System
Identification Using Time Domain Data
What is an Identification
Problem?
First Order
Systems
Step
Response
Impulse
Response
Second
Order Systems
Problems
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Models  Time Response Methods
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What
Is An Identification Problem?
You have a background in linear systems. Often when you are learning
linear systems you encounter problems like the following.

Given a system described
by one of the following

A transfer function,

A linear differential
equation with constant coefficients that relates the input and the output
of the system,

An impulse response,

A set of state equations,

And, given an input to
the system,

Determine the output (response)
of the system.
In
other words, if you are given a system and an input, you should be able
to find the output of the system. That's a problem that could be
difficult, but which has a straight forward approach that will get you
an answer. The approach might be different depending upon how the
system is described, but the approach is always straight forward.
It's not very likely that you will have that problem in your job.
If you do, there will probably be some sort of complication  like a nonlinearity
in the system  that makes your linear systems approach inapplicable.
However, there are many times when you have an inverse problem. You
may not know the system description, and you may need to figure out a description
for the system  transfer function, impulse response, differential equation,
state equations, whatever  knowing a sample of the input and the output.
That's a whole different kettle of fish.
If you think about linear systems, you probably got the idea that you could
always figure out a transfer function for a circuit. But, there are
many systems for which you can't get a good transfer function. Aircraft
may have transfer functions that vary widely with different conditions
(airspeed, altitude, fuel load, atmospheric conditions, etc.) and it may
not be easy to compute those transfer functions from physical data.
Chemical plants are another example of something that you have to control
but where you can't get a good handle on the transfer function of the system.
In situations like that you may need to have some tools that will let you
get a system description from a record of input and output signals.
Let's think about how you would go about that. We'll start with a
simple situation after we examine the goals for this lesson. The
method we will described is shown pictorially on an another page. Click
here for that.
Goals For This Lesson
It's often possible to measure the time response of a system to a controlled
input. In a situation like that you can use that measured data to
calculate a transfer function for a system. In this lesson we look
at some simple systems with these goals in mind.
Given a measurement of the step response of a linear system,
Be able to recognize the responses of first order systems, and to determine
the time constant and DC gain of a system.
Be able to recognize the responsese of second order systems with complex
poles or with two real poles.
Be able to determine DC gain, natural frequency and damping ratio for systems
with complex poles.
Be able to determine DC gain, and values for the two poles for systems
with two real poles.
At
the end of this lesson you will be familiar with first order systems and
second order systems. At that point we will examine some higher order
systems to see how what you have learned about first and second order systems
can be applied there.
A
First Order System
We are going to assume that you have data available from a step response
measurement. Time response measurements are particularly simple.

A step input to a system is simply a suddenly
applied input  often just a constant voltage applied through a switch.

The system output is usually a voltage, or a
voltage output from a transducer measuring the output.

A voltage output can usually be captured in
a file using a C program or a Visual Basic program.
You
can use responses in the time domain to help you determine the transfer
function of a system.
First we will examine a simple situation. Here is the step response
of a system. This is an example of really "clean" data, better than
you might have from measurements. The input to the system is a step
of height 0.4. The goal is to determine the transfer function of
the system.
Now, let's look closely at the response we
have. We will focus on features of the response that are informative.

The overall shape of the
response appears to be a decaying exponential with an added constant value.
Generally, the shape seems to indicate a single decaying exponential which
would indicate one time constant  a first
order system.

At t = 0, the slope of
the response seems to jump immediately to a positive value. That's
an indication that the system could be first
order.
With
these observations it is a reasonable hypothesis that this is the step
response of a first order linear system. To check that hypothesis,
we should do the following.

Determine the DC gain,
G_{dc},
of the system,

Determine the time constant,
t,
of the system,

Form the transfer function,
G(s),

Check the response of
the system and compare it to the response
given in the graph to determine how well the
model we have constructed predicts the measurements we were given.
You need to remember that you are not done until you have some confidence
that your model accounts for the data you started from.
Each
of these different chores has a method you can use to get the results.

To determine the DC gain,
the simplest thing to do is to measure the steady state value of the output
after the transient has died out. Divide the steady state value by
the steady state (constant) value of the input step to get the DC gain.
For the example system above, since the output settles at 1.0 and the input
is 0.5, the DC gain is 2.0.

To determine the time
constant you have a number of options. Here are three of them.

Extrapolate the initial
slope to the steady state value. The time intercept should be the
time constant.

Determine when the transient
has moved 63% of the way from the starting value to the steady state.
The time when that happens is the time constant.

Extract a decaying exponential
from the transient. In the example you can get a decaying exponential
by subtracting the transient from the steady state value. Then do
the following.

You should have a function
that looks like: A e^{t/}^{t}.

Take the log of the data
and you will have ln(A)  t/t.
In other words, if you take the natural logarithm of the data and plot
it you should have a straight line with a negative slope when you plot
that against time, t.

Use any straight line
fit method you like. Mathcad and Matlab have regression functions
that will give you the intercept, ln(A),
and the slope,  1/t.

Form the transfer function.
In other words, use the numerical values you obtained in the steps above
to form:

Calculate the step response
of the system you have found. You can use any number of methods.

Inverse transform the
function you get by multiplying G(s) by 1/s, and plot the result.

Take the step response
of the system and plot it in Mathcad, Matlab or Excel.

Compare your results by
plotting your calculated response on top of the data.
Problems
Here is the step response of a system. The input to the system is
a step of magnitude 2.0.
P1 Is
there any evidence that the system is nonlinear?
P1A Is
the system nonlinear?
P2 If
the input is a step of 2.0, what is the value of the DC gain?
P3 What
is the time constant for this system?
First
Order Systems  Impulse Response
You might be tempted to think that you can't get a transfer function from
an impulse response. That's not true. Imagine that this is
the response to a unit impulse. (Click
here to review how to obtain the impulse response.)
Since you know that this is the response
to a unit impulse, you can get the transfer function from this response.
Consider a first order system transfer function.
G(s) = G_{dc}/(st
+ 1)
The response of this system to a unit impulse
is just the inverse Laplace transform of the transfer function. That
means the impulse response is given by this expression.
g(t) = (G_{dc}/t)e^{t/}^{t}
If the input is a unit impulse, then we can
measure the starting value of the transient.
Starting Value of Transient
= G_{dc}/t
If the impulse is not a unit impulse, we
can still get the same information as long as we know the size of the impulse.
Imagine that you have an impulse, Ad(t).
Then, the response would be A(G_{dc}/t)e^{t/}^{t}.
Then, we would have:
Starting Value
of Transient = AG_{dc}/t
We can always get the ratio of the DC gain
to the time constant. Then, you can use any technique you want to
get the time constant. That will give you both the DC gain and the
time constant, and you'll have the transfer function.
CAUTION:
You may be tempted to assume that you can measure any transient that looks
like the impulse response. For example, imagine that your system
is an oven. Then an impulse input would be a very high heat input
for a very short time. As long as you know the amount of heat input
 the size of the impulse  you can compute the transfer function.
However, if you have an oven that starts out at some temperautre  some
initial condition  and you don't know the amount of heat that was put
into the oven, you could measure a transient like the one above.
However, without knowing the input (and knowing, the initial condition
is not the same as knowing the input.) you can't compute a transfer function.
Knowing
an initial condition is not the same information as knowing an input!
Problems
P4Here
is the impulse response of a system. The input to the system is an
impulse of magnitude 25.0.
P5Is
there any evidence that the system is nonlinear?
P5A Is
the system nonlinear?
P6What
is the time constant for this system?
P7If
the input is an impulse of 25.0, what is the value of the DC gain?
A
Second Order System
Here's a more complex response. This is also the response to a unit
step input. There are interesting aspects to this response also.

The decaying
oscillations tell you that this is at
least a second order system. There is at least
one pair of complex  underdamped  poles. Danger  there could be
more to this system. It could be more than second order, but it surely
is not first order.

If this is a second order
system, then you need to find the following:

DC gain, G_{dc},

The damping ratio
z,

The undamped natural frequency,
w_{n}.
Thus, if you can calculate the DC gain, damping ratio and natural frequency
you have measured the transfer function. The problem is that you
need to get information from the step response  like the one shown above
 that will let you calculate what you need. There are some relationships
that will help you.

The percent
overshoot is related to the damping
ratio. What's more, the relationship
is fairly direct and does not involve the natural frequency or the DC gain.
Click
here for more information.

The
frequency of the decaying oscillations is
related to the natural frequency.
The relationship also involves the damping ratio so you need to be careful
here. Click
here for that.

DC gain
is an easy one. Measure steady state, and divide steady state by
the input. If you're in doubt, click
here.
Another
Second Order System
Here's a puzzling time response

There are no oscillations in the system, so
there are probably no complex poles.

The last part of the response looks like single
timeconstant decay.

There's a glitch in the beginning part of the
transient. Unlike the first order system, the derivative of the output
does not suddenly jump, but increases for a short while.
These
characteristics point to this being a second order system with two real
poles. Let's examine that idea in some more detail.
Here's the same response overlaid with the function:
g(t) = 1.25
e^{t} + 1

You can use the usual
techniques to find this part of the response if you focus on the last part
of the response.

That reinforces the idea
that the end of the response looks like a single timeconstant response.

But, what's the first
part of the response?
Here's
a plot that also includes the difference between the known and unknown
parts of the response.

When we take the difference,
it looks like we have a quickly decaying exponential. That's the
curve shown in black on the figure.

If we can get the parameters
of that response, we may be able to reconstruct the entire response.

Note:
Two time constants implies at least a second order system.
This
example shows how to break down the response of a second order system with
two real poles into its' constituent parts. From here, if you can
determine the time constants you can get the transfer function.
Problems
Links
To Related Lessons
Other Lessons on System
Identification
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