Numerical methods for model predictive control

Abstract

There has been an increased interest in controlling complex systems using Model Predictive Control (MPC). However, the use of resource-constrained computing platforms in these systems has slowed the adoption of MPC. This thesis focuses on increasing the efficiency of numerical methods for MPC in terms of resource usage and solution time, while also simplifying the design process.

We first show how block Toeplitz operators can be used to link the linear MPC matrices to the transfer function of the predicted system, resulting in horizon-independent bounds on the condition number of the condensed Hessian and the upper iteration bound for the Fast Gradient Method (FGM). We derive a horizon-independent preconditioner that produces up to a 9x speedup for the FGM while reducing the preconditioner computation time by up to 50,000x compared to an existing preconditioner. We propose a new method for computing the minimum number of fractional bits needed to ensure the FGM with fixed-point arithmetic is stable, with an example showing decreases of up to 77% in resource usage and 50% in the computational energy when using this method on a Field Programmable Gate Array.

Finally, we present a framework using the derivative-free Mesh Adaptive Direct Search method to solve nonlinear MPC problems with non-differentiable features or quantized variables without the need for complex or costly reformulations. We augment the system dynamics with additional states to compute the Lagrange cost term and the violation of the path constraints along the state trajectory, and then perform a structured search of the input space using a single-shooting simulation of the system dynamics. We demonstrate this framework on a robust Goddard rocket problem with a non-differentiable cost and a quantized thrust input, where we achieve an altitude within 40m of the target, while other methods are unable to get closer than 180m.