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1. Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
Stability Margins

2. Outline of Today’s Lecture
 Review
 Open Loop System
 Nyquist Plot
 Simple Nyquist Theorem
 Nyquist Gain Scaling
 Conditional Stability
 Full Nyquist Theorem
 Is stability enough?
 Margins from Nyquist Plots
 Margins from Bode Plot
 Non Minimum Phase Systems

3. Loop Nomenclature
Reference
Input
R(s)
+-
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
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
( ) ( ) ( )
1 ( ) ( ) ( )
F s C s G s
C s G s H s


4. Open Loop System
++
Output
y(s)
Error
signal
E(s)
Open Loop
Signal
B(s)
Plant
P(s)
Controller
C(s)
Input
r(s)
   
 
 
 
 
( )
The open loop transfer function is ( )
( )
p
c
c p
n s
n s
b s
B s C s P s
r s d s d s
  
Note: Your book uses L(s) rather than B(s)
To avoid confusion with the Laplace transform, I will use B(s)
Sensor
-1
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
 
 
 
 
0
( ) 1
p
c
c p
n s
n s
B i
d s d s
   

5. Simple Nyquist TheoremError
signal
E(s)
++
Output
y(s)
Open Loop
Signal
B(s)
Plant
P(s)
Controller
C(s)
Input
r(s)
Sensor
-1
Simple Nyquist Theorem:
For the loop transfer function, B(i), if B(i) 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.
-1
Real
Imaginary
Plane of the Open Loop
Transfer Function
B(0)
B(i)
( )
B i
-1 is called the
critical point
Stable
Unstable
-B(i)

6. Nyquist Gain Scaling
 The form of the Nyquist plot is scaled by the
system gain

7. Conditional Stabilty
 Whlie most system increase stability by
decreasing gain, some can be stabilized by
increasing gain
 Show with Sisotool
 
2
2
(0.25 0.12 1)
( )
1.69 1.09 1
K s s
B s
s s s
 

 

8. Definition of Stable
 A system described the solution (the response) is
stable if that system’s response stay arbitrarily
near some value, a, for all of time greater than
some value, tf.
( ; ) ( ; ) for all 0
b a x t b x t a t
 
     

9. Full Nyquist Theorem
 Assume that the transfer function B(i) with P
poles has been plotted as a Nyquist plot. Let N be
the number of clockwise encirclements of -1 by
B(i) minus the counterclockwise encirclements
of -1 by B(i)Then the closed loop system has
Z=N+P poles in the right half plane.

10. Determination of Stability
from Eigenvalues
Continuous Time Discrete Time
Unstable
Stable
Asymptotic
Stability
x Ax
 ( 1) ( )
x k Ax k
 
The eigenvectors of A are i i
  
 
If 0 for any simple root
Or 0 for any repeated root
i
i




If 0 for any simple root
Or 0 for any repeated root
i
i




If 0 for any simple root
Or 0 for any repeated root
i
i




0 for all roots
i
 
If 0 for any simple root
Or 0 for any repeated root
i
i




0 for all roots
i
 

11. Is Stability Enough?
 If not Why Not?

12. Margins
 Margins are the range from the current system design
to the edge of instability. We will determine
 Gain Margin
 How much can gain be increased?
 Formally: the smallest multiple amount the gain can be
increased before the closed loop response is unstable.
 Phase Margin
 How much further can the phase be shifted?
 Formally: the smallest amount the phase can be increased
before the closed loop response is unstable.
 Stability Margin
 How far is the the system from the critical point?
0.05
If the gain margin is expressed in dB, then the multiple gain is 10 m
g
G 

13. Gain and Phase Margin Definition
Nyquist Plot
-1
1
m
g

m


14. Example
2
0.05
0.05*10.2
45( 600)
( )
( 6)( 60 900)
From the plot, the gain margin is
10.2 dB
The gain multiple is 10
10 3.2359
m
g
s
G s
s s s
G
G


  

 
Using Matlab command
nyquist(gs)

15. Example
Here the gain from the
previous plot has been
multiplied by 3.2359
The result is that
stability is about to be
lost

16. Example
2
Unstable Example:
270( 600)
( )
( 6)( 60 900)
s
G s
s s s


  
Using Matlab command
nyquist(gs)

17. Gain and Phase Margin Definition
Bode Plots
Positive Gain Margin
Phase Margin
-180
0
Phase,
deg
Magnitude,
dB


Phase Crossover Frequency

18. Example
2
45( 600)
( )
( 6)( 60 900)
s
G s
s s s


  
Using Matlab command
bode(gs)

19. Example
2
3.2359*45( 600)
( )
( 6)( 60 900)
s
G s
s s s


  
Again, stability is about to
be lost.

20. Example
2
270( 600)
( )
( 6)( 60 900)
s
G s
s s s


  
Using Matlab command
bode(gs)

21. Note
 The book does not plot the Magnitude of the
Bode Plot in decibels.
 Therefore, you will get different results than the
book where decibels are required.
 Matlab uses decibels where needed.
20
10 where is measured in dB
m
g
m
Gain g


22. Stability Margin
 It is possible for a system to have relatively large
gain and phase margins, yet be relatively
unstable.
Stability
margin, sm

23. Non-Minimum Phase Systems
 Non minimum phase systems are those systems
which have poles on the right hand side of the plane:
they have positive real parts.
 This terminology comes from a phase shift with sinusoidal
inputs
 Consider the transfer functions
 The magnitude plots of a Bode diagram are exactly the
same but the phase has a major difference:
 
1 2
1 1
( ) and
2 ( 2)
s s
G s G
s s s s
 
 
 

24. Another Non Minimum Phase
System
A Delay
 Delays are modeled by the function which
multiplies the T.F.
( ) ( )
( ) Ts
y t x t T
G s e
 


25. Summary
 Is stability enough?
 Margins from Nyquist Plots
 Margins from Bode Plot
 Non Minimum Phase Systems
Next Class: PID Controls