Showing papers by "Christophe Prieur published in 2007"

Hybrid Feedback Control and Robust Stabilization of Nonlinear Systems

Abstract: In this paper, we show, for any nonlinear system that is asymptotically controllable to a compact set, that a logic-based, hybrid feedback can achieve asymptotic stabilization that is robust to small measurement noise, actuator error, and external disturbance The construction of such a feedback hinges upon recasting a stabilizing patchy feedback in a hybrid framework by making it dynamic with a discrete state, while insisting on semicontinuity and closedness properties of the hybrid feedback and of the resulting closed-loop hybrid system The robustness of stability is then shown as a generic property of hybrid systems having the said regularity properties Auxiliary results give uniformity of convergence and of overshoots for hybrid systems, and give a characterization of asymptotic stability of compact sets

Nonlinear optimal control via occupation measures and LMI-relaxations

Abstract: We consider the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control con- straints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI (lin- ear matrix inequality)-relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assump- tions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments.

Stability analysis for reset systems with input saturation

Abstract: This paper is devoted to the stability analysis for a class of reset systems, such as those including a Clegg integrator or a FORE (first order reset element). The system under consideration is subject to input saturation. Hence, constructive conditions, allowing to characterize the region of stability of the saturated closed-loop system, are proposed based on the use of some suitable Lyapunov functions and a modified sector condition. LMI-based optimization schemes for maximizing the size of the region of stability are then derived. In presence of an additive disturbance, the problem of L2 stability is also addressed.

Output feedback stabilization of a clamped-free beam

Abstract: We consider a Euler–Bernoulli beam, clamped at one extremity and free at the other, to which are attached a piezoelectric actuator and a collocated sensor touching the clamped extremity. We provide an output feedback law and characterize the sensor/actuator lengths for which the strong stabilization holds. Finally, we prove that the energy decreases to zero in a polynomial way for almost all lengths, and in an exponential way for lengths admitting a certain coprime factorization.

On-Ground Aircraft Control Design Using an LPV Anti-windup Approach

Abstract: Based on the LFT model of the on-ground aircraft developed in Chapter 6, an anti-windup control technique is proposed to improve lateral control laws which have been designed using classical methods. The original idea of this work consists in taking advantage of a simplified representation of the nonlinear lateral ground forces, which are approximated by saturation-type nonlinearities. The anti-windup compensator is then implemented on the full nonlinear aircraft model using an on-line estimator of the ground forces. Simulations demonstrate the efficiency of the resulting adaptive controller.

Robust absolute stability and stabilization based on homogeneous polynomially parameter-dependent Lur'e functions

Abstract: This paper provides finite dimensional convex conditions to construct homogeneous polynomially parameter- dependent Lur'e functions which ensure the stability of nonlinear systems with state-dependent nonlinearities lying in general sectors and which are affected by uncertain parameters belonging to the unit simplex. The proposed conditions are written as linear matrix inequalities parameterized in terms of the degree g of the parameter-dependent solution and in terms of the relaxation level d of the inequality constraints, based on an extension of Polya's Theorem. As g and d increase, progressive less conservative solutions are obtained. The results in the paper include as special cases existing conditions for robust stability analysis and for absolute stability. A convex solution for control design is also provided. Numerical examples illustrate the efficiency of the proposed conditions.

Stability analysis for systems with nested backlash and saturation operators

Abstract: This paper is concerned with the problem of stability analysis for a certain class of nonlinear systems resulting from nested backlash and saturation operators. Constructive conditions based on LMIs, to ensure the local or global stability of the system, are proposed by using some suitable Lyapunov functional, generalized sector conditions and polytopic representation. The associated set of all the equilibrium points is precisely defined.

Analyse de stabilité des systèmes avec réinitialisation en présence d'incertitudes

Abstract: This paper is dedicated to the study of specific hybrid systems: reset control systems. A reset system is a system whose states are reset whenever its output and input satisfy an appropriate relationship. Stability analysis and L2-performance are studied for systems (and in particular for systems in closed-loop with reset controller) in presence of perturbations and parametric uncertainties. Based on the use of quadratic and piecewise quadratic Lyapunov functions, constructive conditions into LMI form are proposed. The potentialities of the approach are illustrated on a numerical example borrowed from the literature.

On two hybrid robust optimal stabilization problems

Abstract: We report on some recent results obtained by the authors concerning robust hybrid stabilization of control systems. We stated 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. We investigated the Martinet case, which is a model case in $\R^3$ where singular minimizers appear, and show that such a stabilization result still holds. Namely, in both cases, we prove that 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 errors.

BOUNDARY CONTROL OF A CHANNEL - Last Improvements

Abstract: Different improvements have been developed in regards to the stability and the control of two-by-two non linear systems of conservation laws, and in particular for the Saint-Venant equations and the control of flow and water level on irrigation channel. One stability result based on the Riemann coordinates is presented here and sufficient conditions are given to insure the Cauchy convergence. Another result still based on the Riemann approach is presented too, in the linear case, to improve the feedback control based on the Riemann invariants.