Optimization over state feedback policies for robust control with constraints

This paper is concerned with the optimal control of linear discrete-time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and all allowable disturbance sequences. The cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance-free state and input sequences. The value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic, or a tractable second-order cone program (SOCP) if the disturbance set is given by a 2-norm bound.

[1]  Stephen P. Boyd,et al.  Applications of second-order cone programming , 1998 .

[2]  E. Kerrigan,et al.  AN EFFICIENT DECOMPOSITION-BASED FORMULATION FOR ROBUST CONTROL WITH CONSTRAINTS , 2005 .

[3]  Eduardo F. Camacho,et al.  On the piecewise linear nature of constrained min-max model predictive control with bounded uncertainties , 2003, Proceedings of the 2003 American Control Conference, 2003..

[4]  Aachen,et al.  Stochastic Inequality Constrained Closed-loop Model Predictive Control: With Application To Chemical Process Operation , 2004 .

[5]  F. Blanchini.  Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions , 1994, IEEE Trans. Autom. Control..

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

[7]  Eduardo F. Camacho,et al.  Robust stability of min-max MPC controllers for nonlinear systems with bounded uncertainties , 2004 .

[8]  J. Maciejowski,et al.  Feedback min‐max model predictive control using a single linear program: robust stability and the explicit solution , 2004 .

[9]  David Q. Mayne,et al.  Robust time-optimal control of constrained linear Systems , 1997, Autom..

[10]  Moritz Diehl,et al.  Robust dynamic programming for min-max model predictive control of constrained uncertain systems , 2004, IEEE Transactions on Automatic Control.

[11]  David Q. Mayne,et al.  Robust model predictive control using tubes , 2004, Autom..

[12]  David Q. Mayne,et al.  Control of Constrained Dynamic Systems , 2001, Eur. J. Control.

[13]  Dimitri P. Bertsekas,et al.  Convex Analysis and Optimization , 2003 .

[14]  Diederich Hinrichsen,et al.  Control of Uncertain Systems , 1990 .

[15]  Eric C. Kerrigan,et al.  Efficient robust optimization for robust control with constraints , 2008, Math. Program..

[16]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[17]  D. Bertsekas,et al.  On the minimax reachability of target sets and target tubes , 1971 .

[18]  J. Shamma.  Optimization of the l∞-induced norm under full state feedback , 1996, IEEE Trans. Autom. Control..

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

[20]  O. Bosgra,et al.  A conic reformulation of Model Predictive Control including bounded and stochastic disturbances under state and input constraints , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[21]  Olaf Stursberg,et al.  LMI Approach to Robust Model Predictive Control , 2005 .

[22]  D. Bertsekas.  Infinite time reachability of state-space regions by using feedback control , 1972 .

[23]  Eric C. Kerrigan,et al.  On robust optimization and the optimal control of constrained linear systems with bounded state disturbances , 2003, 2003 European Control Conference (ECC).

[24]  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).

[25]  Tae-Woong Yoon,et al.  A globally stabilizing receding horizon controller for neutrally stable linear systems with input constraints , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

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

[27]  Stephen P. Boyd,et al.  Linear controller design: limits of performance , 1991 .

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

[29]  E. Kerrigan,et al.  RELATIONSHIPS BETWEEN AFFINE FEEDBACK POLICIES FOR ROBUST CONTROL WITH CONSTRAINTS , 2007 .

[30]  David Q. Mayne,et al.  Invariant approximations of the minimal robust positively Invariant set , 2005, IEEE Transactions on Automatic Control.

[31]  D. Limón,et al.  Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive uncertainties , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[32]  Claudio De Persis,et al.  Input-to-state stabilizing MPC for neutrally stable linear systems subject to input constraints , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[33]  Andrew R. Teel,et al.  Discrete-time certainty equivalence output feedback: allowing discontinuous control laws including those from model predictive control , 2005, Autom..

[34]  E. Gilbert,et al.  Theory and computation of disturbance invariant sets for discrete-time linear systems , 1998 .

[35]  T. Georgiou,et al.  l/sup 1/ state-feedback control with a prescribed rate of exponential convergence , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[36]  C. Scherer,et al.  LMI-based closed-loop economic optimization of stochastic process operation under state and input constraints , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[37]  Elana Guslitser.  UNCERTAINTY- IMMUNIZED SOLUTIONS IN LINEAR PROGRAMMING , 2002 .

[38]  Ali Jadbabaie,et al.  ON THE ISS PROPERTY FOR RECEDING HORIZON CONTROL OF CONSTRAINED LINEAR SYSTEMS , 2002 .

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

[40]  H. Witsenhausen.  A minimax control problem for sampled linear systems , 1968 .

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

[42]  T. Alamo,et al.  A convex parameterization for solving constrained min-max problems with a quadratic cost , 2004, Proceedings of the 2004 American Control Conference.

[43]  D. Bertsekas,et al.  Sufficiently informative functions and the minimax feedback control of uncertain dynamic systems , 1973 .

[44]  Roger J.-B. Wets,et al.  On decision rules in stochastic programming , 1974, Math. Program..

[45]  J. Löfberg,et al.  Approximations of closed-loop minimax MPC , 2003, CDC.

[46]  Alberto Bemporad,et al.  The explicit linear quadratic regulator for constrained systems , 2003, Autom..

[47]  R.S. Smith,et al.  Robust model predictive control of constrained linear systems , 2004, Proceedings of the 2004 American Control Conference.

[48]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[49]  Zhong-Ping Jiang,et al.  Input-to-state stability for discrete-time nonlinear systems , 1999 .

[50]  B. Kouvaritakis,et al.  Constrained receding horizon predictive control for systems with disturbances , 1999 .

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

[52]  Franco Blanchini,et al.  Set invariance in control , 1999, Autom..

[53]  Alberto Bemporad,et al.  Min-max control of constrained uncertain discrete-time linear systems , 2003, IEEE Trans. Autom. Control..

[54]  E.C. Kerrigan,et al.  Properties of a new parameterization for the control of constrained systems with disturbances , 2004, Proceedings of the 2004 American Control Conference.

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

[56]  Eduardo D. Sontag,et al.  The ISS philosophy as a unifying framework for stability-like behavior , 2001 .

[57]  Francesco Borrelli,et al.  Constrained Optimal Control of Linear and Hybrid Systems , 2003, IEEE Transactions on Automatic Control.

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