System Dynamics - Response Time

Is Speed Of Response Important?

Measures Of Speed Of Response
Working In The Frequency Domain
Closed Loop Systems
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Is Speed Of Response Important?

        How quickly a system responds to a change in input is an important measure of its' performance.


Goals For This Lesson

 Speed of response is an important measure of how quickly a system responds.  When you evaluate how well a system is performing you need to measure speed of response with some metric.  When you are designing a system you need to be able to predict speed of response.  Your goals for this lesson relate to that.

  Given a system in which you need to predict speed of response,
  Be able to predict speed of response using time domain methods.
  Be able to predict speed of response using frequency domain methods.

Measures Of Speed Of Response

 Speed of response can be a little tricky because we have so many intuitive ideas of what we mean by speed of response.  Let's review some of the ideas that often form a foundation for this concept.

        The concept of a time constant works - and works well - for first order systems because it gives an unambiguous measure of speed of response.  However, even having two time constants complicates the issue.  Let's consider an example with more than one time constant.  Here's a time response.  This response is the response of a linear system to a unit step.

        There are several reasons you can't determine the time constant for this system, but the glaring reason here is this one.         Asking what this transfer function is is not a fair question.  Here's the tranfer function.

        One widely used measure of speed of response is the 10%-90% rise time, or the ten-to-ninety rise time.  For now, we'll just refer to this measure as the rise time, but you need to be aware that it could be defined differently, and sometimes is defined differently.  To measure the rise time do the following.
        At this point, you may want to say that 5.3 seconds is not a good measure of how long it takes this system to respond.  You're right if you said that.  It takes a little while before the system gets going and we probably should account for that when we talk about response time.

        There are other measures, and one of them is the settling time.  We're going to take settling time as the time it takes to get within 10% of the final value, or to 90% of the final value - and, most importantly - stay within that 10%.  That's going to make it interesting if you measure the settling time of a system that has oscillations.

 Now, here's a response that forces us to think harder about what we mean by response time.

We need to be precise in our definitions.


Q1  What is the settling time for the system with the response above?



        We can summarize our discussion on speed of response so far. There are some other things we should note.
 Now, we should get some numbers for at least one common system - the first order system with one time constant.         Finally, we can relate response speed to the position of a pole in the s-plane.  Let's consider a system with a pole at s = -a.  The root locus for this system is shown below for a = 1.

        If we have two complex poles, the situation is not much changed.

What do we note?

What do we conclude?
Working in the Frequency Domain

       Working in the frequency domain is often easier than working in the time domain.  For many system designs, working with Bode' plots is easier than working with a root locus.  The flip side of that is that you may often have to design to time domain specifications like rise time or settling time while you are doing your design in the frequency domain.

        In this section we're going to look at how you can estimate time response parameters from frequency response parameters.  It's not hard, and the connections are more obvious than you think.

        Here are the Bode' plots for two systems.  The transfer functions are shown below.  G1(s) is the red line and G2(s) is the blue line.

        Note the following: Note also the following:         If you're tempted to conclude - from the example systems above -that response time is inversely proportional to bandwidth, that's a pretty good conclusion.         Here's the response of the system with 5 poles at s = -1.  We can examine the frequency response of this system.  That should give us some test data.  The Bode' plot for this system is shown (magnitude plot only) at the left below.

        Here's what we see in this Bode' plot data.
 
3 db point
f3db
rise time
settling time
5 poles@-1
f = .07Hz.
5.3 sec.
7.8 sec
1 pole@-1
f = .159Hz.
2.2 sec.
2.3 sec.

        Now we can multiply the 3 db frequency times the rise time and settling time.  Doing that, we obtain the results in the next table.
 
 

 
f3db*(rise time)
f3db*settling time
5 poles@-1
.371
.546
1 pole@-1
.345
.318

        Are there any conclusions here?  Well, the product of 3 db frequency, f3db, and the rise time seems to work halfway well.  We might be justified placing confidence in an expression like:

f3db*(rise time) = .35

or:

rise time = .35/f3db

        We previously found rise time to be .35/BW for a first order system, and that's exactly what we have here - for another system.

        It's time to try a few more systems.


Closed Loop Systems

        Response time of closed loop systems presents an interesting problem.

        Let's examine a unity feedback system.
        Now, consider two extreme cases.         We can examine the implications of these conclusions.  Take an example Bode' plot.
        Putting all of this together, we can conclude:
The End

        That's it for this lesson.  Obviously, there's a lot of folklore and estimation in this area.  Still, one main point of the lesson is that time response and bandwidth are somehow reciprocally related.  Don't forget that point because it's important in system design.  We'll need to use that idea later.

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