The Root Locus Explained

The simplest form of control is gain control, where you just input a multiple of the error into the system as shown below: Ideally we would solve every problem with gain control. The problem is that it's never really clear what gain to use, or if it's even possible to solve our problem completely from gain control. More specifically, how do we know that a certain level of gain will lead to a stable system? The Root Locus method is a simple way to see what gain control can do for your system. With a root locus plot you can quickly and easily get an idea of what your system will do under different levels of gain control. A root locus plot will often be one of the first steps to take when looking to create a controller for some system.
A root locus plot is a graph of how the closed loop poles of a system change as the gain of the system is changed.
A root locus plot will look like the following: The x's are the open loop poles, the o's are the open loop zeros and the lines show how the closed loop poles change as the gain is increased from 0 to \(\infty\) (starting from the x's and moving to the o's) Using a root locus plot we hope to select a gain that places the closed loop poles in the right spot to get the characteristics that we desire (like rise time, damping ratio etc)
Given the system: Its closed loop transfer function is: $$ \frac{K\cdot G(s)}{1 + K\cdot G(s)} $$ Assuming we can write: \(G(s) = \frac{N(s)}{D(s)}\) $$ \frac{K\frac{N(s)}{D(s)}}{1 + K\frac{N(s)}{D(s)}} $$ $$ \frac{K\cdot N(s)}{D(s) + K\cdot N(s)} $$ To find the closed loop poles at each gain \(K\) we need to solve $$ D(s) + K\cdot N(s) = 0 $$
While today it's simple to get a computer to just find the closed loop poles for thousands and thousands of different gains the Root Locus method was invented back when mass computing power wasn't available. In that vain there are a collection of simple rules that allow anyone to hand draw a root locus plot in just a minute or two. While you won't need to draw these plots when really designing systems, practice in plotting them by hand will teach you how to visualise what happens to a system as the gain is changed so you will know more or less whether to look at gain control as the solution to your problem.
When hand drawing root locus plots we use the equation $$ 1 + K\cdot G(s) = 0 \implies K\cdot G(s) = -1 $$ This means that $$ \lvert K\cdot G(s) \rvert = 1, \quad \angle(K\cdot G(s)) = 180^\circ $$
These are called the magnitude condition and angle condition respectively. So for each point \(s_1\) that is a part of the plot we'll find that plugging \(s_1\) into \(G(s)\) will satisfy both the magnitude condition and the angle condition.
To construct a root locus plot by hand:
  1. Draw the open loop poles and zeros - The root locus branches begin at the open loop poles and end at the open loop zeros (or infinity if their orders don't match)
  2. Determine the root loci on the real axis - Poles and zeros on the real axis result in straight lines along the axis. Pick a point on the real axis, if there are an odd number of poles and zeros (summed together) on the real axis to the right of that point then there is a line there.
  3. Find the asymptotes - as a point moves infinitely far away from the open loop poles and zeros they approach straight lines whose angles are given by $$ \text{Asymptote angles} = \frac{\pm 180^\circ (2k+1)}{n-m}, \quad (k = 0, 1, 2, \ldots) $$ where \(n = \) number of open loop poles \(m = \) number of open loop zeros
  4. The point where all the asymptote lines intersect on the real axis is given by $$ s = -\frac{(p_1 + p_2 + p_3 + \ldots) - (z_1 + z_2 + \ldots)}{n-m} $$ where \(p_x\) is the \(x^{\text{th}}\) open loop pole and \(z_x\) is the \(x^{\text{th}}\) open loop zero.
  5. Find the breakaway and breakin points - These are the points where the root locus enters or leaves the real axis. They are given by $$ \frac{dK}{ds} = \frac{D'(s)N(s) - D(s)N'(s)}{N^2(s)} = 0 $$ If some real point \(s\) satisfies that equation and the root locus passes through that point then it is a breakaway or breakin point. If that point is a complex conjugate point and the root locus passes through it then as long as the corresponding \(K \gt 0\) it is a breakin or breakaway point. All other situations mean the point is NOT a breakin or breakaway point.
  6. Find the angle the root locus makes with a complex pole or zero - $$ \text{Angle of departure from complex pole} = 180^\circ - \text{(sum of angles from other poles)} + \text{(sum of angles from zeros)} $$ $$ \text{Angle of departure from complex zero} = 180^\circ - \text{(sum of angles from other zeros)} + \text{(sum of angles from poles)} $$
  7. Find where it crosses the \(j\omega\) axis - Solve for when \(s = j\omega\) in the characteristic equation for \(\omega\)
That's a lot of steps, and a lot of new stuff. In practice you'll fire up Matlab and have it plot this for you, but it's good to do at least a few root locus plots yourself (and you'll probably be tested on it, so there's that).
practice problems