Numerical Reconstruction of the HOSVD Based Canonical Form of Polytopic Dynamic Models

The main objective of the paper is to introduce how the concept of tensor HOSVD (higher order singular value decomposition) can be carried over to the TP (tensor product) dynamic models. We term this decomposition as HOSVD based canonical form of TP model or polytopic model form. The key idea and the basic concept of this decomposition was proposed recently with the TP model transformation based control design methodology. The novelty of this paper is to present the mathematical background of this concept. The paper shows convergency theorems how the TP model transformation is capable of reconstructing this HOSVD based canonical form numerically.

I. Introduction

The TP (Tensor Product) model transformation was recently proposed in papers [1]–[5]. It is a uniform gateway between a class of non-linear dynamic system models and linear matrix inequality based system control design methodologies. The TP model transformation is capable of decomposing given Linear Parameter Varying (LPV) dynamic models (which are considered in the class of nonlinear dynamic models since the parameter vector contains elements of the state vector) into the parameter varying convex combination of Linear Time Invariant (LTI) systems, where the convex combination is determined by one-variable weighting functions of each element of the parameter vector. This model is termed Tensor Product (TP) model and belongs to the class of polytopic dynamic models. The liner matrix inequality based control design techniques, especially those which are developed under the Parallel Distributed Compensation (PDC) control design framework can immediately be executed on polytopic and, hence, TP models [6].