Recent Advances in Nonlinear Optimal Feedback Control Design

This paper presents a detailed perspective of two constructive feedback strategies characterized by algebraic Riccati equations for solving continuous-time nonlinear deterministic optimal control problems. In each case, the system is first transformed into a linear-like, state-dependent structure. The first method is based on treating the adapted model as an instantaneously linear time-invariant system to approximate it around each point along a trajectory. The control applied at a particular point in state space is determined by solving an infinite-time linearquadratic (LQ) regulator problem using the linear model for the particular point. The resultant control law, which is of feedback form and nonlinear in the state, involves finding the steady-state solution to the State-Dependent Riccati Equation (SDRE) at each point. The SDRE method has become exceptionally well-known within the control community for solving autonomous nonlinear regulator problems. Unfortunately, the undeveloped theory of the infinite-time LQ optimal tracking problem has hindered its application for solving nonlinear tracking problems. Therefore, an approximate approach is introduced in this paper for adapting the SDRE tracking methodology. The second method, still at its infancy, has been introduced recently for solving nonlinear optimal regulator and tracking problems. The algorithm has been inspired by the SDRE concept, and is characterized similarly by solving an Approximating Sequence of Riccati Equations (ASRE) associated with finite-time LQ theory. This paper presents the latest results on SDRE and ASRE theories developed in the literature, and focuses on illustrating the application, computational advantage and validity of each advanced control methodology on a realistic simulation example of a ducted fan engine model for high-performance thrust-vectored aircraft, to fill the gap between theory and practice. The proposed methods overcome many of the difficulties and shortcomings of existing methodologies, and deliver computationally simple, yet effective, algorithms for constructively synthesizing nonlinear optimal feedback controls. Key-Words: Nonlinear control systems; Optimal control; (Non)linear-(non)quadratic problems; Stabilization; Regulation; Tracking; Feedback control design; Continuous-time systems; Aircraft (flight) control systems