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*, 2*T*, 3*T*, 4*T*,…, where *T* is the incremental time of evaluation Δ*τ*. 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 Δ*τ* can be determined. Because Δ*τ* = *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 ...

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