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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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.

        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.

        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.

        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.         Each of these different chores has a method you can use to get the results.
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?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:


P3 What is the time constant for this system?

Enter your answer in the box below,.

Your grade is:


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) = Gdc/(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) = (Gdc/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 = Gdc/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(Gdc/t)e-t/t.  Then, we would have:

Starting Value of Transient = AGdc/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?

Enter your answer in the box below, then click the button to submit your answer.  You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate answer) scale.

Your grade is:


P7If the input is an impulse of 25.0, what is the value of the DC gain?

Enter your answer in the box below, then click the button to submit your answer.

Your grade is:


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.

        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.
Another Second Order System

        Here's a puzzling time response

        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

        Here's a plot that also includes the difference between the known and unknown parts of the response.

        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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