- Introduction
- Properties of IMC
- IMC Designs for No Disturbance Lag
- Design for Processes with No Zeros Near the Imaginary Axis or in the Right Half of the s-Plane
- Design for Processes with Zeros Near the Imaginary Axis
- Design for Processes with Right Half Plane Zeros
- Problems with Mathematically Optimal Controllers
- Modifying the Process to Improve Control System Performance
- Software Tools for IMC Design
- Summary
- Problems
- References

## 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

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

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

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

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).**