An Introduction To Control Systems - Open Loop to Closed Loop

An Introduction To System Identification

Whenever you design a control system you will need to be able to predict the behavior of the system you design. There are a number of ways of predicting behavior, including using a simulation (in Simulink, for example, but any other simulator as well), using root locus or Bode' plot calculations, etc. However you do the prediction, you will need to have a model of the system. In many cases, the only way you can get a model is to take data on the system and try to fit a model to your data.

When you take data on a system, there are three choices.

You can input an impulse, a step or a ramp and analyze the time response to get the transfer function.
You can input a sequence of sinusoids and use gain and phase shift information to determine the transfer function from the frequency response.
You can input random noise (or many other signals) and use FFT techniques to get a frequency response, and then determine a transfer function from the frequency response.

Let's start by looking at a simulated system. If you click here you will get access to a simulator. It simulates a system and shows the unit step response of the system. Your goal is to determine the transfer function of the system. Your first task is to do the following.

Start the simulator.
Input a step. Repeat that experiment, and try changing the input step size.
Print a copy of the graph that results. You will need to pick off some numbers from that graph.

As you get started, here are some ideas for using the lessons, and this introductory lesson in particular.

Print this web page. 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.

Question(s)

Q1. With the step response graph in front of you consider this question.

Is the system first order, second order or some higher order system?

Now, let us assume - for the moment - that the system is second order. The reason for making that assumption is simple.

The system exhibits oscillations that die out. Because of that, the simplest system that exhibits that behavior is a second order system.

In addition, there are no peculiar behaviors that would indicate that this was a higher order system.

Then, if we have a second order system, we need somehow to determine the three parameters of the simplest second order system.

z = Damping Ratio,
w_n= Undamped Natural Frequency,
G_dc = The DC Gain of the System.

In order to calculate those three parameters you need to understand what each parameter influences in the step response. In the link above, you can find the following.

z, the damping ratio, will determine how much the system oscillates as the response decays toward steady state.
w_n,the undamped natural frequency, will determine how fast the system oscillates during any transient response.
G_dc, the DC gain of the system, will determine the size of steady state response when the input settles out to a constant value.

However, you need to know precisely how each of those parameters influences those properties of the response.

We will work first to determine the damping ratio. We know:

z, the damping ratio, will determine how much the system oscillates as the response decays toward steady state.

In particular, one measure of how the response decays is to look at the overshoot in the step response. As a system has more overshoot, that system has less decay per cycle (and that applies to the first cycle where the overshoot occurs) and is indicative of a smaller damping ratio. The link above will take you to an extended discussion of overshoot and damping ratio in second order systems, but the primary result is encapsulated in a plot of overshoot versus damping ratio. That plot is shown below.

Now, let's go back to the problem at hand. You still need to get the system to behave. Can you think of a way to control this system? Try to answer this question before going on, even if your ideas do not correspond with the options below.

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.

A Side Note

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.

Once you have the simulator set, you should run the simulator for several different set points. Try these values, and then we will have some questions.

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

Q4. When you simulated the system, was the response accurate for a set point of 0.5?

Q5. When you simulated the system, was the response accurate for a set point of 1.5?

Q6. When you simulated the system, was the response accurate for a set point of 2.5?

What you should have noticed is that there are certain set points that require such a large control effort that you can't reach the desired output level. When the control effort is limited - as it always is in reality - there is a limit to how large you can make the output level.

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.

What is the downside to ON-OFF control?

Persistent oscillations

What resources will let me learn more about ON-OFF control?

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

That's ON-OFF control in a nutshell

Proportional 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 is a block diagram for a proportional control system. It's very much like the block diagram for the ON-OFF control system. However, here the control effort is shown as:

Control Effort = CE = K_p x 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.

Now you are ready to look at some simulations with proportional control. Once you have the simulator set, you should run the simulator for several different set points and values of proportional gain (K_p).

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

As you run those simulations, note the following and record the data you get. Note the following

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

You should get a strong sense of the following trends.

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.

Continuing On

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.

Now, let's examine another system. If you click here you will get access to another simulator for a system that is similar to the first system. We want you to examine how this system behaves, and we will go through the same steps we went through for the first system.

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.

Q7. Based on what you know already, can you predict how ON-OFF control will work?

Next, look at using ON-OFF control with this system. Do what you did before. The instructions are repeated below.

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.

Once you have the simulator set, you should run the simulator for several different set points. Try these values, and then we will have some questions.

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

When you examined ON-OFF control for the first system, you acquired some expectations. The first system exhibited some oscillations, but they were small.

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.

Next, look at using proportional control with this system.

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!)

An Agenda

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.
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.

These are not really simple questions, and there is a lot to be learned if we want to be able to design systems to meet definitive performance specifications.

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.

Readings