Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
Root Locus Diagrams
Outline of Today’s Lecture
 Review
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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
Actuate
Sense
Block Diagrams
Compute
 Throughout this course, we have used block diagrams to show
different properties
 Here, we will formalize the meaning of block diagrams
Controlle
r
Disturbance
Controller
Plant/Process
Input
r
S
kr
S
u
Output
y
d
x  Ax  Bu
dt
y  Cx  Du
Prefilter
Sensor
x
-K
Plant
State Feedback
State Controller
D
u
S
-1

S
S
c1
c2
z1

…
…
z2
a1
a2
S
S

S
S
cn-1
cn
zn-1

an-1
…
S
zn
an
y
Components
x
x
The paths represent variable values which
are passed within the system
xG(s)
G(s)
x
x+y
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
++
y
x
x
x
Branch points occur when a value is placed on two
lines: no modification is made to the signal
Block Algebra
x
x
Gx
G
Gx
x
Gx
G
H
+-
(G-H)x
G
+-
z
z
x
+-
G
G(x-z)
x
x
Gx-z
1
G
z
G
+-
z
G
Gx
x
G
z
z
(G-H)x
G-H
Hx
Gx
Gx
G
x
x
Gx
G
Gz
G
G
G(x-z)
+-
Gx-z
Loop Analysis
(Very important slide!)
Negative Feedback
Positive Feedback
R(s)
++
E(s)
H(s)
R(s)
Y(s)
+-
E(s)
H(s)
Y(s)
B(s)
B(s)
G(s)
E ( s)  R( s)  B( s )
B( s)  Y ( s)
E ( s)  R( s)  B( s )
B ( s )  G ( s )Y ( s )
Y ( s)  H ( s) E ( s)  H ( s) R( s)  H ( s) B( s)
Y ( s)  H ( s) E ( s)  H ( s) R( s)  H ( s) B( s)
Y  H ( s ) R ( s )  H ( s )Y ( s )
H ( s) R( s)
Y ( s) 
1  H ( s)
Y ( s)
H ( s)
T .F . 

R( s) 1  H ( s)
Y  H ( s ) R ( s )  H ( s )G ( s )Y ( s )
H ( s) R( s)
Y ( s) 
1  H ( s )(G ( s )
Y ( s)
H ( s)
T .F . 

R ( s ) 1  H ( s )G ( s )
Loop Nomenclature
Reference
Input
R(s)
Prefilter
F(s)
+-
Error
signal
E(s)
Disturbance/Noise
Controller
C(s)
Open Loop
Signal
B(s)
+-
Plant
G(s)
Output
y(s)
Sensor
H(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
F ( s )C ( s )G ( s )
1  C ( s )G ( s ) H ( s )
Caveats: Pole Zero Cancellations
 Assume there were two systems that were connected as such
R(s)
1
C ( s)  3
s  8s 2  17 s  10
s 1
G( s)  3
s  12 s 2  47 s  60
 An astute student might note that C( s) 
Y(s)
1
1

s  8s  17s  10  s  1 s  2  s  5
3
2
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.
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.
+
-
2
 When this occurs, it is called an Algebraic Loop. Such loops
are often meaningless and errors in logic.
Gain, Poles and Zeros
G ( s )  C ( sI  A) 1 B  D 
G (0)  D  CA1B 
b( s )
b( s )
K
a( s)
a( s )
yo bm
 s  Ki
uo am
 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
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
3
1
3 
 1
z  1,  
i,  

2 2
2 2 

3
 If n = 4, then
z 1, i, 1, i
f ( z) 
1
z2
where z  a  bi
 If z were real, a hyperbola results
 BUT, if z is a complex number, a
zn  1
 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
 1
3 


i   1 as is

2
2


 Consider a function such as
3
 1
3 


i 1

2
2


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
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
NG ( s)
Output ( s )
KG ( s )
KN G ( s )
DG ( s )



Input ( s ) 1  KG ( s ) 1  K N G ( s ) DG ( s )  KN G ( s )
DG ( s )
K
 The characteristic equation is
1  KG( s)  0  DG ( s)  KNG ( s)
 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
Plotting a Transfer Function Root Locus
 The path is determined from the

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


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)
s3
G( s ) 
s  s  2   s 2  2s  2 
Paths of the Transfer Function
1
s  s  2s  2
K
Closed Loop G ( s ) 
 3
1
s  2s3  2s  K
1 K
s  s 2  2s  2
K
Open Loop TF ( s )  K
2
1
s  s 2  2s  2
The root locus traces the points at which K is a real number
Characteristic Equation is 1 
K
0
s  s 2  2s  2 
K    s3  2s 2  2s 
at the poles, K  0
For s  0.5  0.866i, K  1 which is on the root locus
For s  0.6  0.866i, K  0.8460 - 0.0606i (not on the loci)
K=1
K=3
K=0.1
K=10
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
The effect of placement
on the root locus j
Imaginary axis
jd
Real Axis
  zn

• The magnitude of the vector to
pole location is the natural frequency
of the response, n
The vertical component (the imaginary
part) is the damped frequency, d
The angle away from the vertical is the
inverse sine of the damping ratio, z
Root Locus Factor Responses
j

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
Example
 A radar tracking antenna (Nise, 1995) has the position
control transfer function of
G( s)  K
21
s( s  1.71)( s  100)
 The antenna must have a 5% settling time of less than 2
seconds with an over damped response.
Example
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?
Summary
 Poles and Zeros
 Plotting Functions with Complex Numbers
 Root Locus
 Plotting the Transfer Function
 Effects of Pole Placement
 Root Locus Factor Responses
j
Imaginary axis
jd
Real Axis

  zn
Next: Bode Plots