Stability and Observer Design for a Class of Nonsmooth Dynamical Systems with Nonconvex State Constraints

This paper deals with the stability and observer design for Lur’e systems with multivalued nonlinearities, which are not necessarily monotone or time-invariant. Such differential inclusions model the motion of state trajectories which are constrained to evolve inside timevarying non-convex sets. Using Lyapunov-based analysis, sufficient conditions are proposed for local stability in such systems, while specifying the basin of attraction. If the sets governing the motion of state trajectories are moving with bounded variation, then the resulting state trajectories are also of bounded variation, and unlike the convex case, the stability conditions depend on the size of jumps allowed in the sets. Based on the stability analysis, a Luenberger-like observer is proposed which is shown to converge asymptotically to the actual state, provided the initial value of the state estimation error is small enough. In addition, a practically convergent state estimator, based on the high-gain approach, is designed to reduce the state estimation error to the desired accuracy in finite time for larger initial values of the state estimation error. The two approaches are then combined to obtain global asymptotically convergent state estimates. Key-words: Nonsmooth systems, sweeping process, prox-regular sets, locally bounded variation, Lyapunov stability, observer design, differential measure. ∗ Corresponding author. Email: [email protected] † Team BipOp, INRIA Grenoble, 655 Avenue de l’Europe, 38334 Montbonnot Cedex, France. ‡ Head of Team BipOp. Email: [email protected] § Senior Researcher (CNRS), Gipsa-lab, Department of Automatic Control, 11 Rue des Mathématiques, BP 46, 38402 Saint Martin d’Hères, France. Email: [email protected] ha l-0 07 90 61 8, v er si on 2 9 O ct 2 01 3 Stabilité et Observateurs pour les Systèmes Nonsmooth avec les Contraintes Nonconvexes. Résumé : This paper deals with the stability and observer design for Lur’e systems with multivalued nonlinearities, which are not necessarily monotone or time-invariant. Such differential inclusions model the motion of state trajectories which are constrained to evolve inside timevarying non-convex sets. Using Lyapunov-based analysis, sufficient conditions are proposed for local stability in such systems, while specifying the basin of attraction. If the sets governing the motion of state trajectories are moving with bounded variation, then the resulting state trajectories are also of bounded variation, and unlike the convex case, the stability conditions depend on the size of jumps allowed in the sets. Based on the stability analysis, a Luenberger-like observer is proposed which is shown to converge asymptotically to the actual state, provided the initial value of the state estimation error is small enough. In addition, a practically convergent state estimator, based on the high-gain approach, is designed to reduce the state estimation error to the desired accuracy in finite time for larger initial values of the state estimation error. The two approaches are then combined to obtain global asymptotically convergent state estimates. Mots-clés : Nonsmooth systems, sweeping process, prox-regular sets, locally bounded variation, Lyapunov stability, observer design, differential measure. ha l-0 07 90 61 8, v er si on 2 9 O ct 2 01 3 Stability and Observer Design for Nosmooth Systems 3