- 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
This chapter is from the book
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 )
then q(s) = g-1(s)f(s),
where f(s) = 1/(
s + 1)r,
|
r = the relative order of g(s), |
|
|
|
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., )
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
)
is an all-pass system cascaded with the filter f(s). That is, the poles
of
are at the mirror image of the right half plane zeros of .
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
by including in the controller increasing portions of the model inverse.