Parameterized Solution to a Class of Sylvester Matrix Equations

Yu-Peng Qiao; Hong-Sheng Qi; Dai-Zhan Cheng

doi:10.1007/s11633-010-0530-8

November 2010

Article Contents

Yu-Peng Qiao, Hong-Sheng Qi and Dai-Zhan Cheng. Parameterized Solution to a Class of Sylvester Matrix Equations. International Journal of Automation and Computing, vol. 7, no. 4, pp. 479-483, 2010. doi: 10.1007/s11633-010-0530-8

Cite as: Yu-Peng Qiao, Hong-Sheng Qi and Dai-Zhan Cheng. Parameterized Solution to a Class of Sylvester Matrix Equations. International Journal of Automation and Computing, vol. 7, no. 4, pp. 479-483, 2010. doi: 10.1007/s11633-010-0530-8

Parameterized Solution to a Class of Sylvester Matrix Equations

1.
Center for Control and Optimization, College of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, PRC;
2.
Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PRC

Author Biography:

Hong-Sheng Qi received the Ph.D.degree in systems theory from Academy of Mathematics and Systems Science,Chinese Academy of Sciences,PRC in 2008.
Corresponding author: Yu-Peng Qiao graduated from Harbin Institute of Technology(HIT),PRC in 2003.

Received: 2009-02-20

Fund Project:

supported by National Natural Science Foundation of China (No.60736022, No.60821091)

Abstract

A class of formulas for converting linear matrix mappings into conventional linear mappings are presented.Using them,an easily computable numerical method for complete parameterized solutions of the Sylvester matrix equation AX-EXF=BY and its dual equation XA-FXE=YC are provided.It is also shown that the results obtained can be used easily for observer design.The method proposed in this paper is universally applicable to linear matrix equations.
- Sylvester matrix equation,
- parameterized solution,
- Kronecker product,
- linear matrix equation,
- Luenberger observers

References

[1]	G.R.Duan,N.K.Nichols,G.P.Liu,T.J.Owens.Robust pole assignment in descriptor linear systems via state feed-back.European Journal of Control,vol.8,no.2,pp.136-151,2002.
[2]	R.K.Cavin III,S.P.Bhattacharyya.Robust and well-conditioned eigenstructure assignment via Sylvester s equa-tion.Optimal Control Applications Method,vol.4,no.3, pp.205-212,1983.
[3]	I.Jonsson,Bo Kågström.Recursive blocked algorithms for solving triangular systems—Part I:One-sided and cou-pled Sylvester-type matrix equations.ACM Transactions on Mathematical Software,vol.28,no.4,pp.392-415,2002.
[4]	I.Jonsson,Bo K?agstro¨m.Recursive Blocked Algorithms for Solving Triangular Systems—Part II:Two-sided and generalized Sylvester and Lyapunov matrix equations. ACM Transactions on Mathematical Software,vol.28,no.4, pp.416-435,2002.
[5]	B.H.Kwon,M.J.Youn.Eigenvalue-generalized eigenvec-tor assignment by output feedback.IEEE Transactions on Automatic Control,vol.32,no.5,pp.417-421,1987.
[6]	G.R.Duan.Solution to matrix equation AV+BW=EV F and eigenstructure assignment for descriptor systems.Au-tomatica,vol.28,no.3,pp.639-642,1992.
[7]	G.R.Duan.On the solution to Sylvester matrix equation AV+BW=EV F.IEEE Transactions on Automatic Con-trol,vol.41,no.4,pp.612-614,1996.
[8]	B.Zhou,G.R.Duan.General solutions to the matrix equa-tion AX?EXF=BY and their applications.Control Theory&Applications,vol.24,no.2,pp.193-199,2007. (in Chinese)
[9]	D.Cheng,H.Qi.Semi-tensor Product of Matrices,Theory and Applications,Beijing,PRC:Science Press,2007.(in Chinese)
[10]	D.Cheng,X.Hu,Y.Wang.Non-regular feedback lineariza-tion of nonlinear systems via a normal form algorithm.Au-tomatica,vol.40,no.3,pp.439-447,2004.
[11]	B.Zhou,G.R.Duan.A new solution to the generalized Sylvester matrix equation AV?EV F=BW.Systems and Control Letters,vol.55,no.3,pp.193-198,2006.
[12]	L.Dai.Singular Control Systems,Lecture Notes in Con-trol and Information Science,Berlin,Germany:Springer-Verlag,1989.
[13]	G.R.Duan,G.S.Wang.Eigenstructure assignment in a class of second-order descriptor linear systems:A complete parametric approach.International Journal of Automation and Computing,vol.2,no.1,pp.1-5,2005.
[14]	B.Zhou,G.R.Duan,Y.L.Wu.Parametric approach for the normal Luenberger function observer design in second-order descriptor linear systems.International Jour-nal of Automation and Computing,vol.5,no.2,pp.125-131,2008.

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通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

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Parameterized Solution to a Class of Sylvester Matrix Equations

1. Center for Control and Optimization, College of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, PRC;

2. Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PRC

Corresponding author: Yu-Peng Qiao graduated from Harbin Institute of Technology(HIT),PRC in 2003.

Received: 2009-02-20

Fund Project:

supported by National Natural Science Foundation of China (No.60736022, No.60821091)

Key words:

Abstract: A class of formulas for converting linear matrix mappings into conventional linear mappings are presented.Using them,an easily computable numerical method for complete parameterized solutions of the Sylvester matrix equation AX-EXF=BY and its dual equation XA-FXE=YC are provided.It is also shown that the results obtained can be used easily for observer design.The method proposed in this paper is universally applicable to linear matrix equations.

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Reference (14)

[1]	G.R.Duan,N.K.Nichols,G.P.Liu,T.J.Owens.Robust pole assignment in descriptor linear systems via state feed-back.European Journal of Control,vol.8,no.2,pp.136-151,2002.
[2]	R.K.Cavin III,S.P.Bhattacharyya.Robust and well-conditioned eigenstructure assignment via Sylvester s equa-tion.Optimal Control Applications Method,vol.4,no.3, pp.205-212,1983.
[3]	I.Jonsson,Bo Kågström.Recursive blocked algorithms for solving triangular systems—Part I:One-sided and cou-pled Sylvester-type matrix equations.ACM Transactions on Mathematical Software,vol.28,no.4,pp.392-415,2002.
[4]	I.Jonsson,Bo K?agstro¨m.Recursive Blocked Algorithms for Solving Triangular Systems—Part II:Two-sided and generalized Sylvester and Lyapunov matrix equations. ACM Transactions on Mathematical Software,vol.28,no.4, pp.416-435,2002.
[5]	B.H.Kwon,M.J.Youn.Eigenvalue-generalized eigenvec-tor assignment by output feedback.IEEE Transactions on Automatic Control,vol.32,no.5,pp.417-421,1987.
[6]	G.R.Duan.Solution to matrix equation AV+BW=EV F and eigenstructure assignment for descriptor systems.Au-tomatica,vol.28,no.3,pp.639-642,1992.
[7]	G.R.Duan.On the solution to Sylvester matrix equation AV+BW=EV F.IEEE Transactions on Automatic Con-trol,vol.41,no.4,pp.612-614,1996.
[8]	B.Zhou,G.R.Duan.General solutions to the matrix equa-tion AX?EXF=BY and their applications.Control Theory&Applications,vol.24,no.2,pp.193-199,2007. (in Chinese)
[9]	D.Cheng,H.Qi.Semi-tensor Product of Matrices,Theory and Applications,Beijing,PRC:Science Press,2007.(in Chinese)
[10]	D.Cheng,X.Hu,Y.Wang.Non-regular feedback lineariza-tion of nonlinear systems via a normal form algorithm.Au-tomatica,vol.40,no.3,pp.439-447,2004.
[11]	B.Zhou,G.R.Duan.A new solution to the generalized Sylvester matrix equation AV?EV F=BW.Systems and Control Letters,vol.55,no.3,pp.193-198,2006.
[12]	L.Dai.Singular Control Systems,Lecture Notes in Con-trol and Information Science,Berlin,Germany:Springer-Verlag,1989.
[13]	G.R.Duan,G.S.Wang.Eigenstructure assignment in a class of second-order descriptor linear systems:A complete parametric approach.International Journal of Automation and Computing,vol.2,no.1,pp.1-5,2005.
[14]	B.Zhou,G.R.Duan,Y.L.Wu.Parametric approach for the normal Luenberger function observer design in second-order descriptor linear systems.International Jour-nal of Automation and Computing,vol.5,no.2,pp.125-131,2008.

Parameterized Solution to a Class of Sylvester Matrix Equations

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