(A, B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems

The problem of confining the trajectory of a linear discrete-time system in a given polyhedral domain is addressed through the concept of (A, B)-invariance. First, an explicit characterization of (A, B)-invariance of convex polyhedra is proposed. Such characterization amounts to necessary and sufficient conditions in the form of linear matrix relations and presents two major advantages compared to the ones found in the literature: it applies to any convex polyhedron and does not require the computation of vertices. Such advantages are felt particularly in the computation of the supremal (A, B)-invariant set included in a given polyhedron, for which a numerical method is proposed. The problem of computing a control law which forces the system trajectories to evolve inside an (A, B)-invariant polyhedron is treated as well. Finally, the (A, B)-invariance relations are generalized to persistently disturbed systems.

[1]  H. Witsenhausen.  Sets of possible states of linear systems given perturbed observations , 1968 .

[2]  F. Schweppe,et al.  Control of linear dynamic systems with set constrained disturbances , 1971 .

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

[4]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .

[5]  Per-olof Gutman,et al.  Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states , 1984, The 23rd IEEE Conference on Decision and Control.

[6]  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.

[7]  E. Gilbert,et al.  Computation of minimum-time feedback control laws for discrete-time systems with state-control constraints , 1987 .

[8]  G. Bitsoris.  Positively invariant polyhedral sets of discrete-time linear systems , 1988 .

[9]  C. Burgat,et al.  Regulator problem for linear discrete-time systems with non-symmetrical constrained control , 1988 .

[10]  Elena De Santis,et al.  Techniques of linear programming based on the theory of convex cones , 1989 .

[11]  Franco Blanchini.  Feedback control for linear time-invariant systems with state and control bounds in the presence of disturbances , 1990 .

[12]  Eugênio B. Castelan,et al.  Eigenstructure assignment for state constrained linear continuous time systems , 1992, Autom..

[13]  G. Basile,et al.  Controlled and conditioned invariants in linear system theory , 1992 .

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

[15]  Jean-Claude Hennet,et al.  Discrete Time Constrained Linear Systems , 1995 .

[16]  M. Sznaier,et al.  Persistent disturbance rejection via static-state feedback , 1995, IEEE Trans. Autom. Control..

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

[18]  J.-C. Hennet,et al.  On (A, B)-invariance of polyhedral domains for discrete-time systems , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[19]  J. Hennet,et al.  A geometric approach to the l/sup 1/ linear control problem , 1997, Proceedings of the 36th IEEE Conference on Decision and Control.

[20]  Jean-Claude Hennet,et al.  (A, B)-invariance conditions of polyhedral domains for continuous-time systems , 1997, 1997 European Control Conference (ECC).

[21]  Jean-Claude Hennet,et al.  A Geometric Approach to the l1 Linear Control Problem , 1997 .

[22]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.