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CDS 101/110: Lecture 10.1Limits on Performance
November 28, 2016
Goals: Introduce concept of limits on performance of feedback
systems Introduce Bodes integral formula and the waterbed effect
Show some of the limitations of feedback due to RHP poles and
zeros
Reading: strm and Murray, Feedback Systems, Section 12.6
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2
Loop Shaping: Design Loop Transfer Function
BWGM
PM
C(s) P(s)++
d
e u
-1
r +
n
y Translate specs to loop shape
= ()
Design C(s) to obey constraints
= =1 ( )
=1 ( )
Poles/Zeros from PID Poles/Zeros from
- Lead - Lag
Check the Gang of Four
=1
1 + (); =
()1 + ()
=()
1 + (); =
()1 + ()
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3
Algebraic Constraints on Performance
Goal: keep S & T small S small low tracking error T small
good noise rejection (and
robustness)
Problem: S + T = 1 Cant make both S & T small at the
same frequency Solution: keep S small at low frequency
and T small at high frequency Loop gain interpretation: keep L
large
at low frequency, and small at high frequency
Transition between large gain and small gain complicated by
stability (phase margin)
Sensitivityfunction
Complementarysensitivityfunction
Mag
nitu
de (d
B)
C(s) P(s)++
d
e u
-1
r +
n
y
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4
Bodes Integral Formula and the Waterbed Effect
Bodes integral formula for = 11+()
= = = =
Let be the unstable poles of () and assume relative degree of ()
2
Theorem: the area under the sensitivity function is a conserved
quantity:
Waterbed effect:Making sensitivity smaller over some frequency
range requires increase in sensitivity someplace elsePresence of
RHP poles makes this effect worseActuator bandwidth further limits
what you can doNote: area formula is linear in ; Bode plots are
logarithmic
Frequency (rad/sec)
Mag
nitu
de (d
B)
Sensitivity Function
-40
-30
-20
-10
0
10
100
101
102
103
104
Area below 0 dB + area above 0 dB = Re pk = constant
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5
Example: Magnetic Levitation
System description Ball levitated by electromagnet Inputs:
current thru electromagnet Outputs: position of ball, , (from IR
sensor) States: , Dynamics: = , = magnetic force
generated by wire coilIR
receivierIR
transmitter
Electro-magnet
Ball
System Dynamics
= 2/2
= + 0where:
= current to electromagnet = voltage from IR sensor
Linearization:
=
2 2
Poles at = open loop unstable
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6
Bode Plot of Open Loop System
IRreceivier
IRtransmitter
Electro-magnet
Ball
Real Axis
Imag
inar
y Ax
is
Nyquist Diagram
-4 -2 0 2 4 6 8-1.5
-1
-0.5
0
0.5
1
1.5
Frequency (rad/sec)
Phas
e (d
eg);
Mag
nitu
de (d
B)
Bode Diagram
-100
-50
0
50
100
101
102
103
104-200
-150-100-50
050
Note: RHP pole in L need one net encirclement (CCW)
Linearization:
=
2 2
Poles at = open loop unstable
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7
Control Design
Need to create encirclement To offset RHP pole Loop shaping is
not useful here Flip gain to bring Nyquist plot over -1
point Insert phase to create CCW
encirclement
Can accomplish using a lead compensator Produce phase lead at
crossover Generates loop in Nyquist plot
Real Axis
Nyquist Diagram
-4 -2 0 2 4 6 8-1.5
-1
-0.5
0
0.5
1
1.5
Frequency (rad/sec)
Phas
e (d
eg);
Mag
nitu
de (d
B)
Bode Diagram
-100
-50
0
50
100 101 102 103 104-200-150
-100-50
050
=0=1=0
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8
Control Design
Need to create encirclement
Loop shaping is not useful here Flip gain to bring Nyquist plot
over -1
point Insert phase to create CCW
encirclement
Can accomplish via lead compensator Produce phase lead at
crossover Generates loop in Nyquist plot
Real Axis
Imag
inar
y Ax
is
Nyquist Diagram
-4 -2 0 2 4 6 8-1.5
-1
-0.5
0
0.5
1
1.5
Frequency (rad/sec)
Phas
e (d
eg);
Mag
nitu
de (d
B)
Bode Diagram
-100
-50
0
50
100 101 102 103 104-200-150
-100-50
050
=0=1=0
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9
Performance LimitsNominal design gives low perf
Not enough gain at low frequency Try to adjust overall gain to
improve low
frequency response Works well at moderate gain, but notice
waterbed effect
Bode integral limits improvement
Must increase sensitivity at some point
Frequency (rad/sec)
Mag
nitu
de (d
B)
Sensitivity Function
-40
-30
-20
-10
0
10
100
101
102
103
104
Time (sec.)
Ampl
itude
Step Response
0 0.04 0.08 0.12 0.160
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
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10
Right Half Plane ZerosRight half plane zeros produce non-minimum
phase behavior
Phase vs. frequency has additional lag (not minimum) for a given
magnitude Can cause output to move opposite from input for a short
period of time
Example: vs
Frequency (rad/sec)
Phas
e (d
eg);
Mag
nitu
de (d
B)
Bode Diagrams
-30
-20
-10
0
10
100 101 102-300
-200
-100
0
term
H1
H1
H2
, H2
Time (sec.)
Ampl
itude
Step Response
0 0.2 0.4 0.6 0.8 1 1.2-0.2
0
0.2
0.4
0.6
0.8
1
1.2
H1
H2
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11
Example: Lateral Control of the Ducted Fan
Source of non-minimum phase behavior To move left, need to make
> 0 To generate positive , need 1 > 0 Positive 1 causes fan
to move right
initially Fan starts to move left after short time
(as fan rotates)
Poles: 0, 0, Zeros:
Time (sec.)
Ampl
itude
Step Response
0 0.2 0.4 0.6 0.8 1-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
Fan moves right andthen moves to the left
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12
Stability in the Presence of (RHP) ZerosLoop gain limitations
Poles of closed loop = poles of 1 + L. Suppose () = ()/(), where k
is the
controller gain
For large k, closed loop poles approach open loop zeros RHP
zeros limit maximum gain serious design constraint!
Root locus interpretation Plot location of eigenvalues as a
function of the loop gain k Can show that closed loop poles
go
from open loop poles (k = 0) to openloop zeros ( = )
-7 -6 -5 -4 -3 -2 -1 0 1 2 3-8
-6
-4
-2
0
2
4
6
8
Real Axis
Imag
Axi
s
Original polelocation (k = 0)
Closed loopzeros
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13
-150
-100
-50
0
50
Frequency (rad/sec)
Mag
nitu
de (d
B)
Additional performance limits due to RHP zerosAnother
waterbed-like effect: look at maximum of over frequency range:
Theorem: Suppose that () has a RHP zero at . Then there exist
constants 1 and 2(depending on 1, 2, ) such that 1 log1 + 22 0.
M1 typically
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14
Summary: Limits of PerformanceMany limits to performance
Algebraic: S + T = 1 RHP poles: Bode integral formula RHP zeros:
Waterbed effect on peak of S
Main message: try to avoid RHP poles and zeros when-ever
possible (eg, re-design)
Frequency (rad/sec)
Mag
nitu
de (d
B)
Sensitivity Function
-40
-30
-20
-10
0
10
100
101
102
103
104
0
log | |2
= 1
RHP poles
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Announcements
Homework #8 is due on Friday, 2 pm In class or HW slot (102
STL)
Final exam Out on 5 Dec (Mon.) Due on Fri. December 9, by 5
pm:
- turn in to Sonya Lincoln - 250 Gates-Thomas
Final exam review: December 2 from 2-3 pm, 105 Annenberg
Office hours during study period- 5 Dec (Mon), 3-5 pm- 6 Dec
(Tue), 3-5 pm
15
YouTube: Chicken Head Tracking
CDS 101/110: Lecture 10.1Limits on PerformanceLoop Shaping:
Design Loop Transfer FunctionAlgebraic Constraints on
PerformanceBodes Integral Formula and the Waterbed EffectExample:
Magnetic LevitationBode Plot of Open Loop SystemControl
DesignControl DesignPerformance LimitsRight Half Plane
ZerosExample: Lateral Control of the Ducted FanStability in the
Presence of (RHP) ZerosAdditional performance limits due to RHP
zerosSummary: Limits of PerformanceAnnouncements