Affine Tensor Product Model Transformation

József Kuti,Péter Galambos

doi:10.1155/2018/4073531

Abstract

This paper introduces the novel concept of Affine Tensor Product (TP) Model and the corresponding model transformation algorithm. Affine TP Model is a unique representation of Linear Parameter Varying systems with advantageous properties that makes it very effective in convex optimization-based controller synthesis. The proposed model form describes the affine geometric structure of the parameter dependencies by a nearly minimum model size and enables a systematic way of geometric complexity reduction. The proposed method is capable of exact analytical model reconstruction and also supports the sampling-based numerical approach with arbitrary discretization grid and interpolation methods. The representation conforms with the latest polytopic model generation and manipulation algorithms. Along these advances, the paper reorganizes and extends the mathematical theory of TP Model Transformation. The practical merit of the proposed concept is demonstrated through a numerical example.

Highlights

The importance of polytopic system descriptions is beyond doubt since the development of influential polytopic modelbased analysis and synthesis methods initially introduced by Boyd et al in [1]
Tensor Product (TP) Model Transformation was introduced as a numerical approach to constructing polytopic TP forms of Linear Parameter Varying (LPV)/qLPV models [2] serving as an alternative to analytical procedures such as the sector nonlinearity technique [3]
The section briefly discusses the related concepts of tensor algebra, polytopic LPV/qLPV modeling, and the goals of TP Model Transformation introducing the notations that are used in the followings

Summary

Introduction

The importance of polytopic system descriptions is beyond doubt since the development of influential polytopic modelbased analysis and synthesis methods initially introduced by Boyd et al in [1]. Consolidating the affine geometry-based approach, the main contribution of this paper is the introduction of a new intermediate TP Model (like the HOSVD-based form) that provides a unique description of affine geometric properties serving as direct input for polytopic model construction methods (see [13, 15,16,17]). It reserves all the benefits of the HOSVD-based form: similar uniqueness, compact representation, and capability of complexity reduction.

Notations

Basic Concepts

Affine Decomposition of Univariate Functions

Definition of Affine Tensor Product Form

Numerical Example

Conclusion