1. CH. 6 Root Locus
Chapter6. Root Locus

2. 6.1 Introduction. Root locus by W. R. Evans 1948 Root locus
A graphical presentation of the closed-loop poles as a system parameter is varied.
A powerful method of analysis and design for stability and transient response.
Control system problem
Closed-loop transfer function:
Characteristic equation:
Chapter6. Root Locus

3. Control system problem
Forward transfer function:
Feedback transfer function:
Closed-loop transfer function:
Characteristic equation:
Chapter6. Root Locus

4. Root locus problem Open-loop transfer function: KG(s)
Magnitude condition:
- Phase condition:
Chapter6. Root Locus

5. Vector representation of complex numbers
- Magnitude condition:
- Phase condition:
Chapter6. Root Locus

6. Ex)
Characteristic equation:
Chapter6. Root Locus

7. 6.2 Sketching the Root Locus
General form of root locus problem
1. Number of branches
The number of branches of the root locus equals the number of closed-loop poles
2. Symmetry
Physically realizable systems cannot have complex coefficients in their transfer function
Complex closed-loop poles always exist in conjugate pairs.
The root locus is symmetrical about the real axis.
Chapter6. Root Locus

8. 3. Starting and ending points
The root locus begins at the finite open-loop poles and ends at the finite and infinite open-loop zeros.
Chapter6. Root Locus

9. 4. Real-axis segments
On the real axis, the root locus exists to the left of an odd number of real-axis, finite open-loop poles and/or finite open-loop zeros.
Chapter6. Root Locus

10. 5. Behavior at infinity (asymptotes)
The root locus approaches straight lines as asymptotes as the locus approaches infinity.
For the infinity s, all pole vectors and zero vectors have same angle.
The angle of asymptote
The root locus equation
For n >m +1,
Chapter6. Root Locus

11. For large values of K, m of the closed-loop poles are approximately equal to the open-loop zeros and n – m of the closed-loop poles are from the asymptotic system, whose poles add up to
Chapter6. Root Locus

12. 6. Real-axis break-away and break-in points
Necessary condition for the points
Chapter6. Root Locus

13. 7. The imaginary axis crossing Using Routh table
Using , two equations and two unknown
Chapter6. Root Locus

14. 8. Angles of departure and arrival Using phase condition
Chapter6. Root Locus

15. Two segments come together at 180° and break away at 90°.
9. The other rules
Two segments come together at 180° and break away at 90°.
Three segments approach each other at relative angles of 120° and depart at angles rotated by 60°.
Poles are sources and zeros are sinks.
Chapter6. Root Locus

16. Ex) Open-loop transfer function
General form of transfer function
Locate open-loop poles and zeros
Real-axis segments
Asymptotes
Chapter6. Root Locus

17. Angles of departure and arrival
Real-axis break-away and break-in points
The imaginary axis crossing
Chapter6. Root Locus

18. Ex) Open-loop transfer function
General form of transfer function
Locate open-loop poles and zeros
Real-axis segments
Asymptotes
Chapter6. Root Locus

19. Real-axis break-away and break-in points
The imaginary axis crossing
Chapter6. Root Locus

20. Ex) Open-loop transfer function
General form of transfer function
Locate open-loop poles and zeros
Real-axis segments
Asymptotes
Chapter6. Root Locus

21. The imaginary axis crossing
Real-axis break-away and break-in points
The imaginary axis crossing
- Routh table
Chapter6. Root Locus

22. Angles of departure and arrival
Chapter6. Root Locus

23. 6.3 System analysis using root locus
Estimate stability and performance based on root locus
Ex)
Stable range:
Unstable range:
Conditionally stable system: To achieve strict stability, the other controller is needed.
Chapter6. Root Locus

24. Ex) From the root locus, find the transient response specifications of the closed-loop system (damping ratio of dominant pole, 2% settle time, and steady-state error of unit step response) with respect to the parameter K.
Chapter6. Root Locus

25. Damping ratio of dominant pole:
2% settle time:
Chapter6. Root Locus

26. Steady-state error of unit step response:
Chapter6. Root Locus

27. 6.4 Others Generalized root locus Closed-loop transfer function :
Characteristic equation:
Chapter6. Root Locus

28. Root locus for positive-feedback systems
Chapter6. Root Locus

29. Pole sensitivity
Chapter6. Root Locus

30. Ex) Find the root sensitivity of the system in Figure at s = -9
Ex) Find the root sensitivity of the system in Figure at s = and -5 + j5. Also calculate the change in the pole location for a 10% change in K.
Chapter6. Root Locus

31. The system’s characteristic equation:
Chapter6. Root Locus