Welcome to Control Systems.  Let's just jump right in.  We have a problem for you.  If you click here you will get access to a simulator.  It simulates a system, and when it loads you will be able to put in constant inputs and sinusoidal inputs.  You have one goal to start.

• Your Goal:  To control the system so that the output goes to 1.0 and stays there.

• Start the simulator.
• Input a step.  Repeat that experiment, and try changing the input step size.
• Answer the following questions.

• Print this web page.  It should print out to about 10 pages, but if you print it you will have all of the questions ready for when you run the simulations.
• Print the instructions for the simulator.
• If you take that approach, you should be able to keep the simulator in view the entire time, and use the printed questions and instructions to work from.

Q1.  You have just taken a position at River City Circuits Corporation (RC³), and their control system specialist is on an extended vacaction.  Your supervisor asks you to look at controlling a motor.  That motor is simulated in the simulator you can access above or by clicking here.  You need to specify how to get the motor to 1000 rpm and stay there within 2%.  Note that the simulator shows input and output in units of 1000 rpm, so you want to devise a scheme that gives an output of 1 (within 2%) when the input is exactly 1.

        What is your first thought?

        Actually, you can test to determine whether the system is second order or higher by putting a sine wave into the system.  You can do that, but remember that the output of typical systems falls off at higher frequencies.  To avoid ugly plots, turn off the green plot by clicking for No Second Plot - as soon as the first plot starts.  Then, adjust the input amplitude to be much larger so that you get an output that you can actually see - so that you can see the phase of the output.

• Open Loop Control is not used often because systems in the real world change - in time, from system to system, etc. - and that variation is enough to make Open Loop Control unacceptable in many cases.

• In the simulator, we didn't simulate bearings heating up.  As the motor runs for a long time that would happen, and the motor would slow down slightly.
• If you have many systems - hopefully identical, but not quite - then applying the same control to all of those systems would produce variation in behavior.
• If you design a control system to control the speed of a car, the car might be full of passengers with a full tank of gas, or it might have just a driver with little gas in the tank.  Those extreme cases would have different dynamics that would affect the way the car responds.  The car might be going up a hill or down a hill.  There might be a tail wind or you might be driving into the wind.  All those things cause changes in the behavior using open loop control.
• If you design a control system for an airplane, the airplane responds very differently when it is at 30,000 ft and flying 500 miles an hour compared to how it behaves landing at 180 mph at sea level with the flaps down.
• If anything, the simulator doesn't give you the wide range of variation that you would find in real systems.  Even so, it is enlightening.

Actually, we were trying to get you to the concept of ON-OFF Control.  It's a common way of controlling things like temperature in your house.  The essence of ON-OFF Control is to apply maximum control effort when the system is below where you want it to be, and minimum control effort when the system is above where you want it to be.  In the home thermostat that means your thermostat turns the heat ON when the house is tool cold and it turns the heat OFF when the house is too warm.

        In the simulator, we assume that the motor is driven by an amplifier that saturates in both directions.  So, when the motor is going too slow, the simulator is max positive output - to increase the motor speed.  When the motor is going too fast, the simulator is max negative output to make the motor start to slow down.  Try that in the simulator.  Do the following.

• Click for Closed Loop to show the control loop.
• The control loop has a comparator that checks whether the motor speed is above or below the desired speed and applies the correct voltage to the simulated motor.
• You can set the desired speed using the input text box.
• The control effort is shown in a separate plot.  Note that the scale for the control effort is shown on the right hand side of the plot, and that scale is not the same as the scale for the output.

        In the ON-OFF control system the system is represented pictorially with a block diagram that shows how signals flow within the system and how they are processed.

For an ON-OFF controller, the controller reacts to the error in the system.  The error is the difference between what you want and what you have.  The simplest ON-OFF controller is the home thermostat, which looks at the temperature you want, and the temperature it senses and turns the heating system ON or OFF, depending upon the error in the system.  When the error is positive (i.e. the desired response is larger than the measured response) the controller turns the control effort to full ON.  Otherwise, when the error is negative, the control effort is OFF.  Some ON-OFF systems don't turn the control OFF when the error is negative.  If the system is a motor, and you are trying to get to some predetermined speed, if you go past the desired speed, rather than turning the control effort to OFF, the control effort becomes that maximum possible negative control effort - which will tend to slow the motor down.

        Now, it's time to examine what happens when you implement ON-OFF control.  Our simulator will let us examine what happens.

• Be sure the simulator is in Closed Loop mode.  Click on the Closed Loop button to ensure that is the case.  In Closed Loop mode, the closed loop - including the comparator - shows.  If the controller and the comparator are not showing in a closed loop, then the simulator is not in the Closed Loop mode.
• Select the control you want.
• ON-OFF for ON-OFF Control
• Input the set point/desired response/system input.
• Click the Start button to run the simulation.

• Set Point = 0.5
• Set Point = 1.0
• Set Point = 1.5
• Set Point = 2.0
• Set Point = 2.5

        You also should have seen that ON-OFF control gives oscillations in the output.  Moreover, the oscillations do not die out.  They persist forever.

        What are some of the virtues of ON-OFF control?

• It's simple!  And remember the KISS algorithm.
• In systems like the one in our example it is capable of providing good results.  Oh, and it also seems to work well in home temperature control.
• It can give quick response.  After all, when the system output is below the setpoint, maximum control effort is exerted.

• Persistent oscillations

• Here is a link to A Note on ON-OFF Control.

        There are other kinds of control algorithms that you can use.  The next kind of control is Proportional Control.  It has many similarities to ON-OFF control.

• Both ON-OFF and Proportional Control:
• Have a feedback loop in which the actual response is fed back and compared to the input, i.e. the desired response.
• Compare the input to the actual response and form an error signal.
• Determine a value of the control effort based on the error.

• The difference between ON-OFF control and proportional control is in how the control effort is computed:
• In ON-OFF Control the control effort is always at an extreme value.  The control effort is either at the maximum of minimum possible value.
• In Proportional Control the control effort is proportional to the error.

Here's the block diagram.

        In proportional control, you can get very different behavior.  Let's explore some of that behavior using the simulator.  Here are the instructions for putting the simulator into the correct mode to examine proportional control.

• Be sure the simulator is in Closed Loop mode.  Click on the Closed Loop button to ensure that is the case.  In Closed Loop mode, the closed loop - including the comparator - shows.  If the controller and the comparator are not showing in a closed loop, then the simulator is not in the Closed Loop mode.
• Select the control you want.
• Proportional to use Proportional Control
• Input the set point/desired response/system input.
• Input the proportional gain (K_p) that you wish to use for the simulation.
• Click the Start button to run the simulation.

        First, you will need some data, so try this set of values.

• Set Point = 1.0, K_p = 1.0
• Set Point = 1.0, K_p = 2.0
• Set Point = 1.0, K_p = 5.0

• Note the Steady State value.
• Note how quickly the response gets to the steady state value.

• As the proportional gain becomes higher the steady state response gets closer to the desired output.
• It's time for you to learn just why that is so.  Read this note and work the simulations you can access from there.
• As the proportional gain becomes higher, the system seems to be less and less stable as evidenced by more overshoot, especially.

      Actually, there are numerous questions buried in this example, including questions about what happens if you try to control another system.  Are the conclusions reached above valid for other systems?  Let's review the conclusions, and then look at another system.  What we have noted in the example system above includes the following.

• The example system seemed to have higher than first order dynamics.
• The example system had some random variation in the dynamics of the basic (open-loop) system.  It does not always respond in exactly the same way to a step or sinusoidal input.
• The system has some minor oscillations when you use ON-OFF control.
• Proportional control exhibits oscillations when you get the gain high enough to get the system to have an accurate closed loop step response.

• First, examine the open-loop behavior.
• This system - like the first system - exhibits some variation in the output level when the input is a step.
• This system has a response that looks much like the first system.  It might be a little bit slower responding than the first system.

• Be sure the simulator is in Closed Loop mode.  Click on the Closed Loop button to ensure that is the case.  In Closed Loop mode, the closed loop - including the comparator - shows.  If the controller and the comparator are not showing in a closed loop, then the simulator is not in the Closed Loop mode.
• Select the control you want.
• ON-OFF for ON-OFF Control
• Input the set point/desired response/system input.
• Click the Start button to run the simulation.

• Set Point = 0.5
• Set Point = 1.0
• Set Point = 1.5
• Set Point = 2.0
• Set Point = 2.5

• Using ON-OFF control
• This system exhibits oscillations in the response.  However, unlike the first system, the oscillations in this system are quite large and the excursions around the setpoint are probably unacceptable.

• Using proportional control
• Maybe you think there is a flaw in the simulator, but the system doesn't behave well at all.
• It is impossible to get a decent response (steady state) using proportional control.  As you increase the proportional gain, the system becomes unstable.  (First time you've seen that?  Did they never talk about that in linear systems?  Interesting!)

        There are some interesting issues raised by the simulations you have done, and we need to summarize what we have seen here.

• Real systems have variations in them that you must account for.  Assume you design a controller for a system.
• The system could change with time, and conditions.  If you are controlling a car, the response of the car depends upon how many people are in the car, whether the car is going uphill, level or downhill, etc.
• If you put the same controller into many different copies of the system, all copies will not be the same.  If you are controlling a car, the next car off the production line will perform differently than the car you just put your controller into.
• The dynamics of the system interact with the controller's actions in ways that are not always good.  In the last example, if you increased the gain enough, the system became unstable.  You will want to be able to determine conditions for stability.  There are a lot of systems that cannot be tested adequately.  You won't know if the aircraft controller works properly until you fly the aircraft.  For those kinds of reasons you will need a large toolkit that will enable you to predict performance of your control system with some confidence.
• There are different kinds of control algorithms.  ON-OFF control and proportional control are just two of many, many ways to control a system.  You will need to understand many different kinds of controllers (PID family, various compensators, etc.) in order to be able to design the best control system for your situation.
• When you tried proportional control you found that a higher gain produced less error (and possibly some not-so-good side effects like oscillations and instability).

        There are some serious issues we have raised in this note, and there are clearly some things that we need to understand if we are going to be able to design control systems that behave well.  Here is a partial list of the issues that we need to address.

• What exactly is the difference between the first system and the second system?  Stating the problem as succinctly as possible, we can say that the open loop step responses of the two systems do not look much different, but the closed loop behavior certainly changes - and not for the better - when we move from the first system to the second system.  You need to learn why that happens, and how to design for good behavior.  Actually, you not only need to understand the difference between the ways that first and second order systems behave, but you need to move on to higher order systems - which are all you are going to find in the real world.
• How can we predict behavior when we want to control a system.?
• We could just blindly try all different kinds of control systems, but clearly there needs to be a better way.  You want to do things knowledgeably so that you don't get into a blind alley with no way to get where you want to get.  And, if the problem you are trying to solve is impossible, you need to be able to see why.
• How would you control the second system?  It looks to be a difficult control problem since ON-OFF and proportional control both perform very badly.  We need to understand what it is about that system that is different, and whether there are there any modifications we might make to the control schemes to get better performance.

        Here is an approach to take.  First, we examine the first system in more detail.  If we can figure out what makes that system tick, then perhaps we will be in a position to understand what it is that makes the second system so difficult to control.  In this link you can learn about the fundamental problems that exist in these systems.

• A Note on ON-OFF Control
• A Note on Proportional Control