Abstract. The features of computing of an analytic function of a matrix argument given by a convergent infinite series are considered in the general form. A method is proposed for calculating the transition matrix of a linear stationary system and other functions of matrices, using summation of matrix series. The method is based on the use of the equality of the analytic function of the matrix argument given by an infinite series that converges on the spectrum of the matrix, and the polynomial from the matrix that coincides on the spectrum of the matrix with the analytic function. An example of the calculation of the transition matrix of a linear stationary system using the Laplace transform and the proposed method is considered. The comparison with the method widely used in engineering practice, based on the Laplace transform, demonstrates a much greater simplicity of calculation by the proposed method.  

Keywords: analytical functions of matrix argument, transitional matrix, calculation, summation of matrix series

References:

1. Korshunov A.I. Journal of Instrument Engineering, 1985, no. 29, pp. 16–22. (in Russ.)
2. Bromberg P.V. Matrichnye metody v teorii releynogo i impul'snogo regulirovaniya (Matrix Methods in the Theory of Relay and Pulse Regulation), Moscow, 1967, 324 р. (in Russ.)
3. Hsu J.C., Meyer A.U. Modern control principles and applications, NY, McGraw-Hill, 1968, 769 p.
4. Aoki M., Leondes C.T. Modern control systems theory, NY, McGraw-Hill, 1965.
5. Derusso P.M., Roy R.J., Close Ch.M. State Variables for Engineers, Wiley, 1965.
6. Pontryagin L.S. Obyknovennye differentsial'nye uravneniya (Ordinary Differential Equations), Moscow, 1965, 332 р. (in Russ.)
7. Lankaster P. Theory of matrices. NY, London, 1969.
8. Gantmacher F.R. The Theory of Matrices, AMS Chelsea Publishing: Reprinted by American Mathematical Society, 2000, 660 р.