Design of robust positively invariant set for nonholonomic vehicle

The properties of the (robust) positively invariant sets play a fundamental role in variant techniques of (robust) control, such as the constrained control, robust analysis, synthesis and optimization. In this paper, we consider the robust control for the simultaneous tracking and regulation problem (STRP) of nonholonomic vehicle (whose control input is subject to the saturation constraint and the bounded disturbance), and design a robust positively invariant (RPI) set and an associated feedback controller. Under the associated feedback controller with saturation and bounded disturbance, the nonholonomic vehicle entering into the RPI set will stay in it forever and asymptotically achieve the desired control objective of STRP. These properties are illustrated by both the theoretical proof and the simulation example.

1 Introduction

In the framework of control, such as the constrained control, robust analysis, and model predictive control, the set with (robustly) positive invariance plays a fundamental role. Based on the robust positively invariant (RPI) set, the synthesis of controllers under additional and relatively mild assumptions can guarantee the robust constraint satisfaction, the robust satbility, and the convergence to an proper set, despite the presence of disturbance and hard constraints on the system variables [1]. This attracts more and more attentions on the set invariance theory and its application to control, see, e.g., [2]–[5]. For the theoretical brief of set invariance, most of the existing works paid their attentions on the linear systems (e.g., [1], [6]–[9]), or on the weakly nonlinear systems by linearization near the equilibrium point (e.g., [10], [11]). This paper focuses on the development of the RPI set and its application to the robust control of a specially constrained and perturbed nonlinear system, i.e., a wheeled vehicle formulated by the nonholonomic dynamics with input saturation and disturbance.