Controller design using frequency domain characteristics

The relations ( ) derived in the previous section for the closed loop behaviour of a element, can be applied also to higher-order systems as long as these systems have a dominant pair of poles. For this class of systems an efficient synthesis method exists as shown in the following. The starting point of this method is the representation of the frequency response of the open loop on a Bode diagram. The specifications of the closed loop that must be met are first given as characteristics of the open loop according to the above section. The synthesis requires in the choice of a controller transfer function , which modifies the open-loop transfer function such that the required characteristics are met. The method consists of the following steps:

Step 1: In general, the characteristics of the time response of the closed loop, , and , are given. On the basis of these values from Table 7.1 the gain , from the rule of thumb for according to Eq. (9.23), the crossover frequency and from the phase margin will be determined, and from the damping ratio .

Step 2: First a P element will be chosen as controller such that the gain determined during step 1 will be met. By inserting additional elements in series (often called compensator or correction elements) will be changed such that the other values from step 1, and , can be achieved while the amplitude response decreases by 20 /decade in the vicinity of the crossover frequency .

Step 3: It must be checked whether the response meets the required specifications. This can be performed directly by determining , and by simulation, or indirectly by using the formula in section 9.1 for the resonant peak according to Eq. (A.25) and the bandwidth according to Eq. (9.16). These values must be verified by calculation of the closed-loop frequency domain characteristic

from the open-loop characteristic. In the case of too large deviations from the approximations of and , step 2 must be repeated in a modified form.

This method does not inevitably deliver a proper controller during the first run and it is a trial-and-error method that leads generally to satisfactory results after some recursions.

For the design of this controller the methods given in section 8 for a standard controller are usually not sufficient. The controller must be composed of different elements - as shown above in step 2. In this procedure two special elements are of important interest, which have to perform a phase shift as shown below:

a) The lead element

The increasing phase shift element is used to increase the phase in a certain frequency range. The transfer function of this element is

For the frequency response

follows with the two breakpoint frequencies

and

A further characteristic is the frequency ratio

From Eq. (9.26) the frequency response

follows. The Nyquist plot is shown in Figure 9.12 and it is a semicircle.

Figure 9.12: Nyquist plot of a lead element

The maximum phase shift

can be determined from the condition for

As shown by the Bode plot in Figure 9.13 the lead element has at high frequencies an undesirable increase in the magnitude response of

Figure 9.13: Bode diagram of the lead element

If Eq. (9.26) is broken down as

the lead element consists of a parallel connection of a P element with gain 1 and a element, which is a special controller (compare Eq. (8.9)). For the step response one obtains

which is shown in Figure 9.14.

Figure 9.14: Step response of the lead element

For the practical design of lead elements the normalised phase diagram in Figure 9.15 is helpful. If the frequency is known, from this diagram the frequency ratio can be determined. The lower breakpoint frequency can be either read from the diagram directly or calculated from Eq. (9.32).

Figure 9.15: Normalised phase responses of the lead element:
; ; ;
= lower breakpoint frequency; = upper breakpoint frequency;
normalised phase responses of the lag element:
; ; ;
= lower breakpoint frequency; = upper breakpoint frequency

Example 9.2.1   The phase response of a transfer function must be shifted by at . The maximum of the phase shift of is from Figure 9.15 for and . With follows for the lower breakpoint frequency or from Eq. (9.32) and with Eq. (9.30) for the upper breakpoint frequency .

b) The lag element

The lag element is used to decrease the magnitude response above a certain frequency. Hereby a undesirable decrease of the phase response occurs in a certain frequency range. The transfer function of this lag element is

For and the breakpoint frequencies and the frequency response is

Also in this case a frequency ratio can be defined as

The decrease of the amplitude response at high frequencies is

Figure 9.16 shows the Nyquist plot and Figure 9.17 the Bode diagram of the lag element. The

Figure 9.16: Nyquist plot of the lag element

Figure 9.17: Bode diagram of the lag element

rearrangement of Eq. (9.35) into

shows that the lag element consists of a parallel connection of a P element with gain and a element with gain and time constant . The step response of this lag element follows from Eq. (9.39) as

and is shown in Figure 9.18.

Figure 9.18: Step response of the lag element

It is easy to see that this relation is equal to Eq. (9.34) but with . For practical working the phase diagram of Figure 9.15 can be used, which is in this case in principle the same as that for the lead element but with different parameters and flipped over.

Example 9.2.2   The magnitude response of an open-loop system should be decreased at by 20 , whereby the maximum phase shift must be 10. From Eq. (9.38) it follows that and from this . With and one obtains from the phase response , and with for the breakpoint frequencies and .