Modern Control System Theory and Design, 2nd Edition

2.28.  EVALUATION OF THE STATE TRANSITION MATRIX FROM AN EXPONENTIAL SERIES

The state transition matrix may be evaluated from an exponential series. Several methods have been proposed for its numerical evaluation. References [20] and [21] discuss one type of computational algorithm developed by Faddeev and Faddeeva for accomplishing this. However, this approach requires the Laplace-transform inversion of Φ(s). Unfortunately, this approach is very tedious for matrices of any size. This section presents a straightforward method that evaluates the state transition matrix based on its infinite matrix series definition [22]. Direct application of the series definition gives a very efficient and fast method that depends only on matrix multiplication. It is based on assuming a solution to the homogeneous state equation, as is commonly done in the classical method of solving linear differential equations.

In order to derive the exponential series definition of the state transition matrix [23], let us assume that the solution to the homogeneous state equation

is given by

where

and

We shall ...