JORDI DILMÉ
Transcript of JORDI DILMÉ
Mod
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To my mother, because she is always with me.
To my father and to Toni, because they are always with me, too.
I
Combustion units frequently experience thermoacoustic instabilities, also known as
combustion oscillations, which are consequence of the internal coupling between acoustic
waves and unsteady heat release. Deterioration in system performance, starting by
increased emissions or higher levels of fuel consumption, may occur due to this large‐
amplitude flow oscillations and their associated pressure fluctuations and, under some
circumstances, these can be intense enough to cause structural damage on the installation.
After introducing its physical background, this acoustic phenomenon is modelled in
the time domain using a one‐dimensional linearized analytical approach and then
implemented into Simulink; thus providing a pioneering tool which models combustion
instabilities in an interactive and customizable environment. A first model reproduces the
behaviour of an acoustically excited pipe without combustion, whereas a second one
simulates the performance of a generic combustor with unsteady heat release without
mean flow. The validity of the conceived models is checked through comparison with
acoustics theory and earlier research developed in the Laplace domain. This graphical
modelling aspires to become a faster, more visual alternative to the complex current
approaches to the analysis of combustion oscillations, especially suitable for shorter
projects thanks to its simplicity of use.
As an initial approach to the interruption of combustion oscillations, feedback
control is applied to the modelled combustor. A fixed‐parameter controller is designed in
the time domain using Nyquist and Bode techniques and then implanted into the Simulink
model. Finally, the robustness of the controller to slight changes in the heat release time
delay is assessed.
ABSTRACT
II
I would like to express my sincere thanks my supervisor, Dr Aimee S. Morgans, for
her attention and advice throughout the course of this Master’s project. Her guidance was
crucial and made possible the eventual success of this work.
I would like to extend my gratitude to German Gambon and Damián Álvarez, who
provided priceless help during the first stages of the project to get familiar with the
software involved.
Finally, I am very grateful to my mother, my family and my friends. Without their
encouragement and love I would not have overcome this challenge.
JORDI DILMÉ
London, June 2011
ACKNOWLEDGEMENTS
1
Contents
ABSTRACT ……...……………………………………………………………………………………………………………………………. I
ACKNOWLEDGEMENTS ……………………………………………………………………………………………………………………. II
1. Introduction ………………………………………………………………………………………………………………………….. 3
2. Aim of the project …………………………………………………………………………………………………………………. 4
3. Energy and combustion oscillations ………………………………………………………………………………………. 5
3.1. Physical fundamentals ……………………………………………………………………………………................. 5
3.2. Acoustic analysis ……………………………………………………………………………………………………………. 6
4. Acoustically excited tube without combustion ……………………………………………………………………… 8
4.1. General description of the model and reason for modelling ………………………………………….. 8
4.2. Description of the numerical model ………………………………………………………………………………. 9
4.3. Pressure modelling ………………………………………………………………………………………………………... 10
4.4. Results and model checking …………………………………………………………………………………………… 11
5. Model combustor …………..…………………………………………………………………………………………………….. 15
5.1. General description of the model and reason for modelling ………………………………………….. 15
5.2. Description of the numerical model …………………………………………………………………............... 16
5.3. Pressure modelling ………………………………………………………………………………………………………… 19
5.4. Results and model checking …………………………………………………………………………………………… 20
6. Practical approach to feedback control ……..………………………………………………………………………….. 28
6.1. Fixed‐parameter control applied to the combustor modelled with Simulink ………………. 28
6.1.1. Controller design in Laplace domain ………………………………………………………………………. 28
6.1.2. Practical implementation of the controller into the Simulink model ………………………. 33
6.1.3. Analysis of control robustness ………………………………………………………………………………… 35
7. Future work ………………………………………………………………………………………………………………………….. 37
8. Conclusions …………………………………………………………………………………………………………………………… 38
9. References ……………………………………………………………………………………………………………………………. 39
10. Additional bibliography ……………………………………………………………………………………………………….. 40
Appendices:
A. Simulink block diagram of the acoustically excited tube without combustion
B. Simulink block diagram of the model combustor
C. Simulink block diagram of the model combustor with masked subsystems
D. Simulink block diagram of the model combustor with feedback control
E. Simulink block diagram of the model combustor with feedback control and masked subsystems
2
List of figures
Fig. 1 Control volume of perfect gas within a combustor ……………………………………………………………........ 5
Fig. 2 Schematic of the acoustically excited tube without combustion …………………............................... 8
Fig. 3 Simulink block diagram of the model (acoustically excited tube) ……………………………………………… 11
Fig. 4 Pressure amplitude response to variation of excitation frequency without mean flow …………….. 13
Fig. 5 Pressure amplitude response to variation of excitation frequency with mean flow ………………….. 13
Fig. 6 Mode shapes of pressure amplitude for the first three resonant frequencies ………………………….. 14
Fig. 7 Diagram of the combustor model …………………………………………………………………………………………….. 16
Fig. 8 Simulink block diagram of the combustor model ……………………………………………………………………... 20
Fig. 9 Stability regions of the first mode of G(s) as a function of the heat‐release time delay ……………. 22
Fig. 10 Time evolution of pressure measurements for two different values of ……………………………….. 22
Fig. 11 Stability regions of the time response of the Simulink model (max. step size of 10‐5 s.) ……………. 23
Fig. 12 Stability regions of the i first modes (n = 1…i) of G(s)……………………………………………………………….. 23
Fig. 13 (a) Stability regions of the ten first modes of G(s) obtained from Bode plot analysis ……………….. 24
(b) Stability regions of the time response of the Simulink model (max. step size of 10‐5 s.) ………. 24
Fig. 14 Stability regions of the time response of the Simulink model (max. step size of 10‐4 s.) ……………. 24
Fig. 15 (a) Stability regions of the four first modes of G(s)obtained from Bode plot analysis ………………. 25
(b) Stability regions of the time response of the Simulink model (max. step size of 10‐4 s.) ………. 25
Fig. 16 Gain and phase shift checking for k = 0 (maximum step size = 10‐5 s.) ………………………………………. 26
Fig. 17 Gain and phase shift checking for k = 2 (maximum step size = 10‐5 s.) ………………………………………. 26
Fig. 18 Gain and phase shift checking for k = 1.3 (maximum step size = 10‐5 s.) …………………………………… 27
Fig. 19 Gain and phase shift checking for k = 3.3 (maximum step size = 10‐5 s.) …………………………………… 27
Fig. 20 Generic arrangement for feedback control of combustion oscillations …………………………………….. 28
Fig. 21 Structure of the negative feedback closed‐loop control system ……………………………………………….. 29
Fig. 22 Bode plot of the open‐loop transfer function of G(jω) for k = 0.5 ………………………………………….... 29
Fig. 23 Nyquist diagram of the OLTF from the loudspeaker input to the filtered pressure …………………… 31
Fig. 24 Bode diagram of a generic phase‐lag compensator ………………………………………………………………….. 31
Fig. 25 Nyquist diagram of the controlled system with two anticlockwise encirclements of‐1 point ……. 32
Fig. 26 Simulink block diagram of the controlled system ……………………………………………………………………… 33
Fig. 27 Comparison between pressure measurements with control OFF and ON, respectively ……………. 34
Fig. 28 Pressure time response under varying values of k ……………………………………………………………………. 35
Fig. 29 Pressure measurement with control ON for k = 2.45 ………………………………………………………………… 37
3
1. INTRODUCTION
Combustion units, from gas turbine combustors to rocket motors, frequently experience
thermoacoustic instabilities, also known as combustion oscillations or instabilities. These are
consequence of the internal coupling between acoustic waves and the combustion process itself:
unsteady heat release generates acoustic waves, these propagate along the combustor and reflect from
boundaries to get back to the combustion zone, where they generate more unsteady heat release, for
example through hydrodynamic instabilities [1,2] or local changes in the fuel‐air ratio [3]. Depending on
the phase relationship of this unsteady heat release response, the energy associated to the acoustic
waves may increase rapidly resulting in the occurrence of large‐amplitude instability. When this
instability appears, oscillation amplitudes start to increase exponentially; at some point, some non‐
linearity in the system limits these amplitudes, leading to “self‐excited oscillations” [4, 5, 6].
Deterioration in the system performance may occur due to these almost always unwanted large‐
amplitude flow oscillations and their associated pressure fluctuations and, under some circumstances,
these can be intense enough to cause structural damage on the installation.
Study and control of combustion instabilities is currently a subject of great importance, inasmuch
as their occurrence in the new generation gas turbines, in which reduced emissions of NOx are a priority,
is, at least, frequent [4]: both industrial land‐based gas turbines [7, 8, 9] and aero‐engines [10, 11] are
particularly susceptible to them. Because these gas turbine combustors are operated under lean
premixed conditions, NOx emissions are reduced, but, at the same time, this also makes combustors
especially prone to experience combustion instabilities [12, 13]. This problematic, however, is not
limited to gas turbine combustors: aeroengine afterburners [14, 15, 16], rocket motors [17], ramjets
[18], boilers or furnaces [19] are other systems particularly susceptible to it. On a laboratory‐scale, the
most common device used to reproduce and analyse this phenomenon is a simple open‐ended vertical
tube with a heat source in its lower half, known as Rijke tube [20].
The elimination of thermoacoustic instabilities is achieved by interrupting the coupling between
the acoustic waves and the unsteady heat release. Existing methods used to reach this interruption may
be classified into passive control methods, active control methods or, the most recent alternative, tuned
passive control methods.
The first ones [19, 21] are based on permanent changes and aim to achieve one of these two
objectives: either reduce the susceptibility of the combustion process to acoustic excitation through
hardware design changes, such as modifying the combustor geometry [12, 22, 23] or the fuel injection
system, or remove energy from sound waves using acoustic dampers, such as Helmholtz resonators [24]
or quarter mode tubes [25]. The main inconvenient associated to these mechanisms is that they might
be ineffective at the low frequencies at which some of the most damaging oscillations occur and,
furthermore, required changes of design are usually expensive and time‐consuming [4].
4
On the other hand, active control principle is based on introducing one or more inputs to the
system which are actively varied using an actuator [4, 5, 6]. Active control may be subdivided into open‐
loop, where the mentioned input is independent from measurements on the system, and closed‐loop
(i.e. with feedback), where an actuator modifies a parameter of the system in response to a measured
signal. The aim of closed‐loop active control is to design the control relationship between de sensor and
actuator signal such that the closed loop system is stable. The main advantage of active control,
especially closed‐loop, lies in its power to interrupt oscillations over a range of operating settings.
However, if the controller design is incorrect, it may make the instabilities even more damaging.
As the third option appears the tuned passive control, which has focused recent interest. It
consists of designing the damping devices, such as Helmholtz resonators [24], with a variable geometry
and/or variable forced flow throw them. The fundamental is simple: these dampers offer peak damping
performance at a certain frequency depending on the previously mentioned parameters, so, by tuning
them, high damping and even instability suppression may be achieved across a wider range of
frequency. In comparison to active control, bandwidth requirements are lower, as geometry/flow rate
actuation is only required when operating conditions are modified.
2. AIM OF THE PROJECT
Given the susceptibility of the listed devices to experience these undesired combustion
instabilities, the aim of this project is to model the appearance and behaviour of these large amplitude
flow oscillations in the time domain, in certain different circumstances and settings, using Simulink1
(Matlab). As well as that, the application of closed‐loop control to interrupt the coupling between
acoustic waves and unsteady heat release that gives rise to combustion oscillations is introduced and a
fixed‐parameter controller is designed and applied to the main combustor model. The idea is to provide
a useful but easy to use tool to predict, analyse and, consequently, avoid the occurrence of
thermoacoustic instabilities.
The numerical background of the developed models is based on previous modelling approaches:
Morgans, Evesque and Zhao have developed several Fortran codes in time domain that simulate
different combustor settings, under specific assumptions, whereas Stow and Dowling have implemented
more complicated codes, again in Fortran, combining both frequency and time domains.
This project represents a step forward in combustion oscillations analysis as it seeks to provide
the first tool which models this thermoacoustic phenomenon in an interactive and customizable
graphical environment. This implementation is intended to offer a faster, more visual alternative to the
complex current modelling approaches as none of them offers graphical user interface (GUI). As a result
of this, it is thought to be more suitable for short future projects and researches thanks to the simplicity
of use and the easiness with which it can be customized to satisfy users’ specific requirements.
1 Simulink is an environment for multidomain simulation and Model‐Based Design for dynamic and embedded systems.
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12
12
1. S 3.2
The left‐hand side term represents the rate of change of the sum of the kinetic and potential
energies within volume V. The first term on the right‐hand side represents the exchange of energy
between the combustion and sound waves. Following Rayleigh, when the pressure, p, and the heat
release, q, have a component which is in phase (in other words, when the phase difference lies between
‐90° and +90°), the acoustic energy tends to be increased. Finally, the second term on the right‐hand
side, the surface term, captures energy losses across the bounding surface, S, because the fluid within
this surface does work on the surroundings [4].
Furthermore, it can be observed that acoustic disturbances will grow in magnitude if their gain
from combustion is larger than the energy losses across boundaries. This can be formulated in the
following inequality, which is the generalized form of the Rayleigh’s criterion:
1 . S 3.3
where the overbar denotes an average over one period of the acoustic oscillation. If the inequality is
satisfied, acoustic waves’ amplitude will increase until non‐linear effects limit its growth.
If these nonlinearities appear primarily in the heat release rate with the sound waves remaining
linear and it is assumed that acoustic waves are initially growing, heat release saturation or phase
change effects may tend to equalize the terms in equation (3.3) at a certain pressure amplitude. At this
amplitude, limit cycle oscillations occur [28, 29, 30, 31].
Equation (3.3) does not only explain why combustion instabilities occur and why their size is
limited by non‐linear effects, but also shows that these oscillations may be eliminated by either
decreasing the energy source term, ´ , or increasing the surface loss term, . S. This is what
control methods mentioned in Section 1, both passive and active, pursue.
3.2. Acoustic analysis
The models presented in this project simulate and analyse the behaviour of acoustic waves
travelling along an open‐ended pipe with a constant cross section, A, and a total length of L, under
specific assumptions in each case: different sources and locations of the tube excitation, with or without
mean flow, absence or presence of combustion. Denoting the distance along the tube by x, the
combustion zone (or the harmonically forcing in the absence of combustion) is located at x=0, with the
upstream and downstream open ends at x= ‐xuand x=xd, respectively. Similarly, upstream region is
noted as Region1 and downstream region as Region2.
7
When analysing the system in any of the cases provided, it is assumed that the frequencies of
interest are low enough to ensure that all nonplanar modes (both radial and transversal) are well cut
off, so only one‐dimensional disturbances are important [32], and for the combustion zone to be short
compared to the wavelength [28]. Contributions from entropy waves are neglected, acoustic waves are
assumed to behave linearly to the mean flow [24] and both the mean density, , and the speed of
sound, , are assumed constant all along the tube [4].
Wave strengths (vid Fig. 2 and Fig. 7) are denoted as:
L1(t) for left‐travelling waves in Region1
R1(t) for right‐travelling waves in Region1
L2(t) for left‐travelling waves in Region2
R2(t) for right‐travelling waves in Region2
As pressure obeys the linear wave equation, pressure and velocity upstream the flame, ‐xu<x<0,
can be written as a linear combination of the waves L1 and R1:
,
3.4
,1
3.4
Similarly, downstream the combustion zone, 0>x>xd, pressure and velocity can be expressed as
a linear combination of the waves L2 and R2:
,
3.5
,1
3.5
where an overbar denotes a mean value.
The boundary conditions of the combustor are characterized by upstream and downstream
pressure reflection coefficients Ru and Rd, respectively. Therefore, the reflected waves R1 and L2 are
easily obtained as a function of L1and R2using the following relationships:
3.6
3.6
And hence:
3.7
3.7
8
where τu and τd are, respectively, the upstream and downstream propagation time delays, whose
numerical developments are:
2 1
3.8
2 1
3.8
where / is the mean flow Mach number.
4. ACOUSTICALLY EXCITED TUBE WITHOUT COMBUSTION
4.1. General description of the model and reason for modelling
A first approach to the study of combustion instabilities consists of modelling the acoustically
excited tube shown in Figure 1 and analysing the behaviour of acoustic waves, initially injected by a
loudspeaker located at x=0, travelling along the setup in the absence of combustion, both with and
without mean flow.
The reason why the study firstly models a pipe excited by a loudspeaker instead of directly
introducing a flame as an excitation source is no other than the complexity this second option involves.
An acoustically excited tube is easier to model numerically and implementing this setting is a necessary
stage before proceeding to the sophisticated modelling of a thermoacoustically excited pipe, where the
interaction of combustion and acoustic waves is by no means trivial to describe (vid. Section 5).
In the current model (Fig. 2), two sensors are implemented to measure the pressure fluctuations
along the pipe, each one located at an arbitrary point of each region, these locations denoted as x=‐x1
for the upstream sensor and x=x2for the downstream sensor.
Values for the main parameters used in this first approach are summarized in Table 1.
Fig. 2 Schematic of the acoustically excited tube without combustion
L1 t R2 t
R1 t L2 t
0 u d
p1
LoudspeakerVls t Vsin wt
p2
Region1 Region 2
9
4.2. Description of the numerical model
The device simulated consists of an axial pipe which incorporates the previously mentioned
loudspeaker to generate acoustic waves which propagate along the tube.
In this initial analysis, two different settings or cases regarding the mean flow (ū) are modelled:
a) Assuming negligible mean flow (ū=0)
b) Assuming steady and one‐dimensional mean flow.
The rate of unsteady volume injection, Vls(t), is implemented as a sinusoidal wave:
sin 4.1
where V denotes the amplitude of the wave and ωthe frequency. This wave acts as the acoustic source
which gets the physical system started and our aim is to study and analyse the behaviour of the derived
waves resulting from its interaction with the tube’s ends and the mean flow at different frequencies
over the range of interest. To do so, a matrix system expressing these derived waves (L1(t), R1(t), L2(t)
and R2(t)) as a function of Vls(t)and, by extension, as a function of frequency ω, is required.
The equations of conservation of mass and pressure continuity across the loudspeaker at x=0
are given by:
4.2
0 , 0 , 4.2
Substitution from (3.4) and (3.5) into (4.2a) and (4.2b) leads to:
4.3
4.3
Parameter Value
Effective length or the pipe, L, m 1
Pipe cross-sectional area, A, m2 1.13 x 10-2
Axial distance from the loudspeaker to the upstream end, xu, m 0.2
Axial distance from the loudspeaker to the downstream end, xd, m 0.8
Axial distance from the loudspeaker to the upstream pressure sensor, x1, m 0.1
Axial distance from the loudspeaker to the downstream pressure sensor, x2, m 0.4
Reflection coefficient at upstream end, Ru, - -0.98 / -0.95
Reflection coefficient at downstream end, Rd, - -0.98 / -0.95
Mean density, , kg·m-3 1.2
Mean speed of sound, , m·s-1 350
Loudspeaker sine wave amplitude, V, m3·s-1 1
Table 1 Geometrical and acoustic parameters
10
Making use of the pipe end boundary conditions specified in (3.7), enough information is
provided to solve for each of the four wave strengths in Figure 2. The resulting matrix equation is given
by:
0 4.4
where X and Y are the following coefficient matrices:
1 11 1
; 4.5
Having solved the matrix equation, the expressions for the time evolution of the outgoing waves
L1(t) and R2(t) as a function of ωare obtained:
12
4.6
12
4.6
4.3. Pressure modelling
In order to obtain the time evolution expressions of the pressure fluctuation, theoretically
measured by the pressure sensors at x=‐x1 and x=x2, these points have to be substituted in equations
(3.4a) and (3.5a), respectively, as follows:
,
4.7
,
4.7
Application of the open end boundary conditions and the correspondent time delays leads to:
4.8
4.8
Combining these two expressions with (4.6a) and (4.6b), the graphical block modelling of this
case is implemented with Simulink so that the pressure fluctuation’s behaviour at the pressure sensors,
located at x = ‐ x1 and x=x2, can be analysed when varying the input frequency ωof Vls(t).
The arrangement of blocks in the definitive Simulink model is reproduced in Figure 3 (a larger
view of the model is shown in Appendix A).
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12
First checking: Pressure amplitude behaviour around resonance
If we let λ denote the wave length and L has been previously defined as the total length of the
tube, the acoustic system is in resonance when L is a multiple of half the wavelength, λ/2. That is:
↔ 2; ∀ 4.9
As a result of this, if , resonant frequencies are those that fulfill the following condition:
↔ ; ∀ 4.10
For example, the first resonant frequency (n = 1), also known as first mode frequency or
fundamental frequency, is and the second resonant frequency (n=2) is
.
In order to illustrate the specific behaviour of the pressure’s amplitude around resonance, the
graphs plotting the results of this initial study represent the factor n [‐] from equation (4.9) on the
horizontal axis and pressure amplitude [Pa] on the vertical axis.
Not only two different settings are simulated, without mean flow (Fig. 4) and with mean flow
(Fig. 5), but in each of these cases the results are calculated for two different values of the reflection
coefficients Ru and Rd. As detailed in Table 1, the model is simulated and plotted for Ru = Rd = ‐0.98 and
Ru = Rd = ‐0.95. These values, especially the second one, are slightly less (in magnitude) than the
theoretical value of ‐1 for an open end [32]. Through this assumption, the numerical model is
considering the acoustic energy loss that occurs at both ends of the tube. The lower (in magnitude) the
reflection coefficient is, a greater loss of energy is assumed.
Graphs presented below confirm the expected results: the plotting of pressure amplitude as a
function of frequency presents peak values around resonance (i.e. when index n is an integer; vid. 4.10),
regardless the presence or the absence of mean flow. However, whereas these peaks exactly coincide
with resonant frequencies when no mean flow is considered (Fig. 4[a,b]), they appear slightly before
resonance in both regions of the tube in the second case (Fig. 5[a,b]) due to the interaction between the
unsteady volume injected and the mean flow.
At the same time, it can be observed that the amplitude’s peak values are lower when the
reflection coefficients are lower (in magnitude) (Fig. 4[a] and Fig. 5[a]), and higher when these are closer
to the theoretical value of ‐1 for an open end (Fig. 4[b] and Fig 5[b]). The explanation is simple: as the
reflection coefficients approach to their ideal value, the loss of energy at open ends is reduced and,
consequently, the amplitude of pressure fluctuations remains higher.
Finally, it can be observed that in the four figures, the peak of amplitude for the first mode is
higher in Region2. However, this tendency changes in the third peak thanks to the damping properties
derived from the length of each region.
13
Fig. 4[a,b] Pressure amplitude response to variation of excitation frequency without mean flow
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6x 10
5
n (L=n*/2) [ ]
Am
plitu
de [
Pa]
[ Ru = Rd = -0.95 ] p1
p2
0 0.5 1 1.5 2 2.5 3 3.50
5
10
15x 10
5
n (L=n*/2) [ ]
Am
pltiu
de [
Pa]
[ Ru = Rd = -0.98 ] p1
p2
0 0.5 1 1.5 2 2.5 3 3.50
2
4
6x 10
5
n (L=n*/2) [ ]
Am
plitu
de [
Pa]
[ Ru = Rd = -0.95 ]
0 0.5 1 1.5 2 2.5 3 3.50
5
10
15x 10
5
n (L=n*/2) [ ]
Am
plitu
de [
Pa]
[ Ru = Rd = -0.98 ]
Fig. 5[a,b] Pressure amplitude response to variation of excitation frequency with mean flow ( = 0.1)
The matching between the theoretical results and those obtained from the developed model in
the time domain constitutes a solid validation for this first numerical development and its graphical
implementation with Simulink (Fig. 3).
14
(a) Without mean flow (b) With mean flow
Fig. 6 Mode shapes of pressure amplitude for the first three resonant frequencies (n = 1, 2, 3) with Ru = Rd = -0.98.
-0.2 0 0.2 0.4 0.6 0.80
1
2x 10
6
x [m]
Am
plitu
de [P
a]
-0.2 0 0.2 0.4 0.6 0.80
1
2x 10
6
x [m]
Am
plitu
de [P
a]
-0.2 0 0.2 0.4 0.6 0.80
1
2x 10
6
x [m]
Am
plitu
de [P
a]
-0.2 0 0.2 0.4 0.6 0.80
1
2x 10
6
x [m]
Am
plitu
de [P
a]
-0.2 0 0.2 0.4 0.6 0.80
1
2x 10
6
x [m]
Am
plitu
de [P
a]
-0.2 0 0.2 0.4 0.6 0.80
1
2x 10
6
x [m]
Am
plitu
de [P
a]
n = 1 n = 1
n = 2 n = 2
n = 3 n = 3
Second checking: Mode shapes of the resonant frequencies
The second validation consists of plotting the pressure amplitude along the horizontal axis, from
the upstream end at x= ‐xu to the downstream end at x=xd, for the first three resonant frequencies
(ni=i1,i2,i3) in order to check the agreement between the information this model provides and the
theoretical mode shapes at this frequencies.
Analysing the figures below, it can rapidly be affirmed that the mode shapes perfectly fit the
theory regarding standing waves in air columns in an open ended tube. Precisely, given that the pipe is
open at both ends, the pressure at the ends would theoretically have to be atmospheric, p(t)=0,for
any resonant frequency [32].However, as reflection coefficients are assumed to present values slightly
lower in magnitude than the theoretical values of 1, as possible energy losses across pipe ends are
considered, pressure nodes (i.e. points of zero amplitude) suffer a slight shift from their ideal position
outwards the pipe (Fig. 6).
15
Furthermore, acoustics theory raise that pressure nodes do not only occur at the extremes of an
open ended tube, but at every position of x where:
2 ; ∀ 4.11
(q is a multiplying factor and n is the mode number). At the same time, an antinode of pressure,
corresponding to a point of maximum pressure, is due to appear at every position of x where:
2 14
2 12; ∀ 4.12
Remembering that for a system in resonance, conditions (4.11) and (4.12) clearly match
with the pressure amplitude behaviour shown in Fig. 6 for the first three (n=1,2,3) resonant frequencies.
It must be pointed out that, although it cannot be easily appreciated in the presented figures,
pressure amplitude at pressure nodes all along the tube is not exactly zero. The explanation of this
phenomenon is based on the energy losses which had been previously mentioned when discussing
about the pressure nodes at the extremes of the pipe.
To sum up, the analysis of mode shapes appears to corroborate the validity of this model, as
the agreement between the physical behaviour of the pressure amplitude, the theory of acoustics and
the results plotted is satisfactory.
5. MODEL COMBUSTOR
5.1. General description of the model and reason for modelling
The next step in our way to model combustion instabilities lies in developing a Simulink model
which includes a source of unsteady heat release. To be precise, the following system consists of a
horizontal tube without mean flow, open at both ends, with a heat source contained in its left half and a
loudspeaker in its upstream end which applies a white noise input to get the system started. Figure 7
shows a schematic diagram of the geometry and devices included.
The analysis that follows transfers to the time domain the one that was firstly conceived in the
Laplace domain by Evesque [33], aiming to find the open‐loop transfer function from the loudspeaker
input to the sensor pressure in the absence of a control heat input. This study in the Laplace domain was
later expanded by Dowling and Morgans, so their corresponding paper [4] is taken as reference in this
project.
This modelling approach offers a graphical interface to identify the stability regions of the
uncontrolled system in a specific setting. At the same time, it constitutes the base model for which a
fixed‐parameter controller will be designed in the next step of the current project (vid. Section 6).
16
L1 t R2 t
R1 t L2 t
0u d
HeatreleaseQ’ t
Pref Pressuresensor
Region1 Region2
Loudspeakerinput Vc whitenoise
Fig. 7 Diagram of the combustor model
5.2. Description of the numerical model
Following the assumptions specified in Section 3, equations of conservation of mass, momentum
and energy across the flame at x=0 can be written in the form [28]:
5.1
5.2
12
12
5.3
We firstly seek to obtain a matrix equation that relates the upstream and downstream acoustic
waves to the instantaneous rate of heat release, Q(t), and the loudspeaker signal, i(t). This can be
achieved by combining (5.1), (5.2), (5.3) and the perfect gas equation, and making use of the specific
boundary conditions for this setting.
The first equation needed is obtained directly from substitution of (5.1) intro (5.2), giving:
0 5.4
To obtain the second equation, the left‐hand side of the energy equation (5.3) is expanded in the
form:
12
12
5.5
Use of the perfect gas equation to rewrite Tρ as p/R (where R is the gas constant)
and substitution of (5.1) into (5.5) simplify the previous expression to:
12
12
5.6
Recalling the relationships between specific heat capacities, Cpand Cv, and their ratio, γ, we have:
1 5.7
17
By introducing this equivalence into (5.6), the second equation of the matrix system is obtained:
112
5.8
where Q is the instantaneous rate of heat release, γ is the ratio of specific heat capacities and Acomb is
the combustor cross‐sectional area.
Once these two equations, (5.4) and (5.8), are deduced, substituting their flow variables (p1,p2,
u1, u2) for their expressions, detailed in (3.4a,b) and (3.5a,b), assuming ,making use of the
isentropic condition / ) and the corresponding boundary conditions and linearizing in the
flow perturbations give a matrix system which expresses the time evolution of the outgoing waves L1(t)
and R2(t)generated by the unsteady heat releaseQ(t) (Q(t)=Q’(t),as 0and, then, 0).
However, it is necessary to highlight that, since the mean flow is negligible and due to the
presence of the loudspeaker output, reflected waves R1(t) and L2(t) may be here expressed as a
function of L1(t), R2(t)and the loudspeaker signal,i(t), using the boundary conditions as follows:
5.9
5.9
where i(t) is the white noise injected by the loudspeaker at x=‐xu and 2 / and 2 / are,
respectively, the upstream and downstream propagation time delays in the absence of mean flow.
Ultimately, the resulting matrix expression of the model shows that:
1 111
11
1 1
0′
111
5.10
Similarly to McManus’ formulation for unsteady heat release, known as the (n‐τ) model [5], heat
release is expressed as a function of the fluid velocity just upstream the combustion zone (u1(x=0‐,t)).
Specifically, Dowling and Morgans [4] use the following expression in the Laplace domain for the
instantaneous rate of heat release per unit area, q:
5.11
where H(s) is the flame transfer function. If we transfer this expression to the time domain and develop
the velocity term as a function of the waves in Region1 by using the expression (3.4b), then:
′
1
5.11
Application of the upstream boundary condition (5.9a) gives:
′
5.12
18
At his point, substitution of (5.12) intro (5.10) and grouping of terms, leads to:
1 111
11
1 11 1
01
1 11
5.13
If we then isolate the time evolution of the outgoing waves L1(t) and R2(t),we have:
12
1 22 1 0
12
1 11
12
12 1
5.14
Finally, having solved the matrix equation, the expressions for the time evolution of the outgoing
waves L1(t) and R2(t) as a function of the instantaneousrateofheatrelease expressedthroughH t )
and the acoustic loudspeaker signal, i(t), are obtained:
12
1 2 1 1
5.15
12
2 1 1 2 1
5.15
However, the ultimate aim of the model is to relate pressure measured at the sensor located
downstream, at x=xref, with the loudspeaker input, Vc(t).
The first step to reach this relationship is describing the loudspeaker dynamics through its
transfer function in the Laplace domain, Wac(s):
5.16
where i(s) is the loudspeaker output or signal and Vc(s) is the loudspeaker white noise input.
Nevertheless, the loudspeaker dynamics is assumed to be flat over the low frequencies of interest [4],
consequently, Wac(s) ≈ constant, both in the Laplace and the time domain.
In parallel, once more, Dowling and Morgan’s work [4] is followed when it comes to choose a
specific flame model, H(s): the simplest type, based on a time‐lag concept, is assumed.
∝ 5.17
19
The inclusion of these two last considerations into expressions (5.15a) and (5.15b) results in the
definite expressions of the outgoing waves, L1(t)and R2(t), in the time domain:
12
1 2 1 1
5.18
12
2 1 1 2
1
5.18
5.3. Pressure modelling
The last step before drawing the block diagram of this model consists of deducing the expression
of Pref, the pressure measured at the sensor located at x=xref, as a function of the travelling waves. This
is obtained by substituting this specific location into equation (3.5a) assuming negligible mean flow:
,
5.19
Application of the downstream boundary condition (3.7b) leads to:
,
5.20
where 2 / since negligible mean flow is assumed. So, in the end, we get to the equivalent
expression of sensor pressure as the one previously achieved by Dowling and Morgans [4], but placed in
the time domain:
,
2
5.21
When connecting expressions (5.18a) and (5.18b) with (5.21), enough information is provided to
draw a Simulink block diagram of the model described, which relates the pressure measured at the
sensor, Pref, with the loudspeaker input, Vc(t).
However, before designing the final graphical model, one last improvement should be
introduced: Dowling & Morgans’ analysis in the Laplace domain of this model shows that the stability of
the model strongly depends on the value of the flame model time delay, (5.17); in order to check that
our Simulink model presents the same behaviour (Section 5.4), this time delay is normalized in the same
way it is done in the reference study [4], so the results obtained with our design in the time domain can
be easily compared to the ones presented by Dowling and Morgans. As a result of this, blocks
corresponding to the flame model time delay are implemented as follows:
5.22
where
tube l
stabil
comb
an eq
more
5
betwe
by Eve
menti
graph
micro
Trans
e ω0 is the th
length (vid. 4.
ity map of the
Taking into
ustor with un
Appendix B
uivalent mod
aesthetic view
5.4. Results
The checkin
een the result
esque [33] an
ioned, their p
Essentially,
hical equivalen
ophone pressu
ferring it to th
heoretical first
.10) and k is t
e model.
account all t
nsteady heat r
includes a lar
el structured
w of the grap
and model
ng of the dev
ts derived fro
nd later studie
aper is used a
our model (
nce to the op
ure, Pref, pres
he notation us
Fig. 8 Simul
t mode frequ
the normalizin
these conside
elease and wi
rger view of th
in subsystem
hical model p
l checking
veloped Simu
m our model
ed in depth by
as main refere
Fig. 8), based
pen‐loop trans
sented in sect
sed in our mo
link block diagram
ency (n = 1)
ng parameter
erations, this
ithout mean f
he Simulink b
ms that mask s
resented abo
ulink model (
and the resu
y Dowling and
ence to assure
d on a nume
sfer function,
tion 2.2.1 of
odel, G(s)wou
1
m of the model
corresponding
r used in the f
is the Simul
flow (Fig. 8):
lock diagram,
maller subdiv
ve.
Fig. 9) basica
ults derived fr
d Morgans [4
e the validity o
erical develop
, G(s), from t
Dowling and
uld have the fo
g to a wavele
following sect
ink block diag
whereas App
visions of the d
ally arises fro
om the analy
]. Therefore,
of our model.
pment in the
he loudspeak
Morgans’ rep
ollowing form
1 1
ength of doub
tion to illustra
gram of the
pendix C repro
diagram and
om the comp
ysis firstly pres
as it was prev
time domain
ker input, Vc,
port of their
m:
20
ble the
ate the
model
oduces
offer a
parison
sented
viously
n, is a
to the
study.
5.23
21
where:
Wac(s)is the loudspeaker transfer function:
1
H(s)is flame transfer function, assuming a flame model based on a time‐lag concept:
∝
J is the determinant of a matrix derived from the numerical development in the Laplace domain:
1 1 1 1 1 11
Geometry and flow parameters defining our model are simulated assuming the same values that
are used in [4] to evaluate the open‐loop transfer function, G(s), so the results derived from each
method may be fairly compared. These parameters and their value are summarized in Table 2.
Dowling and Morgans prove in their paper that the stability of the detailed system, expressed
through the transfer function G(s), depends strongly on the value of the flame model time delay, . Our
next aim is, therefore, to confirm that the stability of our Simulink system depends on this parameter in
the same way.
In the reference paper, each mode is considered to be caused by a second‐order transfer
function with the form:
∝2
5.24
∝2
5.24
where is the system’s natural frequency and ξ is the system’s damping ratio. Consequently, at mode
peaks, where , we have:
∝12 2
5.25
Parameter Value
Effective length or the pipe, L, m 0.75
Axial distance from the heat release to the upstream end, xu, m 0.25
Axial distance from the heat release to the downstream end, xd, m 0.5
Axial distance from the heat release to the pressure sensor, xref, m 0.09
Reflection coefficient at upstream and downstream end, Ru and Rd, - -0.95
Loudspeaker transfer function, Wac(s), - 1
Ratio of specific heat capacities, γ, - 1.4
Mean speed of sound, , m·s-1 400
Table 2 Geometrical and flow parameters
22
(a) k = 0 (b) k = 0.5
0 0.02 0.04 0.06 0.08 0.1-10
-5
0
5
10
Time [s]
Am
plitu
de [P
a]
0 0.02 0.04 0.06 0.08 0.1-400
-200
0
200
400
Time [s]
Am
plitu
de [P
a]
Fig. 10 Time evolution of pressure measurements for two different values of the flame model time delay, τH = kΠ/ω0
This means that a positive ξ implies a phase decrease of 180° and, consequently, a stable mode;
and a negative ξ results in a phase increase of 180° and, therefore, an unstable mode. As a result of this,
we can deduce the stability of each mode analyzing the change of phase across the mode peaks in the
Bode plots: a phase decrease of 180° indicates a stable conjugate pair of poles, whereas a phase
increase of 180° indicates an unstable conjugate pair [4].
If we do trace the Bode plots of G(jω), the open loop transfer function from Vc to Pref defined by
Dowling and Morgans (vid. 5.22) for successive values of the flame model time delay, , normalized as
a function of the multiplying factor k (vid. 5.22) and analyze the phase change across the first mode
peak, we obtain the following stability map:
Using the Simulink block diagram (Fig. 8), when analyzing the time evolution of the pressure
measurements in the pressure sensor for different values of the heat‐release time delay, , we also
face two different responses depending on the value of the normalizing factor k: a stable response or an
unstable response. As an example, Figure 10a shows the stable response for k= 0 (which is unsurprising
given that = 0 means that there is no heat release and, by extension, no coupling mechanism to cause
instability), whereas Figure 10b shows the unstable response for k= 0.5, where the coupling between
acoustic waves and unsteady heat release is responsible for the occurrence of combustion instabilities.
Stable
Unstable
Fig. 9 Stability regions of the first mode of G(s), presented by Dowling and Morgans in the Laplace domain, as a function of the heat-release time delay, τH.
0 0.5 1 1.5 2 2.5 3 3.5
k = τHω0/Π
Stability of first mode of G(jω) with flame model time delay
23
Stability of i first modes (n = 1‐i) with flame model time delay
If we simulate (imposing an initially arbitrary maximum step size along each simulation of 10‐5 s.)
and analyze the time response of the Simulink model for the same range of values for the parameter k
as in Figure 9, in order to classify them between unstable and stable following the criterion exemplified
with Figures 10a and 10b, the following stability regions arise:
It may be quickly observed that stability regions derived from Simulink simulations clearly differ
from the stability map presented in Figure 9. However, further research in the interpretation of each
analysis seems to indicate that this difference lies in the fact that higher‐order modes are not
considered in the first case (Fig. 9). Taking this into account, following the steps that led to the stability
map for the first mode, the changes of phase across higher‐order mode peaks are studied and the
coupling, in terms of stability, of an increasing number of modes are captured in the following figure:
n = 1
n = 1‐2
n = 1‐4
n = 1‐6
n = 1‐8
n = 1‐10
Stable
Unstable
Fig. 12 Stability regions of the i first modes (n = 1…i) of G(s), as a function of the heat-release time delay, τH, obtained from Bode plot analysis
Stable
Unstable
0 0.5 1 1.5 2 2.5 3 3.5
k = τHω0/Π
Stability map obtained with Simulink model (max step size = 10‐5 s)
Fig. 11 Stability regions of the time response of the Simulink model as a function of the heat-release time delay, τH. (imposing a maximum step size of 10-5 s. in the simulation)
0 0.5 1 1.5 2 2.5 3 3.5
k = τHω0/Π
24
Stable
Unstable
0 0.5 1 1.5 2 2.5 3 3.5
k = τHω0/Π
Stability map obtained with Simulink model (max step size = 10‐4 s)
Fig. 14 Stability regions of the time response of the Simulink model as a function of the heat-release time delay, τH. (imposing a maximum step size of 10-4 s. in the simulation)
Considering the ten first modes, the stability regions which can be obtained analyzing the Bode
plots of the system’s transfer function, G(s) (vid. 5.23), presented by Dowling and Morgans [4], are
nearly the same as the one resulting from the Simulink model simulations (Fig. 11). Stability evolution in
Figure 12 clearly shows that the higher the number of modes considered to draw the stability maps
above is, the closer they will get to the one obtained through our Simulink model imposing a maximum
step size of 10‐5 s. in the simulations. Our Simulink model is then a useful tool to study the system’s real
behaviour over a determinate range of frequencies of interest. The range of frequency considered by
the model will depend on the maximum step size fixed, so this parameter must be carefully chosen.
Therefore, having permormed our simulations in Simulink imposing such a small maximum step
size, 10‐5 s., a wide range of frequencies are captured in the time response and, therefore, up to ten
modes have to be considered to obtain a similar stability map through Bode plot analysis (Fig . 13).
This statement is verified when observing how the stability map provided by our Simulink model
changes if we impose a significantly wider maximum step size: 10‐4 s.
0 0.5 1 1.5 2 2.5 3 3.5
k = τHω0/Π
Stable
Unstable
(a)
(b)
Comparison of stability maps (I)
Fig. 13 (a) Stability regions of the ten first modes (n=1-10) of G(s) obtained from Bode plot analysis (b) Stability regions of the time response of the Simulink model (maximum step size of 10-5 s.)
25
Figure 14 shows that, after changing the maximum step size of the simulation, the new stability
map obtained by Simulink presents similar stability regions to the one included in Figure 12 in which
only the first six modes were considered (n= 1‐6). These two stability maps are reproduced below for
easier comparison (Fig. 15):
Therefore, by running the simulation with wider step sizes, higher order modes are well cut‐off.
As a result of this, this tool may be used as a “trial‐and‐error” low‐pass filter. Considering this, the
selection of the maximum step size of the simulations is a powerful tool that lets the user restrict his
acoustic study to a determinate range of frequency. In the study of combustion oscillations, the
attention is focused on the low frequencies at which some of the most damaging instabilities occur, so
the second option, a maximum step size of 10‐4 s., appears to be an adequate choice. This hypothesis
will be further discussed in Section 6 when conceiving the feedback control.
At this point, although a clear agreement between the results included in Dowling and Morgans’
paper and the results derived from our Simulink model has been proved, this is yet restricted to stability
regions. Therefore, a deeper checking strategy is performed to ensure total agreement.
To run this ultimate checking, a prior change has to be made in our Simulink model: the white
noise input is substituted by a sine wave block, so the loudspeaker becomes harmonically forced and, as
a result of this, both the input and output waves of the system are sinusoidal. Thanks to this slight
modification, gain and phase shift can be measured at any given frequency for a specific value of
parameter k (used as a normalization of the heat release time delay, ; vid. 5.22). Obviously, this
methodology is restricted to those values of k that stabilize our system, as gain and phase shift can only
be measured in the temporal response when this response is stable.
Having repeated this procedure for a range of “spot check” frequencies and different values of
parameter k, results are scattered on the Bode diagrams that arise from the Laplace analysis, described
in equation (5.23) by Dowling and Morgans [4], for each specific value of parameter k.
0 0.5 1 1.5 2 2.5 3 3.5
k = τHω0/Π
Stable
Unstable
(a)
(b)
Comparison of stability maps (II)
Fig. 15 (a) Stability regions of the four first modes (n=1-6) of G(s) obtained from Bode plot analysis
(b) Stability regions of the time response of the Simulink model (maximum step size of 10-4 s.)
26
It was above demonstrated (Fig. 11), that Simulink model’s response is only stable for k = 0
(without heat release) and for k= 2 (when a maximum step size of 10‐5 s if fixed). So, using these values,
the previously detailed checking method leads to the following diagrams:
The matching between the Bode plots and the scattered points, both for gain and phase,
obtained with our Simulink model with k=0 and k=2 appears to be total.
Fig. 16 Gain and phase shift checking for k = 0 (maximum step size = 10-5 s.)
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
-20
0
20
40Gain and phase shift checking (k=0)
Gai
n [d
B]
[rad/s]
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
-1000
-500
0
Pha
se [d
eg]
[rad/s]
Dowling & Morgans
Simulink Model
Fig. 17 Gain and phase shift checking for k = 2 (maximum step size = 10-5 s.)
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
-20
0
20
40Gain and phase shift checking (k=2)
Gai
n [d
B]
[rad/s]
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
-1000
-500
0
Pha
se [d
eg]
[rad/s]
Dowling & Morgans
Simulink model
27
However, in order to go a step further, not only checking the validity of the Simulink model, but
also the explanation given to Figure 15, the graphs are now plotted for k= 1.3 and k= 3.3, values that
also lead to a stable response in the time domain when the maximum step size imposed is 10‐4 s.
Once more, plots show the expected matching between the two methods. In conclusion, the
reliability of the developed Simulink model (Fig. 8) that relates the pressure measured at a given
downstream location, Pref, to the white noise loudspeaker input, Vc, in presence of heat release and
without mean flow is ensured.
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
-20
0
20
40Open Loop Transfer Function (k=1.3)
Gai
n [d
B]
[rad/s]
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
-1000
-500
0
Pha
se [d
eg]
[rad/s]
Dowling & Morgans
Simulink model
Fig. 18 Gain and phase shift checking for k = 1.3 (maximum step size = 10-4 s.)
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
-20
0
20
40Open Loop Transfer Function (k=3.3)
Gai
n [d
B]
[rad/s]
500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000
-1000
-500
0
Pha
se [d
eg]
[rad/s]
Dowling & Morgans
Simulink Model
Fig. 19 Gain and phase shift checking for k = 3.3 (maximum step size = 10-4 s.)
28
6. PRACTICAL APPROACH TO FEEDBACK CONTROL
Feedback control, also known as closed‐loop control, involves modifying some input to an
unstable combustion system (using a controller and an actuator) in response to an output measurement
(obtained using a sensor). To be precise, feedback control seeks to design the control relationship
between the sensor signal and the actuator signal such that the closed loop comprising combustion
system, the sensor, the controller and the actuator is stable.
Figure 20 shows the feedback layout for a typical combustion system. A sensor measures a time‐
varying output from the combustor, S(t), and feeds it to a controller. This controller produces a signal,
V(t), and this signal drives an actuator to produce a time‐varying input to the combustion system, A(t).
As specified above, the aim of controller design is to choose the relationship between the sensor signal,
S(t), and the actuator signal, A(t), that stabilizes the entire closed‐loop combustion system [4].
The choice of sensor and actuator is one of the most delicate points when implementing
feedback control on practical combustion systems. In spite of this, another main complexity of controller
design for such applications basically lies in the large number of features which have to be taken into
account simultaneously [8, 34].
In 2005, Dowling and Morgans reviewed and summarized all the systematic approaches to
controller design that had been applied to combustion instabilities to date [4]. Schuermans had
presented a similar table in 2003 which also included developments in open‐loop control [35].
6.1. Fixed‐parameter control applied to the combustor modelled with Simulink
6.1.1. Controller design in Laplace domain
As a first stage, previous to much more complex systematic controller design techniques, the
objective of this section is to design a fixed‐parameter controller for the combustor which was modelled
with Simulink in Section 5 (Fig. 8) and implement it adding the feedback control for a given configuration
of this model in the time domain.
Combustionsystem
Sensor
Actuator Controller
ControllersignalV(t)
SensorsignalS(t)
ActuatorsignalA(t)
+
‐
Fig. 20 Generic arrangement for feedback control of combustion oscillations
29
Fig. 21 Structure of the negative feedback closed-loop control system
G(s)+
‐
K(s)
2000 4000 6000 8000 10000 12000-40
-20
0
20
Gai
n in
dB
2000 4000 6000 8000 10000 12000-1000
-500
0
Pha
se in
deg
rees
in rad/s
Because the modelled system is single‐input single‐output (SISO), it is possible to design a robust
controller using straightforward Nyquist and Bode criterions [4]. Once the controller has been designed
in the Laplace domain, it will be added to the Simulink model, which operates in the time domain, in
order to check its effectiveness to interrupt combustion instabilities.
Once more, the open‐loop transfer function (OLTF) of interest, G(s), is the transfer function from
the loudspeaker input, Vc, to the pressure measurement, Pref. We now seek to conceive a controller,
K(s), such that the closed‐loop system illustrated in Figure 21 is stable.
The first step consists of the selection of the specific setting of the model for which the controller
will be designed. Geometry and flow parameters keep the same values that were used in Section 5 and
summarized in Table 2. Choosing a specific heat‐release time delay, , is, however, much more
challenging because the controller design analysis that follows is valid for systems which present only
one unstable mode over the low frequencies of interest.
Taking this into account, Bode plots of G(jω) are drawn for a wide range of values of the
parameter k, taken as the previously detailed normalization of the flame model time delay, τH (vid.
5.22). Considering the phase change across the resonant peaks of G(jω), system’s behaviour for k = 0.5
seems to be susceptible of stabilization using the fixed‐parameter control technique, because only the
first mode (n = 1) is unstable over low frequencies and this instability corresponds to an unstable
conjugate pair of poles (Fig. 22; the blue line remarks the phase change across the first resonant peak).
Fig. 22 Bode plot of the open-loop transfer function of G(jω) for k = 0.5
30
The Nyquist Stability Criterion [36] can be expressed as
Z = N + P
where Z = number of zeros of 1+K(s)G(s) in the right‐half s plane
N = number of clockwise encirclements of the ‐1+j0 point
P = number of poles of K(s)G(s) in the right‐half s plane
Our modelled system with k = 0.5 has two open‐loop unstable poles over the low frequencies of
interest, which means K(s)G(s) presents two poles in the right‐half s plane and, consequently, P = 2.
As a result of this, for a stable control system, we must have Z = 0, or N = ‐P, therefore we must
have P anticlockwise encirclements of ‐1point. In other words, the Nyquist plot for K(jω)G(jω) needs to
encircle the ‐1 point in an anticlockwise direction twice in order to obtain a stable closed‐loop system.
Before drawing the Nyquist plot, however, a remark on the stability of higher‐order modes must
be made: despite Figure 22 clearly shows that in our model only the first mode (n = 1) is unstable over
the low frequencies of interest, the eleventh mode (n = 11), whose frequency (ω11 = 11Π /L rad∙s‐1) is
assumed to be out of the range of interest, is again unstable and could distort the time response.
Considering this, a second‐order low‐pass filter, Gf(s), is added to the open‐loop transfer function
of our system, G(s), so it attenuates high frequencies and the Nyquist plot for Gf(jω)G(jω)is restricted
to the low frequencies of interest, thus highlighting the first mode (n = 1) which causes the instability.
A generic second‐order low‐pass filter, Gf(s), has the following structure:
2 6.1
where = natural frequency of the system
ξ = damping ration (0 < ξ < 1)
= gain of the low‐pass filter
Because we want the filter to attenuate the frequency response just above the unstable mode,
which from the Bode plot in Figure 22 is at a frequency of ω = 1675 rad∙s‐1, a good choice of value for
the low‐pass filter’s natural frequency is = 2000 rad∙s‐1. The importance of the damping ratio’s exact
value is lower, so an arbitrary but realistic value of ξ = 0.5 is chosen. Finally, a unity gain is fixed for the
filter, Kf = 1, because it is thought that if the filter does not modify the response magnitude, the
influence of the controller, K(s), will be later easier to understand and interpret.
After adding the filter, the Nyquist plot for Gf(jω)G(jω) is shown in Figure 23. The anticlockwise
loops correspond to the unstable mode, for positive and negative frequencies, respectively. By
considering the unit circle plotted over the Nyquist diagram, it is clear that there are no encirclements of
the ‐1+j0 point. To achieve these two anticlockwise encirclements, the controller K(s) has to introduce
additional phase lag near the frequency of the unstable mode.
31
-20
-15
-10
-5
Mag
nitu
de (
dB)
101
102
103
104
105
140
160
180
Pha
se (
deg)
Bode Diagram of a phase-lag compensator
Frequency (rad/sec)
Diagram of a phase‐lag compensator
ω [rad/s]
Phase [deg]
Gain [dB]
Fig. 24 Bode diagram of a generic phase-lag compensator
By analyzing the figure above, an interesting choice of controller structure to achieve this phase
lag is that of a phase‐lag compensator with the following form:
1 6.2
The value of the compensator’s should be such that the maximum lag provided by the
compensator appears close to the location of the unstable mode we seek to stabilize, which, as
previously detailed, is at a frequency of ω = 1675 rad∙s‐1 (Fig. 22).
The corner frequencies of the phase lag compensator described in (6.2) are at ω=aand ω=βa
rad·s‐1, so the maximum phase lag occurs at an approximate frequency of rad·s‐1 [4, 36] (Fig. 24).
-2 0 2 4 6 8 10 12-6
-4
-2
0
2
4
6
Nyquist plot for Gf(j)G(j)
Real part
Imag
inar
y P
art
Fig. 23 Nyquist diagram of the OLTF from the loudspeaker input to the filtered pressure (── Gf(jω) G(jω) for positive ω, --- Gf(jω) G(jω) for positive ω, ── unit circle)
32
The exact choice of values for the controller’s parameters responds to the compromise between
maximising the phase and gain margins. Thus, having calculated the phase margin and the gain margin
through Nyquist diagram analysis for several different combinations of a and β that satisfy the condition
≈ 1675 rad∙s‐1, the finally selected values for the controller are a = 853.91 and β = 3.85. The value
of Kc is then chosen to be ‐0.1 to achieve the two required encirclements of the ‐1 point and a
sufficiently wide gain margin.
Considering these values, the ultimate transfer function of the phase lag compensator, K(s), is:
0.1 3.85 853.91853.91
0.1 328.76853.91
6.3
The resulting Nyquist plot for K(jω)Gf(jω)G(jω) is shown in Figure 25.
As the Nyquist Stability Criterion required (vid. 6.1), there are two anticlockwise encirclements of
the ‐1 +j0 point. The gain margin is 4.47 dB and the phase margin is 21.5°; consequently, the closed‐loop
system should be stable once the controller is implemented and, considering the value of these margins,
the controlled system should be robust to slight plant uncertainties or changes.
Fig. 25 Nyquist diagram of the controlled system with two anticlockwise encirclements of-1 point (── Gf(jω) G(jω) for positive ω, --- Gf(jω) G(jω) for positive ω, ── unit circle)
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Nyquist plot for K(j)Gf(j)G(j)
Real part
Imag
inar
y pa
rt
design
a nega
Simul
obtain
system
for k
highe
to an
intere
(highe
of inte
6.1.2. Pract
Once the c
ned using Nyq
ative feedbac
If we imple
ink model (Fi
ned (Fig. 26).
However, b
m for a given
= 0.5), as it h
r‐order mode
alyse the Nyq
est and the o
er‐order unsta
erest), this filt
Fig
tical impleme
ontroller aim
quist and Bod
ck loop and an
ment the pha
ig. 8), the ult
Larger views o
before simula
flame model
happened whe
es needs to be
quist diagram
nly unstable
able modes t
tering will be
g. 26 Simulink b
entation of the
med to stabiliz
e techniques,
nalyse its robu
ase lag compe
imate graphic
of the model
ting this syst
time delay,
en this contro
e made. If in
m (Fig. 23) so
mode (n = 1)
the first one a
here achieved
block diagram of t
e designed co
ze the Simul
the final step
ustness to slig
ensator as th
cal block diag
can be found
tem to prove
(normalized
oller was desi
the controlle
that the ana
) over this ra
at n = 11 , we
d by fixing a s
the controlled sys
ontroller into
ink model de
p consists of im
ht changes in
he transfer fu
gram for the
in Appendice
e the compe
d through para
igned, a ment
r design we h
alysis was foc
nge of freque
re filtered bec
pecific maxim
stem
the Simulink
eveloped in S
mplementing
the heat rele
nction specifi
closed‐loop c
es D and E.
nsator’s pow
ameter k; in t
tion regarding
had to introdu
cused on the
ency was high
cause they we
mum step size
model
Section 5 has
it into the mo
ease time dela
ied in (6.3) in
controlled sys
wer to stabiliz
the current an
g the eliminat
uce a low‐pas
low frequenc
hlighted in th
ere out of the
in the simulat
33
s been
odel as
ay, .
nto the
stem is
ze the
nalysis,
tion of
s filter
cies of
he plot
e range
tions.
34
In Section 5, we traced the stability maps of our model combustor derived from two different
configurations of the simulations: the first one imposing a maximum step size of 10‐5 s and the second
one limiting this simulation parameter to 10‐4 s. As it was then explained, the first setting provided
similar results to ones obtained through Bode plot analysis when the ten first modes were considered,
whereas the second setting led to a stability map really close to the one which could be traced
considering only the first six modes.
Because we now need to restrict our analysis to low frequencies, so that we face a system
arrangement with just one unstable mode over the frequencies of study and, thus, higher‐order
unstable modes have to be well cut‐off, we will perform the following simulations aimed to check the
validity of the designed controller fixing a maximum step size of 10‐4 s. In order words, we are using this
parameter setting as a “trial‐and‐error” low‐pass filter to make sure that out‐of‐interest high
frequencies do not affect the time response of the system, an application which had been previously
suggested in Section 5.
Having done this remark, the Simulink model is simulated without and with the negative
feedback loop so the pressure time response at x= xref for the same settings, specially the value of
factor k (k = 0.5), can be compared with the control OFF and ON, respectively (Fig. 27).
Fig. 27 Comparison between pressure measurements with control OFF and ON, respectively
Pressure measurement with control OFF
Pressure measurement with control ON
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-400
-200
0
200
400
Time [s]
Pre
ssur
e [P
a]
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-5
0
5
Time [s]
Pre
ssur
e [P
a]A
mpl
itude
[Pa]
A
mpl
itude
[Pa]
35
6.1.3. Analysis of control robustness
The last step of the implementation of feedback control to the modelled combustor focuses on
assessing the robustness of the designed controller when the system faces slight changes in the flame
model time delay, . The aim of this analysis is to determine a margin of the parameter k for which the
first frequency mode and, consequently, the whole system, remain stable without modifying the values
of the compensator K(s). Knowing that / , the study simply consists of analyzing the pressure
time response of the system under varying values of the k, starting for the central value k = 0.5 for which
the controller has been specifically designed, and classify it between stable or unstable for each case.
-1
0
1x 10
42
Pre
ssur
e [P
a]
-10
0
10
Pre
ssur
e [P
a]
-10
0
10
Pre
ssur
e [P
a]
-10
0
10
Pre
ssur
e [P
a]
-20
0
20
Pre
ssur
e [P
a]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2
0
2x 10
14
Time [s]
Pre
ssur
e [P
a]
k= 0.39
k= 0.40
k= 0.45
k= 0.50
k= 0.52
k= 0.53
Fig. 28 Pressure time response under varying values of k (normalization of heat-release time delay)
Am
plitu
de [P
a]
Am
plitu
de [P
a]
Am
plitu
de [P
a]
Am
plitu
de [P
a]
Am
plitu
de [P
a]
Am
plitu
de [P
a]
Time [s]
36
By observing Figure 28, it is easy to deduce the interval of k around the face value k = 0.5 for
which the modelled system is stabilized by the designed fixed‐parameter controller. This interval, which
informs about the robustness of the controller to changes in the heat‐release time delay, may be
expressed as:
↔ 0.40 0.52 6.4
If the flame model time delay suffers any change such that the value of its normalization,
expressed as k, is moved out of this interval, in principle a new controller will have to be designed to
stabilize the modeled combustor (some exceptions to this statement will be later explained).
If we examine it in depth, the information provided by Figure 28 does not end here: interval (6.4)
does not only inform about the robustness of the controller but also confirms the usefulness as a low‐
pass filter of a correct setting of the maximum step size of the simulation. In section 6.1.2, it was
explained that, when imposing a maximum simulation step size of 10‐4 s., approximately only the first six
modes (n = 1‐6) of frequency were captured in the time response because higher‐order modes were
filtered and therefore had a negligible influence on the system’s plotted behavior. This assumption is
again corroborated in Figure 28.
Bode plot analysis show that for values from k = 0.15 to k = 1, the first mode of the system is
unstable, but this instability is meant to be interrupted by the designed controller. Then, once the
controller has been added to the model to stabilize this first mode, the new responsible for
desstabilizing the system is the next unstable mode, which will appear at higher frequencies. However,
we only want this second, and higher, unstable modes to affect the system’s response if they are over
the low frequencies of interest and that is why we use the step size of the simulation as a low‐pass filter.
Starting with the face value of k = 0.5 and moving down, for k = 0.45 the second unstable mode is
the eighth (n = 8), but this is filtered when a maximum step size of 10‐4 is imposed and so the system
remains stable. The same happens for k = 0.40: the second unstable mode is the seventh (n = 7), which
is filtered too and consequently a stable response arises. Nevertheless, for k = 0.39, the sixth mode (n =
6) is unstable, so it is not filtered by our simulation settings and the time response is plotted as unstable.
A similar casuistry is observed if the value of k is moved up. For k = 0.52, the second mode (n = 2)
is right on the limit between stability and instability and this is reflected on the time response: peak
values are higher but the response is still stable. For k = 0.53, however, the second (n = 2) and the fourth
(n = 4) modes are clearly unstable and, as these modes are included in the six first modes that are
considered when fixing a maximum step size of 10‐4 s, the response of the simulation becomes unstable.
Figure 29 shows the ultimate example and constitutes one of the exceptions previously
announced. When analyzing the system for k = 2.45, the situation is quite similar to the one derived
from k = 0.5: unstable modes are n = 1, 8, 10… As a result of this, the system response is expected to be
stable because the first unstable mode (n = 1) is stabilized by the designed phase‐lag compensator and
higher‐order unstable modes (n = 8, 10…) are filtered by the simulation’s step size restriction.
37
In conclusion, the designed controller can stabilize any configuration of the model such that only
the first mode is unstable over the low frequencies of interest, in this case, limited up to sixth mode.
Thus, by fixing a determinate maximum step size of the simulation, the user can choose how low the
range of frequency to which he/she wants to restrict the research is. In the current analysis, it was
decided to focus it on, approximately, the six first modes of frequency.
7. FUTURE WORK
At this point, it is clear that research aimed to model combustion instabilities with Simulink
presents several lines of future development, as different stages of the work done up to date may be
improved in order to obtain models that present an acoustic behaviour closer to the performance by
real combustors.
Firstly, some of the assumptions and simplifications specified in Section 3.2 can be modified,
starting for including the contribution of entropy waves in future models, following the analysis by
Morgans, Annaswamy and Chee Su Goh [37,38], or accounting for changes in flow parameters across
the flame axial plane (temperature, density, speed of sound, Mach number, etc.).
Focusing on the model combustor develop in Section 5, the logical next step would imply adding
mean flow to the numerical development of the model. As well as that, future research could select a
more complex flame model (vid. 5.22) aiming to emphasize the effect of combustion oscillations at low
frequencies and, on parallel, accounting for important factors such as combustor geometry, laminar or
turbulent flows, diffusion or premixed flames or combustion efficiency [38].
Finally, regarding the feedback control of the model, it is obvious that the controller designed in
this project is just an initial approach to control techniques and constitutes the base for further
applications of closed‐loop control to Simulink model. Consequently, the complexity of the implemented
feedback control may be widely expanded, for example, through the application of parameters that are
varied in time to respond to measurements on the system.
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-6
-4
-2
0
2
4
6Pressure measurement with control ON for k=2.45
Time [s]
Am
plitu
de [P
a]
Fig. 29 Pressure measurement with control ON for k = 2.45
38
8. CONCLUSIONS
At this point, it seems clear that the goals that were settled at the beginning of the project have
been achieved.
Although a first model simulating the behaviour of an acoustically excited tube has been
designed and the derived results perfectly match the theory of acoustics, this has been conceived as a
prior but necessary step to face the true aim of the study: the modeling of thermoacoustic instabilities
using a graphical interface.
Therefore, a generic combustor experiencing combustion oscillations has been successfully
developed in the time domain using a one‐dimensional linearized approach and then modelled using
Simulink2. Results obtained with this model have been doubled checked with the ones provided by a
similar study carried out by Dowling and Morgans in the Laplace domain [4]: analysis of stability and the
comparison of the gain and phase shifts provided by these two methods for different values of the heat
release time delay are fully satisfactory.
This model constitutes a first version of the easy‐to‐use tool to model thermoacoustic instabilities
in a graphical and customizable environment we aimed to provide when this project was initiated.
Although several improvements may be applied to this first model (vid. Section 7), it represents an
important and firm step to reach the ultimate ambitious objective.
Finally, as an initial approach to the interruption of combustion instabilities, fixed‐parameter
feedback control has been added to the system and its potential to stabilize the system for a specific
given setting with only one unstable mode over the low frequencies of interest has been demonstrated.
This last section of the project has been completed with an analysis of the controller’s robustness to
slight changes in the flame model time delay. Here arises the importance of an adequate configuration
of the simulations in Simulink: the maximum step size setting has been proved as a powerful tool to
restrict the scope of the study to the low frequencies at which combustion oscillations tend to occur.
Over and above the objectives and results attained in this project, it may be affirmed that the
modelling of thermoacoustic instabilities with Simulink presents an important room for improvement. In
this sense, moving towards more accurate and complex numerical developments but preserving the
fundamental idea of providing easily understandable visual models represents an important but
involving challenge for the near future.
2 Simulink is an environment for multidomain simulation and Model‐Based Design for dynamic and embedded systems.
39
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Appendices
A. Simulink block diagram of the acoustically excited tube without combustion
B. Simulink block diagram of the model combustor
C. Simulink block diagram of the model combustor with masked subsystems (aesthetic arrangement)
D. Simulink block diagram of the model combustor with feedback control
E. Simulink block diagram of the model combustor with feedback control and masked subsystems
Append
ix A. Simulink bblock diagram off the acousticallyy excited tube wwithout combustion (Fig. 3)
Appendix B. Simulink bblock diagram off the model combustor (Fig. 8)
Append
ix C. Simulink bblock diagram of f the model combustor with massked subsystemss (aesthetic arraangement)
Append
ix D. Simulink bblock diagram off the model commbustor with feedback control
Appendix E. Simulink block diagram of the model combustor with feeddback control annd masked subsyystems