Introduction

At this point, we have graphed the poles and zeros for a system's closed-loop transfer function to visualize its stability. However, we can actually look deeper into this response by using root locus. Root locus is another way to visualize the stability of our system by isolating the system controller's PID coefficients and determining how each coefficient affects the stability of the system. The way we visualize root locus is by isolating the desired term in the denominator of our closed-loop transfer function.

Isolating RL Form

Let's say for example, we had a closed-loop transfer function as so
$$ G_{cl}(s) = \frac{\frac{\frac{K_i}{s}+K_p}{s+1}}{1 + \frac{\frac{K_i}{s}+K_p}{s+1}} $$
Where the plant is $ P(s) = \frac{1}{s+1} $ and the PI controller is $ C(s) = \frac{K_i}{s}+K_i $ in unity feedback mode. Then what we can do is multiply by $ (s+1) $ to get
$$ G_{cl}(s) = \frac{\frac{K_i}{s}+K_p}{(s+1) + \frac{K_i}{s}+K_p} $$
Now we can multiply by s to get
$$ G_{cl}(s) = \frac{K_i+s K_p}{s(s+1) + K_i + s K_p} $$

The reason we did this is so our transfer function is ready to isolate the desired PI coefficient. Let's say the desired PI coefficient is $ K_i $. Then what we would do is focus on the denominator. In the denominator, we have $ s(s+1) + K_i + s K_p $, or $ s^2 + s(1+K_p) + K_i $ if we simplify. Now we want to get the form of $ 1 + L(s) K_i $ in the denominator because we chose $ K_i $ to isolate. We can do this by dividing by $ s^2 + s(1+K_p) $. That then gives a closed-loop transfer function of
$$ G_{cl}(s) = \frac{\frac{K_i+s K_p}{s^2 + s(1+K_p)}}{1 + K_i \frac{1}{s^2 + s(1+K_p)}} $$
Hence, $ L(s) = \frac{1}{s^2 + s(1+K_p)} $; now we have a transfer function that shows how $ K_i $ affects the stability of the closed-loop transfer function! Now, before we move on, when we put our transfer function in root locus form, all other PID coefficients, other than our targeted PID coefficient, still in $ L(s) $ are set equal to 1. So, $ L(s) $ would actually be
$$ L(s) = \frac{1}{s^2 + 2s} $$
Now we can move on to visualization!

RL in MATLAB

Now there are several ways to visualize this root locus. The first way is via MATLAB. In MATLAB, there is a function called rlocus(), where the input is a transfer function, and it outputs the root locus plot on a real-imaginary graph. In order to do this, let's start out by writing $ L(s) $ as a transfer function in MATLAB

% define L(s) as a TF
Numerator = [ 1 ];
Denominator = [ 1, 2, 0 ];
L = tf( Numerator, Denominator );

We now have $ L(s) $ written as a transfer function in MATLAB using the tf() function. Notice the input to tf() is the numerator's coefficients and the denominator's coefficients. Now that $ L(s) $ has been defined, we can use rlocus() to visualize the graph.

% from before
Numerator = [ 1 ];
Denominator = [ 1, 2, 0 ];
L = tf( Numerator, Denominator );

% use rlocus()
rlocus(L);

And the output from the code above is a graph of the root locus.

Fig. 1 - Plot of root locus for $ L(s) $.

So what MATLAB is doing here, is it's running through the values of $ K_i $ going from zero to $ \infty $, and is plotting all of the zeros and poles. Now, for this example's $ L(s) $, we did not have any zeros, but we had two poles (this is because the denominator was a second order polynomial). MATLAB even shows us the paths of each pole, that is why there are two slightly different colored lines.

RL by Hand

Next we will learn some of the tips and tricks for plotting root locus by hand! Here are the rules/recommendations to keep in mind.

• For the form in the denominator of a closed-loop transfer function, $ 1 + K \frac{J(s)}{H(s)} $, there are n lines, where n is the maximum degree between $ J(s) $ and $ H(s) $.
• As k increases from 0 to $ \infty $, the roots of the closed-loop transfer function move from the poles to the zeros.
• The same root (path) will never cross over its own path (no loops).
• The portion of the real axis to the left of an odd number of open loop poles and zeros are part of the loci.
• Lines break and enter the real axis at 90°.
• If there are not enough poles or zeros to make pair, they will go to or come from $ \infty $.
• Lines go to $ \infty $ along asymptotes.
• If there are at least two roots that come from $ \infty $, then the sum of the roots is constant.
• K going from 0 to $ -\infty $ can be drawn by reversing rule 4 and adding 180° to the asymptote angles.

Now we have a better understanding of what root locus are, why they are important, and how they can be graphed using MATLAB, and by hand!