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This chapter is from the book

This chapter is from the book

3.10 Summary

This chapter presents IMC design techniques for inherently stable linear processes where the disturbance enters directly into the process output (i.e., the disturbance lag is either unity or has very fast time constants relative to the process time constants). The controller design methods differ depending on the location of the zeros of the numerator of the process transfer function.

Controllers for processes with no right half plane zeros, or zeros near the imaginary axis, can be obtained simply by inverting the entire model except any multiplicative delay. That is, if 060fig01.gif then q(s) = g-1(s)f(s),

where f(s) = 1/(s + 1)r,

r = the relative order of g(s),


= an adjustable filter time constant.


We recommend choosing the filter time constant to avoid excessive high frequency noise amplification by using the criterion


where N is between 10 and 20.

If the term g(s) has complex zeros with small damping ratios, then the controller, q(s), might amplify noise more than is desirable at some midrange frequency (i.e., 060fig03.gif for some ωc). When this occurs, we recommend either (1) not inverting the zeros that cause the noise amplification, or (2) increasing the damping ratio of the denominator terms in the controller q(s) that arise from inverting zeros with small damping ratios.

If the term g(s) has right half plane zeros, then the ISE optimal IMC controller inverts all of g(s) except the right half plane zeros, and poles are added to the controller q(s) so that the loop transmission 060fig02.gif is an all-pass system cascaded with the filter f(s). That is, the poles of 060fig02.gif are at the mirror image of the right half plane zeros of 060fig02.gif. As usual, the filter order is the relative order of the model, g(s).

Finally, if the ISE optimal IMC controller described in the previous paragraph is overly complex (as might occur when the right half plane zeros of the model are due to transcendental terms such as (p1(s) + p2(s)e-Ts), where p1(s) and p2(s) are polynomials in s, or if the loop response is undesirable because of excessive oscillations or overshoot, then we recommend rejecting the ISE optimal IMC controller in favor of a simpler controller. A systematic method of obtaining such a controller is to start with a controller that is just the inverse of the model gain, then attempt to improve the loop response 060fig02.gif by including in the controller increasing portions of the model inverse.

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