Convex hull generation methods for polytopic representations of LPV models

The paper focuses on the control design of LPV and qLPV models via TS fuzzy model representation and Linear Matrix Inequality (LMI) based design under the Parallel Distributed Compensation control design framework. The related literature reports considerable research on how to manipulate with LMI's to achieve better control performance. This paper shows that this effort is not enough, the optimization of the control performance must include the convex hull manipulation on the TS fuzzy model beside manipulating with the LMIs. The LMI's guaranty the optimal solution for a given convex hull, but the convex hull representation is not invariant. Furthermore, the solutions by LMI's are very sensitive for the convex hull. The paper proposes a systematic concept for various convex hull manipulation method.

I. Introduction

The Parallel Distributed Compensation (PDC) concept proposed by Tanaka et al. [1], and developed to Takagi-Sugeno (TS) fuzzy models is a widely adopted control design framework typically applied to quasi Linear Parameter Varying (qLPV) models. This concept starts with the fuzzy rules of a given TS fuzzy model and determines the feedback consequents to these rules to from a TS fuzzy controller. The determination of the controller's fuzzy rules is typically based on the feasibility of Linear Matrix Inequalities (LMI) (derived from the Lyapunov stability criteria) capable of optimizing various multi-objective control performances. Because of the theoretical similarity we can find quite a considerable overlap between the PDC framework and other non fuzzy type modern polytopic and affine model and LMI based control design methodologies, see the a detailed and broad overview by Scheerer [2].