Modern Control System Theory and Design, 2nd Edition

2.24.  DIGITAL COMPUTER EVALUATION OF THE TIME RESPONSE

The state-variable representation of a system’s dynamics easily lends itself to analysis by means of a digital computer. The technique involves the division of the time axis into sufficiently small increments t = 0, T, 2T, 3T, 4T,…, where T is the incremental time of evaluation Δτ. This time increment must be made small enough for accurate results. Round-off errors in the computer, however, limit how small the time increment can be.

To illustrate the procedure, let us consider the equation

By definition of a derivative,

Utilizing this definition, the value of x(t) when t is subdivided into the increments Δτ can be determined. Because Δτ = T, we can say (approximately) that

Substituting Eq. (2.243) into Eq. (2.241), we obtain

Equation (2.244) may be solved for x(t + T) as follows

This equation can be written as

To generalize this ...