True derivative action is neither possible nor desirable. If a true derivative controller saw a step change in a measurement it would have to produce an infinite output. Thus a small but sharp noise *spike* , very common on practical electrical measurements, could produce a very large, sudden and spurious, change in a process adjustment. this is obviously highly undesirable, and so the derivative term on any controller must be modified to prevent this happening.

Derivative action is sometimes omitted altogether. In previous tuning examples we have use only PI control. In practice derivative action should be used only when very precise control is required on *measurements which are known to be reliable* . If in doubt, use PI only.

All real controllers use what is called *compensated rate action* . This introduces another parameter which is used to *damp* the derivative action to minimise the effect of noise spikes.

Most controllers nowadays are implemented digitally. To obtain derivatives a computer based controller would require numerical differentiation. This is an unsatisfactory procedure, since it also introduces noise. The rate compensated controller is described entirely by o.d.es, and so is implemented using only numerical *integration* .

Here is the theory of the compensated controller.

We require:

Without performing numerical differentiation.

Let:

Solve for **z** by numerical *integration* .

Then for small :

is called the rate compensation parameter, and can either have a fixed value or be made, e.g. **0.1 * deadtime of process** .

This corresponds to a form of *filtering* of the derivative term which prevents it goint to infinity in the presence of a step disturbance.