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
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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
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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.
Quaternion Field, Lyapunov Equation, Tridiagonal-Arrow Matrix, Kronecker Product, Optimal Approximation
To cite this article
Wang Yun, Huang Jingpin, Lan Jiaxin, 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. doi: 10.11648/j.ajam.20190701.15
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