Root Locus - 4
Two Rules to Use to Draw Approximate Root Locus
Before the advent of the great computer analysis tools for the root locus that we find in LabVIEW, all calculations had to be done manually, with a calculator. In our textbook you will find procedures for determining breakout points, angles of approach, angles of departure, etc. You do not need to learn many of these procedures because they can be done approximately and more easily using the Matlab root locus GUI rltool( ). I would like for you to be able to draw an approximate root locus by hand, so that you can get a general idea of how a system's behavior changes with K without going to a computer. For this you will need to learn only two rules.
Rule 1 - Which Parts of the Root Locus Are on the Real Axis?
Plot the open-loop poles and zeros. Any parts of the real axis with an odd number of open-loop poles and zeros to the right of them are part of the root locus. Let's look at the previous example. The pole/zero plot for
is
The pole/zero furthest to the right on the real axis is the zero at 0. The numbers of poles/zeros to its right is 0. 0 is an even number. So the part of the real axis to the right of this 0 is not on the root locus. Move to the next pole/zero, the pole at -1. To its right is one zero. Every point on the real axis between -1 and 0 has one zero to its right. So this portion of the real axis is on the root locus. For the portion of the real axis between -4 and -1, there is one pole and one zero to the right, two poles/zeros altogether. Since this is an even number, the -4 to -1 section of the real axis is not on the root locus. If we continue like this we will discover that in this case, all portions with a zero to their right are on the root locus.
In fact, for this example, since there are the same number of finite poles and zeros, there are no infinite zeros. So this is the entire root locus. Every open-loop pole has been "captured" by an open-loop zero.