An Introduction To The Root Locus

Introduction

Goals
The Root Locus
Rules For Constructing The Root Locus
Using The Root Locus
To Make Performance Predictions
Links to other pages
 The Root Locus Arcade
You are at:  Design Techniques - The Root Locus - In Continuous Systems
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Why Do You Need To Know About Root Locus?

        Control systems can be very complex.  An aircraft control system, for example, needs to control a system with six degrees of freedom (Up/down, left/right, forward/backward, roll pitch & yaw, and you have velocity and acceleration in each variable.).  That means that the dynamics of the system are very complex, and there are at least 12 poles in the system.  If you want to control a system with complex dynamics you need tools that will let you design controllers for the system.

        Many systems do not behave well without a controller.  An airplane is a good example.  A paper airplane will often exhibit pitch instability or pitch oscillations when you fly it (like the plane at the right!).  If you haven't done that in a while, do it now and observe what happens.The kinds of behaviour that your paper airplane exhibits is a sample of what a real airplane will exhibit - only the real airplane will be worse.  A real airplane may need to fly at 500 miles/hour at 30,000 ft and at 180 miles/hour 50 feet above the runway.  It has severe differences in flying conditions, and the aircraft has to behave in any conditions.  Ensuring that an aircraft will perform well despite severe changes in conditions is an example of the problems faced by control engineers.  Control engineers need a large arsenal of weapons to use on these problems.


Goals For The Root Locus

        In this lesson you will learn about a control system analysis and design tool called the root locus.  The root locus is a way of presenting graphical information about a system's behavior when the controller is working.  The root locus is a widely used tool for design of closed loop systems, and it has the virtue of being a good design tool for continuous time systems (where you work in the s-plane) and for sampled (computer controlled) systems (where you work in the z-plane).

        You should also know where the root locus rules come from.  We'll discuss that within the lesson.

lGiven an open loop transfer function for a system,
 uBe able to determine the root locus for the system.

lGiven a root locus for a system,
 uBe able to use root locus information to predict
       aspects of the system's closed loop behavior,
       including speed of response.


Root Locus

      We will begin by examining a simple control system.  We will use a controller that is just a gain - a proportional contoller - and some simple feedback to control a first order system.  That system is shown in the diagram below.  It has an adjustable gain, K, and a single pole at s = -a.

        The closed loop transfer function is:

                    CLTF(s) = K/(s + a + K).

We are interested in the behaviour of the closed loop system, and the primary indicator of closed loop behaviour is the closed loop pole.

      The closed loop system has a pole at s = - a - K.

        Here is a movie of how the closed loop pole moves as the gain, K, changes (for a = 1 for this example).  In this movie, the gain starts near zero and increases until the pole moves off the graph.  Then the gain resets and the movie repeats.  This movie - and others - will point out applications of rules we will develop later in the lesson.

      What the movie shows is really a root locus for our system.  A root locus is the locus of the closed loop poles (roots of the denominator polynomial) plotted as the open loop gain constant changes.  The movie is not a root locus because it does not show - at one time - all the locations that the poles can have for different values of gain.  Here is the root locus with points filled in and with an "x" marked at the location of the pole for this system.

        The concept of a root locus sounds simple enough, but the problem is that closed loop poles can exhibit many strange and wonderful perambulations in the s-plane when the system is even a little bit complex.  The example is simple, and the pole motion is simple.  It's even simple to calculate the closed loop pole as a function of gain.  But what happens when you have two poles, as in a motor? - five poles, as in a subway car? - twenty nine poles, as in an airplane? - or even hundreds of poles, as in a satellite with flexible solar collectors?

 We'll examine a few more systems, each one a little more complex as we go along.

        Now, here is a system with two open loop poles.

Again, we can compute the closed loop transfer function.  If we do that, the result is:

Closed Loop Transfer Function  = 

Closed Loop Denominator  = 

        Now, if we compute the closed loop poles (using the quadratic formula) we obtain roots at:

        Note that we can make the closed loop roots complex if we make the gain large enough.  This video clip shows how the roots move as the gain, K, varies.  In the video K starts at low values (near K = 0) and increases.





        Notice in the video that the two poles are real for low values of gain, and that when the gain, K, is increased beyond a certain point, the poles become complex.

        The two example root loci given so far are fairly simple.  More complex systems will have more complex root loci.  Here is a link to a lesson on some root locus rules.
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Rules For Construction Of The Root Locus

        What is given here a brief version of the root locus rules with accompanying commentary.  After a few rules we will visit the Root Locus Museum and Arcade and observe how the rules work.  Here's the first rule.

Rule 0:  The Root Locus has as many branches (closed loop poles migrating in the s-plane) as there are open loop poles."It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was an open loop pole, it was a closed loop pole", (falsely attributed to Charles Dickens)

        What you have to remember is that it is the closed loop poles that wander around in the s-plane, but there are as many closed loop poles are there are open loop poles.

        Here is the root locus for the first order system.  Notice that there is one open loop pole at s = -1, and that there is one closed loop pole moving to various locations in the s-plane as the gain varies.

        Now, examine the root locus for the second order system.  It has two branches.  There are two closed loop poles that move around in the s-plane.

        You will need to know more about how and where the closed loop poles migrate within the s-plane.  There are more rules that determine where and how the closed loop poles migrate.

Rule 1:The locus of 1 + KG(s)H(s) = 0 begins at the open loop poles (of G(s)H(s)) for K=0, and ends at the open loop zeroes (of G(s)H(s)) as K goes becomes infinite.  Yea, and each pole shall leave its open loop pole parent and cling to the open loop zero unless there be not enough zeroes, and then the closed loop pole will wander in the wasteland forever.

        As the gain in the system varies, the closed loop poles move.  They start moving - for low gain - from the open loop poles.  There's something here that invites confusion, so you need to remember to distinguish between open loop poles and closed loop poles.  It's the closed loop poles that move.  They start at the open loop poles.  Some eventually end up on open loop zeros as the gain becomes infinitely large.  The rest go to infinity.

          Let's look at the two systems we saw above.  The root locus for the first system is shown here again.  Notice that the open loop pole here is at -1.  For low gain values the closed loop pole starts moving away from the open loop pole at -1, and goes off to the left as the gain increases.  The clip shows pole location and prints the gain value as well, so you can see how the closed loop pole moves as the gain is changed. Compare the plot (above) - which shows many of the points on the root locus, with the movie clip which shows how the closed loop poles move when the gain changes.

There are two important points to understand before you leave this area:

        We can also look at the root locus for the second order system.

        The closed loop poles start at the open loop poles at -1 and -4, and then move from there.  Here is the movie again that shows how they move.

        The most interesting point is that the two poles start at -1 and -4 and move toward each other.  Eventually, the two closed loop poles meet at s = -2.5, and after they collide, they begin to move vertically, so they become complex - and acquire an imaginary part.  Remember this because we're going to see a more complex example where you will find similar - but more complex - behavior.

        More complex systems will demand more thought.  Let use consider a slightly more complex system with three open loop poles and one open loop zero.  Here is a system with three poles and one zero.  The transfer function is:

G(s) = K(s + 2)/(s + 1)(s + 4)(s + 6)

In the frame you can click to see where the poles are - where the root locus starts for low gain - and where the zeroes are - where the root locus segments end.


Rule 2:The Root Locus lies on the real axis to the left of an odd number of singularities - poles and zeroes.  Sinister survival! - Actually, sinister means "left", and left handed people were once considered "sinister".  Be careful how you use the word "sinister" from now on.

       From the looks of this rule, there are usually going to be some places on the real axis that the closed loop poles will go through for some gain values.  Equally important is the fact that there are places - to the left of an even number of singularities - where the root locus will not go.

        You should have noticed on the two and three pole loci that branches went to infinity at different angles.  After a short example, we have a rule for how that happens - what the angles are.  However, going back to the issue of where the root locus lies on the real axis, consider this plot of the poles and zeroes for a system with three poles (marked by x's at -1, -4 and -6) and one zero (marked by "o" at -2).  This is the same example you saw above.

Click here to see why this rule holds.


Rule 3: The Root Locus goes to infinity at angles of (2k+1)180o/(#P-#Z) for a sequence of integer k's.Gentlemen, please go to your corners.

Note that #P is the number of open loop poles and #Z is the number of open loop zeroes.

       Click here to take a short quiz to see if you understood the rules to this point.

        Next you can go to the Root Locus Arcade to see a number of different root loci.  There you can examine those root loci and see how these rules are satisfied for various pole-zero constellations.  Listed below are a number of "thinking points" - things you should think about as you experience the root loci in the arcade.



Click here to go to the Root Locus Arcade.  Archived there are numerous ancient root loci.  As you play the root locus movies in the arcade remember to think about:         Several Root Loci in the Root Locus Arcade show how the rules given above apply.  Also, you may find some behavior that you need the next rule to explain.  Again, try to predict the behavior each root locus in the Root Locus Arcade before you play the clip.  Remember the questions we asked above.

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        Now, it is time to consider another root locus rule.

Rule 4:The Root Locus goes to infinity at angles given by Rule 3, radiating from a centroid located at (Spoles - Szeroes)/(#P-#Z) Finally, we know the location of the Big Bang.  (Note, the centroid can also be referred to as the center of gravity.)

        Note that Spoles is just the sum of all the poles.  Add them up.  If you have poles at -1 and -2, the sum is -3.

        What this rule means is that you can draw lines going to infinity - starting at the centroid as computed above - and that the root locus will be asymptotic to those lines.

Example:
        Let's take the 3-pole-1-zero system with this transfer function.

G(s) = K(s + 4)/(s + 1)(s + 3)(s + 5)

To compute the centroid do the following:

Sum of the poles = (-1) + (-3) + (-5) = -9

Sum of the zeroes = -4

There are 3 poles and 1 zero, so:

#Poles - #Zeroes = 3 - 1
                           = 2
                           = Number of branches that go to infinity.
Then the centroid is:

Centroid = [(-9) - (-4)]/2 = -5/2 = -2.5

Here is the root locus.

Now be sure you can answer the following questions.

        Now we're ready to take a look at some other kinds of systems that give some thought provoking root loci.  One system is shown below.  Here we'll assume that the poles are at -1 and -4, but that the zero is at -5.

G(s) = K(s + 5)/[(s + 1)(s + 4)]

 There's a little problem here.  Try sketching the root locus.  Do you see the problem?

        What you should have observed is that there are two places on the root locus that are to the left of an odd number of singularities.

        We have let the movie run for just a few frames, to how the poles start to move, so you can then think about what can happen so that the rules we have discussed so far can all be satisfied.  What's going to happen on this root locus?  Consider the following:

        If you can't answer these questions - or even if you can - check out the complete root locus below.
 
 




        Now, let's play a little "What If?".  What if we take this system with two poles (-1 and -4) and a zero (-5) and add a third pole - somewhere much further out on the negative real axis.  Here is the root locus.

NOTE THE FOLLOWING ABOUT THE ROOT LOCUS BEHAVIOR IN THIS PLOT.

        Think about how the poles obey the rules for this system.

        Here is that root locus again.  Play it again, and make sure that you understand exactly how the rules are applied to all of the details in the locus that we have just discussed.


Rule 5: At any point on the locus, the gain can be calculated as the product of distances from the point to the poles divided by the product of distances from the point to the zeroes.  (Note, if there are no zeroes, the denominator is 1.)

        This rule is best exemplified by an example.  Shown below is the first system we will consider.

This system has two poles, and the root locus for this system is shown below.

        As the expression to the right above shows, the distance from one point (either point since things are symmetrical!) is the square root of 10, or about 3.16.  Thus the gain at the point s= - -1 + 3j is 10 - the product of the two distances.  If we wanted the closed loop poles to be at that point, the gain would have to be set to 10.

Caveat:  There is one caveat in this calculation.  The calculation assumes that the G(s) term has a numerator such that the highest power of s is 1.  This is paraphrased as saying the leading coefficient in the numerator is 1.  Similarly, the leading coefficient of the denominator must also be 1.  If that is not true, then you will have to do some algebra to put the system into that form.  Otherwise, the calculated gain will not be correct!

        Now, here is another example system.  This system has two poles and one zero.

        The poles are at:        s = - 1,    and    s = - 4.

        The zero is at:          s = - 5.

Of course, that pole-zero structure leads to an interesting root locus that leaves the real axis to form a circle.  The example we want to investigate is to calculate the gain at the point indicated on the plot.  Before you click the buttons and read on, try to determine what you have to do to calculate that gain.  Note that you are trying to calculate the gain for a point on the root locus that is at the center of the rectangle on the root locus plot.

        One result given above is the product of the pole distances.  You need to divide the product of the pole distances by the zero distance in this case.  Doing that, you should obtain:

       Root Locus Gain = 5.862/1.972 = 2.973.

        And that's all there is to this second example.  Now it is time for you to try some problems to see how well you have learned this material. Click here to go to the problem list.

        There are other rules, and they are tabulated in many control systems texts.  However, today it is simpler to compute root loci using mathematical analysis programs like Mathcad and Matlab, and, for the most part, those other rules - while important - do not help as much in sketching and understanding as the rules presented here.  Our additude is that you should sketch a root locus to understand it, but that details are best left to a computer.


Using The Root Locus To Predict System Performance

        You still need to learn how to use the root locus to make predictions about how systems will perform and how to use the root locus to help you set system parameters to achieve performance you specify.

        In this section you will begin using the root locus to set gains that give different kinds of performance.  here are the goals for this section.  We're going to work some examples that will help us achieve these goals.

Given a feedback control system and the root locus for the system,
  Be able to determine the root locus for the system.
  Be able to use root locus information to predict aspects of the system's closed loop behavior, including speed of response.
  Determine the proportional controller gain(s) that will produce a given damping ratio in complex poles.
        In all cases, in this section we will assume that we have a system like the one shown below.  When we refer to poles and zeroes they belong to G(s), and when we refer to a gain, it will be the adjustable gain, K.

        Let's consider a system with a pole at s = -1.  The root locus for this system is shown below for a = 1.  How large should the gain be for a settling time of 1 second?

        At this point you know where the pole needs to be, but you also need to know how to put the pole in a position that satisfies the time response specification.  However, you know that the root locus specifies where the pole moves as the proportional gain, K, is varied, and you can compute K for any pole position.  So, the question now is to determine the gain that gives an acceptable pole position.
 


Now, let's consider a more complex system.


        To compute the root locus gain for instability,


 For this example, we can compute those distances


4.805 * 5.576 * 6.862 = 183.851

        Now, you should check your understanding of this kind of calculation by doing a problem.  Click here for the problem.  This problem will give you a chance to compute a gain, and if you do not get it correctly, it will guide you through the process. 


Conclusions

        In this lesson you have started to learn what a root locus looks like, and how to do gain calculations for points on the locus.  However, the root locus is not an isolated thing.  Rather, it is a tool that you use for a number of purposes.  Some of those purposes are:

        To use the root locus effectively, you also need to know things like how pole position influences response, how pole position is related to damping ratio, how pole position is related to measures of response time and how gain influences system behavior, particularly steady state error.



Links to Topics in this Lesson
Introduction
Goals
The Root Locus
Rules For Constructing The Root Locus
Using The Root Locus
To Make Performance Predictions


Links to other pages

     The Root Locus Arcade


Problems        1
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Problems