Computing Solutions to the Polynomial-Polynomial Regulator Problem

We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, requiring lower control costs, and 2) for some problems they can provide a larger region of stabilization. We derive explicit formulas for the polynomial approximation to the value function that solves the optimal control problem; we also provide scalable algorithms and software that exploit the tensor structure of the equations for general purpose practical use. The method is illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control. Then we demonstrate the scalability of the approach on a semidiscretization of a PDE, for which the PPR control reduces the control cost by approximately 75% compared to LQR.

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Recommended citation: N. A. Corbin and B. Kramer, “Computing Solutions to the Polynomial-Polynomial Regulator Problem,” in 2024 Conference on Decision and Control (CDC), Dec. 2024.