Matrix spectrum dichotomy and generalized Lyapunov matrix equation

2008 ◽  
pp. 95-102 ◽  
Author(s):  
Aider Ya. Bulgakov
Keyword(s):  
Matrix Equation ◽  
Lyapunov Matrix Equation ◽  
Lyapunov Matrix ◽  
Matrix Spectrum ◽  
Spectrum Dichotomy

Solutions to the matrix equation X − AXB = CY+R and its application

2016 ◽  
Vol 40 (3) ◽  
pp. 995-1004 ◽  
Author(s):  
Caiqin Song ◽  
Guoliang Chen
Keyword(s):  
Linear Systems ◽  
Matrix Equation ◽  
Controller Design ◽  
Equivalent Form ◽  
Closed Form Solutions ◽  
Lyapunov Matrix Equation ◽  
Special Cases ◽  
The Matrix ◽  
Lyapunov Matrix ◽  
Low Dimensional

The solution of the nonhomogeneous Yakubovich matrix equation [Formula: see text] is important in stability analysis and controller design in linear systems. The nonhomogeneous Yakubovich matrix equation [Formula: see text], which contains the well-known Kalman–Yakubovich matrix equation and the general discrete Lyapunov matrix equation as special cases, is investigated in this paper. Closed-form solutions to the nonhomogeneous Yakubovich matrix equation are presented using the Smith normal form reduction. Its equivalent form is provided. Compared with the existing method, the method presented in this paper has no limit to the dimensions of an unknown matrix. The present method is suitable for any unknown matrix, not only low-dimensional unknown matrices, but also high-dimensional unknown matrices. As an application, parametric pole assignment for descriptor linear systems by PD feedback is considered.

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