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.
| Published in | American Journal of Applied Mathematics (Volume 7, Issue 1) |
| DOI | 10.11648/j.ajam.20190701.15 |
| Page(s) | 30-36 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2019. Published by Science Publishing Group |
Quaternion Field, Lyapunov Equation, Tridiagonal-Arrow Matrix, Kronecker Product, Optimal Approximation
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APA Style
Wang Yun, Huang Jingpin, Lan Jiaxin. (2019). A Hybrid Structure Solution of Quaternion Lyapunov Equation and Its Optimal Approximation. American Journal of Applied Mathematics, 7(1), 30-36. https://doi.org/10.11648/j.ajam.20190701.15
ACS Style
Wang Yun; Huang Jingpin; Lan Jiaxin. A Hybrid Structure Solution of Quaternion Lyapunov Equation and Its Optimal Approximation. Am. J. Appl. Math. 2019, 7(1), 30-36. doi: 10.11648/j.ajam.20190701.15
AMA Style
Wang Yun, Huang Jingpin, Lan Jiaxin. A Hybrid Structure Solution of Quaternion Lyapunov Equation and Its Optimal Approximation. Am J Appl Math. 2019;7(1):30-36. doi: 10.11648/j.ajam.20190701.15
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Download@article{10.11648/j.ajam.20190701.15,
author = {Wang Yun and Huang Jingpin and Lan Jiaxin},
title = {A Hybrid Structure Solution of Quaternion Lyapunov Equation and Its Optimal Approximation},
journal = {American Journal of Applied Mathematics},
volume = {7},
number = {1},
pages = {30-36},
doi = {10.11648/j.ajam.20190701.15},
url = {https://doi.org/10.11648/j.ajam.20190701.15},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20190701.15},
abstract = {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.},
year = {2019}
}
TY - JOUR T1 - A Hybrid Structure Solution of Quaternion Lyapunov Equation and Its Optimal Approximation AU - Wang Yun AU - Huang Jingpin AU - Lan Jiaxin Y1 - 2019/06/15 PY - 2019 N1 - https://doi.org/10.11648/j.ajam.20190701.15 DO - 10.11648/j.ajam.20190701.15 T2 - American Journal of Applied Mathematics JF - American Journal of Applied Mathematics JO - American Journal of Applied Mathematics SP - 30 EP - 36 PB - Science Publishing Group SN - 2330-006X UR - https://doi.org/10.11648/j.ajam.20190701.15 AB - 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. VL - 7 IS - 1 ER -
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