An Introduction To Control Systems - Open Loop to Closed Loop

An Introduction To Root Locus

Using This Lesson

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

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.

If you are doing this lesson, we assume that you have examined the introductory material at this link. In that lesson you encountered some systems that exhibited interesting behavior. In particular, you should have noted the following.

There were two systems that - at first glance - looked to be very similar. What we mean by that is this:

If you input a step to the two systems (with no control loop) the step response looked very much the same for the two systems.

However, when you applied a control to the systems, they behaved very differently.

The first system seemed to behave well when using ON-OFF control. There were some initial oscillations that eventually got to be pretty small. The second system had fairly severe oscillations using ON-OFF control.
The first system never went unstable when you used proportional control. In the second system, a sufficient increase in the proportional gain caused the system to exhibit growing oscillations. In other words, the second system was unstable.

Clearly, it is important to be able to understand what happens in those two systems. Why are two systems so different in their responses when the step responses of the two systems don't look that much different? What is going on in those two systems?

In this lesson we will examine the root cause of the behavior of those two systems. In order to understand that behavior you will need to be able to apply some basic knowledge from linear systems. In particular, you will need to know the following.

Given a transfer function, you should be able to find the poles and zeroes of the transfer function, and you should be able to predict general features of responses of the system from those poles.

From pole location you should be able to sketch step response, including determining whether the system exhibits oscillations and you should be able to calculate how quickly those oscillations decay (or grow).
From the transfer function you should be able to determine general features of the Bode' plot (frequency response) of the system including DC gain, resonant peaks, high frequency gain and phase asymptotes, etc.

Now, we are going to give you a little experience with a simple control system. In this second order system you can specify the poles by typing values into the simulator.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.

Your first task is to do the following.

Start the simulator.
Be sure that the simulator is in the Open Loop mode.
Input a step. Use the default values for the system parameters (gain and pole values). Repeat that experiment, and try changing the input step size.
Answer the following questions.

Question(s)

Q1. Using the default values (Gain = 1, both poles at s = -1), does the system have real poles?

Q2. Again, using the default values, do you expect the system to exhibit any oscillations?

Q3. With real poles can you ever have oscillations in the system?

OK, you have looked at the open-loop system and hopefully you have a clear understanding of what goes on there. (If not, you may wish to review some material on the step response of second order systems with real poles.) It's time to examine some control strategies. The first control strategy you will examine is proportional control. (Proportional control is the simplest control used. ON-OFF is usually used only on very simple systems where good accuracy is not expected. Your wall thermostat is ON-OFF, and you don't control to a fraction of a degree with it.)

Proportional Control

What we want you to do is this.

Start the simulator.
Set the simulator for Closed Loop.
Set the control mode for Proportional Control.
Set the proportional gain (the text block inside the controller) to 1.0.
Click the Start button and observe the simulation.

Q4. In the simulation, does the system seem to have real poles?

Now, we want you to change the proportional gain. When the proportional gain is 1.0 the system has a substantial error. By increasing the gain to 10 the error should decrease. (Click here for a short note that gives you a derivation of how the error depends upon the system gain.(s).)

Q5. In the simulation, does the error decrease when you increase the gain? In other words, does the output match the desired output (input) more closely?

Q6. In the simulation, does the system seem to have real poles?

Now, we have a dilemma. Cleary - if you followed the directions - the poles that you set in the system were real - and not complex. Equally clearly, the behavior of the closed loop control system shows that the system has complex poles. We MUST resolve that dilemma. In order to resolve the dilemma we need to be very, very careful with our terms. First, let's ask you what you think.

Q7. The reason for the oscillating behavior when we enter two real pole values and use a high gain is (choose one of the buttons).

There isn't any way out. The poles of the closed loop system must be complex. However, the poles of the system we are controlling are real. It's an assumption/guess on your part if you think that the poles of the closed loop system are the same as the poles that you typed in. That's where any problem is. Here's what you need to remember.

The poles of a closed loop control system are not at the same location as the poles of the system being controlled.

That simple statement is really quite profound because it encapsulates everything important about the root locus. The root locus is a systematic way of describing how the poles of a system move when you close the control loop for the system and adjust the gain in a proportional controller. Let's look at some specific things you can do to help you understand what's going on here.

You can pick a system and examine it in some detail.

The system should be simple - so that you can do calculations and make specific predictions.
The system should exhibit dynamics that are complex enough to show some of the behavior that you have seen in the example systems used in this lesson.

We have assembled some material like that in this link.

Readings