Lectures 5 - Frequency Response
Introduction
How to describe dynamics
K. J. strm
The Giant Table - One way to view dynamics
1. Introduction
The heritage of electrical engineering
2. Frequency response
Fits block diagrams
3. Nyquist curves and stability
Makes it possible to deal with systems having a large
number of states.
4. Bode plots
5. The concept of minimum phase
6. Summary
Theme: The input-output view of dynamical systems. Fouriers
idea: sinusoidal inputs. Graphical representations of frequency response. Nyquist and Bode plots. The concepts of minimum and non-minimum phase.
Bode: Electronic feedback amplifiers are much more
complex than steam engines, systems have orders 50100 rather than 2-4. (A lot of capacitors!)
Synonyms: input-output models, external descriptions,
black boxes.
Experimental determination of dynamics
The Idea of Black Boxes
Input
System
Linear Time Invariant Systems
Linearity: Let (u1 , y1) och (u2 , y2) be input-output pairs.
Then (au1 + bu2, a y1 + a y2) is also an input-output pair,
superposition.
Output
Consider a system as a black box. Forget about the internal
details and focus on the input-output behavior of the system.
Make a Giant Table over all pairs of inputs and outputs
A stroke of luck: A few entries suffice for linear timeinvariant systems
Steps (step response), reaction curve
Impulses (impulse response)
Sinusoids (Fourier, frequency responses)
Time-invariance: Let t be an operator that shifts a signal
t time units forward and let (u, y) be an input-output pair. A
system is time-invariant if ( t u, t y) is also an input-output
pair.
Consequences: The table can be simplified drastically for
linear time-invariant systems. It is enough to give one pair only.
If all initial conditions are zero the output for all inputs is given
by the transfer function of the system.
Z t
y(t) =
g(t s)u(s)ds, Y (s) = G (s) U (s)
0
c K. J. strm August, 2001
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Frequency Response
Interpretation of Frequency Responses
Fouriers idea: An LTI system is completely determined by
its response to sinusoidal signals.
The complex number G (i ) tells how a sinusoid propagates
through the system in steady state. If the input is u(t) = sin t,
then the output is
y(t) = h G (i )h sin t + arg G (i )
Implications for the Table
Transmission of sinusoids given by G (i )
The j -method
The number h G (i )h is called gain ratio or simply gain and the
number arg G (i ) is called phase of the transfer function.
Analytic continuation
The transfer function G (s) uniquely given by its values on
the imaginary axis
Experimental determination of the frequency response
Notice Steady State Responses
Proof
Output y
0.25
0.2
0.15
0.1
0.05
0
0.05
0.1
10
15
Input u
This corresponds to the time function
X
y(t) = G (i 0 ) ei 0 t +
0.5
0.5
Consider a system with transfer function G (s) having distinct
stable poles k . Let the input be u(t) = ei 0 t = cos 0 t +
1
i sin 0 t. The Laplace transform of the input is U (s) =
.
s i 0
The Laplace transform of the output is
X Rk
1
1
1
Y ( s) = G ( s)
= G (i 0 )
+
s i 0
s i 0
s k k i 0
10
15
Rk
e k t
k i 0
Since all k are negative the first terms go to zero and y(t)
G (i 0 ) ei 0 t for large t.
c K. J. strm August, 2001
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Nyquists Stability Theorem
The Nyquist Curve
So far focus on the characteristic equation
Difficult to see how the characteristic equation is influenced by controller.
How to change the controller to make unstable system
stable?
H. Nyquist was born in Sweden. Emigrated to the US and
made his career at Bell Laboratories. The Nyquist curve
represents the transfer function by showing a graph of the
complex number G (i ) as a function of frequency.
Physical interpretation!
Im G(i )
Nyquists results was a major paradigm shift
Investigate propagation of sinusoids around the loop
Ultimate point
Based on transfer functions (Always useful to have different ways to look at a problem!)
Re G(i )
a
Strong practical implications
Possibilities to introduce stability margins.
Nyquists Stability Theorem
Conditions for Oscillations
y
So far focus on the characteristic equation
Difficult to see how the characteristic equation is influenced by controller.
How to change the controller to make unstable system
stable?
L(s)
Nyquists results was a major paradigm shift
Investigate propagation of sinusoids around the loop
Based on transfer functions (Always useful to have different ways to look at a problem!)
Strong practical implications
Possibilities to introduce stability margins.
Cut the loop. Let u be a sinusoid. If y is a sinusoid with the
same amplitude and phase, then the loop can be closed and
the oscillation will be maintained. The condition for this is
L(i ) = 1
where L = PC is the loop transfer function. The condition
implies that the Nyquist curve of L goes through the point 1
(the critical point)!
c K. J. strm August, 2001
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The Complete Nyquist Curve
Nyquists Stability Theorem
The complete Nyquist curve is image of the contour C under
the map L(s)
s-plane
L(s)-plane
In the special case when the loop transfer function does not
have poles in the right half plane the closed loop system is
stable if the complete Nyquist curve does not encircle the
critical point.
There are more general result which also covers the case
where the loop transfer function has poles in the RHP.
Use pencil and string to determine encirclements in tricky
situations.
There is some really beautiful mathematics behind this!
5
5
3(s + 1)2
Conditional Stability L(s) =
s(s + 6)2
Example
5
0.5
400
0.4
300
0.3
L(s) =
1
s(s + 1)2
200
0.2
100
L no poles in RHP.
No encirclements
Stable
0.1
0.1
100
0.2
200
0.3
300
0.4
5
5
400
400
300
200
100
100
200
300
400
0.5
9
Loop transfer function has no zeros in the RHP. No encirclements. Closed loop system stable. Notice counter-intuitive.
c K. J. strm August, 2001
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Stability Margins
Stability Margins
Stability as it has been defined
is black and white. In practice
there is often a need to have
concepts like degrees of stability. Some useful concepts
are
Gain margin gm (2-6)
Stability as it has been defined
is black and white. In practice
there is often a need to have
concepts like degrees of stability. Some useful concepts
are
Gain margin gm (2-6)
Phase margin m (45 -60 )
Phase margin m (45 -60 )
Shortest distance to critical
point d (0.5-0.8)
Notice d is safe but only one of
gm or m is not!
Shortest distance to critical
point d (0.5-0.8)
Notice d is safe but only one of
gm or m is not!
The Bode Plot
gm
pc
m
gc
Gain
10
Bode was a researcher at Bell Laboratories. The complex
function G (s) can also be represented by two graphs, one
for the gain curve, h G (i )h, and one for the phase curve,
arg G (i ). It is tradition to use logarithmic scales for frequency
and gain and linear scales for the phase. A nice consequence
of this is that the curves have asymptotes that are very easy
to obtain! The gain curve is sometimes calibrated in dB (20 dB
equals a factor of 10).
10
10
10
10
2
10
10
10
w
10
10
Phase
50
Making the plot
Interpreting the plot
50
Extends the intuitive argument that small s correspond
to large t to all frequencies. Gives a quick view of the
behavior of the system for all frequencies.
100
150
2
10
c K. J. strm August, 2001
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10
10
w
10
10
Matlab
Sketching Bode Plots
There are excellent tools in Matlab for drawing Nyquist curves
and Bode plots in the Control System Toolbox in Matlab.
NYQUIST Nyquist frequency response of LTI models.
BODE Bode frequency response of LTI models.
Use the help function to figure out how the commands work.
Try a few examples.
Notice that Matlab uses decibel (dB) as a standard unit for
amplitude, where 20 dB corresponds to a factor 10. If you think
dB is an unnecessary complication it is easy to make your own
plots with other units.
It is easy to sketch Bode plots because with the right scales
they have linear asymptotes. This is useful in order to get a
quick estimate of the behavior of a system. It is also a good
way to check numerical calculations.
Consider first a transfer function which is a polynomial G (s) =
B (s)/ A(s). We have
log G (s) = log B (s) log A(s)
Since a polynomial is a product of terms:
s,
s2 + 2 as + a2
s + a,
it suffices to be able to sketch Bode diagrams for these terms.
The Bode plot of a complex system is then obtained by composition.
Integrator
Differentiator
Bode Diagrams
Bode Diagrams
From: U(1)
From: U(1)
20
20
15
15
arg G ( i ) = /2
Phase (deg); Magnitude (dB)
log h G ( i )h = log
5
10
15
20
91
90.5
To: Y(1)
G ( s) = s
We have G (i ) = i
log h G ( i )h = log
90
arg G ( i ) = /2
89.5
89
1
10
10
10
5
0
Phase (deg); Magnitude (dB)
1
s
We have G (i )
i 1
G ( s) =
5
10
15
20
89
89.5
To: Y(1)
10
90
90.5
91
1
10
10
Frequency (rad/sec)
10
10
Frequency (rad/sec)
Matlab:
Matlab:
sys=tf([1 0],1)
bode(sys,{0.1,1})
sys=tf(1,[1 0])
bode(sys,{0.1,1})
c K. J. strm August, 2001
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First Order System G (s) = s + a
We have G (i ) = a + i , hence h G (i )h =
arg G (i ) = arctan /a, hence
log h G (i )h =
1
log ( 2 + a2 ),
2
Gain
10
2 + a2 and
1
10
arg G (i ) = arctan /a
0
log a
log h G ( i )h log a + log 2
log
arg G ( i ) 4 + 2 log a
10
2
10
10
if << a,
if = a,
10
10
80
if >> a
60
if << a,
if a,
/a
Phase
10
40
20
0
2
10
if >> a
Second Order System G (s) = s2 + 2 a + a2
G ( i ) = a2 2 + 2i a
1
log h G ( i )h = log ( 4 + 2a2 2 (2 2 1) + a4 )
2
arg G ( i ) = arctan 2 a /( a2 2 )
10
/a
10
10
10
Gain
10
10
10
10
2 log a
if << a,
if = a, ,
log h G ( i )h 2 log a + log 2
2 log
if >> a
0
if << a,
a
+
if = a, ,
arg G ( i )
2
a
if >> a
10
/a
Phase
10
10
150
100
50
0
1
10
c K. J. strm August, 2001
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/a
10
10
Sketching Bode Plots
Gain
10
System
10
G ( s) =
200(s + 1)
1+s
=
s(s + 10)(s + 200)
10s(1 + 0.1s)(1 + 0.01s)
Determine break points (poles and zeros) sort them in
increasing frequency
1
Start with low frequencies ( G (s)
)
10s
Draw the low frequency asymptote
Go over all break points and note the slope changes
A crude sketch of the phase curve is obtained by using the
relation that, for systems with no RHP poles or zeros, one
unit slope corresponds to a phase of 90
10
10
10
Gain
10
10
10
100
150
1
10
10
10
10
10
The Concept of Minimum Phase
A system is called a minimum phase system if all its poles
and zeros are in the left half plane. Minimum phase systems
are easy to control.
For minimum phase systems the phase curve is given by the
gain curve and vice versa. An approximate relation is
10
arg G (i )
10
Phase
10
50
Gain and Phase Margins in Bode Plots
Make a Bode plot of the loop transfer function L = PC
10
10
10
Phase
A slope of one for the gain curve corresponds to 90 phase.
The exact relations are called Bodes relations. Systems that
are not minimum phase are called non-minimum phase. The
property of non-minimum phase imposes severe limitations to
what can be achieved by control.
100
120
140
160
180
200
1
10
d log h G (i )h
,
2
d log
10
c K. J. strm August, 2001
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Bodes Relations between Amplitude and Phase
let G (s) be a transfer function with all poles and zeros in the
left half plane. Introduce
The Smoothing Kernel
Z
We have
2 0
log h G ( i )h log h G ( i 0 )h
d
0
2 20
Z
+
1
d log h G ( i )h
d log h G ( i )h
0
=
log
d
0
d log
0
2
d log
Weight
6
5
4
y
arg G ( i 0 ) =
log h G ( i )h
2 2
= 0
log h G ( i 0 )h
1
1 arg G ( i )
0 arg G ( i 0 )
d
2
2 0
0
Z
+
2 20 d 1 arg G ( i )
0
=
log
d
0
d
0
RHP Zero G (s) =
as
a+s
We have h G (i )h = 1 and arg G (i ) = 2 arctan
1
2
1
0
3
10
10
/a
10
10
10
10
10
Laplace transform of the step response h is
Z
G ( s)
=
est h(t)dt
s
0
If G has a RHP zero at s = > 0 we have
Z
G ( )
0=
=
e t h(t)dt
10
10
Step Response for System with RHP Zero
Gain
10
+
2
0
log
d
=
0
2
10
/a
Phase
0
10
10
Since the integral is zero the step response must assume both
positive and negative values
50
0.5
100
150
1
10
/a
0
10
0.5
1
10
c K. J. strm August, 2001
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0.5
1.5
2.5
RHP Pole G (s) =
Time Delay G (s) = esL
We have h G (i )h = 1 and arg G (i ) = L
Gain
s a
s+ a
We have h G (i )h = 1 and arg G (i ) = 2 arctan b =
+ 2 arctan b
Gain
10
10
10
10
1
10
10
L
Phase
0
10
10
10
10
100
/a
Phase
0
10
10
/a
10
200
50
300
100
400
1
10
L
0
10
10
150
1
10
10
Airplanes
Power Systems
The transfer function from elevon to height of an airplane with
elevons in the rear are always non-minimum phase. The Wright
brothers avoided this by an elevon in front.
Level dynamics in boilers is non-minimum phase because
of the shrink and swell phenomena
Modern fighter planes have canards in the front and even jet
thrusters to avoid the problem.
The transfer function from tube opening to power is for a
hydro electric power system
X-29 is an experimental aircraft. In one operating condition the
system is approximately described by the transfer function
Gnmp(s) =
P ( s)
P0 1 2sT
=
A(s)
A0 1 + sT
s 26
s6
One pole and one zero in the right half plane. This plane is
difficult to control well.
c K. J. strm August, 2001
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Bicycles
Summary
Front wheel steering
The input-output view of dynamical systems
Describe a system by making a table of all input-output
pairs
V0
( s)
amQ V0
a
=
( s)
bJ s2 mgQ
J
s+
Fouriers idea: look at steady state propagation of sinusoids
Non-minimum phase because of the right half plane pole
Rear wheel steering
Frequency response G (i )
Graphical representations, very useful for intuition
V0
( s)
amQ V0 s + a
=
( s)
bJ s2 mgQ
J
Both poles and zeros in the right half plane
Bode and Nyquist plots
The concept of non-minimum phase
Systems which are non-minimum phase have severe
performance limitations
c K. J. strm August, 2001
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