Optimized robust control invariance for linear discrete-time systems: Theoretical foundations

This paper introduces the concept of optimized robust control invariance for discrete-time linear time-invariant systems subject to additive and bounded state disturbances. A novel characterization of two families of robust control invariant sets is given. The existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear-convex case and a standard linear/quadratic program when the constraints are polyhedral or polytopic. The solution of the same optimization problem yields the corresponding feedback control law that is, in general, set-valued. A procedure for selection of a point-valued, nonlinear control law is provided.

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

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

[3]  R. Ruth,et al.  Stability of dynamical systems , 1988 .

[4]  J. P. Lasalle The stability of dynamical systems , 1976 .

[5]  David Q. Mayne,et al.  REGULATION OF DISCRETE-TIME LINEAR SYSTEMS WITH POSITIVE STATE AND CONTROL CONSTRAINTS AND BOUNDED DISTURBANCES , 2005 .

[6]  Maria Domenica Di Benedetto,et al.  Computation of maximal safe sets for switching systems , 2004, IEEE Transactions on Automatic Control.

[7]  F. Blanchini,et al.  Constrained stabilization with an assigned initial condition set , 1995 .

[8]  R. Schneider Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

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

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

[11]  Ilya Kolmanovsky,et al.  Fast reference governors for systems with state and control constraints and disturbance inputs , 1999 .

[12]  Mato Baotic,et al.  Multi-Parametric Toolbox (MPT) , 2004, HSCC.

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

[14]  A. Garulli,et al.  Robustness in Identification and Control , 1989 .

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

[16]  P. Caravani,et al.  Invariant Equilibria of Polytopic Games via Optimized Robust Control Invariance , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

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

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

[19]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[20]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[21]  Frank Allgöwer,et al.  State and Output Feedback Nonlinear Model Predictive Control: An Overview , 2003, Eur. J. Control.

[22]  S. Tarbouriech,et al.  INVARIANCE AND CONTRACTIVITY OF POLYHEDRA FOR LINEAR CONTINUOUS-TIME SYSTEMS WITH SATURATING CONTROLS , 1999 .

[23]  Elena De Santis,et al.  Doubly invariant equilibria of linear discrete-time games , 2002, Autom..

[24]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

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

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

[27]  J. Hennet,et al.  (A, B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems , 1999 .

[28]  D. Mayne,et al.  OPTIMIZED ROBUST CONTROL INVARIANT SETS FOR CONSTRAINED LINEAR DISCRETE-TIME SYSTEMS , 2005 .

[29]  Elena De Santis,et al.  A polytopic game , 2000, Autom..

[30]  David Q. Mayne,et al.  Nonlinear Control of Constrained Dynamic Systems , 1997 .