Stochastic MPC with Imperfect State Information and Bounded Controls

This paper addresses the problem of output feedback Model Predictive Control for stochastic linear systems, with hard and soft constraints on the control inputs as well as soft constraints on the state. We use the so-called purified outputs along with a suitable nonlinear control policy and show that the resulting optimization program is convex. We also show how the proposed method can be applied in a receding horizon fashion. Contrary to the state feedback case, the receding horizon implementation in the output feedback case requires the update of several optimization parameters and the recursive computation of the conditional probability densities of the state given the previous measurements. Algorithms for performing these tasks are developed.

[1]  Pravin Varaiya,et al.  Stochastic Systems: Estimation, Identification, and Adaptive Control , 1986 .

[2]  Dimitri P. Bertsekas,et al.  Dynamic Programming and Optimal Control, Two Volume Set , 1995 .

[3]  David K. Smith,et al.  Dynamic Programming and Optimal Control. Volume 1 , 1996 .

[4]  Alberto Bemporad,et al.  Robust model predictive control: A survey , 1998, Robustness in Identification and Control.

[5]  E. Kamen,et al.  Introduction to Optimal Estimation , 1999 .

[6]  J. W. Nieuwenhuis,et al.  Boekbespreking van D.P. Bertsekas (ed.), Dynamic programming and optimal control - volume 2 , 1999 .

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

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

[9]  G. Picci,et al.  Silverman algorithm and the structure of discrete-time stochastic systems☆ , 2002 .

[10]  O. Bosgra,et al.  A full solution to the constrained stochastic closed-loop MPC problem via state and innovations feedback and its receding horizon implementation , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[11]  J. Löfberg.  Minimax approaches to robust model predictive control , 2003 .

[12]  A. Ben-Tal,et al.  Adjustable robust solutions of uncertain linear programs , 2004, Math. Program..

[13]  I Ivo Batina,et al.  Model predictive control for stochastic systems by randomized algorithms , 2004 .

[14]  Christian P. Robert,et al.  Monte Carlo Statistical Methods , 2005, Springer Texts in Statistics.

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

[16]  Basil Kouvaritakis,et al.  Stochastic MPC with inequality stability constraints , 2006, Autom..

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

[18]  Maarten H. van der Vlerk,et al.  Integrated Chance Constraints: Reduced Forms and an Algorithm , 2006, Comput. Manag. Sci..

[19]  Stephen P. Boyd,et al.  Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems , 2006, Math. Program..

[20]  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.

[21]  Dimitris Bertsimas,et al.  Constrained Stochastic LQC: A Tractable Approach , 2007, IEEE Transactions on Automatic Control.

[22]  L. Blackmore,et al.  Optimal, Robust Predictive Control of Nonlinear Systems under Probabilistic Uncertainty using Particles , 2007, 2007 American Control Conference.

[23]  Basil Kouvaritakis,et al.  Probabilistic Constrained MPC for Systems with Multiplicative and Additive Stochastic Uncertainty , 2008 .

[24]  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.

[25]  Pablo A. Parrilo,et al.  Optimality of affine policies in multi-stage robust optimization , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[26]  Stephen P. Boyd,et al.  Performance bounds for linear stochastic control , 2009, Syst. Control. Lett..

[27]  John Lygeros,et al.  On convexity of stochastic optimization problems with constraints , 2009, 2009 European Control Conference (ECC).

[28]  Alberto Bemporad,et al.  Scenario-based model predictive control of stochastic constrained linear systems , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[29]  Basil Kouvaritakis,et al.  Model predictive control for systems with stochastic multiplicative uncertainty and probabilistic constraints , 2009, Autom..

[30]  Stephen P. Boyd,et al.  Design of Affine Controllers via Convex Optimization , 2010, IEEE Transactions on Automatic Control.

[31]  John Lygeros,et al.  Stochastic Receding Horizon Control With Bounded Control Inputs: A Vector Space Approach , 2009, IEEE Transactions on Automatic Control.