Imaginary
-1 is called the
-1
critical point
Plane of the Open Loop
Transfer Function
B(i)
Stable
Unstable
-B(iw)
B(0)
Real
B(iw)
Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
Loop Transfer Function
Outline of Today’s Lecture
 Review
 Partial Fraction Expansion
 real distinct roots
 repeated roots
 complex conjugate roots
 Open Loop System
 Nyquist Plot
 Simple Nyquist Theorem
 Nyquist Gain Scaling
 Conditional Stability
 Full Nyquist Theorem
Partial Fraction Expansion
 When using Partial Fraction Expansion, our objective is to
turn the Transfer Function
K  i 1 ( s  zi ) i 1 ( s  2 iwni s  wni 2 )
m
G( s) 
k
s r  i 1 ( s  pi ) i 1 ( s  2 iwni s  wni 2 )
n
q
into a sum of fractions where the denominators are the
factors of the denominator of the Transfer Function:
Bq ( s)
K A1 ( s) A2 ( s)
An ( s)
B1 ( s)
G( s )  r 

 ... 

 ... 
s s  p1 s  p2
s  pn s  2 1wn1s  wn12
s  2 qwnq s  wnq 2
Then we use the linear property of Laplace Transforms and
the relatively easy form to make the Inverse Transform.
Case 1: Real and Distinct Roots
G( s) 
...
s i 1 ( s  pi )
n
Put the transfer function in the form of
a0
a1
a2
an
G( s)  

 ... 
s s  p1 s  p2
s  pn
where the ai are called the residue at the pole pi
and determined by
a0  sG  s  s 0
a3  ( s  p3 )G ( s ) s  p
3
a1  ( s  p1 )G  s  s  p
...
1
a2   s  p2  G  s  s  p
2
an   s  pn  G  s  s  p
n
Case 1: Real and Distinct Roots
Example
G( s) 
 s  2  s  4 
s  s  1 s  5
a0
a
a
 1  2
s s 1 s  5
 s  2  s  4   a0  s  1 s  5  a1s  s  5  a2 s  s  1
G( s) 
s 2  6s  8  a0  s 2  6s  5  a1  s 2  5s   a2  s 2  s 
 a0  a1  a2  1
 a1  a2  0.6
a1  0.75

6
a

5
a

a

6


 0


1
2
5
a

a


3.6
 a2  0.15
5a  8  a  1.6  1 2
0
 0
1.6 0.75 0.15


s s 1 s  5
g (t )  1.6  0.75e  t  0.15e 5t
G( s) 
Case 2: Complex Conjugate Roots
G( s) 
...
... i 1 ( s 2  2 iwi s  wi 2 )
q
We can either solve this using the method of matching coefficients
which is usually more difficult or by a method similar to that
previously used as follows:


s  2 1w1s  w12  s   1w1  w1  12  1 s   1w1  w1  12  1
then the term
A( s )
a1
a2


2
s  2 iwi s  wi
s   1w1  w1  12  1 s   1w1  w1  12  1
2
proceeding as before

 s   w  w

 1 G  s 
a1  s   1w1  w1  12  1 G  s  s  w w
a2
1 1

1
 12
1 1
1
12 1
s 1w1 w1  12 1
Case 3: Repeated Roots
G( s) 
...
...( s  pi ) n ...
Form the equation with the repeated terms expanded as
an
an 1
a1
G ( s )  ... 


...

...
n
n 1
( s  pi )
( s  pi )
s  pi
an  ( s  pi ) n G ( s ) s  p
i
d
( s  pi ) n G  s 
s  p
ds
d2
an 2  2 ( s  pi ) n G  s 
s  p
ds
d3
an 3  3 ( s  pi ) n G  s 
s  p
ds
...
an 1 
d n 1
a1  n 1 ( s  pi ) n G  s 
s  p
ds
Heaviside Expansion


n


 A  bi   bi t
As
Heaviside Expansion Formula: L1 
 

e
 B  s   i 1 d


 B  bi  
 ds

where bi are the n distinct roots of B( s)
15( s  2)
s( s 2  2 s  25)
Roots of the denominator are 0,  1  i4.899, and  1  i 4.899
Example:
G( s) 
d
B  s   ( s 2  2 s  25)  2 s 2  2 s  3s 2  4 s  25
ds
3
 15( s  2) 
si t
1
L G  s      2
 e
i 1  3s  4 s  25  s  si
g (t ) 
30 0t
15. 73.485i  1i 4.899 t
15. 73.485i  1i 4.899 t
e 
e

e
25
48.00  9.798i
48.00  9.798i
g (t )  1.2   0.6  1.408i  e
1i 4.899 t
  0.6  1.408i  e
1i 4.899 t
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 )
Closed Loop System
Input
r(s)
++
Error
signal
E(s)
Controller
C(s)
Open Loop
Signal
B(s)
Plant
P(s)
Output
y(s)
-1
The closed loop transfer function is
nc  s  n p  s 
dc  s  d p  s 
nc  s  n p  s 
C s P s
y( s)
G yr ( s ) 



n  s  n p  s  d c  s  d p  s   nc  s  n p  s 
r( s) 1  C  s  P  s 
1 c
dc  s  d p  s 
The characteristic polynomial is
 ( s )  1  C  s  P  s   d c  s  d p  s   nc  s  n p  s 
For stability, the roots of  ( s ) must have negative real parts
While we can check for stability, it does not give us design guidance
Note: Your book uses L(s) rather than B(s)
To avoid confusion with the Laplace transform, I will use B(s)
Open Loop System
Input
r(s)
++
Error
signal
E(s)
Controller
C(s)
Open Loop
Signal
B(s)
Sensor
-1
Plant
P(s)
Output
y(s)
nc  s  n p  s 
b( s)
The open loop transfer function is B( s) 
 C s P s 
r( s)
dc  s  d p  s 
If in the closed loop, the input r(s) were sinusoidal and if the signal were
to continue in the same form and magnitude after the signal were disconnected,
it would be necessary for
B(iw0 ) 
nc  s  n p  s 
 1
dc  s  d p  s 
Open Loop System
Nyquist Plot
Error
signal
E(s)
Input
r(s)
++
Controller
C(s)
Open Loop
Signal
B(s)
nc  s  n p  s 
B(iw0 ) 
 1
dc  s  d p  s 
Imaginary
-1
B(i)
B(-iw)
Sensor
-1
Plane of the Open Loop
Transfer Function
B(0)
B(iw)
-1 is called the
critical point
Plant
P(s)
Real
Output
y(s)
Simple Nyquist Theorem
Error
signal
E(s)
Input
r(s)
++
Controller
C(s)
Open Loop
Signal
B(s)
Imaginary
-1 is called the
-1
critical point
Plant
P(s)
Sensor
-1
-B(iw)
Plane of the Open Loop
Transfer Function
B(i)
Stable
B(0)
Real
B(iw)
Unstable
Simple Nyquist Theorem:
For the loop transfer function, B(iw), if B(iw) has no poles in the right
hand side, expect for simple poles on the imaginary axis, then the
system is stable if there are no encirclements of the critical point -1.
Output
y(s)
Example
 Plot the Nyquist plot for B( s) 
B(iw ) 

1
iw  iw   2wi  2
2
B(0)  i
1
s s 2  2s  2



-1
Re
B(1i )  0.4  0.2i
B(1i )  0.4  0.2i
B(2i )  0.1  0.05i
B(2i )  0.1  0.05i
Im
Stable
Example
 Plot the Nyquist plot for B( s) 
10
s  s  2s  2 
Im
20   20  10w  i
B (iw ) 

2
w4  4
iw  iw   2wi  2

10

B (0)  i
B (1i )  4  2i
B ( 1i )  4  2i
-1
B (2i )  1  0.5i
B ( 2i )  1  0.5i
B (4i )  0.077  0.135i
B ( 4i )  0.077  135i
Unstable
Re
Nyquist Gain Scaling
 The form of the Nyquist plot is scaled by the system gain
B( s ) 
K
s  s  2s  2 
 Show with Sisotool
Conditional Stabilty
 While most system increase stability by decreasing gain,
some can be stabilized by increasing gain
 Show with Sisotool
K (0.25s 2  0.12 s  1)
B( s ) 
s 1.69 s 2  1.09 s  1
Full Nyquist Theorem
 Assume that the transfer function B(iw) with P poles has
been plotted as a Nyquist plot. Let N be the number of
clockwise encirclements of -1 by B(iw) minus the
counterclockwise encirclements of -1 by B(iw)Then the
closed loop system has Z=N+P poles in the right half plane.
 Show with Sisotool
B( s ) 
K ( s  5  2i )( s  5  2i )
s  s  .5  2i  s  .5  2i  s  2  6i  s  2  6i 
Summary
Im
 Open Loop System
 Nyquist Plot
 Simple Nyquist Theorem
 Nyquist Gain Scaling
-1
 Conditional Stability
 Full Nyquist Theorem
Unstable
Next Class: Stability Margins
Re