An Introduction To Control Systems
Goals
What You Will Need
What Is A Control System?
Examples
What If?
Simulators
First Order
Second Order
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Why Use Feedback Control?

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.

1. 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).
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.) 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:

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
Links To Related Lessons

Other Introductory Lessons

More Advanced Material On Proportional Control
Send us your comments on these lessons.