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 Δu, the output response is

Equation 3.48

A dimensionless output can be defined by dividing Equation (3.48) by k_pΔu, and t/τ_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 τ_n/τ_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).

Figure 3-12. Step responses of the lead-lag example.

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