Root Locus

The graph below shows the root locus for a characteristic equation of the form

The roots of d(s) are indicated by x. The roots of n(s) are indicated by o. Drag the x's and o's to change the polynomial. The graph show the corresponding root locus for 0 ≤ k ≤ ∞.

Experiment with different patterns and notice how the root locus pattern is similar for similar configurations of x's and o's.

Questions:

1. What is the root locus (RL) pattern for two x's on the real axis - note the break away point? (answer)
   The root locus starts at the x's, meets midway between them and then is a vertical line.
2. What does the RL shape for system with two real x's tell you about the minimum settling time for that system regardless of the value of k? (answer)
   The minimum settling occurs once the RL leaves the real axis. After that the real part of the roots are unchanged.
3. What is the RL pattern for two real x' and one real o? (answer)
   This depends on the

4. What can you say about the root locus when a x is near an o? (answer)
   When a o is near a x the o attracts the roots from the x.
5. What happens the system for large k if order is d(s) - order of n(s) ≥ 3 (answer)
   When order is d(s) - order of n(s) ≥ 3 there will be three roots at ∞ and at least two will be in the right-half plane. The system will be unstable for large k.
6. Determine the root locus for s³ + (6+k)s² + (11+5k)s + 6 (answer)
   s³ + (6+k)s² + (11+5k)s + 6
   =
   s³ + 6 s² + 11 s + 6 + k ( s² + 5 s)

   so roots of d(s) = -1, -2, -3 and n(s) = 0,-5