Static properties of the closed loop

Frequently the behaviour of an open loop (according to Figure 7.4 and Eq. (7.5)) can be described by a generalised transfer function of the form

where the constant denotes the type of transfer function . is the gain of the open loop. Therefore shows for

;	delayed proportional behaviour	(delayed P behaviour)
;	delayed integral behaviour	(delayed I behaviour)
;	delayed double integral behaviour	(delayed  behaviour) .

We assume now that the term of the rational fraction in Eq. (7.15) contains only poles in the left half plane. For the different types of transfer functions with different forms of the command signal or of the disturbance the steady state of the closed loop for can be analysed.

With

from Eqs. (7.1) and (7.6) it follows for the control error

Under the assumption, that the limit of the control error for exists, one obtains by using the final value theorem of the Laplace transform (see section 2.3) the steady-state value of the control error

For the case of all disturbances being related to the plant output from Eq. (7.17) it follows that - sign apart - both types of inputs, command or disturbance, can be treated equally. Hence in the following to represent both types of input signals the term is chosen as the input signal. Using both Eqs. (7.17) and (7.18) the steady-state values of the control error for the different signal types of and for different types of transfer functions of the open loop can be obtained. These values characterise the behaviour of the control loop. They are obtained consecutively for the most important cases.

For further treatment the following test signals according to Figure 7.7 are used:

a)

Step input signal:

where is the height of the step.

b)

Ramp input signal:

where describes the slope of the ramp signal .

c)

Parabolic input signal:

where is a measure of the acceleration of the parabolic signal .

Figure 7.7: Different input signals , which are frequently used for the disturbance and command input : (a) step, (b) ramp and (c) parabolic input signal

Following Eq. (7.17) the control error is obtained by

where the difference between command and disturbance behaviour is only in the sign of (disturbance: ; command: ). Inserting this relation into Eqs. (7.19) to (7.21) the corresponding control error can be obtained for different types of transfer functions . This will be demonstrated in the following.