A Hybrid Structure Solution of Quaternion Lyapunov Equation and Its Optimal Approximation
American Journal of Applied Mathematics
Volume 7, Issue 1, February 2019, Pages: 30-36
Received: Apr. 22, 2019;
Published: Jun. 15, 2019
Views 473 Downloads 99
Wang Yun, College of Science, Guangxi University for Nationalities, Nanning, China
Huang Jingpin, College of Science, Guangxi University for Nationalities, Nanning, China
Lan Jiaxin, College of Science, Guangxi University for Nationalities, Nanning, China
Recently, the establishment of a multi-structure control system has demonstrated vital significance in practice. Its stability analysis are mostly determined by Lyapunov matrix equation. Tridiagonal-arrow matrix (TA matrix for short) is a special matrix with hybrid structure. In this paper, the problem of TA constraint solution to continuous Lyapunov equation A*X+XA=C over quaternion field is discussed. By using the representation of vectors of a TA matrix and Kronecker product of matrices, a constrained problem will be transformed into an unconstrained equation. Then the necessary and sufficient conditions for the equation with TA and self-conjugate TA solutions as well as the expression of general solution are obtained. Meanwhile, when the solution set is nonempty, by using invariance of Frobenius norm of orthogonal matrix product, the optimal approximation solution with minimal Frobenius norm for a given TA matrix is derived. Finally, two numerical examples are provided to verify the algorithm.
A Hybrid Structure Solution of Quaternion Lyapunov Equation and Its Optimal Approximation, American Journal of Applied Mathematics.
Vol. 7, No. 1,
2019, pp. 30-36.
Z. Gajic, and M. T. J. Qureshi, Lyapunov Matrix Equation in System Stability and Control, Academic Press, 1995.
Y. M. Fu, and G. R. Duan, “Robust stability analysis for uncertain systems with time-varying delays,” J Nat Sci Heilongjiang Univ, vol. 22, pp. 655-658, May 2005.
Z. F. Feng, “Study on network of temperature sensors via Lyapunov optimization,” Master Thesis, South China University of Technology, 2017.
Q. Hui, and J. M. Berg, “Semistability theory for spatially distributed systems,” Syst Control Lett, 2013, vol. 62, pp. 862-870.
A. L. Afflitto, W. M. Haddad, and Q. Hui, “Optimal control for linear and nonlinear semistabilization,” J Franklin I, vol. 352, pp. 851-881, March 2015.
V. Simoncini, “A new iterative method for solving large-scale Lyapunov matrix equations,” SIAM J Sci Comp, 2007, vol. 29, pp. 1268-1288.
S. F. Xu, and M. S. Cheng, “On the solvability for the mixed-type Lyapunov equation,” IMA J Appl Math, vol. 71, pp. 287-294.
Y. Deng, and J. P. Huang, “The subpositive definite solution of the mixed-type Lyapunov equation over quaternion field,” Acta Sci Natur Univ Nankai, 2011, vol. 44, pp. 41-46.
J. P. Huang, Y. S. Lu, and K. J. Xu, “On cyclic matrix solution of unified algebraic Lyapunov equation and its optimal approximation over quaternion field,” J. Southwest China Normal Univ (Nat Sci Ed), vol. 41, pp. 1-5, April 2016.
G. H. Golub, and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, 1996.
K. Atkinson, and W. M. Han, Theoretical Numerical Analysis, New York: Springer-Verlag, 2009.
Y. Yu, J. H. Yuan, and J. Qian, “Cubic spline interpolation with new boundary conditions,” Comput Eng Softw, 2016, vol. 37, pp. 25-28.
H. B. Li, and E. J. Zhong, “A note on semi-discrete Difference schemes of heat conduction equation,” Math Num Sin, vol. 37, pp. 401-414, April 2015.
L. L. Ma, and J. P. Huang, “On the construction of arrow-like quaternion matrices from two right eigenpairs,” ICMTA2010, Shanghai, China, July, 2010, pp. 280-284.
J. P. Huang, J. X. Lan, and L. Y. Mao, “The solutions of quaternion sylvester equation with arrowhead matrix constraint,” J Math Practice Theory, 2018, vol. 48, pp. 264-271.
F. J. Duan, T. Fang, and F. Yuan, “Inverse eigenvalue problems for a class of special matrices,” J Math, 2018. DOI: 10.13548/j.sxzz.20181206.001.