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 moasew@unimelb.edu.au,manziec@unimelb.edu.au, mjbrear@unimelb.edu.au

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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Est

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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