Contraction Region Estimate for State-Dependent Riccati Equation-Based Controllers and its Application to a Two-Wheeled Inverted Pendulum

Abstract

The state–dependent Riccati equation (SDRE) offers a systematic technique for controller design applicable to a wide range of nonlinear processes, especially to complex systems of higher order with inherently fast dynamics. Despite the numerous benefits of the SDRE technique, an open issue remains in providing stability regions for the regulated system as the closed-loop dynamics are not explicitly known. Standard techniques, such as Lyapunov’s direct method, do not allow to infer global properties from local analysis. However, the recently developed contraction theory enables the study of closed-loop dynamics exclusively known pointwisely, which suggests its applicability to SDRE-controlled systems. Thus, this paper presents a novel technique for computing contraction region estimates for nonlinear stabilisation using SDRE-based controllers. By solving an optimisation problem, the region estimate is generated by a smooth Riemannian metric which assures exponential convergence towards the origin. Moreover, a guaranteed lower bound of the contraction rate is explicitly given. To highlight the benefits of the proposed method, numerical simulations of a Two-wheeled inverted pendulum (TWIP) robot are provided. Thus, this paper presents a novel technique for computing contraction region estimates for nonlinear stabilisation using SDRE-based controllers. By solving an optimisation problem, the region estimate is generated by a smooth Riemannian metric which assures exponential convergence towards the origin. Moreover, a guaranteed lower bound of the contraction rate is explicitly given. To highlight the benefits of the proposed method, numerical simulations of a Two-wheeled inverted pendulum (TWIP) robot are provided.