On the solvability for the mixed-type Lyapunov equation

Xu, Shengyuan; Cheng, Mingsong

doi:10.1093/imamat/hxh099

Cited by 3 publications

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“…The stability of many control systems in modern control theory is ultimately analyzed through the solution to Lyapunov matrix equation, thus enjoying an extensive application [1][2][3]. According to different problems in practice, Lyapunov equation can be classified into mixed, continuous and discrete equations, which have generated fruitful results in the complex field and quaternion field [4][5][6][7][8][9]. For example, by using the complex representation operator of the quaternion matrices, the sub-positive definite solutions and iterative algorithm with parameter of mixed quaternion Lyapunov equation were discussed [8].…”

Section: Introductionmentioning

confidence: 99%

A Hybrid Structure Solution of Quaternion Lyapunov Equation and Its Optimal Approximation

Wang¹

2019

AJAM

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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.

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Section: Introductionmentioning

confidence: 99%