Open Access Publications from the University of California

## New Techniques in Linear Parameter-Varying Systems

Abstract

Linear Parameter-Varying (LPV) techniques provide a convenient extension of linear systems theory to a rich class of systems - including uncertain, switched and non-linear systems. LPV systems theory also allows for the analysis of gain-scheduled controllers - where a controller is designed to perform over multiple operating points. The arrival of interior-point methods in the 1990s brought LPV systems and the analysis of LPV systems to the attention of many as a large subclass of LPV design conditions can be expressed as Linear Matrix Inequalities (LMIs).

This dissertation makes several contributions to LPV systems theory - both in terms of the analysis of this class of systems and new approaches for controller and filter design.

We start by revisiting the issue of quadratic gain-scheduled and robust state-feedback. The goal of this analysis is to explore to what extent solvability of certain LMIs for gain-scheduled control also implies solvability of the corresponding robust control inequalities. One issue investigated in detail is the use of pre-filters to handle uncertainty appearing in the input matrix. We show that this technique is rarely productive in that the solvability of certain gain-scheduled control design problems for the original system augmented with a pre-filter often implies existence of a robust control for the original system.

Following this, we introduce new conditions for the ${H}_{\infty}$ synthesis of discrete-time gain-scheduled state feedback controllers and LPV state estimators in the form of LMIs. A distinctive feature of the proposed conditions is the ability to handle time-variation in both the dynamics and the input or output matrices without resorting to pre-filtering or conservative iterative procedures. We show that these new conditions contain existing poly-quadratic conditions as a particular case and illustrate by way of numerical examples their superiority to many existing conditions.

To conclude, we introduce a strategy for combining these state-feedback and state-estimation conditions for the $H_{\infty}$ synthesis of output feedback controllers. This strategy allows us to design output-feedback controllers where time-variation is present in the dynamics and the input or output matrices. To our knowledge, no techniques presently exist to solve this problem - even when the input and output matrices are held fixed.