Computationally Improved State-Dependent Riccati Equation Scheme for Nonlinear Benchmark System

Abstract

This article presents new analytical results that substantially improve the computational performance of the state-dependent Riccati equation (SDRE) scheme to control a nonlinear benchmark problem. The analysis formulates the equivalent applicability condition in a reduced-dimensional system space, which is in terms of the pointwise solvability of SDRE but generally deemed challenging/impossible. It starts with a unified coverage of the ${\\boldsymbol{\\alpha }}$-parameterization method, which has been widely utilized to exploit the flexibility of the state-dependent coefficient (SDC) matrix in the SDRE scheme. When specializing to a practically meaningful SDC, the analysis further sheds light on a much simpler equivalent condition by virtue of a novel categorization of the entire state space. This largely alleviates the dominant computational burden pointwise at each time instant or system state, which is supported by complexity analysis, and validated through simulations. In addition, it enlarges the domain of interest in the previous design, which was constrained due to the numerical implementation. Notably, the generality of the analytical philosophy also includes robustness to parameter values of this benchmark application, and a variety of nonlinear control systems within and beyond the SDRE design framework.