Claude Iung

Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach

This paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete-time domain. The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. The first one is classical while the second is new and uses a slack variable, which makes it useful for design problems. These two conditions are proved to be equivalent for stability analysis. Investigating the static output feedback control problem, we show that the second condition is, in this case, less conservative. The reduction of the conservatism is illustrated by a numerical evaluation.

Stabilization of Arbitrary Switched Linear Systems With Unknown Time-Varying Delays

We consider continuous time switched systems that are stabilized via a computer. Several factors (sampling, computer computation, communications through a network, etc.) introduce model uncertainties produced by unknown varying feedback delays. These uncertainties can lead to instability when they are not taken into account. Our goal is to construct a switched digital control for continuous time switched systems that is robust to the varying feedback delay problem. The main contribution of this note is to show that the control synthesis problem in the context of unknown time varying delays can be expressed as a problem of stabilizability for uncertain systems with polytopic uncertainties

Linear quadratic optimization for hybrid systems

A general optimal hybrid control problem which enables a direct maximum principle approach is presented. Necessary conditions at switching time for controlled and autonomous switching are derived in the case where a linear quadratic criteria is used. Further extra condition for stability of switched systems is addressed. Finally, an illustrative example of optimal hybrid control problem is treated.

Analysis and control of LTI and switched systems in digital loops via an event-based modelling

This paper is dedicated to the modelling of LTI continuous time systems in digital control loops. We consider the digital control problem on non-uniform sampling periods. Moreover, we assume that time varying delays that may have a variation range larger than a sampling period affect the closed-loop. Our goal is to present a unique model that is able to include these problems simultaneously and that can be handled by classical control synthesis tools. We present a new event based discrete-time model (an exponential uncertain system with delay) and we show that the stabilizability of this system can be achieved by finding a control for a switched polytopic system with an additive norm bounded uncertainty. The methodology is extended to the case of switched system.

A poly-quadratic stability based approach for linear switched systems

In this paper, a link between poly-quadratic stability and stability of arbitrary switching systems is established. Polyquadratic stability aims to check asymptotic stability of a polytopic system by means of polytopic quadratic Lyapunov functions. The necessary and sufficient condition of poly-quadratic stability proposed in Daafouz and Bernussou (2001) is shown to be immediately applicable to a class of switched control and observer design problems. Chaos synchronization for which the transmitter is described by a piecewise linear map is presented as an application.

Practical optimal state feedback control law for continuous-time switched affine systems with cyclic steady state

In this article, a method for computing an optimal state feedback control law for continuous-time switched affine systems exhibiting cyclic behaviour in steady state is presented. The hybrid solutions are deduced from the Fillipov solutions. It is shown that the optimal trajectory synthesis implies to determine singular arcs. Algebraic conditions are given to obtain these particular arcs of the trajectory. A numerical procedure is then proposed to generate optimal trajectories on a given state space area avoiding the classical two-point boundary value problem occurring in optimal control synthesis. The interpolation of the solutions set, through a neural network, yields a state feedback control law. Several examples in the power converters field show the feasibility and the efficiency of the method.

Switched Affine Systems Using Sampled-Data Controllers: Robust and Guaranteed Stabilization

The problem of robust and guaranteed stabilization is addressed for switched affine systems using sampled state feedback controllers. Based on the existence of a control Lyapunov function for a relaxed system, we propose three sampled-data controls. Global attracting sets, computed by solving a sequence of optimization problems, guarantee practical and global asymptotic stabilization for the whole system trajectories. In addition, robust margins with respect to parameters uncertainties and non uniform sampling are provided using input-to-state stability. Finally, a buck-boost converter is considered to illustrate the effectiveness of the proposed approaches.

Robust Steering Control of Hot Strip Mill

In this paper, a robust steering control is proposed as a means to guarantee the stability of a hot strip finishing mill and improve its performances. The aim of the rolling process is to obtain a metal strip which is uniform and at the desired thickness. The lateral movement of the strip may reduce the quality of the product and damage the rolls. Hence, this displacement must be limited to improve process reliability. Since a hot strip mill treats products with very heterogeneous characteristics, a significant database was created and divided into different families of products. A method for reducing the numerical complexity of the problem exploiting the relation between the different product parameters is also presented. This method yields a convex stabilisation problem. Hence, for each family, a different robust controller is designed. Simulation and experimental results concerning the ArcelorMittal hot strip mill of Eisenhüttenstadt are presented.

Stability and Stabilization of Two Time Scale Switched Systems in Discrete Time

In this technical note, stability and stabilization of two time scale switched linear systems in the singular perturbation form are addressed in discrete time. We show that, under an arbitrary switching rule, stability of the slow and fast switched subsystems is not sufficient to assess stability of the original two time scale switched system, even if the singular perturbation parameter tends to zero. Therefore, we propose LMI based conditions that guarantee the asymptotic stability of the two time scale switched system using switched quadratic Lyapunov functions. These conditions express the fact that the coupling between fast and slow subsystems has to be taken into account in addition to stability properties of the two subsystems, when the switching rule is arbitrary. The presented conditions are extended to state feedback control design. A numerical example illustrates the features of the proposed approach.

Stabilization of continuous-time singularly perturbed switched systems

In this article, stability of continuous-time switched linear systems in the singular perturbation form is investigated. We show that the stability of slow and fast switched subsystems is not a sufficient condition for stability of the corresponding two-time scale switched system, under an arbitrary switching law. Thus, LMI conditions to design a state-feedback control law stabilizing continuous-time singularly perturbed switched linear systems are proposed.