1. Root Locus Diagrams Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
Root Locus Diagrams

2. Outline of Today’s Lecture
Review
The Block Diagram
Components
Block Algebra
Loop Analysis
Block Reductions
Caveats
Poles and Zeros
Plotting Functions with Complex Numbers
Root Locus
Plotting the Transfer Function
Effects of Pole Placement
Root Locus Factor Responses

3. Sense
Compute
Actuate
Block Diagrams
Throughout this course, we have used block diagrams to show different properties
Here, we will formalize the meaning of block diagrams
Controller
Plant
Sensor S
Disturbance
Controller
Plant/Process
Input r
Output y x -K kr
State Feedback
Prefilter
State Controller u D c1 c2
cn-1 cn -1 a1 a2
an-1 an S … u y z1 z2
zn-1 zn

4. Components The paths represent variable values which
G(s) x
xG(s) + y
x+y
The paths represent variable values which
are passed within the system
Blocks represent System components which
are represented by transfer functions and multiply
their input signal to produce an output
Addition and subtraction of signals are represented
by a summer block with the operation indicated
on the arrow
Branch points occur when a value is placed on two
lines: no modification is made to the signal

5. Block Algebra G x Gx G x Gx G x H Hx + - Gx (G-H)x G-H (G-H)x x G Gx-zGz
G(x-z) G
G(x-z) + - x z

6. Loop Analysis (Very important slide!)
H(s) + -
R(s)
Y(s)
E(s)
B(s)
Negative Feedback
G(s)
Positive Feedback
H(s) +
R(s)
Y(s)
E(s)
B(s)

7. Loop Nomenclature Disturbance/Noise Reference Error Input signal+ -
Output
y(s)
Error
signal
E(s)
Open Loop
Signal
B(s)
Plant
G(s)
Sensor
H(s)
Prefilter
F(s)
Controller
C(s)
Disturbance/Noise
Reference
Input
R(s)
The plant is that which is to be controlled with transfer function G(s)
The prefilter and the controller define the control laws of the system.
The open loop signal is the signal that results from the actions of the
prefilter, the controller, the plant and the sensor and has the transfer function
F(s)C(s)G(s)H(s)
The closed loop signal is the output of the system and has the transfer function

8. Caveats: Pole Zero Cancellations
Assume there were two systems that were connected as such
An astute student might note that and then want to cancel the (s+1) term
This would be problematic: if the (s+1) represents a true system dynamic, the dynamic would be lost as a result of the cancellation. It would also cause problems for controllability and observability. In actual practice, cancelling a pole with a zero usually leads to problems as small deviations in pole or zero location lead to unpredictable dynamics under the cancellation.
R(s)
Y(s)

9. Caveats: Algebraic Loops
The system of block diagrams is based on the presence of differential equation and difference equation
A system built such the output is directly connected to the input of a loop without intervening differential or time difference terms leads to improper block interpretations and an inability to simulate the model.
When this occurs, it is called an Algebraic Loop. Such loops are often meaningless and errors in logic. + - 2

10. Gain, Poles and Zeros
The roots of the polynomial in the denominator, a(s), are called the “poles” of the system
The poles are associated with the modes of the system and these are the eigenvalues of the dynamics matrix in a state space representation
The roots of the polynomial in the numerator, b(s) are called the “zeros” of the system
The zeros counteract the effect of a pole at a location
The variable s is a complex number:
The value of G(0) is the zero frequency or steady state gain of the system

11. Plotting functions on the Complex Plane
Plotting functions on the complex plane is more complicated than the real plane because of unexpected forms that occur
Consider an equation such as
If z is limited to real numbers, z must be 1 for any n
BUT, this is not the case if z is allowed to be a complex number
if n = 3, then
If n = 4, then
Consider a function such as
If z were real, a hyperbola results
BUT, if z is a complex number, a totally different result occurs
Both a and b vary with results in surface rather than a curve
The result of the function could be either real or complex
Therefore, visualization is difficult

12. Root Locus
The root locus plot for a system is based on solving the system characteristic equation
The transfer function of a linear, time invariant, system can be factored as a fraction of two polynomials
When the system is placed in a negative feedback loop the transfer function of the closed loop system is of the form
The characteristic equation is
The root locus is a plot of this solution for positive real values of K
Because the solutions are the system modes, this is a powerful design tool
While we focus here on the gain, K, we can plot any parameter this way

13. Plotting a Transfer Function Root Locus
The path is determined from the open loop transfer function by varying the gain
‘s’ as used in a transfer function is a complex number
Poles will be marked with X
Zeros with be marked with an O
Each path represents a branch of the transfer function in the complex plane
All paths
start at poles and
end at zeros
mirror across the real axis
There must be a zero for each pole
Those that are not shown on the plot are at infinity
Matlab command rlocus(sys)

14. Paths of the Transfer Function
K=1
K=3
K=0.1
K=10

15. Paths of the Transfer Function
The real values of the gain move the poles along the root loci
Notice that the placement of the gain moved poles dictates the output response of the system
Poles in the right half plane are unstable responses
K=1
K=3
K=0.1
K=10

16. The effect of placement on the root locus
Imaginary axis
Real Axis jw s
jwd wn
s = -zwn
sin-1(z)
The magnitude of the vector to
pole location is the natural frequency
of the response, wn
The vertical component (the imaginary
part) is the damped frequency, wd
The angle away from the vertical is the
inverse sine of the damping ratio, z

17. Root Locus Factor Responsesjw s
A complete system will sum all
of these effects that are present in
the system’s response
The dominating effects will be from the poles closest to the origin
Real Axis

18. Example
A radar tracking antenna (Nise, 1995) has the position control transfer function of
The antenna must have a 5% settling time of less than 2 seconds with an over damped response.

19. Example

20. Example
Current system can not meet either requirement with gain alone:
By adding a zero at -1.34, a pole at -11 and a gain of 271, we get
Is this the best controller?

21. Summary Poles and Zeros Plotting Functions with Complex Numbers
Root Locus
Plotting the Transfer Function
Effects of Pole Placement
Root Locus Factor Responses
Imaginary axis
Real Axis jw s
jwd wn
s = -zwn
sin-1(z)
Next: Bode Plots