To understand what root locus plots are, and why they are important, let's examine the behavior of a system when it is in a control system. Assume that the system is defined by the transfer function:
We'll control this system with a very simple proportional controller in which the input to the system to be controlled is proportional (with gain, K) to the difference between the input, R(s), and the output, C(s).
The loop gain is K·G(s), so the closed loop gain is given by
We want to examine how the behavior of the system varies as K changes, so let's try several values of K. Let's arbitrarily try K=1, 10 and 100 so that we have a wide range of K values.
|K||Xfer Function||Step Response|
The response with K=1 was too slow, the response with K=100 was too oscillatory, and the response with K=10 is almost just right, though we may want to adjust K to get a little bit less overshoot. Clearly this method is rather "hit-or-miss" and it may take us a long time to find a suitable value for K.
A more analytical method might involve finding the poles of the closed loop transfer function. Since the transfer function is second order, we can factor the denominator using the quadratic equation. The roots of the denominator are at:
This will give overdamped response for 9>4K, underdamped response for 9<4K, and critically damped response for 9=4K. If we specify that we want an underdamped response with ζ=1/√2 we must set the magnitudes of the real and imaginary parts of the roots equal to each other. So starting from
we know that the real and imaginary parts must have equal magnitude
and we can solve for K.
Let's try this value. If we use K=4.5 we get the transfer function
and the step response is shown below. As you can see, it is superior to our other attempts; it is reasonably fast with fairly small overshoot.
While this technique is suitable for simple problems, like the one given, it quickly becomes untenable for more complicated systems. For example what would we do if
We will need another approach to deal with such systems. This is introduced in the next section (though in the context of the simple system we have been dealing with).
For the very simple problem described above, it was possible to calculate the precise location of the roots, and choose a value of K that gave us a good response. For more complicated systems it is not so straightforward so we need a more general method for approaching the problem. This more general method is called the "root locus" method. With this technique we make a plot of the path of the roots as a parameter (usually the gain, as above) varies. (Note).
Consider the simple example from above,
If we want to plot the path of the roots as K varies we can calculate the roots of the equation
for many values of K by using the quadratic equation.
|4||-1.5 ± j1.32|
|10||-1.5 ± j2.78|
|20||-1.5 ± j4.21|
|40||-1.5 ± j6.14|
|100||-1.5 ± j9.89|
These values are plotted below (for complex conjugate roots, the value of K is only shown for the root with a positive imaginary part).
There is a lot of information in this diagram. It tells us that the system starts out overdamped for small values of K, and becomes underdamped as K increases, and becomes increasingly underdamped as K continues to increase.
A root locus plot is a variation on this kind of plot. It shows the path of the roots as K is varied, but does not show the actual values of K. This kind of plot is sufficiently important that Matlab has a command specifically for generating these plots. A root locus plot for this system is shown below, along with the Matlab used to create it.
>> G=tf(1,[1 3 0]) %Define gain of system in the loop Transfer function: 1 --------- s^2 + 3 s >> rlocus(G) >> axis([-4 0 -10 10])
The starting points for the roots, when K=0 (note), are shown by the two small "x" marks at s=0 and s=-3. As K increases, the two roots move horizontally towards each other, meet at s=-1.5, and then move vertically away from each other.
Though the value of K isn't plotted on the graph it is easily found. For example, if we want to know the value of K at s=-1.5 we can use the fact that the characteristic equation of
is given by the denominator,
This equation is true for any point on the root locus (more on this on the next page), and in particular it is true at s=-1.5. Since at s=-1.5
We can use this information to calculate the desired value of K:
For the very simple system of this problem, there were many ways to find how the roots varied as we varied the gain of the system. For a more complicated system this is not easy. The root locus plot gives us a graphical way to observe how the roots move as the gain, K, is varied. The next page gives a description of techniques for sketching the location of the closed loop poles of a system for systems that are much more complicated.
The root locus plot indicates how the closed loop poles of a system vary with a system parameter (typically a gain, K).
Derive Rules for Sketching the Root Locus