Robust receding horizon control of uncertain fuzzy systems

The optimal control of uncertain fuzzy systems with constraints is still an open problem. One candidate to deal with this problem is robust receding horizon control (RRHC) schemes, which can be formulated as a differential game. Our focus concerns numerically solving Hamilton–Jacobi–Issac (HJI) equations derived from RRHC schemes for uncertain fuzzy systems. The developed finite difference approximation scheme with sigmoidal transformation is a stable and convergent algorithm for HJI equations. Accelerated procedures with boundary value iteration are developed to increase the calculation accuracy with less time consumption. Then, the state-feedback RRHC controller is designed for some class of uncertain fuzzy systems with constraints. The value function calculated by numerical methods acts as the design parameter. The closed-loop system is proven to be asymptotically stable. An engineering implementation of the controller is discussed.

[1]  N. El‐Farra,et al.  Bounded robust control of constrained multivariable nonlinear processes , 2003 .

[2]  N. L. Ricker,et al.  Adaptive optimal control: The thinking man's GPC : R. R. Bitmead, M. Gevers and V. Wertz , 1993, Autom..

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

[4]  Qingshuo Song.  Convergence of Markov chain approximation on generalized HJB equation and its applications , 2008, Autom..

[5]  R. Bellman.  Dynamic programming. , 1957, Science.

[6]  Li-Xin Wang,et al.  A Course In Fuzzy Systems and Control , 1996 .

[7]  Alberto Isidori,et al.  Nonlinear control systems: an introduction (2nd ed.) , 1989 .

[8]  T. Başar,et al.  Stochastic and differential games : theory and numerical methods , 1999 .

[9]  Manfred Morari,et al.  Robust Model Predictive Control , 1987, 1987 American Control Conference.

[10]  J. Richalet,et al.  Industrial applications of model based predictive control , 1993, Autom..

[11]  Chin-Teng Lin,et al.  Optimal fuzzy controller design: local concept approach , 2000, IEEE Trans. Fuzzy Syst..

[12]  James B. Rawlings,et al.  Nonlinear Model Predictive Control: A Tutorial and Survey , 1994 .

[13]  Kazuo Tanaka,et al.  Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach , 2008 .

[14]  Michael A. Henson,et al.  Nonlinear model predictive control: current status and future directions , 1998 .

[15]  Chonghui Song,et al.  Numerical optimization method for HJB equations with its application to receding horizon control schemes , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[16]  D. Mayne,et al.  Receding horizon control of nonlinear systems , 1990 .

[17]  E. Polak,et al.  Moving horizon control of nonlinear systems with input saturation, disturbances and plant uncertainty , 1993 .

[18]  Hong Chen,et al.  Nonlinear Model Predictive Control Schemes with Guaranteed Stability , 1998 .

[19]  HAROLD J. KUSHNER,et al.  Numerical Approximations for Stochastic Differential Games , 2002, SIAM J. Control. Optim..

[20]  Panagiotis D. Christofides,et al.  Predictive control of switched nonlinear systems with scheduled mode transitions , 2005, IEEE Transactions on Automatic Control.

[21]  H. Michalska,et al.  Receding horizon control of nonlinear systems , 1988, Proceedings of the 28th IEEE Conference on Decision and Control,.

[22]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[23]  Tamer Basar,et al.  H∞-Optimal Control and Related , 1991 .

[24]  Jay H. Lee,et al.  Model predictive control: past, present and future , 1999 .

[25]  M. James.  Controlled markov processes and viscosity solutions , 1994 .

[26]  N. El‐Farra,et al.  Integrating robustness, optimality and constraints in control of nonlinear processes , 2001 .

[27]  R. Elliott,et al.  The Existence Of Value In Differential Games , 1972 .

[28]  P. Souganidis.  Two-Player, Zero-Sum Differential Games and Viscosity Solutions , 1999 .

[29]  J. Rawlings,et al.  The stability of constrained receding horizon control , 1993, IEEE Trans. Autom. Control..

[30]  A. Isidori.  Nonlinear Control Systems: An Introduction , 1986 .

[31]  Derong Liu,et al.  Generalized Receding Horizon Control of Fuzzy Systems Based on Numerical Optimization Algorithm , 2009, IEEE Transactions on Fuzzy Systems.

[32]  P. Souganidis,et al.  Differential Games and Representation Formulas for Solutions of Hamilton-Jacobi-Isaacs Equations. , 1983 .

[33]  Tianyou Chai,et al.  Comment on "Discrete-time optimal fuzzy controller design: global concept approach" , 2005, IEEE Trans. Fuzzy Syst..

[34]  Gang Feng,et al.  Analysis and design for a class of complex control systems Part I: Fuzzy modelling and identification , 1997, Autom..

[35]  J. Quadrat.  Numerical methods for stochastic control problems in continuous time , 1994 .

[36]  J. Richalet,et al.  Model predictive heuristic control: Applications to industrial processes , 1978, Autom..

[37]  Jinchun Ye,et al.  Optimal Life Insurance Purchase, Consumption and Portfolio Under an Uncertain Life , 2006 .

[38]  Chin-Teng Lin,et al.  Optimal fuzzy controller design in continuous fuzzy system: global concept approach , 2000, IEEE Trans. Fuzzy Syst..

[39]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[40]  L X Wang,et al.  Fuzzy basis functions, universal approximation, and orthogonal least-squares learning , 1992, IEEE Trans. Neural Networks.

[41]  Panagiotis D. Christofides,et al.  Robust hybrid predictive control of nonlinear systems , 2005, Autom..

[42]  D. Mayne,et al.  Robust receding horizon control of constrained nonlinear systems , 1993, IEEE Trans. Autom. Control..

[43]  V. Wertz,et al.  Adaptive Optimal Control: The Thinking Man's G.P.C. , 1991 .

[44]  Optimal Life Insurance, Consumption and Portfolio under Uncertainty: Martingale Methods , 2007, 2007 American Control Conference.

[45]  Marko Bacic,et al.  Model predictive control , 2003 .

[46]  J. H. Leet,et al.  Worst-case formulations of model predictive control for systems with bounded parameters , 1997, Autom..