Application of the design method

The design procedure for compensators described in the previous sections will be applied in this concluding section to an example already introduced in section 10.3. It is the unstable plant with the transfer function

(see example 10.3.4).

Disturbance at the plant input

If a disturbance at the plant input is assumed, the degree of the numerator and denominator polynomial of the free selectable disturbance transfer function according to Eqs. (10.58) and (10.59) is given by and . The disturbance transfer function will be obtained as

and the the controller transfer function formally according to Eq. (10.55) as

For the realisability condition of the controller it follows that

The final realisable controller transfer function is then

Furthermore, the above equation for provides for the disturbance transfer function

The procedure for the design of the pre-filter for disturbances at the plant input starts with the given order of the transfer function for command input according to Eq. (10.69). For the command behaviour and is chosen. According to that the transfer function for command input is

The synthesis of the controller is required to be performed such that the behaviour for a command input will be characterised by approximately 10% maximum overshoot and settling time. In this case of a second-order system these demands are met by a natural frequency and by a damping ratio according to Figures 9.3 and 9.5. From this the transfer function for command input follows as

If for the inherent behaviour of the closed loop the same coefficients are taken for the disturbance case, then

and therefore the disturbance transfer function is

and the controller transfer function is

The pre-filter design, which is given by

is simplified by to

Step responses for the disturbance and command inputs are shown in Figure 10.10.

Figure 10.10: Step responses for the design for a step disturbance at the plant input:
(a) controlled variable and manipulated variable for
(b) controlled variable and manipulated variable for
(c) controlled variable and manipulated variable for the simultaneous application of and

Disturbance at the plant output

If the closed loop is designed for a disturbance at the plant output, for the unstable plant the controller transfer function

from section 10.5.2 is significant. With

and with the approach

it follows - as in the previous case - for

that

The realisability condition for the controller is thus

and if for simplicity is set equal to zero, then

Figure 10.11: Step responses for the design for a step disturbance at the plant output:
(a) controlled variable and manipulated variable for
(b) controlled variable and manipulated variable for
(c) controlled variable and manipulated variable for the simultaneous application of and

the controller transfer function is

as before. The corresponding disturbance transfer function is

The transfer function of the pre-filter is calculated according to Eq. (10.67), that is

With the assumptions

the pre-filter is simplified to

again as before. The step responses for this case are shown in Figure 10.11.

Problem 10.1   Compensator design - two questions