### Lecture 24: Stability Margins

Professor Walter W. Olson
Department of Mechanical, Industrial and Manufacturing Engineering
University of Toledo
Stability Margins
Outline of Today’s Lecture
 Review
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Open Loop System
Nyquist Plot
Simple Nyquist Theorem
Nyquist Gain Scaling
Conditional Stability
Full NyquistTheorem
 Is stability enough?
 Margins from Nyquist Plots
 Margins from Bode Plot
 Non Minimum Phase Systems
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 )
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(i0 ) 
nc  s  n p  s 
 1
dc  s  d p  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(i)
Plane of the Open Loop
Transfer Function
B(i)
Stable
B(0)
Real
B(i)
Unstable
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.
Output
y(s)
Nyquist Gain Scaling
 The form of the Nyquist plot is scaled by the system gain
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
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.
b  a    x(t; b)  x(t; a)   for all t  0
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.
Determination of Stability
from Eigenvalues
The eigenvectors of A are i    i
Continuous Time
x  Ax
Unstable
Stable
Asymptotic
Stability
Discrete Time
x(k  1)  Ax(k )
If i  0 for any simple root
If i  0 for any simple root
Or i  0 for any repeated root
Or i  0 for any repeated root
If i  0 for any simple root
If i  0 for any simple root
Or i  0 for any repeated root
Or i  0 for any repeated root
i  0 for all roots
i  0 for all roots
Is Stability Enough?
 If not, why not?
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.
If the gain margin is expressed in dB, then the multiple gain is G  100.05gm
 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 system from the critical point?
Gain and Phase Margin Definition
Nyquist Plot
1

gm
-1
m
Example
45( s  600)
( s  6)( s 2  60 s  900)
From the plot, the gain margin is
10.2 dB
G( s) 
The gain multiple is G  100.05 gm
G  100.05*10.2  3.2359
Using Matlab command
nyquist(gs)
Example
Here the gain from the
previous plot has been
multiplied by 3.2359
The result is that
stability is about to be
lost
Example
Unstable Example:
270( s  600)
G( s) 
( s  6)( s 2  60s  900)
Using Matlab command
nyquist(gs)
Magnitude, dB
Gain and Phase Margin Definition
Bode Plots
0
Positive Gain Margin
Phase, deg

-180
Phase Margin

Phase Crossover Frequency
Example
G( s) 
45( s  600)
( s  6)( s 2  60s  900)
Using Matlab command
bode(gs)
Example
G( s) 
3.2359 * 45( s  600)
( s  6)( s 2  60s  900)
Again, stability is about to
be lost.
Example
G( s) 
270( s  600)
( s  6)( s 2  60s  900)
Using Matlab command
bode(gs)
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.
Gain  10
gm
20
where gm is measured in dB
Stability Margin
 It is possible for a system to have relatively large gain and
phase margins, yet be relatively unstable.
Stability
margin, sm
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
s 1
s 1
G
(
s
)

and
G

2
 Consider the transfer functions 1
s  s  2
s( s  2)
 The magnitude plots of a Bode diagram are exactly the same but the phase
has a major difference:
Another Non Minimum Phase System
A Delay
 Delays are modeled by the function which multiplies the T.F.
y (t )  x (t  T )
G( s)  eTs
Summary
 Is stability enough?
 Margins from Nyquist Plots
 Margins from Bode Plot
 Non Minimum Phase Systems
Next Class: PID Controls