Stochastic Model Predictive Control using a combination of randomized and robust optimization

In this paper, we focus on Stochastic Model Predictive Control (SMPC) problems for systems with linear dynamics and additive uncertainty. One way to address such problems is by means of randomized algorithms. Typically, these algorithms require substituting the chance constraint of the SMPC problem with a finite number of hard constraints corresponding to samples of the uncertainty. Earlier approaches toward this direction lead to computationally expensive problems, whose solutions are typically very conservative in terms of cost. To address these limitations, we follow an alternative methodology based on a combination of randomized and robust optimization. We show that our approach can offer significant advantages in terms of both cost and computational time. Both the open-loop MPC formulation (i.e. optimizing over input sequences), as well as optimization over policies using the affine disturbance feedback formulation are considered. We demonstrate the efficacy of the proposed approach relative to standard randomized techniques on a building control problem.

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

[2]  Manfred Morari,et al.  Model predictive control: Theory and practice - A survey , 1989, Autom..

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

[4]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

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

[6]  David Q. Mayne,et al.  Robust model predictive control of constrained linear systems with bounded disturbances , 2005, Autom..

[7]  Manfred Morari,et al.  Stochastic Model Predictive Control for Building Climate Control , 2014, IEEE Transactions on Control Systems Technology.

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

[9]  Melvyn Sim,et al.  Tractable Approximations to Robust Conic Optimization Problems , 2006, Math. Program..

[10]  Peter Kall,et al.  Stochastic Programming , 1995 .

[11]  D. Mayne,et al.  Min-max feedback model predictive control for constrained linear systems , 1998, IEEE Trans. Autom. Control..

[12]  Luigi Chisci,et al.  Systems with persistent disturbances: predictive control with restricted constraints , 2001, Autom..

[13]  Marco C. Campi,et al.  A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality , 2011, J. Optim. Theory Appl..

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

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

[16]  John Lygeros,et al.  Convexity and convex approximations of discrete-time stochastic control problems with constraints , 2011, Autom..

[17]  James B. Rawlings,et al.  Postface to “ Model Predictive Control : Theory and Design ” , 2012 .

[18]  Eric C. Kerrigan,et al.  Input-to-state stability of robust receding horizon control with an expected value cost , 2008, Autom..

[19]  James A. Primbs,et al.  Stochastic Receding Horizon Control of Constrained Linear Systems With State and Control Multiplicative Noise , 2007, IEEE Transactions on Automatic Control.

[20]  Giuseppe Carlo Calafiore,et al.  Randomized Model Predictive Control for stochastic linear systems , 2012, 2012 American Control Conference (ACC).

[21]  John Lygeros,et al.  On the Road Between Robust Optimization and the Scenario Approach for Chance Constrained Optimization Problems , 2014, IEEE Transactions on Automatic Control.

[22]  Jan M. Maciejowski,et al.  Predictive control : with constraints , 2002 .

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

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

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

[26]  G. Andersson,et al.  A probabilistic framework for security constrained reserve scheduling of networks with wind power generation , 2012, 2012 IEEE International Energy Conference and Exhibition (ENERGYCON).

[27]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[28]  Alberto Bemporad,et al.  Reducing conservativeness in predictive control of constrained systems with disturbances , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[29]  John Lygeros,et al.  A randomized approach to Stochastic Model Predictive Control , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

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