Lyapunov-based Stochastic Nonlinear Model Predictive Control: Shaping the State Probability Density Functions

—Stochastic uncertainties in complex dynamical systems lead to variability of system states, which can in turn degrade the closed-loop performance. This paper presents a stochastic model predictive control approach for a class of nonlinear systems with unbounded stochastic uncertainties. The control approach aims to shape probability density function of the stochastic states, while satisfying input and joint state chance constraints. Closed-loop stability is ensured by designing a stability constraint in terms of a stochastic control Lya- punov function, which explicitly characterizes stability in a probabilistic sense. The Fokker-Planck equation is used for describing the dynamic evolution of the states’ probability density functions. Complete characterization of probability density functions using the Fokker-Planck equation allows for shaping the states’ density functions as well as direct computation of joint state chance constraints. The closed-loop performance of the stochastic control approach is demonstrated using a continuous stirred-tank reactor.

[1]  David Q. Mayne,et al.  Model predictive control: Recent developments and future promise , 2014, Autom..

[2]  A. Mesbah,et al.  Stability for receding-horizon stochastic model predictive control , 2014, 2015 American Control Conference (ACC).

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

[4]  Marcello Farina,et al.  A probabilistic approach to Model Predictive Control , 2013, 52nd IEEE Conference on Decision and Control.

[5]  Prashant Mhaskar,et al.  Lyapunov-based model predictive control of stochastic nonlinear systems , 2012, Autom..

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

[7]  M Morari,et al.  Energy efficient building climate control using Stochastic Model Predictive Control and weather predictions , 2010, Proceedings of the 2010 American Control Conference.

[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]  S. Narayanan,et al.  Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems , 2006 .

[10]  O. Bosgra,et al.  Stochastic closed-loop model predictive control of continuous nonlinear chemical processes , 2006 .

[11]  J. F. Forbes,et al.  Control design for first-order processes: shaping the probability density of the process state , 2004 .

[12]  Alberto De Santis,et al.  Stabilization in probability of nonlinear stochastic systems with guaranteed region of attraction and target set , 2003, IEEE Trans. Autom. Control..

[13]  Alison L Gibbs,et al.  On Choosing and Bounding Probability Metrics , 2002, math/0209021.

[14]  R. LeVeque Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[15]  M. Krstić,et al.  Output-feedback stabilization of stochastic nonlinear systems driven by noise of unknown covariance ( , 2000 .

[16]  Dorin Comaniciu,et al.  Real-time tracking of non-rigid objects using mean shift , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

[17]  Hong Wang,et al.  Robust control of the output probability density functions for multivariable stochastic systems with guaranteed stability , 1999, IEEE Trans. Autom. Control..

[18]  Guy Jumarie,et al.  Tracking control of non-linear stochastic systems by using path cross-entropy and Fokker-Planck equation , 1992 .

[19]  J. Elgin The Fokker-Planck Equation: Methods of Solution and Applications , 1984 .

[20]  T. Kailath The Divergence and Bhattacharyya Distance Measures in Signal Selection , 1967 .

[21]  John Lygeros,et al.  Stochastic receding horizon control with output feedback and bounded controls , 2012, Autom..

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

[23]  Xiongzhi Chen Brownian Motion and Stochastic Calculus , 2008 .

[24]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

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

[26]  R. Chabour,et al.  On a universal formula for the stabilization of control stochastic nonlinear systems , 1999 .

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

[28]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[29]  F. D. Garber,et al.  The Quality of Training Sample Estimates of the Bhattacharyya Coefficient , 1990, IEEE Trans. Pattern Anal. Mach. Intell..