Abstract
Representations of linear time-invariant discrete-time systems are discussed. A system is defined as a behavior, that is, as a family of trajectoriesmapping the time axis into the signal space. The following characterizations are equivalent: (i) the system is linear, time-invariant, and complete, (ii) the behavior is linear, shiftinvariant, and closed, (iii) the behavior is kernel of a linear difference operator with a polynomial symbol, (iv) the system allows a linear input/output representation in terms of polynomial matrices, (v) the system allows a linear constant coefficient input/state/output representation, and (vi) the behavior is kernel of a linear difference operator with a rational symbol. If the system is controllable, then the system also allows (vii) an image representation with a polynomial symbol, and an image representation with a rational symbol.
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Willems, J.C.: From time series to linear system — Part I. Finite dimensional linear time invariant systems, Part II. Exact modelling, Part III. Approximate modelling. Automatica, 22, 561−580 and 675−694 (1986), 23, 87−115 (1987)
Willems, J.C.: Paradigms and puzzles in the theory of dynamical systems. IEEE Transactions on Automatic Control, 36, 259−294 (1991)
Polderman, J.W., Willems, J.C.: Introduction to Mathematical Systems Theory: A Behavioral Approach. Springer-Verlag (1998)
Willems, J.C.: Thoughts on system identification. In: Francis, B.A., Smith, M.C., Willems, J.C. (eds), Control of Uncertain Systems: Modelling, Approximation and Design, Springer Verlag Lecture Notes on Control and Information Systems, vol. 329, 389−416 (2006)
Willems, J.C.: The behavioral approach to open and interconnected systems, Modeling by tearing, zooming, and linking. Control Systems Magazine, 27, 49−99 (2007)
Willems, J.C., Yamamoto, Y.: Behaviors defined by rational functions. Linear Algebra and Its Applications, 425, 226−241 (2007)
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Willems, J.C. (2009). Linear Systems in Discrete Time. In: Hof, P., Scherer, C., Heuberger, P. (eds) Model-Based Control:. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0895-7_1
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DOI: https://doi.org/10.1007/978-1-4419-0895-7_1
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