H∞Loop Shaping Control of Input Saturated Systems with Norm-Bounded Parametric Uncertainty

Renan Lima Pereira,Karl Heinz Kienitz

doi:10.1155/2015/383297

Abstract

This paper proposes a gain-scheduling control design strategy for a class of linear systems with the presence of both input saturation constraints and norm-bounded parametric uncertainty. LMI conditions are derived in order to obtain a gain-scheduled controller that ensures the robust stability and performance of the closed loop system. The main steps to obtain such a controller are given. Differently from other gain-scheduled approaches in the literature, this one focuses on the problem ofH∞loop shaping control design with input saturation nonlinearity and norm-bounded uncertainty to reduce the effect of the disturbance input on the controlled outputs. Here, the design problem has been formulated in the four-blockH∞synthesis framework, in which it is possible to describe the parametric uncertainty and the input saturation nonlinearity as perturbations to normalized coprime factors of the shaped plant. As a result, the shaped plant is represented as a linear parameter-varying (LPV) system while the norm-bounded uncertainty and input saturation are incorporated. This procedure yields a linear parameter-varying structure for the controller that ensures the stability of the polytopic LPV shaped plant from the vertex property. Finally, the effectiveness of the method is illustrated through application to a physical system: a VTOL “vertical taking-off landing” helicopter.

Highlights

In recent years, input saturation and model uncertainty problems have been extensively studied in the control system literature, where much attention has been focused on the problems of robust stabilization and performance [1,2,3,4,5,6,7,8,9]
Applying the procedure described in [19, 20, 23] for the LMIs conditions above, we obtain the gain-scheduled H∞ loop shaping controller that ensures the robust stability of the closed loop system subject to constraint of input saturation nonlinearity
The parametric uncertainty is considered in form ΔA = FΔ(t)E, where F and E are known constant matrices with appropriate dimensions and Δ(t) is an unknown matrix with Lebesgue measurable elements such that ‖Δ(t)‖2 ≤ 1

Summary

Introduction

Input saturation and model uncertainty problems have been extensively studied in the control system literature, where much attention has been focused on the problems of robust stabilization and performance [1,2,3,4,5,6,7,8,9]. When a system is subject to input saturation, two main issues arise: the guarantee of stability and the containment of performance degradation [7] To solve this problem, there exist two approaches: two-step and onestep designs [7]. Recent work focused on employing gain-scheduling controllers designed with an H∞ approach [19] This resulted in an advantageous technique using LMI for an H∞ loop shaping controller design with input saturation, derived from a four-block. Sufficient conditions for the existence of the gain scheduled parametric H∞ loop shaping controller are given in terms of an LMI framework, which provides the Lyapunov matrix ensuring the stability and robust performance of the LPV controlled system from of the vertex property [20, 21].

Problem Statements and Preliminaries

A Ki CKi

Main Results

Numerical Example

Conclusions