9.7 Summary

We have shown how the IMC procedure can be used to design PID-type feedback controllers. If the process has no time delay and the inputs do not hit a constraint, then the IMC-based PID controllers will have the same performance as does IMC. If there is dead time, then the IMC-based PID controllers will not perform as well as IMC because of the Padé approximation for deadtime.

Table 9-3. PID Tuning Parameters for Unstable Processes

g_p(s)
g_CL(s)
k_c
t_I
t_D
Notes^[a]
A
g

1
B
g + t_p
1
C
g

1, 2

^[a] 1, . 2, PI cascaded with a lead-lag filter, .

It is interesting to note that the IMC-based PID controllers for all of the transfer functions shown in Table 9-1 could have been designed using the direct synthesis method of Chapter 6. The key issue in the direct synthesis method is the specification of the closed-loop response characteristic. If the process has a RHP zero, then the specified closed-loop response must also have a RHP zero. The IMC-based PID procedure provides a clear method for handling this. Also, note that the standard IMC filter design results in good setpoint response performance, but other filter designs must be used for good input disturbance rejection.

The IMC design method of Chapter 8 was modified to handle unstable processes. The standard IMC structure cannot handle unstable processes, so the controller for an unstable process must be implemented in standard PID feedback form.

For a good example application of the IMC-based PID procedure, work through Module 7 on Biochemical Reactors.

IMC-Based PID Procedure Summary

Find the IMC controller transfer function, q(s), which includes a filter, f(s). The controller, q(s), may be semiproper or even improper to give the resulting PID controller derivative action. If a process model has a time delay, use an approximation, such as a first- or second-order Padé.
For good setpoint tracking, a filter with the following form is generally used

For improved disturbance rejection, or for integrating and open-loop unstable processes, a filter with the following form is used:

For disturbance rejection, g is selected so that the term inside the brackets cancels a slow pole in the distubance transfer function [often g_d(s) = g_p(s) for input disturbances],

For unstable processes, a value of g is found that satisfies the filter requirement f(s = p_u) = 1.
Find the IMC-based PID controller using the transformation

Write this in the form of a ratio between two polynomials.
Show this in PID form and find k_c, t_I, t_D. Sometimes this procedure results in an ideal PID controller cascaded with a first-order filter, with a filter time constant (t_F):
Perform closed-loop simulations for both the perfect model case and the cases with model mismatch. Adjust the value of l on-line as a tradeoff between performance and robustness (sensitivity to model error).

[ Team LiB ]