Foundations of System Theory: The Hankel Matrix

Abstract After introducing the notion of “dynamical interpretation functor” to provide a general methodology for nonlinear state-space description, we define the Hankel matrix for an arbitrary adjoint system. This returns the usual definition for linear systems, but also applies to sequential machines, group machines, and bilinear machines. We provide row-recurrence and column-recurrence criteria for the “finite-dimensional” realizability of a Hankel matrix. Finally, we introduce the concept of subquotient ascendancy to define an absolute notion of finite rank in terms of which we analyze partial realizations of Hankel matrices.

[1]  Michael A. Arbib.  Fondations de la théorie de systèmes: Systèmes décomposablesGrundlagen der systemtheorie: Dekomponierbare systemeФopмиpoвaниe тeopии cиcтeм: дeкoмпoзaбeльнe cиcтeмы , 1974 .

[2]  H. Paul Zeiger,et al.  Ho's Algorithm, Commutative Diagrams, and the Uniqueness of Minimal Linear Systems , 1967, Inf. Control..

[3]  Joseph A. Goguen,et al.  Discrete-Time Machines in Closed Monoidal Categories. I , 1975, J. Comput. Syst. Sci..

[4]  Michael A. Arbib,et al.  Intertwined Recursion, Tree Transformations, and Linear Systems , 1979, Inf. Control..

[5]  Michael A. Arbib,et al.  Foundations of system theory: Decomposable systems , 1974, Autom..

[6]  A. Isidori.  Direct construction of minimal bilinear realizations from nonlinear input-output maps , 1973 .

[7]  J. R. Isbell,et al.  Some Remarks Concerning Categories and Subspaces , 1957, Canadian Journal of Mathematics.

[8]  M. Arbib,et al.  MACHINES IN A CATEGORY: AN EXPOSITORY INTRODUCTION* , 1974 .

[9]  M. Arbib.  A Common Framework for Automata Theory and Control Theory , 1965 .

[10]  G. Marchesini,et al.  Algebraic realization theory of bilinear discrete-time input-output maps☆ , 1976 .

[11]  Hartmut Ehrig,et al.  Universal Theory of Automata , 1976 .

[12]  Michel Fliess.  Un codage non commutatif pour certains systèmes échantillonnés non linéaires , 1978, Inf. Control..

[13]  J. Goguen.  Minimal realization of machines in closed categories , 1972 .

[14]  Michael A. Arbib,et al.  Foundations of system theory: multidecomposable systems☆ , 1976 .

[15]  S Maclane,et al.  Groups, Categories and Duality. , 1948, Proceedings of the National Academy of Sciences of the United States of America.

[16]  T. Tarn,et al.  Realization of discrete-time internally bilinear systems , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[17]  Joseph A. Goguen,et al.  Initial Algebra Semantics and Continuous Algebras , 1977, J. ACM.

[18]  M. Arbib,et al.  Arrows, Structures, and Functors: The Categorical Imperative , 1975 .

[19]  S. Lane.  Categories for the Working Mathematician , 1971 .

[20]  M. Arbib,et al.  Adjoint machines, state-behavior machines, and duality☆ , 1975 .

[21]  Michael A. Arbib,et al.  Generalized Hankel Matrices and System Realization , 1980 .

[22]  Michael A. Arbib.  Coproducts and Decomposable Machines , 1973, J. Comput. Syst. Sci..