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
.
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
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).
:
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
 |
as shown in Figure 9.3.
:
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
 |
must be evaluated
numerically. One gets a function of the form
 |
. This
relationship is shown in Figure 9.4.
:
Using Eq. (9.3) the
decay of the amplitude to a value less than
for
can be estimated from
the envelope of the response
 |
is chosen, one
gets
that will be shown
later in Figure 9.9.
Comparing the results from Figures 9.3 to 9.5 one can summarise as follows:
depends only on 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
.
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
 |
with
the natural
frequency
 |
the required natural
frequency is
 |
is the sharper requirement.
Therefore,
must be chosen. For the
pair
from Eq. (A.24) the
resonant peak frequency
 |
 |
 |
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
 |
and
that
,
and
are shown in Figure 9.6.
By approximation of
,
and
the following 'rules of thumb' can be determined:
 for  |
(9.15) |
and
, the bandwidth
can be determined either from Eq. (9.14) as
 |
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:
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
 |
for
from
 |
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
 |
 |
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
. It is easy to check that this curve can be
described in the range of
by the approximation
for the
phase margin as
 |
 |
|
 |
with
one can show
that in the range of the mainly interesting values of the damping
the approximation
 |
,
,
.
(in %) relative to
as function of the damping ratio 
(normalised rise time) as a function of the damping ratio 
and normalised rise time
,
and
depending on the damping ratio 
for 
for 
,
.
from
in the
Bode diagram
for 
or 
