Approximate Optimal Control Design for a Class of Nonlinear Systems by Lifting Hamilton-Jacobi-Bellman Equation

We consider optimal control design of a class of affine nonlinear systems whose right-hand sides are analytic functions. We build upon ideas from Carleman linearization, which is a nonlinear procedure to transform (lift) a finite-dimensional nonlinear system into an infinite-dimensional linear system with no loss, and lift a Hamilton-Jacobi-Bellman (HJB) equation into an infinite-dimensional quadratic form that resembles the familiar algebraic Riccati equation. Then, we propose an efficient method to calculate solution of the resulting infinite-dimensional equation using one algebraic Riccati equation (of the same dimension as the original nonlinear system) and a series of linear matrix equations in an iterative manner. One can obtain arbitrarily near-optimal solution using finite truncations. It is shown that the resulting approximate solutions are symmetric. Using these approximate solutions to the HJB equation, we construct approximate optimal control laws. Our simulation results assert that our method enjoys high accuracy in compared to the actual optimal feedback control laws and the accuracy increases as higher-order truncations are used.