What is an Identification Problem?

First Order Systems
Step Response
Impulse Response

Second Order Systems

Problems

        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.

        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.

        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.

        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.

        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.

• 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.

• 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:
• G(s) = G_dc/(st + 1)
• 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.

        Here is the step response of a system.  The input to the system is a step of magnitude 2.0.

P1A Is the system nonlinear?

        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.

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.

If the input is a unit impulse, then we can measure the starting value of the transient.

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:

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!

P4Here is the impulse response of a system.  The input to the system is an impulse of magnitude 25.0.

P5A Is the system nonlinear?

        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.

• 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.

        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 time-constant 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.

        Here's the same response overlaid with the function:

• 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 time-constant response.
• But, what's the first part 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.

• Problem Ident1P1 - Step Response, 1st Order System
• Problem Ident1P2 - Impulse Response, 1st Order System
• Problem Ident1P3 - Impulse Response, 1st Order System
• Problem Ident1P4 - Impulse Response, 1st Order System
• Problem Ident1P5 - Impulse Response, 1st Order System
• Problem Ident1P6 - Step Response, 1st Order System
• More Complex Problems
• A guided problem in system identification
• Al Dente's Oven

• System Identification in the Time Domain
• System Identification in the Frequency Domain