Pierre Riedinger

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.

Comparison of Hybrid Control Techniques for Buck and Boost DC-DC Converters

Five recent techniques from hybrid and optimal control are evaluated on two power electronics benchmark problems. The benchmarks involve a number of practically interesting operating scenarios for fixed-frequency synchronous dc-dc converters. The specifications are defined such that good performance can only be obtained if the switched and nonlinear nature of the problem is accounted for during the design phase. A nonlinear action is featured in all methods either intrinsically or as external logic. The designs are evaluated and compared on the same experimental platform. Experiments show that the proposed methods display high performances, while respecting circuit constraints, thus protecting the semiconductor devices. Moreover, the complexity of the controllers is compatible with the high-frequency requirements of the considered application.

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.

An Optimal Control Approach for Hybrid Systems

In this paper optimal control for hybrid systems will be discussed. While defining hybrid systems as causal and consistent dynamical systems, a general formulation for an optimal hybrid control problem is proposed. The main contribution of this paper shows how necessary conditions can be derived from the maximum principle and the Bellman principle. An illustrative example shows how optimal hybrid control via a set of Hamiltonian systems and using dynamic programming can be achieved. However, as in the classical case, difficulties related to numerical solutions exist and are increased by the discontinuous aspect of the problem. Looking for efficient algorithms remains a difficult and open problem which is not the purpose of this contribution.

Time optimal control of hybrid systems

We discuss optimal control for hybrid systems. First, we present a general formulation of an optimal control problem for hybrid systems. Such a formulation allows us to use straightaway the maximum principle of Pontryagin. We focus on time optimal control for hybrid systems defined by a finite family of controlled linear systems and show how switching time can be achieved. An illustrative example ends the paper.

On the Convergence of Linear Switched Systems

This paper investigates sufficient conditions for the convergence to zero of the trajectories of linear switched systems. We provide a collection of results that use weak dwell-time, dwell-time, strong dwell-time, permanent and persistent activation hypothesis. The obtained results are shown to be tight by counterexample. Finally, we apply our result to the three-cell converter.

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.

Brief paper: On the algebraic characterization of invariant sets of switched linear systems

In this paper, a suitable LaSalle principle for continuous-time linear switched systems is used to characterize invariant sets and their associated switching laws. An algorithm to determine algebraically these invariants is proposed. The main novelty of our approach is that we require no dwell time conditions on the switching laws. By not focusing on restricted control classes we are able to describe the asymptotic properties of the considered switched systems. Observability analysis of a flying capacitor converter is proposed as an illustration.

Hybrid control of a three-level three-cell dc-dc converter

This paper compares three synthesis methods for controlling a three-level three-cell dc-dc converter. The main contribution of this paper is to analyse different strategies: i) The Passivity Based Control that uses the notion of average model, ii) A stabilizing method in which a unique Lyapunov function is introduced and iii) A new predictive control approach, which relies on the use of optimization procedures.