A new approach is presented for obtaining the solutions to Yakubovich- -conjugate quaternion matrix equation based on the real representation of a quaternion matrix. Compared to the existing results, there are no requirements on the coefficient matrix . The closed form solution is established and the equivalent form of solution is given for this Yakubovich- -conjugate quaternion matrix equation. Moreover, the existence of solution to complex conjugate matrix equation is also characterized and the solution is derived in an explicit form by means of real representation of a complex matrix. Actually, Yakubovich-conjugate matrix equation over complex field is a special case of Yakubovich- -conjugate quaternion matrix equation . Numerical example shows the effectiveness of the proposed results. 1. Introduction The linear matrix equation , which is called the Kalman-Yakubovich matrix equation in [1], is closely related to many problems in conventional linear control systems theory, such as pole assignment design [2], Luenberger-type observer design [3, 4], and robust fault detection [5, 6]. In recent years, many studies have been reported on the solutions to many algebraic equations including quaternion matrix equations and nonlinear matrix equations. Yuan and Liao [7] investigated the least squares solution of the quaternion -conjugate matrix equation (where denotes the -conjugate of quaternion matrix ) with the least norm using the complex representation of quaternion matrix, the Kronecker product of matrices, and the Moore-Penrose generalized inverse. The authors in [8] considered the matrix nearness problem associated with the quaternion matrix equation by means of the CCD-Q, GSVD-Q, and the projection theorem in the finite dimensional inner product space. In addition, Song et al. [9, 10] established the explicit solutions to the quaternion -conjugate matrix equation , , but here the known quaternion matrix is a block diagonal form. Wang et al. in [11, 12] investigated Hermitian tridiagonal solutions and the minimal-norm solution with the least norm of quaternionic least squares problem in quaternionic quantum theory. Besides, in [13, 14], some solutions for the Kalman-Yakubovich equation are presented in terms of the coefficients of characteristic polynomial of matrix or the Leverrier algorithm. The existence of solution to the matrix equation , which, for convenience, is called the Kalman-Yakubovich-conjugate matrix equation, is established, and the explicit solution is derived. Several necessary and sufficient conditions for the existence of a unique
References
[1]
R. R. Bitmead, “Explicit solutions of the discrete-time Lyapunov matrix equation and Kalman-Yakubovich equations,” IEEE Transactions on Automatic Control, vol. 26, no. 6, pp. 1291–1294, 1981.
[2]
B. H. Kwon and M. J. Youn, “Eigenvalue-generalized eigenvector assignment by output feedback,” IEEE Transactions on Automatic Control, vol. 32, no. 5, pp. 417–421, 1987.
[3]
D. G. Luenberger, “An introduction to observers,” IEEE Transactions on Automatic Control, vol. 16, pp. 596–602, 1971.
[4]
C.-C. Tsui, “New approach to robust observer design,” International Journal of Control, vol. 47, no. 3, pp. 745–751, 1988.
[5]
J. Chen, R. J. Patton, and H.-Y. Zhang, “Design of unknown input observers and robust fault detection filters,” International Journal of Control, vol. 63, no. 1, pp. 85–105, 1996.
[6]
J. Park and G. Rizzoni, “An eigenstructure assignment algorithm for the design of fault detection filters,” IEEE Transactions on Automatic Control, vol. 39, no. 7, pp. 1521–1524, 1994.
[7]
S. Yuan and A. Liao, “Least squares solution of the quaternion matrix equation with the least norm,” Linear and Multilinear Algebra, vol. 59, no. 9, pp. 985–998, 2011.
[8]
S. F. Yuan, A. P. Liao, and G. Z. Yao, “The matrix nearness problem associated with the quaternion matrix equation ,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 133–144, 2011.
[9]
C. Song and G. Chen, “On solutions of matrix equation and over quaternion field,” Journal of Applied Mathematics and Computing, vol. 37, no. 1-2, pp. 57–68, 2011.
[10]
C. Q. Song, G. L. Chen, and X. D. Wang, “On solutions of quaternion matrix equations and ,” Acta Mathematica Scientia, vol. 32, no. 5, pp. 1967–1982, 2012.
[11]
S. Ling, M. Wang, and M. Wei, “Hermitian tridiagonal solution with the least norm to quaternionic least squares problem,” Computer Physics Communications, vol. 181, no. 3, pp. 481–488, 2010.
[12]
M. H. Wang, M. S. Wei, and Y. Feng, “An iterative algorithm for least squares problem in quaternionic quantum theory,” Computer Physics Communications, vol. 179, no. 4, pp. 203–207, 2008.
[13]
T. S. Jiang and M. S. Wei, “On a solution of the quaternion matrix equation and its application,” Acta Mathematica Sinica, vol. 21, no. 3, pp. 483–490, 2005.
[14]
T. Jiang and M. Wei, “On solutions of the matrix equations and ,” Linear Algebra and Its Applications, vol. 367, pp. 225–233, 2003.
[15]
L. Huang, “The quaternion matrix equation ,” Acta Mathematica Sinica, vol. 14, no. 1, pp. 91–98, 1998.
[16]
J. H. Bevis, F. J. Hall, and R. E. Hartwig, “Consimilarity and the matrix equation ,” in Current Trends in Matrix Theory, pp. 51–64, North-Holland, New York, NY, USA, 1987.
[17]
J. H. Bevis, F. J. Hall, and R. E. Hartwig, “The matrix equation and its special cases,” SIAM Journal on Matrix Analysis and Applications, vol. 9, no. 3, pp. 348–359, 1988.
[18]
B. Hanzon, “A faddeev sequence method for solving Lyapunov and Sylvester equations,” Linear Algebra and Its Applications, vol. 241, pp. 401–430, 1996.
[19]
A.-G. Wu, G.-R. Duan, and H.-H. Yu, “On solutions of the matrix equations and ,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 932–941, 2006.
[20]
A.-G. Wu, Y.-M. Fu, and G.-R. Duan, “On solutions of matrix equations and ,” Mathematical and Computer Modelling, vol. 47, no. 11-12, pp. 1181–1197, 2008.
[21]
B. Hanzon and R. M. Peeters, “A Feddeev sequence method for solving Lyapunov and Sylvester equations,” Linear Algebra and Its Applications, vol. 241–243, pp. 401–430, 1996.
[22]
A.-G. Wu, H.-Q. Wang, and G.-R. Duan, “On matrix equations and ,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 690–698, 2009.