3.8 Lead-Lag Behavior

Lead-lag transfer functions have the same order numerator polynomial as denominator polynomial. This occurs when the input has a direct effect on the output variable. In terms of the state space model (3.2), this means that D 0.

Consider a lead-lag transfer function where the numerator and denominator polynomials are first order:

Equation 3.47

For a step input of magnitude Du, the output response is

Equation 3.48

A dimensionless output can be defined by dividing Equation (3.48) by k_pDu, and t/t_p is a natural dimensionless time. The responses to a step input at t = 0 are shown in Figure 3-12. Notice that there is an immediate response that is equal to the t_n/t_p ratio. This can also be found by applying the initial value theorem to Equation (3.47"/>) for a step input change (see Exercise 3).

It is rare for processes to exhibit lead-lag behavior, but many controllers exhibit such behavior.