System Identification Using Frequency Response Methods
What is an Identification Problem?
Why Use Frequency Methods?
A More Complex Example
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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.  (The author of this page spent a few hours in an airplane with function generators on the floor of the plane feeding low frequency sine waves to the ailerons, etc.  It's a good recipe for producing a queasy stomach.)  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 frequency response of a system using a sinusoidal input.  Those measurements can be used to produce a Bode' plot of the frequency response.  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 Bode' plot for a system,
To be able to determine the resonant frequency and damping
  ratio for complex poles in the system.
    To be able to determine the DC gain for the system.
    To be able to determine the resonant frequency and damping
  ratio for complex poles in the system.
        We assume that you are familiar with frequency response ideas and plots.  (Click here to review that material.)  At the end of this lesson you will be even more familiar with Bode' plots for 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.

Why Use Frequency Techniques?

        Frequency response methods are widely used.

        Frequency response methods are widely used.  Often you can get a Bode' plot for a system, and you need to determine what the system is.  If you can achieve these goals you're on your way to doing that.

        One widely used way of getting a transfer function is the following.

A First Order Example System

       Here's an example plot.  Look at the features that would help you identify the system.  Note that the complete plot includes both magnitude and frequency plots.

        Why can't we assume that this system is first order?  (Actually it is first order, and the transfer function is 1/(s + 1) - a particularly simple one that lets us make a few points we need to make.)
        Here's another example.  Let's examine this system the same way.  This system is a little more complex and presents a few other interesting features.
There are other things we can learn about this example system.         Since this system is a second order system, it not only has a natural frequency, but there is a damping ratio to be determined.  We can use the resonant peak to get the damping ratio. It's not a correct computation.  The correct formula is: And that's right!

        Now, we have enough information to compute the transfer function.

That gives us a transferfunction of:

G(s) = 7.11/(s2 + 3.77 s + 14.29)

A More Complex Example

       Here's another example.  This one looks fairly horrible.  What can we conclude?  What evidence do we see?

We can note the following.
        We need to answer the question about the magnitude slope.         The best bet is that there is a real pole that causes the phase to begin getting negative around f = .01Hz, and the magnitude to dip there as well.         For a final example, here's a deceptive system.

        This system is really a third order system.  Here's the transfer function.

G(s) = (5 s + 1)/[(s + 1)3]

        Can we conclude anything from this?  Here are some items to note.
What If?

        There are some questions to ponder?

Problems Return to Table of Contents

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