On Solutions of Inverse Problem for Hermitian Generalized Hamiltonian Matrices

Abstract:

Hamiltonian matrices have many applications to design automation and autocontrol, in particular in the linear-quadratic autocontrol problem. This paper studies the inverse problems of generalized Hamiltonian matrices for matrix equations. By real representation of complex matrix, we give the necessary and sufficient conditions for the existence of a Hermitian generalized Hamiltonian solutions to the matrix equations, and then derive the representation of the general solutions.

• Periodical: Advanced Materials Research (Volumes 860-863)
• Pages: 2727-2731
• DOI: https://doi.org/10.4028/www.scientific.net/AMR.860-863.2727
• Online since: December 2013
• Authors: Kai Fu Liang, Ming Jun Li, Ze Lin Zhu
• Keywords: Hamiltonian Matrices, Inverse Problem, Skew Involution

References

[1] William F. Trench, Minimization problems for (R, S)-symmetric and (R, S)-skew symmetric matrices, Linear Algebra Appl. 389(2004)23-31.

DOI: 10.1016/j.laa.2004.03.035

[2] T. Kailath, Inear Systems, Information and System Sciences, Prentice-Hall, (1980).

[3] V. L. Mehrmann, Lecture Notes in Control and Information Sciences, Springer-Verlag.

[4] K. Zhou, J. C. Doyle, K. Glover, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ, (1995).

[5] R. Coleman, On the construction of real canonical forms of Hamiltonian matrices whose spectrum is an imaginary, Math. Comput. Simul. 46(1998) 117-155.

DOI: 10.1016/s0378-4754(98)00073-1

[6] N. Del Buono, L. Lopez, T. Politi, computation of functions of Hamiltonian and skew-symmetric matrices, Math. Comput. Simul. 79(2008) 1284-1297.

DOI: 10.1016/j.matcom.2008.03.011

[7] V. Mehrmann, C. Schr¨oder, D. S. Watkins, A new block method for computing the Hamiltonian Schur form, Linear Algebra Appl. 431(2009)350-368.

DOI: 10.1016/j.laa.2009.01.026

[8] Zhongzhi Zhang, Xiyan Hu and Lei Zhang, Least-Squares Solutions of Inverse Problem for Hermitian Generalized Hamiltonian Matrices, Appl. Math. Lett. 17(2004)303-308.

DOI: 10.1016/s0893-9659(04)90067-5

[9] Y. Wang, C. Li, D. Cheng, Generalized Hamiltonian realization of time-invariant nonlinear system, Automat. 39(2003)1473-1443.

DOI: 10.1016/s0005-1098(03)00132-8

[10] T. Jiang, M. Wei, On solutions of the matrix equations and , Linear Algebra Appl. 367(2003)225-233.

[11] A. Wu, G. Duan, H. Yu, On solutions of the matrix equations and Appl. Math. Comput. 183(2006)932-941.

[12] A. Wu, G. Feng, G. Duan, W. Wu, iterative solutions to copuled Sylvester-conjugate matrix equations, Comput. Math. Appl. 60(2010)54-66.

DOI: 10.1016/j.camwa.2010.04.029