- 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

## 3.4 Design for Processes with No Zeros Near the Imaginary Axis or in the Right Half of the *s*-Plane

When *N*(*s*) has no zeros in the right half of the s-plane or near the imaginary axis, the inverse of the model is stable and not overly
oscillatory. In this case, the IMC controller for Eq. (3.14) can be chosen as

where r = the relative order^{4} of *N*(*s*)/*D*(*s*).

From Eq. (3.13), the filter time constant in Eq. (3.15) must satisfy

As before, the limit given by Eq. (3.16) ensures that the high frequency gain of the controller is not more than 20 times its low frequency gain. The actual values of the filter time constant will more often be dictated by modeling errors and will be computed as shown in Chapter 7.

The form of the filter (i.e., 1/(*s* + 1)^{r}) is somewhat arbitrary. It was chosen because it is the simplest form with a single adjustable parameter, , that provides
an overdamped response^{5} and makes *q*(s) realizable. Such a filter has the great merit of simplicity at the possible price of being suboptimal. There is also no
incentive to use a filter order, *r*, greater than the minimum required to make the IMC controller realizable, because when there are modeling errors, higher
order filters lead to slower responses, as shown in Chapter 7. Choosing a filter whose order is the same as the relative order of the model leads to a controller, *q*(*s*), whose relative order is zero.