On the sample size of randomized MPC for chance-constrained systems with application to building climate control

X. Zhang, S. Grammatico, G. Schildbach, P. J. Goulart and J. Lygeros

@inproceedings{ZGSetal:2014,
  author = {X. Zhang and S. Grammatico and G. Schildbach and P. J. Goulart and J. Lygeros},
  title = {On the sample size of randomized MPC for chance-constrained systems with application to building climate control},
  booktitle = {European Control Conference},
  year = {2014},
  pages = {478-483},
  url = {http://dx.doi.org/10.1109/ECC.2014.6862498},
  doi = {10.1109/ECC.2014.6862498}
}

We consider Stochastic Model Predictive Control (SMPC) for constrained affine systems with additive disturbance and affine disturbance feedback (ADF) policies. One way of solving the chance constrained optimization problem associated with the SMPC formulation is by means of randomization, where the chance constraints are replaced by a number of sampled hard constraints, each corresponding to a disturbance realization. The ADF formulation leads to a quadratic growth in the number of decision variables with respect to the prediction horizon, which results in a quadratic growth in the sample size. This leads to computationally expensive problems with solutions that are conservative in terms of both cost and violation probability. We address these limitations by proposing a bound on the sample size which scales linearly in the prediction horizon. The new bound is obtained by explicitly computing the maximum number of active constraints, leading to significant advantages both in terms of computational time and conservatism of the solution. The efficacy of the new bound relative to the existing one is demonstrated on a building control case study.