Dynamics - The Effect of Pole Location on Response
treatise on the effect of pole location on second order response.
You are at Basic Concepts -
Time Response - Second Order Systems - Effect of Pole Location on
Second Order System Response
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of the Responses of Second Order Systems
It is important to understand the response of second order systems because
the response of higher order systems is composed of responses that look
like first order and second order responses. Second order responses
are more complex than first order responses, and you need to spend a little
more time examining them. We begin by examining the response of a
particular second order system. Here is the differential equation
of the system we will use.
This system is the
simplest second order system because the transfer function of this system
has no zeroes. (The RHS of the differential equation is a constant
multiplying the input, u(t).)
We are going to focus on the step response of this system. If the
input is a unit step, the expression for the step response is given below.
This expression is valid when the damping ratio, z,
is less than one.
That response would
look something like the one shown below. The overshoot might be more
or less, the oscillation frequency might not be what is shown, and the
DC gain could change, but this shows the general form of this step response.
that you need to know about this response are the following.
There are other
things that you need to know about this system.
There is a constant, Gdc,
in the response, and when the rest of the response dies out, that is what
remains. If the input is a unit step, then it should be obvious that
Gdc is the DC gain.
The rest of the response
dies out. The rest of the response is a sinusoid multiplied ("modulated")
by a decaying exponential.
The frequency of the sinusoid
is close to the undamped natural frequency, wn.
The exact frequency of
The decaying exponential
has a "time constant" of 1/zwn.
There will be some overshoot
in the response. The overshoot is a function of the damping ratio
- and nothing else. Here is a graph of the overshoot as a function
of the damping ratio. (This
link takes you to the explanation for this function.)
Now, focus on a particular case, and let us see how we can use this information.
Imagine that we have a second order system. It will have a denominator
with this general form.
The poles of the system
are given by:
The poles of the system
give you information about how the system responds because the pole encode
all of the information about the natural frequency and the damping ratio.
Here is a plot of the
poles in the s-plane.
Note the following.
The vertical location
of the pole is the frequency of the oscillations in the response.
The horizontal location
of the pole is the reciprocal of the time constant of the exponential decay.
So, the further the pole
is to the left in the s-plane, the faster the response dies out.
Get familiar with measures
of response time.
The distance of the pole
from the origin in the s-plane is the undamped natural frequency.
(Or, you can interpret that as the length of the vector from the origin
to the pole.)
The damping ratio is given
= cos (Angle of the pole off the horizontal) (That angle is f
in the diagram above.)
Let us examine a particular
+ 8s +
32 = 0
If you use the quadratic
formula on this polynomial, you should find that the roots are:
Now, examine the conclusions
we can draw from this.
So, now you might be wondering what happens if the poles are not complex.
(That would be when the damping ratio is greater that one.) In that
case, the form of the response is shown below, where the s-values are the
poles - using the quadratic formula expression we had earlier.
When you have that expression,
you get a different form for the response.
Since the poles are at
a 45o angle off the horizontal, the damping ratio is
Since the damping ratio
is 0.7, we should expect about a 5% overshoot for a step input.
Since the natural frequency
is around 5. 7 rad/sec or just a little under 1 Hz at 0.9 Hz, we expect
oscillations with a period of around 1.1 second period, but we don't expect
to see them real well since they die out so quickly. (Remember that
That response could
look something like the one shown below. It looks something like
a single time constant system, but if you examine it closely it starts
out with zero derivatie. That's something
for you to explore.