Root Locus Design Method

The Main Idea

I want to choose the desired closed loop pole locations since that affects performance.
How can I find the pole locations when I do something to my controller (vary gain), I want to know all possible locations of these closed loop poles…

The design process involves modifying the gain (as mentioned) and possibly adding compensators (lead or lag) to shape the root locus for desired performance

Tip

Probably on exam (given an empty complex plane, draw the root locus)examquestion

• Root locus is the trace of closed-loop poles drawn in the s-plane with respect to different values of P (proportional) gains
• Mirror image for the real axis.
• Starts at open-loop poles $(K=0)$ and ends at open-loop zeros $(K=\infty )$.
• When a pair of poles change from imaginary to real (or vice versa), the root locus breaks into (or away from) the real axis.
• Why do we learn this?
  • Root locus gives a big picture of how closed-loop poles move for each gain.
• Our goal:
  • Given the open-loop system, draw the root locus.

Zeroes

For a transfer function, there are an equal amount of zeroes and poles where the zeroes are not always finite (s in the numerator) but could be a number of infinite zeroes

The Six Steps to Draw

!! Important !!

1. Identify open-loop poles (starting points) and zeros (ending points)
2. Draw real axis branches
3. Draw asymptotes
4. Identify break-in and/or break-away points
5. Compute departure/arrival angles
6. Determine $j\omega$ axis crossing

Angle principle

For $G\left(s\right)=\frac{1}{K}$, $<G\left(s\right)=-180°$. This needs to hold for a pole to be on the R.L We can find branches by finding values of s that hold this condition and then they approach a real zero of an infinite zero.

Rule 1

• Poles are starting points and zeroes (including the infinite zeroes) are ending points

Rule 2

• If the real-axis possesses any pole or zero, there exists a branch of root locus in the real-axis
• If # of poles and zeros to the right of a point on the real-axis is odd, it belongs to the root locus
• Any conjugate pole pair cancels each other out

Rule 3

• If $m<n$, there exist $n-m$ asymptotes (to $\infty$) where $n$ and $m$ are the order of the denominator and numerator
• Angle(s) of asymptote ${\varphi }_i=\frac{2i+1}{n-m}\cdot 180°$
• Center of asymptote $\alpha =\frac{\Sigma p_i-\Sigma z_j}{n-m}$

Rule 4

• If two poles (or two zeros) on real-axis are next to each other, there is a break point
• $\frac{dG\left(s\right)}{ds}=0$
• The roots of this equation are potential break points
• The direction of the poles when they approach infinity doesn’t exist

Rule 5

• For complex poles or zeros, we can compute the initial direction of the branch…
• Departure angle for a pole
  • ${\varphi }_{k,\text{dep}}={\sum }_{i=1}^m\angle (p_k-z_i)-{\sum }_{\begin{array}{c}j=1\\ j\ne k\end{array}}^n\angle (p_k-p_j)-180^{\circ }(2\ell +1)$
• Departure angle for a zero
  • ${\vartheta }_{k,\text{arr}}={\sum }_{j=1}^n\angle (z_k-p_j)-{\sum }_{\begin{array}{c}i=1\\ i\ne k\end{array}}^m\angle (z_k-z_i)+180^{\circ }(2\ell +1)$
• $l$ is just some arbitrary number to make your angle nice

Rule 6

• Crossing $j\omega$ axis means change of stability
• Find the value of gain $K$ to change stability (Routh Test)
• Using this $K$, solve $1+KG\left(j\omega \right)=0$