Infinite horizon nonlinear quadratic cost regulator

Abstract

Infinite horizon optimal control has been a leading methodology for both linear and nonlinear systems. The Hamilton-Jacobi-Bellman (HJB) approach is a very effective approach for infinite horizon optimal control which involves solving the associated nonlinear partial differential equation known as the HJB equation. Because of the importance and high difficulty of solving the HJB equation, different techniques and approximations to solve the HJB equation are proposed in the literature. In the case of linear systems, the HJB equation becomes the known Reccati equation which provides the well known and powerful Linear Quadratic Regulator (LQR). Therefore, the focus of this research is to generalize the idea of the LQR and develop a Nonlinear Quadratic cost Regulator (NLQR) based on the solution of the HJB equation for the infinite horizon problem. We present a novel and an efficient technique based on Taylor series expansion for the HJB equation around an equilibrium point. Utilizing a set of minimal polynomial basis functions that includes all possible combinations of the states, a nonlinear matrix equation similar to the Riccati equation is constructed from the HJB equation. Solving this nonlinear matrix equation term by term renders the associated value function (i.e, optimal cost-to-go) and the optimal controller with a prescribed truncation order. The computational complexity of this approach is shown to have only a polynomial growth rate with respect to the series order. The developed HJB based equation can be solved independently of the current states and hence the optimal nonlinear control can be obtained a-priori offline for smooth nonlinear control affine systems. A general recursive closed form procedure to find the coefficients of high order control laws is provided. Set of examples are presented with different systems natures and nonlinearities including inputs saturation.