Effect of numerical integration on the poles and zeros of a transfer function

An expression for the transfer function of a numerical integrator is derived, and is illustrated by a mapping in the complex plane. Analysing a complete system using numerical integration is shown to be equivalent to displacing the poles and zeros of its transfer function. A simple graphical method for determining this displacement is described. The implicit Euler, trapezoidal and Shichman integration formulas are considered in detail. It is shown that these are special cases of a general formula, which possesses A stability. The trapezoidal formula is the most accurate A -stable equation, although it requires a reliable method of step-length adjustment. The implicit Euler and Shichman formulas sacrifice accuracy for better stability properties, when using large step lengths.