The first question is really "Why do you need a control system at all?"
Consider the following.
What good is an airplane
if you are a pilot and you can't make it go where you want it to go?
What good is a chemical
products production line if you can't control temperature, pressure and
pH in the process and you end up making tons of garbage?
What good is an oven if
you can't control the temperature? (And, does it matter if it's an
oven in a kitchen or an oven in a heat-treating department that is used
to harden metal parts?)
What good is a pump if
you can't control the flow rate it produces? (And, there are many
times when the flow rate must be controlled.)
The
common denominator in all of these questions is that there is some physical
quantity that must be somehow controlled in a way that ensures that the
physical quantity takes on the value that is specified. There are
even times when the physical quantity should take on some pre-determined
values that follow a function of time. (An example of that would
be landing an airplane where you want the plane to meet the ground following
a specified curve.) We need to think about how to control physical
quantities in general, and to determine what can be done - in a general
way - to implement any schem we devise.
What is clear is that if you want to control a system, you need to know
what you want it to do, and you need to know how well it is doing.
That implies a couple of things. First, you need to know what you
want the system to do. There are lots of ways you can do that.
For example, in your home you set a temperature by dialing it into the
thermostat. That's the way you tell the system what you want it to
do. When an airplane is landing there is a radar beacon at the far
end of the runway that tells the aircraft if it is too high or too low,
too far right or too far left, and how much in all those cases. There
are any number of ways you can tell a system what you want it to do.
You can turn a dial, type a number into a computer program, or you can
use some other physical quantity. (An example of that is trying to
point an antenna at a weather or communications satellite. The satellite's
position - which might be predictable with an astronomical formula - gives
the system the information it needs on where the antenna has to point.)
One way or another, the control system has to know what it has to do.
The other thing that the control system has to know is how well the system
is doing. That radar antenna at the end of the runway when the airplane
is landing tells the airplane what to do, but it also tells the airplane
where it is at (up/down and left/right) and how far off the desired position
the aircraft is. The thermostat tells the system whether the temperature
is above or below where it is desired to be. You can use temperature
sensors, pressure sensors, tachometers and many other sensors that measure
physical variables to get a handle on system performance. One way
or the other, the system has to measure or monitor its performance.
Once you have the information on how well the system is performing, you
have to do something with that information. The problem the control
system designer faces is to determine how to use the information available
to develop and apply a control signal that will make the system do what
he or she wants it to do. At this point in these lessons that what
you are just starting to learn.
As you think about what you have to do to control a system, you realize
that the information about how well a system is performing - usually taken
at the output of the system - has to be fed
back around the system to the input and compared somehow with the input
- the information about what you want the system to do - and that comparison
gives you the information you need to produce/develp and apply a control
signal. Feeding back that performance information is what gives us
the idea of feedback and feedback control
systems.
Feedback control systems are very important. You've used them.
Did you drive in this morning? Imagine you're going to drive to Toledo
to see the Mudhens. You get in your car, and you use feedback.
You didn't think about
it, but you did look to see where you were on the road, and you used that
information.
If you didn't use that
information, you were effectively driving blindfolded.
We're not suggesting an
experiment here. But if you did drive blindfolded you wouldn't get
any feedback about where your car was on the road, and you wouldn't get
where you want to go unless you want to have an accident.
In
a feedback control system, information about performance is measured and
that information is used to correct how the system performs. It's
common. It's used in the human body over and over again to correct
body temperature, the amount of light that hits your retina, and lots of
things you never have to think about. But feedback systems don't
exist only in the natural world. They're ubiquitous in the man-made
world also. You'll find feedback control systems in chemical process
plants, plants that package food, plants that make steel, in transportation
vehicles to keep the vehicle on course at a desired speed. They're
everywhere, and they don't always happen naturally, so you need to learn
about how to design them.
The individuals who design control systems are a special group. You
will often find electrical engineers who design control systems for aircraft
of chemical plants. Designing control systems takes a person who
can bring together various disparate aspects of a system and make them
work together, and often that process is highly analytical and mathematical.
You will need to learn how to use all the things you know about systems
and bring them together to produce a good design.
In this lesson we'll start to examine feedback control systems and how
you design them. First, we will look at your learning goals.
Goals
For This Lesson
This lesson introduces you to control systems.
Here's what you should get from this lesson. Look for it, and look
for goals in every lesson so you can stay tuned in to what is important.
Given a control system,
Identify the system components and their function, including the comparator,
controller, plant and sensor.
Given a variable to be controlled,
Determine the structure of a system that will control that variable.
Given a control system design problem,
To appreciate and understand that the complexity of most systemsm makes
it difficult to predict their behavior.
Along
the way in this course you should also gain an appreciation of the idea
that control systems are designed and the designers
predict
the behavior of some very complex systems, and people
trust those designers. You trust them when ride in an airplane,
astronauts trust them when they go to the moon or fly the space shuttle.
A control system designer often has to consider the safety or even the
lives of the people using the systems they design.
It's not always easy to predict how a system will behave. Analysis
tools are not perfect, and systems are not always completely understood.
Despite that, if the systems are going to be used, you - if you are the
control system designer - need to do the best job that you can to ensure
that your system performs well.
Some
Examples
We want you to imagine that you have a job - actually several jobs.
These jobs will involve designing control systems.
Imagine that you have
been hired by a company that produces specialty metals. They are
setting up a new production line for a new kind of magnetic material that
they are producing for shielding rooms from magnetic fields produced by
high currents. The production of this metal involves heating it to
very specific temperatures and holding those temperatures for specified
periods of time. Actually, the rates of temperature increase and decrease
between the set points are also critical. If the temperatures vary
too much from what is required the metal produced will have to be scrapped.
Are you ready for this job? It's not tough conceptually, but can
you guarantee the company that they can go forward with confidence that
the temperature control systems will work to produce the required temperature
vs time profile?
Here's another situation.
Imagine that are part
of an aircraft control system design team. Your company has just
bid on the control of a new supersonic aircraft. Your team will need
to design the autopilot systems for the aircraft. In other words,
you need to design systems that will keep the aircraft at the same altitude
and on the same heading when the pilot is not actuating the controls.
Can you design a system and guarantee that the system will work?
Remember also that you need to design this system so that it works when
the fuel tanks are full and when they are nearing empty at the end of a
flight, and that the system has to work in all kinds of conditions including
heavy cross winds.
There are numerous other situations we could dream up. These situations
could exist. They do exist. They're not really made up.
There are many situations which involve a need for a design of a control
system. What do you need to do to design a system? (More
Goals)
You need to understand the general schemes that can be used to control
a system.
You will need to understand the system you're trying to control.
If you're an electrical engineer and you're in on the design of a
control system for a chemical plant or an aircraft, then you'll
need to use the material from some of those courses you
thought you'd never have to deal with again.
You need to develop your ability to predict how a system
behaves, and that means that you will need to work on some
mathematical techniques that involve differential equation
solution.
What
You'll Need
Here are some of the topics you'll
need along with some links to those topics.
You need to understand
the general schemes that can be used to control a system. That includes
proportional control, integral control, and combinations of proportional
and integral (plus derivative?) control. That's something you will
get to soon enough.
You will need to understand
the system you're trying to control. If you're an electrical engineer
and you're in on the design of a control system for a chemical plant or
an aircraft, then you'll need to use the material from some of those courses
you thought you'd never have to deal with again. (And, we hope that
you did not sell back those books!) We discuss some of those issues
in the lessons on modelling systems.
You need to develop your
ability to predict how a system behaves, and that means that you will need
to work on some mathematical techniques that involve differential equation
solution.
What
Is A Control System?
In most systems there will be an input and an output. This block
diagram represents that. (Control system designers and engineers
use block diagrams to represent systems. Get used to them.)
Signals flow from the input, through the system and produce an output.
The input will usually
be an ideal form of the output. In other words the input is really
what we want the output to be. It's the desired output.
The output of the system
has to be measured. In the figure below, we show the system
we are trying to control - the "plant" - and
a sensor that
measures what the controlled system is doing.
The input to the plant
is usually called the control effort,
and the output of the sensor is usually called the measured
output, as shown below in the figure.
For example, if we want the output to be 100oC,
then that's the input.
If we want to control the output, we first need to measure the output.
Within the whole system is the system we want to control - the plant -
along with a sensor that measures what the
output actually is.
In our block diagram representation,
we show the output signal being fed to the sensor which produces another
signal that is dependent upon the output.
A sensor might be an LM35,
which produces a voltage proportional to temperature - if the output signal
is a temperature.
We need the sensor in the system to measure what the system is doing.
The sensor measures the
output of the system we are controlling.
It often converts the
output into a variable we can use. If the output is a temperature,
we might want to have a voltage
we can use to control a heater, for example. The LM35 temperature
sensor, for example, produce .01 volts for every 1.0oC
change.
To control the system
we need to use the information provided by the sensor.
Usually, the output, as
measured by the sensor is subtracted from the input (which is the desired
output) as shown below. That forms an error
signal that the controller can use to control
the plant.
The device which performs
the subtraction to compute the error, E, is a comparator.
Finally, the last part
of this system is the controller.
The controller
acts on the error signal and uses that information to produce the signal
that actually affects the system we are trying to control.
The controller has to
provide enough power to drive the system. You don't want to try to
control a large motor with a 741 operational amplifier. You just
can't do that, so the controller has to be able to compute the control
signal, and it has to be able to drive the system you are trying to control.
Thus, the controller has
two things that it has to achieve.
The controller has to
compute
what the control errort should be.
The controller has to
apply
the computed control effort.
Consider how this controller
works.
If the gain in the forward
path, from the error to the output, is large, then a
small error can produce a much larger ouput.
There is a certain logic
to that strategy. You want a small error, but you need a control
effort large enough to control the system. That seems to imply that
the gain of the controller should be large.
It looks like a good strategy
would be for the controller to be a high gain power amplifier (for many
control situations) because then a small error could produce the output
we want, or something very close to what we want - because the error would
be small.
Now, let's start to refine our model.
Let's assume that the
system we are trying to control is a linear
system.
To account for the linear
dynamics, we'll show the transfer function
of the system. That transfer function will be G(s).
Once
we realize that we can describe the system we are controlling, the plant,
we realize that we can describe all of the components in the sytem with
a transfer function description.
The sensor most likely
has an output - typically a voltage - that is proportional to the physical
variable it measures. That means that the transfer function is just
a constant - a gain. We'll denote that by Ks.
The
last item in our system is the controller. Controllers come in many
varieties. The simplest - but certainly not the only one used - is
a proportional controller. That's what
we will consider here, but remember there are also integral controllers,
and controllers that blend integral, proportional and derivative control
and lots of others.
In a proportional controller,
the control action is proportional to the error, and we can represent the
controller as a gain, Kp.
That
completes a verbal and algebraic description of the system, but there is
also a diagrammatic representation for the system. The block diagram
shown below captures all of the information about the system as we have
developed it above. Note, in this system, we are assuming that all
of the signals are Laplace transform versions of the time signals we have
been discussing, and the descriptions of the blocks in the block diagram
are really transfer functions.
With this block diagram, let's review what we hope happens in this system.
There is an input, u(t),
to the system, which we assume starts from rest. In the block diagram,
that is represented by U(s).
The output of the system,
Y(s), is measured with a sensor that has a transfer function Ks.
That transfer function could have a time constant, etc., but for now we
will examine it as though it is a constant.
There is an error, E(s),
developed, particularly because the controlled system, G(s), cannot respond
immediately and the feedback signal that is subtracted from the input to
form the error is zero.
The error that is developed
acts through the proportional controller, Kp, to
start to move the output of the system to where we want it to be.
As the system continues
to operate, the output of the system (described by G(s)) rises, reducing
the error so that the control effort from the proportional controller gets
smaller.
Even though the error
gets smaller, if the gain of the proportional controller is large it will
still provide enough output to drive the system close to where we want
it to be.
This
kind of system is referred to as a closed loop system,
since there is a feedback signal that "closes the loop" in the system.
That's a little jargon you need to learn and remember.
But we're not done yet. We need to take our description and use it
to determine how this system behaves. That's the next section.
There we will look at a simple system and apply our analytical abilities
in order to get a better idea of how it all works. But first, we
are going to look at a few simulations of systems using the kind of system
described above.
Example/Experiment
E1
By clicking
here, you can get to a simulator for the system below. When you
click, you will get instructions for operating the simulator, as well as
a link to the simulator which eventually opens in a separate window.
You can return to this window, and keep the simulator window open.
In the simulator, we assume
that G(s) is a first order system.
G(s) = Gdc/(st
+ 1)
In the simulator, the
following items can be set.
Gdc
- The DC gain for G(s)
t - The
time constant for G(s)
K - The proportional gain
in the controller
The Desired output, u,
which corresponds to U(s) in the diagram above.
The system preloads with
a gain of 5 for the controller, with a gain of 2 in the system being controlled.
Run the simulator with the preloaded gains and parameter values and note
how quickly the system responds, and how accurately it responds.
Accuracy is determined by examining the steady state error (SSE).
The SSE is the difference between the desired output (preloaded as 2.0)
and the actual output (which will be displayed as the system runs).
Now, double the gain -
from 5 to 10 - by entering a new value in the gain text box, and run the
simulation again (You will have to clear the previous plot to do that.)
and observe the final value again.
Compare the results and
determine if the claims above about getting a small error with a large
gain are true.
Does the system perform
more accurately with the higher gain?
Now
you should have seen that the system performs better with a higher gain.
It is more accurate, and - if you didn't notice - it is also faster for
the higher gain. It's tempting to conclude that you always want higher
gain because you will get better performance. We will check that
on a second order system later.
First, we need to point out another thing that happens in a closed loop
system. Let's get back to our original system and examine another
detail in the system's performance. What we will look at is how the
error changes in time. There are some things we can learn from that.
Example/Experiment
E2
In this system we will examine how the error changes in time. First,
we have the same block diagram for the system.
In the simulator, we assume
that G(s) is a first order system.
G(s) = Gdc/(st
+ 1)
In the simulator, the
following items can be set.
Gdc
- The DC gain for G(s)
t
- The
time constant for G(s)
K - The proportional gain
in the controller
The Desired output, u,
which corresponds to U(s) in the diagram above.
Now, the simulator below
also shows how the error changes as the system operates.
Run the simulator.
(Note that the green plot is the plot of the error in this simulation.)
You should notice the following.
As the system runs, the
error is initially very large.
When the error is large
the control effort in the system is large. That means that there
is a larger input that is driving the system that is being controlled,
probably causing it to respond quickly.
As the system runs, the
error gets smaller - although it never gets to zero.
As the error gets smaller,
the control effort becomes smaller. When you get close to the desired
output, you don't need to "push" the system toward the desired output,
and you only need a control effort large enough to keep the system at a
constant value.
Run the system again (Reset
the system first.) with a larger gain, Kp. At larger
gains a smaller error produces the same control effort, and it takes a
smaller error to produce enough control effort to keep the system at the
desired output level. The net result is a smaller steady state error.
Example/Experiment
E3
In this simulator, the system is the one shown in the block diagram below.
It's the same configuration that we had before.
In the simulator, we assume
that G(s) is a second order system.
G(s) = Gdc/(s2
+ 2zwn
+ wn2)
In the simulator, the
following items can be set.
Gdc
- The DC gain for G(s)
z - The
damping ratio for G(s)
wn
- The undamped natural frequency
for G(s)
K - The proportional gain
in the controller
The Desired output, u.
To operate the simulator,
You can start by just
using the values that are pre-loaded into the simulator.
Click the Start
button. A plot will be generated.
If you want to change
anything, enter the new data, then click the Reset
button which appears when the plot is complete. That clears the plot
and brings back the start button.
The output is indicated
as the simulation runs.
Now, double the gain -
from 5 to 10 - by entering a new value in the gain text box, and run the
simulation again (You will have to clear the previous plot to do that.)
and observe the final value again.
Compare the results and
determine if the claims above about getting a small error with a large
gain are true.
Does the system perform
more accurately with the higher gain?
Does the system perform
better with the higher gain?
The
second system points out an interesting conundrum. The system gets
better one way, but it deteriorates in another way. Go back and try
to increase the gain still further and notice what happens. You can
do that in the simulator below, which has been modified to show the error
as well as the output.
Example/Experiment
E4
In this simulator, the system is the one shown in the block diagram below.
It is the same configuration that we had before.
In the simulator, we assume
that G(s) is a second order system.
G(s) = Gdc/(s2
+ 2zwn
+ wn2)
In the simulator, the
following items can be set.
Gdc
- The DC gain for G(s)
z - The
damping ratio for G(s)
wn
- The undamped natural frequency
for G(s)
K - The proportional gain
in the controller
The Desired output,
u.
In this simulator you can adjust the proportional gain.
Run the simulator with
the gain as shown. You should note that there is a time when the
error becomes almost exactly zero, yet the system continues to run and
settles out at a non-zero error.
Run the simulator with
a proportional gain (Kp) of 10. (The simulator
should pre-load with a proportional gain of 5.0.) Now, notice that
the error actually becomes negative.
There is another interesting point to observe here. The system that
is pre-loaded is a system with a damping ratio of 2, which means that the
system has two real poles. Real poles can't produce oscillations.
Oscillations can only come from complex poles. With a little experimentation
you should be able to see that the system does not exhibit oscillations
- so it has real poles - at low gain values, and that you only get oscillations
at larger gain values. Try it now.
You should also observe that the error behaves in some interesting ways.
With a proportional gain of 10.0, the system can exhibit a transient negative
error, which implies that the control effort - which is proportional to
the error - becomes negative. If this is a temperature control system,
that would mean turning the heater off and running an air-conditioning
unit to take heat out of the controlled space. If you are trying
to control the level of liquid in a tank, the negative control effort means
you are trying to remove liquid from the tank - running a pump backwards.
In both of these cases the linear model we have assumed might not really
be a good representation for the system that you actually build.
At this point things are starting to get interesting. You should
realize that predicting how a system behaves is not going to be simple,
and that we will need to be able to develop tools that can help us predict
behavior, especially in more complex systems. We haven't used the
most complex systems. Even a simple model of an airplane might have
twenty or more poles. But, the tools that we will develop have been
used successfully to design systems that large and much larger. We
need to consider what we have to do and we will start by looking at the
block diagram representation of our system. We are working
toward an explanation of what is happening, particularly what happens in
the simulated systems for starters.
Getting
the Closed Loop Transfer Function
The
block diagram we have developed shows how the signals within the system
interact. Actually, we can think of the block
diagram as a way of representing algebraic relationships that exist within
the system. Each item in the block diagram
represents some algebraic relationship that exists in the system.
We can use those relationships to get a relationship between the input
and output signals. Let's look at a simple system. In this
system, the sensor has a transfer function of 1.
The output of
the system is related to the error signal.
y(t)
=
Output
Y(s) =
G(s)*W(s)
That is the algebraic relationship that exists
between the input to the system we are controlling - denoted by W(s) in
the diagram - and the output of the system, Y(s). The essence of
that relationship is the transfer function, G(s), which might describe
some very complicated dynamics in that system.
There is another block, the controller, which is assumed here to be just
a gain, K. The relationship that block sets up is:
W(s)
=
K*E(s)
Continuing
along this avenue, we can substitute for the error.
The error signal, E, is
the difference between the input and the sensor signal.
u(t) = Input
E(s) = U(s) - Y(s)
Finally, we can note that this equation
lets us compute the relationship between input and output.
E(s)
=
U(s)
- Y(s)
and, we can get a relationship between E(s)
and the system output, Y(s) by combining the first two relationships we
have.
W(s)
=
K*E(s)
Y(s) =
G(s)*W(s)
= G(s)*K*E(s)
so, we have:
E(s)
=
Y(s)/[K*G(s)]
=
U(s)
- Y(s)
solve for the output to
obtain
Y(s)
=
U(s)
* K * G(s)/[1 + K*G(s)]
The
ratio of output to input for the closed loop system is referred to as the
Closed
Loop Transfer Function (CLTF). There are very few things in
control systems that you should memorize, but you should remember the form
for the closed loop transfer function. This is what you have to remember.
Closed Loop Transfer
Function = KG(s)/[1 + KG(s)]
A
First Order System Example
Now, let's examine a particular case where G(s) is a first order system.
In that case we would have:
G(s) = Gdc/(st
+ 1)
Then, the closed loop transfer function - Y(s)/U(s),
given above - can be computed in detail:
GCL(s)
= KGdc/(st
+ 1 + KGdc) = Gdc,CL/[stCL
+ 1]
This expression is put into a standard form
at the right. That expression has two parameters, the closed loop
DC Gain, Gdc,CL, and the
closed loop time constant, tCL.
Those parameters are given by:
Gdc,CL=
KGdc/(1 + KGdc)
tCL
=t/(1
+ KGdc)
This is
a particularly interesting result. The closed loop system does not
have the same parameters as the original system. Both
the time constant and the DC gain have changed as a result of having
the feedback loop in the system.
Question
Q1.
In the proportional control system described above, you want to be sure
that the output matches the input as well as possible. Ideally, the
closed loop DC gain would be 1.0. If Gdc
= 1, and you determine that the system is
not accurate enough. Would you need to increase K
or would you need to decrease K?
Something interesting happens here, but before we look at that, let's introduce
a little terminology.
The Open
Loop DC Gain for this system is the product
of the DC gain of the controller - K
- and the DC gain of the system being controlled - Gdc.
Now,
we can note the following for the closed loop system.
As the Open Loop DC gain
changes, the Closed Loop DC gain also changes - but it approaches 1.0 as
the gain gets large.
As the Open Loop DC gain
changes, the Closed Loop Time Constant also changes. It just gets
smaller and smaller as the gain gets large.
There are some interesting implications of these
changes.
What this means is that
the steady state output gets closer and closer to the value of the input
when the input is a contstant.
It also means is that
the steady state is reached faster in the closed loop system.
Here is the first order simulator again.
Here you can check the predictions we have just made. Follow the
instructions below.
Example/Experiment/Problem
E5
In this simulator, the system is the one shown in the block diagram below.
To simplify things we have used a sensor with a gain of 1, and shown the
feedback path as a gain of one.
In the simulator, we assume
that G(s) is a first order system.
G(s) = Gdc/(st
+ 1)
In the simulator, the
following items can be set.
Gdc
- The DC gain for G(s)
t - The
time constant for G(s)
K - The proportional gain
in the controller
The Desired output, u,
which corresponds to U(s) in the diagram above.
To operate the simulator,
Set the time constant
to 20 seconds.
Set the DC gain of the
plant - the controlled system - to 1.
Set the controller gain
to 1.
Predict the closed loop
time constant, so that you are sure of what you expect. Use the graded
response form below to check your answer.
P1 Enter
your answer in the box below, then click the button to submit your answer.
You will get a grade on a 0 (completely wrong) to 100 (perfectly accurate
answer) scale.
Your grade is:
Predict the closed loop
DC gain, so that you are sure of what the final value should be.
P2 Enter
your answer in the box below, then click the button to submit your answer.
Your grade is:
Click the Start
button in the simulator below, get the plot, and determine the closed loop
time constant and the closed loop DC gain experimentally. Compare
your measured results to the experimental results.
Problems
P3
In this system, you need a closed loop time constant of .5 seconds or less.
Determine the gain, K, that produces a time constant of 0.5 seconds.
(And, if you want, you can use the simulator above - at least to check
your answer.)
What
If?
So far, you've seen that feedback can have some really good effects when
the system being controlled is a first order linear system. What
if the system is different? There are lots of other situations you
could encounter.
The system might be second
order or even a higher order system.
The system could be nonlinear.
The system might be computer
controlled. The system we have been looking at has been a system
with an analog controller.
In
all of these cases, something different is going to happen - but's that's
a subject for another discussion.
More complex systems are just that - more complex. That complexity
means that that design techniques can not be limited to first or second
order systems. Models of aircraft might involve twentieth order differential
equations or higher - twenty or more poles if you are looking at things
using transfer functions. Satellites are orders of magnitude more
difficult in that sense.
The techniques you are going to learn are going to permit you to design
those complex systems and to predict their performance. They will
be based on what you know about simpler systems, but they will extend you
in the process. To get on with it, you can look at a basic type of
control system, the proportional control system. Click here to get
to the introductory lessons on proportional control.
You need to learn a number of things
to work in control system.
You need to learn about
system dynamics - how systems behave in time and how to model them.
You need to learn how
to use models of systems - transfer functions, block diagrams, etc.
There's a long list of what you need,
but we're going to stop here.
Problems
Given a control system,
Identify the system components and their function, including the comparator, controller, plant and sensor.
