Extremum Seeking Control
for Real-Time Optimization
Miroslav Krstic
UC San Diego
IEEE Advanced Process Control Applications for
Industry Workshop
Vancouver, 2007
Example of Single-Parameter Maximum Seeking
f * *


asin  t

f ''
f ( )  f 
  *
2
*
ˆ
k
s


Plant

y
2
s
sh
sin  t
1
Example of Single-Parameter Maximum Seeking
*
f - unknown!
f ( (t))
2
Topics - Theory
• History
• Single parameter ES, how it works, and stability analysis by
averaging
• Multi-parameter ES
• ES in discrete time
• ES with plant dynamics and compensators for performance
improvement
• Internal model principle for tracking parameter changes
• Slope seeking
• Limit cycle minimization via ES
3
Topics - Applications
• PID tuning
• Internal combustion (HCCI) engine fuel consumption minimization
• Compressor instabilities in jet engines
• Combustion instabilities
• Formation flight
• Fusion reflected RF power
• Thermoacoustic coolers
• Beam matching in particle accelerators
• Flow separation control in diffusers
• Autonomous vehicles without position sensing
4
History
• Leblanc (1922) - electric railways
• Early Russian literature (1940’s) - many papers
• Drapper and Li (1951) - application to IC engine spark timing tuning
• Tsien (1954) - a chapter in his book on Engineering Cybernetics
• Feldbaum (1959) - book Computers in Automatic Control Systems
• Blackman (1962 chap. in book by Westcott) - nice intuitive presentation of ES
• Wilde (1964) - a book
• Chinaev (1969) - a handbook on self-tuning systems
• Papers by[Morosanov], [Ostrovskii], [Pervozvanskii], [Kazakevich], [Frey, Deem,
and Altpeter], [Jacobs and Shering], [Korovin and Utkin] - late 50s - early 70’s
• Meerkov (1967, 1968) - papers with averaging analysis
• Sternby (1980) - survey
• Astrom and Wittenmark (1995 book) - rates ES as one of the most promising
areas for adaptive control
5
Recent Developments
• Krstic and Wang (2000, Automatica) - stability proof for single-parameter
general dynamic nonlinear plants
• Choi, Ariyur, Wang, Krstic - discrete-time, limit cycle minimization, IMC for
parameter tracking, etc.
• Rotea; Walsh; Ariyur - multi-parameter ES
• Ariyur - slope seeking
• Tan, Nesic, Mareels (2005) - semi-global stability of ES
• Other approaches: Guay, Dochain, Titica, and coworkers; Zak, Ozguner, and
coworkers; Banavar, Chichka, Speyer; Popovic, Teel; etc.
• Applications not presented in this workshop:
o
o
o
o
o
Electromechanical valve actuator (Peterson and Stephanopoulou)
Artificial heart (Antaki and Paden)
Exercise machine (Zhang and Dawson)
Shape optimization for magnetic head in hard disk drives (UCSD)
Shape optimization of airfoils and automotive vehicles (King, UT Berlin)
6
ES Book
7
Tutorial Topics Covered in the Book
• Introduction, history, single-parameter stability analysis
• Plant dynamics, compensators, and IMC for tracking parameter changes
• Limit cycle minimization via ES
• Multi-parameter ES
• ES in discrete time
• Slope seeking
• Compressor instabilities in jet engines
• Combustion instabilities
• Formation flight
• Anti-skid braking
• Bioreactor
• Thermoacoustic coolers
• Internal combustion engines
• Flow separation control in diffusers
• Beam matching in particle accelerators
• PID tuning
• Autonomous vehicles without position sensing
8
Basic Extremum Seeking - Static Map
f * *

ˆ

k

s
asin  t
f *  minimum of the map
f "  second derivative (positive - f ( ) has a min.)
ˆ  estimate of  *


y
2

s
sh
sin  t
k  adaptation gain (positive) of the integrator
y  output to be minimized
 *  unknown parameter

f ''
f ( )  f 
  *
2
*
Plant
1
s
a  amplitude of the probing signal
  frequency of the probing signal
h  cut-off frequency of the "washout filter"
/  modulation/demodulation
s
sh
9
How Does It Work?
f * *



f ''
f ( )  f 
  *
2
*
ˆ
asin  t
Estimation error:
k

s


Plant

y
2
s
sh
sin  t
   *  ˆ
2
2
a
f
"
f
"
a
f"
2
y  f* 
 %
 af "%sin  t 
cos 2 t
4
2
4
Loc. Analysis - neglect
quadratic terms:
2
2
a
f
"
f
"
a
f"
2
y  f* 
 %
 af "%sin  t 
cos 2 t
4
2
4
10
How Does It Work?
f * *

ˆ

asin  t

f ''
f ( )  f 
  *
2
*
k

s


Plant

y
2
s
sh
sin  t
2
2
a
f
"
a
f"
*
%
y f 
 af " sin  t 
cos 2 t
4
4
2
s
a2 f "
a
f"
*
[y]  f 
 af "%sin  t 
cos 2 t
sh
4
4
11
How Does It Work?
f * *


asin  t

f ''
f ( )  f 
  *
2
*
ˆ
k

s


Plant

y
2
s
sh
sin  t
Demodulation:
2
s
a
f"
2
  sin  t
[y]  af "%sin  t 
cos 2 t sin  t
sh
4
a2 f " % a2 f " %
a2 f "


 cos 2 t 
sin  t  sin 3 t 
4
4
8
12
How Does It Work?
f * *


f ''
f ( )  f 
  *
2
*
ˆ

Plant
asin  t
k

s


y
2
s
sh

sin  t
Since
   *  ˆ
then
  ˆ
&
13
How Does It Work?
f * *


asin  t

f ''
f ( )  f 
  *
2
*
ˆ
k

s


Plant

y
2
s
sh
sin  t

k  a2 f " % a2 f " %
a2 f "
  

 cos2t 
sin t  sin 3t 
s 4
4
8

high frequency terms - attenuated by integrator
14
How Does It Work?
f * *



f ''
f ( )  f 
  *
2
*
ˆ
asin  t
k

s


Plant

y
2
s
sh
sin  t
ka 2 f " %


4
Stable because k,a, f "  0
15
Stability Proof by Averaging
f * *

f ( )  f * 
Plant

f ''
  *
2

   *  ˆ
y
2
e f* 

asin  t
ˆ
k

s

  t
s
sh

h
y 
sh
sin  t
Full nonlinear time-varying model:
d % k  f" %
 
  asin 

d
 2


2

 e sin 

2
d
h
f" %
e   e 
  asin  


d

2


16
Stability Proof by Averaging
f * *

f ( )  f * 

f ''
  *
2
Plant

y
2
   *  ˆ
e f* 

ˆ

k

s
asin  t

s
sh
  t
h
y 
sh
sin  t
Average system:
d %
kaf " %
 av  
 av
d
2
d
h
f "  %2 a 2  
eav   eav    av   
d

2 
2 
Average equilibrium:
 av  0
a2 f "
eav  
4
17
Stability Proof by Averaging
f * *

f ( )  f * 

f ''
  *
2
Plant

y
2
   *  ˆ
e f* 

asin  t
ˆ
k

s


s
sh
  t
h
y 
sh
sin  t
Jacobian of the average system:
 kaf "


0
 2

J av  

h
 0
 

 
18
Stability Proof by Averaging
f * *

f ( )  f * 

f ''
  *
2
Plant

y
2
   *  ˆ
e f* 
ˆ

asin  t
k

s


s
sh
  t
h
y 
sh
sin  t
Theorem. For sufficiently large  there exists a unique exponentially stable
periodic solution of period 2/ and it satisfies
a2 f "
 1
2 / (t)  e2 / (t) 
 O   , t  0

4
Speed of convergence proportional to 1/, a2, k, f "
19
Stability Proof by Averaging
f * *

f ( )  f * 

f ''
  *
2
Plant

y
2
   *  ˆ
e f* 

asin  t
ˆ
k

s


s
sh
  t
h
y 
sh
sin  t
Output performance:
 1

y  f *  f "O  2  a 2 


20
PID Tuning Using ES
Based on contributions by: Nick Killingsworth
Background & Motivation
Proportional-Integral-Derivative (PID) Control
• Consists of the sum of three control terms
- Proportional term:
- Integral term:
- Derivative term:
uP (t )  Ke(t )
K t
u I (t )   e( s)ds
TI
de(t )
u D (t )  KTD
dt
e(t) = r(t) – y(t)
r(t) reference signal
y(t) measured output
• Often poorly tuned (Astrom [1995], etc.)
22
Background – PID
We use a two degree of freedom controller
The derivative term only acts on y(t)
• This avoids large control effort when there is a step change in
the reference signal
r
Cr
u
+

-
y
G
Cy

1 

Cr  K 1 
 TI s 


1
C y  K 1 
 TD s 
 TI s

23
Tuning Scheme
Step
function
r(t)
Continuous Time
Cr  k 


-
u(t)
J(k)
y(t)
G
C y  k 
k
Extremum
Seeking
Algorithm
Discrete Time
24
Extremum Seeking
 (k )

 cos(k )
y (k )
J ( )

z 1

z 1
zh
 cos(k )
Simple - three lines of code
25
Extremum Seeking Tuning Scheme
Implementation
1.
2.
Run Step response
experiment with ZN PID
parameters
Calculate J
T
1
2
J ( k ) 
e( k ) dt

T  t0 t0
26
Extremum Seeking Tuning Scheme
Implementation
1.
2.
3.
Run Step response
experiment with ZN PID
parameters
Calculate J
Calculate next set of PID
parameters using discrete
ES tuning method
 (k )  h (k  1)  J (k  1)
ˆi (k  1)  ˆi (k )   i i cos(i k )J (k )  (1  h) (k )
 i (k  1)  ˆi (k  1)   i cos(i (k  1))
27
Extremum Seeking Tuning Scheme
Implementation
1.
2.
3.
4.
Run Step response
experiment with ZN PID
parameters
Calculate J
Calculate next set of PID
parameters using discrete
ES tuning method
Run another step
response experiment with
new PID parameters
28
Extremum Seeking Tuning Scheme
Implementation
1.
2.
3.
4.
5.
Run Step response
experiment with ZN PID
parameters
Calculate J
Calculate next set of PID
parameters using discrete ES
tuning method
Run another step response
experiment with new PID
parameters
Repeat 2-4 set number of
times or until J falls below a
set value
Repeat
29
Implementation – Cost Function
Cost Function J(k)
Used to quantify the controller’s performance
Constructed from the output error of the plant and the control effort
during a step response experiment
Has discrete values at the completion of each step response
experiment
T
1
2
J ( k ) 
e
(

)
dt
k

T  t0 t0
where T is the total sample time of each step response
experiment
 is a vector containing the PID parameters:   K , TI , TD 
30
Implementation – Cost Function
Cost Function J(k)
t0 is the time up until which zero weightings are placed on the error.
This shifts the emphasis of the PID controller from the transient
phase of the response to that of minimizing the tracking error
after the initial transient portion of the response
T
1
2
J ( k ) 
e
(

)
dt
k

T  t0 t0
to
31
Example Plants
Four systems with dynamics typical of some industrial
plants have been used to test the ES PID tuning method
1.
Time delay
G1 ( s ) 
2.
3.
1
e 5 s
1  20 s
Large time delay
1
G2 ( s ) 
e  20 s
1  20 s
Single pole of order eight
G3 ( s ) 
4.
1
(1  10s )8
Unstable zero
G4 ( s) 
1  5s
(1  10s)(1  20s)
32
Results
•
Ziegler-Nichols values used as initial conditions in
the ES tuning algorithm
•
Results compared to three other popular PID
tuning methods:
- Ziegler-Nichols (ZN)
- Internal model control (IMC)
- Iterative feedback tuning (IFT, Gevers, ‘94, ‘98)
33
Results -
1
G1 ( s ) 
e 5 s
1  20 s
a) Evolution of Cost Function
b) Evolution of PID Parameters
c) Step Response of output
d) Step Response of controller
34
Results -
G2 ( s ) 
a) Evolution of Cost Function
c) Step Response of output
1
e  20 s
1  20 s
b) Evolution of PID Parameters
d) Step Response of controller
35
Results -
G3 ( s ) 
1
(1  10s )8
a) Evolution of Cost Function
b) Evolution of PID Parameters
c) Step Response of output
d) Step Response of controller
36
Results -
G4 ( s) 
a) Evolution of Cost Function
c) Step Response of output
1  5s
(1  10s)(1  20s)
b) Evolution of PID Parameters
d) Step Response of controller
37
Results – Cost Function Comparison
Step Response of output
The following cost functions
were minimized using ES:
T
1
ISE   e( k ) 2 dt
T 0
38
Results – Cost Function Comparison
Step Response of output
The following cost functions
were minimized using ES:
T
1
ISE   e( k ) 2 dt
T 0
T
1
ITSE   te( k ) 2 dt
T0
39
Results – Cost Function Comparison
Step Response of output
The following cost functions
were minimized using ES:
T
1
ISE   e( k ) 2 dt
T 0
T
1
ITSE   te( k ) 2 dt
T0
T
1
IAE   | e( k ) | dt
T0
40
Results – Cost Function Comparison
Step Response of output
The following cost functions
were minimized using ES:
T
1
ISE   e( k ) 2 dt
T 0
T
1
ITSE   te( k ) 2 dt
T0
T
1
IAE   | e( k ) | dt
T0
T
1
ITAE   t | e( k ) | dt
T0
41
Results – Cost Function Comparison
Step Response of output
The following cost functions
were minimized using ES:
T
1
ISE   e( k ) 2 dt
T 0
T
1
ITSE   te( k ) 2 dt
T0
T
1
IAE   | e( k ) | dt
T0
T
1
ITAE   t | e( k ) | dt
T0
T
1
2
Window 
e
(

)
dt
k

T  t0 t0
42
Actuator Saturation
Saturation of 1.6 applied to control signal for plant G1
1
G1 ( s ) 
e 5 s
1  20 s
ES and IMC compared with and without the addition of an anti
windup scheme
Tracking anti-windup scheme
43
Actuator Saturation
Step response of output
Control signal during step response
1.6
1.2
1.4
IMC
IMCtracking
1.2
ES
EStracking
1
1
u(t)
y(t)
0.8
0.6
0.8
0.6
0.4
IMC
IMC
ES
EStracking
0.2
0
0.4
tracking
0
20
40
Time(sec)
60
80
0.2
100
0
0
20
40
60
80
100
Time(sec)
44
Effects of Noise
Band-limited white noise has been added to output
Power spectral density = 0.0025
Correlation time = 0.2
Independent noise signal for each iteration
Simulations on plant G1
1
5 s
G1 ( s) 
e
1  20s
45
Effects of Noise
a) Evolution of Cost Function
c) Step Response of output
b) Evolution of PID Parameters
d) Step Response of controller
46
Selecting Parameters of ES Scheme
Must select
, perturbation step size
, adaptation gain
, perturbation frequency
h, high-pass filter cut-off frequency
 (k )

ˆ(k )
i
J (k )
J ( )
i
 i
z 1
 (k )

z 1
zh
 cos(k )
Looks like have more parameters to pick than we started out with!
However, ES tuning is less sensitive to parameters than PID controller.
47
Selecting Parameters of ES Scheme
G2 ( s) 
1
e  20s
1  20s
ES Tuning
Parameters
  0.06,0.30,0.20T
 ,
  2500,2500,2500
T
i  0.8i 
h  0.5
 ,
2
,
10
 ,
2 10
K
Ti
Td
1.01
31.5
7.16
1.00
31.1
7.6
1.01
31.3
7.54
1.01
31.0
7.65
48
Selecting Parameters of ES Scheme
Need to select an adaptation gain  and perturbation
amplitude  for EACH parameter to be estimated
In the case of a PID controller,  = [K, Ti, Td], so we need
three of each.
The modulation frequency is determined by:
i  ai *
where 0 < a < 1
The highpass filter (z-1)/(z+h) is designed with 0<h<1
with the cutoff frequency well below the modulation
frequency  .
i
Convergence rate is directly affected by choice of  and ,
as well as by cost function shape near minimizer.
49
Example of ES-PID tuner GUI
Evolution of step response under ES tuning
1 iteration (ZN)
5 iterations
10 iterations
50 iterations
1.1
Step tracking response
1.05
1
0.95
0.9
0.85
0.8
0
10
20
30
40
50
Time (sec)
60
70
80
90
100
50
Punch Line
ES yields performance as good as the best of the other
popular tuning methods
Can handle some nonlinearities and noise.
The cost function can be modified such that different
performance attributes are emphasized
51
Control of HCCI Engines
Based on contributions by: Nick Killingsworth (UCSD),
Dan Flowers and Sal Aceves (Livermore Lab),
and Mrdjan Jankovic (Ford)
HCCI = ?
HCCI = Homogeneous Charge Compression Ignition
Low NOx emissions like spark-ignition engines
High efficiency like Diesel engines
More promising in near term than fuel cell/hydrogen engines
53
HCCI Engine Applications
Distributed power generation
Automotive hybrid powertrain
54
What is the difference
between Spark Ignition,
Diesel, and HCCI engines?
Categories of Engines
Compression
Ignition
Spark ignition
Homogeneous
charge
HCCI
Spark ignition
engine
Inhomogeneous
charge
Diesel
Direct injection
engine
56
Spark Ignition Engine
Basic engine thermodynamics: engine efficiency increases as the
compression ratio and =cp/cv (ratio of specific heats) increase
engine indicated efficiency
0.7
0.6
1.4
0.5
SI engines
0.4
1.3
0.3
0.2
1
Engine Efficiency  1 
CR  1
0.1
 = 1.35 for fuel and air mixture
0.0
 = 1.4 for air
1
3
5
7
9
11
13
15
17
19
compression ratio
57
Diesel Engine
Highly efficient because they compress only air ( is high) and are
not restricted by knock (compression ratio is high)
engine indicated efficiency
0.7
Diesel engines
0.6
1.4
0.5
SI engines
0.4
1.3
0.3
1
Engine Efficiency  1 
CR  1
0.2
 = 1.4 for air
0.1
0.0
 = 1.35 for fuel and air mixture
1
3
5
7
9
11
13
15
17
19
compression ratio
58
HCCI Engine
Compression ratio not restricted by “knock” (autoignition of gas ahead
of flame in spark ignition engines) a efficiency comparable to Diesel
engine indicated efficiency
0.7
0.6
1.4
0.5
Diesel and
HCCI engines
SI engines
0.4
1.3
0.3
Engine Efficiency
0.2
 1
 = 1.4 for air
0.1
0.0
1
CR  1
 = 1.35 for fuel and air mixture
1
3
5
7
9
11
13
15
17
19
compression ratio
59
HCCI Engine
Potential for high efficiency (Diesel-like)
Low NOx and PM (unlike Diesel)
BUT, no direct trigger for ignition - requires feedback
to control the timing of ignition!
60
Experiment at Livermore Lab
Caterpillar 3406 natural
gas spark ignited engine
converted to HCCI
Set up for stationary
power generation (not
automotive)
61
Actuators
Mixing “Tees”(x 6)
Combustion timing (output)
is very sensitive to
intake temperature (input)
Cooled
Intake Air
Valve Actuators
Heated
Intake Air
Controlled Intake Temperature
to Individual Cylinders
Cold Manifold
Hot Manifold
62
Overall Architecture: Sensors and Software
User interface
Valve Position
Real-Time Controller
PC running Labview RT OS
Cylinder Pressure
Crank Angle Position
63
ES used to MINIMIZE FUEL CONSUMPTION of HCCI engine
by tuning combustion timing setpoint
+ S e CA50
-
PI
CA50SP
Tintake
HCCI
Engine
CA50
.
mfuel
Extremum
Seeking
64
ES delays the combustion timing 6 crank angle
degrees, reducing fuel consumption by > 10%
65
Larger adaptive gain: ES finds same minimizer,
but much more quickly
4.8
16
CA50 ave
CA50_setpoint
mass flow rate of fuel
4.7
12
4.6
10
4.5
8
4.4
6
4.3
4
4.2
2
4.1
0
4
0
50
Fuel Consumption [g/s]
Combustion Timing CA50 [CAD ATDC]
14
100 150 200 250 300 350 400 450 500 550
Time (sec)
66
Axial Flow (Jet Engine-Like) Compressor Control
•
Active controls for
rotating stall only reduce
the stall oscillations but
they do not bring them to
zero nor do they
maximize pressure rise.
•
Extremum seeking to
optimize compressor
operating point.
Experimental Results
Extremum seeking stabilizes the maximum pressure rise.
Pressure Rise
Problem Statement
time
bleed valve
Caltech
COMPRESSOR
Motivation
Smaller, lighter
compressors;
higher payload in
aircraft
Pressure rise
EXTREMUM
SEEKER
Air Injection
Stall Controller
1
s
sin t
washout
filter
Combustion Instability Control
Problem Statement
•
•
Experiment on UTRC 4MW combustor
Rayleigh criterion-based
controllers, which use phaseshifted pressure
measurements and fuel
modulation, have emerged as
prevalent
ext. seeking suppresses oscillations
The length of the phase
needed varies with operating
conditions. The tuning method
must be non-model based.
time
Pressure
Motivation
fuel
•
•
Tuning allows operation
with minimum
oscillations at lean
conditions
Reduced engine size,
fuel consumption and NOx
emissions
COMBUSTOR
Phase-Shifting
Controller

1
s
sin t
Frequency/
amplitude
observer
washout
filter
EXTREMUM SEEKER
Formation Flight Engine Output Minimization
Tune reference inputs yref and zref to the autopilot
of the wingman to maximize its downward pitch
angle or to minimize its engine output
69
Simulation of C-5 Galaxy transport airplane
for a brief encounter of “clear air turbulence”
70
Thermoacoustic Cooler (M. Rotea)
Electric energy
Acoustic energy
Hot-end heat
exchangers
Stack
Cold-end heat
exchangers
Heat pumping
Resonance tube
Pressurized He-Ar mixture
Electro-dynamic
driver
Standing sound wave creates the
refrigeration cycle
Solid surface
(stack plate)
Gas particle in a
standing wave
QH
3
2
QL
1
4
Heat Pumping
1-2: adiabatic compression and displacement
2-3: isobaric heat transfer (gas to solid)
3-4: adiabatic expansion and displacement
4-1: isobaric heat transfer (solid to gas)
71
Thermoacoustic Cooler
Neck (mass)
Volume (stiffness)
Moving piston
(varying resonator’s
stiffness)
Electro-dynamic
Driver
Heat
exchangers
Helmholtz resonator
Tuning Variables
• Piston position (acoustic impedance)
• Driver frequency
72
ES with PD compensator
ax sin(xt   x )
sin(xt )
LPF
Integrator
PD
+
POS Command
a f sin( f t   f )
sin( f t )
LPF
HPF
+
Integrator
Qc
Cooling
Power
Calculation
PD
+
Tunable
Cooler
+ FREQ Command
1
Tf s  1
1
Tx s  1
73
Experiment – Fixed Operating Condition
4
141
22.65
142
29.92
143
35.67
144
28.63
145
21.25
142
144
15.89
34.12
145
39.68
146
35.12
148
19.34
140
4.95
142
9.00
144
18.55
145
146
23.86
35.99
147
41.28
148
38.00
149
30.36
150
19.36
146
16.34
148
33.34
149
41.21
150
40.70
151
153
34.69
19.63
151
32.16
152
35.60
153
31.74
5
6
7
8
Cooling Performance with ESC
Cooling Power (Watt)
POWER W
60
Piston Position (in)
FREQ Hz
8
40
20
0
ESC ON
0
50
100
150
200
250
300
0
150
50
100
150
200
250
300
50
100
150
Time (sec)
200
250
300
6
4
2
Driving Frequency (Hz)
POS in.
145
140
0
ESC quickly finds optimum operating point
(41.3W, 147Hz, 6.2in)
74
Cooling Power (Watt)
Experiment – Varying Operating Condition
100
50
0
0
50
100
150
200
250
300
350
400
0
50
100
150
200
250
300
350
400
150
200
250
300
350
400
350
400
Piston Position (in)
12
10
8
6
Driving Frequency (Hz)
4
155
150
ESC ON
145
140
0
50
100
Flow Rate Change
100
Flow Rate (ml/s)
ESC tracks optimum
after cold-side flow rate
is increased
80
Cold Side
Hot Side
60
40
20
0
50
100
150
200
250
300
Time (sec)
75
ES for the Plasma Control in the
Frascati Fusion Reactor
Contribution by Luca Zaccarian (U. Rome, Tor Vergata)
Optimization Objective
Framework:
Additional Radio Frequency
heating injected in the
plasma by way of Lower
Hybrid (LH) antennas:
plasma reflects some power
Goal:
Optimize coupling between
the Lower Hybrid antenna
and tha plasma, during the
LH pulse
Reflected Power Map
Reflected power:
Convex fcn of edge density
Convex fcn of edge position
Possible approaches to optimize:
1. Move the antenna
(too slow!)
2. Move the plasma
(viable – adopted here)
Probing not Allowed - Modified ES Scheme
Knob
Extracted
Input sinusoid
Extracted
Output sinusoid
Experimental results with medium gain
K = 300
Safety
saturation
limits
performance
Control action
is quite
aggressive.
Experimental results with lower gain
K = 200
(Antenna has
been moved)
Graceful
convergence
to the
minimum
reflected
power
Gain too high - instability
K = 350
Instability
Gain is
too large
Experiments - Summary
Input/output plane representation:
K = 300: saturation prevents reaching the minimum
K = 200: graceful convergence to minimum (slight overshoot)
K = 350: gain too high – all the curve is explored