A novel approach to the computation of the maximal controlled invariant set for constrained linear systems

In this paper, the problem of the determination of the maximal controlled invariant set of linear systems subject to polyhedral input and state constraints, together with the corresponding state-feedback control law is investigated. Instead of computing one-step reachable sets or maximizing the volume of a specific invariant set, the proposed method consists of the iterative expansion of an initial “small” invariant set by adding new vertices to its convex hull. This is achieved by minimizing the distance of each new vertex from the vertices of the polyhedral set defining the state constraints. This approach, established for both continuous-time and discrete-time systems, does not require invertibility of matrix A, open-loop stability or symmetry of the constraints.

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

[2]  Mark Cannon,et al.  Asymmetric constraints with polytopic sets in MPC with application to coupled tanks system , 2004 .

[3]  D. Limon,et al.  Enlarging the domain of attraction of MPC controllers , 2005, Autom..

[4]  M. Cwikel,et al.  Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states , 1984 .

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

[6]  K. T. Tan,et al.  Linear systems with state and control constraints: the theory and application of maximal output admissible sets , 1991 .

[7]  E. Misawa,et al.  Semi-ellipsoidal controlled invariant sets for constrained linear systems , 2002 .

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

[9]  George Bitsoris,et al.  Design techniques for the control of discrete-time systems subject to state and control constraints , 1999, IEEE Trans. Autom. Control..

[10]  Franco Blanchini,et al.  Dynamic augmentation and complexity reduction of set-based constrained control , 2008 .

[11]  M. Cwikel,et al.  Convergence of an algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states , 1985, 1985 24th IEEE Conference on Decision and Control.

[12]  Sophie Tarbouriech,et al.  Control of linear systems subject to input constraints: a polynomial approach , 2001, Autom..

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

[14]  J. Hennet,et al.  On invariant polyhedra of continuous-time linear systems , 1993, IEEE Trans. Autom. Control..

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

[16]  Luca Benvenuti,et al.  Invariant polytopes of linear systems , 1998 .

[17]  Zongli Lin,et al.  On enlarging the basin of attraction for linear systems under saturated linear feedback , 2000, Proceedings of the 2000 American Control Conference. ACC (IEEE Cat. No.00CH36334).

[18]  B. De Moor,et al.  Polytopic invariant sets for continuous-time systems , 2007, 2007 European Control Conference (ECC).

[19]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[20]  Jean B. Lasserre,et al.  Reachable, controllable sets and stabilizing control of constrained linear systems , 1993, Autom..

[21]  Jean-Claude Hennet,et al.  COMPUTATION OF MAXIMAL ADMISSIBLE SETS OF CONSTRAINED LINEAR SYSTEMS , 1996 .

[22]  G. Bitsoris.  On the positive invariance of polyhedral sets for discrete-time systems , 1988 .