## 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.

### MSC:

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

  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  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  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  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  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  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  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  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  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  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  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.  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  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  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  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  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.  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.  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  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.  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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