Feedback stabilization of MIMO 3-D linear systems

The authors solve the open problem of the existence of double coprime factorizations for a large class of multi-input/multi-output (MIMO) three-dimensional (3-D) linear systems. It is proven that if all the unstable zeros of the contents associated with left and right matrix fraction descriptions of a given feedback stabilizable causal MIMO 3-D plant are simple, then the plant has a double coprime factorization. The authors then give a parameterization of all stabilizing compensators for a MIMO 3-D system in this class. The key result developed in the paper is a novel and constructive technique of "replacing" an unstable polynomial with a stable polynomial step by step. An illustrative example is also provided.

[1]  Dante C. Youla,et al.  The Quillen - Suslin theorem and the structure of n-dimensional elementary polynomial matrices , 1984 .

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

[3]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[4]  Virendra Sule,et al.  Feedback Stabilization Over Commutative Rings: The Matrix Case , 1994 .

[5]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[6]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[7]  Dante C. Youla,et al.  Modern Wiener-Hopf Design of Optimal Controllers. Part I , 1976 .

[8]  Venkat Anantharam.  On stabilization and the existence of coprime factorizations , 1985 .

[9]  Li Xu,et al.  Output feedback stabilizability and stabilization algorithms for 2D systems , 1994, Multidimens. Syst. Signal Process..

[10]  Zhiping Lin.  On matrix fraction descriptions of multivariable linear n-D systems , 1988 .

[11]  N. Bose.  Multidimensional Systems Theory , 1985 .

[12]  Zhiping Lin,et al.  Feedback Stabilizability of MIMO n-D Linear Systems , 1998, Multidimens. Syst. Signal Process..

[13]  Jiang Qian Ying,et al.  Conditions for Strong Stabilizabilities of n-Dimensional Systems , 1998, Multidimens. Syst. Signal Process..

[14]  Zhiping Lin.  On the elementary factors of linear systems over unique factorization domains , 1999 .

[15]  R. Cuninghame-Green,et al.  Applied Linear Algebra , 1979 .

[16]  N. Bose.  Applied multidimensional systems theory , 1982 .

[17]  Graham Goodwin,et al.  Discrete time multivariable adaptive control , 1979, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[18]  Zhiping Lin,et al.  Feedback stabilization of MIMO nD linear systems , 2000, IEEE Trans. Autom. Control..

[19]  J. Geromel.  Convex analysis and global optimization of joint actuator location and control problems , 1989 .

[20]  Dante C. Youla,et al.  Notes on n-Dimensional System Theory , 1979 .

[21]  Sun-Yuan Kung,et al.  New results in 2-D systems theory, part I: 2-D polynomial matrices, factorization, and coprimeness , 1977, Proceedings of the IEEE.

[22]  H. Weber,et al.  Analysis and optimization of certain qualities of controllability and observability for linear dynamical systems , 1972 .

[23]  M. J. Carpenter,et al.  Optimal Redesign of Linear Systems , 1993, 1993 American Control Conference.

[24]  Carlos A. Berenstein,et al.  I-Inverness for polynomial matrices of non-constant rank , 1986 .

[25]  Ettore Fornasini,et al.  nD Polynomial Matrices with Applications to Multidimensional Signal Analysis , 1997, Multidimens. Syst. Signal Process..

[26]  J. Guiver,et al.  Causal and Weakly Causal 2-D Filters with Applications in Stabilization , 1995 .

[27]  Zhiping Lin,et al.  Notes on n-D Polynomial Matrix Factorizations , 1999, Multidimens. Syst. Signal Process..

[28]  J. Guiver,et al.  Polynomial matrix primitive factorization over arbitrary coefficient field and related results , 1982 .

[29]  Zhiping Lin.  On primitive factorizations for 3-D polynomial matrices , 1992 .

[30]  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.

[31]  José Claudio Geromel,et al.  H2-norm optimization with constrained dynamic output feedback controllers: decentralized and reliable control , 1999, IEEE Trans. Autom. Control..

[32]  Zhiping Lin,et al.  Feedback stabilization of multivariable two-dimensional linear systems , 1988 .

[33]  Shin-Ju Chen,et al.  Robustness analysis of uncertain linear singular systems with output feedback control , 1999, IEEE Trans. Autom. Control..

[34]  C. Byrnes,et al.  Frequency domain and state space methods for linear systems , 1986 .

[35]  P. Peres,et al.  On a convex parameter space method for linear control design of uncertain systems , 1991 .

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