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**Solutions to a family of matrix equations by using the Kronecker matrix polynomials.**
*(English)*
Zbl 1181.15020

Using the standard technique of Kronecker products, the authors give closed form solutions to the real generalized Sylvester matrix equation \(\sum_i A_iXF^i + \sum_k B_kYF^k = \sum_j E_jRF^j\), where \(A_i\), \(B_k\), \(E_j\) and \(F\), \(R\) are known matrices and the matrix pair \((X,Y)\) is the solution. A generalization of this equation is also considered.

Reviewer: Mihail M. Konstantinov (Sofia)

### Keywords:

Sylvester matrix equations; closed form solutions; Kronecker matrix product; Kronecker matrix polynomials; coprime; linear system theory
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\textit{B. Zhou} et al., Appl. Math. Comput. 212, No. 2, 327--336 (2009; Zbl 1181.15020)

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### References:

[1] | Duan, G. R., Solutions to matrix equation \(AV + BW = VF\) and their application to eigenstructure assignment in linear systems, IEEE Transactions on Automatic Control, 38, 2, 276-280 (1993) · Zbl 0775.93098 |

[2] | Duan, G. R., The solution to the matrix equation \(AV + BW = EVJ + R\), Applied Mathematics Letters, 17, 1197-1204 (2004) · Zbl 1065.15015 |

[3] | Duan, G. R., Parametric eigenstructure assignment in high-order linear systems, International Journal of Control, Automation and Systems, 3, 3, 419-429 (2005) |

[4] | Duan, G. R., Two parametric approaches for eigenstructure assignment in second-order linear systems, Journal of Control Theory and Applications, 1, 59-64 (2003) · Zbl 1260.93073 |

[5] | Duan, G. R.; Liu, G. P., Complete parametric approach for eigenstructure assignment in a class of second-order linear systems, Automatica, 38, 4, 725-729 (2002) · Zbl 1009.93036 |

[6] | Duan, G. R.; Liu, G. P.; Thompson, S., Eigenstructure assignment design for proportional-integral observers: continuous-time case, IEE Proceedings: Control Theory & Applications, 148, 3, 263-267 (2001) |

[7] | G.R. Duan, B. Zhou, Solution to the equation \(\mathit{MVF}^2 + \mathit{DVF} + \mathit{KV} = \mathit{BW} \); G.R. Duan, B. Zhou, Solution to the equation \(\mathit{MVF}^2 + \mathit{DVF} + \mathit{KV} = \mathit{BW} \) |

[8] | Duan, G. R.; Zhou, B., Solution to the second-order Sylvester matrix equation \(MVF^2 + DVF + KV = BW\), IEEE Transactions on Automatic Control, 51, 5, 805-809 (2006) · Zbl 1366.15011 |

[9] | Zhou, B.; Duan, G. R., An explicit solution to the matrix equation \(AX - XF = BY\), Linear Algebra and Its Applications, 402, 345-366 (2005) · Zbl 1076.15016 |

[10] | Zhou, B.; Duan, G. R., A new solution to the generalized Sylvester matrix equation \(AV - EVF = BW\), Systems & Control Letters, 55, 3, 193-198 (2006) · Zbl 1129.15300 |

[11] | Zhou, B.; Duan, G. R., On the generalized Sylvester mapping and matrix equations, Systems & Control Letters, 57, 3, 200-208 (2008) · Zbl 1129.93018 |

[12] | B. Zhou, G.R. Duan, Parametric approach for the normal Luenberger function observer design in second-order linear systems, in: Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 1423-1428.; B. Zhou, G.R. Duan, Parametric approach for the normal Luenberger function observer design in second-order linear systems, in: Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 1423-1428. |

[13] | Lei, Y.; Liao, A.-P., A minimal residual algorithm for the inconsistent matrix equation \(AXB = C\) over symmetric matrices, Applied Mathematics and Computation, 188, 1, 499-513 (2007) · Zbl 1131.65038 |

[14] | Kaabi, A.; Kerayechian, A.; Toutounian, F., A new version of successive approximations method for solving Sylvester matrix equations, Applied Mathematics and Computation, 186, 1, 638-645 (2007) · Zbl 1121.65044 |

[15] | Gavin, K. R.; Bhattacharyya, S. P., Robust and well-conditioned eigenstructure assignment via Sylvester’s equation, Optimal Control Application and Methods, 4, 205-212 (1983) · Zbl 0512.93035 |

[16] | Kwon, B. H.; Youn, M. J., Eigenvalue-generalized eigenvector assignment by output feedback, IEEE Transactions on Automatic Control, 32, 5, 417-421 (1987) · Zbl 0611.93030 |

[17] | Luenberger, D. G., An introduction to observers, IEEE Transactions on Automatic Control, 16, 596-602 (1971) |

[18] | Park, J.; Rizzoni, G., An eigenstructure assignment algorithm for the design of fault detection filters, IEEE Transactions on Automatic Control, 39, 1521-1524 (1994) · Zbl 0825.93822 |

[19] | Tsui, C. C., New approach to robust observer design, International Journal of Control, 47, 745-751 (1988) · Zbl 0636.93030 |

[20] | Tsui, C. C., A complete analytical solution to the equation \(TA - FT = LC\) and its applications, IEEE Transactions on Automatic Control, 32, 742-744 (1987) · Zbl 0617.93009 |

[21] | Tsui, C. C., On the solution to matrix equation \(TA - FT = LC\) and its applications, SIAM Journal of Matrix Analysis and Applications, 14, 1, 34-44 (1993) · Zbl 0768.15009 |

[22] | Wang, Q.-W.; Chang, H.-X.; Ning, Q., The common solution to six quaternion matrix equations with applications, Applied Mathematics and Computation, 198, 1, 209-226 (2008) · Zbl 1141.15016 |

[23] | F. Rincon, Feedback stabilization of second-order models, Ph.D. dissertation, Northern Illinois University, De Kalb, Illinois, USA, 1992.; F. Rincon, Feedback stabilization of second-order models, Ph.D. dissertation, Northern Illinois University, De Kalb, Illinois, USA, 1992. |

[24] | Kim, Y.; Kim, H. S., Eigenstructure assignment algorithm for mechanical second-order systems, Journal of Guidance, Control and Dynamics, 22, 5, 729-731 (1999) |

[25] | Chu, E. K.; Datta, B. N., Numerically robust pole assignment for second-order systems, International Journal of Control, 64, 4, 1113-1127 (1996) · Zbl 0850.93318 |

[26] | Inman, D. J.; Kress, A., Eigenstructure assignment algorithm for second-order systems, Journal of Guidance, Control and Dynamics, 22, 5, 729-731 (1999) |

[27] | Ding, F.; Liu, P. X.; Ding, J., Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Applied Mathematics and Computation, 197, 1, 41-50 (2008) · Zbl 1143.65035 |

[28] | Ramadan, M. A.; El-Sayed, E. A., On the matrix equation \(XH = HX\) and the associated controllability problem, Applied Mathematics and Computation, 186, 1, 844-859 (2007) · Zbl 1126.93012 |

[29] | Wu, A. G.; Duan, G. R., Kronecker maps and Sylvester-polynomial matrix equations, IEEE Transactions on Automatic Control, 52, 5, 905-910 (2007) · Zbl 1366.93190 |

[30] | Huang, L., The explicit solutions and solvability of linear matrix equations, Linear Algebra and its Applications, 311, 195-199 (2000) · Zbl 0958.15008 |

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