Carleman State Feedback Control Design of a Class of Nonlinear Control Systems

Abstract

We consider optimal feedback control design for nonlinear control systems with polynomial right-hand sides. The control objective is to minimize a quadratic cost functional over all state feedback control laws with polynomial structure subject to dynamics of the system. First, we utilize ideas from Carleman linearization to lift a given finite-dimensional nonlinear system into an infinite-dimensional linear system. Finite-order truncations of the resulting infinite-dimensional linear system are investigated and connections between (local) stability properties of the original nonlinear system and its finite-order truncations are established. We show that the optimal feedback control design can be approximated and cast as an optimization problem with bilinear matrix equation constraints. Through several simulations, we show that this approximate method can be efficiently implemented using the Alternating Direction Method of Multipliers (ADMM) methods.