- 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
3.6 Design for Processes with Right Half Plane Zeros
When N(s) in Eq. (3.14) has factors of the form (-τs+1) or (τ^{2}s^{2} – 2τζs + 1), with τ and ζ greater than zero, its inverse is unstable. In this case the IMC controller cannot be formed as given by Eq. (3.15). The integral square error (ISE)^{7} optimal choice of controller for such cases is to invert that portion of the model which has zeros in the left half plane and add poles at the mirror image of the right half plane zeros (Morari & Zafiriou, 1989). That is, we assume that the model given by Eq. (3.14) can be rewritten as
where |
N_{-}(s) contains only left half plane zeros, none of which have small damping ratios. |
N_{+}(s) contains only right half plane zeros, and can be written as |
Notice that the gain of N_{+}(s) is one.
Before designing the IMC controller, we strongly recommend that the model be put in time constant form (i.e., the numerator and denominator are factored into products of the form (±τs + 1), (τ^{2}s^{2} ± 2τζs + 1) so that it is easy to form N_{+}(s) and N_{-}(s). The MATLAB functions tcf and tfn provided with IMCTUNE were developed specifically to put transfer functions into time constant form, and to facilitate their manipulation in this form. There are also other software programs that can be used to accomplish the desired factorization as described in Section 3.9.
The ISE optimal IMC controller for Eq. (3.19a) is
where the zeros of N_{+}(–s) are all in the left half plane and are the mirror images of the zeros of N_{+}(s). r = relative order of N(s)/D(s) as before.
The choice of controller given by Eq. (3.20) results in a loop response given by
The loop response given in Eq. (3.21) is optimal in an ISE sense for a filter time constant of zero, and is suboptimal for finite . Also, when is zero, the loop transfer function given by Eq. (3.21) is called all-pass, since the magnitude of the frequency response is one over all frequencies.
Example 3.3 One Right Half Plane Zero
The process model is
Putting Eq. (3.22a) in time constant form yields
The IMC controller is
The resulting loop response is
Figure 3.6 compares the step response of the ISE optimal loop transmission given by Eq. (3.21) with = 0 to step responses of suboptimal responses obtained by increasing and decreasing the controller time constant. Notice that the faster response obtained with pq(s) = (–s + 1)/(.5s + 1) comes at the expense of a more negative initial response. Thus, for this simple example, the ISE optimal response is also qualitatively the best compromise between a more sluggish response and a faster response with a more negative initial response.
Figure 3.6. Response of processes with one right half plane zero.
The transfer function given by the controller of Eq. (3.22c) does not result in an optimal response to a step setpoint change unless is zero. We could get closer to an optimal transfer function by selecting the IMC controller as
The controller given by Eq. (3.22e) was obtained by forcing the coefficient of the linear term of its expanded denominator to be one, which is the same as the linear term in the denominator of Eq. (3.22c) when is zero. The loop response then becomes that given by Eq. (3.22f).
The cubic and quadratic terms in s in the denominator of Eq. (3.22f) are small relative to its linear term so that Eq. (3.22f) approaches the optimal transfer function given by Eq. (3.22d) with equal to zero. Notice, however, that Eq. (3.22e) is valid only for ≤ .5. For larger values of the filter time constant, Eq. (3.22e) is not stable.
The approach used to obtain the controller given by Eq. (3.22e) can be used to develop nearly optimal controllers for arbitrary nonminimum^{8} phase processes. However, such controllers will generally be useful only for small filter time constants.
Example 3.4 Two Right Half Plane Zeros
When the initial process response to a step is in a direction opposite to that of the final steady state, as in the previous example, the process is said to exhibit an inverse response. Processes with an odd number of right half plane zeros exhibit inverse responses. Processes with an even number of right half plane zeros do not have inverse responses, since the initial value at time zero plus is always in the direction of the steady-state^{9} as shown in Figure 3.7 for the loop response given by
Figure 3.7. Response of processes with two right half plane zeros.
with τ= .5, 1, and 2.
Notice that the ISE optimal response in Figure 3.7 is again that which also gives the qualitatively best compromise between a sluggish response and a response with too much initial overshoot.