Determining continuous-time state equations from discrete-time state equations via the principal<tex>q</tex>th root method

Abstract

Fast computational methods are developed for finding the equivalent continuous-time state equations from discrete-time state equations. The computational methods utilize the direct truncation method, the matrix continued fraction method, and the geometric-series method in conjunction with the principal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> th root of the discrete-time system matrix for quick determination of the approximants of a matrix logarithm function. It is shown that the use of the principal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> th root of a matrix enables us to enlarge the convergence region of the expansion of a matrix logarithm function and to improve the accuracy of the approximants of the matrix logarithm function.