Diophantine equations in control - A survey

Abstract This survey is also a tutorial whose aim is to explain the role of diophantine equations in the synthesis of feedback control systems. These are linear equations in a ring and result from a fractional representation of the systems involved. The cornerstone of the exposition is a simple parametrization of all stabilizing controllers for a given plant. One can then choose, in principle, the best controllers for various applications. These ideas evolved from early attempts to use polynomial equations in the design of discrete-time linear systems. By now they have been extended to continuous-time, infinite-dimensional, time-varying and non-linear systems.

[1]  M. Sebek,et al.  Polynomial solution of 2D Kalman-Bucy filtering problem , 1992 .

[2]  C. Desoer,et al.  Algebraic theory for robust stability of interconnected systems: Necessary and sufficient conditions , 1982, 1982 21st IEEE Conference on Decision and Control.

[3]  Stephen P. Boyd,et al.  Linear controller design: limits of performance , 1991 .

[4]  Vladimír Kučera,et al.  Stationary LQG control of singular systems , 1986 .

[5]  V. Peterka,et al.  Predictor-based self-tuning control , 1982, Autom..

[6]  Václav Peterka,et al.  On steady state minimum variance control strategy , 1972, Kybernetika.

[7]  O. Zariski,et al.  Commutative Algebra, Vol. I , 1959 .

[8]  Vladimír Strejc,et al.  Synthese von Regelungssystemen mit Prozessrechner , 1967 .

[9]  T. Kaczorek.  Two-Dimensional Linear Systems , 1985 .

[10]  M. Šebek.  n-D polynomial matrix equations , 1988 .

[11]  F. J. Kraus,et al.  FIFO Stable Control Systems , 1993 .

[12]  V. Kučera.  Exact model matching, polynomial equation approach , 1981 .

[13]  L. Pernebo.  An algebraic theory for design of controllers for linear multivariable systems--Part I: Structure matrices and feedforward design , 1981 .

[14]  Jacob Hammer,et al.  Fraction representations of non-linear systems: a simplified approach , 1987 .

[15]  Vladimír Kucera,et al.  Closed-loop stability of discrete linear single-variable systems , 1974, Kybernetika.

[16]  Michael J. Grimble,et al.  Optimal multivariable LQG control using a single diophantine equation , 1987 .

[17]  Vladimír Kucera.  Algebraic theory of discrete optimal control for multivariable systems [I.] , 1974, Kybernetika.

[18]  Vladimír Kucera.  Transfer-function solution of the Kalman-Bucy filtering problem , 1978, Kybernetika.

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

[20]  M. Vidyasagar,et al.  Algebraic and topological aspects of feedback stabilization , 1980 .

[21]  Bruce A. Francis,et al.  The internal model principle of control theory , 1976, Autom..

[22]  L. Pernebo.  An Algebraic Theory for the Design of Controllers for Linear Multirate Systems Part I: Structure Matrices and Feedforward Design , 1981 .

[23]  G. Stein,et al.  Multivariable feedback design: Concepts for a classical/modern synthesis , 1981 .

[24]  Vladimír Kučera.  Shortest correlation control strategy , 1977 .

[25]  V. Kučera.  Stability of Discrete Linear Feedback Systems , 1975 .

[26]  Vladimír Kucera.  Algebraic theory of discrete optimal control for single-variable systems. I. Preliminaries , 1973, Kybernetika.

[27]  M. Vidyasagar.  Control System Synthesis : A Factorization Approach , 1988 .

[28]  C. Desoer,et al.  The feedback interconnection of lumped linear time-invariant systems☆ , 1975 .

[29]  M. M. Newmann,et al.  Polynomial approach to Wiener filtering , 1988 .

[30]  Anders Ahlén,et al.  Wiener filter design using polynomial equations , 1991, IEEE Trans. Signal Process..

[31]  C. Desoer,et al.  Multivariable Feedback Systems , 1982 .

[32]  R. Kálmán.  Mathematical description of linear dynamical systems , 1963 .

[33]  M. Grimble.  Polynomial systems approach to optimal linear filtering and prediction , 1985 .

[34]  Michael J. Grimble,et al.  Implicit and explicit LQG self-tuning controllers , 1984, Autom..

[35]  A. S. Morse.  Ring Models for Delay-Differential Systems , 1974 .

[36]  Michael Sebek,et al.  A polynomial solution to regulation and tracking. II. Stochastic problem , 1984, Kybernetika.

[37]  J. Hammer.  Stabilization of non-linear systems† , 1986 .

[38]  Vladimír Kucera.  Internal properness and stability in linear systems , 1986, Kybernetika.

[39]  M. Heymann,et al.  Linear Feedback—An Algebraic Approach , 1978 .

[40]  Michael Sebek,et al.  A polynomial solution to regulation and tracking. I. Deterministic problem , 1984, Kybernetika.

[41]  Kenneth J. Hunt.  Stochastic Optimal Control Theory with Application in Self-Tuning Control , 1989 .

[42]  K. J. Hunt,et al.  The standard ℋ2-optimal control problem: a polynomial solution , 1992 .

[43]  Vladimír Kučera,et al.  Analysis and design of discrete linear control systems , 1991 .

[44]  E. Kamen,et al.  A transfer function approach to linear time-varying discrete-time systems , 1982, 1982 21st IEEE Conference on Decision and Control.

[45]  David Clarke,et al.  Self-tuning control , 1979 .

[46]  N. Jacobson.  Lectures In Abstract Algebra , 1951 .

[47]  Michael Sebek,et al.  Polynomial design of stochastic tracking systems , 1982 .

[48]  Vladimír Kucera,et al.  Algebraic theory of discrete optimal control for single-variable systems. II. Open-loop control , 1973, Kybernetika.

[49]  Vladimír Kucera,et al.  Linear quadratic control. State space vs. polynomial equations , 1983, Kybernetika.

[50]  C. Desoer,et al.  An algebra of transfer functions for distributed linear time-invariant systems , 1978 .

[51]  Vladimír Ku Era.  A dead-beat servo problem , 1980 .

[52]  H. Kushner.  Introduction to Stochastic Control Theory (Karl J. Astrom) , 1972 .

[53]  Michael J. Grimble,et al.  Optimal H∞ robustness and the relationship to LQG design problems , 1986 .

[54]  C. Desoer,et al.  Feedback Systems: Input-Output Properties , 1975 .

[55]  F. Lewis.  A survey of linear singular systems , 1986 .

[56]  Michael J. Grimble.  H/sub infinity / filtering problem and the relationship between polynomial and state-space results , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[57]  H. Kwakernaak.  The polynomial approach to H ???-optimal regulation , 1991 .

[58]  Julius T. Tou,et al.  Digital and sampled-data control systems , 1963 .

[59]  Bruce A. Francis,et al.  Algebraic and topological aspects of the regulator problem for lumped linear systems , 1983, Autom..

[60]  Björn Wittenmark,et al.  On Self Tuning Regulators , 1973 .

[61]  Hartmut Logemann,et al.  Multivariable feedback design : J. M. Maciejowski , 1991, Autom..

[62]  R. Chabour,et al.  Stabilization of nonlinear systems: A bilinear approach , 1993, Math. Control. Signals Syst..

[63]  Dante C. Youla,et al.  Modern Wiener--Hopf design of optimal controllers Part I: The single-input-output case , 1976 .

[64]  John B. Moore,et al.  On the Youla-Kucera parametrization for nonlinear systems , 1990 .

[65]  Jan Jezek.  An algebraic approach to the synthesis of control for linear discrete meromorphic systems , 1989, Kybernetika.

[66]  E. Kamen,et al.  Stabilization of time-delay systems using finite-dimensional compensators , 1985, IEEE Transactions on Automatic Control.

[67]  Edoardo Mosca,et al.  On the polynomial equations for the MIMO LQ stochastic regulator , 1990 .

[68]  Vladimír Kučera,et al.  Stochastic multivariable control: A polynomial equation approach , 1980 .

[69]  C. Desoer,et al.  Feedback system design: The fractional representation approach to analysis and synthesis , 1979, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[70]  H. Kwakernaak.  Minimax frequency domain performance and robustness optimization of linear feedback systems , 1985 .

[71]  Jan C. Willems,et al.  From time series to linear system - Part I. Finite dimensional linear time invariant systems , 1986, Autom..

[72]  T. Kailath,et al.  A generalized state-space for singular systems , 1981 .

[73]  Michael Sebek.  Characteristic polynomial assignment for delay-differential systems via 2-D polynomial equations , 1986, 1986 25th IEEE Conference on Decision and Control.

[74]  Jan C. Willems,et al.  Models for Dynamics , 1989 .

[75]  Vladimír Kučera,et al.  Disturbance rejection: A polynomial approach , 1983 .

[76]  Kameshwar Poolla,et al.  Stabilizability and stable-proper factorizations for linear time-varying systems , 1987 .

[77]  Karl Johan Åström,et al.  Robustness of a design method based on assignment of poles and zeros , 1980 .

[78]  V. Kučera,et al.  Discrete Linear Control: The Polynomial Equation Approach , 1981, IEEE Transactions on Systems, Man, and Cybernetics.

[79]  Minghua Chen,et al.  Algebraic theory for robust stability of interconnected systems: Necessary and sufficient conditions , 1982, CDC 1982.

[80]  Michael Sebek,et al.  On deadbeat controllers , 1984 .

[81]  M. Fliess.  Some basic structural properties of generalized linear systems , 1991 .

[82]  Anders Ahlén,et al.  Optimal deconvolution based on polynomial methods , 1989, IEEE Trans. Acoust. Speech Signal Process..

[83]  M. Hautus.  The Formal Laplace Transform for Smooth Linear Systems , 1976 .

[84]  W. Wonham.  Linear Multivariable Control: A Geometric Approach , 1974 .