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April  2017, 13(2): 623-631. doi: 10.3934/jimo.2016036

Parametric solutions to the regulator-conjugate matrix equations

Lei Zhang 1, , Anfu Zhu 2, , Aiguo Wu 3, and  Lingling Lv 2,,

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

* Corresponding author: Lingling Lv

Received  August 2015 Revised  December 2015 Published  May 2016

Fund Project: This work is supported by the Programs of National Natural Science Foundation of China (Nos. 61402149,11501200, U1604148), Innovative Talents of Higher Learning Institutions of Henan (No. 17HASTIT023), Program for Innovative Research Team in University of Henan Province (No. 16IRTSTHN017).

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.

Keywords: Parametric solutions, regulator-conjugate matrix equations, nonhomogeneous equations, conjugate operations, superposition principle.
Mathematics Subject Classification: 15A24.
Citation: Lei Zhang, Anfu Zhu, Aiguo Wu, Lingling Lv. Parametric solutions to the regulator-conjugate matrix equations. Journal of Industrial & Management Optimization, 2017, 13 (2) : 623-631. doi: 10.3934/jimo.2016036
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.  Google Scholar

[2]

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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