Re-parametrising Cost Matrices for Tuning Model Predictive Controllers

In control systems engineering, the selection of controller parameters play an important role in obtaining optimal controller performance. However, it is often not possible to obtain closed-form relationships between the parameters and performance, making the selection process difficult. This paper presents an automated tuning strategy for Model Predictive Controllers (MPC) whereby a meta-cost function is introduced to penalise undesirable behaviour, and subsequently optimised over using black-box search algorithms. To this end, we propose a method of re-parametrising the cost matrices in MPC. This approach results in a box-constrained parametrisation for the matrices, as well as a reduction in the search dimension. The procedure is demonstrated on a diesel engine case study, where we compare the tuning of MPC using the proposed parametrisation to an unbounded parametrisation on a test suite of optimisation algorithms: Simulated Annealing (SA), Particle Swarm Optimisation (PSO), Genetic Algorithms (GA), Nesterov's gradient-free algorithm (NGF) and Covariance Matrix Adaptation Evolution Strategy (CMA-ES). We find that the proposed parametrisation provides a statistically significant advantage on all algorithms tested except CMA-ES, for which the performance was similar. We discuss this latter empirical result in relation to the theoretical invariance properties of CMA-ES.