An Introduction To The Root Locus

       Here is the Root Locus Arcade.  Archived here are many ancient root loci.  The open loop systems have poles at

System 1: One pole at -1
System 2: Two poles at -1 and -4
System 3: Three poles at -1, -3 and -5
System 4: Three poles at -1, -3 and -5 and a zero at -4.
System 5:Three poles - 2 of which are complex, and a zero.
System 6:Like System 5, but with a different zero placement.
System 7:Three poles - 2 of which are complex, and a zero.
System 6:Five (5!) poles (2 complex) and two complex zeroes

        There are also some examples of how root loci change with parameter changes (i.e. when poles and zeroes change).  Click here to check those out also.

 As you play the root locus movies remember to think about:

        Now, click these hotwords above to see and play the various root loci.  For each root locus there will be some comments/observations that will direct you to interesting points for each locus.

One Real Pole

        The root locus for the system with one real pole is the starting point for gaining understanding of root loci.  Even though it is the simplest system with a root locus, it still has two of the root locus rules that apply - as noted above.

        The next root locus is just slightly more complex.


Two Real Poles

        The root locus for a system with two real poles has another rule that comes into play.  Not only do the pole move off at +90o and -90o, they ultimately appear to move away from the centroid.  The next root locus has more poles, and applying the centroid rule is a little bit more subtle.
Three Real Poles

        The three root loci above are interesting, but they all have one thing in common.  The open loop transfer functions all have poles but no zeroes.  With no open loop zeroes some interesting behavior may not be seen.  Here is a root locus with three poles and one zero.


Three Real Poles, One Real Zero

        Notice how the system looks somewhat like a second order system, but the centroid is not located at the mid-point of the real axis segment from which the branches emanate - the ones that eventually go to infinity.
Four Poles - Two Complex, One Real Zero

        This root locus has four poles and one zero, so it has three branches going to infinity.  However, the "extra" pole and zero make life interesting before that happens!
Four Poles - Two Complex, One Real Zero

      Compare this locus to the previous one.


Four Poles - Two Complex, Two Complex Zeroes


Five (5!) poles (2 complex) and two complex zeroes

      This one is here for fun, and to get you to think some more about all of the root locus rules!


Here are links to the various systems (by number)    1   2  3   4  5   6  7   8

Some General Observations
  • You may have observed that the roots move more quickly when they approach each other on the real axis.
    • That means that the root position is more sensitive to small changes in the gain.
    • It would be harder to adjust gain to put the roots there compared to other locations.
  • When a zero is on the root locus, it appears to attract the locus toward it.
Root Locus and Parameter Variations

        It is also interesting to see how root loci change when parameters in a system - a pole or a zero, for example - change.  In this section you can see some root loci that give vivid demonstrations of changing root loci.

        This system has two poles at -1 and -2, and a third pole which can move.  There is also a zero in this system located at -3.  The movable pole starts at s = -3 and moves to s = -19 in the movie.  Notice how the nature of the root locus changes.  When the movable pole is somewhere between s = -14 and s = -15 the complex branches of the locus move the the real axis, and when the movable pole is further to the left, those poles travel along the negative real axis.

In this second system, there is a movable zero.  Note the dramatic change in the form of the locus as the zero changes position.