This example examines the effects that adding either a pole or a zero to
the open-loop system has on the step response of the standard second-order
system. The open-loop and closed-loop transfer functions of the standard
second-order system are shown below, and the step response for damping
ratio = 0.5 and undamped natural frequency = 4 r/s is shown. The
closed-loop poles are located at s = -2 +/- j3.46.
Step Response: zeta = 0.5, wn = 4 r/s
The standard second-order system has no zeros in the transfer function.
If the system is modified to include a zero in it, the overshoot and
settling time are greatly affected by the location of that zero. Assume
that the open-loop system is modified as shown below by the addition of a
zero. The gain of the transfer function is also modified by the 1/z term
so that the velocity error constant is not changed from its original value
as the zero is moved. This normalizes the problem that is being studied.
The values of zeta and wn will be unchanged.
Six different values will be used for the location of the zero. Each
of them places the zero in the left-half of the s-plane. The locations of
the zero are: s = -0.2, -0.5, -1, -2, -5, -10. The two open-loop poles
are fixed at s = 0 and s = -4. The step responses for the closed-loop
system with the different values for z are shown in the next figure. They
should be compared with that of the standard second-order system. The
locations of the closed-loop poles are given in the following table.
Compare the real value of the pole closest to the jw axis with the
settling time for the corresponding step response. Also note that for the
first four values of z (0.2, 0.5, 1, 2), both closed-loop poles are real
and there is no overshoot in the step response. This is similar to the
overdamped case for the standard second-order system. For the last two
values of z (5, 10), the closed-loop poles are complex conjugates and
there is some overshoot in the step reponse. For z=5, the effective
damping ratio of the complex poles is 0.9, which would correspond to an
overshoot of 0.15%. For z=10, the effective damping ratio of the complex
poles is 0.7, which would correspond to an overshoot of 4.6%. Although
the actual overshoots are larger than that in each case, the concept of
relating the time domain step response characteristics with the effective
damping ratio and natural frequency still gives a good "ball park" figure.
The rise time increases as the zero is moved further to the left in the
s-plane. When the zero is close to the jw axis, one of the closed-loop
poles is also close to the jw axis, and the settling time is longer than
in the standard case.
Step Response with
Added Zero
Table of Closed-Loop Pole
Locations with Added Zero
The standard second-order system has two poles in the transfer function.
If the system is modified to include a third pole in it, the overshoot and
settling time are again greatly affected by the location of that pole,
particularly in terms of whether it is to the left or the right of the
original open-loop pole at s = -4. Assume that the open-loop system is
modified as shown below by the addition of a third pole. The gain of the
transfer function is also modified by the p term so that the velocity
error constant is again unchanged from its original value.
Five different values will be used for the location of the extra pole.
Each of them places the pole in the left-half of the s-plane. The
locations of the pole are: s = -1, -2, -5, -10, -20. The two original
open-loop poles are still fixed at s = 0 and s = -4. The step responses
for the closed-loop system with the different values for p are shown in
the next figure. They should be compared with that of the standard
second-order system. The locations of the closed-loop poles are given in
the following table. Note that there are now three closed-loop poles.
Again, compare the real value of the pole closest to the jw axis (complex
in each case) with the settling time for the corresponding step response.
Also compare the ratio of imaginary part to real part for the closed-loop
poles closest to the jw axis. This ratio is related to the effective
damping ratio by zeta_eff = cos(atan(imag(clp)/abs(real(clp)))). Note
that the overshoot and settling time decrease as the extra pole is moved
to the left. However, even with the extra pole at s = -20, the overhoot
and settling time are still larger than the original second-order system.
Step Response with Added Pole
Table of Closed-Loop Pole Locations with Added
Pole
The ability to analyze the
behavior of higher-order systems and systems with zeros based on
second-order approximations is very useful. Although the calculations
developed for the standard second-order system will not yield exact
results for other system, they can provide an acceptable approximation in
many cases. The most accurate calculation will be for settling time; that
will be 4 divided by the absolute value of the pole (or real part of
complex poles) closest to the jw axis in most cases. This is called the
dominant pole (or pair of poles).
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Lastest revision on Wednesday, June 7, 2006 12:05 PM
