On the parametric solution to the second-order Sylvester matrix equation \(EVF^{2} - AVF - CV=BW\). (English) Zbl 1160.15017

For a real diagonalizable matrix \(F\), a controllable matrix triple \((E,A,B)\), and a given \(C\), the authors describe a parametric solution \(V,W\) for the second order Sylvester equation
\[ EVF^2 - AVF - CV = BW. \]
If \(F\) is not explicitly given, the solution is found by elementary transformations performed on respective matrix pencils and the solution depends on the eigenvalues of \(F\), as well as on another set of parameters. If the eigenvalues of \(F\) are prescribed, then the solution method involves the singular value decomposition.


15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
93B40 Computational methods in systems theory (MSC2010)
93B05 Controllability
Full Text: DOI


[1] L. H. Keel, J. A. Fleming, and S. P. Bhattacharyya, “Minimum norm pole assignment via Sylvester/s equation,” in Linear Algebra and Its Role in Systems Theory, vol. 47 of Contemporary Mathematics, pp. 265-272, American Mathematical Society, Providence, RI, USA, 1985. · Zbl 0574.93025
[2] S. P. Bhattacharyya and E. de Souza, “Pole assignment via Sylvester/s equation,” Systems & Control Letters, vol. 1, no. 4, pp. 261-263, 1982. · Zbl 0473.93037
[3] C.-C. Tsui, “A complete analytical solution to the equation TA - FT=LC and its applications,” IEEE Transactions on Automatic Control, vol. AC-32, no. 8, pp. 742-744, 1987. · Zbl 0617.93009
[4] G.-R. Duan, “Solution to matrix equation AV+BW=EVF and eigenstructure assignment for descriptor systems,” Automatica, vol. 28, no. 3, pp. 639-642, 1992. · Zbl 0775.93083
[5] G.-R. Duan, “Solutions of the equation AV+BW=VF and their application to eigenstructure assignment in linear systems,” IEEE Transactions on Automatic Control, vol. AC-38, no. 2, pp. 276-280, 1993. · Zbl 0775.93098
[6] G.-R. Duan, “On the solution to the Sylvester matrix equation AV+BW=EVF,” IEEE Transactions on Automatic Control, vol. AC-41, no. 4, pp. 612-614, 1996. · Zbl 0855.93017
[7] A. Saberi, A. A. Stoorvogel, and P. Sannuti, Control of Linear Systems with Regulation and Input Constraints, Communications and Control Engineering Series, Springer, London, UK, 2000. · Zbl 0977.93001
[8] A. Jameson, “Solution of the equation AX+XB=C by inversion of an M\times M or N\times N matrix,” SIAM Journal on Applied Mathematics, vol. 16, no. 5, pp. 1020-1023, 1968. · Zbl 0169.35202
[9] E. de Souza and S. P. Bhattacharyya, “Controllability, observability and the solution of AX - XB=C,” Linear Algebra and Its Applications, vol. 39, pp. 167-188, 1981. · Zbl 0468.15012
[10] R. E. Hartwig, “Resultants and the solution of AX - XB= - C,” SIAM Journal on Applied Mathematics, vol. 23, no. 1, pp. 104-117, 1972. · Zbl 0222.15007
[11] J. Jones Jr. and C. Lew, “Solutions of the Lyapunov matrix equation BX - XA=C,” IEEE Transactions on Automatic Control, vol. AC-27, no. 2, pp. 464-466, 1982.
[12] J. Z. Hearon, “Nonsingular solutions of TA - BT=C,” Linear Algebra and Its Applications, vol. 16, no. 1, pp. 57-63, 1977. · Zbl 0368.15007
[13] V. Sreeram and P. Agathoklis, “Solution of Lyapunov equation with system matrix in companion form,” IEE Proceedings Control Theory and Applications D, vol. 138, no. 6, pp. 529-534, 1991. · Zbl 0754.34051
[14] B. Zhou and G.-R. Duan, “An explicit solution to the matrix equation AX - XF=BY,” Linear Algebra and Its Applications, vol. 402, pp. 345-366, 2005. · Zbl 1076.15016
[15] G.-R. Duan and G.-S. Wang, “Two analytical general solutions of the matrix equation EVJ2 - AVJ - CV=BW,” Journal of Harbin Institute of Technology, vol. 37, no. 1, pp. 1-4, 2005. · Zbl 1081.15523
[16] G.-S. Wang and G.-R. Duan, “Robust pole assignment via P-D feedback in a class of second-order dynamic systems,” in Proceedings of the 8th International Conference on Control, Automation, Robotics and Vision (ICARCV /04), vol. 2, pp. 1152-1156, Kunming, China, December 2004.
[17] G.-R. Duan and 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.
[18] G.-R. Duan and G.-P. Liu, “Complete parametric approach for eigenstructure assignment in a class of second-order linear systems,” Automatica, vol. 38, no. 4, pp. 725-729, 2002. · Zbl 1009.93036
[19] G.-R. Duan, G.-S. Wang, and L. Huang, “Model reference control in second-order dynamic systems,” in Proceedings of the Control Conference (UKACC Control /04), Bath, UK, September 2004.
[20] G.-S. Wang, B. Liang, and G.-R. Duan, “Reconfiguring second-order dynamic systems via P-D feedback eigenstructure assignment: a parametric method,” International Journal of Control, Automation and Systems, vol. 3, no. 1, pp. 109-116, 2005.
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