Frequency characteristics
Characteristics in frequency and time domain
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.
(in %) relative to 
as function of the damping ratio 
(normalised rise time) as a function of the damping ratio 
and normalised rise time 
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.
of a closed loop with a dominant pair of poles should show a maximum overshoot of 
, a rise time of 
and a settling time of 
. How must the damping ratio and the natural frequency be chosen?
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.
, 
and 
depending on the damping ratio 
By approximation of , and the following 'rules of thumb' can be determined:
for 
 for  |
(9.15) |
for 
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 is
or
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
 |
from 
in the Bode diagram
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
for 
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.
 |
Next: Controller design using frequency Up: Design of controllers using Previous: Design of controllers using Contents Christian Schmid 2005-05-09
