Newton-Like Extremum-Seeking Part II: Simulations and Experiments
William H. Moase, Chris Manzie and Michael J. Brear
Abstract— In the first instalment of this paper, a Newton-likeextremum-seeking (ES) scheme was developed for applicationto problems involving optimisation of plants for which thecurvature of input-output relationship is operating conditiondependent. Strong operating condition dependence of the plantmap curvature can be seen, for example, when using a phase-shift controller to reduce the limit-cycle pressure oscillations ina premixed gas turbine combustor experiencing thermoacousticinstability. In this paper, the behaviour of Newton-like ES isfurther explored in simple numerical simulations before beingexperimentally demonstrated in a phase-shift controller for thereduction of thermoacoustic oscillations in a model premixedcombustor.
I. INTRODUCTION
Consider a plant with an output, y, which is an unknown
function of an input, �. If perturbation based ES is used to
tune � in order to minimise y, then the convergence rate and
stability of the scheme about the minimum is typically very
sensitive to the curvature of y(�). In applications where the
curvature of y(�) is operating condition dependent, changes
in the operating condition may result in either an undesirable
reduction of the convergence rate or destabilisation of the ES
scheme. As discussed in [1], this sensitivity to the curvature
of y(�) arises from the use of a regular gradient descent
adaptation law. In [1] a perturbation based ES scheme using
a Newton-like step was developed, and it was analytically
shown that the local convergence and stability properties of
the scheme are curvature-independent. However, the influ-
ence of various parameters used in Newton-like ES remains
largely unexplored. Furthermore, Newton-like ES has not
been demonstrated either numerically or in an engineering
application.
In Section II of this paper, the Newton-like ES scheme
given in [1] is repeated, and the selection of some key
parameters is discussed. In Section III, the behaviour of
Newton-like ES is further explored in numerical simulations.
The simulations demonstrate the influence of the adaptation
gain as it deviates from the ‘small’ limit assumed in [1]. The
behaviour of the scheme is investigated for different, fixed
dither signal amplitudes as well as amplitudes dynamically
scaled using the dither signal amplitude schedule (DSAS) de-
veloped in [1]. Simulations are also performed to investigate
the influence of measurement noise and plant dynamics.
In Section IV, Newton-like ES is demonstrated in an
experimental application: the reduction of thermoacoustic
This research was partially supported under Australian Research Coun-cil’s Discovery Projects funding scheme (project number DP0984577).
W. H. Moase, C. Manzie and M. J. Brear are with theDepartment of Mechanical Engineering, The University ofMelbourne, 3010, Victoria, Australia [email protected],[email protected], [email protected]
oscillations in a lean premixed combustor. The use of lean,
premixed combustion in gas turbines is now widespread due
to their low NOx emissions. Such systems are, however,
susceptible to a phenomenon called thermoacoustic instabil-
ity, which occurs as a result of unstable coupling between
the combustion chamber acoustics and the flame. It can lead
to large amplitude pressure oscillations within a combustor
at frequencies in the hundreds of hertz. These pressure
oscillations can result in unacceptably large noise levels,
flame blow-out, reduced performance and fatigue failure of
the combustor walls.
In [2], a linear proportional valve was used to harmonically
perturb the mass flow-rate of fuel to an industrial premixed
combustor. Let y be the amplitude of the pressure fluctuations
and � be the phase of the fuel addition with respect to
the pressure oscillations. It was shown in [2] that y(�) is
a smooth function with a unique minimum (modulo 2πradians). An ES scheme was used to tune � in order to
minimise y, however, it was found that the adaptation gain
had to be varied by a factor of approximately six between
high- and low-power operating conditions.
The necessity to tune the adaptation gain in [2] arose from
the use of a regular gradient descent adaptation law on a
plant where the curvature of y(�) was heavily dependent
upon the operating condition. Because in [1], Newton-like
ES was shown to achieve local convergence rates that are
independent of the curvature of y(�), then it would seem
that it is a sensible choice of scheme for the minimisation
of thermoacoustic oscillations in a premixed gas turbine
combustor. In Section IV, the proposed scheme is experi-
mentally demonstrated to minimise the thermoacoustic limit-
cycle pressure oscillations in a model premixed combustor
and its performance is compared to an ES scheme similar to
that used in [2].
II. PROPOSED SCHEME AND PARAMETER
SELECTION
Fig. 1 shows a schematic of the proposed scheme. The
plant is subject to the input,
� = �0 + a sin (!�t) , (1)
where !� > 0. The quantity �0 is progressed according to
the adaptation law:
d�0dt
=
⎧⎨⎩−k�!� y′0
/y′′0 if
∣∣∣y′0∣∣∣ < �aminy′′0 ,
−k�!��amin sgn(y′0
)otherwise,
(2)
where �, k�, amin > 0 are dimensionless quantities, y′ =
dy/d�, y′′ = d2y/d�2 and ( ) denotes an estimate. As
Joint 48th IEEE Conference on Decision and Control and28th Chinese Control ConferenceShanghai, P.R. China, December 16-18, 2009
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discussed in [1], k� should be a relatively small number
in order to provide stability. The gradient and curvature
estimates are gained from:
dx
dt= !�Ax+ !�L (y − y) , (3a)
y = Cx, (3b)
ay′0 = C′x, (3c)
a2y′′0 = C′′x, (3d)
where
x =
⎛⎜⎜⎜⎜⎝
y0 +14a
2y′′0y′0a sin(!�t)y′0a cos(!�t)y′′0a
2 sin(2!�t)y′′0a
2 cos(2!�t)
⎞⎟⎟⎟⎟⎠
, A =
⎛⎜⎜⎜⎜⎝
0 0 0 0 00 0 1 0 00 −1 0 0 00 0 0 0 20 0 0 −2 0
⎞⎟⎟⎟⎟⎠
,
C =(1 1 0 0 − 1
4
),
C′ =
(0 sin(!�t− �1) cos(!�t− �1) 0 0
),
C′′ =
(0 0 0 sin(2!�t− �2) cos(2!�t− �2)
),
L ∈ ℝ5 is a non-dimensional gain vector and (�1, �2) ∈ ℝ
2.
In [1] it is shown that the proposed scheme can be made
stable when (A−LC) is Hurwitz. Even under this restriction,
there is a substantial degree of freedom in choosing the
gradient estimator gain vector, L. The ‘best’ choice of
L will be application dependent and is likely to involve
consideration of the noise-sensitivity, stability margins, and
transient response of the estimator. The value of L selected
for this study is solely chosen to demonstrate the capabilities
of the proposed ES scheme. Referring to the definition of x
given in (3a) and (3b), then the estimator states, x2 and x3,
are expected to oscillate at !� rad/s. Similarly, x4 and x5
are expected to oscillate at 2!� rad/s. Ideally, information at
other frequencies would be attenuated. This can be enforced,
to some extent, by considering the transfer function from the
plant output to the estimator states,
G (s) = (sI−A+ LC)−1
L, (4)
where s = s/!� and ( ) denotes a non-dimensional quantity.
The oscillating states can be made most sensitive to the
frequencies at which they are intended to oscillate (compared
to nearby frequencies) if the following conditions hold:
d
d!∣G2 (i!)∣ =
d
d!∣G3 (i!)∣ = 0, at ! = 1, (5a)
d
d!∣G4 (i!)∣ =
d
d!∣G5 (i!)∣ = 0, at ! = 2, (5b)
where ! = !/!� and Gn denotes the n-th element in G.
Solving (5a) and (5b) gives
L =(1 + k 1
2 (1− k) 12 (1− k) 4k −4k
)T, (6)
for some k ∈ ℝ. This choice of L achieves 20 dB/dec roll-off
in ∣G2∣, ∣G3∣, ∣G4∣ and ∣G5∣ at high- and low-frequencies. In
order to ensure that (A − LC) is Hurwitz, then k ∈ (0, 1).Increasing k within this range increases the attenuation in
∣G2∣ and ∣G3∣ for frequencies outside of ! = 1 but decreases
Gradient
estimator
+∫
Plant
θ
y
×sin(ωθt)
×
Adaptation
law
DSASa
θ0
ay′0
a2y′′0
Fig. 1. Basic schematic of proposed ES.
the attenuation in ∣G4∣ and ∣G5∣ for frequencies outside of
! = 2. A choice of k ≈ 0.271 achieves a reasonable transient
response by minimising max5n=1 Re(pn), where pn are the
poles of G. For all tests, the state vector x is initialised to
zero.
The dither signal amplitude, a, may be constant (a =amin) or tuned using DSAS:
da
dt= ka!� (�− a) , (7a)
� = amin + amin�
(1
amin!�k�
d�0dt
), (7b)
where ka > 0, � : ℝ → ℝ>0 and �(z) ≤ ∣z∣ for all z and
some ≥ 0. In this study,
� (z) = max ( ∣z∣ − 1, 0) . (8)
By substituting (8) into (7a), then
� = max
(∣∣∣∣
k�!�
d�0dt
∣∣∣∣ , amin
). (9)
In other words, the schedule attempts to scale a with ∣d�0/dt∣when operating far from the extremum but prevents a from
dropping below amin.
III. SIMULATION RESULTS
A. Noiseless plant with no dynamics or DSAS
Let ( )∗
denote a quantity evaluated at � = �∗ where �∗ is
the input which minimises y. The behaviour of the scheme
is first considered for the most simple of scenarios — a
noiseless plant with y = 12y
′′
∗�2 and no dynamics. Before the
Newton-like ES scheme is tested in a closed-loop simulation,
the performance of the gradient estimator is investigated in
an open-loop test with �0 and a set to constants. Fig. 2
shows that the normalised errors in the gradient and curvature
estimates converge towards zero, and are negligible within a
few cycles of the dither signal. Although results for y′′∗= 1
and 10 are shown, the curves are indistinguishable. Similarly,
the normalised errors are independent of a, although it is
important to note that �0 ∝ a for this particular example.
In Fig. 3, the loop is closed by progressing �0 according
to (2), while the dither signal amplitude, a, remains constant.
Fig. 3 shows the progression of �0/amin. The parameter � has
been set to a large value in order to minimise the occurrence
of saturation in the rate of change of �0. For k� = 10−3, �0
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imate
erro
rs100
50
0
− 500 1 2 3 4 5 6
ωθt/(2π)
Fig. 2. Error in gradient and curvature estimates. Open loop test withy = 1
2y′′∗�2, a = amin, k = 0.271, �0 = −ae3 for all t, y′′
∗∈ {1, 10}
and a ∈ 1, 10. a(y′0− y′
0)/y′′
∗(dashed) and (y′′
0− y′′
0)/y′′
∗(solid).
0 3 6 9
0
3
kθωθt
−3
−6
0.001 0.01 0.02 0.03
ln|θ
0/a|
Fig. 3. Progression of �0 with y = 1
2y′′∗�2, a = amin, k = 0.271,
� = e3, �0(0) = −ae3, �1 = �2 = 0, y′′∗
∈ {1, 10} and a ∈ {1, 10}.k� given on contours.
exponentially converges to zero with a time constant equal
to 1/(k�!�), except during the early stage of the simulation
when the gradient estimator transients are settling. Some
deviation from this behaviour is observed for larger k�. As
observed for the open-loop tests, the normalised behaviour
of the scheme is independent of both y′′∗
and a.
Some of the results observed for the parabolic plant map,
such as a-independence of �0/a and arbitrarily accurate
convergence of (�0, y′0, y′′
0 ) → (�∗, y′
0, y′′
0 ), will not be
observed for higher order plant maps. Fig. 4 shows the
behaviour of the scheme with,
y = 12y
′′
∗�2
(1 + 1
3�). (10)
As with the parabolic map, changing y′′∗
has no effect on
the progression of �0. However, it should be noted that,
under (10), y(3)∗ scales with y′′
∗, where y(3) = d3y/d�3. If
y′′∗
was changed and y(3)∗ was held constant, then different
behaviour would be observed. For the plant map given
in (10), y′′ < 0 for � < −1. In the simulations, �0 = −1.5at t = 0, so y′′0 is initially negative. If the scheme was to
progress �0 according to a Newton step, then the scheme
would seek the local maximum at � = −2 rather than the
desired local minimum at � = 0. For this reason, (2) instead
forces �0 to initially follow a sign-of-gradient descent. It
0 5 10 15kθωθt
0 5 10 15
0
−4
−8
−12
0.04
0
5
−5
−10
0.08
DSAS
DSAS
ln|θ
0|
ln((
a−
am
in)/
am
in)
Fig. 4. Progression of �0 and a with y = 1
2y′′∗�2(1 + 1
3�), k = 0.271,
k� = 0.03, �0(0) = −1.5, y′′∗∈ {1, 10}, �amin = 1, and �1 = �2 = 0.
a = amin (with a given on contours), and DSAS with a(0) = amin =0.01, ka = k� , and = 0.5.
is not until the intermediate stages of the simulation that
�0 follows an approximated Newton step. Eventually �0converges to a non-zero value because y(3) is non-zero.
B. Influence of dither signal amplitude
According to Theorem 1 given in [1], for constant
dither signal amplitudes, �0 converges to an O(a2y(3))-neighbourhood of �∗. Therefore, small a are desirable for
accurate convergence of �0 → �∗ and y → y∗. Furthermore,
consider the evolution equation (adapted from [1]) for the
state error vector, x = x− x:
dx
dt= !� (A− LC) x+ Lℎ−
∂x
∂�0
d�0dt
−∂x
∂a
da
dt(11)
where ℎ consists of terms of O(a3y(3)). When a and �0are constant or changing very slowly, then the equilibrium
solution of (11) will give x of O(ℎ), which is consistent
with Theorem 1 from [1]. However, when �0 is far from �∗,
the second last term in (11), due to changing �0, can have a
significant influence on the solution to (11). ∂x/∂�0 consists
of O(y′0) and O(ay′′0 ) terms. In contrast, the terms in the
state vector used for estimating the gradient and curvature
are O(ay′0) and O(a2y′′0 ) respectively. It follows that large
a are desirable for accurate gradient and curvature estimates
when operating away from the extremum. Thus, in selecting
a, there is trade-off between performance near and far from
the extremum. This is demonstrated in Fig. 4 which shows
that decreasing a improves the accuracy of convergence of
�0 → �∗, but does so at the cost of slower convergence during
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TABLE I
CASES CONSIDERED IN FIG. 5.
� �1 �2 k�/k�A 0 0 0 1B π/5 0 0 cos(π/5)/ cos(2π/5)C π/5 π/5 2π/5 1
0 3 6 9
0
3
kθωθt
ln|θ
0/a|
−3
−6
Fig. 5. Progression of �0 with output dynamics Fo(s) = exp(−s�),static map f = 1
2�2 and ES parameters: k = 0.271; k� = 0.001; a =
amin = 1; �0(0) = − exp(3); and � = exp(3). Cases A (dotted, butbarely distinguishable from case C), B (dashed), and C (solid).
large ∣�0 − �∗∣ (even though !�k�amin�, and therefore the
maximum allowable value of ∣d�0/dt∣, is held constant).
Fig. 4 also demonstrates the influence of DSAS. As shown,
a is initialised to amin, but is rapidly increased by the DSAS
during the early stages of the simulation. During the later
stages of the simulation, when it is not necessary to rapidly
vary �0, the DSAS decreases a. As a result of using DSAS,
the scheme simultaneously achieves faster and more accurate
convergence to the extremum than either of the test cases
using statically defined dither signal amplitudes.
C. Influence of output dynamics
Consider a plant with a transport lag, Fo(s) = exp(−s�),at the output of simple parabolic map f(�) = 1
2�2. According
to Theorem 2 in [1], then the local convergence of �0 to
�∗ can be described by a first order, linear, time-invariant
transfer function with a single pole at s = −k�, where
k� = k�cos (�1 − �)
cos (�2 − 2�).
Table I describes three cases considered. For each case, Fig. 5
shows that �0 locally converges to �∗ at an exponential rate
with a time-constant equal to 1/(k�!�), as predicted by
Theorem 2. It is also of interest to note that the non-local
behaviour in case B is quite different to that of the other two
cases. This is because the larger value of k� in case B causes
the adaptation law to initially saturate ∣d�0/dt∣.
D. Influence of measurement noise
The influence of measurement noise is briefly investigated
by adding white noise, �, to the plant output, y. Estimation
of the plant map curvature requires measurement of O(a2)effects on the plant output, whereas estimation of the gradient
3 6 9 12 15
0
0.1
0.2
kθωθt
θ 0/a
min
-0.1
-0.2
Fig. 6. Progression of �0 with y = 1
2�2(1 + 1
3�), k = 0.05, ka =
k� = 0.03, �0(0) = −1.5, �amin = 1, a(0) = amin = 0.1, = 0.5and �1 = �2 = 0. � = 0 (dashed) and �2rms = 10[y(amin sin(!�t))]
2
rms
(solid).
requires measurement of O(a) effects. It follows that it is
considerably more important to filter out noise on G4 and
G5 than on G2 and G3, so k is set to 0.05. The noise
power is set an order of magnitude larger than the power
of y(amin sin(!�t)), the fluctuating part of the output when
�0 = �∗ and a = amin. As shown in Fig. 6, �0 fluctuates
about �∗ with an amplitude of about 0.2amin. The dither
signal amplitude (not shown) is relatively unaffected by the
noise, and has a response similar to that shown in Fig. 6.
Therefore the contribution of non-zero �0 − �∗ to y − y∗ is
an order of magnitude smaller than that of the dither signal.
IV. EXPERIMENTAL RESULTS
As discussed in the introduction, ES has previously been
used in the suppression of thermoacoustic instability in
a premixed, gas turbine combustor [2]. However, in this
application, the effectiveness of a traditional gradient descent
adaptation law has been hindered by the variation in plant
map curvature between different combustor power settings.
Newton-like ES is unlikely to have this limitation since it
has been demonstrated to have convergence properties that
are independent of plant map curvature. In this section,
Newton-like ES is used to tune a phase-lag controller for
the suppression of thermoacoustic instability in a laboratory
premixed combustor and its performance is compared to an
ES scheme similar to that used in [2].
Fig. 7 shows a simplified sectional view of the combustor
used in the study. The majority of the rig is constructed
from 50 mm diameter (inner) stainless steel pipe. Air is
supplied to the combustor from a 60 hp screw compressor.
The primary fuel (gaseous phase, liquefied petroleum gas)
is mixed with air in the upstream section of the rig before
passing through a flame trap and choke plate and eventually
entering the working section. The working section is 1 m
long and approximately axisymmetric. The flame is stabilised
by a bullet-shaped bluff body located approximately in the
middle of the working section. At the outlet of the working
section is a nozzle which contracts to a 32 mm diameter.
Further details of the rig design are provided in [3], [4].
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primaryfuel
air
flametrap
burstingdisk
chokedinlet
pressuretransducer
holderflame
secondaryfuel
injector
outletnozzle
Fig. 7. Cross-sectional view of model premixed combustor.
Frequencytrackingobserver
Computer
Extremumseekingscheme
θ
y
Combustor
Fuelinjector
Pressuretransducer
Fig. 8. Basic schematic of control system.
Figure 8 shows a basic schematic of the control system.
A Kulite WCT-312M fast response pressure transducer mea-
sures the pressure slightly upstream of the flame holder. A
Keihin on-off gaseous LPG injector (driven by a National
LM1949 injector drive controller) is mounted downstream
of the flame-holder allowing additional fuel to be injected
directly into the flame. The pressure transducer and in-
jector driver are connected to a computer (via a National
Instruments PCI-6025E) running MATLAB xPC Target. An
extended Kalman filter (EKF) acting as a frequency tracking
observer approximates the amplitude, frequency and phase
of the dominant pressure mode within the combustor. This
allows the injector to be driven by a square-wave signal
which lags the pressure fluctuations by �.
The experimentally determined relationship between � and
y is shown in Fig. 9 for two different inlet Mach num-
bers: 0.034 (operating condition A), and 0.040 (operating
condition B). In both cases, the equivalence ratio based on
the primary fuel supply is 0.9. Despite the relatively small
power difference between the operating conditions, y′′∗
in
case A is approximately twice that in case B. For a fixed
operating condition, the difference between the maximum
and minimum of y(�) is relatively small compared to that
achieved in [2]. This is largely due to the amount of noise on
y and the poor authority achievable with the on-off injector at
the frequency of the oscillations (≈ 100 Hz). Nonetheless,
the test rig serves as a suitable platform for experimental
comparison of the proposed ES scheme with one using a
regular gradient descent adaptation law.
Fig. 10(a) shows the behaviour of the proposed scheme
(without DSAS) at both operating conditions. For the pur-
pose of comparison, a similar ES scheme to that used in [2]
is also tested, and its behaviour is shown in Fig. 10(b). This
reference ES scheme uses the adaptation law,
d�0dt
= −k�!�y′0, (12)
0 90 180 270 3600.9
1.1
1.3
1.510
4
θ (deg.)
y(P
a)
×
Fig. 9. Plant maps for operating conditions A (solid) and B (dashed).Averaged measurements of y (∘) and fitted curve (line).
and a gradient estimator as given in (3a)–(3c) but with:
C′ =
(0 sin(!�t) cos(!�t)
), C =
(1 1 0
),
x =
⎛⎝
y0y′0a sin(!�t)y′0a cos(!�t)
⎞⎠ , A =
⎛⎝0 0 00 0 10 −1 0
⎞⎠ , L =
⎛⎝122
⎞⎠ .
As is evident in Fig. 10, the convergence rate of the pro-
posed ES scheme is relatively independent of the operating
condition whereas the convergence rate of the reference ES
scheme changes by a factor of approximately two between
the different operating conditions. Less predictably, both
schemes are more sensitive to noise for operating condition
B since it has a flatter corresponding plant map. This is of
particular practical significance to the proposed ES scheme.
Despite being relatively insensitive to the plant map in a
noise-free environment, the presence of noise can introduce
some amount of sensitivity to the plant map when �0 − �∗is sufficiently small.
Further details of the experiments performed on the model
combustor, including details of the EKF and justification for
the selected ES parameters are provided in [4].
V. CONCLUSIONS
Numerical simulations and experiments were performed
to further investigate the behaviour of the Newton-like ES
scheme developed in [1]. As well as supporting the theorems
developed in [1], the simulations demonstrated:
∙ Large a are desirable for allowing rapid convergence
of �0 to �∗ whereas small a are desirable for accurate
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0 5 10 15 20
20
60
100
t (s)
θ ∗−
θ 0(d
eg.)
0 5 10 15 20
20
60
100θ ∗
−θ 0
(deg
.)
(b)
(a )
−20
−20
Fig. 10. Comparison of controller behaviour at operating condition A(solid) and operating condition B (dashed) with a = amin = 30∘, !� =1 Hz and �0(0) = �∗ − 100∘. (a) shows proposed ES scheme with � = 3,k� = 5 × 10−2, k = 0.05 and �1 = �2 = 0. (b) shows reference ESscheme with k� = 5× 10−5.
convergence of � to �∗. By using DSAS, it was possi-
ble to simultaneously achieve both rapid and accurate
convergence of � to the extremum.
∙ With appropriate selection of amin, it was shown that
the influence on y − y∗ of fluctuating �0 due to noisy
estimates of y′0 and y′′0 was small compared to the
influence of the dither signal, and the influence of the
dither was small compared to the influence of the noise.
When applied to the problem of thermoacoustic limit-cycle
amplitude reduction in a laboratory premixed gas-turbine
combustor, the proposed ES scheme was demonstrated to
be less plant sensitive than an ES scheme using a more
typical gradient descent adaptation law. However, it was also
demonstrated that high process and measurement noise levels
in the experiment introduced some amount of sensitivity to
the plant map curvature once the control input had converged
to a small neighbourhood of the extremum.
This research also presents a number of areas for further
research. These include:
∙ Extension to multivariable ES. One might go about
extending the proposed ES scheme to multivariable
optimisation problems by using a vector of sinusoids
with different frequencies as the dither (as is done in [5]
for a steepest gradient descent law) and an observer to
track the magnitudes of the fluctuating components of y
corresponding to elements of the gradient and Hessian.
However, it is important to note that the number of
observer states scales with the square of the number of
inputs (in order to estimate the Hessian). Thus, such a
scheme may only be practical for a moderate number of
inputs. Rigorous analysis of such a scheme would need
to be performed and coupled with a method for selecting
the observer gains, L, in order to achieve acceptable
performance.
∙ Quantifying the influence of noise. Mathematical anal-
ysis of the effect of measurement and process noise
on the closed-loop response of the proposed scheme
would likely aid in the selection of parameters such
as amin and L. Such a study might follow an analysis
similar to that used in [6], but would be complicated
by the division of y′0 by y′′0 in the adaptation law. For
plants with independent, Gaussian measurement/process
noise, it would be worthwhile investigating the effect of
using a Kalman filter instead of a state-space observer
to estimate the gradient and curvature of the plant map
(following a similar approach to those used in [7], [8]).
∙ More general forms of the adaptation law. The proposed
adaptation law determines �0 by integrating a saturated
estimate of the Newton step. Although such an adapta-
tion law is sufficient when �∗ is a constant or step func-
tion, it may result in significant tracking errors for more
complicated dynamic behaviour of �∗. Following similar
arguments to those presented in [6], it is expected that
these tracking errors could be reduced through more
careful selection of the dynamical relationship between
�0 and the estimated Newton step.
∙ More detailed analysis of the proposed ES scheme. It
would be beneficial to more closely study the influence
of the ES parameters on the region of attraction and
convergence rate of �0 to �∗.
REFERENCES
[1] W. H. Moase, C. Manzie, and M. J. Brear, “Newton-like extremum-seeking part I: theory,” in Proceedings of the IEEE Conference on
Decision and Control, 2009.[2] A. Banaszuk, K. B. Ariyur, M. Krstic, and C. A. Jacobsen, “An adaptive
algorithm for control of thermoacoustic instability,” Automatica, vol. 40,pp. 1965–1972, 2004.
[3] P. A. Hield and M. J. Brear, “Comparison of open and choked premixedcombustor exits during thermoacoustic limit cycle,” AIAA J., vol. 46,no. 2, pp. 517–526, 2008.
[4] W. H. Moase, “Dynamics and control of thermoacoustic instability,”Ph.D. dissertation, Dept. of Mechanical Engineering, The University ofMelbourne, 2009.
[5] M. A. Rotea, “Analysis of multivariable extremum seeking algorithms,”in Proceedings of the American Control Conference, 2000, pp. 433–437.
[6] M. Krstic, “Performance improvement and limitations in extremumseeking control,” Syst. Control Lett., vol. 39, pp. 313–326, 2000.
[7] L. Henning, R. Becker, G. Feuerbach, R. Muminovic, R. King,A. Brunn, and W. Nitsche, “Extensions of adaptive slope-seeking foractive flow control,” P. I. Mech. Eng. I-J. Sys, vol. 222, pp. 309–322,2008.
[8] D. F. Chichka, J. L. Speyer, C. Fanti, and C. G. Park, “Peak-seekingcontrol for drag reduction in formation flight,” J. Guid. Control Dynam.,vol. 29, no. 5, pp. 1221–1230, 2006.
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