Showing papers by "Christophe Prieur published in 2006"

Quasi-optimal robust stabilization of control systems

Abstract: In this paper, we investigate the problem of semi-global minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closed-loop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator noise.

Control of a clamped-free beam by a piezoelectric actuator

Abstract: We consider a controllability problem for a beam, clamped at one boundary and free at the other boundary, with an attached piezoelectric actuator. By Hilbert Uniqueness Method (HUM) and new results on diophantine approximations, we prove that the space of exactly initial controllable data depends on the location of the actuator. We also illustrate these results with numerical simulations.

Quasi-Optimal Robust Stabilization of Control Systems

Abstract: In this paper, we investigate the problem of semiglobal minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that in the absence of minimal time singular trajectories, the solutions of the closed-loop system converge to the origin in quasi-minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances, and actuator noise.

On the robustness to measurement noise and unmodeled dynamics of stability in hybrid systems

Abstract: Results on robustness to measurement noise and unmodeled dynamics of stability in hybrid systems are presented. We show that arbitrarily small measurement noise can lead to lack of existence of solutions in hybrid systems. One solution to this problem is to pass the measurements through a filter. Robustness to measurement noise using this filtering is shown explicitly. We also study the effect of unmodeled sensor/actuator dynamics in the closed loop and we demonstrate that stability is robust to a class of singular perturbations. The results are illustrated for the inverted pendulum on a cart system when attempting to globally asymptotically stabilize the inverted position of the pendulum and the neutral cart position.

Smooth patchy control Lyapunov functions

Abstract: A smooth patchy control Lyapunov function for a nonlinear system consists of an ordered family of smooth local control Lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy a decrease condition when the domains overlap. We prove that such a control Lyapunov function exists for any asymptotically controllable nonlinear system. We also show a construction, based on such a Lyapunov function, of a stabilizing hybrid feedback that is robust to measurement noise.

Robust stabilization of nonlinear control systems by means of hybrid feedbacks.

Abstract: The general problem under study in this paper is the robust asymptotic stabilization of nonlinear systems. Different types of systems are considered: the Artstein’s circles, the systems which are asymptotically controllable to the origin and also the chained systems. We recall some results on the stabilization by means of hybrid feedback laws (namely controller with a mixed continuous/discrete component) and by means of discontinuous feedbacks. We note that hybrid feedbacks yield a robust stabilization property with respect to measurement noise, actuator errors and exogeneous disturbances, for a larger class of solutions than those obtained with discontinuous state-feedbacks.

Stability Analysis and Stabilization of Systems Presenting

Abstract: This note addresses the problems of stability analysis and stabilization of systems presenting nested saturations. Depending on the open-loop stability assumption, the global stability analysis and stabiliza- tion problems are considered. In the (local) analysis problem, the objective is the determination of estimates of the basin of attraction of the system. Considering the stabilization problem, the goal is to design a set of gains in order to enlarge the basin of attraction of the closed-loop system. Based on the modelling of the system presenting nested saturations as a linear system with dead-zone nested nonlinearities and the use of a generalized sector condition, linear matrix inequality (LMI) stability conditions are formulated. From these conditions, convex optimization strategies are proposed to solve both problems. Index Terms—Linear matrix inequality (LMI), nested saturations, sta- bility regions, stabilization.

L2-Performance Analysis for Sandwich Systems with Backlash

Abstract: This paper addresses the problems of performance and stability analysis for a certain class of nonlinear systems resulting from a sandwiched backlash operator in the connection of a plant and an actuator device, and subject to Lscr2-limited time-derivative disturbance. Constructive conditions based on LMIs to ensure both the internal stability and the trajectories boundedness of the sandwich system are proposed by using some suitable Lyapunov functionals and generalized sector conditions. The associated set of all the admissible equilibrium points is precisely defined. Some extensions to the case where some parameters of the backlash nonlinearity are uncertain are also provided

Hybrid robust stabilization in the Martinet case

Abstract: In a previous work, Prieur, Trelat (2006), we derived a result of semi-global minimal time robust stabilization for analytic control systems with controls entering linearly, by means of a hybrid state feedback law, under the main assumption of the absence of minimal time singular trajectories. In this paper, we investigate the Martinet case, which is a model case in IR 3 , where singular mini- mizers appear, and show that such a stabilization result still holds. Namely, we prove that the solutions of the closed-loop system con- verge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small mea- surement noise, external disturbances and actuator errors.