PID Controllers, 2nd Edition

Chapter 7.4.2 - Control Paradigms: A General Controller Structure

A General Controller Structure

The system in Figure 7.9 uses feedforward to improve command signal following. It is possible to combine this with feedforward from measured disturbances as discussed in Section 7.3. We will then obtain the general controller structure shown in Figure 7.10. In this case feedforward is used both to improve setpoint response and to reduce

07_4-3.jpg

Figure 7.10 Block diagram of a system that combines feedback and feedforward.

the effects of a measurable disturbance.

The properties of the system are analyzed here. If the subsystems are linear and time invariant, we find that the Laplace transform of the control error is given by

07_4-4.jpg

where Gff1 and Gff2 are transfer functions for feedforwards from the command signal and from the disturbance. Gfbis the transfer function for the feedback. The process has the transfer function

07_4-5.jpg

It follows from Equation (7.6) that the transfer function from command signal to error is given by

07_4-6.jpg

This transfer function is small if the loop transfer function Gl = GpGfb is large even without feedforward. The loop transfer function is typically large for frequencies smaller than the servo bandwidth. Therefore, the transfer function G is small for low frequencies even without feedforward. By choosing the feedforward so that

07_4-7.jpg

we find that the transfer function becomes zero for all frequencies, irrespective of the feedback used. Since we are combining feedback and feedforward, we can let the feedback handle the low frequencies and use feedforward compensation only to deal with the high frequencies. This means that Equation (7.9) need only be satisfied for higher

frequencies. This makes the feedforward compensator simpler. Similarly it follows from Equation (7.6) that the transfer function from the disturbance to the control error is given by

07_4-8.jpg

In an analogy with the previous discussion, we find that this transfer function will be zero if

07_4-9.jpg

holds. The transfer function will also be small if the loop transfer function Gl is large.

These simple calculations illustrate the differences between feedback and feedforward. In particular they show that feedforward reduces disturbances by canceling two terms, while feedback reduces the disturbances by dividing them by a large number. This clearly demonstrates why feedforward is more sensitive than feedback.

Tuning Feedforward Controllers

Feedforward controllers must be well tuned. Unfortunately, it is difficult to tune such controllers. The main difficulty is that it is often not possible to change the disturbances in order to investigate the disturbance response. Therefore, it is necessary to wait for a natural disturbance before the performance of the feedforward can be observed. This makes tuning very time consuming.

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