Unified parametrization for the solutions to the polynomial Diophantine matrix equation and the generalized Sylvester matrix equation

The polynomial Diophantine matrix equation and the generalized Sylvester matrix equation are important for controller design in frequency domain linear system theory and time domain linear system theory, respectively. By using the so-called generalized Sylvester mapping, right coprime factorization and Bezout identity associated with certain polynomial matrices, we present in this note a unified parametrization for the solutions to both of these two classes of matrix equations. Moreover, it is shown that solutions to the generalized Sylvester matrix equation can be obtained if solutions to the Diophantine matrix equation are available. The results disclose a relationship between the polynomial Diophantine matrix equation and generalized Sylvester matrix equation that are respectively studied and used in frequency domain linear system theory and time domain linear system theory.

[1]  W. A. Wolovich,et al.  The Canonical Diophantine Equations with Applications , 1983, 1983 American Control Conference.

[2]  Shankar P. Bhattacharyya,et al.  Robust and well‐conditioned eigenstructure assignment via sylvester's equation , 1983 .

[3]  J. Feinstein,et al.  The solution of the matrix polynomial equation A(s)X(s) + B(s)Y(s) = C(s) , 1984 .

[4]  Myung-Joong Youn,et al.  Eigenvalue-generalized eigenvector assignment by output feedback , 1987 .

[5]  Chia-Chi Tsui,et al.  A complete analytical solution to the equation TA - FT = LC and its applications , 1987 .

[6]  Y. Lai,et al.  An algorithm for solving the matrix polynomial equation B(s)D(s)+A(s)N(s)=H(s) , 1989 .

[7]  Michael Sebek,et al.  Two-sided equations and skew primeness for n -D polynomial matrices , 1989 .

[8]  Chun Hsiung Fan,et al.  A novel approach for solving Diophantine equations , 1990 .

[9]  K. J. Hunt,et al.  Implied polynomial matrix equations in multivariable stochastic optimal control , 1991, Autom..

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

[11]  C. Fang,et al.  A simple approach to solving the Diophantine equation , 1992 .

[12]  Huibert Kwakernaak,et al.  Robust control and H∞-optimization - Tutorial paper , 1993, Autom..

[13]  G. Duan.  Solutions of the equation AV+BW=VF and their application to eigenstructure assignment in linear systems , 1993, IEEE Trans. Autom. Control..

[14]  K. J. Hunt,et al.  Polynomial LQ optimization for the standard control structure: Scalar solution , 1993, Autom..

[15]  Vladimír Kučera,et al.  Diophantine equations in control - A survey , 1993, Autom..

[16]  G. Rizzoni,et al.  An eigenstructure assignment algorithm for the design of fault detection filters , 1994, IEEE Trans. Autom. Control..

[17]  M. B. Estrada,et al.  Solving the right diophantine equation in a geometric way , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[18]  Manabu Yamada,et al.  On solving Diophantine equations by real matrix manipulation , 1995, IEEE Trans. Autom. Control..

[19]  Biswa Nath Datta,et al.  Numerically robust pole assignment for second-order systems , 1996 .

[20]  Guang-Ren Duan,et al.  On the solution to the Sylvester matrix equation AV+BW=EVF , 1996, IEEE Trans. Autom. Control..

[21]  Yasuyuki Funahashi,et al.  Stability robustness for linear state space models—a Lyapunov mapping approach , 1997 .

[22]  Michael Sebek,et al.  Symmetric Matrix Polynomial Equation: Interpolation Results , 1998, Autom..

[23]  Juan Carlos Martínez-García,et al.  Fixed poles and disturbance rejecting feedback synthesis , 1997, Autom..

[24]  A. B. Ogunye.  Solution of unilateral and bilateral Diophantine equations using symbolic computation , 1999, Proceedings of the 1999 IEEE International Symposium on Computer Aided Control System Design (Cat. No.99TH8404).

[25]  V. Kučera,et al.  A bridge between state-space and transfer-function methods , 1999 .

[26]  Daizhan Cheng,et al.  An algorithm for common quadratic Lyapunov function , 2000, Proceedings of the 3rd World Congress on Intelligent Control and Automation (Cat. No.00EX393).

[27]  Sophie Tarbouriech,et al.  Control of linear systems subject to input constraints: a polynomial approach , 2001, Autom..

[28]  R. Patton,et al.  Robust fault detection using Luenberger-type unknown input observers-a parametric approach , 2001 .

[29]  Huang Liping,et al.  To solve matrix equation ΣA i XB i =C by the Smith normal form , 2002 .

[30]  Liping Huang,et al.  To solve matrix equation ΣAiXBi=C by the Smith normal form , 2002 .

[31]  GuangrenDUAN.  Two parametric approaches for eigenstructure assignment in second-order linear systems , 2003 .

[32]  Dimitri Peaucelle,et al.  Positive polynomial matrices and improved LMI robustness conditions , 2003, Autom..

[33]  Feng Ding,et al.  Iterative least-squares solutions of coupled Sylvester matrix equations , 2005, Syst. Control. Lett..

[34]  G. Duan,et al.  An explicit solution to the matrix equation AX − XF = BY , 2005 .

[35]  Eugênio B. Castelan,et al.  On the solution of a Sylvester equation appearing in descriptor systems control theory , 2005, Syst. Control. Lett..

[36]  M. Malabre,et al.  Solving the Diophantine equation by state space inversion techniques: an illustrative example , 2006, 2006 American Control Conference.

[37]  Bin Zhou,et al.  A new solution to the generalized Sylvester matrix equation AV-EVF=BW , 2006, Syst. Control. Lett..

[38]  Guang-Ren Duan,et al.  Parametric approach for the normal Luenberger function observer design in second-order descriptor linear systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[39]  Mohamed Darouach.  Solution to Sylvester equation associated to linear descriptor systems , 2006, Syst. Control. Lett..

[40]  Guang-Ren Duan,et al.  Solution to the second-order Sylvester matrix equation MVF/sup 2/+DVF+KV=BW , 2006, IEEE Transactions on Automatic Control.

[41]  P. A. Tzekis,et al.  A new algorithm for the solution of a polynomial matrix Diophantine equation , 2007, Appl. Math. Comput..

[42]  Guang-Ren Duan,et al.  Solutions to generalized Sylvester matrix equation by Schur decomposition , 2007, Int. J. Syst. Sci..

[43]  Guang-Ren Duan,et al.  An explicit solution to the matrix equation AV+BW=EV J , 2007 .

[44]  Guang-Ren Duan,et al.  On the generalized Sylvester mapping and matrix equations , 2008, Syst. Control. Lett..