Control Systems Problem

An Introductory Root Locus Problem
Problem RLocus1IP00Int

This is a guided problem to help you with basic root locus concepts. If you click here you can get a calculator for a root locus with two real poles. Use that calculator for the first questions.

Problems

You have a mathematical model for a closed loop control system. The block diagram for the system is shown below.

The system being controlled (actuator + plant) has a transfer function, G(s).

G(s) = 1/(s + 2)(s + 4)

The first chore for you is this:

Calculate the closed loop transfer function of the system. When you do that calculation, be sure to include the proportional gain, K, where necessary.
Once you have the closed loop transfer function, determine the poles of the closed loop system.

Take notice that you are asked for the poles of the closed loop system - not the system being controlled, i.e. the open loop system, G(s).
You already know (or should realize) that the open loop poles are at -2 and -4. But, those are not the closed loop poles. We will refer to them as the open loop poles.
Finally, your answer should be a function of the proportional gain, K.

And, your answer will also depend upon where the open loop poles are. You would get better understanding of that if you do the problem symbolically using p₁ and p₂ for the open loop poles. Then the result will be a function of the proportional gain, K, and the two open loop pole locations, p₁ and p₂.

Next, answer these questions.

Q1 Are the closed loop poles always real? (In other words, are they real for any value of proportional gain, K, whatsoever?)

Q2 Is there a critical value for the proportional gain, K? In other words, is there a value of proportional gain that separates real poles from complex poles.

Finally, there is indeed a critical value for the proportional gain. The only problem is "What is that value?".

P1 Determine the proportional gain that gives closed loop poles just verging on becoming complex. (HINT: That value of gain will produce two poles at the same location, the proverbial "double pole".) Assume the values above for the open loop poles.

Now, it seems clear that the poles, indeed, do move when the control loop is closed. Henceforth (a good word that will remind us this is important!), you need to distinguish clearly between the open loop poles, and the closed loop poles. You need to make that distinction because the closed loop poles and the open loop poles are not the same.

Now, we are not saying that the closed loop poles are completely unrelated to the open loop poles. We are saying, however, that they are not the same. Not only are they not the same, but the closed loop poles move! How they move (where they start, where they finish, the pattern(s) of their movement) are determined by two items: the open loop poles (and zeroes in more complex systems) and the proportional gain.

P2 - For possible hand-in. Check with your instructor. Using the expression you derived above, sketch the locus of the closed loop poles as the proportional gain increases, starting with K = 0, running to an infinite K.

What you have sketched is called the root locus of the closed loop system. The root locus can be a very powerful tool because a lot of information about closed loop system performance is encoded into the closed loop pole locations. (You may want to review the information at this link. Or, you may want to read the entire lesson on second order system response.) The closed loop pole locations determine the following response characteristics.

The speed of the response is determined by how quickly the response due to the pole dies out. That is determined entirely by the distance of the pole into the LHP. The further the pole is to the left, the faster the response due to that pole dies out.
The frequency of any oscillations is determined by the vertical distance (imaginary part) of the pole(s). If the pole is real, there are no oscillations. For a pair of complex poles, the further up (the larger the imaginary part) they are, the higher the frequency of oscillations.
The amount of overshoot is determined by the damping ratio. Here is a separate link for that information.

With all of that in mind, we want you to put the values of the poles for our example system (s = -2 and s = -4) into a special root locus calculator. Here is a link to the calculator page. Do the following.

Input the pole values and click Start.
Observe how the poles move as the gain is varied.

In this root locus calculator, the first points plotted are for the lowest gains, and the gain increases as more points are plotted. You should note how the closed loop poles move as the gain increases.

Closed loop poles start at the open loop poles and move away from the open loop poles as the gain increases.
Closed loop poles can go to infinity as the gain increases indefinitely.

Now, after the simulation has run, and while the points that have been calculated are still showing (And, they will disappear if you hit the Clear button, so don't do that.), you can input a gain value for the proportional gain. Do the following.

Input a value of 0.1 for the proportional gain and observe where the closed loop poles are. You will be treated to little blinking symbols showing the location of the closed loop poles.
Input a value of 1.0 for the proportional gain and observe where the closed loop poles are.
Input a value of 10 for the proportional gain and observe where the closed loop poles are.
Answer the questions below.

Q3 Which value of gain produces closed loop poles closest to the open loop poles?

And, here are a few questions for you. Your instructor may wish you to hand these in when you complete them. Whether your instructor requires them or not, you should be able to answer these questions.

P3 - For possible hand-in. Check with your instructor. What value of gain produces the fastest system?

You need to determine which system is fastest using analytical techniques. This link discusses the relationship between pole position and speed of response.

Now, from the root locus you can see the following in this system.

As the gain starts from zero, the two poles approach each other on the real axis. Eventually, they coalesce, and then break away - becoming complex.
As the complex poles move away from the real axis, they move vertically (one up and one down). In the process, they stay in the left half of the s-plane, so the system never exhibits growing oscillations.

Recall the first system you examined (at this link). That system also exhibited oscillations for a high enough proportional gain, but those oscillations, also, did not grow.

From all of this, there are still some unresolved questions you should have. In the example problem there were two systems. The second one did not behave as nicely as the first one. The second system became unstable for a high enough proportional gain. Clearly something different is going on in the second system. Something different has to be going on for the poles to enter the right half of the s-plane to make the system unstalbe. Clearly, the poles in the second system are somehow moving differently than they were in the first system.

It is this different behavior of the closed loop poles that we need to understand. What is needed is to look at another system that can exhibit the kind of behavior that we found in the second system. We're going to try looking at a third order system. Here's what we want you to do.

The system being controlled (actuator + plant) has a transfer function, G(s).

G(s) = 1/(s + 1)(s + 2)(s + 4)

The first chore for you is this:

Calculate the closed loop transfer function of the system. When you do that calculation, be sure to include the proportional gain, K, where necessary.
Ok, you have the closed loop transfer function. Can you determine the poles of the closed loop system.

Take notice that you are asked for the poles of the closed loop system - not the system being controlled, i.e. the open loop system, G(s).
You already know (or should realize) that the open loop poles are at -1, -2 and -4. But, those are not the closed loop poles. We will refer to them as the open loop poles.
When you had a second order system, you could answer this question using analytical techniques. Here you need to factor a third order polynomial to get the closed loop poles, and the answer will be a function of the proportional gain, K. It's not impossible to do this, but it is pretty miserable. It would be even more difficult if we had four (4) or five (5) open loop poles. Perhaps we should look at other options. After we do that, we can examine questions like the ones we had for the second order system.

Questions about the closed loop poles being real, values for proportional gain that change the poles from real to complex, etc. are very difficult to answer for this system.

The key to getting information about how this system behaves lies with the root locus. You have to use it and understand the information it gives. (Click here for a link that gets you the root locus calculator for a third order system.)

We're going to exercise that root locus calculator a little bit. Do the following.

Input the pole values into the root locus calculator. Poles are at -1, -2 and -4, so you enter a "1" a "2" and a "4" in the appropriate text boxes. Then, Click Start.
Observe how the poles move as the gain is varied.

In this root locus calculator, the first points plotted are for the lowest gains, and the gain increases as more points are plotted. You should note how the closed loop poles move as the gain increases.

Closed loop poles start at the open loop poles and move away from the open loop poles as the gain increases.
Closed loop poles can go to infinity as the gain increases indefinitely.
Comparing this system to the second order system.

In the second order system, as the complex poles move away from the real axis, they move vertically (one up and one down). In the process, they stay in the left half of the s-plane, so the system never exhibits growing oscillations.
In the third order system, as the complex poles move away from the real axis, they tend to move toward the right (as the imaginary part gets larger). They can eventually move into the right half of the s-plane and the system can become unstable (and it will exhibit growing oscillations).

Clearly, the closed loop poles in the third order system behave differently. We observe the following.

There are several critical gain values.

The first critical gain value is the gain that just produces two real poles at the same point. For larger gain values than this value the poles become complex.
The second critical gain value is the gain the produce two purely imaginary poles (on the imaginary axis). For larger gain values, the poles move into the right half of the s-plane and the system becomes unstable.

The three closed loop poles move simultaneously. While the complex poles are moving to the right, the real pole simultaneously moves to the left. If you need to know where all three poles are for any given gain, then you have an interesting calculation to do. If you want to control the real pole independently, you are not going to be able to do that.

We are going to give you some short problems to solve using the root locus calculator.

P2 Determine the proportional gain that gives closed loop poles just verging on becoming complex. (HINT: That value of gain will produce two poles at the same location, the proverbial "double pole".) Assume the values above for the open loop poles. Note that the easiest way to do this is trial and error. Adjust the gain to give two equal real poles.

And here is another, perhaps more important, problem - and a related problem.

P3 Determine the proportional gain that gives closed loop poles just verging on entering the right half of the s-plant.. (HINT: That value of gain will produce two poles with a real part equal to zero!) Assume the values above for the open loop poles. Note that the easiest way to do this is trial and error. Adjust the gain to give two poles just right on the imaginary axis.

P4 When the poles are exactly on the imaginary axis there is a third pole - one that is real and on the negative real axis. Where is that pole?

You should notice that you cannot set the third pole independently, and that leads us to ask this question.

Q4 If you want the third pole at -10, will the closed loop system be stable?

There are some even more intriguing questions we could ask, but it is getting time to summarize some of the things you should have seen. Then, we will point out an intriguing little observation we have made about this system, and point you to what you need to learn. Here is what you have seen.

Different systems can have wildly different behavior. The second order system we studied never became unstable, while the third order system became unstable.
When you plot the poles in the root locus, the closed loop poles can go to infinity, but they don't always do that in the same direction. The second order system closed loop poles went to infinity and +90^o, and -90^o. In the third order system, the closed loop poles went to infinity at +60^o, -60^o, and -180^o. (And, we hope you remembered that third pole. Now we want you to think about it considering the angle it goes to infinity.)

There are some invalid conclusions that you could come to. Here are a few, and it's not a complete list by any means.

In a second order system the poles always end up moving vertically. NOT TRUE! We will look at some systems later where that doesn't happen. It will take a system with a zero as well as two poles, but it will still be second order.
In a third order system, the system will always become unstable eventually. NO TRUE! It's going to take a zero here as well.

In order to understand the behavior of the root locus you have some learning to do. Here is a link to the first full scale root locus lesson. However, we would like to get you thinking about some things that happen here. Do these problems and we will explain - after the problems - what is going on here.

P5 Using the pole values we used above (-1, -2 and -4) calculate the pole locations for a gain of 0.4. Then add up all the poles. (And, remember that the imaginary parts are going to cancel out.) What is the sum of the poles?

P6 Using the pole values we used above (-1, -2 and -4) calculate the pole locations for a gain of 5. Then add up all the poles. (And, remember that the imaginary parts are going to cancel out.) What is the sum of the poles?

P7 Using the pole values we used above (-1, -2 and -4) calculate the pole locations for a gain of 90. Then add up all the poles. (And, remember that the imaginary parts are going to cancel out.) What is the sum of the poles?

P8 Using the pole values we used above (-1, -2 and -4) calculate the pole locations for a gain of 120. Then add up all the poles. (And, remember that the imaginary parts are going to cancel out.) What is the sum of the poles?

Pretty amazing isn't it. All the answers work out the same, and they are all -7. Why is that? Will it always be so? (It's a pretty good bet that it works that way for this system, but what about other systems?)

Let's examine the denominator polynomial for the closed loop system. Here it is. However, you should multiply it out, and focus on the coefficient of the s² term in the result.

(s + 1)(s + 2)(s + 4) + K

= s³ + (1 + 2 + 4)s² + 14s + 8 + K

= s³ + 7s² + 14s + 8 + K

Whenever you have a polynomial, the coefficient of the next to the highest power is always the negative sum of the roots. Check how your multiplication happens, and you will see that will always be the case.

Now, think about what that means in the third order system. When the real pole moves to the left, the complex poles have to move to the right to keep the sum constant. Maybe we should have expected that behavior?

Well, the truth of the matter is that there are a lot of casual facts floating around, and we just brought that one to bear on this problem. You weren't stupid if you didn't think of it. You have a lot of those facts buried somewhere in your memory, and perhaps a few of them will have to be resurrected to understand why the root locus behaves as it does. And, maybe it's time for you to move on to that. Here's the link.