Linear Systems Over Rings: From R. E. Kalman to the Present

The paper begins with a brief history of the role played by R.E. Kalman in the establishment of the field of linear systems over a commutative ring. The theory of systems over rings is motivated by considering integer systems, systems with time delays, parameter-dependent systems, and multidimensional systems including spatially-distributed systems. Then a brief survey is given of existing work on the control of systems over rings along with some discussion of open research problems.

[1]  Edward W. Kamen,et al.  Module Structure of Infinite-Dimensional Systems with Applications to Controllability , 1976 .

[2]  R. Kálmán.  ALGEBRAIC STRUCTURE OF LINEAR DYNAMICAL SYSTEMS, I. THE MODULE OF Sigma. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[3]  P. Fatou.  Séries trigonométriques et séries de Taylor , 1906 .

[4]  John W. Bunce,et al.  Linear Systems over Commutative Rings , 1986 .

[5]  Allen Tannenbaum,et al.  Polynomial rings over arbitrary fields in two or more variables are not pole assignable , 1982 .

[6]  Christopher Byrnes.  Realization theory and quadratic optimal controllers for systems defined over Banach and Frechét algebras , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[7]  Michael A. Arbib,et al.  Topics in Mathematical System Theory , 1969 .

[8]  R.W. Brodersen,et al.  Applications of CCD and switched capacitor filter technology , 1979, Proceedings of the IEEE.

[9]  E. Emre,et al.  Regulation of linear systems over rings by dynamic output feedback , 1983 .

[10]  A. Stephen Morse.  Ring models for delay-differential systems , 1976, Autom..

[11]  Eduardo D. Sontag,et al.  REMARKS ON THE POLE-SHIFTING PROBLEM OVER RINGS , 1981 .

[12]  R. Brockett,et al.  Toda flows, inverse spectral transform and realization theory , 1991 .

[13]  R. W. Newcomb,et al.  ON THE REALIZATION OF MULTIVARIABLE TRANSFER FUNCTIONS. , 1966 .

[14]  C. Byrnes,et al.  On the stabilizability of linear control systems depending on parameters , 1979, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[15]  Edward W. Kamen,et al.  An operator theory of linear functional differential equations , 1978 .

[16]  Bostwick F. Wyman,et al.  Linear Dynamical Systems over Integral Domains , 1974, J. Comput. Syst. Sci..

[17]  Pramod P. Khargonekar,et al.  Regulation of split linear systems over rings: Coefficient-assignment and observers , 1982 .

[18]  R. Eising,et al.  Realization and stabilization of 2-d systems , 1978 .

[19]  R. E. Kalman,et al.  REALIZATION OF CONTINUOUS-TIME LINEAR DYNAMICAL SYSTEMS: RIGOROUS THEORY IN THE STYLE OF SCHWARTZ , 1972 .

[20]  E. Kamen.  Stabilization of Linear Spatially-Distributed Continuous-Time and Discrete-Time Systems , 1985 .

[21]  Eduardo D. Sontag.  On Split Realizations of Response Maps over Rings , 1978, Inf. Control..

[22]  E. Kamen,et al.  Asymptotic stability of linear shift-invariant two-dimensional digital filters , 1980 .

[23]  G. Naudé,et al.  Comments on pole assignability over rings , 1985 .

[24]  Yutaka Yamamoto,et al.  Module structure of constant linear systems and its applications to controllability , 1981 .

[25]  Paul-Jean Cahen,et al.  Éléments quasi-entiers et extensions de fatou , 1975 .

[26]  Pramod P. Khargonekar,et al.  On the relation between stable matrix fraction factorizations and regulable realizations of linear systems over rings , 1981, CDC 1981.

[27]  Eduardo D. Sontag,et al.  On first-order equations for multidimensional filters , 1978 .

[28]  Eduardo Sontag.  Linear Systems over Commutative Rings. A Survey , 1976 .

[29]  A. Olbrot,et al.  On reachability over polynomial rings and a related genericity problem , 1982 .

[30]  W. Schmale.  Three-dimensional feedback cyclization over C y , 1989 .

[31]  V. Zakian,et al.  A Ring of Delay Operators with Applications to Delay-Differential Systems , 1977 .

[32]  Pramod P. Khargonekar.  On Matrix Fraction Representations for Linear Systems over Commutative Rings , 1982 .

[33]  W. Schmale.  Feedback cyclization over certain principal ideal domains , 1988 .

[34]  E. Kamen,et al.  Stabilizability of Linear Systems Over a Commutative Normed Algebra with Applications to Spatially-Distributed and Parameter-Dependent Systems , 1985 .

[35]  R. P. Kurshan,et al.  Digital single-tone generator-detectors , 1976, The Bell System Technical Journal.

[36]  Christopher I. Byrnes,et al.  Algebraic and Geometric Aspects of the Analysis of Feedback Systems , 1980 .

[37]  P. K. Sharma.  Some results on pole-placement and reachability , 1986 .

[38]  E. Kamen.  Lectures on algebraic system theory: Linear systems over rings , 1978 .

[39]  W. Wonham,et al.  On pole assignment in multi-input controllable linear systems , 1967, IEEE Transactions on Automatic Control.

[40]  Edward W. Kamen.  A Distributional-Module Theoretic Representation of Linear Dynamical Continuous-Time Systems. , 1971 .

[41]  E. Emre,et al.  On Necessary and Sufficient Conditions for Regulation of Linear Systems Over Rings , 1982 .

[42]  R E Kalman,et al.  Algebraic Structure of Linear Dynamical Systems. III. Realization Theory Over a Commutative Ring. , 1972, Proceedings of the National Academy of Sciences of the United States of America.

[43]  Giuseppe Conte,et al.  On the zeros and poles of a transfer function , 1989 .