Abstract
In the behavioral approach to (discrete-time) multidimensional linear systems, one views solution trajectories simply as the set of all solutions of a homogeneous linear system of difference equations. In this setting the Oberst transfer matrix is identified as the unique rational matrix function H satisfying Q = PH where R = [−Q P] is a partitioning of the kernel representation R for the behavior such that P has full column rank equal to the rank of R. This Oberst transfer matrix can be seen as a more fundamental and unifying formalism capturing the transfer functions associated with the older Givone-Roesser and Fornasini-Marchesini input/state/output approaches to multidimensional linear systems. A quite different type of input/state/output linear system having original motivation from operator theory is the Livšic linear system, where the state-evolution equations are overdetermined and lead to compatibility constraints on the input and output signals; the result is that the admissible input signals are not free but form their own nontrivial behavior. The main point of the present work is to identify how Livšic systems fit into the behavioral framework. In particular, we extend the Oberst transfer matrix to the setting of autonomous behaviors lacking any free variables (in which case the standard Oberst transfer matrix is trivial with no columns) by letting the reduced ring (the quotient of the polynomial ring by the behavior-annihilator ideal) act on the behavior. We then make explicit identifications between the Oberst transfer matrix over the reduced ring and the Livšic Joint Characteristic Function.
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Ball, J.A., Boquet, G.M. & Vinnikov, V. A behavioral interpretation of Livšic systems. Multidim Syst Sign Process 23, 17–48 (2012). https://doi.org/10.1007/s11045-010-0145-9
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DOI: https://doi.org/10.1007/s11045-010-0145-9