- 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

## Problems

**3.1** Derive all the transfer functions from Eq. (3.1) to Eq. (3.6).

**3.2** Use SIMULINK or other simulation software to obtain the step responses for each of the examples in the chapter. Note that
only the transfer functions of the loop response need be simulated, since all of the examples are for perfect model responses
to a step setpoint change.

**3.3** Simulate a step disturbance response for Example 3.1 with *p*_{d}(s) = 1. How does this response relate to the step setpoint response?

**3.4** Find an IMC controller for each of the following process models. The controller must satisfy

with equality holding at some frequency (possibly infinity). State the rationale for your choice of the part of the model
that the controller inverts. Recall the statement in Section 3.6 that "*Before designing the IMC controller, we strongly recommend that the model be put in time constant form*."

**3.5** Find an IMC controller for each of the following process models. The IMC controllers should satisfy the same noise amplification
criterion as that for Problem 3.4. Note, however, that the inverse of the numerator in each of the following process models can be inverted using a simple
negative feedback loop that has unity in the forward path and *Ke ^{-Ts}* in the feedback path, with appropriate values of

*K*and

*T*. This is not to say, however, that such inverses will not be very oscillatory, or even unstable. Thus, in developing an IMC controller for each of the following, the student should decide whether or not to invert the numerator exactly; approximate it by replacing the exponential with a Padé approximation, and then invert all or part of the numerator; or perhaps simply invert the numerator gain. To complicate matters further, the filter time constant necessary to satisfy the same noise amplification criterion as that for Problem 3.4 depends on which option is selected, and if a Padé approximation is used, it even depends on whether highest order terms in the Padé polynomials are odd or even. We suggest that before designing a controller for the following problems, the student get a step response of the process model.

The student should note that, as in most engineering situations, there is no single correct answer. In such cases the simplest controller is often the best controller. To compare controllers, we recommend that the student simulate the output response to a step setpoint change.

**3.6** Why do the IMC controller filter time constants in the IMC controllers for all the processes of Problem 3.5 change depending on whether an even or odd order Padé approximation for the exponential term is used in forming the controller?