Polynomial Solutions to the Matrix Equation

Solutions are constructed for the Kalman-Yakubovich-transpose equation . The solutions are stated as a polynomial of parameters of the matrix equation. One of the polynomial solutions is expressed by the symmetric operator matrix, controllability matrix, and observability matrix. Moreover, the explicit solution is proposed when the Kalman-Yakubovich-transpose matrix equation has a unique solution. The provided approach does not require the coefficient matrices to be in canonical form. In addition, the numerical example is given to illustrate the effectiveness of the derived method. Some applications in control theory are discussed at the end of this paper. 1. Introduction In the control area, the Kalman-Yakubovich-transpose matrix equation occurs in fault detection [1], control with constrains systems [2], eigenstructure assignment [3], and observer design [4]. In order to obtain explicit solutions, many researchers have made much efforts. Braden [5] studies the Lyapunov-transpose matrix equation via matrix decomposition. Liao et al. [6] propose an effective method to obtain the least square solution of the matrix equation using GSVD, CCD and projective theorem. Piao et al. [7] investigate the matrix equation by the Moore-Penrose generalized inverse and give the explicit solutions for the Sylvester-tranpose matrix equation. Song et al. [8, 9] establish the explicit solution of the quaternion matrix equation and , where denotes the -conjugate of the quaternion matrix. Moreover, other matrix equations such as the coupled Sylvester matrix equations and the Riccati equations have also been found numerous applications in control theory. For more related introduction, see [10, 11] and the references therein. The matrix equation is considered by the iterative algorithm [12, 13]. In [14, 15], the following linear equation is considered, where , , , , and are some known constant matrices of appropriate dimensions and is a matrix to be determined. And the least squares solutions and least square solutions with the minimal-norm have been obtained. In [16], using the hierarchical identification principle, authors consider the following more general coupled Sylvester-transpose matrix equation: where , , , , , , and are the given known matrices and and are the matrices to be determined. In addition, the generalized discrete Yakubovich-transpose matrix equation has important applications in dealing with complicated linear systems, such as large scale systems with interconnections, linear systems with certain partitioned structures or extended models, and second order