Constrained quadratic stabilization of discrete-time uncertain nonlinear multi-model systems using piecewise affine state-feedback

Abstract

In this paper a method for nonlinear robust stabilization based on solving a bilinear matrix inequality (BMI) feasibility problem is developed. Robustness against model uncertainty is handled. In different non-overlapping regions of the state-space called clusters the plant is assumed to be an element in a polytope which vertices (local models) are affine systems. In the clusters containing the origin in their closure, the local models are restricted to be linear systems. The clusters cover the region of interest in the state-space. An affine state-feedback is associated with each cluster. By utilizing the affinity of the local models and the state-feedback, a set of linear matrix inequalities (LMIs) combined with a single nonconvex BMI are obtained which, if feasible, guarantee quadratic stability of the origin of the closed-loop. The feasibility problem is attacked by a branch-and-bound based global approach. If the feasibility check is successful, the Liapunov matrix and the piecewise affine state-feedback are given directly by the feasible solution. Control constraints are shown to be representable by LMIs or BMIs, and an application of the control design method to robustify constrained nonlinear model predictive control is presented. Also, the control design method is applied to a simple example

Robust Stabilization of Discrete-Time Multi-Model Systems Using Piecewise Affine State-Feedback

“…If, in addition, there exists 2 (0 1) s u c h that for a given setR Ã R A f x j x T P x g N(0) then the origin is said to be a quadratically stable equilibrium for system (9) with a region of attraction associated withR A of at least fx j x T P x g. The proof (Slupphaug and Foss, 1998b) proceeds by using the so-called S-procedure and Schur complements (Boyd et al, 1994), and some other results involving matrix inequalities.…”

“…A short discussion on the outer approximations below i s g i v en in (Slupphaug and Foss, 1998b). Here, we only note that the given approximations exist for the given sets, and save them for later reference.…”

Computation and approximation of piecewise affine control laws via binary search trees

“…Evaluation of PWA functions is also of interest with other PWA control structures than explicit MPC control (e.g. [10,11,12,13,14]). The most immediate way of evaluating a PWA function is to do a sequential search through the regions representing the PWA function (see Algorithm 1 below).…”

Evaluation of piecewise affine control via binary search tree

“…All of these approaches lead to PWA state feedback laws. Evaluation of PWA functions is also of interest with other PWA control structures than explicit MPC control (see for example (Sontag, 1981;Rantzer and Johansson, 2000;Hassibi and Boyd, 1998;Slupphaug and Foss, 1999)). The most immediate way of evaluating a PWA function is to store the linear inequalities representing every polyhedral region of the PWA function defining the state feedback law, and do a sequential search (see Algorithm 1) through these to find the region where the state belongs.…”