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Parametric solutions to the regulator-conjugate matrix equations
1. | Institute of Data and Knowledge Engineering, Henan University, Kaifeng 475004, China |
2. | Institute of electric power, North China University of Water Resources and Electric Power, Zhengzhou 450011, China |
3. | Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China |
The problem of solving regulator-conjugate matrix equations is considered in this paper. The regulator-conjugate matrix equations are a class of nonhomogeneous equations. Utilizing several complex matrix operations and the concepts of controllability-like and observability-like matrices, a special solution to this problem is constructed, which includes solving an ordinary algebraic matrix. Combined with our recent results on Sylvester-conjugate matrix equations, complete solutions to regulator-conjugate matrix equations can be obtained by superposition principle. The correctness and effectiveness are verified by a numerical example.
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A. Wu, G. Duan and H. Yu,
On solutions of the matrix equations $XF-AX= C$ and $XF-A\bar{X}= C $, Applied Mathematics and Computation, 183 (2006), 932-941.
doi: 10.1016/j.amc.2006.06.039. |
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C. Yang, J. Liu and Y. Liu,
Solutions of the generalized Sylvester matrix equation and the application in eigenstructure assignment, Asian Journal of Control, 14 (2012), 1669-1675.
doi: 10.1002/asjc.448. |
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K. F. C. Yiu, K. L. Mak and K. L. Teo,
Airfoil design via optimal control theory, Journal of Industrial & Management Optimization, 1 (2005), 133-148.
doi: 10.3934/jimo.2005.1.133. |
[12] |
B. Zhou and G. Duan,
A new solution to the generalized Sylvester matrix equation AV-EVF= BW, Systems & Control Letters, 55 (2006), 193-198.
doi: 10.1016/j.sysconle.2005.07.002. |
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B. Zhou, G. Duan and Z. Li,
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show all references
References:
[1] |
P. Benner, J. R. Li and T. Penzl,
Numerical solution of large scale Lyapunov equations, Riccati equations, and linear quadratic optimal control problems, Numerical Linear Algebra with Applications, 15 (2008), 755-777.
doi: 10.1002/nla.622. |
[2] |
J. Bevis, F. Hall and R. Hartwig,
The matrix equation $A\bar{X}-XB=C$ and its special cases, SIAM Journal on Matrix Analysis and Applications, 60 (2010), 95-111.
|
[3] |
Y. Hong and R. Horn,
A canonical form for matrices under consimilarity, Linear Algebra and its Applications, 102 (1988), 143-168.
doi: 10.1016/0024-3795(88)90324-2. |
[4] |
T. Jiang and M. Wei,
On solutions of the matrix equations $X-AXB=C$ and $X-A\overline{X} B=C$, Linear Algebra and its Applications, 367 (2003), 225-233.
doi: 10.1016/S0024-3795(02)00633-X. |
[5] |
X. Jiang and Y. Zhang,
A smoothing-type algorithm for absolute value equations, Journal of Industrial & Management Optimization, 9 (2013), 789-798.
doi: 10.3934/jimo.2013.9.789. |
[6] |
A. Wu, G. Feng, G. Duan and and W. Wu,
Closed-form solutions to Sylvester-conjugate matrix equations, Computers & Mathematics with Applications, 60 (2010), 95-111.
doi: 10.1016/j.camwa.2010.04.035. |
[7] |
A. Wu and G. Duan,
Solution to the generalised Sylvester matrix equation AV+ BW= EVF, IET Control Theory & Applications, 1 (2007), 402-408.
doi: 10.1049/iet-cta:20050390. |
[8] |
A. Wu, L. Lv, G. Duan and W. Liu,
Parametric solutions to Sylvester-conjugate matrix equations, Computers & Mathematics with Applications, 62 (2011), 3317-3325.
doi: 10.1016/j.camwa.2011.08.034. |
[9] |
A. Wu, G. Duan and H. Yu,
On solutions of the matrix equations $XF-AX= C$ and $XF-A\bar{X}= C $, Applied Mathematics and Computation, 183 (2006), 932-941.
doi: 10.1016/j.amc.2006.06.039. |
[10] |
C. Yang, J. Liu and Y. Liu,
Solutions of the generalized Sylvester matrix equation and the application in eigenstructure assignment, Asian Journal of Control, 14 (2012), 1669-1675.
doi: 10.1002/asjc.448. |
[11] |
K. F. C. Yiu, K. L. Mak and K. L. Teo,
Airfoil design via optimal control theory, Journal of Industrial & Management Optimization, 1 (2005), 133-148.
doi: 10.3934/jimo.2005.1.133. |
[12] |
B. Zhou and G. Duan,
A new solution to the generalized Sylvester matrix equation AV-EVF= BW, Systems & Control Letters, 55 (2006), 193-198.
doi: 10.1016/j.sysconle.2005.07.002. |
[13] |
B. Zhou, G. Duan and Z. Li,
A Stein matrix equation approach for computing coprime matrix fraction description, IET Control Theory & Applications, 3 (2009), 691-700.
doi: 10.1049/iet-cta.2008.0128. |
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