Infinite Horizon Nonlinear Quadratic Cost Regulator

This paper develops a Nonlinear Quadratic cost Regulator (NLQR) through an efficient Taylor series expansion of the Hamilton-Jacobi-Bellman (HJB) equation. 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. A recursive closed form procedure to find the coefficients of the series is presented. The computational complexity of this approach is shown to have only a polynomial growth rate with respect to the series order. The developed algorithm, which may be implemented offline, is applied to two nonlinear systems with different types of nonlinearities including actuator saturation.