Abstract
In this paper we consider in a behavioral setting the subclass of dis-crete-time, linear, finite-dimensional systems, which can be represented by autoregressive (AR) equations. It will be shown that, with respect to the convergence of all coefficients in an AR representation, there exist continuously dependent input-state-output (i/s/o) representations, under the condition that some specified degree remains constant. This continuous i/s/o representation is given by the Fuhrmann realization.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Brodmann,Algebraische Geometrie. Eine Einführung, Birkhäuser, 1989.
C. I. Byrnes and T. E. Duncan, On certain topological invariants arising in system theory,New Directions in Applied Mathematics (P. J. Hilton and G. S. Young, eds.), pp. 29–71. Springer-Verlag, 1981.
J. De Does, H. Glüsing-Lüerßen, and J. M. Schumacher, Connectedness properties of spaces of linear systems,Proceedings SINS'92 International Symposium on Implicit and Nonlinear Systems, pp. 210–215, Arlington, Texas, 1992.
P. A. Fuhrmann,Linear Systems and Operators in Hilbert space, McGraw-Hill, 1981.
H. Glüsing-Lüerßen, On various topologies for finite-dimensional linear systems, Report 273, Institut für Dynamische Systeme, Universität Bremen, 1992.
M. Hazewinkel, (Fine) moduli (spaces) for linear systems: What are they and what are they good for?Geometrical Methods for the Theory of Linear Systems, (C. I. Byrnes and C. F. Martin, eds.), pp. 125–193, D. Reidel, 1980.
M. Kuijper and J. M. Schumacher, Realization of autoregressive equations in pencil and descriptor form,SIAM J. Control and Optim.,28(5) (1990), 1162–1189.
C. Martin and R. Hermann, Applications of algebraic geometry to systems theory: The McMillan degree and Kronecker indices of transfer functions as topological and holomorphic system invariants,SIAM J. Control and Optim.,16 (1978), 743–755.
J. W. Nieuwenhuis, More about continuity of dynamical systems.Syst. Control Lett.,14 (1990), 25–29.
J. W. Nieuwenhuis and J. C. Willems, Continuity of dynamical systems: A system theoretical approach,Math. Control Sign. Syst.,1 (1988), 147–165.
Li Qiu and E. J. Davison, Pointwise gap metrics on transfer matrices.IEEE Trans. Automat. Control,AC-37 (1992), 741–758.
M. S. Ravi and J. Rosenthal, A smooth compactification of the space of transfer functions with fixed McMillan degree,Acta Appl. Math. (1994), to appear.
M. S. Ravi and J. Rosenthal, A general realization theory for higher order linear differential equations, Preprint, 1994.
M. Vidyasagar, H. Schneider, and B. A. Francis, Algebraic and topological aspects of feedback stabilization,IEEE Trans. Automat. Control,AC-27 (1982), 880–894.
J. C. Willems, From time series to linear systems: Part 1. Finite dimensional linear time invariant systems,IEEE Trans. Automat. Control,22 (1986), 561–580.
J. C. Willems, Models for dynamics,Dynamics Reported,2 (1989), 171–269.
J. C. Willems, Paradigms and puzzles in the theory of dynamical systems,IEEE Trans. Automat. Control,AC-36 (1991), 259–294.
G. Zames and A. K. El-Sakkary, Unstable systems and feedback: the gap metric,Proceeding 16th Allerton Conference, pp. 380–385, 1980.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Glüsing-Lüerßen, H. Continuous state representations for AR Systems. Math. Control Signal Systems 8, 82–95 (1995). https://doi.org/10.1007/BF01212368
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01212368