The Linear Constrained Control Problem for Discrete-Time Systems: Regulation on the Boundaries

International audienceThe chapter deals with the problem of regulation of linear systems around an equilibrium lying on the boundary of a polyhedral domain where linear constraints on the control and/or the state vectors are satisfied. In the first part of the chapter, the fundamental limitations for constrained control with active constraints at equilibrium are exposed. Next, based on the invariance properties of polyhe-dral and semi-ellipsoidal sets, design methods for guaranteeing convergence to the equilibrium while respecting linear control constraints are proposed. To this end, Lyapunov-like polyhedral functions, LMI methods and eigenstructure assignment techniques are applied

Stabilizability and control co-design for discrete-time switched linear systems

International audienceIn this work we deal with the stabilizability property for discrete-time switched linear systems. First we provide a constructive necessary and sufficient condition for stabilizability based on set-theory and the characterization of a universal class of Lyapunov functions. Such a geometric condition is considered as the reference for comparing the computation-oriented sufficient conditions. The classical BMI conditions based on Lyapunov-Metzler inequalities are considered and extended. Novel LMI conditions for stabilizability, derived from the geometric ones, are presented that permit to combine generality with convexity. For the different conditions, the geometrical interpretations are provided and the induced stabilizing switching laws are given. The relations and the implications between the stabiliz-ability conditions are analyzed to infer and compare their conservatism and their complexity. The results are finally extended to the problem of the co-design of a control policy, composed by both the state feedback and the switching control law, for discrete-time switched linear systems. Constructive conditions are given in form of LMI that are necessary and sufficient for the stabilizability of systems which are periodic stabilizable

Impact of average-dwell-time characterizations for switched nonlinear systems on complex systems control

It is well known, present day theory of switched systems is largely based on assuming certain small but finite time interval termed average dwell time. Thus it appears dominantly characterized by some slow switching condition with the average dwell time satisfying a certain lower bound, which implies a constraint nonetheless. In cases of nonlinear systems, there may well appear certain non-expected complexity phenomena of particularly different nature when switching becomes no longer useful. A fast switching condition with average the dwell time satisfying an upper bound is explored and established. A comparison analysis of these innovated characterizations via slightly different overview yielded new results on the transient behaviour of switched nonlinear systems, while preserving the system stability. The approach of multiple Lyapunov functions is used in current analysis and the switched systems framework is believed to be extended slightly. Thus some new insight into the underlying, switching caused, system’s complexities has been achieved