On Approximating Polynomial-Quadratic Regulator Problems

Feedback control problems involving autonomous polynomial systems are prevalent, yet there limited algorithms and software for approximating their solution. This paper represents a step forward by considering the special case of regulator problem where state equation has nonlinearity, costs quadratic, feedback is approximated low-degree polynomials. As this natural extension linear-quadratic (LQR) quadratic-quadratic (QQR) problems, we denote class as polynomial-quadratic (PQR) problems. The present approach amenable to approximations with low degree polynomials modest model dimension. setting can be achieved in many using modern reduction methods. Al’Brekht algorithm, when applied nonlinearities represented Kronecker products leads an elegant formulation. terms lead large linear that effectively solved N-way generalization Bartels-Stewart algorithm. We demonstrate our algorithm numerical examples include Lorenz equations, ring van der Pol oscillators, discretized version Burgers equation. described here available on Github.