# One-Degree of Freedom Internal Model Control

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## 3.5 Design for Processes with Zeros Near the Imaginary Axis

If the term N(s) in Eq. (3.14) contains complex roots with low damping ratios,6 then such terms can cause an IMC controller formed like that given in Equations (3.15) and (3.16) to amplify noise excessively at intermediate frequencies. There are two relatively simple options to reduce such excessive noise amplification. The first option is to increase the filter time constant sufficiently to reduce the peak to an acceptable level. This option generally requires large filter time constants, excessively increasing the settling time of the control system, and is therefore not recommended. The second, and recommended, option is to not invert low damping ratio zeros. Rather, form a controller similar to that given by Eq. (3.15), but with the damping ratio in the original polynomials in N(s) modified so as to be sufficiently large to avoid excessive noise amplification at intermediate frequencies. An example should help clarify the suggested design procedure.

## Example 3.2 A Process with Low Damping Ratio Zeros

Consider the process given by

Equation 3.17a

By Equations (3.15) and (3.16), the controller and filter time constant would be

Equation 3.17b

Equation 3.17c

The frequency response of q(s) given by Equations (3.17b) and (3.17c) is in shown Figure 3.4. The magnitude of the peak in Figure 3.4 is actually 3810. A filter time constant of 20 would be required to reduce the noise amplification at frequencies around 1.0 to a factor of 20. The settling time of the control system with such a controller exceeds 100 units. As we shall see, a controller with the same form as that given by Eq. (3.17b), but with a different damping ratio in the denominator, gives a much faster response. Figure 3.4. Frequency response of q(s) = (s + 1)4/((s2 + 2ζs + 1)(.22s + 1)2).

A better controller than that of Eq. (3.17b) is

Equation 3.18a

To reduce the controller frequency response peak so that its magnitude is only 20 (with = .22, as before) requires a damping ratio, ζ, of 0.1. The resulting loop response pq(s) is

Equation 3.18b

Figure 3.5 shows the loop response given by Eq. (3.18b) for a damping ratio of 0.1 as well as a damping ratio of 0.5. As is apparent from the figure, a controller damping ratio of .5 gives a less oscillatory response than that given by a controller damping ratio of .1. On the other hand, the response for ζ = .5 is more sluggish than that for ζ = .1. In such cases the engineer has to use process knowledge to select the most appropriate controller Figure 3.5. Perfect model loop response for p(s) = (s2 + .001s + 1)/(s + 1)4 with q(s) = (s + 1)4/((s2 + 2ζs + 1)(.22s + 1)2).