The method of Truxal and Guillemin

For the closed loop shown in Figure 10.1

Figure 10.1: Block diagram of the closed loop to be designed

the behaviour is described by the transfer function

where the numerator and denominator polynomials and must have no common roots. Furthermore, is normalised to and must be valid.

It is assumed that is stable and minimum phase. For the controller to be designed, the transfer function

is chosen and normalised to . Because of the realisability of the controller the relation must be valid. Now, the controller must be designed such that the closed loop behaves like a given transfer function for Eq. (10.1), whereby should be freely chosen under the condition of the realisability of the controller. From the closed-loop transfer function

one obtains the controller transfer function

or with the numerator and denominator polynomials given above

The condition of realisability for the controller is

or

The pole excess () of the desired closed-loop transfer function must be larger than or equal to the pole excess () of the plant. Within these constraints the order of is free. According to Eq. (10.8) the controller contains the inverse plant transfer function . This is a total compensation of the plant as shown in the block diagram of Figure 10.2. For the realisation

Figure 10.2: Compensation of the plant

of the controller Eq. (10.9) is used, not the controller structure as shown in this figure with the plant inverse . As the controller implicitly contains the plant inverse, i.e.  the plant zeros are in the set of the controller poles and the plant poles are in the set of the controller zeros, the plant must be stable and minimum phase as mentioned at the beginning. Otherwise, the manipulated variable and/or the controlled variable will show unstable behaviour.

Example 10.3.1   The plant transfer function is given as

The pole excess of the plant is . According to (10.10) the pole excess of the desired closed-loop transfer function must be

The coefficients of the transfer function that obeys the realisability condition (10.10) are subjected to practical constraints, like the maximum range of the manipulated variable, plant parameter errors and measurement noise in the controlled variable, which is disturbing the controller output. The procedure for the design of will be demonstrated by the following example.

Example 10.3.2   For a plant with the transfer function

a controller should be designed such that the closed loop shows optimal behaviour in the sense of the performance index and has a rise time of .

First, it follows from the realisability condition Eq. (10.10) and from that the pole excess of the desired transfer function is

Inspecting Table A.2 one obtains from the form for and the standard polynomial

From the associated step response it follows from Table A.2 that the normalised rise time

and with this value from the specified rise time the relative frequency is . Eq. (10.13) is now

As for the chosen standard form for the numerator polynomial is , so it follows from Eq. (10.9) that the compensator transfer function is

or

This controller contains an integrator. The time responses are shown in Figure 10.3.

Figure 10.3: Closed-loop behaviour for the example 10.3.2: step response of the controlled variable on step in the set point, step response of the associated controlled variable, step response of the uncontrolled plant

If as a further example the plant given by Eq. (10.11) instead of Eq. (10.12) is taken, then for the same the controller is

For these two very different plants the same closed-loop behaviour for the controlled variable can be achieved.

In the considerations of this section it has been hitherto assumed that is stable and minimum phase. For plants that do not have this properties this design method cannot be applied in this form. The method must be extended to the following:

A direct compensation of the plant poles and zeros by the controller must be avoided, otherwise stability problems would arise. In these cases, the closed-loop transfer function cannot be arbitrary. For a stable non-minimum phase plant the transfer function must be given such that the zeros of contain the right-half-plane zeros of . Whereas for an unstable plant the zeros of the transfer function must contain the right-half-plane poles of . Of course, this restricts the choice of as the following examples demonstrate.

Example 10.3.3   For an all-pass plant with the transfer function

a controller is to be designed such that the closed loop has the desired transfer function

Using Eq. (10.9) one gets for the controller transfer function

This controller gives a direct compensation (cancellation) of the plant zero. This is undesirable as already discussed above, and must be selected as

With Eq. (10.9) one obtains the controller transfer function as

Because of this choice of , the closed loop shows also all-pass behaviour. This effect is more intense the smaller the time constant . Figure 10.4 shows the time responses of this control system.

Figure 10.4: Closed-loop behaviour of the example 10.3.3: step response of the controlled variable on step in the set point, step response of the associated controlled variable, step response of the uncontrolled plant ( ; )

Example 10.3.4   The transfer function of the unstable plant

is given and a controller is required for which fulfills the realisability condition and for which the zeros of must contain the plant pole . This is expressed by the approach

whereby is chosen such that

is valid. In the present case should be chosen such that . From this follows . In order to have a stable one can take

and obtain

Observing the realisability condition, it follows that

Comparing the coefficients and taking one obtains

and finally

The parameters and are still free and may now be chosen such that taking an acceptable behaviour of the manipulated variable into account, a given damping ratio and natural frequency for can be reached. Without going into details,

will be chosen for the present case and from this it follows that

The desired closed-loop transfer function will be

and

The conditions for the design are fulfilled and for the controller transfer function one obtains from Eq. (10.8) or Eq. (10.9)

This design obviously produces a PI controller. The time responses of this control system are shown in Figure 10.5 for . The relatively large maximum overshoot cannot be avoided with an acceptable behaviour of the manipulated variable.

Figure 10.5: Closed-loop behaviour of the example 10.3.4: step response of the controlled variable on step in the set point, step response of the associated controlled variable