Negative Feedback
R(s)
+-
E(s)
H(s)
Y(s)
B(s)
G(s)
Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
Block Diagrams
Outline of Today’s Lecture
 Review
 A new way of representing systems
 Coordinate transformation effects
 hint: there are none!
 Development of the Transfer Function from an ODE
 Gain, Poles and Zeros






The Block Diagram
Components
Block Algebra
Loop Analysis
Block Reductions
Caveats
Alternative Method of Analysis
 Up to this point in the course, we have been concerned about the




structure of the system and discribed that structure with a state
space formulation
Now we are going to analyze the system by an alternative method
that focuses on the inputs, the outputs and the linkages between
system components.
The starting point are the system differential equations or
difference equations.
However this method will characterize the process of a system
block by its gain, G(s), and the ratio of the block output to its
input.
Formally, the transfer function is defined as the ratio of the
Laplace transforms of the Input to the Output:
TF ( S ) 
Output ( s )
for s a complex number and the Laplace Variable
Input ( s )
Coordination Transformations
x1
d
d
 x  Ax  Bu  z  Az  Bu
  dt
 dt

 y  Cz  Du
 y  Cx  Du

z2
x2
for A  TAT 1, B  TB and C  CT 1
Since the ouput y is unchanged by the transformation
z1
y  C ( sI  A) 1 B  D
and
y  C ( sI  A) 1 B  D
y  G ( s )u(t )  G ( s )u(t )  G ( s )  G  s 
 Thus the Transfer function is invariant under coordinate
transformation
Linear System Transfer Functions
General form of linear time invariant (LTI) system is expressed:
(n)
 n1
 n 2 
 m
 m1
 m2 
y  a1 y  a2 y  ...  an2 y  an1 y  an y  b0 u  b1 u  b2 u  ...  bn2u  bn1u  bnu
For an input of u(t)=est such that the output is y(t)=y(0)est
s
n
 a1s n 1  a2 s n 2  ...  an 2 s 2  an 1 2  an  y0e st   b0 s m  b1s m 1  b2 s m 2  ...  bn 2 s 2  bn 1s  bn  e st
b s bs


s  a s
m
y (t )  y (0)e
st
0
m 1
1
n 1
n
1
 b2 s m 2  ...  bn 2 s 2  bn 1s  bn 
 a2 s n 2  ...  an 2 s 2  an 1 2  an 
e st
The transfer function form is then
b0 s m  b1s m1  b2 s m2  ...  bn 2 s 2  bn 1s  bn b( s )
y ( t )  G ( s )u ( t )  G ( s )  n

s  a1s n 1  a2 s n 2  ...  an 2 s 2  an 1 2  an
a(s)
Note that the transfer function for a simple ODE can be written as the ratio
of the coefficients between the left and right sides multiplied by powers of s
The order of the system is the highest exponent of s in the denominator.
Simple Transfer Functions
Differential
Equation
Transfer
Function
Name
yu
s
Differentiator
yu
1
s
1
s2
1
sa
Integrator
yu
y  ay  u
y  2n y  n 2 y  u
y  k pu  kd u  ki  udt
2nd order Integrator
1st order system
1
s  2n s  n
2
k p  kd s 
ki
s
2
Damped Oscillator
PID Controller
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

K
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 value of G(s) is the zero frequency or steady state gain of the
system
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-y
+-
y
-
y
x
x-y
+-
x
G
+
z
x
-
z-x+y
x
z-x
-+
+
z
y
x-y
+
+
xGH
H
x
y
GH
z-x+y
++
x
y
z
xG
z-x+y
x
H
xGH
xH
G
xGH
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
Block Algebra
x
x
Gx
G
Gx
G
x
1
G
x
y
x-y
x
x-y
+-
x
y
y
+-
x-y
+-
y
x
x-y
+
G
H
x
1
H
y
+-
H
G
Closed Loop Systems
r
y
++
H
A positive feedback system
r
y
+-
H
A negative feedback system
Loop Analysis
(Very important slide!)
Negative Feedback
R(s)
E(s)
+-
H(s)
Y(s)
B(s)
G(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 ) R ( s )  H ( s )G ( s )Y ( s )
Y  s   H ( s )G ( s )Y ( s )  H ( s ) R ( 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 Analysis
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 )
Block Reduction
Example
x
y
++
D
C
B
A
+-
+-
+-
E
F
First, uncross signals where possible
G
E
x
+-
+-
A
+-
G
F
B
C
D
++
y
Block Reduction
Example
x
D
C
B
A
+-
+-
+-
E
++-
y
G
F
E
BC
Next: Reduce Feed Forward Loops where possible
x
+-
+-
A
+-
G
F
B
C
D
++-
y
Block Reduction
Example
x
+-
+-
A
+-
E
BC
B
C
D
++-
y
G
F
Next: Reduce Feedback Loops starting with the inner most
x
+-
A
1  AG
+F
B
C
BCD  E
BC
y
Block Reduction
Example
x
A
1  AG
+-
+-
B
C
C
BCD  E
BC
BCD  E
BC
y
F
AB
AB
1  AG 
FAB
1  AG  ABF
1
1  AG
x
+-
AB
1  AG  ABF
ABC
ABC
1  AG  ABF 
ABC
1  AG  ABF  ABC
1
1  AG  ABF
x
ABC
1  AG  ABF  ABC
BCD  E
BC
y
y
Block Reduction
Example
x
ABC
1  AG  ABF  ABC
x
BCD  E
BC
AB 2C 2 D  ABCE
BC  ABCG  AB 2CF  AB 2C 2
y
y
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.
x
Summary
x
 The Block Diagram
xG(s)
G(s)
 Components
 Block Algebra
x
 Loop Analysis
 Block Reductions
x+y
++
y
 Caveats
x
x
Negative Feedback
R(s)
+-
E(s)
H(s)
x
Y(s)
B(s)
G(s)
Next: Bode Plots