Affine Tensor Product Model Transformation

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 samplingbased 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.

Cyclical inverse interpolation: An approach for the inverse interpolation of black‐box models using tensor product representations

“…Based on the literature, TP models can be useful whenever the goal is to represent and transmit functions that are difficult to express as an analytic formula in a canonical form [13]; approximate functions using a smaller set of dimensions such that the deterioration in accuracy is minimal in a linear algebraic sense [1,22]; reduce the complexity of models, such as fuzzy rule bases [23,24]; identify and reduce noise in measurement data [20] estimate values for missing data points in measurement data [25]; in control theory, express parameter‐varying (and event time‐varying) models with multiple inputs as a convex combination of univariate linear time‐invariant models, upon which linear matrix inequality (LMI) and a variety of other well‐known control design approaches are directly applicable [1,14,26–30]. …”