with bounded variation we denote it with rcbv. It is locally rcbv if this holds for any bounded interval

[a, b] ⊂ R. If f (•) is BV on [a, b]
then it has countable number of jump discontinuities. Moreover, it has right and left limits everywhere. The right and left limits of a function at t are denoted by f (t + ) := lim s t f (s) and f (t -) := lim s t f (s), respectively, provided they exist. In this notation, right continuity of f (•) in t, means that

f (t + ) = f (t).
For an interval I ⊆ R, we denote by L 1 (I, R n ; µ) and L 1 loc (I, R n ; µ) the space of integrable and locally integrable functions, respectively, from I to R n with respect to the measure µ. If the measure is not specified then the integration is with respect to the Lebesgue measure. An absolutely continuous

(AC) function f : [a, b] → R n is a function that can be written as f (x) -f (a) = a x ḟ (t)dt for any x ≥ a for a function ḟ (•) ∈ L 1 ([a, b], R n ), which is considered as its derivative. The space of continuously differentiable functions is denoted by C 1 (R n , R m ), for m, n ∈ N.
If v : I → R n is a function of bounded variation, then one can associate with it a Lebesgue-Stieltjes measure or the socalled differential measure dv on I.

Also, if v(•) is rcbv on [a, b], then we have the relation that v(b) = v(a) + (a,b] dv.
The density of the measure dv with respect to a positive Radon measure µ over an interval I is defined as:

dv dµ (t) := lim ε→0 dv(I(t, ε)) µ(I(t, ε)) , (2)
where

I(t, ε) := I ∩ [t -ε, t + ε].
Similarly, one can define the density of the Lebesgue measure dt with respect to the Radon measure µ. A Radon measure ν is absolutely continuous with respect to µ if for every measurable set A, µ(A) = 0 implies that ν(A) = 0. Further, the measure ν is absolutely continuous with respect to µ if and only if the density function dν dµ (•) is well-defined (finite µalmost everywhere) and is dµ integrable.

For a set V ⊂ R n , we will denote its interior by int V , and the boundary of this set is denoted by ∂V . If V is convex, then N V (v) denotes the normal cone to V at v ∈ V and is defined as:

N V (v) := {w ∈ R n | w, x -v ≤ 0 ∀x ∈ V }, (3)
where •, • denotes the usual inner product in R n .

III. SOLUTION FRAMEWORK

In this section, we will describe the dynamics of nonsmooth Lagrangian systems using differential inclusions and briefly talk about their solutions. The observer will then be designed using this formalism.

A. Mathematical model

We consider mechanical systems with n-degrees of freedom that are subjected to some unilateral constraints. The position variable q ∈ R n is thus assumed to evolve in a set that admits the following form:

Φ := {q ∈ R n | h i (q) ≥ 0, i = 1, 2, . . . , m}. (4)
The geometry of the set Φ is determined by the functions h i (•), and the only condition we will impose on the functions h i (•) is that they are continuously differentiable so that ∇h i (•) is continuous for each i. This allows us to model a large number of closed domains which may even be nonconvex.

The convex polyhedral tangent cone V (q) to the region Φ at a point q is given by:

V (q) := {v ∈ R n | v ∇h i (q) ≥ 0, ∀i ∈ J (q)} (5)
where the set J (q) denotes the set of active constraints at q, i.e., J (q) := {i ∈ {1, . . . , m} : h i (q) ≤ 0}.

One can think of the set V (q) as the set of admissible velocities that keep the position variable q inside the set Φ. The closed convex polyhedral cone V • (q), polar to V (q) with respect to usual inner product on R n , is given by:

V • (q) := {w ∈ R n : w v ≤ 0, ∀v ∈ V (q)}, (6)
that is, V • (q) is the outward normal cone to Φ at q, and is generated by the vectors ∇h i (q), i ∈ J (q). It is noted that

V (q) = R n and V • (q) = {0} if q ∈ int Φ.
We now formulate the dynamics of system (1) as a measure differential inclusion:

dq = vdt (7a) M (q)dv + F (t, q, v)dt ∈ -N V (q) (v e ) ⊆ V • (q) (7b)
where

v e (t) := v(t + ) + ev(t -) 1 + e , (7c)
and e ∈ [0, 1] is the coefficient of restitution. The initial condition is assumed to satisfy q 0 := q(0) ∈ Φ, and v 0 := v(0) ∈ V (q 0 ). Since V (q(t)) is convex, the normal cone N V (q(t)) at each t is defined as in (3). To make connection to Lagrangian dynamical systems, in (7b), it is assumed that F (t, q, v) = C(q, v)v + f (t, q, v) where C(q, v)v denote the Coriolis and centrifugal torques.

The formulation for constrained mechanical systems, as in [START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF], was pioneered by J. J. Moreau [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF], and the MDI ( 7) is called a second order sweeping process (because the constraint set for velocities appearing in (7b) depends on the state variable q(•)). The motivation for working with the MDI is that we are seeking a solution to the evolution problem in the space of locally rcbv functions to deal with possible collisions with the boundary of the admissible set. Functions which are locally rcbv possess generalized derivatives that can be identified with Stieltjes measure and equation (7b) precisely describes the inclusion of the measure dv, associated with v(•), into a normal cone described by the constraint set Φ. It is also noted that the post-impact velocity determined according to Moreau's collision rule (or Newton's impact law) is directly encoded in the MDI [START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF]. Further details on inclusions of type [START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF] and comparisons with other modeling frameworks could be found in [START_REF] Brogliato | Nonsmooth Mechanics[END_REF]. For our purpose, it is seen that the observer design given in Section IV is partially aided by this compact formulation.

B. Existence of Solution

The solution of MDI [START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF] is considered in the following sense:

Definition 1: A solution to the Cauchy problem [START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF] with initial data (q 0 , v 0 ) ∈ Φ × V (q 0 ), over an interval I = [0, T ], is a pair (q, v) such that v(•) is rcbv on I; q(t) = q 0 + t 0 v(s)ds; and furthermore, there exists a positive measure (represented by) dµ such that both dt and dv possess densities with respect to dµ, denoted by dt/dµ, and dv/dµ respectively, and satisfy

M (q) dv dµ (t) + F (t, q, v) dt dµ (t) ∈ N V (q(t)) (v e )
, dµ-a.e. on I.

(8) The choice of the measure dµ is not unique since the righthand side of ( 7) is a cone. However, by Lebesgue-Radon-Nikodym theorem, the functions dt/dµ(•) ∈ L 1 (I, R; dµ) and dv/dµ(•) ∈ L 1 (I, R n ; dµ) are uniquely determined for a given dµ.

The problem of existence of solutions for evolution problems (1) has been studied for a long time. Earlier results on this problem dealt with the single constraint case (m = 1) and one may refer to [13, Chapter 3], [START_REF] Paoli | A numerical scheme for impact problems I and II[END_REF] for results in this direction. The basic idea in these works is to introduce a time discretization scheme, either at position level [START_REF] Paoli | A numerical scheme for impact problems I and II[END_REF] or velocity level [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, volume 9 of Progress in Nonlinear Differential Equations and their Applications[END_REF] to construct a sequence of approximate solutions which is shown to converge as the sampling interval converges to zero. For several unilateral constraints (m ≥ 2), the existence and uniqueness has been proved in [START_REF] Ballard | The dynamics of discrete mechanical systems with perfect unilateral constraints[END_REF] under analytic assumptions on the data using the solution theory for differential equations and variational inequalities. The most relaxed conditions, under which the existence of solutions has been proved using discretization at velocity level, have appeared recently in [START_REF] Dzonou | A convergence result for a vibro-impact problem with a general inertia operator[END_REF], [START_REF] Paoli | A proximal-like method for a class of second order measuredifferential inclusions describing vibro-impact problems[END_REF]. Based on the work of [START_REF] Paoli | A proximal-like method for a class of second order measuredifferential inclusions describing vibro-impact problems[END_REF], the following regularity assumptions are required on the system data for the existence of solution, and are also needed for the observer design:

(H1) The function F (•, •, •) is continuous and is continuously differentiable (C 1 ) with respect to its second and third arguments.

(H2) The mapping M (•), from R n to the set of symmetric positive definite matrices, belongs to class C 1 and

λ M |v| 2 ≤ v M (q)v ≤ λ M |v| 2 ∀ (q, v) ∈ Φ × R n . (9)
(H3) For each i ∈ {1, . . . , m}, the function h i ∈ C 1 (R n , R), ∇h i (q) is locally Lipschitz continuous and does not vanish in a neighborhood of {q ∈ R n : h i (q) = 0}.

(H4) The active constraints are functionally independent, i.e., {∇h i (q)} i∈J (q) is linearly independent for all q ∈ Φ.

Without recalling the formal result on existence and assuming that a solution exists in the sense of Definition 1 under hypotheses (H1) -(H4), we only collect the properties of the solutions to system (7) which provide more insight.

C. Solution Characteristics

1) The post-impact velocity: The condition, q(t) ∈ Φ, for all t, allows for jumps in the velocity when q hits the boundary of Φ. At that instant, one must have v(t -) ∈ -V (q(t)), and v(t + ) ∈ V (q(t)). According to (7b), the jumps in v(•) satisfy:

M (q(t))(v(t + ) -v(t -)) ∈ -N V (q(t)) (v e (t)),
If there is only one active constraint, i.e., J (q(t)) = {i}, for some fixed i ∈ {1, . . . , m}, then V (q) = R + ∇h i (q(t)), or equivalently, N V (q(t)) (v(t)) = R -∇h i (q(t)). These relations imply that

v(t + ) = -ev(t -) + (1 + e) proj M (q(t)) (V (q(t)), v(t -)), (10
) where proj M (q) (V (q)) denotes the projection on V (q) according to the kinetic metric at q, which is defined by the inner product v, w M (q) = v, M (q)w = M (q)v, w .

2) Regularity of state trajectories: The function q is absolutely continuous, but not necessarily everywhere differentiable. The velocity v(•) is a locally rcbv function, for which the left and right limits are defined everywhere. The acceleration is represented by the measure dv and can be decomposed as a sum of two measures: an atomic measure dµ a and Lebesgue-integrable function v(•), i.e., dv = dµ a + vdt.

3) Countably many impacts: The set of impact times, at which v(•) is discontinuous, is countable. One may simply take

dµ a = k≥0 [v(t + k ) -v(t - k )]δ t k
, where δ t k is the Dirac impulse at time t k and {t k } k≥0 is an ordered sequence of impact times. Thus, the formulation [START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF] does not exclude the Zeno phenomenon (with left accumulation point). However, if e = 1, then there exists a constant ρ > 0 such that t k+1t k > ρ, for each k ≥ 0.

4) Non-uniqueness and Continuity of solutions: The solution of system ( 7) is unique if the system data is analytic [START_REF] Ballard | The dynamics of discrete mechanical systems with perfect unilateral constraints[END_REF], but in general, it may not be the case. Even under the analyticity assumption, the solutions may not vary continuously with respect to initial conditions under the hypotheses (H1) -(H4). For this to hold, there is an additional condition on the set {∇h i (q)} i∈J (q) which states that for e = 0, the active constraints must satisfy ∇h i (q), M -1 (q)∇h j (q) ≤ 0, and for e ∈ (0, 1], ∇h i (q), M -1 (q)∇h j (q) = 0. The work of [START_REF] Forni | Follow the bouncing ball: Global results on tracking and state estimation with impacts[END_REF] assumes this condition because their design is based on closeness of solutions (in graphical sense) with respect to initial conditions. However, our observer design doesn't require this property and hence no such condition is imposed in our results.

IV. OBSERVER DESIGN

We now address the problem of designing observers for the systems considered in Section III. It will be assumed that the position q(•) is the measured variable, and the objective is then to design an estimator for the velocity v(•) of the moving particle. In addition, we suppose that the following assumption holds:

Assumption 1: The velocity v(•) obtained as a solution to (7) stays bounded, that is,

v(t) ∈ B v := {v ∈ R n : |v| ≤ C v } ∀t ≥ 0, (11)
and furthermore, there exists a constant C M > 0 s.t.

C(q, v) ≤ C M |v|, ∀ v ∈ B v . (12)
Before describing the observer dynamics, define the function v → g(q, v) = C(q, v)v and let f (t, q, •) and ḡ(q, •) denote the Lipschitz extensions 1 of f (t, q, •) and g(q, •) from B v with Lipschitz constants C f (t, q) and C g (q), respectively, and let F (t, q, •) = f (t, q, •) + ḡ(q, •). The observer we propose is:

M (q)dz + F (t, q, v)dt -M (q)Lvdt ∈ -N V (q) (v e ) (13a) where v(t) := z(t) -Lq(t), (13b) ve (t
) := v(t + ) + ev(t -) 1 + e (13c)
and q(•) in (13a) and (13b) is an absolutely continuous function of time which is obtained from [START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF] as the measured output. The initial condition v0 := v(0) ∈ B v . The design criteria for the output injection matrix L will appear later in Section IV-B. The solution to system (13) is interpreted in the sense similar to Definition 1.

In the sequel, we will show that the proposed observer (13) has the following two properties: 1 For a locally Lipschitz function

F : B → R n , the function F : R n → R n is called the Lipschitz extension of F (•) from B ⊂ R n if F (•) is globally Lipschitz over R n and F (v) = F (v) for all v ∈ B.
• Well-posedness: For each absolutely continuous function q(•), there exists a unique locally rcbv function v(•) that satisfies [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, volume 9 of Progress in Nonlinear Differential Equations and their Applications[END_REF]. • Error convergence: The estimate v(•) converges to v(•) asymptotically, i.e., lim t→∞ |v(t) -v(t)| = 0. In the following two subsections, it is proved that both these properties indeed hold. Before proceeding, note that the original system may not have unique solutions, but the observer has the property that it generates a unique trajectory corresponding to the each function q(•) observed as an output of system [START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF]; see [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: A passivity approach[END_REF]Remark 3.3] for further explanation along these lines.

A. Observer Well-posedness

It is noted that the function q(•) is seen as an external "input" by the observer and hence V (q(•)) is seen as a timeparameterized multi-valued function that does not depend on any of the internal states of the estimator. This makes the observer (13) a sweeping process of first order and the solutions for such systems are relatively easier to study. We have the following result:

Proposition 1: Assume that hypotheses (H1) -(H4) and Assumption 1 hold, then the system ( 13) is well-posed, that is, there exists a unique locally rcbv solution z : [0, ∞) → R. Moreover, it holds that ve (t) ∈ V (q(t)) ∀ t ≥ 0.

(14) The basic idea in proving the existence of solutions of such systems is to construct a sequence of approximate solutions using the time-discretization scheme and show that the limit of that sequence is a solution to [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, volume 9 of Progress in Nonlinear Differential Equations and their Applications[END_REF]. These arguments are quite technical and the proof has been omitted due to space limitations. The relation [START_REF] Moreau | Unilateral contact and dry friction in finite freedom dynamics[END_REF], however, automatically holds for any solution of [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, volume 9 of Progress in Nonlinear Differential Equations and their Applications[END_REF]. We remark that, unlike [START_REF] Castaing | BV periodic solutions of an evolution problem associated with continuous moving convex sets[END_REF], the uniqueness of solutions holds for system [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction, volume 9 of Progress in Nonlinear Differential Equations and their Applications[END_REF].

B. Error Analysis

The main result on convergence of error now follows: Proposition 2: Assume that hypotheses (H1) -(H4) and Assumption 1 hold. If there exist a constant β > 0 and a matrix L such that, for all (t, q) ∈ R + × Φ,

M (q)L+L M (q) ≤ -(2C M C v +2C f (t, q)+2C g (q)+β)I n , (15)
where I n ∈ R n×n denotes the identity matrix, then the velocity estimate v(•) given by ( 13) converges to v(•) exponentially, that is, for some c > 0,

|v(t) -v(t)| ≤ c e -βt |v(0) -v(0)|.
(16) Proof: Define the state estimation error as follows:

ṽ := v -v.
Then the dynamics for velocity estimation error are: M (q)dṽ = M (q)dv -M (q)dv (17a) = M (q)dv -M (q)dz + M (q)Ldq (17b)

= M (q)Lṽdt -(F (t, q, v) -F (t, q, v))dt -(η -η) (17c)
where

η ∈ N V (q) (v e ) and η ∈ N V (q) (v e )
and v e , ve are defined as in (7c) and (13c), respectively. In what follows, we fix dµ = dt + dµ a + dμ a , where the atomic measure dµ a (respectively dμ a ) is supported at the time instants where v(•) (respectively v(•)) is discontinuous. It is seen that dt + dµ a , and dt + dμ a are absolutely continuous with respect to dµ and hence the densities dv dµ (•) and dv dµ (•) are well-defined.

Let W : [0, ∞) → R + be defined as W (t) := ṽ(t) M (q(t))ṽ(t), then W (•) is locally rcbv using the chain rule [START_REF] Moreau | A chain rule involving vector functions of bounded variation[END_REF]Theorem 3], and its differential of the composition function is computed as follows:

dW dµ (t) = (ṽ(t + ) + ṽ(t -)) M (q(t)) dṽ dµ (t) + ∂ ∂q (ṽ(t + ) M (q(t)ṽ(t + )) dq dµ (t). (18)
dW dµ (t k ) = (ṽ + + ṽ-) M (q(t k )) dṽ dµ (t k ) (19)
= (ṽ + + ṽ-) M (q(t k ))(ṽ + -ṽ-),

where we used the notation ṽ+ = ṽ(t + k ), ṽ-= ṽ(t - k ). By definition, it follows that η ∈ -N V (q(t)) (v e ) ⇐⇒ η, v e -ve ≥ 0

η ∈ -N V (q(t)) (v e ) ⇐⇒ η, v e -ve ≤ 0

which in turn implies that η -η, v e -ve ≥ 0 or equivalently,

η -η, ṽ+ + eṽ - 1 + e ≥ 0. (23)
Substituting η -η = -M (q(t k ))(ṽ + -ṽ-), ( 23) is rewritten as

M (q(t k ))(ṽ + -ṽ-), ṽ+ + eṽ - 1 + e ≤ 0. (24)
In general, for e ∈ [0, 1], a, b ∈ R n , and a symmetric positive definite matrix M , one can show that,

M (a -b), (a + eb) (1 + e) ≤ 0 ⇐⇒ a M a -eb M b ≤ (1 -e)a M b ⇐⇒a M a -eb M b ≤ 2(1 -e)a M b -a M a + eb M b ⇐⇒(1 + e)(a M a -b M b) ≤ -(1 -e)(a -b) M (a -b) ⇐⇒ |a| 2 M -|b| 2 M ≤ - (1 -e) (1 + e) |a -b| 2 M
where | • | M denotes the norm with respect to the matrix M . Using the fact that M (q(t k )) is a symmetric positive definite matrix, ( 24) is equivalent to

dW dµ (t k ) = M (q(t k ))(ṽ + -ṽ-), ṽ+ + ṽ- = |ṽ + | 2 M (q(t k )) -|ṽ -| 2 M (q(t k )) ≤ - (1 -e) (1 + e) |ṽ + -ṽ-| 2 M (q(t k )) ≤ 0. (25)
If t = t k , then dt dµ (t) = 1, and we let ∂ ∂q (ṽ(t)M (q(t))ṽ(t)) dq dt (t) = ṽ(t) Ṁ (q(t), v(t))ṽ(t)

which results in dW dµ (t) = dW dt (t) = ṽ (M (q)L + L M (q))ṽ -ṽ (η -η)

-2 ṽ (F (t, q, v) -F (t, q, v)) + ṽ Ṁ (q(t), v(t))ṽ

where ṽ and q are evaluated at time t and the argument has been suppressed for conciseness. In the above expression, ṽ (η -η) ≥ 0 because v(t) = v e (t) and v(t) = ve (t) are contained in V (q(t)) for all continuity points t ≥ 0, due to which η, v -v ≥ 0, and η, v -v ≤ 0. Adding and subtracting the term 2ṽ C(q, v)ṽ from the right-hand side, and using the fact that Ṁ (q, v) -2C(q, v) is a skew symmetric operator, we obtain dW dµ (t) = ṽ (M (q)L + L M (q))ṽ -2ṽ (F (t, q, v) -F (t, q, v) -C(q, v)ṽ)

where we rewrite the norm of the last term as follows:

|F (t, q, v) -F (t, q, v) -C(q, v)ṽ| = |g(q, v) -ḡ(q, v) -C(q, v)ṽ + f (t, q, v) -f (t, q, v)| = C g (q)|ṽ| + C M C v |ṽ| + C f (t, q)|ṽ| It follows under condition (15) that dW dµ (t) ≤ -βW (t), t = t k .

Since we fixed dµ = dt + dµ a + dμ a , and W (•) is nonincreasing at the atoms of dµ a and dμ a because of [START_REF] Tanwani | Observability of switched differentialalgebraic equations for general switching signals[END_REF], and decreasing exponentially with respect to Lebesgue measure due to [START_REF] Tanwani | An observer for switched differentialalgebraic equations based on geometric characterization of observability[END_REF]. This results in W (t) ≤ e -βt W (0) ⇒ |ṽ(t)| ≤ λ M λ M e -βt |ṽ(0)|

showing that the error ṽ(•) decays exponentially.

Example 1: We now apply our result to a bouncing ball which is a one degree of freedom system with a single constraint. We let m(q) = 1, h(q) = q ≥ 0, so that V (q) = R if q > 0, and V (q) = R + if q ≤ 0. This leads to N V (q) (v) = {0}, if q > 0, or v > 0, R -, if q ≤ 0, and v ≤ 0. In the form of ( 7), the dynamics could be described by dq = v and dv + gdt ∈ -N V (q) (v).

If there is a discontinuity in v(•) at t k , then v(t - k ) < 0, and the impact occurs when q = 0, and at that instant

v(t + k ) -v(t - k ) ∈ -N R+ v(t + k ) + ev(t - k ) 1 + e .
This is an implicit relation for v(t + k ), for which the only solution is v(t + k ) = -ev(t - k ). The estimated velocity for this system is given by v(t) = z(t) -lq(t), where z(•) is obtained from

dz + gdt -lzdt + l 2 qdt ∈ -N V (q) (v e )
with l < 0. The proposed observer has been simulated in SICONOS [START_REF] Acary | An introduction to SICONOS[END_REF] and the simulation results are given in Figure 1. It is seen that the estimated velocity respects the same constraints as imposed on the actual velocity of the system. However, not every impact of the ball with the ground causes a discontinuity in v(•).

V. CONCLUSIONS

The problem of designing asymptotically convergent state estimators for nonsmooth mechanical systems was considered in this paper. The formalism of differential inclusions was used for both the system and observer dynamics. For the later, we proved the existence and uniqueness of solutions. The error analysis (for the convergence of velocity estimate) was based on generalizing the Lyapunov techniques to functions of locally bounded variation.

Fig. 1 .

 1 Fig. 1. Simulation of velocity estimator for bouncing ball with e = 0.9.

ACKNOWLEDGEMENTS

The first author would like to thank Prof. Manuel D. P. Monteiro-Marques for useful discussions related to the proof of Proposition 1.


Bernard Brogliato

Maurice Heemels

Observer Design for Lur'e Systems With Multivalued Mappings: A Passivity Approach

Keywords: Differential inclusion, Lur'e system, maximal monotone mapping, multivalued system, normal cone, observer, passivity, positive realness

This paper considers the design of state observers for Lur'e systems with multivalued mappings in the feedback path. In particular, we focus on maximal monotone mappings that that do not require any compactness and local boundedness properties and include various models for relays, friction characteristics and complementarity conditions. We propose two types of observers that are constructed by rendering a suitable operator passive in an appropriate sense. The well-posedness properties of the observer dynamics are carefully analyzed and the global asymptotic stability of the observation error is formally proven.

I. INTRODUCTION

Differential inclusions are ubiquitous in many engineering fields such as mechanics (mechanical systems with unilateral constraints and/or friction), electrical engineering (switched circuits), hybrid systems (relay control systems, discontinuous dynamical systems adopting generalized solutions concepts), economics (projected dynamical systems describing oligopolistic markets or traffic networks) and so on, see, e.g., [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF], [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides}, ser. Mathematics and its Applications[END_REF], and [START_REF] Van Der Schaft | The complementary-slackness class of hybrid systems[END_REF]. In this paper, we focus on the class of Lur'e systems with multivalued mappings in the feedback path. Such systems consist of the interconnection of a linear system given by _ x(t) = Ax(t) 0 Gw(t) + Bu(t) z(t) = Hx(t) y(t) = Cx(t) (1a) with a multivalued nonlinearity of the form

w(t) 2 (z(t)) (1b)
where for each z 2 IR l (z) is a subset of IR l containing possibly an infinite number of elements, x(t) 2 IR n and y(t) 2 IR p denote the state and the measured output, respectively, z(t) 2 IR l and w(t) 2 IR l are the variables going into and coming out of the nonlinearity, respectively, and u(t) 2 IR m is the control input at time t 2 IR. Systems of the considered type may arise as a natural consequence of modelling (e.g. friction models in mechanical systems, descriptions of ideal diodes in electrical circuits, etc.) or as a consequence of using generalized solution concepts for discontinuous dynamical systems (e.g., Filippov solutions [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides}, ser. Mathematics and its Applications[END_REF]). Examples include various important classes of nonsmooth dynamical systems such as certain piecewise linear systems, linear relay systems, mechanical systems with friction, linear complementarity systems [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF], [START_REF] Heemels | Linear complementarity systems[END_REF], and electric circuits with switching elements and MOS transistors [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF], [START_REF] Vasca | A new perspective for modeling power electronics converters: complementarity framework[END_REF]. In this paper we consider the problem of observer design for such Lur'e systems with multivalued B. Brogliato is with the INRIA, Bipop Team-Project, ZIRST Montbonnot, Saint-Ismier 38334, France (e-mail: bernard.brogliato@inrialpes.fr).

W. P. M. H. Heemels is with the Department of Mechanical Engineering, Eindhoven University of Technology Eindhoven, 5600 MB, The Netherlands (e-mail: M.Heemels@tue.nl). mappings in the feedback path. In particular, we will focus on the case where [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] is maximal monotone [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF]. The requirement that the map- ping is monotone is an extension of the usually considered concept of continuous, sector bounded nonlinearity.

Only few works on observability and observer design methodologies for differential inclusions are available in the literature. The observability properties of a class of differential inclusions, different from those studied in this paper, are examined in [START_REF] Aubin | Observability of systems under uncertainty[END_REF], while in [START_REF] Camlibel | Conewise linear systems: Non-zenoness and observability[END_REF] observability for a class of linear complementarity systems is studied, however, without considering the problem of observer design. The only exception in the literature that really considers observer design for multivalued systems is formed by the work in [START_REF] Osorio | Dissipative design of observers for multivalued nonlinear systems[END_REF]. However, the differential inclusions considered in [START_REF] Osorio | Dissipative design of observers for multivalued nonlinear systems[END_REF] are different from those considered in this paper, as compactness and local boundedness properties of the multivalued maps are assumed. These conditions are typically not satisfied by the multivalued right-hand sides that we work with.

II. PRELIMINARIES

A. Notation and Definitions

For a symmetric matrix A, we denote its smallest eigenvalue by min (A) and its largest eigenvalue by max (A). A (not necessarily symmetric) square matrix P 2 IR n2n is called positive definite (denoted by P 0), if x > P x > 0 for all x 2 IR n , x 6 = 0. Similarly, it is called nonnegative definite (denoted by P 0), if x > P x 0 for all x 2 IR n . For a nonnegative definite and symmetric matrix P , we denote its square root by R = P 1=2 , which satisfies R = R > 0 and R 2 = P . The matrix triple (A; B; C), where B has full column rank, is called strictly passive, if there exist a P = P > 0 and a

Q = Q > 0 such that P A + A > P = 0 Q (2a) B > P = C: (2b)
The material that follows is taken from [START_REF] Hiriart-Urruty | Fundamentals of Convex Analysis[END_REF] and [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems[END_REF]. A mapping : IR l ,! IR l is said to be multivalued if it assigns to each element x 2 IR l a subset (x) IR l (which may be empty). The domain of the mapping (1), dom() is defined as dom() = fx 2 IR l j(x) 6 = ;g. We define the graph of the mapping (1) as Graph = f(x; x 3 )jx 3 2 (x)g.

A multivalued mapping (1) is said to be monotone, if for all x 1 , x 2 2 dom() and all x 3 1 2 (x1) and x 3 2 2 (x2) it holds that hx 3 1 0x 3 2 ; x10 x2i 0, where h1; 1i is the inner product in IR l . In addition, k1k denotes the corresponding Euclidean norm. A multivalued mapping (1) is said to be maximal monotone, if (1) is monotone and no enlargement of its graph is possible without destroying monotonicity, or more precisely, if for every pair (x 1 ; x 3 1 ) 6 2 Graph there exists a point (x 2 ; x 3 2 ) 2 Graph with hx 3 1 0 x 3 2 ; x1 0 x2i < 0. For two (multivalued) functions F : IR k ,! IR l and G : IR l ,! IR m , G F denotes their composition, i.e. for x 2 IR k we define G F (x) = G(F (x)) := y2F (x) G(y).

For an interval I IR, we denote by L 1 The normal cone to a convex set S IR n at a point x 2 S is N (S; x) = fz 2 IR n j hz; y 0 xi 0; for all y 2 Sg. A convention is that N (S; x) = ; if x 6 2 S. Next we present a technical lemma that will be of importance in the remainder of the paper and can be derived from [START_REF] Rockafellar | Variational Analysis[END_REF]Theor. 12.43].

Lemma 2.1: Let H : I R n ! IR l be the affine mapping given by H(x) = Hx + h for all x 2 IR n corresponding to the matrix H 2 IR l2n and the vector h 2 IR l and suppose that H has full row rank l. Then the following statements hold. If : I R l ,! IR l is a maximal monotone mapping with dom 6 = ;, then the mapping

x 7 ! H > (Hx + h) is also maximal monotone.

B. Monotone Differential Inclusions

Let us consider the differential inclusion (DI) _ x(t) 2 0F (x(t)) + f (t); x(t 0 ) = x 0 2 dom(F ):

(3) Definition 2.2: A locally AC trajectory x : [t 0 ; 1) ! IR n is called a solution to the DI and the initial condition in (3), if x(t 0 ) = x 0 , for all t 2 [t0; 1) x(t) 2 dom(F ), and for almost all t 2 [t0;

1) _ x(t) 2 0F (x(t)) + f (t).
The following result is a generalization of the Hille-Yosida Theorem [12, Theor. 3.7.1].

Theorem 2.3: Let F : IR n ,! IR n be a maximal monotone operator and f : IR ! IR n be locally AC. Then the DI and the initial condition in (3) with x 0 2 dom(F ) possess a unique locally AC solution x(1) on [t 0 ; +1) in the sense of Definition 2.2.

C. Perturbed Moreau's Sweeping Process

Let us now deal with another class of DIs consisting of so-called perturbed Moreau's sweeping processes, see [START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF], in which the multivalued mapping (1) can possibly be time-varying. In particular, we consider (t; x) = N (S(t); x) the normal cone to the set S(t) at x. The resulting DIs are then given by 0 _

x(t) 2 N (S(t); x(t)) + f (t; x(t)) a.e. t 2 [t 0 ; +1), x(t 0 ) = x 0 2 S(t 0 ) (4) in case one is interested in locally AC solutions. The difference between (3) and ( 4) is reflected by the fact that in (3) the multivalued mapping F (1) depends on x only and f (1) on the time only, while in (4) both the multivalued mapping and f (1; 1) depend both on the time t and the state x. In addition, the solution trajectories of (4) might have to be considered in a larger function class consisting of functions of local bounded variation when the set S(1) is not varying in a locally AC manner (see (A2) below). In that case, (4) has to be interpreted as a measure differential inclusion, a term coined by J.J. Moreau (see e.g. [START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF], [START_REF] Leine | Stability and Convergence of Mechanical Systems With Unilateral Constraints[END_REF] and [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] for an introduction to Moreau's sweeping process) in which the solutions are of bounded variation and can allow for discontinuities in the state trajectory x(1). Some of the well-posedness and observer design techniques presented below can be extended to this case as well, see [START_REF] Brogliato | Design of Observers and Certainty Equivalence Controllers for Differential Inclusions: A Passivity Approach INRIA Research[END_REF] for more details.

It is noteworthy that when S(t) is closed, convex and nonempty for each t, then N (S(t); 1) defines a maximal monotone mapping for each fixed t as proven in [START_REF] Rockafellar | Variational Analysis[END_REF]Coroll. 12.18] Let us now present existence and uniqueness results for the inclusions of the form (4). Let f : I 2IR n ! I R n be a separately measurable map on I = [t0; t1] with t 1 < +1 such that:

• For every > 0 there exists a nonnegative function k (1) 2 L 1 (I ; I R) such that for all t 2 I and for any (x; y) 2 B(0; ) 2 B(0; ) one has kf(t; x) 0 f (t; y)k k (t)kx 0 yk; • there exists a nonnegative function (1) 2 L 1 (I ; I R) such that, for all t 2 I and for all x 2 s2I S(s), kf(t; x)k (t)(1 + kxk). Then for any x 0 2 S(t 0 ) the inclusion (4) has a unique AC solution

x(1) on I (in a sense similar to Definition 2.2).

The first condition is a kind of local Lipschitz continuity property in the second variable of f (1; 1) and the second condition is a natural growth condition. In case t 1 = 1 then the theorem provides a result on the existence and uniqueness of a locally AC solution in a straightforward manner (in which k ( 1) and (1) become L 1 loc -functions and S(1) varies in a locally AC manner).

III. PROBLEM STATEMENT Consider the system (1), where a time-independent multivalued map (1) is in the feedback loop, which yields the following DI:

_ x(t) = Ax(t) 0 Gw(t) + Bu(t) w(t) 2 (H x(t)) y(t) = Cx(t); x(0) = x0 2 dom( H); (5)
where A 2 n2n , B 2 n2m , G 2 n2l , H 2 l2n and C 2 p2n . Moreover, we assume that G has full column rank, the mapping : l ,! l is maximal monotone and u( 1) is locally AC. To make the observer design meaningful, it is natural to assume that the system (5) whose state x(1) is to be estimated by an observer, allows solutions for all relevant initial states and input functions.

Assumption 3.1: The system in (5) possesses a locally AC solution

x(1) on [0; +1) for any x(0) 2 dom( H) and locally AC input functions u(1). Such conditions can be verified by results in the mathematical literature on DIs [START_REF] Deimling | Multivalued Differential Equations, ser. Nonlinear Analysis and Applications[END_REF], [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides}, ser. Mathematics and its Applications[END_REF] (in case of local boundedness of (1)) or the results in Section II above, which also apply when (z) is an unbounded set for some z 2 I R l . The first proposed observer ("basic" observer scheme) for the system (5) has the following form:

_ x(t) = (A 0 LC)x(t) 0 G ŵ(t) + Ly(t) + Bu(t) ŵ(t) 2 (H x(t)) (6)
where L 2 n2p is the observer gain and H x(0) 2 dom(). The second proposed observer ("extended" observer scheme) has the following form:

_ x(t) = (A 0 LC)x(t) 0 G ŵ(t) + Ly(t) + Bu(t) ŵ(t) 2 ((H 0 KC)x(t) + Ky(t)) (7)
where L 2 n2p and K 2 l2p are the observer gains and x(0) is such that (H 0KC)x(0)+Ky(0) 2 dom(). The basic observer is a special case of the extended observer with K = 0. The idea of the extended observer was also used for Lipschitz continuous (single-valued) systems in [START_REF] Arcak | Observer based control of systems with slope-restricted nonlinearities[END_REF]. Although the basic observer is much more natural and closer to the observer structures for linear systems, there exist multivalued systems as in [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] for which no basic observer can be constructed by the design procedure given below, but one can still find an extended observer as will be demonstrated in Example 4.1 below.

Problem 3.2: The problem of observer design consists in finding the gain L for the basic observer or the gains L and K for the extended observer, such that:

• Observer well-posedness: for each solution x(1) to the observed plant ( 5) there exists a unique solution x (1) to the observer dynamics on [0; 1).

• Asymptotic state recovery: x(1) asymptotically recovers x(1),

i.e. lim t!1 [x(t) 0 x(t)] = 0.

Remark 3.3:

As we only assume the existence of solutions in Assumption 3.1, it might be the case that the observed plant (5) allows for multiple solutions given an initial condition x(0) = x 0 . However, as the solution trajectory x(1) enters the observers ( 6) and ( 7) through the measured output y(1), the observer dynamics has different terms in the right-hand side corresponding to the different solutions. Indeed, let x a (1) and x b (1) be two solutions of the observed plant for x(0) = x 0 and let y a = Cx a and y b = Cx b be the corresponding outputs. When the observer dynamics produces unique solutions given an initial condition and given the driving inputs u and y, as imposed in Problem 3.2, the observer has for each individual solution trajectory of the observed plant a unique response. To be more precise, for initial condition In essence, this is not different than for observer design for linear or smooth nonlinear systems with unique trajectories given an initial condition and exogenous inputs: the observer just recovers asymptotically the state trajectory that corresponds to the input and output trajectories that are actually fed to it. From a practical point of view, the given interpretation is meaningful, because the actual physical plant (for which ( 5) is only a model) typically behaves according to only one of the possible solutions as allowed by the model, for instance, due to the presence of small disturbances.

IV. MAIN RESULTS

In this section we will prove that if the gains L and K are chosen such that the triple (A 0LC;G; H) (respectively (A 0LC; G; H 0KC))is strictly passive, then the obtained observer ( 6) [ [START_REF] Brogliato | Design of Observers and Certainty Equivalence Controllers for Differential Inclusions: A Passivity Approach INRIA Research[END_REF], respectively] will satisfy the requirements mentioned in Problem 3.2. To compute the gains L and K such that (A 0 LC; G; H 0 KC) is strictly passive, one can solve the matrix (in)equalities: 0 (A 0 LC) > P 0 P (A 0 LC) 0 P = P > 0 G > P = H 0 KC [START_REF] Camlibel | Conewise linear systems: Non-zenoness and observability[END_REF] where Q in (2) can then be taken as Q := 0(A 0 LC) > P 0 P (A 0 LC). Condition ( 8) is a linear matrix inequality (LMI) in P , K, L > P , which can be solved efficiently. For necessary and sufficient conditions for the existence of solutions to [START_REF] Camlibel | Conewise linear systems: Non-zenoness and observability[END_REF], see for instance [START_REF] Arcak | Observer based control of systems with slope-restricted nonlinearities[END_REF].

To motivate the introduction of the extended observer next to the basic observer, consider the following example. and B arbitrary. As A is unstable, the triple (A; G; H) is not (strictly) passive by itself. If we try to find a matrix L such that (A 0LC;G; H) is strictly passive, we have to satisfy 0(A0LC) > P 0P(A0LC) 0; P = P > 0 and G > P = H

for some matrix P . Since the condition G > P = H requires (1 0)P = (0 1), we can conclude that the first row of P must be equal to (0 1), which obstructs the positive definiteness of P . Hence, one cannot make (A0LC; G; H) strictly passive by suitable choice of L. Consequently, we will not be able to find a basic observer using the results below as they require (A 0LC; G; H) to be strictly passive. However, with L = (2 0)

> and K = 01, we get A 0 LC = 0I 2 and H 0 KC = (1 1).

Since the matrix P = (

1 1 1 2
) is positive definite and symmetric and satisfies Q := 0(A 0 LC) > P 0 P (A 0 LC) 0 and G > P = H 0 KC, (A 0 LC; G; H 0 KC) is strictly passive and an extended observer can be designed based on the theory presented next.

A. Well-Posedness of the Observer

To prove that strict passivity of (A 0 LC; G; H 0 KC) guarantees the proper behavior of the observer, we start with two lemmas on wellposedness. We will start with the case K = 0 (the basic observer) Lemma 4.2: [Time-independent [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF], basic observer, AC solutions] Consider the system ( 5) and the basic observer [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF]. We assume that the triple (A 0 LC; G; H) is strictly passive, G has full column rank, (1) is maximal monotone, and Assumption 3.1 holds. Let u(1) be a locally AC input function and x(1) a corresponding locally AC so- lution to ( 5) with output trajectory y(1) for some x(0) 2 dom( H).

Then the corresponding observer dynamics ( 6) has a unique locally AC solution on [0; 1) for any initial state x(0) 2 dom( H). = Rx; [START_REF] Edmond | Relaxation of an optimal control problem involving a perturbed sweeping process[END_REF] where R = P 1=2 , transforms (6) into _ (t) = R(A 0 LC)R 01 (t)0RG ŵ(t)+RBu(t)+RLy(t) ŵ(t) 2 HR 01 (t) : 12) and ( 6) follow now from Theorem 2.3.

In the following lemma we address the question of well-posedness of the extended observer scheme. Since in this case the multivalued mapping in ( 7) is time-dependent, we will consider a particular class of mappings (1) that corresponds to Section II-C. Actually, in this case it will turn out that the second and third condition in (8), i.e. the existence of a symmetric positive definite matrix P such that G > P = H 0 KC suffices to prove well-posedness.

Lemma 4.3: [Time-independent (1) = N(S; 1), extended observer, AC solutions] Consider the system (5) and the extended ob-server ( 7) with (1) = N(S; 1), where the set S IR l is assumed to be nonempty, closed and convex. Suppose that Assumption 3.1 holds and assume that there exists a symmetric positive definite matrix P such that G > P = H 0KC and G has full column rank. Let the signal u(1) be locally AC and let x(1) be a corresponding locally AC solution to [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF] with output trajectory y(1) for some x(0) with Hx(0) 2 S. Then the corresponding observer dynamics (7) has a unique locally AC solution on [0; 1) for each x(0) with (H 0 KC)x(0) + Ky(0) 2 S = dom().

Proof: Let us introduce the change of variable ( 10) for [START_REF] Brogliato | Design of Observers and Certainty Equivalence Controllers for Differential Inclusions: A Passivity Approach INRIA Research[END_REF], where as before, R = P 1=2 . In the same way as in the proof of Lemma 4.2, ( 7) is transformed into

_ (t)2R(A0LC)R 01 (t)0R 01 (H0KC) >
2 (H 0KC)R 01 (t)+Ky(t) +RBu(t)+RLy(t): [START_REF] Heemels | Observer design for Lur'e systems with monotonic multivalued mappings[END_REF] Let S 0 (t) = f 2 IR n j (H 0 KC)R 01 + Ky(t) 2 Sg IR n . Since G has full column rank l, (H 0 KC)R 01 = G > R has full row rank l. As S is nonempty by the hypothesis, this implies that S 0 (t) is nonempty for each t. Indeed, S is a subset of IR l and the full row rank of (H 0KC)R 01 implies that for any l-dimensional vector z of S. one can find at least one such that (H 0KC)R 01 = z0Ky(t). By [15, Prop. 1.2.4], the closedness and convexity of S carry over to S 0 (t) for each t. Hence, condition (A1) of Theorem 2.4 is satisfied. In addition, the fact that y(1) is locally AC implies that S 0 (1) varies in a locally AC manner as in (A2). Consider now N(S 0 (t); x). By applying Lemma 2.1 (for fixed t) we obtain that N(S 0 (t); ) = R 01 (H0KC) > N(S; (H0 KC)R 01 + Ky(t)). Therefore, one can rewrite (13) as 0 _ (t) + R(A 0 LC)R 01 (t) 0 RBu(t) 0RLy(t) 2 N S 0 (t); (t) (14) where (0) = Rx(0) 2 S 0 (0). The description [START_REF] Heemels | Linear complementarity systems[END_REF] fits within (4) with S 0 (1) satisfying the conditions of Theorem 2.4. Since u(1) and y(1) are locally AC, the result follows now from Theorem 2.4.

B. Asymptotic Recovery of the State

The following theorem states the main result of the paper. Theorem 4.4: Consider the observed system (5) and either the basic observer [START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF] or the extended observer [START_REF] Brogliato | Design of Observers and Certainty Equivalence Controllers for Differential Inclusions: A Passivity Approach INRIA Research[END_REF], where (A 0 LC; G; H) or (A 0 LC; G; H 0 KC), respectively, is strictly passive with corre- sponding positive definite and symmetric matrices P and Q satisfying [START_REF] Arcak | Observer based control of systems with slope-restricted nonlinearities[END_REF]. Assume also that the additional conditions of Lemma 4. Proof: Using Lemma 4.2 or Lemma 4.3 for the basic and extended observers, respectively, it follows that for each locally AC solution to the observer plant [START_REF] Brogliato | Some perspectives on the analysis and control of complementarity systems[END_REF], the observer also has a locally AC solution x(1) provided H x(0) 2 dom() or (H 0 KC)x(0) + Ky(0) 2 dom() = S, respectively. Hence, the observation error e(1) = x(1)0

x(1) is also locally AC and satisfies for the extended observer (7) almost everywhere the error dynamics, obtained by subtracting ( 5) and ( 7)

_ e(t) = (A 0 LC)e(t) 0 G (w(t) 0 ŵ(t)) (16a) w(t) 2 (Hx(t)) (16b) ŵ(t) 2 (Hx(t) 0 (H 0 KC)e(t))): (16c)
Note that the error dynamics for the basic observer is obtained as a special case of ( 16) by taking K = 0. We consider now the candidate Lyapunov function V (e) = (1=2)e > Pe. Since e is locally AC, t 7 ! V (e(t)) is also locally AC, and the derivative _ V (e(t)) exists for almost all t. Hence, _

V (e(t)) satisfies for almost all t _ V (e(t)) = e > (t)P _ e(t) = e > (t)P ((A 0 LC)e(t) 0 G (w(t) 0 ŵ(t))) = 0 1 2 e > (t)Qe(t) 0 e > (t)(H 0 KC) > (w(t) 0 ŵ(t)) [START_REF] Monteiro Marques | Differential Inclusions in Nonsmooth Mechanical Problems[END_REF] for some w(t), ŵ(t) satisfying (16b) and (16c). Since e > (t)(H 0 KC)

As e > Qe min (Q)e > e ( min (Q)= max (P ))e > Pe= (2 min (Q)= max (P ))V (e) for all e 2 IR n , we have that _ V (e(t)) 0 min (Q) max (P ) V (e(t)) :

Clearly, this implies that min (P ) ke(t)k 2 V (e(t)) e 0 t V (e(0)) max(P)e 0 t ke(0)k 2 :

This proves the exponential recovery of the state. The condition ( 15) is obtained by taking the square root of the inequality above.

Two remarks on the above results are in order. Remark 4.5: The above derived methodologies were successfully applied to the observer design problem for an experimental setup of deep sea oil drilling equipment in the preliminary version [START_REF] Heemels | Observer design for Lur'e systems with monotonic multivalued mappings[END_REF] of this paper. The multivalued mapping was caused by the presence of dry friction with Stribeck effect that could be well described by a multivalued friction characteristic. The underlying linear model was not passive itself. Only the extended observer turned out to be applicable in this case to exponentially recover the state of the drilling equipment. See [START_REF] Heemels | Observer design for Lur'e systems with monotonic multivalued mappings[END_REF] for more details.

Remark 4.6: Possible extensions of the observer design methodologies to other classes of differential inclusions are possible. For instance, extensions of Theorem 4.4 to more general systems of the form _ x(t) = Ax(t) 0 Gw(t) + Bu(t) w(t) 2 (Hx(t) 0 Dw(t) + Eu(t)) y(t) = Cx(t) [START_REF] Van Der Schaft | The complementary-slackness class of hybrid systems[END_REF] may be obtained provided one can prove the well-posedness property of the corresponding observer. The stability proof can be carried over mutatis mutandis, but the hard part of the proof would be obtaining existence and uniqueness of solutions x (1) for the observer dynamics. Another direction for extension of Lemma 4.3, for which partial results are reported in [START_REF] Brogliato | Design of Observers and Certainty Equivalence Controllers for Differential Inclusions: A Passivity Approach INRIA Research[END_REF], is the consideration of perturbed Moreau's sweeping processes in which the set S becomes time-varying as well (possibly not in a locally AC manner). This results in state trajectories that possess discontinuities and as such, the observer design problem has to be tackled for systems with state jumps, which is known to be a highly nontrivial problem. For polyhedral sets, one can find observer design methods in the spirit of this paper in the accompanying report [START_REF] Brogliato | Design of Observers and Certainty Equivalence Controllers for Differential Inclusions: A Passivity Approach INRIA Research[END_REF].

V. CONCLUSIONS

In this paper, we presented novel methodologies for the observer design for a class of multivalued systems consisting of Lur'e systems with maximal monotone multivalued mappings in the feedback path. As the considered class of systems is nonsmooth and multivalued, the existing literature on nonlinear observer design does not apply to this setting. We proposed two observer structures (the basic and the extended observer) and carefully examined the existence and uniqueness of solutions given an initial condition and input function (well-posedness), as this is not a priori guaranteed. The observer design is constructive in nature as it is based on rendering the linear part of the observation error dynamics strictly passive by choosing appropriate observer gains. Under the natural assumption that the observed system has a solution, when the control input belongs to a certain admissible class, it is shown that there exists a unique solution for the estimated state, and that the observer recovers the state of the original system exponentially.

  f 2 L 1 ([a; b]; IR n ), which is considered as its derivative. An absolutely continuous function is almost everywhere differentiable. A function f : I ! IR n is locally AC if it is AC for any bounded interval [a; b] I.

Theorem 2 . 4 :

 24 [10, Theor. 1] Let S(1) satisfy the following assumptions. (A1) For each t t0, S(t) is a nonempty, closed and convex subset of IR n . (A2) S(1) varies in an AC way, i.e. there exists an AC function v such that for any y 2 IR n and s, t t 0 jd (y; S(t)) 0 d (y; S(s))j kv(t) 0 v(s)k ; where d(y; S) = inffky 0 xk j x 2 Sg.

  x(t0) = x0 and external inputs u(1) and y(1), where the latter can be y = y a = Cx a or y = y b = Cx b , the observer has two different solutions, say xa (1) and xb (1), respectively. The problem definition above should be interpreted in the sense that it requires in this case that both lim t!1 [x a (t) 0 x a (t)] = 0 and lim t!1 [x b (t) 0 x b (t)] = 0.

  Proof: Since the triple (A 0 LC; G; H) is strictly passive and G has full column rank there exist positive definite and symmetric matrices P and Q that satisfy (2) with K = 0. Applying the change of variables introduced in[START_REF] Brogliato | Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings[END_REF]

( 11 )

 11 Since H x(0) 2 dom(), we have HR 01 (0) 2 dom(). Define the mapping : n , ! n as () = R 01 H > (HR 01 ). Using the strict passivity condition that yields G > P = H,[START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides}, ser. Mathematics and its Applications[END_REF] can be rewritten as _ (t) 2 R(A 0 LC)R 01 (t) 0 ((t)) + RBu(t) + RLy(t)[START_REF] Goeleven | Variational and Hemivariational Inequalities: Theory, Methods and Applications, ser. Nonconvex Optimization and its Applications[END_REF] where (0) 2 dom(). From the strict passivity condition (2) (with K = 0) and the full-column rank of G, it follows that H = G > P and HR 01 = G > R have full row rank. Together with the fact that (1) is maximal monotone we have that (1) is maximal monotone as well due to Lemma 2.1. From the strict passivity condition (8) it follows that R 01 (A 0 LC) > R + R(A 0 LC)R 01 is negative definite, which means that the mapping 7 ! 0R(A0LC)R 01 is monotone by def- inition. Maximality of the mapping 7 ! 0R(A 0 LC)R 01 follows from linearity, see [4, Prop. 2.3]. Hence, the mapping 7 ! 0R(A 0 LC)R 01 + () is maximal monotone as the sum of maximal monotone mappings is maximal monotone again [19, Coroll. 12.44]. Since the signal u(1) is locally AC, and y(1) is locally AC due to Assumption 3.1, existence and uniqueness of locally AC solutions to (

2 or Lemma 4 . 3 ,

 43 respectively, are satisfied. Let x(1) be a locally AC solution to (5) for x(0) 2 dom( H) and locally AC input u : [0; +1) ! IR m . Then the observer (6) or[START_REF] Brogliato | Design of Observers and Certainty Equivalence Controllers for Differential Inclusions: A Passivity Approach INRIA Research[END_REF], respectively, has for each x(0) with H x(0) 2 dom() or (H 0 KC)x(0) + Ky(0) 2 dom() = S, respectively, a unique locally AC solution x(1), which exponentially recovers the state x in the sense that the observation error e(t) := x(t) 0 x(t) satisfies the exponential decay bound ke(t)k max(P ) min(P ) ke(0

  (I; IR n ) and L 1 loc (I; IR n ) the Lebesgue space of integrable and locally integrable functions, respectively, from I to IR n . A function f : I 2 IR n

	! IR n is separately
	measurable if f (1; x) : I

! IR n is measurable for all x 2 IR n . An absolutely continuous (AC) function f : [a; b] ! IR n is a function that can be written as f (x) 0 f (a) =

x a _ f (t)dt for any x a for a function _