Stabilizing stochastic MPC without terminal constraints

The stability proofs of Model Predictive Control without terminal constraints and/or cost are tightly based upon the principle of optimality, which does not hold in most currently employed approaches to Stochastic MPC. In this paper, we first provide a stability proof for Stochastic Model Predictive Control without terminal cost or constraints under the assumption of optimization over feedback laws and propagation of the probability density functions of predicted states. Based thereon, we highlight why the proof does not remain valid if approximations such as parametrized feedback laws or relaxations on the chance constraints are employed and provide tightened assumptions that are sufficient to establish closed-loop stability. General statements valid for nonlinear systems are provided along with examples and computational simplifications in the case of linear systems.

[1]  L. Grüne,et al.  Nonlinear Model Predictive Control : Theory and Algorithms. 2nd Edition , 2011 .

[2]  Frank Allgöwer,et al.  Constraint-Tightening and Stability in Stochastic Model Predictive Control , 2015, IEEE Transactions on Automatic Control.

[3]  Basil Kouvaritakis,et al.  Explicit use of probabilistic distributions in linear predictive control , 2010, Autom..

[4]  Basil Kouvaritakis,et al.  Stochastic tubes in model predictive control with probabilistic constraints , 2010, Proceedings of the 2010 American Control Conference.

[5]  Basil Kouvaritakis,et al.  Model Predictive Control , 2016 .

[6]  Dimitri P. Bertsekas,et al.  Stochastic optimal control : the discrete time case , 2007 .

[7]  Sebastian Engell,et al.  Multi-stage nonlinear model predictive control applied to a semi-batch polymerization reactor under uncertainty , 2013 .

[8]  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.

[9]  Frank Allgöwer,et al.  Stochastic MPC with offline uncertainty sampling , 2016, Autom..

[10]  Basil Kouvaritakis,et al.  Model Predictive Control: Classical, Robust and Stochastic , 2015 .

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

[12]  Andrew R. Teel,et al.  Model predictive control: for want of a local control Lyapunov function, all is not lost , 2005, IEEE Transactions on Automatic Control.

[13]  Sten Bay Jørgensen,et al.  Unreachable Setpoints in Model Predictive Control , 2008, IEEE Transactions on Automatic Control.

[14]  Masahiro Ono,et al.  A Probabilistic Particle-Control Approximation of Chance-Constrained Stochastic Predictive Control , 2010, IEEE Transactions on Robotics.

[15]  H. Kushner.  Stochastic Stability and Control , 2012 .

[16]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[17]  Michel Verhaegen,et al.  Robustly asymptotically stable finite-horizon MPC , 2006, Autom..

[18]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[19]  Andrew R. Teel,et al.  Nominally Robust Model Predictive Control With State Constraints , 2007, IEEE Transactions on Automatic Control.

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

[21]  A. Mesbah,et al.  Stochastic Model Predictive Control: An Overview and Perspectives for Future Research , 2016, IEEE Control Systems.

[22]  Richard D. Braatz,et al.  Stochastic nonlinear model predictive control with probabilistic constraints , 2014, 2014 American Control Conference.