In the following the most important characteristics in the frequency domain of the open and closed loop for command inputs for a closed loop having a transfer function with two complex poles will be given. This section is based on the relationship between the frequency characteristics and the performance indices in the time domain for the closed loop introduced in section 7.3.1.

A closed loop showing a step response according to Figure 7.8 has a frequency response with a peak as qualitatively shown in Figure 9.1. For describing this behaviour the following characteristics mentioned earlier can be used:

These characteristics are shown in Figure 9.1.

The closed loop for a element has the transfer function

according to Eq. (4.54) with .

The *
natural frequency*
and the *
damping ratio*
characterise the control behaviour completely. This can be
used as a good approximation for other transfer functions if they contain a
*dominant
pair of poles*
according to Figure 9.2.

This pair of poles is assumed to be the closest pair to the axis in the domain and therefore it describes the slowest mode and influences the dynamical behaviour of the system very strongly provided the other poles are sufficiently far away on the left-hand side of the plane.

The step response for the transfer function of Eq. (9.1) is

and according to Eq. (A.31) it can be put into the more suitable form

where for or is valid. From Eq. (9.3) the weighting function, which follows by differentiation, is

Therewith the conditions are accomplished in order to determine the maximum overshoot, rise time and settling time that depends on the characteristics in the frequency domain, e.g. natural frequency and damping ratio . With and the interesting items and can be calculated directly by the Eqs. (A.24) and (A.25).

- a)
*Determination of the maximum overshoot*:For calculation of the time will be determined at which will be first zero according to Eq. (9.4). This is when the the function in Eq. (9.4) has

This gives

From Eq. (9.2) and (9.3) it follows that the maximum overshoot is

The maximum overshoot is therefore only a function of the damping ratio as shown in Figure 9.3.- b)
*Determination of the rise time*:In the following the rise time will not be calculated by the tangent at the turning point, but by the tangent at time (see Figure 7.8), where reaches 50% of the stationary value . So the time must be determined, for which according to Eq. (9.2) and (9.3) is valid. From Eq. (9.2) it follows that

This equation for the product must be evaluated numerically. One gets a function of the form

From Eq. (9.4) it follows that

and from this together with Eq. (9.7) the normalised rise time is

which also only depends on the damping ratio . This relationship is shown in Figure 9.4.- c)
*Determination of the settling time*:Using Eq. (9.3) the decay of the amplitude to a value less than for can be estimated from the envelope of the response

From this the normalised settling time

follows. If is chosen, one gets

This relationship is shown in Figure 9.5 together with the normalised rise time that will be shown later in Figure 9.9.

Comparing the results from Figures 9.3 to 9.5 one can summarise as follows:

- The maximum overshoot
depends only on the damping
ratio .
- A change of the damping ratio in the range of
approximately
behaves contrary to the settling time
compared to the rise time
,
i.e. an increase of the damping ratio in order to obtain a
smaller settling time
increases the rise time
.
- For a fixed damping ratio the parameter
determines the speed of the control loop. A large value of
shows a small settling and rise time.

For the practical application of the diagrams in Figures 9.3 to 9.5 the following example is given.

With the given value of one obtains from Figure 9.3 the damping ratio

For this value of with the natural frequency

follows from Figure 9.4. But from Figure 9.5 for the required natural frequency is

The rise time of is the sharper requirement. Therefore, must be chosen. For the pair from Eq. (A.24) the resonant peak frequency

and from Eq. (A.25) the resonant peak

giving

respectively, can be determined.

In order to estimate the bandwidth for a given damping ratio , the relationship between these two parameters is often needed. Based on the bandwidth as defined in Figure 4.19, that is

it follows after a short calculation using Eq. (9.1) for and that

and

Furthermore, one obtains using Eqs. (9.8) and Eq. (9.11)

The graphs of the functions , and are shown in Figure 9.6.

By approximation of , and the following 'rules of thumb' can be determined:

for | (9.15) |

Applying these rules to Example 9.1.1 with and , the bandwidth can be determined either from Eq. (9.14) as

or with from Eq. (9.16) as

The Bode plot of a typical corresponding open-loop frequency response is shown in Figure 9.7. From this and from Eqs. (5.19) and (5.20) one can use the characteristics:

Since the closed-loop transfer function has been assumed to be approximated by Eq. (9.1), the corresponding open-loop transfer function isor

with and . The frequency response of Eq. (9.17) and (9.18) is shown in Figure 9.8. This Bode plot is considerably different from that of Figure 9.7. The system in Figure 9.7 does not have an integrator. Furthermore, it is of order higher than two, as the phase characteristic exceeds the value of -180. But close to the crossover frequency , both Bode plots show a similar behaviour. If for the magnitude response of a given system is valid for and for , then can often be approximated in the vicinity of the crossover frequency by Eqs. (9.17) and (9.18). The associated transfer function contains a dominant conjugate complex pair of poles. In order to transfer the known performance indices of a second-order system to control systems of higher order, the design must be performed such that the magnitude response decreases by 20 /decade in the vicinity of . For Eq. (9.18) this is only possible if is valid (compare with Figure 9.8). From Eq. (9.17) one obtains under the condition

after a short calculation

With for from

the condition follows. When for the damping ratio a value of is chosen, then it is guaranteed that the magnitude response of the open loop falls off in the vicinity of the crossover frequency by 20 /decade. Figure 9.9 shows that only the interval is a range of suitable damping ratios, since both, the rise time and the maximum overshoot, show acceptable values from the performance index point of view. This also means that the phase and gain margin and show proper values.

From these considerations one can conclude that for control systems with minimum-phase behaviour, which can be approximately described by a element, the magnitude response of the open loop must decrease by 20 /decade in the vicinity of the crossover frequency if a good performance is to be achieved, i.e. a sufficient large phase margin .

As already mentioned in section 5.3.6, the crossover frequency is an important performance index of the dynamical behaviour of the closed loop. The larger , the larger is the bandwidth of in general, and the faster is the reaction to set-point changes. For the frequency response for set-point changes one gets approximately

From this, the asymptote of the magnitude response of can be determined (Figure 9.10). If decreases in the vicinity of by 20 /decade, then for this range

is valid, and thus it follows that

behaves in this range as a
element. As generally known, the
magnitude response of a
element decreases by 3
at
the
breakpoint frequency (here
). Therefore, the crossover
frequency
of the open loop is just the bandwidth
of the closed loop, i.e.
. From this it follows that for minimum phase
systems the frequency response of
can be determined piecewise from
according to Figure 9.10. Thereby
for fulfilling Eq. (9.20) in the *lower
frequency range* the value of
and therefore
also the
loop gain must be large to hold the
steady-state error as small as possible.
This lower frequency range of
is
responsible for the steady-state behaviour of
the closed loop, whereas the *middle frequency range* is essential for the transient behaviour
and is characteristic for the damping. In order to avoid
non-suppressable high-frequency
disturbances of the set point in the closed loop,
and therefore also
must decrease quickly in the *upper
frequency range*.

From these ideas it is now possible to specify besides Eq. (9.19) additional important relationships between the characteristics of the time response of the closed loop and the characteristics of the frequency response of the open and partly of the closed loop. Using Eq. (9.8) and Eq. (9.19) it follows immediately that

Figure 9.11 shows the graphical representation of . It is easy to check that this curve can be described in the range of by the approximation

or

A further relationship may be determined from the crossover frequency for the phase margin as

which yields

Figure 9.11 also shows this function. By superposition of with one can show that in the range of the mainly interesting values of the damping the approximation

is valid. This 'rule of thumb' can only be applied for values of the variables with the given dimensions in squared brackets.