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Extremum seeking control
Dragan Nešić
The University of Melbourne
Acknowledgements: Y. Tan, I. Mareels, A. Astolfi, G. Bastin, C. Manzie; A. Mohammadi; W. Moas Australian Research Council.
Outline
Motivating examples Background Ad hoc designs Black box:
- Problem formulation- Systematic design
Gray box:- Problem formulation- Systematic design
Conclusions & future directions
A Prelude
This is an approach for online optimisation of the steady-state system behaviour.
A standing assumption is that the plant model or the cost is not known.
The controller finds the extremum in closed-loop fashion.
Motivating examples
Continuously Stirred Tank (CST) Reactor
Substrate Product
u=Vol. flow rate Performance output y:
Productivity JP
Yield JY
Inflow Outflow
Overall JT
J T := ¸J P +(1¡ ¸)J Y ; ¸ 2 (0;1)
Single enzymatic reactionMichaelis-Menten Kinetics
Productivity and yield Total cost
is typically unknown!!
In steady-state, we would typically want to operate around u¤J T (¹u)
G. Bastin, D. Nešić, Y. Tan and I. Mareels, “On extremum seeking in bioprocesses with multivalued cost functions”, Biotechnology Progress, 2009.
Raman amplifiers
Fibre span
Power sensors
Pump lasers
u=laser power
P.M. Dower, P. Farrell and D. Nešić, “Extremum seeking control of cascaded optical Raman amplifiers”, IEEE Trans. Contr. Syst. Tech., 2008.
CostPerformance output y:
•Spectral flatness (equalization)• Desired power
¸
p
Pi (pi ¡ pd)2
Other engineering examples
Plant Performance output
Turbine Generated power
Solar cell Generated power
Variable cam timing Fuel consumption
Tokamak Reflected power during Lower Hybrid (LH) plasma heating experiments
Non-holonomic vehicles Distance from a source of a signal
Paper machine Retention of fines and fibers in the sheet
Ultrasonic/Sonic Driller/Corer Distance from resonance
Human Exercise Machine The user’s power output
ABS Magnitude of friction force
Examples from biology
E. Coli bacteria search for food in a similar manner to an extremum seeking algorithm (M. Krstic et al).
Some fish search for food in a similar manner to extremum seeking (M. Krstic et al).
k
kv
Background
Classification of approaches
NLP based ESC [Popović, Teel,…]
Adaptive ESC [Krstić, Ariyur, Guay, Tan, Nešić,…]
Deterministic Stochastic
Adaptive ESC [Krstić, Manzie,…]
NLP based ESC[Spall,..]
Also continuous-time versus discrete-time.
Brief history (deterministic):
1922 1950 2000
Firs
t ESC
?
1960 1970 2009
Vibra
nt re
sear
ch a
rea
Man
y ne
w s
chem
es p
ropo
sed
Espec
ially
Ada
ptiv
e
Firs
t loc
al s
tabi
lity
resu
lt fo
r ada
ptiv
e ESC
Syste
mat
ic d
esig
n di
scre
te-ti
me
NLP
.
Syste
mat
ic d
esig
n ad
aptiv
e
Schem
es.
Beginning Ad-hoc designs Rigorous analysis and design
Åströ
m &
Witt
enm
ark:
“one
of t
he m
ost
prom
isin
g ad
aptiv
e co
ntro
l tec
hniq
ues”
.
1995
Ad hoc adaptive designs
Adaptive ESC [Krstić & Wang 2000]
µ
asin(! t)
Ks
_x = f (x;u)
y = h(x)
y
+ x
asin(! t)
Extremum seeking controller
Wh(s)Wl(s)
u
Parameters:
a; K ; !
Static scalar case (gradient descent)
µ
u= µ+asin(! t)
asin(! t)
Ks
y
+ xµ
sin(! t)
Extremum seeking controller
a; K ;!Parameters:
Y. Tan, D. Nešić and I. Mareels, “On non-local stability properties of
extremum seeking control”, Automatica, 2006.
_x = f (x;u)
y = h(x)
Comments
Many similar adaptive algorithms proposed. Case-by-case convergence analysis. No clear relationship with optimization.
A unifying design approach is unavailable. A unifying convergence analysis is missing.
A unifying approach exists for another class of schemes [Teel and Popovic, 2000].
Black Box Approach
Problem formulation (black box)
Extremum Seeking
Controller
Assumption 1:
- Q(.) has an extremum (max)
- Q(.) is unknowny=Q(u) yu _x = f (x;u)
y = h(x)Dynamic case:
Problem:
Design ESC so that limsupt! 1 jy(t) ¡ y¤j ¼0
y¤ := Q(u¤) ¸ Q(u); 8u
9 (̀¢) ) 0 = f ( (̀u);u)
Q(u) := h± (̀u)
Assumption: u(t) ´ ¹u =) y(t) ! Q(¹u)
Systematic design(derivatives estimation)
D. Nešić, Y. Tan, W. Moas and C. Manzie, “A unifying approach to extremum seeking:adaptive schemes based on derivatives estimation”, IEEE Conf. Dec. Contr. 2010.
Continuous optimization (offline)
y=Q(u)
• No inputs & outputs
• Q(.) is known, so all derivatives of Q(.) are known
_µ= F (DN (µ)) limt! 1 jµ(t) ¡ u¤j = 0
DN (u) := [Q(u) DQ(u) D2Q(u) :: : DNQ(u)]
Examples
Gradient method
Continuous Newton method
_µ=DQ(µ) :
_µ= ¡ DQ(µ)D 2Q(µ) :
Extremum seeking (online)
y=Q(u)y
• Inputs & outputs available
• Q(.) is unknown
=µ+asin(t)
_µ= ²F (dDN (µ))
limsupt! 1 jµ(t) ¡ u¤j ¼0
dDN Derivativesestimator
asin(t)
+
• a, !L, ² are positive controller parameters
! L
u
Systematic design (use the previous block diagram)
Step 1: Choose an optimization scheme.
Step 2: Use an estimator for DN Q(¢).
Step 3: Adjust the controller parameters.
Estimator design
Estimating DQ(µ)
y=Q(u)yu= µ+asin(t)
! Ls+! L
£sin(t)
limt! 1 »1(t) ¼ a2DQ(µ)
»1
dDQ(µ) = 2a»1
Analysis
Q(µ+asin(t)) sin(t) ¼Q(µ) sin(t) +aDQ(µ) sin2(t) + a2
2 D2Q(µ) sin3(t)
where µ is assumed constant.
Average the right hand side of the model.
_»1 ¼¡ ! Lh»1 ¡ DQ(µ)a2
i
Model of the system:
_»1 = ¡ ! Lh»1 ¡ Q(µ+asin(t)) sin(t)
i
limt! 1 »1(t) ¼ a2DQ(µ)
Estimating D2Q(µ)
y=Q(u)yu= µ+asin(t)
! Ls+! L
£sin2(t)
limt! 1 »2(t) ¼ 12Q(µ) +
3a2
16 D2Q(µ)
»2
! Ls+! L
»0
limt! 1 »0(t) ¼Q(µ) + a2
4 D2Q(µ)
dD2Q(µ) = 8a2 (2»2 ¡ »0)
Higher order derivatives
y=Q(u)yu= µ+asin(t)
! Ls+! L
£sin(t)
! Ls+! L
g(a;»0; : : :;»N )
...
dDN (µ)
! Ls+! L
£sin(t)
......
»0
»1
»N
Convergence analysis
Model of the overall system
_̂µ = ²! L F (g(a;»0; : : : ;»N ))_»i = ¡ ! L (»i ¡ ³i (t;µ;a)); i = 0;1;: : : ;N
µ = µ̂+asin(t)
³i (t;µ;a) = Q(µ+asin(t)) sini (t)
!L, ² and a are controller parameters that need to be tuned to achieve appropriate convergence properties.
Slow:
Fast:
Assumption 1 (global max)
There exists a global maximum
DQ(µ) = 0 ( ) µ= µ¤
D2Q(µ¤) < 0
Assumption 2 (robust optimizer)
The solutions of
satisfy
for sufficiently small w(t).
_µ= F (DNQ(µ) +w(t))
limsupt! 1 jµ(t) ¡ µ¤j ¼0
Theorem
Suppose Assumptions 1-2 hold. Then
8(¢ ;º); 9(! ¤L ;a¤)
+
8! L 2 (0;! ¤L ); a 2 (0;a¤); 9²
+
j(µ(t0) ¡ µ¤;»(t0))j · ¢
+
limsupt! 1 j»(t) ¡ ¹ (µ(t);a)j · º;
limsupt! 1 jµ(t) ¡ µ¤j · º
Tuning guidelines
Geometrical interpretation
µ
»(»0;µ0)
Fast transient (estimator)
»= ¹ (µ;a)
Slow transient optimization
» ! L
» ²! L
lim supt! 1
jµ(t) ¡ µ¤j · º =) lim supt! 1
jy(t) ¡ y¤j · º1
B¢
Exist !L, ², a
Bº
µ¤
For any ¢, º
Comments
A systematic design approach proposed. Rigorous convergence analysis provided. Controller tuning proposed in general. Dynamic plants treated in the same way. Multi-input case is treated in a similar way. Averaging and singular perturbations used. Tradeoffs between the domain of attraction,
accuracy and speed of convergence!
Bioreactor example
All our assumptions hold – gradient method used.
Gray Box Approach
Problem formulation (gray box)
Extremum Seeking
Controller
Assumption 1:
- Q(.,p) has an extremum (max)
- Q(.,.) is known; p is unknowny=Q(u;p) yu _x = f (x;u;p)
y = h(x;p)Dynamic case:
Problem:
Design ESC so that limsupt! 1 jy(t) ¡ y¤j ¼0
y¤ := Q(u¤;p) ¸ Q(u;p); 8u
9 (̀¢;p) ) 0 = f ( (̀u);u;p)
Q(u;p) := h( (̀u;p);p)
Assumption: u(t) ´ ¹u =) y(t) ! Q(¹u;p)
Systematic design(parameter estimation)
D. Nešić, A. Mohamadi and C. Manzie, “A unifying approach to extremum seeking:adaptive schemes based on parameter estimation”, IEEE Conf. Dec. Contr. 2010.
Extremum seeking (online)
yu= µ+asin(t)
• Inputs & outputs available
• p is unknown
y=Q(u;p)
_µ= ²F (DN (µ; p̂))
limsupt! 1 jµ(t) ¡ u¤j ¼0
p̂Parameterestimator
asin(t)
+
• a, !L, ² are controller parameters
Comments
Similar systematic framework in this case. Similar convergence analysis holds. Classical adaptive parameter estimation
schemes can be used. Dynamic plants dealt with in the same way. Persistence of excitation is crucial for
convergence. Tradeoffs between domain of attraction,
accuracy and convergence speed.
Example
Consider the static plant:
We used the continuous Newton method.
Classical parameter estimation used.
Values p1=9 and p2=8 used in simulations.
y= u21+p1u1+p2u22
Simulations
Performance output Control inputs Parameters
Final remarks
Several tradeoffs exist; convergence slow. Many degrees of freedom: dither shape,
controller parameters, optimization algorithm, estimators.
Some global convergence results available (similar to simulated annealing).
Summary
A systematic design framework presented for two classes of adaptive control schemes.
Precise convergence analysis provided. Controller tuning and various tradeoffs
understood well. Applicable to a range of engineering and non-
engineering fields.
Future directions
Tradeoffs: convergence speed, domain of attraction and accuracy.
Various extensions: non-compact sets, global results, non-smooth systems, multi-valued cost functions.
Schemes robust although no formal proofs. Tailor the tools to specific problems. Exciting research area.
Thank you!