The Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems

Feedback control problems involving autonomous quadratic systems are prevalent, yet very few software tools are available for approximating their solution. This paper represents a step forward in the special case where both the state equation and the control costs are quadratic. As it represents the natural extension of the linear-quadratic regulator (LQR) problem, we describe this setting as the quadratic-quadratic regulator (QQR) problem. We describe an algorithm that exploits the structure of the QQR problem that arises when implementing Al'Brekht's method. As we show, this well-known algorithm has an elegant formulation when written using Kronecker products and produces linear systems with a special structure that can take advantage of modern tensor-based linear solvers. We demonstrate this formulation on a random test problem then apply it to a discretized distributed parameter control problem that fits the QQR framework. This approach is amenable to low degree polynomial feedback laws in systems with modest model dimensions, for example, systems produced by modern model reduction methods. Comparisons to linear feedback control laws show a benefit in using the QQR formulation.