基于矩阵半张量积求解弱双四元数调节方程

兰州理工大学学报 ›› 2023, Vol. 49 ›› Issue (1): 152-157.

基于矩阵半张量积求解弱双四元数调节方程

袭沂蒙, 李莹^*, 刘志红, 孙建华

聊城大学数学科学学院, 山东聊城 252000

收稿日期:2022-03-03 出版日期:2023-02-28 发布日期:2023-03-21
通讯作者: 李莹(1974-),女,山东聊城人,博士,教授. Email:liyingld@163.com
基金资助:
国家自然科学基金(62176112),山东省自然科学基金(ZR2020MA053)

Solving reduced biquaternion regulating equation based on semi-tensor product of matrices

XI Yi-meng, LI Ying, LIU Zhi-hong, SUN Jian-hua

School of Mathematical Sciences, Liaocheng University, Liaocheng, 252000 China

Received:2022-03-03 Online:2023-02-28 Published:2023-03-21

摘要/Abstract

摘要： 基于矩阵半张量积及弱双四元数的实向量表示,将弱双四元数调节方程A₁X－A₂XB=C转化为无约束的实矩阵方程,利用实矩阵方程得到弱双四元数调节方程的(anti-)Hermitian解,通过数值实验检验了此方法的有效性,并将此方法应用于时变线性系统的连续归零动力学设计.

关键词: 调节方程, 矩阵半张量积, 实向量表示, 弱双四元数矩阵

Abstract: Based on the semi-tensor product of matrices and the real vector representation of reduced biquaternion, the problem of reduced biquaternion regulating equations A₁X－A₂XB=C is transformed into a matrix equation on real number field without constraint, and then the (anti-)Hermitian solution of reduced biquaternion regulating equations is obtained by using real matrix equation. The effectiveness of this method is verified by numerical experiments. Finally, this method is applied to the continuous zeroing dynamics design of time-varying linear systems.

Key words: regulating equation, semi-tensor product of matrices, real vector representation, reduced biquaternion matrix

中图分类号:

O241.6

袭沂蒙, 李莹, 刘志红, 孙建华. 基于矩阵半张量积求解弱双四元数调节方程[J]. 兰州理工大学学报, 2023, 49(1): 152-157.

XI Yi-meng, LI Ying, LIU Zhi-hong, SUN Jian-hua. Solving reduced biquaternion regulating equation based on semi-tensor product of matrices[J]. Journal of Lanzhou University of Technology, 2023, 49(1): 152-157.

参考文献

[1] WANG G,XU Z.Solving a kind of restricted matrix equations and Cramer rule [J].Applied Mathematics and Computation,2005,162(1):329-338.
[2] 王飞,王婷,王慧萍.基于负荷比值的计量供热间连网质调节方程 [J].太原理工大学学报,2008,39(3):316-319.
[3] 赵林明.水轮机流量调节方程的应用研究 [J].华北水利水电学院学报,1990(3):57-69.
[4] 王飞,王婷.基于负荷系数的固定供水温度量调节方程 [J].建筑热能通风空调,2008,27(4):46-48.
[5] GUO B Z,MENG T.Robust output regulation for Timoshenko beam equation with two inputs and two outputs [J].International Journal of Robust and Nonlinear Control,2020,31(4):1245-1269.
[6] CORROCHANO E B.A survey on quaternion algebra and geometric algebra applications in engineering and computer science 1995-2020 [J].IEEE Access,2021,9:104326-104355.
[7] PEI S C,CHANG J H,DING J J.Commutative reduced biquaternions and their Fourier transform for signal and image processing applications [J].IEEE Transactions Signal Processing,2004,52(7):2012-2031.
[8] HIDAYET H K.Least-squares solutions of the reduced biquaternion matrix equation AX=B and their applications in colour image restoration [J]. Journal of Modern Optics, 2019,66(18):1-9.
[9] KYRCHEI I I.Cramer's rule for quaternionic systems of linear equations [J].Journal of Mathematical Sciences,2008,155(6):839-858.
[10] 程代展,齐洪胜.矩阵半张量积的基本原理与适用领域 [J].系统科学与数学,2012,32(12):1488-1496.
[11] LU J Q,LI H T,LIU Y,et al.Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems [J].IET Control Theory and Applications,2017,11(13):2040-2047.
[12] LI H T,WANG Y Z,LIU Z B.On the observability of free Boolean networks via the semi-tensor product method [J].Journal of Systems Science and Complexity,2014,4:666-678.
[13] FU F Q,WEI W,ZHOU Y,et al.Research on semi-tensor product solution method for optimal decision of a class of multi-valued control networks [J].Advances in Applied Mathematics,2018,7(10):1308-1316.
[14] 葛爱冬,王玉振,魏爱荣.基于矩阵半张量积方法的随机模糊系统控制器设计 [J].山东大学学报(工学版),2013,43(3):30-37.
[15] 丁文旭,李莹,王栋,等.基于矩阵半张量积求解弱双四元数矩阵方程AX=B [J].数学的实践与认识,2021,51(8):253-259.
[16] 程代展,夏元清,马宏宾,等.矩阵代数、控制与博弈 [M].北京:北京理工大学出版社,2016.
[17] 戴华.矩阵论 [M].北京:北京科学出版社, 2001.