# Refined proportional control

The offset which occurs with simple proportional control may be removed to a large extent by introducing reset. The set point of a proportional controller is altered in a way which is related either to the deviation itself or to the rate at which the deviation is changing.

A controller the output signal of which varies at a rate which is proportional to the deviation is called a proportional controller with integral action.

When the output signal from a proportional controller is proportional to the rate of change of the deviation, the controller is said to have derivative action. Since derivative action is not proportional to deviation, it cannot be used alone but must be combined with another control action. Such a combination is then termed ‘compound control action’.

Figure 13.8 illustrates some of the behaviours of different methods of control action. Deviation is plotted against time. Curve A, for proportional control plus derivative action,



Reaches a steady state quite rapidly but some offset is present: Curve B shows the case of simple proportional control; there is a larger maximum deviation than with curve A and the offset is greater. Curve C is for proportional plus integral control; the maximum deviation is greater than for the other two curves and the value of the controlled variable oscillates for some time before it settles down to a steady value without any offset at all.

Proportional plus derivative plus integral control is not shown. Its behaviour is similar to curves A and C but the controlled variable settles down more quickly. There is no offset. In terms of potential correction, integral action may be defined by

 (13.2) ^ — -*.e

At

Whence

\$ = — kj Q dt + C2

(13.3)

Where k, is the integral control factor and C2 is a constant of integration. This shows that the potential correction is proportional to the integral of the deviation over a given time.

(Incidentally, floating control can be regarded as a form of integral action that applies to a potential correction whenever any deviation occurs, the magnitude of the correction being independent of the size of the deviation, depending only on its sign, positive or negative.)

Proportional plus integral control (often abbreviated as P + I) is much used and can be expressed by the sum of equations (13.1) and (13.3):

<|> = — kpQ -kjQdt+C (13.4) Where C is the combined constant. Derivative action may be similarly defined by (13.5)

Where kA is the derivative control factor.

Proportional plus integral plus derivative (abbreviated P +1 + D) control can be expressed by adding equations (13.4) and (13.5):

 (13.6) (j> = — kpQ — k, /0 dt — kd d0/d? + C

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