Solutions to the linear transpose matrix equations and their application in control
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Solutions to the linear transpose matrix equations and their application in control Caiqin Song1,2 · Wenli Wang1 Received: 8 June 2020 / Revised: 17 September 2020 / Accepted: 19 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we study the solutions to the linear transpose matrix equations AX + X T B = C and AX + X T B = CY , which have many important applications in control theory. By applying Kronecker map and Sylvester sum, we obtain some necessary and sufficient conditions for existence of solutions and the expressions of explicit solutions for the Sylvester transpose matrix equation AX + X T B = C. Our conditions only need to check the eigenvalue of B T A−1 , and, therefore, are simpler than those reported in the paper (Piao et al. in J Frankl Inst 344:1056–1062, 2007). The corresponding algorithms permit the coefficient matrix C to be any real matrix and remove the limit of C = C T in Piao et al. Moreover, we present the solvability and the expressions of parametric solutions for the generalized Sylvester transpose matrix equation AX + X T B = CY using an alternative approach. A numerical example is given to demonstrate that the introduced algorithm is much faster than the existing method in the paper (De Terán and Dopico in 434:44–67;2011). Finally, the continuous zeroing dynamics design of time-varying linear system is provided to show the effectiveness of our algorithm in control. Keywords Kronecker map · Jameson’s Theorem · Transpose matrix equation · Time-varying linear system AMS subject classification 15A24 · 15A21
Communicated by Jinyun Yuan. This work is supported by the National Natural Science Foundation of China (No. 11501246 and No. 11801216) and Shandong Natural Science Foundation (No. ZR2017BA010).
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Caiqin Song [email protected]
1
School of Mathematical Sciences, University of Jinan, Jinan 250022, People’s Republic of China
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Department of Mathematics and Statistics, University of Nevada, Reno 89503, USA 0123456789().: V,-vol
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C. Song, W. Wang
1 Introduction In this paper, we consider the explicit solutions to the equations AX + X T B = C,
(1)
AX + X T B = CY ,
(2)
and
where A, B, C ∈ Rn×n are given n × n real matrices and X , Y ∈ Rn×n are unknown n × n real matrices to be determined. There are named as the Sylvester transpose matrix and the generalized Sylvester transpose matrix equation, respectively. If X = X T , Eqs. (1) and (2) reduce to the well-known Sylvester (or generalized Sylvester) matrix equation, which have many important applications in control theory. Therefore, many researchers focused on the solutions to these two equation, for details, please see Frank (1990); Fletcher et al. (1986); Dai (1989); Wang and Zhang (2008); Desouza and Bhattacharyya (1981); Ma (1966). Because (1) and (2) are obtained by changing the second unknown matrix of the (generalized ) Sylvester matrix equation, they are called as the (generalized ) Sylvester transpose matrix equation. The tradition
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