Controllability and reachability of 2-D positive systems: a graph theoretic approach

When dealing with two-dimensional (2-D) discrete state-space models, controllability properties are introduced in two different forms: a local form, which refers to single local states, and a global form, which instead pertains the infinite set of local states lying on a separation set. In this paper, these concepts are investigated in the context of 2-D positive systems by means of a graph theoretic approach. For all these properties, necessary and sufficient conditions, which refer to the structure of the digraph, are provided. While the global reachability index is bounded by the system dimension n, the local reachability index may far exceed the system dimension. Upper bounds on the local reachability index for some special classes of positive systems are finally derived.

[1]  H. Maeda,et al.  Positive realization of difference equations , 1981 .

[2]  M. E. Valcher.  Controllability and reachability criteria for discrete time positive systems , 1996 .

[3]  M. Bisiacco.  State and output feedback stabilizability of 2-D systems , 1985 .

[4]  Ettore Fornasini,et al.  Global properties and duality in 2-D systems , 1982 .

[5]  Ettore Fornasini,et al.  Primitivity of Positive Matrix Pairs: Algebraic Characterization, Graph Theoretic Description, and 2D Systems Interpretation , 1998, SIAM J. Matrix Anal. Appl..

[6]  Ettore Fornasini,et al.  ON THE SPECTRAL AND COMBINATORIAL STRUCTURE OF 2D POSITIVE SYSTEMS , 1996 .

[7]  Tadeusz Kaczorek,et al.  Reachability and controllability of 2D positive linear systems with state feedbacks , 1999 .

[8]  Ettore Fornasini,et al.  State models and asymptotic behaviour of 2D positive systems , 1995 .

[9]  S. Rinaldi,et al.  Positive Linear Systems: Theory and Applications , 2000 .

[10]  Ettore Fornasini,et al.  A 2-D systems approach to river pollution modelling , 1991, Multidimens. Syst. Signal Process..

[11]  Maria Elena Valcher,et al.  On the internal stability and asymptotic behavior of 2-D positive systems , 1997 .

[12]  Ettore Fornasini,et al.  Directed graphs, 2D state models and characteristic polynomials of irreducible matrix pairs , 1997 .

[13]  Ettore Fornasini,et al.  Doubly-indexed dynamical systems: State-space models and structural properties , 1978, Mathematical systems theory.

[14]  T. Kaczorek.  REALIZATION PROBLEM FOR POSITIVE 2-D ROESSER TYPE MODEL , 1997 .

[15]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[16]  W. Marszalek.  Two-dimensional state-space discrete models for hyperbolic partial differential equations , 1984 .

[17]  Elena Sánchez,et al.  Canonical forms for positive discrete-time linear control systems☆ , 2000 .

[18]  Ettore Fornasini,et al.  On the Positive Reachability of 2D Positive Systems , 2003, POSTA.

[19]  R. Roesser.  A discrete state-space model for linear image processing , 1975 .