1. Chapter 8 Root Locus and Magnitude-phase Representation
§ Root Locus in Control Problem
§ Magnitude-phase Representation
§ Evan’s Root-locus Method
§ Root-locus Method in Design

2. § 8.1 Root Locus in Control Problem (1)
Proportional Control Problem:
K: Can be controller parameter or plant parameter
C-L System:
Zeros of T(s): Consist of the zeros of G(s) and poles of H(s)
Poles of T(s): Changes with K
Problem: Investigate the effect of varied K on controlling system dynamics
through the poles of closed-loop system.

3. § 8.1 Root Locus in Control Problem (2)
Fundamental of Root Locus:
Definition of root locus:
Graphical representation of the path of the closed-loop poles as one or more parameters of the open-loop transfer function are varied (usually ).
Usage of root locus:
Provide relationship among open-loop system parameter, closed-loop pole location, and output transient response.

4. § 8.1 Root Locus in Control Problem (3)
Root Locus of 2nd-order Closed-loop System:
Root locus:

5. § 8.2 Magnitude-phase Representation (1)
Vector Representation of Complex Numbers:
Cartesian
Coordinate
Polar Coordinate

6. § 8.2 Magnitude-phase Representation (2)
(1) Absolute magnitude and phase
As a free vector
Any horizontal reference line
(2) Relative magnitude and phase

7. § 8.2 Magnitude-phase Representation (3)
Ex: Find residues in partial fraction expansion for
by pole-zero representation.
Sol:
pole-zero representation

8. § 8.2 Magnitude-phase Representation (4)
Ex: If s is a complex variable, solve
Sol:
poles: s=0, s=-1
In complex plane
From eq.(1)

9. § 8.2 Magnitude-phase Representation (5)
Where is the “s” to satisfy eqs. (2a) and (2b) in complex plane?
(A) From phase relationship (2b)
(B) From magnitude relationship (2a)
i.e. s is on the bisection line
Conclusions: From (A) and (B) s=-0.5, -0.5

10. § 8.2 Magnitude-phase Representation (6)
Ex: Find the root locus of a DC-servo position control system with varied K.
Sol: C-L system,
characteristic eq.
(A) Direct solution method
O-L System
pole-zero representation
C-L system is stable with varied K.
C-L system is underdamping when K>0.25.
Root locus with varied K

11. § 8.2 Magnitude-phase Representation (7)
(B) Use magnitude-phase representation
characteristic eq.
From the example in section 8.2:
(a) Phase relationship s is on the bisection line between 0 and –1.
(b) Magnitude relationship
Root locus

12. § 8.2 Magnitude-phase Representation (8)
Step response
Key parametric values of K in root locus:
(1) Intersection between root locus and axis
Critical K of Instability
(2) Intersection between root locus and -axis
Critical K of Oscillation

13. § 8.3 Evan’s Root-locus Method (1)
1. Parametrilization of closed-loop system with K varied as:
The highest power in both the numerator and denominator of G(s)H(s) are
normalized to unity.
2. Obtain characteristic equation of the closed-loop poles: 1+KG(s)H(s)=0
3. From magnitude-phase representation to obtain criteria for poles and zeros:
4. Sketch shape with scale and parameter K:
The shape of root locus is determined entirely by the phase criterion.
The magnitude criterion is used only to assign scale and parameter K of the
locus.

14. § 8.3 Evan’s Root-locus Method (2)
Terminologies and Symbols About Root Locus

15. § 8.3 Evan’s Root-locus Method (3)
Five Rules for Sketching the Root Locus
1. Number of branches: The number of branches of the locus is equal to
the order of the characteristic polynomial.
2. Symmetry: The locus is symmetrical about the real axis.
3. Real-axis segments: For K>0, the locus on the real axis exists to the
left of an odd number of real-axis open-loop poles
plus zeros.
4. Starting and ending points: The locus begins at the poles of G(s)H(s)
with K=0 and terminate with , either at
the zeros of G(s)H(s) or at infinity.
5. Behavior at infinity: The locus approaches straight line asymptotes as the
locus approaches infinity.

16. § 8.3 Evan’s Root-locus Method (4)
Refining the Sketch
Root locus calibration:
Breakaway (Break-in) points:
K attains local maximum (minimum) on the real axis.
Points of imaginary-axis crossings:
Found by using Routh-Hurwitz criterion or substituting into the equation of root locus and solving the equations.
Angles of departure (arrival):
Obtain by choosing an arbitrary point infinitesimally close to the pole (zero)
and applying the angle criterion.

17. § 8.3 Evan’s Root-locus Method (5)
Ex: Find root locus for a DC motor position servo as
Sol: C-L poles:
O-L poles: s=0, s=-1, # p=2
zeros: none, # z=0
Real-axis segment: lies between the poles of s=0 and s=-1.
Asymptotes:
Breakaway point:

18. § 8.3 Evan’s Root-locus Method (6)
The closed-loop position servo is stable for any positive proportional gain.

19. § 8.3 Evan’s Root-locus Method (7)
Ex: Find stability conditions for various K in the characteristic equation:
Sol: Standard form
C-L poles:
O-L poles: s=0, s=-1, s=-2, # p=3
zeros: none, # z=0
Real-axis segment: lies between the poles of s=0 and s=-1.
lies to the left of pole s=-2.
Asymptotes:
Imaginary axis crossing:

20. § 8.3 Evan’s Root-locus Method (8)
Breakaway point:
Root locus and stability conditions
Stability conditions
(1)
three negative real roots
stable
(2) <K<6
two complex roots, one negative real root
(3) K>6
two complex roots in RHP, one negative
real root
unstable

21. § 8.3 Evan’s Root-locus Method (9)
Ex: Consider a system includes lightly damped flexible modes near
imaginary axis, find the root for
Sol: (1) (2)
Stability insensitive
to varied K
Stability sensitive
to varied K

22. Cascade and Feedback Compensation
§ Root-locus Method in Design (1)
Design Problems
Adjust parameter and / or introduce the pole(s) and zero(s) of the dynamic
compensator to alter the root locus so that the performance specifications
can be satisfied.
Performance specs: Stability Margin, Transient Response, Steady State Error.
Configurations of Compensation
Cascade Compensation
Feedback Compensation
Cascade and Feedback Compensation
O-L Transfer Function:
Static Compensator:

23. § 8.4 Root-locus Method in Design (2)
Dynamic Compensators
Passive compensator --- Mainly RC type network
Active compensator --- Mainly OP and RC circuit
Require external power
Basic Gain and Phase Compensation
Passive Compensator
Active Compensator
PD controller
PI controller
Phase lead
Phase lag
PID controller or lag-lead compensator can be used to improve transient
response, s.s. error and trade off stability margin.

24. § 8.4 Root-locus Method in Design (3)
Design Constraints
Feasible region of 1st / 2nd-order dominant poles
Optimal region of 2nd-order dominant poles

25. § 8.4 Root-locus Method in Design (4)
Addition of Poles to G(s)H(s)
The effect of adding a pole to G(s)H(s) is to push the root loci toward the
RHP.

26. § 8.4 Root-locus Method in Design (5)
Addition of Zeros to G(s)H(s)
The effect of adding a left-half plane zero to G(s)H(s) is to move and bend
the root loci toward the LHP.

27. § 8.4 Root-locus Method in Design (6)
Practical Compensator Realized by OP:

28. § 8.4 Root-locus Method in Design (7)
Root Sensitivity Robust K of characteristic roots
Breakaway (breakin) point:
Avoid selecting the value of K to operate at the breakaway (breakin) points.
For characteristic eq. 1+KG(s)H(s)=0,
Root locus is less sensitive to changes in gain at the lower value of K.
Root sensitivity provides information of changes in both magnitude and direction of specific root for designer.

29. § 8.4 Root-locus Method in Design (8)
Ex: For a DC-servo with position and tacho feedbacks, find K1 and K2 to satisfy the
control specs: (1) settling time sec, (2) dominant poles with ,
(3) ramp-input steady state error
Sol:
Characteristic eq.
From ramp input
From setting time
Root locus for b=K1=20

30. § 8.4 Root-locus Method in Design (9)
Feasible region and root locus