An Introduction To System Dynamics - First Order Systems
Introduction
Goals
The System
Impulse Response
Problems
You are at Basic Concepts - Time Response - 1st Order Step and Impulse Response
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Why Worry About Time Response Behavior?

        Time behavior of a system is important.  When you design a system, the time behavior may well be the most important aspect of its' behavior.  Points you might worry about include the following.

        These examples are intended to show you that the ability to predict details of how a system responds is important when you design systems.  These are but a few of many different aspects of time behavior of a system that are important in control system design.  The examples above really are talking about aspects like:         When you design systems or circuits you often need to worry about these aspects of the system's time behavior.  Before you build your system, you want to know how it will perform.  You need to make predictions.

        In this lesson, we will begin to examine how it is possible to predict aspects of the time behavior of a system.  We'll do that by starting with a first order system and examining the parameters of that kind of system that control their time behavior.  We'll do that in ways that will let us generalize concepts to more complex systems - and there are lots of more complex systems you'll be worrying about.  With that under your belt, you will have the knowledge you need to predict how first order systems behave.  That will set the stage for learning about more complex systems.


Goals For This Lesson

        There are a number of goals for you in this lesson.

 First, if you have a first order system, you need to be able to predict and understand how it responds to an input, so you need to be able to do this.

  Given a first order system,
Determine the impulse and step response of the system.
        Secondly, you may go into a lab and measure a system, and if it is first order, you need to be able to do this.
Given the step response of a first order system,
Determine the parameters - DC gain and time constant of the system.
This second goal is considerably different from the first.  In the first goal, you are given information about the system and the input to the system, and have to determine how the system responde.  In the second goal, you are given information about the input and output of a system and have to determine what the system is.  That's a completely different kind of problem, but in both cases you will need to learn the material in the rest of this lesson.  There is a separate lesson on system identification where you look at input and output and work to determine a model for the system.  Click here to go to that lesson.


The System - And Some Examples

       The simplest possible dynamic system is one which satisfies a first order, linear, differential equation.  Here's a generic form of the differential equation.  A block diagram representation of the system is given at the right.
tdx/dt + x(t) = Gdcu(t)

x(t)  =  Response of the System,
u(t)  =  Input to the System,
t  =  The System Time Constant,
Gdc  =  The DC Gain of the System.

        The parameters you find in a first order system determine aspects of various kinds of responses.  Whether we are talking about impulse response, step response or response to other inputs, we will still find the following relations.         You also need to note that a system that satisfies the differential equation above has a transfer function of the form:
G1(s) = X(s)/U(s) = Gdc/(st + 1)


Some Examples of First Order Systems

        The differential equation describes many different systems of many different types.  We can look at some of the systems whose behavior is described by a first order differential equation like the one above.

        The parameters you find in a first order system determine aspects of various kinds of responses. Whether we are talking about impulse response, step response or response to other inputs, we will still have the following quantities and system parameters.

x(t)  =  Response of the System,
u(t)  =  Input to the System,
t  =  The System Time Constant,
Gdc  =  The DC Gain of the System.
Every system will have an input which we can call u(t), and a response we will denote by x(t).  Each system will also have a time constand and a DC gain. Now, let's look at some example systems.  The first system is not entirely whimsical.

A Cartoon Biplane

.

        At the right is a movie of an airplane - actually a biplane - in which the pilot suddenly changes the controls so that the altitude of the biplane changes.  The new steady state altititude is higher than the previous altitude.  This system shows time constant behavior as the airplane changes altitude.  You can click the button at the lower right to see the path followed by the biplane.  Click on the button and release the mouse outside the button to let the path be shown continuously.  This is an example of a time-constant you can see.

A Siren

        An old-fashioned siren is an example of a time constant you can hear.  Listen to a siren as it starts up.

A Thermal System

        Take a pair of cotton or wool gloves and put them in a refrigerator for a half hour.  Then take them out and put them on your hands.  As they warm up you will experience a time constant you can feel.

A Resistor-Capacitor Circuit

        In this system, applying Kirchoff's Laws and the voltage-current relations for a resistor and capacitor produces a first order linear differential equation relating the output voltage to the input voltage.

 The circuit diagram          The circuit equation

        Here's another system that satisfies  first order differential equation.

A Simple Thermal System

        Here is a heated Space with Insulation.  In this system heat flows into a heated space and the temperature within the heated space follows a first order linear differential equation.

 The system diagram            The system equation

        Here's one more system that satisfies a first order differential equation.

Your memory

        Psychologists tell us that memory obeys the same kind of differential equation as the previous two systems.  If you learn information, what you retain satisfies a first order differential equation.


        As you think about the systems above they come from very diverse places, including aerodynamics, themal dynamics, circuit theory and psychology.  However, there is a common mathematical description for all of those systems.  That's what you need to focus on in this lesson.  When you learn about first order system dynamics you are learning a topic that:
Has applicability to a wide variety of areas
Is a good introduction to more complex system dynamics, like second order systems and more complex systems of higher order.
        Learning about first order systems is important for these reasons.  We'll start learning about first order systems by learning how a first order system responds to two inputs, the unit impulse and the unit step.  We choose these inputs because they are basic kinds of inputs, and the characteristics of impulse responses and step responses for systems give insight into how the system will behave for other kinds of inputs.
Impulse Response Of A First Order System

        The impulse response of a system is an important response.

The impulse response is the response to a unit impulse.
The unit impulse has a Laplace transform of unity (1).  That gives the unit impulse a unique stature
         If a system has a unit impulse input, the output transform is G(s), where G(s) is the transfer function of the system.  The unit impulse response is therefore the inverse transform of G(s), i.e. g(t), the time function you get by inverse transforming G(s).  If you haven't begun to study Laplace transforms yet, you can just file these last statements away until you begin to learn about Laplace transforms.  Still there is an important fact buried in all of this.
Knowing that the impulse response is the inverse transform of the transfer function of a system can be useful in identifying systems (getting system parameters from measured responses).
        In this section we will examine the shapes/forms of several impulse responses.  We will start with simple first order systems, and give you links to modules that discuss other, higher order responses.

        A general first order system satisfies a differential equation with this general form.

If the input, u(t), is a unit impulse, then for a short instant around t = 0 the input is infinite.

        Let us assume that the state, x(t), is initially zero, i.e. x(0) = 0.  We will integrate both sides of the differential equation from a small time, e, before t = 0, to a small time, e, after t = 0.  We are just taking advantage of one of the properties of the unit impulse.

        The right hand side of the equation is just Gdc since the impulse is assumed to be a unit impulse - one with unit area.  Thus, we have:

        We can also note that x(0) = 0, so the second integral on the right hand side is zero.  Also the leftmost integral becomes:

x(e) - x(-e)

       In other words, what the impulse does is it produces a calculable change in the state, x(t), and this change occurs in a negligibly short time (the duration of the impulse) after t = 0.  Later we should realize that for this to happen, the duration of the impulse should be much less than the time constant, t.  In any event, we can calculate the change in x as:

Dx = Gdc / t

So, the way we have to view the effect of the impulse in this system is that the impulse changes the value of the state, x(t), in a short time right after t = 0.  In effect, the system is now going to run from a new initial condition, Dx (given above), and there will be no input after t = 0 because the impulse has stopped - gone away - become identically zero.

        That leads us to a simple strategy for getting the impulse response.

  • Calculate the new initial condition after the impulse passes.
  • Solve the differential equation - with zero input - starting from the newly calculated initial condition.
    If u(t) = 0, and it x(0-) = 0, the solution (when the input is a unit impulse) is:

x(t) = x(0+)e-t/t = (Gdc/t)e-t/t

Another viewpoint

        We can also look at this problem in the transform domain.

  • If we have this differential equation:
  • Then the system has a transfer function:
  • Then, if the input is a unit impulse, the impulse has a transform of 1, so that the transfer function is the transform of the output.  Takine the inverse transform of the transfer function, we find that the impulse response is:
x(t) = (Gdc/t)e-t/t

        And, this is exactly what we found on the earlier using a different method!

        Let us summarize what we now know about impulse responses for first order systems.

  • First order systems satisfy this generic differential equation.
  • For a unit impulse input, the response is:
x(t) = (Gdc/t)e-t/t
  • Since the system is linear, larger impulses will produce proportionally larger responses.
  • The impulse response is the inverse transform of the transfer function of the system with the differential equation above:

        Now, we need to examine what the impulse response looks like.  Let us look at an example.


Example

E1   Consider a system with the following parameters.

  • t  =  0.1 sec
  • Gdc  20
The problem is to determine the impulse response of a system that has these parameters.  We know the form of the impulse response:

x(t) = (Gdc/t)e-t/t

With the parameters above, the impulse response is:

x(t) = (Gdc/t)e-t/t

x(t) = (20/.1)e-t/.1

x(t) = 200e-10t

And even though the DC gain is only 20, the impulse response starts at a value of 200!


E2:

        At the right is the impulse response of a system - i.e. the response to a unit impulse.  The system starts with an initial condition of zero just before the impulse comes along at t = 0, so x(0-) = 0.  Here the problem is the inverse of the problem above.  We are given the impulse response, and we need to compute the parameters of the system.

        We can see that the impulse response immediately jumps to a value of 20.  Then, if the form of the impulse response is given by:

x(t) = (Gdc/t)e-t/t
we must have:
Gdc/t = 20

That's part of what we need.  If we can now find either the DC gain or the time constant, then we can compute both parameters.  If we can find the DC gain, we can use the equation just above to find the time constant and vice-versa.

        The problem is that there doesn't seem to be an easy way to get the DC gain.  However, we might be able to compute the time constant because the time constant is what determines the rate of decay.  Let's try to use the other information in the response curve.  At 2 seconds, it looks like the response has decayed to 8 or maybe a little under 8.  That looks like a good point to use because we can get the values relatively accurately.  (And, if that's not possible, you'll have to use whatever you think will work the best.)  Anyhow, at 2 seconds we have:

x(t) = (Gdc/t)e-t/t

x(2) = 8 = 20e-2/t

Now, we can rearrange this last equation:
e-2/t= (8/20) = 0.4
So
-2/t = ln(0.4) = -0.9163
or
t = 2/0.9163 = 2.2 sec
Finally, we can use this information to determine the DC gain.  Previously we found:
Gdc/t= 20
so:
Gdc = 20*t =  20*2.2 = 44
And, we can draw a few conclusions from these two examples.
  • Calculating the impulse response is straight-forward.  Given the system parameters it is not difficult to calculate - predict - the response of the system.
  • The inverse problem is somewhat more difficult.  Given a response, you will have to be more inventive to determine what the system was that produced the given response - the  system identification problem.
  • The underlying theory is the same.  You use the same general principles to solve both problems, but the way you have to use the information makes the identification problem more difficult.
        Now, let us move on to the step response of this system.
Step Response Of A First Order System

        A standard first order linear system will satisfy this differential equation.

A first order linear system will almost always have this form - or can be put into this form.  The variables and parameters of this system are:

x(t)  =  Response of the System,
u(t)  =  Input to the System,
t  =  The System Time Constant,
Gdc  =  The DC Gain of the System.
        Next, we are going to investigate what the response of this system is when the input is a constant.

        We are going to solve the differential equation describing a first order system and we are going to assume that that input is a constant.  That's an important special case for this system.  Let's take the differential equation, and rearrange it so that we can integrate it.

Now, we can rearrange to integrate.

And, we can integrate this fairly easily, obtaining:

which becomes:
x(t) - Gdcu(t)= e-t/t(x(0) -Gdcu(t))
or:

x(t) = e-t/t(x(0) -Gdcu(t))+ Gdcu(t)

    Now, let's look at the details of this expression,

  • We assumed the input, u(t), was constant.
  • When t = 0, i.e. when the system starts, the value of x(t) as given by this expression is just x(0), exactly what it should be.
  • As time goes on, the value of x(t) approaches GdcU.

Examples

E3   Consider a system with the following parameters and inputs.

x(t)  =  Response of the System and x(0) = -2
u(t)  =  Input to the System, and u(t) = 5 for t > 0
t  =  The System Time Constant = 1 second
Gdc  =  The DC Gain of the System = 1
Put those numbers into the expression above, and you will get this plot of the response of the system.

  • It starts at -2!
  • It approaches 5.
  • The time constant is one second.  That may not be obvious, so try to check it out.


E4   Here is a movie that shows how the step response of a system changes as the DC Gain changes.  Here are the parameters for this system.
x(t)  =  Response of the System and x(0) = 0
u(t)  =  Input to the System, and u(t) = 1 for t > 0
t  =  The System Time Constant = 5 seconds
Gdc  =  The DC Gain of the System (Adjustable)


E5   Here is a movie that shows how the step response of a system changes as the DC Gain changes.  Here are the parameters for this system.
x(t)  =  Response of the System and x(0) = 0
u(t)  =  Input to the System, and u(t) = 1 for t > 0
t  =  The System Time Constant (Adjustable)
Gdc  =  The DC Gain of the System = 1



Experiment/Example

E6   Finally, here is a simulation of a general first order system.  In this simulation, you can change the DC gain and the time constant, giving you a chance to experiment with a first order system.


Some Observations on First Order Systems

        There are some important points to note about the step response of a first order linear system.

  • When the step is applied, the derivative of the output changes immediately.
    • To check this observation, move back up to the videos and note how the derivative changes when the step is applied.
    • The size of the derivative change depends upon the size of the step, but as long as the step is non-zero, the derivative will have a jump.
  • To get the steady state value, multiply the input step size by the DC Gain.
    • If the input is not a step but if it does reach a steady state value, the output will be the DC Gain multiplied by the steady state value of the input.
        That's pretty much it for the step response of a first order system.  Now that you know what it looks like it's time to start looking at how you can use this concept.

Encountering First Order Systems

        Once you know how a first order system responds to impulse and step inputs, there are several different ways you can use that information.

  • If you have a first order system, with either a step or impulse input, you can compute the output response of the system. That is an analysis problem.
  • If you have an unknown system, and you have input and output data, and your data set resembles an impulse input and a first order impulse response, or a step input and a first order step response, then you can use what you know to determine what the system is.  That is a system identification problem.

Problems         You might also want to examine this problem.
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