9.6 IMC-Based PID Controller Design for Unstable Processes

The IMC procedure must be modified for unstable processes. Rotstein and Lewin (1991) have used the procedure developed by Morari and Zafiriou (1989) to find IMC-based PID controllers for unstable processes. The modification to the procedure shown in Sections 9.2 and 9.3 is to use a slightly more complicated filter transfer function.

Find the IMC controller transfer function, q(s), which includes a filter, f(s), to make q(s) semiproper. An additional requirement is that the value of f(s) at s = p_u (where p_u is an unstable pole) must be 1. That is,

Morari and Zafiriou (1989) recommend a filter transfer function that has the form

where n is chosen to make q(s) proper (usually semiproper). A value of g is found that satisfies the filter requirement f(s = p_u) = 1.
Find the equivalent standard feedback 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 a PID controller cascaded with a lag term (t_F):
Perform closed-loop simulations for both the perfect model case and cases with model mismatch. Choose the desired value for l as a tradeoff between performance and robustness.

Example 9.4 illustrates this procedure for a first-order unstable process.

Example 9.4: IMC-Based PID Design for a First-Order Unstable Process

Find the IMC-based PID controller for a first-order unstable process

Equation 9.25

where t_u is given a positive value. The pole, p_u, is 1/t_u, which is positive, indicating instability.

Step 1. Find the IMC controller transfer function, q(s),

Equation 9.26

Note that we have selected a second-order polynomial filter to make the controller, q(s), semiproper. Now we solve for g so that f(s = p_u) = 1.

graphics/09equ26a.gif

graphics/09equ26b.gif

Solving for g, we find

Equation 9.27

Step 2. Find the equivalent standard feedback controller using the transformation

After a lengthy bit of algebra,

Equation 9.28

Step 3. This is in the form of a PI controller, where

Equation 9.29

graphics/09equ29.gif

As a numerical example, consider . The closed loop output responses for various values of l are shown in Figure 9-7a, while the manipulated variable responses are shown in Figure 9-7b. Notice that we do not achieve the nice overdamped-type of closed-loop output responses that we were able to obtain with open-loop stable processes. The reader should show that the closed-loop relationship for this system is

which has overshoot if g > l [this is always the case for this system; see Equation 9.27)].

In Figure 9-7 we notice that the closed-loop response had overshoot. A response to a setpoint change, without overshoot, can be obtained by including a setpoint filter, as shown in Figure 9-8. The setpoint filter is

Figure 9-7. Responses for a step setpoint change, for various values of l. (a) Output; (b) manipulated input.

graphics/09fig07.gif

Figure 9-8. FBC with a setpoint filter.

graphics/09fig08.gif

which yields the following closed-loop response for a setpoint change.

The results for several unstable process transfer functions are shown in Table 9-3. See Rotstein and Lewin (1991) for a discussion of the effect of dead time and model uncertainty on the control of unstable processes.

Summary of IMC-Based PID Controller Design for Unstable Processes

A major tuning consideration by the student is that there are both upper and lower bounds on l to assure stability of an unstable process. This is in contrast to stable processes, where the closed loop is guaranteed to be stable under model uncertainty, simply by increasing l to a large value (detuning the controller).

[ Team LiB ]