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- Performance Measures - Speed of Response
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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.
In an airplane, if the
pilot wants the engines to go full speed, s/he doesn't want to wait fifteen
minutes for that to happen. How quickly the engine speed control
system responds affects the safety of passengers and crew.
In your home, you surely
consider it important to have the temperature come up to your setting when
you return from a trip having turned the heat down before you left.
It's almost always true
that you want a system to respond as quickly as possible whenever you change
the setting or desired state of the system.
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.
First order systems have
time constants. Clearly, in those systems we can take the time constant
as a measure of speed of response.
For two first order systems,
the system with the smaller time constant will respond more quickly to
a step input or any other input.
Knowing the time constant
allows us to estimate aspects of a response. For example, a first
order system with a time constant, t,
will respond to a step input so that the system is within 5% of the final
value in 3t
seconds - i.e. three time constants.
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.
Can you determine a time
constant for this system?
There
are several reasons you can't determine the time constant for this system,
but the glaring reason here is this one.
A first order system -
with a single time constant - has a response that changes slope immediately
after the step is applied. The slope of this system does not change
immediately, and the response seems to take a long time to get started.
This response seems to
take 2 or 3 seconds to get started, but once it does get started, it's
over by 8 seconds.
Can you guess what the
transfer function of this system might be?
Asking
what this transfer function is is not a fair question. Here's the
tranfer function.
This system does have
a time constant of 1 second.
The problem is that it
has 5 time constants, all of 1 second.
If you do encounter a
system like this, you will need to have a numerical way to measure the
speed of response. Next, we'll propose a few different ways.
On the other hand, if you encounter a system like this one, you may want
to run away.
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.
Determine the time at
which the response reaches 10%. Click the button on the left, and
you should see that time is about 2.5 seconds.
Determine the time at
which the response reaches 90%. Click the button on the right, and
you should see that time is about 7.8 seconds.
Subtract the two.
7.8 - 2.5 = 5.3 seconds.
So the ten-to-ninety rise
time is 5.3 sec.
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.
Determine the time at
which the response reaches 90%. Click the hotword, and you should
see that time is about 7.8 seconds.
The settling time is 7.8
seconds.
With a little imagination,
you may realize that we need to be somewhat more precise than we have been.
Time of response has some subtleties that we need to take into consideration.
Now, here's a response that forces us
to think harder about what we mean by response time.
Clearly this system has
a short rise time. It looks to be just a few seconds.
The settling time is also
just a few seconds, since the response gets to within 90% in that time.
The problem is that the
response doesn't stay within 10% of the final value. It just passes
through that range on its way to oscillation after oscillation.
We need to be precise
in our definitions.
Rise time is going to
be tough, so we're going to leave that definition as it stands.
Settling time
can be defined as the time it takes to get and stay within 10% of
the final value.
Click the button below
to get a picture of the 10% range, then estimate the settling time.
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's no substitute
for knowing where are the poles are zeroes are in a system. Knowing
a system has five poles at s = -1 is more information than knowing rise
time because you can plot the response and compute rise time and more.
(Root locus analysis will help you determine where the poles are located
in a closed loop system.
Ten-to-Ninety rise time
is the time it takes to go from 10% to 90% of the final value. It
can be misleading if the system oscillates or if there is a delay getting
started.
Settling time can
be a good way to measure response time as long as care is taken to ensure
that the response stays within 10% (or 5%?) of the final, steady-state
value.
There are some other things we should note.
There are going to be
times when you are working in the frequency domain. In that situation,
you will need to get some measure of speed of response from frequency domain
data. We'll discuss that in the next section.
There are plenty of systems
that have peculiar - maybe perverse - responses.
Consider each case individually,
and use good judgement. Here's a peculiar response.
Much of what we have discussed
does not apply here. This is just a response that is an aberration
- not included in what we have discussed to this point. However,
you may well encounter a system with this sort of response, or - even worse
- you could end up inadvertently designing a system like this one.
Now, we should get some numbers for at
least one common system - the first order system with one time constant.
The formula for the unit
step response of this system is:
Response(t) = G_{dc}
* (1 - e^{-t/}^{t})
We can calculate the time
it takes to get to 0.9*Gain.
Doing
that we find the settling time = 2.3t.
Similarly, the rise time
(10%-90%) is 2.2t.
(It has to be less since both computations, rise time and settling time,
use response to 90%.).
For later work, note that
the bandwidth of this system is 1/2pt,
so we have:
rise time = 2.2t
= 2.2/(2pBW)
= .35/BW.
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.
As the gain is increased,
the closedloop pole moves to the left.
We know that settling
time = 2.3t,
so we can say that the smaller the time constant, the smaller the settling
time.
A settling time of 1 second
would require t
of 1/2.3 or 0.435 seconds.
That would require a pole
to the left of s = - 1/.435 or s = -2.3.
That can be seen by clicking
the button at the right of the root locus in the plot below.
If
we have two complex poles, the situation is not much changed.
How
far into the left half plane the pole is defines how quickly the system
will respond.
Here are two responses.
The oscillatory system has two complex poles (only one of two shown), and
the exponential system has one real pole.
On the figure below there
are hot spots around the pole locations. Move the mouse over those
hotspots for more information.
What do we note?
The real part of the two
poles are the same for the two systems.
The settling times are
the same for the two systems.
What do we conclude?
The speed of response
of a system is determined by how far into the left half plane the poles
of the system are.
The distance into the
left half plane is -1/t
since the pole is located there.
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. G_{1}(s) is the red line and G_{2}(s)
is the blue line.
G_{1}(s) = 1/(s
+ 1)
G_{2}(s) = 1/(.1s
+ 1)
Note
the following:
G_{1}(s) has a
one second time constant.
G_{2}(s) has a
.1 second time constant - ten times smaller.
Note also the following:
G_{1}(s) has a
bandwidth of .159 Hz.
GG_{2}(s) has
a bandwidth of 1.59 Hz, ten times larger.
Bandwidth is calculated
as the frequency at which the Bode' plot is 3 db down from the DC gain.
That's the 3 db bandwidth.
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.
In general, the wider
the system bandwidth, the faster the system responds!
We may be able to develop
a rule of thumb that will allow you to make reasonably accurate estimates
of response time from system bandwidth.
To get a rule of thumb,
we should examine a few more systems.
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.
The 3 db point is at approximately
f = .07Hz.
We have examined the time
response of this system earlier in this lesson, and we concluded that rise
time = 5.3s and settling time = 7.8s.
Now, let's examine the
frequency response data for the system. Here's the Bode' plot for
the system.
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.
Many times it is much easier to design in the
frequency domain.
There is often some type of specification on
response time for the closed loop system.
It is valuable to have some measure of closed
loop response time that can be estimated from open loop frequency response
data - the kind of data that you would use for the design.
Let's examine a unity feedback system.
Here's the closed loop
transfer function.
From the closed loop transfer
function, we can compute the closed loop frequency response by substituting
jw
for s.
It is valuable to have
some measure of closed loop response time that can be estimated from open
loop frequency response data - the kind of data that you would use for
the design.
Now,
consider two extreme cases.
There are frequencies
- usually low frequencies - for which the magnitude of the frequency response
function, |KG(jw)|,
is large.
When |KG(jw)|
is large, the closed loop frequency response, |CLTF(jw)|,
is close to one in value.
There are frequencies
- usually high frequencies - for which the magnitude of the frequency response
function, |KG(jw)|,
is small.
When |KG(jw)|
is small, the closed loop frequency response, |CLTF(jw)|,
is close to |KG(jw)|
because the denominator in |CLTF(jw)|
is essentially 1.
We
can examine the implications of these conclusions. Take an example
Bode' plot.
At low frequencies, the
closed loop frequency response, |CLTF(jw)|,
is close to one - i.e. zero db - in value.
Click the top green button
to see the closed loop frequency response at low frequencies.
At high frequencies, the
closed loop frequency response, |CLTF(jw)|,
is close to |KG(jw)|.
Click the lower green
button to see the closed loop frequency response at high frequencies.
At the frequencies in
between, we find the zero db crossing of the open loop response, and we
know the 3db closed loop bandwidth is in the same region.
Putting
all of this together, we can conclude:
The closed loop bandwidth
is close to the open loop zero db crossing.
We also know that:
rise time = .35/f3db ~
.35/fopen loop zero db crossing.
We conclude that we can
make a closed loop system respond faster if we can make the open loop zero
db crossing higher.
Remember, all of these
conclusions are approximate, but still good estimates to use in design.
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.