## Section: New Results

### Control

#### Observer design

Participant : Bernard Brogliato.

The general problem of state observation for nonsmooth dynamical systems, or hybrid dynamical systems, remains largely open, in particular for systems whose trajectories may jump. In [16] , [18] solutions are proposed for the design of asymptotic observers for various classes of nonsmooth systems (differential inclusions, complementarity systems). The problem of “closing the loop” (the separation principle) is also solved in particular cases.

#### Trajectory tracking

Participants : Bernard Brogliato, Tran Anh Tu Nguyen.

In these works [25] , [24] the problem of extending the so-called passivity-based controllers to Lagrangian systems with unilateral constraints is considered. The first work [25] treats fully actuated rigid systems. The second work [24] deals with the case when joint flexibilities are present. This is thought to be quite important since impacts are likely to excite vibrational modes and possibly destabilize the closed-loop system. We first derive a suitable stability criterion, then we design a switching control algorithm and numerical simulations are performed with the Moreau's time-stepping scheme of the siconos platform.

#### Optimal control

Participant : Bernard Brogliato.

The problem of quadratic optimal control with state inequality constraints is studied in [15] , where the Pontryagin's necessary consitions take the form of a linear complementarity system (LCS). We take advantage of the formalism of the higher order Moreau's sweeping process [32] , that is a distribution differential inclusions, to analyze this LCS. The work of ten Dam on the geometrical analysis of the positive invariance of systems with inequality state constraints is also used. Both frameworks allow us to better study the qualitative properties of the optimal trajectories.

#### Digital sliding mode control

Participants : Vincent Acary, Bernard Brogliato.

The problem of difital sliding mode controllers is a long-standing issue not yet satisfactorily solved. We propose in [12] ideas which are inspired from the numerical methods of contact mechanics [1] and which permit a) to suppress the numerical chattering, b) to obtain a smooth stabilization on the sliding surfaces. The work is continued together with Yury Orlov in more general cases where the system is acted upon by disturbances and a disturbance estimation is added.