Model Predictive Control of Polynomial Systems

This chapter describes the design of nonlinear model predictive control (MPC) for polynomial systems. Polynomial systems arise in many applications, including in power generation, automotives, aircraft, magnetic levitation, chemical reactors, and biological networks. Furthermore, general nonlinear dynamical systems can usually be rewritten exactly as polynomial systems or approximated as polynomial systems using Taylor series. MPC for discrete-time polynomial systems is formulated as a polynomial program. Hierarchical semidefinite programing relaxation methods are discussed for solving these polynomial programs to global optimality. Then, the methods for fast polynomial MPC are described, including convexification formulations for input-affine systems and explicit algorithms using algebraic geometry methods. Methods are then described for converting general nonlinear dynamical systems into polynomial systems using Taylor’s theorem, and an illustrative simulation example is presented for the practical implementation of Taylor’s theorem for bounding control trajectories. Finally, future directions for research are proposed, including real-time, output-feedback, and robust/stochastic polynomial MPC.