A Robust Time-Varying Riccati-Based Control for Uncertain Nonlinear Dynamical Systems

Abstract

Riccati equation-based control approaches such as linear-quadratic regulator (LQR) and time-varying LQR (TVLQR) are among the most common methods for stabilizing linear and nonlinear systems, especially in the context of optimal control. However, model inaccuracies may degrade the performance of closed-loop systems under such controllers. To mitigate this issue, this paper extends and encompasses Riccati-equation based controllers through the development of a robust stabilizing control methodology for uncertain nonlinear systems with modeling errors. We begin by linearizing the nonlinear system around a nominal trajectory to obtain a time-varying linear system with uncertainty in the system matrix. We propose a modified version of the continuous differential Riccati equation (MCDRE), whose solution is updated based upon the estimates of model uncertainty. An optimal least squares (OLS) algorithm is presented to identify this uncertainty and inform the MCDRE to update the control gains. The unification of MCDRE and OLS yields a robust time-varying Riccati-based (RTVR) controller that stabilizes uncertain nonlinear systems without the knowledge of the structure of the system's uncertainty a priori. The convergence of the system states is formally proven using a Lyapunov argument. Simulations and comparisons to the baseline backward-in-time Riccati-based controller on two real-world examples verify the benefits of our proposed control method.