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

We consider Stochastic Model Predictive Control (SMPC) for constrained linear systems with additive disturbance, under affine disturbance feedback (ADF) policies. One approach to solve the chance-constrained optimization problem associated with the SMPC formulation is 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 establishing 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 climate control case study.

[1]  Sergio Grammatico,et al.  A Scenario Approach for Non-Convex Control Design , 2014, IEEE Transactions on Automatic Control.

[2]  Sergio Grammatico,et al.  On the sample size of random convex programs with structured dependence on the uncertainty , 2015, Autom..

[3]  Sergio Grammatico,et al.  A scenario approach to non-convex control design: Preliminary probabilistic guarantees , 2014, 2014 American Control Conference.

[4]  J. Lygeros,et al.  Randomized Nonlinear MPC for Uncertain Control-Affine Systems with Bounded Closed-Loop Constraint Violations , 2014 .

[5]  John Lygeros,et al.  Stochastic Model Predictive Control using a combination of randomized and robust optimization , 2013, 52nd IEEE Conference on Decision and Control.

[6]  Manfred Morari,et al.  Scenario-based MPC for energy-efficient building climate control under weather and occupancy uncertainty , 2013, 2013 European Control Conference (ECC).

[7]  Giuseppe Carlo Calafiore,et al.  Robust Model Predictive Control via Scenario Optimization , 2012, IEEE Transactions on Automatic Control.

[8]  Lorenzo Fagiano,et al.  Randomized Solutions to Convex Programs with Multiple Chance Constraints , 2012, SIAM J. Optim..

[9]  Manfred Morari,et al.  Use of model predictive control and weather forecasts for energy efficient building climate control , 2012 .

[10]  Giuseppe Carlo Calafiore,et al.  Random Convex Programs , 2010, SIAM J. Optim..

[11]  R. Tempo,et al.  On the sample complexity of randomized approaches to the analysis and design under uncertainty , 2010, Proceedings of the 2010 American Control Conference.

[12]  Basil Kouvaritakis,et al.  Probabilistic Constrained MPC for Multiplicative and Additive Stochastic Uncertainty , 2009, IEEE Transactions on Automatic Control.

[13]  Manfred Morari,et al.  A tractable approximation of chance constrained stochastic MPC based on affine disturbance feedback , 2008, 2008 47th IEEE Conference on Decision and Control.

[14]  Marco C. Campi,et al.  The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs , 2008, SIAM J. Optim..

[15]  Marco C. Campi,et al.  The exact feasibility of randomized solutions of robust convex programs , 2008 .

[16]  Giuseppe Carlo Calafiore,et al.  The scenario approach to robust control design , 2006, IEEE Transactions on Automatic Control.

[17]  Eric C. Kerrigan,et al.  Optimization over state feedback policies for robust control with constraints , 2006, Autom..

[18]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[19]  Manfred Morari,et al.  Robust constrained model predictive control using linear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.