1. Lec 10. Root Locus Analysis II
More about the root locus
Breakin and Breakaway Points
Departure and Arrival Angles
Cross Points of j! Axis
Magnitude Condition
Pole Zero Cancelation
Root Locus with Positive Feedback
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2. Breakin/Breakaway Points
Breakin and breakaway points: where two or more branches of the root locus cross (repeated closed-loop poles)
Breakaway points
Breakin points

3. Finding Breakin/Breakaway Points
Find K so that the characteristic equation 1+K L(s)=0 has repeated roots
1. Rewrite the characteristic equation as (A(s), B(s) are polynomials):
2. Solve the characteristic equation for K as:
3. Solve the equation:
solutions are candidates of breakin and/or breakaway points

4. Determine Breakin/Breakaway Points
Need to find K so that the characteristic equation 1+K L(s)=0 has repeated roots
Write 1+K L(s) = 0 as:
where A(s) and B(s) are polynomials independent of K
If s is a root of f(s), then
If s is a repeated root of f(s), then we have in addition
So,
solutions are candidates of breakin
and breakaway points

5. Example
Characteristic equation:

6. Example +

7. Breakin and Breakaway Points

8. Multiple Breakin/Breakaway Points+

9. Complex Breakin/Breakaway Points+

10. Departure Angles Departure angle: Example:
Angle along which the root loci leave the open-loop poles
Example:
Test point s is on the root locus if
For a test point s very close to p1, this implies
1 is the departure angle from p1
Departure angle from p2?

11. Arrival Angles Arrival angle: Example:
Angle along which the root loci enter the open-loop zeros
Example:
Angle condition:

12. General Departure/Arrival Angle
A general point s is on the root locus if and only if
Departure angle i from an open loop pole pi: (choose a test point close to pi)
Arrival angle fi from an open loop zero zi: (choose a test point close to zi) +

13. Repeated Zeros/Poles Case
If we have repeated open loop zeros/poles, then there are multiple arrival/departure angles associated with them
Example: +

14. Points Where the Root Locus Crosses the j Axis
Determining the points where the root locus crosses the j axis is important because it gives on bound on K for the stability of the system +
cross points?
corresponding K?

15. Direct Method
To find K so that the characteristic equation has solutions on the
imaginary axis, we let s=j

16. Method One: Routh’s Criterion
Closed loop poles are solutions of the characteristic equation
Cross points occur correspond to boundary value of K for stability
Routh’s array:

17. Angle Condition (review)
Closed loop poles are solutions of characteristic equation
s is on the root locus if and only if +

18. Magnitude Condition
For a point s known to be on the root locus, what is the corresponding K?

19. Pole-Zero Cancellation+
Root locus suggest the closed loop system is stable for all K>0

20. Pole-Zero Cancellation (cont.)
In practice, parameter inaccuracy may result in a slightly different system +

21. Pole Zero Cancellation (cont.)
Correct root locus
a branch starting and ending at s=1 + +

22. Root Locus with Positive Feedback+
Characteristic equation
Angle condition: a point s is on the root locus if and only if

23. Rules for Plotting a Root Locus with Positive Feedback

24. Example +