Extremum Seeking Feedback Tools for
Real-Time Optimization
Miroslav Krstic
University of California, San Diego
Large Scale Robust Optimization, Sandia, August 2005
Example of Single-Parameter Maximum Seeking
f * θ*
θ
+
a sin ω t
(
f ''
f (θ ) = f +
θ − θ*
2
*
θ̂
k
s
ξ
×
Plant
)
y
2
s
s+h
sin ω t
1
Example of Single-Parameter Maximum Seeking
2
Theory
• History
• Single parameter ES, how it works, and stability analysis by
averaging
• ES with plant dynamics and compensators for performance
improvement
• Internal model principle for tracking parameter changes
• Multi-parameter ES
• Slope seeking
• ES in discrete time
• Limit cycle minimization via ES
3
Applications
• Anti-skid braking
• Compressor instabilities in jet engines
• Combustion instabilities
• Flow separation control in diffusers
• Thermoacoustic coolers
• Automotive engine mapping
• Beam matching in particle accelerators
• Formation flight
• Bioreactors
• PID tuning
• 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 outside of my group:
o
o
o
o
o
Electromechanical valve actuator (Peterson and Stephanopoulou)
Artificial heart (Antaki and Paden)
Exercise machine (Zhang and Dawson)
Chemical reactors and petroleum processing (several research groups)
Shape optimization for magnetic head in hard disk drives (UCSD)
6
ES Book
7
Basic Extremum Seeking - Static Map
f * θ*
θ
θ̂
+
k
−
s
a sin ω t
f * = minimum of the map
f " = second derivative (positive - f (θ ) has a min.)
θ̂ = estimate of θ
*
ξ
)
y
2
×
s
s+h
sin ω t
k = adaptation gain (positive) of the integrator
y = output to be minimized
θ * = unknown parameter
(
f ''
f (θ ) = f +
θ − θ*
2
*
Plant
a = amplitude of the probing signal
ω = frequency of the probing signal
h = cut-off frequency of the "washout filter"
+/× = modulation/demodulation
1
s
s
s+h
8
How Does It Work?
f * θ*
θ
+
(
f ''
f (θ ) = f +
θ − θ*
2
*
θ̂
a sin ω t
Estimation error:
k
−
s
ξ
×
Plant
)
y
2
s
s+h
sin ω t
θ = θ * − θ̂
2
2
a
f
"
f
"
a
f"
y = f* +
+ θ 2 − af "θ sin ω t +
cos 2ω t
4
2
4
Loc. Analysis - neglect
quadratic terms:
2
2
a
f
"
f
"
a
f"
y ≈ f* +
+ θ 2 − af "θ sin ω t +
cos 2ω t
4
2
4
9
How Does It Work?
f * θ*
θ
θ̂
+
a sin ω t
(
f ''
f (θ ) = f +
θ − θ*
2
*
k
−
s
ξ
×
Plant
)
y
2
s
s+h
sin ω t
2
2
a
f
"
a
f"
*

y≈ f +
− af "θ sin ω t +
cos 2ω t
4
4
2
s
a2 f "
a
f"
*
[y] ≈ f +
− af "θ sin ω t +
cos 2ω t
s+h
4
4
10
How Does It Work?
f * θ*
θ
+
a sin ω t
(
f ''
f (θ ) = f +
θ − θ*
2
*
θ̂
k
−
s
ξ
×
Plant
)
y
2
s
s+h
sin ω t
Demodulation:
2
s
a
f"
2
ξ = sin ω t
[y] ≈ −af "θ sin ω t +
cos 2ω t sin ω t
s+h
4
a2 f "  a2 f " 
a2 f "
ξ≈−
θ+
θ cos 2ω t +
(sin ω t − sin 3ω t )
4
4
8
11
How Does It Work?
f * θ*
θ
θ̂
+
(
f ''
f (θ ) = f +
θ − θ*
2
*
a sin ω t
k
−
s
ξ
Plant
)
×
y
2
s
s+h
sin ω t
Since
θ = θ * − θ̂
then
̂
θ = −θ
12
How Does It Work?
f * θ*
θ
+
a sin ω t
(
f ''
f (θ ) = f +
θ − θ*
2
*
θ̂
k
−
s
ξ
×
Plant
)
y
2
s
s+h
sin ω t
2
2
2
⎡
⎤
k
a
f
"
a
f
"
a
f"



θ ≈ ⎢−
θ+
θ cos 2ω t +
(sin ω t − sin 3ω t )⎥
s⎣
4
4
8
⎦
high frequency terms - attenuated by integrator
13
How Does It Work?
f * θ*
θ
+
(
f ''
f (θ ) = f +
θ − θ*
2
*
θ̂
a sin ω t
k
−
s
ξ
×
Plant
)
y
2
s
s+h
sin ω t
2
ka f " 
θ ≈ −
θ
4
Stable because k, a, f " > 0
14
Stability Proof by Averaging
f * θ*
θ
f (θ ) = f * +
Plant
(
f ''
θ − θ*
2
)
θ = θ * − θ̂
y
2
e= f* −
+
a sin ω t
θ̂
k
−
s
ξ
τ = ωt
s
s+h
×
h
[ y]
s+h
sin ω t
Full nonlinear time-varying model:
d  k ⎛ f" 
θ= ⎜
θ − a sin τ
⎝
dτ
ω 2
(
)
2
⎞
− e⎟ sin τ
⎠
2⎞
d
h⎛
f" 
e = ⎜ −e −
θ − a sin τ ⎟
⎝
⎠
dτ
ω
2
(
)
15
Stability Proof by Averaging
f * θ*
θ
f (θ ) = f * +
(
f ''
θ − θ*
2
Plant
)
y
2
θ = θ * − θ̂
e= f* −
+
θ̂
ξ
k
−
s
a sin ω t
×
s
s+h
τ = ωt
h
[ y]
s+h
sin ω t
Average system:
d 
kaf " 
θ av = −
θ av
dτ
2ω
d
h⎛
f " ⎛  2 a2 ⎞ ⎞
eav = ⎜ −eav − ⎜ θ av + ⎟ ⎟
dτ
ω⎝
2 ⎝
2 ⎠⎠
Average equilibrium:
θav = 0
a2 f "
eav = −
4
16
Stability Proof by Averaging
f * θ*
θ
f (θ ) = f * +
(
f ''
θ − θ*
2
Plant
)
y
2
θ = θ * − θ̂
e= f* −
+
a sin ω t
θ̂
k
−
s
ξ
×
s
s+h
τ = ωt
h
[ y]
s+h
sin ω t
Jacobian of the average system:
⎡ kaf "
⎤
−
0
⎢ 2ω
⎥
J av = ⎢
⎥
h
⎢ 0
− ⎥
⎢⎣
ω ⎥⎦
17
Stability Proof by Averaging
f * θ*
θ
f (θ ) = f * +
(
f ''
θ − θ*
2
Plant
)
y
2
θ = θ * − θ̂
e= f* −
θ̂
+
a sin ω t
k
−
s
ξ
×
s
s+h
τ = ωt
h
[ y]
s+h
sin ω t
Theorem. For sufficiently large ω there exists a unique exponentially stable
periodic solution of period 2π/ω and it satisfies
2
a
f"
⎛ 1⎞

θ 2 π /ω (t) + e2 π /ω (t) −
≤ O ⎜ ⎟ ,→→ ∀t ≥ 0
⎝ω⎠
4
Speed of convergence proportional to 1/ω, a2, k, f "
18
Stability Proof by Averaging
f * θ*
θ
f (θ ) = f * +
(
f ''
θ − θ*
2
Plant
)
y
2
θ = θ * − θ̂
e= f* −
+
a sin ω t
θ̂
k
−
s
ξ
×
s
s+h
τ = ωt
h
[ y]
s+h
sin ω t
Output performance:
⎛ 1
⎞
y − f * → f "O ⎜ 2 + a 2 ⎟
⎝ω
⎠
19
Extremum-Seeking Control of
Combustion Instabilities
Andrzej Banaszuk
United Technologies Research Center, E. Hartford, CT, U.S.A.
Acknowledgements:
Youping Zhang, Numerical Technologies
Kartik Ariyur, Honeywell
Miroslav Krstic, UCSD
Jeff Cohen, Pratt & Whitney
Clas Jacobson, UTRC
Workshop on Real-Time Optimization by Extremum-Seeking Control, ACC 05
Partially sponsored by AFOSR
Thermo-acoustic instability
Coupling of acoustics with heat release results in pressure oscillations
Fluctuating acoustic pressure and
velocity driven by unsteady heat release
Air
Acoustics
Fuel
Fuel/air mixing nozzle
Combustor
Combustion
Fluctuating heat release driven
by fluctuating velocity
Thermo-acoustic instabilities affect cost and
performance of gas turbines and rocket engines
25MW Pratt & Whitney FT8 gas turbine engine
•Power generation: pressure oscillations
prevent low emission operation
• Rockets: passive fixes increase
weight, limit payload
100
90
80
Combustion
70
Instability
NOx
60
50
40
30
20
10
0
0.400.420.440.460.480.500.520.540.560.580.60
NOx
•Military aeroengines: afterburner
screech and rumble limit performance,
passive fixes increase weight,
maintenance costs
Fuel/Air Ratio
Active fuel control can suppress oscillations
Fuel/air mixing nozzle
Combustor
Acoustics
Air
Control
Fuel
Pressure
measurement
Fuel valve
Combustion
Controller
Experiment in UTRC 4MW Single Nozzle Rig
14
• Numerous demonstrations at universities and
industrial rigs
12
• Rolls-Royce demonstrated control in afterburner
• Phase-shifting control effective, but no models to
guide selection of parameters
6X reduction in amplitude
Decrease in NOX & CO emissions
8
Amplitude
(psi)
• Siemens implemented control in 260MW gas
turbine engine
Uncontrolled
10
6
Controlled
4
2
0
0
200
400
600
800
1000
Summary of results: two extremum seeking schemes were successfully
demonstrated on UTRC 4MW single nozzle rig in August 1998.
Excellent performance at high power
Improved performance at low power
file r56p37 ,Date & Time of Capture: Wed Aug 19 12:00:34 1998
file r54p18 ,Date & Time of Capture: Thu Aug 13 11:57:35 1998
100
50
Optimal phase
control phase
control phase
Control
phase in
degrees
0
−50
−100
15
20
25
30
5
4
Optimal level
3
2
1
5
10
15
20
25
Performance measure 1 = 8.0355, Performance measure 2 = 20.6
15
20
25
30
6
4
2
0
30
0
5
10
15
20
25
Performance measure 1 = 27.9418, Performance measure 2 = 79.3275
30
file r56p37 ,Date & Time of Capture: Wed Aug 19 12:00:34 1998
7
11
Phase-magnitude map
5
4
3
Uncontrolled level
2
10
pressure and its magnitude
6
pressure and its magnitude
10
8
file r54p18 ,Date & Time of Capture: Thu Aug 13 11:57:35 1998
Pressure
magnitude
in psi
Time in5 seconds
10
Uncontrolled level
6
0
magnitude of bulk mode
magnitude of bulk mode
7
0
0
0
−50
Time in seconds
5
10
−150
0
Pressure
magnitude
in psi
50
9
8
7
6
5
Optimal level
1
−200
−150
−100
−50
0
control phase
Control phase in degrees
50
100
150
200
4
−200
−150
−100
−50
0
control phase
Control phase in degrees
50
100
150
200
Formation Flight Optimization
(Acknowledgement: Paolo Binetti)
1
Benefits of Formation Flight
• Reduction in power demand
• Alleviation of airspace
congestion
Obstacles to Attainment
• Sensitivity to positioning
• Difficulty in precise
measurements
• Absence of precise
modeling
2
Physics of Aircraft Wakes
and Wake Modeling
• Wake structure—modeled as two
NASA counter-rotating vortices
Γ
r2
W
!
V# (r ) =
, Γ=
, bred = b
2
2
2!r r + rc
"V∞bred
4
(
)
• Wake transport—
a modeling uncertainty
• Wake decay—neglected
6
Selection of the Problem
Why the C-5?
• Large savings in fuel consumption
• Representative of large transports
• They will be in service for the next 40 years
• Availability of wake data
5
Extremum Seeking for Formation Flight
Estimator
zref
yref
Aircraft with
formation autopilot
in the wake
+
!w
sin ("1t − ϕ1 )
s
s + h2
k
− 2
s
a2 sin ("2t )
- Objective:
s
s + h1
k
− 1
s
a1 sin ("1t )
!z
sin ("2t − ϕ 2 )
13
Simulation for brief CAT Encounter …
18
Extremum Seeking Control of
Thermoacoustic Coolers
Mario Rotea
June 7, 2005
Purdue University
[email protected]
REF. Extremum Seeking Control of Tunable Thermoacoustic Coolers, Y. Li, M. Rotea, G. Chiu, L.
Mongeau, I. Paek, IEEE Transactions on Control
Systems Technology, Vol. 13, No. 4, July 2005
Thermoacoustic Cooling
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)
Thermoacoustic Cooling
• Benefits
– Environmentally benign (inert gases)
– Simple, easy to maintain configuration
• Limitations
– Very hard to model ! hard to tune key parameters (driver, stack
location, duct geometry) for best performance
• Existing prototypes
–
–
–
–
–
Acoustic Stirling Heat Engine (Los Alamos National Lab)
Triton 10 kW refrigerator (Penn State)
Space Thermo-acoustic Refrigerator (NASA)
Purdue 300 W standing wave unit
Miniature thermo-acoustic cooler (Rockwell Science Center)
Tunable Thermoacoustic Cooler
Neck (mass)
Volume (stiffness)
Moving piston
(varying resonator’s
stiffness)
Electro-dynamic
Driver
Heat
exchangers
Helmholtz resonator
Tuning Variables
Performance and efficiency
sensitive to variations in
– Piston position (acoustic impedance)
– Driver frequency
Extremum Seeking Controller
ax sin(! xt # " x )
sin(!x t )
LPF
Integrator
PD
+
POS Command
a f sin(! f t # " f )
sin(! f t )
LPF
HPF
+
Integrator
Q! c
Cooling
Power
Calculation
ESC Parameters
– High pass filter (HPF)
– Low pass filters (LPF)
– Dither frequencies, amplitudes
and phases (!x,ax,"x; !x,ax,"x)
– PD gains (4 gains)
PD
+
Tunable
Cooler
+ FREQ Command
1
Tf s #1
1
Tx s # 1
(Tuning mechanism)
Cooling Power (Watt)
Experiment – Varying Operating Condition
100
50
0
0
50
100
150
200
250
300
350
400
Piston Position (in)
12
10
8
6
Driving Frequency (Hz)
4
0
50
100
150
200
250
300
350
400
150
200
250
300
350
400
350
400
155
150
ESC ON
145
140
0
50
100
Flow Rate Change
Flow Rate (ml/s)
100
80
Cold Side
Hot Side
60
40
20
0
50
100
150
200
Time (sec)
250
300
ESC tracks optimum
after cold-side flow rate
is increased
Extremum-Seeking Control of Flow
Separation in a Planar Diffuser
Andrzej Banaszuk
United Technologies Research Center, E. Hartford, CT, U.S.A.
Acknowledgements:
Satish Narayanan, UTRC: experiment
Youping Zhang, Numerical Technologies: algorithm
Workshop on Real-Time Optimization by Extremum-Seeking Control, ACC 05
Partially sponsored by AFOSR
Objective of pressure recovery control
Control effort
Performance
pexit " pinlet
C p (t ) # 1
2
2 !U inlet
pinlet
2
C $ (t ) #
u neck h
2
U inlet
H
pexit
H
U inlet
h
u neck
Cp
C$
Speaker command
=
bang
buck
Experimental Setup
Pressure recovery as function of diffuser angle (no control)
C p (t )
no stall
unsteady
stall
fully-developed
2D stall
jet flow
UTRC diffuser rig
•fully turbulent BL
•40,000 < ReH < 140,000
•Re%e > 300; M < 0.1
•Actuation: C$ ~ 0.001
%
% & 14
o
•Optimum uncontrolled performance
•Insignificant improvement with control
% & 18
o
•Poor uncontrolled performance
•Significant improvement with control
Need: control algorithm to optimize performance
Two frequency control law: U(t)=A1*(sin(2*"*f*t) + sin(2*"*2f*t-!))
U = 20m/s
50
U = 30m/s
Optimal operating point: f=31Hz, ! =60, Cp =0.21
45
0.05
0.05
0
Frequency f
0.15
0.1
40
0.05
35 0.1
35
+
30
+
0.15
0
0.2
30
0.1
25
20
Optimal operating point: f=36Hz, ! =60, Cp =0.16
45
Frequency f
40
50
0.05
25
0.15
20
0.1
0
0.1
15
15
0.05
10
-150
-100
-50
Phase !
-0.05
0
50
100
150
10
-150
-100
Phase !
-50
0
50
Objective:
•Optimize performance without exhaustive search
Challenges:
•Noisy measurement
•Flow transients
•Keeping up with operating condition change
100
150
Adaptive control used to optimize performance
Filter noise, wait for transient
to settle, adapt parameters to
improve performance
C p (t )
Measure performance
Pressure
Pressurerecovery
recovery
calculation
calculation
Extremum-seeking
Extremum-seeking
algorithm
algorithm
Adjustable parameters
Multi-frequency
Multi-frequency
forcing
forcing
Speaker command U (t ) $
N
% A sin(2"f t # ! )
i $1
Ai , f i ,! i , i $ 1..N
Excite multiple vortices,
explore their interactions
i
C & (t ) $ const
i
i
Automatic Control Parameter Tuning to Optimum Values
On-line optimization of pressure recovery using extremum-seeking algorithm demonstrated.
Optimal pressure recovery
reached & held
Transient behavior
Dynamics determines
transient time!
0.4
0.2
Pressure
recovery
0
0
10
20
30
40
50
Optimal pressure recovery:
Cp =0.21
optimal control parameters
f=31Hz
! =60
60
60
50
40
Frequency
20
40
0
10
20
30
40
50
60
200
Control
phase
0.05
0.15
0.1
35
Frequency
0
45
0.2
30
25
20
0
0.15
0.1
0.1
15
0.05
-200
10
-150
0
10
20
30
Time in seconds
40
50
60
-100
-50
0
50
Phase
•Mean pressure recovery, control frequency, and phase in four independent adaptive control experiments.
•The control frequency and phase initialized away from the optimal values.
100
150
Autonomous Vehicle Control
via Extremum Seeking
Antranik A. Siranosian and Miroslav Krstic
Model
Unicycle
Unicycle Dynamics
x = v cos(θ )
y = vsin(θ )
θ = ω
Unicycle Inputs
v, ω
System is linearly Uncontrollable (from inputs v,w) and
Unobservable (from output f(x,y) at its peak)
2
Control Inputs
Unicycle
3
Block Diagram
Unicycle
4
Simulation Results – Trajectory
Unicycle
5
Simulation Results – ES Value
Unicycle
6
BEAM MATCHING IN
PARTICLE ACCELERATORS
Eugenio Schuster
Complex Control Systems Laboratory
Mechanical Engineering and Mechanics
LEHIGH
U N I V E R S I T Y
Workshop: Real Time Optimization by Extremum Seeking Control
American Control Conference 2005
June 7, 2005
PARTICLE ACCELERATORS
The Spallation Neutron Source (SNS), being built in Oak Ridge,
Tennessee, by the U.S. Department of Energy with a cost of $1.4 billion,
is the most intense accelerator-based neutron source in the world.
SPALLATION NEUTRON SOURCE
BEAM MATCHING CHANNEL
0
L
z
xini
Drift Lens Drift Lens Drift Lens Drift Lens Drift
#1
#2
#3
#4
x fin
) X2
2K
X (( & # ! z " X &
& 3 %0
X 'Y X
) Y2
2K
Y (( ' # ! z "Y &
& 3 %0
X 'Y Y
# !z "
$# 1
$# 3
L
$# 2
$# 4
Lq
Ld
z
BEAM MATCHING OPTIMIZATION
0
L
z
xini
Drift Lens Drift Lens Drift Lens Drift Lens Drift
#1
#2
#3
#4
x fin
xini
/ X fin ,
/ X ini ,
/ X tar ,
-X ( *
-X ( *
-X ( *
fin
ini
* , x % - tar *
* , x fin % %- Y fin * tar - Ytar *
- Yini *
- ( *
- ( *
- ( *
Y
Y
fin
. ini +
. Ytar +
.
+
J % k1 J 1 ' k 2 J 2 ' k 3 J 3
J 1 % K X ! X fin & X tar " ' K Y !Y fin & Ytar "
2
2
( " ' K dY !Y fin
( & Ytar
( "
J 2 % K dX ! X (fin & X tar
2
J3 %
L
2o
0
2
1
w ! z " K iX ! X ( z ) & X des ( z ) " ' K iY !Y ( z ) & Ydes ( z ) " dz
2
2
BEAM MATCHING OPTIMIZATION
#*
# !k "
7
J*
Plant
J !# !k ""
J !# "
#^ !k " & 3
q &1
a cos !4 k " Low-Pass
Filter
8 !k "
J f !k " q & 1
6
q'h
b cos !4 k & 5 " High-Pass
Filter
J f !k " % & hJ f !k & 1" ' J !k " & J !k & 1"
8 !k " % J f !k "b cos !4 k & 5 "
#ˆ !k ' 1" % #ˆ !k " & 38 !k "
# !k ' 1" % #ˆ !k ' 1" ' a cos !4 ( k ' 1) "
BEAM MATCHING OPTIMIZATION – 4D
TERMINAL CONSTRAINTS ONLY
K X ! 2000 , KY ! 1000 , K dX ! 1, K dY ! 1, K iX ! K iY ! 0 ,
k1 ! 1, k2 ! 1, k3 ! 0.
Cost Function
a1,2 ,3 ,4 ! )2.25 2 1.75 1.5* if - 40dB , J
−10
−15
a1,2 ,3 ,4 ! )0.25 0.75 0.5 0.5* if -50dB , J + -40dB
−20
a1,2 ,3 ,4 ! )0.1 0.25 0.1 0.1* if -55dB , J + -50dB
J [dB]
−25
−30
a1,2 ,3 ,4 ! )0.05 0.05 0.05 0.05* if -60dB , J + -55dB
−35
a1,2 ,3 ,4 ! )0.01 0.01 0.025 0.025* if J + -60dB
−40
−45
−50
−55
50
100
150
200
250
300
350
400
450
500
Iteration
Parameter Evolution
Beam profile for estimated θ
−3
50
40
10
5
0
4.5
30
x 10
X
Y
Xtar
Ytar
4
−10
20
3.5
−20
10
θ1
θ2
θ3
θ4
0
−10
−20
J [dB]
Filtered θ
xtar
' 0 .001094 $
' 0 .001070 $
% ( 0 .007864 "
% ( 0 .006730 "
" x fin ! %
"
!%
% 0 .003290 "
% 0 .003289 "
%
"
%
"
& 0 .011726 #
& 0 .011034 #
3
−30
2.5
−40
2
1.5
−50
−30
1
−60
−40
−50
50
100
150
200
250
300
Iteration
350
400
450
500
−70
0
0.5
5000
10000
Case Number
15000
0
0
0.2
0.4
0.6
z
0.8
PID Tuning using Extremum
Seeking
Extremum Seeking Tuning Scheme
Step
function
r(t)
!"#$%#&"&'()%*+
Cr #" k $
!
!
-
u(t)
J("k)
y(t)
G
C y #" k $
"k
Extremum
Seeking
Algorithm
,%'-.+$+()%*+
1 " 5s
G4 ( s ) #
(1 ! 10 s )(1 ! 20s )
a) Evolution of Cost Function
b) Evolution of PID Parameters
c) Step Response of output
d) Step Response of controller