PARAMETRIC SOLUTIONS TO THE GENERALIZED SYLVESTER MATRIX EQUATION AX ‐ XF = BY AND THE REGULATOR EQUATION AX ‐ XF = BY + R

In this paper, explicit parametric solutions to the generalized Sylvester matrix equation AX - XF = BY and the regulator matrix equation AX - XF = BY + R are proposed without any transformation and factorization. The proposed solutions are presented in terms of the Krylov matrix of matrix pair (A, B), a symmetric operator and the generalized observability matrix of matrix pair (Z, F) where Z is an arbitrary matrix and is used to denote the degree of freedom in the solution. Due to its elegant form and convenient computation, these proposed solutions will play an important role in solving and analyzing these types of equations in control systems theory.

[1]  B. Porter,et al.  Design of linear multivariable continuous-time tracking systems incorporating feedforward and feedback controllers , 1975 .

[2]  Shankar P. Bhattacharyya,et al.  Controllability, observability and the solution of AX - XB = C , 1981 .

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

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

[5]  Basil G. Mertzios.  Pole assignment of two-dimensional systems for separable characteristic equations , 1984 .

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

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

[8]  Shankar P. Bhattacharyya,et al.  Robust and Well Conditioned Eigenstructure Assignment via Sylvester's Equation , 1982 .

[9]  Guang-Ren Duan,et al.  The solution to the matrix equation AV + BW = EVJ + R , 2004, Appl. Math. Lett..

[10]  S. Nash,et al.  Approaches to robust pole assignment , 1989 .

[11]  R. J. Patton,et al.  Robust fault detection in linear systems using Luenberger observers , 1998 .

[12]  W. Q. Liu,et al.  Robust Model Reference Control for Multivariable Linear Systems: A Parametric Approach , 2000 .

[13]  Guo-Ping Liu,et al.  Eigenstructure assignment design for proportional-integral observers: continuous-time case , 2001 .

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

[15]  Guang-Ren Duan,et al.  Robust eigenstructure assignment via dynamical compensators, , 1993, Autom..

[16]  Guang-Ren Duan,et al.  Complete parametric approach for eigenstructure assignment in a class of second-order linear systems , 1999, Autom..

[17]  Ali Saberi,et al.  Control of Linear Systems with Regulation and Input Constraints , 2000 .

[18]  Chia-Chi Tsui,et al.  On the solution to the matrix equation TA−FT = LC and its applications , 1993 .

[19]  Jie Chen,et al.  Design of unknown input observers and robust fault detection filters , 1996 .

[20]  Guang-Ren Duan,et al.  Robust fault detection in descriptor linear systems via generalized unknown input observers , 2002, Int. J. Syst. Sci..

[21]  R. Bhatia.  Matrix Analysis , 1996 .

[22]  Paul Van Dooren.  Reduced order observers: A new algorithm and proof , 1984 .

[23]  Mihail M. Konstantinov,et al.  Synthesis of linear systems with desired equivalent form , 1980 .

[24]  G. Duan,et al.  Simple algorithm for robust pole assignment in linear output feedback , 1991 .

[25]  Chia-Chi Tsui,et al.  New approach to robust observer design , 1988 .

[26]  N. Loh,et al.  Design of observers for two-dimensional systems , 1985, 1985 24th IEEE Conference on Decision and Control.