FEEDBACK CONTROL OF OSCILLATIONS IN
COMBUSTION AND CAVITY FLOWS
Simon J. Illingworth
Magdalene College
University of Cambridge
A dissertation submitted for the degree of
Doctor of Philosophy
November 2009
ii
iii
PREFACE
Except where indicated below, this dissertation is the result of my own work and in-
cludes nothing which is the outcome of work done in collaboration. This dissertation
is not substantially the same as any that I have submitted for a degree or diploma or
other qualification at any other University.
The work reported in chapters 5 & 6 on cavity oscillations was carried out in col-
laboration with Professor Clarence Rowley at Princeton University.
On the subject of spelling, the author sides with the Oxford English Dictionary,
who justify their consistent use of the termination -ize as opposed to -ise by observing
that “some have used the spelling -ise in English, as in French. . . But the suffix itself,
whatever the element to which it is added, is in its origin the Greek -izein, Latin -izare;
and, as the pronunciation is also with z, there is no reason why in English the special
French spelling should be followed, in opposition to that which is at once etymological
and phonetic.”
This dissertation contains 50 figures, 3 tables and approximately 32,000 words.
Simon Illingworth
11th November 2009
Cambridge, U.K.
iv
vACKNOWLEDGEMENTS
Dr Aimee Morgans has been a fantastic supervisor, and I cannot thank her enough for
her support, guidance, patience, and for being so generous with her time. Her contri-
butions both to this thesis and to my happiness at Cambridge these last few years are
very gratefully acknowledged. I also thank my advisor Professor Dame Ann Dowling
for her time and support, and Dr Simon Stow for help with his low order model.
Over the course of my PhD, it was my great pleasure to spend two months at
Princeton University, where Professor Clarence Rowley shared his time, his expertise
on cavity oscillations, and his DNS code. I also thank Dr Anuradha Annaswamy, who
kindly hosted me at the Massachusetts Institute of Technology and shared her great
insight into adaptive control.
Financial support from the EPSRC and Rolls-Royce plc is gratefully acknowl-
edged.
I have met some wonderful people here in Cambridge. I am very grateful to my
friends for their encouragement, and for making my time in Cambridge so happy and
memorable.
This thesis is dedicated to my Mum, for loving me and supporting me in everything
that I have wanted to do.
vi
vii
ABSTRACT
This thesis considers the control of combustion oscillations, motivated by the suscep-
tibility of lean premixed combustion to such oscillations, and the long and expensive
development and commissioning times that this is giving rise to. The controller used
is both closed-loop, employing an actuator to modify some system parameter in re-
sponse to a measured signal, and adaptive, meaning that it is able to maintain control
over a wide range of operating conditions. The controller is applied to combustion
systems with annular geometries, where instabilities can occur both longitudinally and
azimuthally, and which require multiple sensors and multiple actuators for control.
One of the requirements of Lyapunov-based adaptive control which is particularly
troublesome for combustion systems is then addressed: that the sign of the high-
frequency gain of the open-loop system is known. We address it by using an adaptive
controller which employs a Nussbaum gain, and successfully apply it experimentally
to combustion oscillations in a Rijke tube.
Another type of fluid-acoustic resonance is then considered: the compressible flow
past a shallow cavity. We start by finding a linear model of the cavity flow’s dynamics,
or its ‘transfer function’, which we identify from direct numerical simulations. We
compare this measured transfer function to that given by a conceptual model which
is based on the Rossiter mechanism, and which models each component of the flow
physics separately.
We then look at using closed-loop control to eliminate these cavity oscillations.
We start by designing a robust H2 controller based on a balanced reduced order model
of the system, the model being provided by the Eigensystem Realization Algorithm
(ERA). The robust controller provides closed-loop stability over a much wider Mach
number range than seen in previous studies. Finally, we look at the suitability of the
adaptive controller, earlier developed for combustion oscillations, for the cavity. Based
on some general properties of the cavity flow, and by using collocated control, the
oscillations are eliminated at all Mach numbers tested in the range 0.4≤M ≤ 0.8.
viii
CONTENTS
1 INTRODUCTION 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 COMBUSTION OSCILLATIONS AND THEIR CONTROL 5
2.1 Historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Physics of combustion oscillations . . . . . . . . . . . . . . . . . . . . . 5
2.3 Control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Feedback control of combustion oscillations . . . . . . . . . . . . . . . . 9
2.4.1 Adaptive feedback control . . . . . . . . . . . . . . . . . . . . . 11
2.5 Practical implementation and challenges of feedback control . . . . . . . . 12
2.5.1 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.2 Actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.3 Time delays . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.4 The origin of secondary peaks . . . . . . . . . . . . . . . . . . . 14
2.6 General properties of longitudinal combustion systems . . . . . . . . . . . 17
3 ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS 21
3.1 Introductory remarks on circumferential modes . . . . . . . . . . . . . . 21
3.2 Open-loop transfer function for control purposes . . . . . . . . . . . . . . 22
3.2.1 Modal open-loop transfer function . . . . . . . . . . . . . . . . . 24
3.3 Lyapunov adaptive control . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 An adaptive controller for a class of combustion systems . . . . . . 25
3.3.2 Adaptation rates . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Modal adaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Description of the thermoacoustic network model . . . . . . . . . . . . . 35
3.6 Adaptive control of an annular combustor in LOTAN . . . . . . . . . . . . 37
3.6.1 Combustor geometries . . . . . . . . . . . . . . . . . . . . . . 37
3.6.2 Single circumferential mode . . . . . . . . . . . . . . . . . . . . 38
3.6.3 Changing operating conditions . . . . . . . . . . . . . . . . . . 40
3.6.4 Multiple unstable modes . . . . . . . . . . . . . . . . . . . . . 41
3.7 Control using fewer transducers and actuators . . . . . . . . . . . . . . . 43
3.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
ix
x CONTENTS
4 ADAPTIVE CONTROL FOR UNKNOWN CONTROL DIRECTION 49
4.1 Importance of the high-frequency gain . . . . . . . . . . . . . . . . . . . 49
4.1.1 Physical significance of the high-frequency gain . . . . . . . . . . 50
4.1.2 Importance for feedback control . . . . . . . . . . . . . . . . . . 51
4.1.3 Importance for Lyapunov-based adaptive control . . . . . . . . . . 52
4.2 Adaptive control using a Nussbaum gain . . . . . . . . . . . . . . . . . . 52
4.3 Nussbaum gain adaptive control applied to a Rijke tube . . . . . . . . . . . 54
4.3.1 Experimental arrangement . . . . . . . . . . . . . . . . . . . . 54
4.3.2 Nussbaum adaptive controller . . . . . . . . . . . . . . . . . . . 55
4.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1 Fixed operating conditions . . . . . . . . . . . . . . . . . . . . 57
4.4.2 Varying operating conditions . . . . . . . . . . . . . . . . . . . 57
4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 DYNAMICS OF CAVITY OSCILLATIONS 61
5.1 Introduction and historical perspective . . . . . . . . . . . . . . . . . . . 61
5.2 Cavity geometry and numerical method . . . . . . . . . . . . . . . . . . 63
5.3 System identification of the cavity flow . . . . . . . . . . . . . . . . . . 64
5.4 Linear models for control of cavity oscillations . . . . . . . . . . . . . . . 67
5.4.1 Constituent transfer functions . . . . . . . . . . . . . . . . . . . 68
5.4.2 Overall linear model . . . . . . . . . . . . . . . . . . . . . . . 73
5.4.3 High-frequency dynamics . . . . . . . . . . . . . . . . . . . . . 76
5.5 Properties of the cavity transfer function . . . . . . . . . . . . . . . . . . 79
5.5.1 Transfer function zeros . . . . . . . . . . . . . . . . . . . . . . 80
5.5.2 Relative degree . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.6 Suitability of the adaptive controller for cavity oscillations . . . . . . . . . 85
5.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 FEEDBACK CONTROL OF CAVITY OSCILLATIONS 89
6.1 Reduced order modeling for fluids . . . . . . . . . . . . . . . . . . . . . 89
6.2 Balanced reduced order modeling . . . . . . . . . . . . . . . . . . . . . 91
6.2.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2.2 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.2.3 Eigensystem realization algorithm . . . . . . . . . . . . . . . . . 93
6.3 Application of the ERA to the cavity flow . . . . . . . . . . . . . . . . . 96
6.4 LQG control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.4.1 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4.3 Gain scheduling . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.5 Adaptive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
CONTENTS xi
7 CONCLUSIONS 115
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . 117
A GENERAL PROPERTIES OF COMBUSTION SYSTEMS 119
B X AND Y MATRICES 125
C PROOF OF CLOSED-LOOP STABILITY FOR A NUSSBAUM GAIN 127
D ADAPTATION RATES FOR THE NUSSBAUM GAIN 129
BIBLIOGRAPHY 132
LIST OF FIGURES
2.1 A control volume of perfect gas within a combustor . . . . . . . . . . . . 6
2.2 Classification of control strategies . . . . . . . . . . . . . . . . . . . . . 8
2.3 Typical feedback control arrangement for a combustion system . . . . . . . 10
2.4 Examples of secondary peaks seen in experimental studies . . . . . . . . . 15
2.5 General combustion system with a feedback controller . . . . . . . . . . . 18
3.1 Circumferential mode shapes for increasing values of n . . . . . . . . . . . 23
3.2 Open-loop transfer function for an annular combustion system . . . . . . . 23
3.3 Block diagram of the adaptive controller . . . . . . . . . . . . . . . . . . 29
3.4 Modification of the data vector . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Modal adaptive control algorithm . . . . . . . . . . . . . . . . . . . . . 35
3.6 Annular combustor geometry . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Bode diagram of the modal transfer function P±1(s) . . . . . . . . . . . . 39
3.8 Pressure mode shape of the n =±1 spinning mode . . . . . . . . . . . . . 40
3.9 Adaptive control of the n =±1 unstable mode . . . . . . . . . . . . . . . 41
3.10 Adaptive control of the n =±1 unstable mode: zoomed-in section . . . . . 42
3.11 Adaptive control for changing flame transfer function . . . . . . . . . . . . 43
3.12 Bode diagram of the modal transfer functions Po(s) and P±1(s) . . . . . . . 44
3.13 Adaptive control of the n = 0 and n =±1 unstable modes . . . . . . . . . 45
3.14 Adaptive control using fewer transducers and actuators . . . . . . . . . . . 46
4.1 Feedback arrangement using a feedback gain . . . . . . . . . . . . . . . . 51
4.2 Experimental set-up of the Rijke tube . . . . . . . . . . . . . . . . . . . 55
4.3 Experimental results for fixed operating conditions . . . . . . . . . . . . . 58
4.4 Experimental results for varying tube length . . . . . . . . . . . . . . . . 59
5.1 Basic configuration of a cavity flow . . . . . . . . . . . . . . . . . . . . 62
5.2 System identification arrangement for the cavity . . . . . . . . . . . . . . 65
5.3 Transfer functions measured from direct numerical simulations . . . . . . . 66
5.4 Linear model of the cavity flow . . . . . . . . . . . . . . . . . . . . . . 68
5.5 Shear layer transfer function G(s) . . . . . . . . . . . . . . . . . . . . . 70
5.6 Acoustic transfer function A(s) . . . . . . . . . . . . . . . . . . . . . . 71
5.7 Scattering transfer function S(s) . . . . . . . . . . . . . . . . . . . . . . 72
xii
5.8 Receptivity transfer function R(s) . . . . . . . . . . . . . . . . . . . . . 73
5.9 Bode diagram comparison of the open-loop transfer function . . . . . . . . 74
5.10 Nyquist diagram comparison of the open-loop transfer function . . . . . . . 75
5.11 Modified linear model including a body force transfer function . . . . . . . 78
5.12 Bode diagram comparison of the modified open-loop transfer function . . . 79
5.13 Collocated control arrangement for adaptive control . . . . . . . . . . . . 87
6.1 Open-loop Bode diagram comparison for the ERA . . . . . . . . . . . . . 97
6.2 Closed-loop cavity arrangement . . . . . . . . . . . . . . . . . . . . . . 98
6.3 Bode diagram of the LQG regulator . . . . . . . . . . . . . . . . . . . . 101
6.4 Nyquist diagram of the LQG regulator . . . . . . . . . . . . . . . . . . . 102
6.5 Closed-loop control using the LQG regulator at M = 0.60 . . . . . . . . . 103
6.6 Performance of the LQG regulator at off-design Mach numbers . . . . . . . 104
6.7 Time delay phase portraits of the LQG regulator . . . . . . . . . . . . . . 105
6.8 Performance of the gain-scheduled LQG regulator . . . . . . . . . . . . . 107
6.9 Adaptive control for M = 0.40 . . . . . . . . . . . . . . . . . . . . . . . 110
6.10 Adaptive control for M = 0.50 . . . . . . . . . . . . . . . . . . . . . . . 111
6.11 Adaptive control for M = 0.60 . . . . . . . . . . . . . . . . . . . . . . . 112
6.12 Adaptive control for M = 0.70 . . . . . . . . . . . . . . . . . . . . . . . 113
6.13 Adaptive control for M = 0.80 . . . . . . . . . . . . . . . . . . . . . . . 114
A.1 Disturbances in a combustion chamber . . . . . . . . . . . . . . . . . . . 120
LIST OF TABLES
3.1 Summary of unstable combustor’s key parameters . . . . . . . . . . . . . 37
3.2 Frequencies and growth rates of the two unstable modes . . . . . . . . . . 43
6.1 Corresponding fixed controller at each of the five Mach numbers . . . . . . 109
xiii
xiv
CHAPTER 1
INTRODUCTION
1.1 MOTIVATION
Combustion oscillations can occur in any system where combustion takes place within
an acoustic resonator, and are caused by a coupling between unsteady heat release
and acoustic waves. Unsteady combustion is an efficient acoustic source (Dowling &
Ffowcs Williams, 1983) and combustors tend to be highly resonant systems, which
together can lead to self-excited oscillations by the following mechanism. Pressure
waves, generated by unsteady heat release at the flame, reflect from the combustor
boundaries and arrive back at the flame. If the phase relationship is suitable, their
interaction with the flame gives rise to further unsteady heat release, which in turn
generates more pressure waves.
Gas turbines, aeroengine afterburners, ramjets, boilers and furnaces are all sus-
ceptible to combustion oscillations, with the most common laboratory-scale example
being an open-ended vertical tube with a heat source in the lower half, known as a Rijke
tube (Rayleigh, 1945). Combustion oscillations are of particular concern, however, for
the new generation of gas turbines operating under lean premixed prevaporized (LPP)
conditions. LPP combustion reduces emissions of oxides of Nitrogen (NOx), but is
especially prone to combustion oscillations (Richards & Janus, 1998).
To eliminate combustion oscillations, the coupling between unsteady heat release
and pressure waves must be interrupted, and this can be achieved in a number of ways.
Using feedback control – where an actuator modifies some system parameter in re-
sponse to some measured signal – is one option, and allows many powerful tools from
1
2 CHAPTER 1: INTRODUCTION
control theory to be applied to the problem.
This thesis focuses on eliminating combustion oscillations using feedback control.
The emphasis is on robust and adaptive feedback control, motivated by the requirement
that any control scheme be effective over a wide range of combustor operating condi-
tions. We then look at using feedback control methods to eliminate another type of
self-excited oscillations: those induced by the compressible flow past an open cavity.
1.2 OVERVIEW
This thesis considers two quite different self-excited systems. What binds it together
is the common challenges faced in applying feedback control – the overwhelming ma-
jority of literature for which is concerned with linear, finite-dimensional systems – to
systems that are governed by the Navier-Stokes equations, a set of non-linear, partial
differential equations.1
The main contributions of this thesis are as follows:
• The development of an adaptive feedback controller for unstable combustors
with annular geometries, where both longitudinal and circumferential modes are
possible.
• The adaptive control – demonstrated experimentally – of combustion oscillations
for unknown sign of the high-frequency gain (or ‘control direction’) by using a
Nussbaum gain.
• The identification of the linear frequency response (or ‘transfer function’) of the
compressible flow past a two-dimensional cavity which is useful for feedback
control design, and its comparison with a simple linear model.
• The determination of a balanced low-order model of the cavity flow, and its
utilization to form a robust controller using H2 robust control methods.
• The adaptive control of cavity oscillations using a Lyapunov-based adaptive con-
troller.
1Clearly, combustion oscillations introduce the added complexity of flame dynamics.
1.2. OVERVIEW 3
Chapters 2–4 look at feedback control of combustion oscillations, whilst chapters
5 & 6 (which are self-contained) look at feedback control of cavity oscillations.
Chapter 2 collects together relevant material from the literature. We start by look-
ing at the physics of combustion oscillations, before summarizing and categorizing
methods for their control. In the latter half of the chapter we focus specifically on
feedback control methods for combustion oscillations. We first look at the challenges
and fundamental limitations of feedback control, and then at some general properties of
combustion systems that are important for the adaptive feedback controller of chapter
3.
Chapter 3 focuses on adaptive feedback control of combustion systems with an-
nular geometries. We first look at a Lyapunov-based adaptive controller previously
developed for longitudinal combustion systems, before developing it further for appli-
cation to annular geometries. The adaptive controller is then applied to a number of
annular combustor geometries in a low-order thermoacoustic network model.
Chapter 4 looks at one of the fundamental requirements of Lyapunov-based adap-
tive control: that the sign of the high-frequency gain of the open-loop system is known.
We first give physical significance to the high-frequency gain of a system. A novel
method for adaptive stabilization without knowledge of the sign of the high-frequency
gain is then introduced, before being applied experimentally to a Rijke tube.
In chapter 5 we turn our attention to feedback control of cavity oscillations. In this
chapter we focus on linear models that are useful for feedback control purposes, and
we do this in two ways. First, direct numerical simulations of the cavity flow are used
to characterize its linear dynamics (or ‘transfer function’). Second, we compare the
transfer function found in direct numerical simulations to that given by a simple linear
model.
Chapter 6 first builds on chapter 5 by forming a reduced order model of the cavity
flow. This model is then used for the design of anH2 robust controller, which provides
stability over a larger Mach number range than seen previously in the literature. Fi-
nally, we apply adaptive feedback control to the cavity, and see that it provides stability
over an even larger range of free-stream Mach numbers.
Chapter 7 contains concluding remarks and suggestions for future work.
4 CHAPTER 1: INTRODUCTION
CHAPTER 2
COMBUSTION OSCILLATIONS AND THEIR CONTROL
2.1 HISTORICAL PERSPECTIVE
Rayleigh (1945) was the first to describe the mechanism for combustion oscillations.
Rayleigh pointed out that an acoustic wave will gain energy if heat is added in phase
with pressure, but will lose energy if heat is added out of phase with pressure, and
these observations constitute Rayleigh’s criterion.
Many of the earliest studies of combustion oscillations were for liquid-propellant
rocket motors in the 1950s and 1960s (Crocco & Cheng, 1956; Crocco, 1965), research
that was motivated by the stability problems encountered in these devices. More re-
cently, the problem of combustion stability has been encountered in low NOx gas tur-
bine applications, where using lean premixed combustion makes the combustor espe-
cially prone to combustion oscillations (Correa, 1998; Richards & Janus, 1998), and
this has motivated a great deal of research on the subject (Keller, 1995; Fleifil et al.,
1996; Lieuwen et al., 1998; Broda et al., 1998).
More details about combustion oscillations in general can be found in the review
articles of McManus et al. (1993), Candel (2002) and Ducruix et al. (2003).
2.2 PHYSICS OF COMBUSTION OSCILLATIONS
We now look at a generalization of Rayleigh’s criterion made by Chu (1964), which
looks at the energy balance within a thermoacoustic system, and includes the effect
of boundary conditions. The inequality that we find provides insight into combustion
oscillations and methods for their control, and was earlier used by Dowling & Morgans
5
6 CHAPTER 2: COMBUSTION OSCILLATIONS AND THEIR CONTROL
unsteady
combustion
surface, Scontrol volume, V
acoustic
waves
ρ¯ , c¯
FIGURE 2.1: A control volume of perfect gas within a combustor.
(2005) for the same purpose.
Consider a perfect gas contained in a volume V and surface S, shown in figure 2.1.
For simplicity, assume that the gas is linearly disturbed from rest (no mean flow), that
there is no mean heat release, and that viscous forces are negligible. We use ρ , u, p, c,
q and γ to denote the density, particle velocity, pressure, speed of sound, heat release
rate per unit volume and ratio of specific heat capacities respectively. An over-bar
denotes a mean value, while a prime denotes a perturbation.
An inhomogeneous wave equation can then be obtained from the equations of con-
tinuity of mass and momentum:
1
c2
∂ 2 p′
∂ t2
− ∂
2 p′
∂x2
=
∂ 2
∂ t2
(
p′
c2
−ρ ′
)
. (2.1)
Expressing the source term on the right-hand side of (2.1) in terms of the heat release,
which occurs in the combustion zone only
∂ 2
∂ t2
(
p′
c2
−ρ ′
)
=
γ−1
c2
∂q
∂ t
,
integrating with respect to time, and multiplying by p′/cρ , one arrives at a local bal-
ance equation for the acoustic energy:
∂
∂ t
(
p′2
2ρc2
+
1
2
ρu2
)
+
∂ p′u
∂x
=
γ−1
ρc2
p′q.
2.2. PHYSICS OF COMBUSTION OSCILLATIONS 7
A balance equation for the whole control volume is obtained by integrating over that
volume, giving
∂
∂ t
∫
V
(
1
2
ρ¯u2+
p′2
2ρc2
)
dV =
∫
V
(γ−1)p′q
ρc2
dV −
∫
S
p′udS. (2.2)
The left-hand side of (2.2) is the rate of change of the sum of the kinetic and potential
energies in the volume V . The first term on the right-hand side represents the exchange
of energy between combustion and acoustic waves, while the final term corresponds
to energy lost by the gas in doing work on the bounding surface S. There is another
source of energy loss: that due to the momentum and thermal boundary layers of
the combustor (Landau & Lifshitz, 1959; Matveev, 2003) – whose influence could be
important in a Rijke tube, for example – but we neglect this here.
We now see Rayleigh’s criterion in a more quantitative form: for the energy of the
gas to increase with time, the product p′q must be positive and such that
∫
V
(γ−1)p′q
ρc2
dV >
∫
S
p′udS, (2.3)
where the over-bar denotes an average over one period of oscillation. Therefore the
phase difference between p′ and q must lie in the range [−90◦,+90◦], and their product
over a cycle must be sufficiently large to overcome damping, for self-excited oscilla-
tions to occur. This simplified example gives valuable insight into both the causes of
combustion instabilities and ways in which they can be mitigated.
The range of phenomena leading to combustion instabilities in real gas turbines
is much richer, including vortex dynamics, hydrodynamic instabilities and pulsations
of the flame front – although the underlying coupling mechanism still comes from
pressure waves in the vast majority of cases. For a review of these processes, see
Candel (1992).
Combustion instabilities will grow until their amplitude is such that non-linear ef-
fects become important. The dominant non-linearity occurs in the heat release rate,
which essentially saturates: the amplitudes of the pressure fluctuations are sufficiently
small that the acoustic waves remain linear (Dowling, 1997). In some combustion sys-
8 CHAPTER 2: COMBUSTION OSCILLATIONS AND THEIR CONTROL
tems, however, such as solid rocket motors, non-linearity of the acoustic waves can
also be important (Culick, 1971).
2.3 CONTROL STRATEGIES
From (2.3), it is clear that there are two broad strategies for eliminating combustion
oscillations: one can either target the left-hand side of (2.3) by modifying the phase
relationship between p′ and q, or target the right-hand side by increasing the level of
damping in the system.
Techniques to suppress combustion oscillations can be categorized in many dif-
ferent ways, and this can lead to some ambiguity in the terms ‘passive control’ and
‘active control’. Here we choose the classification shown in figure 2.2, which is con-
sistent with that used in active noise and vibration control. According to this defi-
nition, active control provides external energy to the system via an actuator, whilst
passive control techniques do not. Active control can be further divided into open- and
closed-loop strategies. By its very definition closed-loop control involves a feedback
loop, where an actuator modifies some system parameter in response to some mea-
sured signal.1 Open-loop control, although incorporating an actuator, involves no such
feedback loop.
passive control
control strategies
open-loop closed-loop (feedback)
active control
FIGURE 2.2: Classification of control strategies.
In practice, passive control is achieved either by reducing the susceptibility to in-
1Within closed-loop schemes, a further distinction can be made based on the time scales of the
controller response (McManus et al., 1993). We make no such distinction here, instead assuming that
the controller is acting on the same time scales as the oscillations, unless otherwise stated.
2.4. FEEDBACK CONTROL OF COMBUSTION OSCILLATIONS 9
stability via, for example, modifications to the combustor geometry or to the fuel injec-
tion system (Richards & Janus, 1998; Steele et al., 2000), or by removing energy from
the acoustic waves using, for example, acoustic liners (Eldredge & Dowling, 2003) or
Helmholtz resonators (Gysling et al., 2000; Bellucci et al., 2004). The disadvantages
of passive control are that it tends to be effective only over a limited range of oper-
ating conditions, and so instability could still occur at off-design conditions; and it is
ineffective at the low frequencies at which some combustion oscillations can occur.
There have been relatively few studies of active, open-loop control of combustion
oscillations. Those that exist typically use an oscillatory signal with a fixed amplitude
and frequency, and the open-loop actuation is provided either by modulation of the fuel
mass flow rate (Richards et al., 1997; Prasanth et al., 2002) or by direct excitation of
the shear layer (McManus et al., 1990). For a review of open-loop (and early closed-
loop) control studies, see McManus et al. (1993).
It is important to understand that the mechanism of suppression used by open-loop
control approaches is fundamentally different to that used by closed-loop control. The
dynamics of a linear system cannot be changed by open-loop control, and so any open-
loop actuator that modifies combustion oscillations must do so at finite amplitudes,
typically at amplitudes similar to those of the oscillations being suppressed.
By contrast, closed-loop control can act to change the dynamics of a linear sys-
tem, which implies that very small-amplitude actuation can be used, especially once
the initial large-amplitude oscillations have been suppressed. (This is shown clearly by
Tachibana et al. (2007), who compare open-loop and closed-loop control strategies us-
ing secondary fuel injection.) Furthermore, a rich literature exists on robust and adap-
tive feedback control, whose techniques can be used to design feedback controllers
that are effective over a wide range of operating conditions. There are many studies
of closed-loop control in the literature, and we now look at this particular strategy in
more detail.
2.4 FEEDBACK CONTROL OF COMBUSTION OSCILLATIONS
Figure 2.3 shows the effect of a generic closed-loop control scheme on a combustion
system: an actuator modifies some system parameter in response to some measured
10 CHAPTER 2: COMBUSTION OSCILLATIONS AND THEIR CONTROL
signal. In this way, the dynamics of the overall system are modified and an unstable
combustor can be stabilized.
combustor sensor
controlleractuator
measurementinput
FIGURE 2.3: Typical feedback control arrangement for a combustion system.
Many studies of closed-loop control of combustion oscillations have been per-
formed, and no attempt is made to list them all here. Rather, an overview of closed-
loop control strategies in the literature is given. For an excellent recent review of the
subject, see Dowling & Morgans (2005).
The concept of using closed-loop, feedback control to eliminate combustion oscil-
lations is not new, dating back to the 1950s when it was applied theoretically to low-
frequency oscillations in liquid-propellant rockets (Tsien, 1952). Closed-loop control
was first applied experimentally to the Rijke tube by Dines (1983) and then by Lang
et al. (1987) and Heckl (1988), and shortly afterwards to an afterburner (Bloxsidge
et al., 1988; Langhorne et al., 1990) and to a turbulent ducted flame (Poinsot et al.,
1989). (See also Paschereit et al. (1998) for application to a swirl-stabilized combus-
tor.)
Those early studies all involved very simple phase-shift or time-delay control loops.
Gulati & Mani (1992) showed the inadequacies of such controllers, and demonstrated
the improved closed-loop performance of a controller designed using the frequency
response of the system and Nyquist techniques. Since then, model-based control tech-
niques have been promoted (Annaswamy et al., 2000; Morgans & Dowling, 2007) and
many such studies have been performed. In any model-based control scheme, there
are two pertinent issues to address. First, a model of the system must be available,
and this can come from a physics-based model (Hathout et al., 1998; Chu et al., 2003;
Campos-Delgado et al., 2003a,b) or, less commonly, directly from experiments us-
2.4. FEEDBACK CONTROL OF COMBUSTION OSCILLATIONS 11
ing system identification techniques (Tierno & Doyle, 1992; Murugappan et al., 2003;
Kjær et al., 2006; Niederberger et al., 2009). (The dynamical processes involved in
large- and full-scale systems, such as combustion and turbulence, are too complex for
physics-based models to be useful, and so system identification techniques are cru-
cial for model-based control at these scales.) Such system identification techniques
are useful for describing the linear dynamics of the combustion system. Recently, a
unified framework for incorporating flame non-linearities into a model of combustion
oscillations has been proposed (Noiray et al., 2008). Second, a framework for the con-
troller design must be chosen, and this has typically involved Linear Quadratic control
or some variant (Hathout et al., 1998; Murugappan et al., 2003) or H∞ loop-shaping
techniques (Tierno & Doyle, 1992; Hong et al., 2000; Chu et al., 2003). More recently,
model predictive control techniques have been applied (Gelbert et al., 2008).
2.4.1 ADAPTIVE FEEDBACK CONTROL
A particularly attractive type of control scheme is adaptive control, where the controller
adapts to changes in the system, thereby maintaining control over a wide range of
operating conditions.
Perhaps the earliest adaptive control study for combustion oscillations was per-
formed by Brouwer et al. (1991), but here the characteristic timescales of the control
signal were much longer than those of the instability – a type of feedback control often
referred to as trim adjustment (McManus et al., 1993).
The first ‘dynamic’ adaptive controllers for combustion oscillations, using con-
comitant sensing and actuation timescales, were Least Mean Squares (LMS) con-
trollers (Billoud et al., 1992). Since then neural networks (Liu & Daley, 1999; Blon-
bou et al., 2000) and observer-based adaptive controllers (Neumeier & Zinn, 1996;
Sattinger et al., 2000) have also been considered.
More recently, a class of self-tuning regulators has been considered (Krstic et al.,
1999). The adaptive tuning of the parameters of the controller is based on finding a
Lyapunov function that reduces in amplitude with time when the control parameters
are updated in the right way. Annaswamy et al. (1998) looked at such an adaptation
scheme for a specific combustion system, and Evesque et al. (2003b) followed by
12 CHAPTER 2: COMBUSTION OSCILLATIONS AND THEIR CONTROL
considering a scheme that was valid for a whole class of combustion systems.
Determining an accurate model of a combustion system which is useful for feed-
back control purposes is one of the most difficult aspects of model-based feedback
control, and so the advantages of adaptive control are clear. No such model is re-
quired, and instead only minimal information about the system is needed. This is
particularly true of the self-tuning regulators of Annaswamy et al. (1998) and Evesque
et al. (2003b), since only some very general assumptions about the combustion system
are made. We will look at this class of self-tuning regulators in more detail in chapter
3.
2.5 PRACTICAL IMPLEMENTATION AND CHALLENGES OF FEEDBACK CONTROL
The practical implementation of feedback control in combustion systems presents many
challenges. These include turbulence and combustion, which are too complex to be
modeled accurately; significant time delays brought about by the combustion process
and by acoustic propagation times; and non-linear dynamics. Despite these challenges,
both large- and full-scale demonstrations of closed-loop control already exist (Seume
et al., 1998; Moran et al., 2000; Johnson et al., 2001).
Two of the defining features of any feedback control system are the measurement
of a feedback signal and the actuation of some system parameter, and the sensing and
actuation that these necessitate are two of the greatest challenges of closed-loop control
of combustion oscillations. We now look at the subjects of sensing and of actuation in
more detail.
2.5.1 SENSORS
Experimental feedback control studies have most commonly used microphones and
pressure transducers for sensing. Since pressure waves propagate throughout the entire
combustor, they do not need to be placed near the high temperatures of the heat release
zone (although increasing this distance will increase the acoustic time delay). They
offer high bandwidth and reasonable robustness. Examples of their use range from the
early experiments of Heckl (1988) and Lang et al. (1987) to full-scale demonstrations
2.5. PRACTICAL IMPLEMENTATION AND CHALLENGES OF FEEDBACK CONTROL 13
by Seume et al. (1998) and Moran et al. (2000).
An alternative sensing approach is is to measure the light emitted by certain progress
chemicals in the flame. Such sensors are independent of mode shape unlike pressure
sensors, but require optical access to the system, and may be susceptible to changes
in the flame location. Examples of this approach include the earliest experimental
demonstration on a Rijke tube by Dines (1983) and a diesel-fueled turbulent combus-
tor (Hermann et al., 1996).
For a review of sensors for combustion control, see Docquier & Candel (2002).
2.5.2 ACTUATORS
Satisfactory actuation is one of the greatest challenges for feedback control of combus-
tion oscillations. The ideal actuator is robust and durable; has low power requirements;
has high control authority to affect limit cycle oscillations; has a linear response; and
has a high bandwidth to excite the system at the frequencies of the unstable modes. In
reality it is very difficult to meet all of these requirements with a given actuator.
Many of the early feedback control studies used loudspeakers for actuation (Dines,
1983; Lang et al., 1987; Heckl, 1988; Poinsot et al., 1989). These provide a linear
response and high bandwidth, but suffer from insufficient robustness and prohibitive
power requirements for full-scale applications.
Exploiting the chemical energy released in combustion is an efficient means of
meeting power requirements for feedback control, and so actuation of the fuel mass
flow rate has been employed in more recent studies. The first fuel modulation actuators
were based on on-off fuel injectors (Langhorne et al., 1990) and were thus non-linear.
Solenoid valves have a linear response and are thus preferable, and have been used in
large- and full-scale applications (Seume et al., 1998). Their main drawback is their
limited bandwidth and control authority. Magneto-restrictive valves have increased
bandwidth (Neumeier & Zinn, 1996; Niederberger et al., 2009), and show promise
for application at full-scale, provided that control authority can be maintained at high
frequencies.
14 CHAPTER 2: COMBUSTION OSCILLATIONS AND THEIR CONTROL
2.5.3 TIME DELAYS
To see the effect of time delays, we look at a simple second-order system, which cor-
responds to the dynamic model of low-frequency liquid-propellant rocket instabilities
that was proposed in the earliest studies of combustion oscillations (Crocco & Cheng,
1956):
x¨(t)+2ζωox˙(t)+ω2o x(t) =−Fx(t− τ), (2.4)
where ωo and ζ characterize the resonance frequency and damping of the system and
F is a constant. In the rocket instabilities model, x(t) corresponds to the perturbation
of the injection rate, whose characteristic equation is forced by the injection rate at an
earlier time x(t− τ). Taking a Taylor series of the right-hand side to first order, which
is more easily justified when one considers that the frequencies of interest are low, and
therefore that the time scales of the characteristic equation are long, (2.4) becomes
x¨(t)+(2ζωo−Fτ)x˙(t)+(ω2o +F)x(t) = 0.
Thus for positive F , the time delay reduces the effective damping of the system, and
for Fτ > 2ζωo the system is unstable. This simple example helps to highlight the
open-loop destabilizing effect that time delays can introduce.
Time delays also present difficulties when feedback control is used, perhaps un-
surprisingly when the actuation provided by the closed-loop controller is now based
on old information. Banaszuk et al. (1999) use the presence of large time delays in
combustion systems as an explanation of the peak splitting phenomenon seen in some
closed-loop experiments. We now look at the origins of secondary peaks in more de-
tail.
2.5.4 THE ORIGIN OF SECONDARY PEAKS
Secondary peaks have been observed in many closed-loop studies on combustion oscil-
lations: two examples from Bloxsidge et al. (1988) and from Langhorne et al. (1990)
are shown in figure 2.4. One observes that although the original oscillation is signif-
icantly reduced, new frequencies appear in the spectra when control is activated. For
many of the early studies, this phenomenon can be attributed to the very simple con-
2.5. PRACTICAL IMPLEMENTATION AND CHALLENGES OF FEEDBACK CONTROL 15
trollers used, and this was acknowledged by Langhorne et al. (1990), made clear via
experiments by Gulati & Mani (1992), and studied in some detail for analytical models
of combustors by Fleifil et al. (1998).
We now explain the phenomenon of secondary peaks – purely by way of a linear
analysis – by considering the general feedback arrangement of a plant P(s), a controller
K(s) and an actuator L(s), where we include a disturbance d at the input, and noise n
at the output:
− P (s)
L(s)
p
K(s)
d
n
Pcl(s) =
P(s)
1+K(s)L(s)P(s)
. (2.5)
Here s= iω is the Laplace variable. We look at two possible reasons for secondary
peaks. The first involves looking at the closed-loop transfer function (2.5): the closed-
loop system can exhibit peaks in its frequency response if the denominator of (2.5) is
close to zero at some frequency ωp. That is, setting s = iωp, if
K(iωp)L(iωp)P(iωp)≈−1. (2.6)
(a) (b)
FIGURE 2.4: Examples of secondary peaks seen in experimental studies: (a) taken from
Bloxsidge et al. (1988) (closed-loop case given by dashed line); and (b) taken from
Langhorne et al. (1990) (closed-loop case given by solid line).
16 CHAPTER 2: COMBUSTION OSCILLATIONS AND THEIR CONTROL
This secondary peak could correspond either to an unstable mode – which would then
be limited in amplitude by non-linearities – or to a stable, lightly damped mode that
is driven by external disturbances. This is essentially the explanation of secondary
peaks given by Fleifil et al. (1998), and motivates the use of both a good model of the
system and sophisticated controller design: if these are both available, then the ‘loop
gain’ K(s)L(s)P(s) can be shaped to make sure that the condition given by (2.6) is
not approached. (Although when large time delays are present in the system, avoiding
(2.6) at all frequencies is particularly challenging (Banaszuk et al., 1999).)
There is a second, less obvious reason for secondary peaks that was first explained
by Banaszuk et al. (1999) and by Cohen & Banaszuk (2003), who show that closed-
loop amplification of disturbances will necessarily occur at some frequencies for nar-
row bandwidth controllers. To explain this second cause, first consider how feedback
affects the amplification of disturbances d. Without control, the transfer function from
a disturbance to the measured pressure is simply P(s), whilst with feedback it becomes
P(s)/(1+K(s)L(s)P(s)). Therefore feedback modifies the open-loop transfer function
by the amount
S(s) =
1
1+K(s)L(s)P(s)
, (2.7)
which is known as the sensitivity function. If |S(iω)| < 1, then disturbances are at-
tenuated relative to open-loop, whereas if |S(iω)|> 1, then feedback amplifies distur-
bances. Ideally, one would like to design a compensator K(s) such that disturbances
are attenuated at all frequencies, |S(iω)|< 1, and one may hope that with a good model
of the system and a well-designed controller, this is possible. Unfortunately, Bode’s in-
tegral rule dictates that it is not. Under some quite general assumptions, Bode showed
that a decrease in the sensitivity over one frequency range must be balanced by an
increase in sensitivity over some other frequency range. More precisely, for a system
with a relative degree of at least two,1 Bode’s integral rule states that∫ ∞
0
log |S(iω)|dω = pi∑
k
Re(pk), (2.8)
where pk are the unstable poles of K(s)L(s)P(s). Therefore for a stable plant, any
1The relative degree of a transfer function is defined in § 2.6 (equation (2.9)).
2.6. GENERAL PROPERTIES OF LONGITUDINAL COMBUSTION SYSTEMS 17
negative area (|S(iω)|< 1) in the log-linear plot of S(iω) versusω must be balanced by
a positive area (|S(iω)|> 1) at some other frequency. For an unstable plant,∑Re(pk)>
0, and the situation is even worse.
Bode’s area rule (2.8) is a fundamental limit of feedback control and, whilst not a
problem in itself, is made a problem by actuators with limited bandwidth. For an ac-
tuator with unlimited bandwidth, (2.8) poses no problem, since the positive sensitivity
can be spread over a large frequency range. If the actuator L(s) has limited band-
width, however, then log |S(iω)| necessarily approaches zero at high frequencies, and
any positive log |S(iω)| must occur within the bandwidth of the actuator. As a result
any actuator, as well as providing excitation at the frequency of open-loop oscillations,
ideally needs sufficient bandwidth so that the implications of (2.8) can be ‘spread out’
over a large frequency range.
If this is not possible, then secondary peaks can occur, even if an accurate model of
the combustor is available, and even if a well-designed controller is used. This second
source of secondary peaks, unlike the first, is not well-documented in the literature,
but it is likely to be a problem when limited-bandwidth fuel flow actuators are used for
control.
2.6 GENERAL PROPERTIES OF LONGITUDINAL COMBUSTION SYSTEMS
We now look at some general properties of longitudinal combustion systems which are
useful for feedback control purposes. These properties were derived by Evesque et al.
(2003b), and we will use them again in chapter 3 for the development of an adaptive
controller for annular combustor geometries.
Figure 2.5 shows a general (longitudinal) combustion system fitted with a closed-
loop controller. For feedback control, we assume that the feedback signal is a pressure
measurement at some point in the combustor, and that modulation of the fuel mass
flow rate is used for actuation.
Including the actuator in the open-loop transfer function of such a combustor,
Evesque et al. (2003b) made four non-restrictive assumptions about the combustor’s
open-loop dynamics:
(i) the pressure reflection coefficients at the upstream and downstream boundaries
18 CHAPTER 2: COMBUSTION OSCILLATIONS AND THEIR CONTROL
Ru Rd
p
Q′
Vc
Flame
Controller
Actuator Transducer
FIGURE 2.5: General combustion system with a feedback controller.
of the combustor satisfy
|Ru(s)|< 1, |Rd(s)|< 1,
i.e. the amplitude of a reflected wave at a boundary is smaller than the amplitude
of the incoming wave.
(ii) the flame has stable dynamics, i.e. instability only occurs because of interactions
between the flame and the combustor acoustics, and not because of any hydrody-
namic instabilities of the flame itself.
(iii) the flame response has a limited bandwidth.
(iv) the actuator dynamics has a relative degree not exceeding two and has no right-
half plane zeros.
By using a wave expansion of the acoustic field in the combustor, and by approx-
imating the time delays in the system using Padé approximants, Evesque formed an
open-loop transfer function for a longitudinal combustor of the form
P(s) = Po(s)e−sτ ,
where Po(s) is a rational approximation to the delay-free part of P(s) and can be written
as the ratio of two monic polynomials:
Po(s) = go
NP(s)
DP(s)
.
2.6. GENERAL PROPERTIES OF LONGITUDINAL COMBUSTION SYSTEMS 19
Here go is the high-frequency gain of Po(s). The relative degree n∗ of Po(s) is then
defined as
n∗[Po(s)] = deg[DP(s)]−deg[NP(s)], (2.9)
which corresponds to the number of poles of Po(s) minus the number of its zeros.
Three properties of any actuated combustion system meeting assumptions (i–iv)
were then derived by Evesque:
PROPERTY 2.1. The zeros of P(s) are stable (i.e. in the left half-plane).
PROPERTY 2.2. The sign of the high-frequency gain of P(s) is given by the sign of the
high-frequency gain of the actuator transfer function.
PROPERTY 2.3. The relative degree of P(s) is given by the relative degree of the actuator
transfer function.
More details of the approach taken by Evesque et al. (2003b) – together with some
development of the approach for annular combustor geometries which is important for
chapter 3 – is given in appendix A.
20 CHAPTER 2: COMBUSTION OSCILLATIONS AND THEIR CONTROL
CHAPTER 3
ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
In this chapter, an adaptive feedback controller derived using Lyapunov’s direct method
is developed for axisymmetric annular combustion systems. For such systems, mul-
tiple sensors and multiple actuators are needed and thus straightforward single-input
single-output approaches for longitudinal modes are no longer valid. The adaptive con-
troller is well-suited to real engine environments for three reasons: it is able to identify
and control both longitudinal and circumferential modes in annular combustors; it can
adapt to large changes in the engine’s operating conditions; and it does not require
prior characterization of the system.
The adaptive feedback controller is implemented computationally in a low-order
thermoacoustic network model (LOTAN) of an annular combustor and demonstrated
in time domain simulations. The adaptive controller stabilizes two different combus-
tion systems: one for which a circumferential instability only is present, and one with
both a circumferential and a longitudinal instability. The controller retains control
following a large change in the combustor operating conditions by re-tuning its adap-
tive parameters. Finally, a more easily implemented control arrangement with fewer
transducers and actuators is investigated, and it is seen that closed-loop stability is still
achieved.
3.1 INTRODUCTORY REMARKS ON CIRCUMFERENTIAL MODES
For a longitudinal combustor geometry, only plane waves transport acoustic energy,
and the pressure perturbation field (for example) can be written as a linear combination
21
22 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
of upstream- and downstream-traveling waves:
p′(x, t) = A±eiωt+ik±x,
where x is the axial coordinate. A+ represents a downstream-propagating wave and A−
represents an upstream-propagating wave, and k+ and k− are the corresponding wave
numbers. For such a geometry, one actuator and one sensor suffice for feedback control
purposes. (The sensor must be axially well-placed to avoid nodes of the pressure mode
shape.)
For an annular combustor geometry, we must also consider azimuthal and radial
perturbations. For the annular geometries of interest in this chapter, the annular gap is
small, and the radial dependence can be ignored (Stow et al., 2002). Then the pressure
perturbation field can be described by
p′(x,θ , t) = A±eiωt+ik±x+inθ ,
where θ is the azimuthal angle and n is the circumferential mode number. This in-
troduces the possibility of circumferential modes: typical mode shapes are shown in
figure 3.1. An n = 0 mode corresponds to a longitudinal mode with no circumferential
dependence, and increasing n corresponds to increasing circumferential wave number.
For circumferential modes, the pressure mode shape may spin in time, or may be lo-
cated at a fixed azimuthal angle, and therefore one cannot guarantee that a single sensor
is not at a node. Similarly, one cannot guarantee that a single actuator will find itself in
a useful position for control. From these observations it is clear that multiple sensors
and multiple actuators are required for circumferential modes.
3.2 OPEN-LOOP TRANSFER FUNCTION FOR CONTROL PURPOSES
Figure 3.2 shows the open-loop linear transfer function for a combustion system in-
cluding actuator (where again s = iω is the Laplace variable). Although this transfer
function is not needed explicitly by the adaptive controller that we will develop, we
introduce it here to make clear the Multi-Input Multi-Output (MIMO) nature of the
3.2. OPEN-LOOP TRANSFER FUNCTION FOR CONTROL PURPOSES 23
n = 0 n = ±1 n = ±2
FIGURE 3.1: Circumferential mode shapes for increasing values of n.
problem for annular combustors, before introducing a modal equivalent of this transfer
function in § 3.2.1.
Fuel valves will be used for actuation, since these are the most practical actuators
for large- and full-scale applications, and pressure transducers will be used for sens-
ing. For the annular combustor geometries that we will consider, several fuel valves
and several pressure transducers are spaced azimuthally around the combustor’s cir-
cumference. Then the overall open-loop transfer function is between the vector of
actuator voltages V= [V1 . . . VB]T and the vector of transducer pressure measurements
p = [p1 . . . pT ]T , where B is the number of burner ducts and T is the number of pres-
sure transducers. Referring to figure 3.2, f∇, φ∇, Q∇ and a∇ represent the fractional
change in the fuel mass flow rate, equivalence ratio, heat release rate and air mass flow
rate respectively. Bold type is used (as it is for V and p) to denote the vector-valued
parameters f∇, φ∇, Q∇ and a∇ (all of dimension B). Since all of the system inputs and
outputs are vectors, the four constituent transfer functions are matrices: L(s), F(s), and
H(s) are of dimension B×B, whilst G(s) is of dimension T×B. The overall open-loop
acoustics
H(s)
P (s)
L(s)
actuator
F (s)
flame
G(s)
acoustics
V f∇ φ∇
a∇
Q∇ p
FIGURE 3.2: Open-loop transfer function for an annular combustion system.
24 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
transfer function is then
P(s) = G(s)(I+F(s)H(s))−1F(s)L(s) (3.1)
and has dimension T×B. For small perturbations, the fractional change in equivalence
ratio φ∇ is related to the fractional change in fuel mass flow rate f∇ and the fractional
change in air mass flow rate a∇ by φ∇ = f∇−a∇.1
3.2.1 MODAL OPEN-LOOP TRANSFER FUNCTION
The adaptive controller that we will develop can be applied to Single-Input Single-
Output (SISO) systems. The transfer function given by figure 3.2 and equation (3.1),
however, has potentially many inputs and many outputs, and we remedy this by devel-
oping a modal transfer function and a corresponding modal adaptive controller. This
simplifies the control problem from one for the MIMO system in figure 3.2 to one for a
SISO system, and relies on the combustor’s being axisymmetric and therefore having
decoupled modes. Morgans & Stow (2007) show that this modal transfer function is
given by
Pn(s) =
pn(s)
V n(s)
=
Lb(s)Fb(s)Gn(s)
B[1+Fb(s)Hn(s)]
, (3.2)
Fb(s) is the flame transfer function for a given burner b (which are forcibly all identical
for an axisymmetric combustor), whilst Lb(s) is the actuator transfer function for a
given burner b (assumed to be all identical).
Equation (3.2) for the modal transfer function Pn(s) is very similar to that given by
(3.1) for the single-input single-output case, which becomes
P(s) =
p(s)
V (s)
=
L(s)F(s)G(s)
1+F(s)H(s)
. (3.3)
It is for this single-input single-output case that Evesque et al. (2003b) derived the
three properties of longitudinal combustion systems (summarized in § 2.6) that are
useful for feedback control purposes, and upon which the proof of stability for the
1When normalized, φ = ( f/a)/( f/a)stoich becomes 1+φ∇ = (1+ f∇)/(1+ a∇), which for small
perturbations is simply φ∇ = f∇−a∇.
3.3. LYAPUNOV ADAPTIVE CONTROL 25
adaptive controller to be introduced relies.
Therefore before developing an adaptive controller for circumferential modes, we
must first ask if the modal transfer function (3.2) satisfies those same properties that
are satisfied in the SISO case (3.3). This question is addressed in appendix A, where
it is shown that the three properties derived by Evesque et al. (2003b) still hold for
the modal transfer function, if the additional assumption is made that the frequencies
of interest are well above the cut-off frequency.1 This extra assumption is used to
show that all of the zeros of Pn(s) are in the left half-plane. (Furthermore, for the two
combustor geometries that we will consider, we will see that all zeros of the open-loop
systems are in the left half-plane, and so Evesque’s property 2.1 concerning the zeros
is satisfied.)
3.3 LYAPUNOV ADAPTIVE CONTROL
The term adaptive control can be used to refer to many different control strategies
(Åström, 1987), but they all share the common goal of controlling dynamical systems
whose characteristics are not completely known.
Lyapunov-based adaptive control – so-called because of its foundations in Lya-
punov stability theory – is one such scheme for the adaptive stabilization of unknown
systems, and has found diverse applications including ship steering, flight control sys-
tems and flexible structures (Åström, 1983; Harris & Billings, 1981).
We now look at a Lyapunov-based adaptive controller for the adaptive stabiliza-
tion of combustion oscillations. For completeness, we first look at the specific case of
adaptive control of the class of combustion systems studied by Evesque et al. (2003b).
This adaptive controller is valid for longitudinal combustors, where one pressure trans-
ducer and one actuator suffice for control. We then develop the controller further for
application to annular combustion systems with axisymmetric geometries.
3.3.1 AN ADAPTIVE CONTROLLER FOR A CLASS OF COMBUSTION SYSTEMS
Annaswamy et al. (1998) looked at self-tuning regulators (also known as model-based
1The cut-off frequency is the frequency below which the acoustic modes are attenuated with distance
and carry no acoustic power – see appendix A for more details.
26 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
adaptive controllers) for a specific longitudinal combustion system: for such systems,
one burner valve and one axially well-placed pressure transducer suffice for control. It
was then shown (Evesque et al., 2003b) that this control structure could be applied to
a whole class of combustion systems. Evesque et al. (2003a) extended the algorithm
to account for large time delays in the system by including a Smith-type controller
(Smith, 1959): we will confine our attention to the algorithm without Smith controller
here.
We consider an unstable open-loop transfer function (including actuator) P(s)which
can be stabilized using feedback control, and we denote this closed-loop stabilized
transfer function by P∗cl(s). It can be shown from root-locus arguments that, for the
class of combustion systems studied by Evesque et al. (2003b), a first-order compen-
sator of the form
K(s) = kc
s+ zc
s+ pc
(3.4)
is sufficient to provide closed-loop stability. The adaptive feedback controller that
we now consider finds stabilizing parameters for this first-order compensator. The
derivation of such an adaptive control algorithm relies on P∗cl(s) being Strictly Positive
Real (SPR).
If P∗cl(s) is SPR, it is possible to form a positive-definite Lyapunov function which
depends on the states (in a state-space sense) of P∗cl(s) and a vector of control param-
eters. Because the Lyapunov function is positive-definite, it can be thought of as an
energy function. The time derivative of the Lyapunov function is guaranteed to be
negative if the time derivative of the control parameters obeys a certain rule: this pro-
vides the updating rule for the control parameters. When this updating rule is used,
the Lyapunov function is guaranteed to decrease monotonically in time and the control
parameters are guaranteed to converge to a stabilizing set. The existence of such a
Lyapunov function – which depends on P∗cl(s) being SPR – is crucial for obtaining an
adaptation algorithm which guarantees stability.
P∗cl(s) being SPR requires that:
(i) P∗cl(s) has no right half-plane zeros (i.e. the closed-loop system is minimum
phase).
3.3. LYAPUNOV ADAPTIVE CONTROL 27
(ii) The high-frequency gain of P∗cl(s) is positive.
(iii) Re[P∗cl(s)] > 0, ∀s, which is equivalent to the phase of P∗cl(s) lying in the range
−90◦ < ∠[P∗cl(s)]<+90◦.
For a more rigorous definition of an SPR transfer function, see Narendra & An-
naswamy (1989).
These requirements on the closed-loop stabilized system P∗cl(s) translate into re-
quirements on the open-loop system P(s). We now make these requirements on P(s)
clear, and show how they relate to the general properties of combustion systems that
were derived by Evesque et al. (2003b) and that were summarized in § 2.6.
(i) P∗cl(s) has no right half-plane zeros. Since feedback has no effect on the zeros of
a transfer function, this means that P(s) must also have no right half-plane ze-
ros. Evesque et al. (2003b) showed that this was met for longitudinal combustors
(property 2.1), and in appendix A it is shown that this is also true of circumferen-
tial modes when the frequencies of interest are well above the cut-off frequency.
We will also see that the transfer functions of the combustors studies in this chap-
ter contain no right half-plane zeros. (For extension to combustion systems with
right half-plane zeros, see Morgans & Annaswamy (2008).)
(ii) The high-frequency gain of P∗cl(s) is positive. For any feedback controller, P(s)
and P∗cl(s) have the same high-frequency gain. Therefore the high-frequency gain
of P(s) must also be positive, and this corresponds to Evesque’s property 2.2. For
adaptive control, however, we will see that it is enough to know the sign of the
high-frequency gain of P(s). (The case where the sign of the high-frequency gain
is unknown is the focus of chapter 4.)
(iii) Re[P∗cl(s)] > 0, ∀s. This places an upper bound on the relative degree of P∗cl(s).
Since P(s) and P∗cl(s) have the same relative degree, this upper bound also applies
to P(s). An SPR transfer function has a relative degree of at most one. By mod-
ifying the adaptive controller, however, it is possible to apply it to systems with
a relative degree not exceeding two, and this corresponds to Evesque’s property
2.3.
28 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
We now look at the adaptive control problem for two cases: first for systems with a
relative degree of one; and then for systems with a relative degree not exceeding two,
where the relative degree n∗ of a transfer function P(s) is defined as the number of
poles of P(s) minus the number of zeros (see equation (2.9)).
Relative degree n∗ = 1
When n∗[P(s)] = 1 then n∗[P∗cl(s)] = 1 as well and P
∗
cl(s) is SPR.
1 The adaptive con-
troller for such a system is shown in figure 3.3(a). We now define the control vector
k(t) = [k1(t) k2(t)]T and the data vector d(t) = [p(t) Vz(t)]T , where Vz(t) =
V (t)
s+zc
, with
zc chosen and fixed. By defining k˜ = k−k∗, where k∗ denotes the k for which the
closed-loop system is stabilized, the total feedback signal can be written as kT d =
k∗T d+ k˜T d. Using this expression, the closed-loop system can be represented as
shown in figure 3.3(b). Therefore the closed-loop system Pcl(s) at time t is identi-
cal to the closed-loop stabilized system P∗cl(s) with input disturbance −k˜T d, which
means that p(t) = P∗cl(s)[−k˜T d(t)]. In state-space form, this can be written as
x˙(t) = Ax(t)+B[−k˜T (t)d(t)]
p(t) =Cx(t).
(3.5)
From lemma 2.4, Narendra & Annaswamy (1989), the strict positive realness of
P∗cl(s) assures the existence of a matrix Q = Q
T > 0 which satisfies
AT Q+QA =−R
QB =CT ,
(3.6)
where R also satisfies R = RT > 0.
We now want to find a Lyapunov function Λ(t) that is strictly positive and that
decays in time when the control parameters k(t) are updated correctly. The existence
of such a function will be used to prove the asymptotic stability of the closed-loop
1For the third condition Re[P∗cl(s)]> 0, the pole and zero pairs must also interlace in frequency.
3.3. LYAPUNOV ADAPTIVE CONTROL 29
(a)
P (s)
1
s+ zc
V
Vz
p
k1
k2
−−
P ∗cl(s)
−k˜Td
P (s)
1
s+ zc
V
Vz
p
k∗1
k∗2
− −
(b)
FIGURE 3.3: Block diagram of the adaptive controller: (a) the original block diagram, and
(b) the equivalent closed-loop stabilized system P∗cl(s) with input disturbance −k˜T d.
system. We choose this function to be quadratic in both x(t) and k˜(t):
Λ(t) = x(t)T Qx(t)+ k˜T (t)k˜(t)≥ 0.
Evaluating the time derivative of Λ(t) using (3.5) and (3.6) gives
Λ˙(t) =−xT (t)Rx(t)+2k˜T (t)[ ˙˜k(t)− p(t)d(t)],
and −xT (t)Rx(t)≤ 0, so choosing the updating rule and control law
k˙(t) = p(t)d(t) (3.7a)
V (t) =−kT (t)d(t) (3.7b)
ensures that Λ˙(t)≤ 0. It follows that k˜(t) and x(t) are bounded. Now (3.5) shows that
x˙(t) is bounded. Finally lemma 2.12, Narendra & Annaswamy (1989) shows that x(t)
and therefore p(t) are guaranteed to converge asymptotically to zero.
In all of the above we have assumed that the high-frequency gain of the open-loop
30 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
1
s+ λ
s+ λkT
−d −dλ V
FIGURE 3.4: Modification of the data vector d.
system is positive. If the high-frequency gain of the open-loop system is negative, then
the SPR arguments can be applied to −P∗cl(s) instead. Then a more general updating
rule for the adaptive parameters k(t) is given by
k˙(t) = sgn(go)p(t)d(t), (3.8)
where go is the high-frequency gain of P(s).
Relative degree n∗ = 2
The third SPR condition Re[P∗cl(s)]> 0 can only be met by n
∗ = 1 systems. Therefore
for n∗ = 2 systems, P∗cl(s) is not SPR. For the right choice of λ , however, the transfer
function (s+λ )P∗cl(s) is SPR. Therefore Annaswamy et al. (1998) proposed modifying
the data vector d(t) as shown in figure 3.4. This modification makes the closed-loop
system SPR (whilst avoiding explicit differentiation of the data vector d(t)), and results
in a modified updating rule and control law
k˙(t) = sgn(go)p(t)dλ (t) (3.9a)
V (t) =−kT (t)d(t)− k˙T (t)dλ (t), (3.9b)
where dλ (t) is the modified data vector dλ (t) = 1s+λ d(t).
If k were held fixed, then both controllers (3.7) and (3.9) would correspond to the
first-order compensator
K(s) = k1
s+ zc
s+(k2+ zc)
.
Comparing this with (3.4), we see that k1(t) acts to tune the gain of the adaptive con-
troller, that k2(t) acts to tune the position of the controller’s pole, and that choosing zc
3.3. LYAPUNOV ADAPTIVE CONTROL 31
corresponds to specifying the controller’s zero.
This adaptive controller has been applied previously to plane modes in a longi-
tudinal combustor, where one fuel valve and one pressure transducer suffice for con-
trol (Evesque, 2000; Evesque et al., 2003b; Riley et al., 2004). We will develop this
controller for circumferential instabilities in axisymmetric annular combustors, where
multiple fuel valves and multiple pressure transducers are required for control. First,
though, we derive expressions for the controller’s adaptation rates.
3.3.2 ADAPTATION RATES
For the adaptive control of real systems, some command over the rate of adaptation of
the controller is required: this can be achieved by introducing adaptation rates in (3.9a)
to give
k˙(t) = Γsgn(go)p(t)dλ (t), (3.10)
where Γ= diag(γ1 γ2) is a diagonal matrix containing the adaptation rates. Adaptive
control may at this point appear to be a Sisyphean task when the requirement to specify
k1 and k2 has been replaced by the requirement to choose two new parameters γ1 and γ2.
However, the permissible range of values for γ1 and γ2 is several orders of magnitude
greater than that for k1 and k2. Furthermore, expressions for γ1 and γ2 which are
functions of known parameters only will now be developed.
The adaptation rates can be set by specifying the desired speed of adaptation (Yildiz
et al., 2007). If the adaptive parameters start at zero, then this corresponds to 3τn for a
5 % band around the set point (where τn is the period of the unstable mode):
∣∣k˙1(t)∣∣= |k∗1|3τn =
∣∣∣∣γ1 p(t) p(t)s+λ
∣∣∣∣∣∣k˙2(t)∣∣= |k∗2|3τn =
∣∣∣∣γ2 p(t)Vz(t)s+λ
∣∣∣∣ .
32 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
Therefore at the instability’s frequency, s = iωn,
γ1 =
|k∗1||iωn+λ |
3τn pˆ2
γ2 =
|k∗2||iωn+λ |
3τn|pˆVˆz|
,
where pˆ and Vˆz are characteristic values of p(t) and Vz(t) respectively. Using an order
of magnitude analysis, one can arrive at expressions for the adaptation rates in terms of
the known parameters k∗1, ωn, zc and pˆ. (It suffices to know only the order of magnitude
of each of these parameters.)
λ andωn will be comparable in size, so we make the approximation |iωn+λ | 'ωn,
giving
γ1 =
|k∗1|ω2n
6pi pˆ2
.
Determining γ2 is a little more involved, since expressions for k∗2 and Vˆz are re-
quired. Using
Vz(t) =
V (t)
s+ zc
=
V (t)
iωn+ zc
at the instability frequency, and
V (t) = k1
s+ zc
s+(zc+ k2)
p(t)
Vˆ ∼ k∗1 pˆ,
together with
k2 = pc− zc
k∗2 ∼ ωn
(where−pc is the pole of the final stabilizing controller) gives a final expression for γ2
of
γ2 =
ω3n
√
ω2n + z2c
6pi|k∗1|pˆ2
.
A final note on the choice of pˆ: this will depend on whether or not a limit cycle
3.4. MODAL ADAPTIVE CONTROLLER 33
is already present when control is activated. If the pressure perturbation is still grow-
ing when control is activated, pˆ is given by the (order of magnitude of the) pressure
perturbation that one wants the controller to allow: previous pressure perturbations of
smaller orders of magnitude than this will give rise to negligible rates of change of the
adaptive parameters in (3.10). If the pressure perturbation has already established a
limit cycle when control is activated, however, then the choice of pˆ is instead imposed
by the limit cycle’s amplitude.
These expressions provide a systematic way of choosing the adaptation rates, which
were previously chosen by trial-and-error.
3.4 MODAL ADAPTIVE CONTROLLER
We now look at adaptive control of circumferential modes in annular combustors. This
requires multiple pressure measurements and multiple fuel valves for feedback control,
and therefore MIMO control (Morgans & Stow, 2007).
Instead of using MIMO control techniques, the adaptive controller that we now
develop draws on the SISO adaptive controller already described, and relies on the
combustor having an axisymmetric geometry. (For a large number of equally-spaced,
identical burner valves, the combustor can still be considered axisymmetric with the
burners in place – see Akamatsu & Dowling (2001) or Evesque & Polifke (2002).)
The combustor’s axisymmetric geometry allows each unstable mode to be treated
separately as a single-input single-output system, with open-loop modal transfer func-
tion Pn(s) given by (3.2). Since the modal adaptive controller works in terms of modal
pressure pn(t) and modal voltage V n(t), we must find expressions relating the indi-
vidual actuator voltages Vb(t) and individual transducer pressures pt(t) to their modal
equivalents. For a combustor with B burners (with actuation at all burner locations),
and if the pressure perturbation at some downstream location is measured using T pres-
sure transducers equally-spaced around the combustor circumference, then the modal
34 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
voltage and pressure can be found using
pn(t) =
1
T
T
∑
t=1
pt(θ , t)Φn1(θ) pt(θ , t) =
N
∑
n=−N
pn(t)Φn2(θ) (3.11)
V n(t) =
B
∑
b=1
Vb(θ , t)Φn1(θ) Vb(θ , t) =
1
B
N
∑
n=−N
V n(t)Φn2(θ), (3.12)
where
Φn1(θ) =

1 n = 0
2cos(nθ) n > 0
2sin(nθ) n < 0
Φn2(θ) =

1 n = 0
cos(nθ) n > 0
sin(nθ) n < 0.
(3.13)
The modal parameters pn(t) and V n(t) represent the discrete spatial Fourier trans-
forms of the transducer pressures pt(t) and actuator voltages Vb(t). A standing wave
representation is used because the controller parameters are updated in the time domain
(Stow & Dowling, 2009).1
From (3.13), the +n and −n modes are 90 degrees out of phase with each other for
a given n. Since Pn(s) is a SISO transfer function, the modal equivalent of the updating
law (3.10) and feedback control law (3.9b) can now be used:
k˙n(t) = Γnsgn(gno)p
n(t)dnλ (t) (3.14a)
V n(t) =−[kn]T (t)dn(t)− [k˙n]T (t)dnλ (t). (3.14b)
Figure 3.5 shows the adaptive control procedure in pictorial form: at some time t,
the adaptive controller is provided with a vector of pressure measurements p(t). From
this, and using (3.11), a vector of modal pressures pn(t) is calculated. For each mode n,
the modal pressure is used in the updating law for the adaptive parameters (3.14a), and
the modal voltage resulting from this controller is calculated using (3.14b). Finally, the
vector of modal voltages Vn(t) is used to calculate the vector V(t) using (3.12), which
gives the voltage required at each burner.
1At a given moment in time, one cannot determine if a given mode shape is spinning in time or
not. Therefore although one could use a spinning wave decomposition, we choose standing waves here,
which make more sense conceptually.
3.5. DESCRIPTION OF THE THERMOACOUSTIC NETWORK MODEL 35
Adaptive 
controller
p(t) pn(t) Vn(t) V(t)1
B
B∑
b=1
V nΦn2
1
T
T∑
t=1
ptΦn1
FIGURE 3.5: Modal adaptive control algorithm.
3.5 DESCRIPTION OF THE THERMOACOUSTIC NETWORK MODEL
LOTAN is a low order thermoacoustic network model developed by Dowling & Stow
(2003) for the simulation of longitudinal and annular combustion systems. Low order
modeling techniques are particularly attractive for annular combustors, owing to the
high cost of computational fluid mechanics and of large-scale experiments for such
geometries. The low order modeling approach is based on the fact that the main non-
linearity is in the combustion response to flow perturbations (Dowling, 1997). LOTAN
has been verified experimentally against both a sector rig (Stow & Dowling, 2001) and
an atmospheric test rig (Stow & Dowling, 2004). The combustion system is modeled
as a series of interconnected modules. Longitudinal ducts, annular ducts, combustion
zones and area changes are amongst the module types that can be modeled.
The model decomposes the flow into a steady mean axial component and small per-
turbations. The perturbations throughout the combustor are related via wave propaga-
tion, in which acoustic waves, entropy waves and vorticity waves are all included. The
flow conservation equations are used to track the evolution of these acoustic waves,
vorticity waves and entropy waves. A thin annulus is assumed: therefore axial and
azimuthal variations are considered, but radial variation of the flow parameters is ig-
nored. The connecting modules are modeled as acoustically compact, meaning that
their axial length is short in comparison to the acoustic wavelengths of interest. The
acoustic boundary conditions at the inlet and outlet of the combustor are assumed to
be known.
Combustion is assumed to take place at one axial location and to be acoustically
compact. The flame dynamics are modeled using a flame transfer function relating the
unsteady heat release Q to the fluctuation in equivalence ratio φ . This is motivated
by the fact that, for lean premixed prevaporized combustion, fluctuations in the inlet
fuel-air ratio have been shown to be the dominant cause of unsteady combustion – see,
36 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
for example Richards & Janus (1998) or Lieuwen & Zinn (1998). Saturation bounds
are applied to the unsteady heat release: this saturation leads to non-linear behaviour
of the flame and the development of a limit cycle.
Calculations can be performed in both the frequency domain and the time domain
(Stow & Dowling, 2009). Frequency domain calculations can be either linear or non-
linear. For linear calculations, the values of complex frequency that satisfy both the
inlet and outlet boundary conditions are found: this complex frequency gives the fre-
quency of oscillation and growth rate of a given mode. Non-linear calculations can
be performed to find the frequency and amplitude of the limit cycle. In this case, a
non-linear flame transfer function (where the unsteady heat release rate saturates) is
used whilst keeping the rest of the model linear. The resulting limit cycle’s frequency
and amplitude can then be calculated.
In time domain simulations, the transfer functions (excluding the flame) are con-
verted to a Green’s function and combined with the non-linear time domain model of
the flame. Noise can be introduced into the system in the form of white noise super-
imposed on to the heat release at each burner, which acts to simulate turbulence and
promotes the growth of unstable modes.
The two combustor geometries considered are simple representations of a Lean
Premixed Prevaporized (LPP) aero-engine combustor, depicted in figure 3.6. Each
combustor is made up of an annular plenum connected to an annular combustor by a
number of equally-spaced premix ducts. Both combustor geometries are axisymmetric
and therefore modal coupling is absent – see Akamatsu & Dowling (2001), where pre-
mix ducts leads to coupling of radial modes only, and Evesque & Polifke (2002), where
circumferential modal coupling only occurs for non-identical premix ducts. Having
uncoupled modes means that forcing the valve voltage in circumferential mode n gives
rise to pressure fluctuations in circumferential mode n only.
For closed-loop control in LOTAN, actuation can be provided by fuel valves at
each of the ducts, and sensing can be achieved using pressure transducers positioned
downstream of the combustion zone.
3.6. ADAPTIVE CONTROL OF AN ANNULAR COMBUSTOR IN LOTAN 37
3.6 ADAPTIVE CONTROL OF AN ANNULAR COMBUSTOR IN LOTAN
Using LOTAN, time domain simulations of an annular combustor with the modal adap-
tive controller (3.14) are now investigated. Control is first demonstrated for a com-
bustor with a circumferential n = ±1 unstable mode only. It is then shown that the
adaptive controller maintains control following a large change in combustor operating
conditions. Then by modifying the combustor geometry, an additional n= 0 longitudi-
nal instability is introduced. The adaptive controller stabilizes this modified system by
simultaneously controlling the n = 0 and n = ±1 modes. Finally, a more practicable
feedback control arrangement with fewer transducers and actuators is investigated. It
is observed that the modal adaptive controller still provides stability with only three
transducers and three actuators in place.
3.6.1 COMBUSTOR GEOMETRIES
The combustor geometries used are the same as those used by Morgans & Stow (2007).
Details of the geometry can be found in figure 3.6, whilst key parameters of the com-
bustor are summarized in table 3.1. Twenty burners (B = 20) and twenty pressure
transducers (T = 20) are positioned uniformly around the circumference of the an-
nulus. A choked inlet and choked outlet are used for the upstream and downstream
boundary conditions.
TABLE 3.1: Summary of unstable combustor’s key parameters.
Air mass flow rate, a¯ = 100 kg/s
Combustion temperature, Tc = 2500 K
Number of fuel valves, B = 20
Number of pressure transducers, T = 20
Actuator transfer function,1 Lb(s) = 0.1e−2.0×10
−4s
Flame transfer function, Fb(s) = 4.0e−1.5×10
−3s
Flame saturation properties: Qsat = 0.4 Q¯
1 The fuel valve can saturate at some points in the cycle if the fuel
flow rate fluctuation exceeds the mean (at which point the overall
fuel flow rate can reach zero).
38 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
pressure
pressure
transducer
burner
valve chokedchokedinlet outlet
fuel
flame flow
plenum
premix
ducts
sensing
0.100.10
0.15
combustor
lc
FIGURE 3.6: Annular combustor geometry. All dimensions in metres. lc = 0.30 m for the
circumferential mode only case, and lc = 0.71 m for the multiple mode case. The plenum
inner and outer radii are 0.22 m and 0.38 m respectively. The combustion chamber inner
and outer radii are 0.25 m and 0.35 m respectively. The locations of the actuators and
transducers used in § 3.7 are shown in black.
A simple gain/time delay flame transfer function of the form Fb(s) = k f e−τ f s is
used with saturation bounds. The flame transfer function used is very similar in form
to that used by Bellucci et al. (2005). The gain k f represents the flame’s linear gain,
the time delay τ f represents the convection time from fuel input to combustion, and
the saturation bound Qsat represents the maximum fractional change in heat release
that can occur. According to this model, the heat release responds linearly to flow per-
turbations for small oscillations. For unstable modes, though, the growing amplitude
causes the heat release to saturate. This saturation results in non-linear behaviour of
the flame and a limit cycle. Armitage et al. (2004) discuss flame models in more detail.
3.6.2 SINGLE CIRCUMFERENTIAL MODE
The Bode diagram of the open-loop n=±1 transfer function P±1(s) is shown in figure
3.7 for frequencies up to 2000 Hz (note the linear frequency scale). This transfer func-
tion is found using LOTAN: referring to equation (3.2), Lb(s) and Fb(s) are prescribed,
whilst the modal transfer functions Gn(s) and Hn(s) are found in LOTAN by forcing
3.6. ADAPTIVE CONTROL OF AN ANNULAR COMBUSTOR IN LOTAN 39
20
40
60
80
100
0 200 400 600 800 1000 1200 1400 1600 1800 2000
−1080
−900
−720
−540
−360
−180
0
180
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
Frequency [Hz]
FIGURE 3.7: Open-loop Bode diagram of the modal transfer function P±1(s).
their respective inputs over a discrete set of frequencies for the mode n of interest.
The poles of the system (3.2) correspond to the peaks of the Bode diagram: a phase
increase of 180◦ across a peak indicates an unstable conjugate pair of poles, whilst a
phase decrease of 180◦ indicates a stable conjugate pair (Dorf & Bishop, 2005). The
frequency and growth rate of the unstable mode are 521 Hz and 56.1 rad s−1. Figure
3.8 shows the limit cycle mode shape of the instability. The mode spins in time during
the limit cycle – an effect caused by the modal coupling introduced by the non-linearity
which is discussed in more detail by Schuermans (2003).1 From figure 3.7 it can be
seen that all of the zeros of the open-loop system are in the left half-plane. (Left
half-plane zeros induce an increase in phase, whilst right half-plane zeros induce a
reduction in phase.) Therefore assumption (i) concerning the position of the system’s
zeros is satisfied.
The parameter values used for the n = ±1 adaptive controller (in this and all
other test cases) are z±1c = 11000 rad s−1, γ
±1
1 = 5.0× 10−14, γ±12 = 1.0× 103, and
sgn(g±10 ) = +1. The time domain simulation of the closed-loop system with adaptive
controller is shown in figure 3.9. White noise is superimposed on to the mean heat
release at each burner – the amplitude of the noise is 20 % of the mean heat release
at each burner. Control is activated at 0.5 seconds, by which time a limit cycle has
1This modal coupling is only present during the non-linear limit cycle.
40 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
FIGURE 3.8: Pressure mode shape of the n =±1 spinning mode (arbitrary scale).
established itself. The adaptive controller finds stabilizing modal control parameters
k±1 within approximately 0.2 seconds, and the pressure measurements are reduced to
the background noise level. The adaptive controller arrives at identical modal con-
trollers for the n =−1 (k−1) and n =+1 (k+1) modes: this corresponds to there being
equal amounts of each mode present in the spinning limit cycle. Figure 3.10 shows a
zoomed-in section of the time domain simulation for 0.70≤ t ≤ 0.72. Only three of the
twenty pressure measurements and actuated fuel mass flow rates are shown for clarity.
From the phase difference observed between these three pressure measurements, it is
clear that the mode shape is spinning in time when control is activated.
3.6.3 CHANGING OPERATING CONDITIONS
One of the distinct advantages of adaptive control is its ability to deal with large
changes in the plant transfer function. This is now demonstrated by varying the com-
bustor’s flame transfer function F(s) with time once control of the nominal plant has
been achieved. Specifically, the flame transfer function gain k f (refer to table 3.1) is
varied linearly from 4.0 to 5.2 over a 0.2 second period. Figure 3.11 shows the modal
adaptive controller’s response to this change in operating conditions. The adaptive pa-
rameters k±1(t) re-adapt and find a new stabilizing controller, and closed-loop stability
is maintained despite the change. Note that a fixed controller with the same parameters
3.6. ADAPTIVE CONTROL OF AN ANNULAR COMBUSTOR IN LOTAN 41
0 0.2 0.4 0.6 0.8 1 1.2
−5
0
5
x 105
0 0.2 0.4 0.6 0.8 1 1.2
−0.05
0
0.05
0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
x 10−5
0 0.2 0.4 0.6 0.8 1 1.2
−10000
−5000
0
p′
[P
a]
f∇
[-]
kn 1
kn 2
Time [s]
FIGURE 3.9: Adaptive control of the n = ±1 unstable mode. The upper two plots show
the measured pressure perturbation p′ and the actuated fractional fuel mass flow rate f∇.
The lower two plots show the adaptation of the modal control vector kn(t) for n = +1
(——) and n =−1 (−4−).
as the original adaptive controller would have lost control following this change.
3.6.4 MULTIPLE UNSTABLE MODES
By increasing the combustion chamber’s length from 0.30 m to 0.71 m (see figure 3.6),
an n = 0 longitudinal instability is introduced, with the frequency and growth rate of
the n =±1 mode changing slightly. Table 3.2 shows the frequencies and growth rates
of the two modes, and figure 3.12 shows the two modes’ open-loop Bode diagrams.
Again, and for both transfer functions, all of the zeros are in the left half-plane, and so
assumption (i) concerning the position of the system’s zeros is satisfied.
The parameter values used for the n= 0 adaptive controller are z0c = 11000 rad s
−1,
γ01 = 1.5×10−15, γ02 = 3.0×102 and sgn(g00) = +1. The time domain simulation for
42 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
0.7 0.702 0.704 0.706 0.708 0.71 0.712 0.714 0.716 0.718 0.72
−4
−2
0
2
4
x 105
0.7 0.702 0.704 0.706 0.708 0.71 0.712 0.714 0.716 0.718 0.72
−0.04
−0.02
0
0.02
0.04
0.7 0.702 0.704 0.706 0.708 0.71 0.712 0.714 0.716 0.718 0.72
1.04
1.06
1.08
x 10−5
0.7 0.702 0.704 0.706 0.708 0.71 0.712 0.714 0.716 0.718 0.72
−7600
−7400
−7200
p′
[P
a]
f∇
[-]
kn 1
kn 2
Time [s]
FIGURE 3.10: Adaptive control of the n = ±1 unstable mode: zoomed-in section for
0.70 ≤ t ≤ 0.72. The upper two plots show the measured pressure perturbation p′ and
the actuated fractional fuel mass flow rate f∇. Data for only three pressure transducers
and three actuators at azimuthal angles of θ = 0◦ (——), θ = 126◦ (−−−) and θ = 234◦
(—·—) are shown for clarity. The lower two plots show the adaptation of the modal control
vector kn(t) for n =+1 (——) and n =−1 (−−−).
the multiple unstable mode case is shown in figure 3.13. For no control whatever, the
longitudinal mode with the higher growth rate grows first and dominates the response
in the limit cycle. Thus only the longitudinal mode is present when control is activated
at 0.5 seconds. The modal adaptive controller arrives at stabilizing adaptive parameters
ko(t) after approximately 0.2 seconds. Following this the circumferential mode grows:
the adaptive parameters k±1(t) adapt accordingly and find a stabilizing controller. The
combustion system is completely stabilized.
3.7. CONTROL USING FEWER TRANSDUCERS AND ACTUATORS 43
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−5
0
5
x 105
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.06
−0.03
0
0.03
0.06
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
x 10−5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−10000
−5000
0
p′
[P
a]
f∇
[-]
kn 1
kn 2
Time [s]
FIGURE 3.11: Time domain simulation of adaptive control of the n =±1 unstable mode.
Flame transfer function gain k f increased linearly from 4.0 at t = 1.0 s to 5.2 at t = 1.2 s.
Legend same as figure 3.9.
3.7 CONTROL USING FEWER TRANSDUCERS AND ACTUATORS
To ensure that the axisymmetry assumption still holds with the adaptive feedback con-
troller in place, twenty transducers and twenty actuators were used. From practical
considerations, though, this is a large number. Therefore it is interesting to investigate
TABLE 3.2: Frequencies and growth rates of the two unstable modes.
n Frequency [Hz] Growth rate [rad s−1]
0 616 16.0
±1 511 11.3
44 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
20
40
60
80
100
120
0 200 400 600 800 1000 1200 1400 1600 1800 2000
−1080
−900
−720
−540
−360
−180
0
180
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
Frequency [Hz]
FIGURE 3.12: Open-loop Bode diagram of the two modal transfer functions: Po(s) (——)
and P±1(s) (−−−).
the behaviour of the closed-loop system with fewer transducers and actuators in place.1
To construct the modal pressure vector pn = [p−N . . . p+N ]T from the vector of
measured pressures p = [p1 . . . pT ]T , the minimum number of transducers needed is
given by the maximum circumferential mode of interest N. Since one must ensure
that the transducers do not find themselves all simultaneously at pressure nodes of the
combustor, the minimum number of transducers required Tmin is given by Tmin = 2N+
1. A similar argument can be applied to the number of actuators to give Amin = 2N+1.
(It is worth noting that reducing the number of measurement points will not affect the
combustor’s axisymmetry, whilst reducing the number of actuation points will.)
Since it was assumed previously that actuation occurred at all burner locations
(i.e. A = B), the control voltage vector is now of reduced dimension V = [V1 . . . VA]T .
In this simpler case, (3.11) still holds: the modal pressure amplitudes pn are real quan-
tities and can still be deduced from a reduced number of pressure measurements. For
the control voltage, though, the modal voltage Vn is not being deduced from the ac-
tuator voltages V. Rather, for a given Vn, which is provided by the modal adaptive
controller (figure 3.5), we want to find the combination of actuator voltages V which
1Although the number of actuators is reduced, the combustor still has twenty burners: the reduction
means that closed-loop actuation is only taking place at some rather than all burner locations.
3.7. CONTROL USING FEWER TRANSDUCERS AND ACTUATORS 45
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
−5
0
5
x 105
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
−0.02
0
0.02
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
0.5
1
x 10−5
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
−6000
−4000
−2000
0
p′
[P
a]
f∇
[-]
kn 1
kn 2
Time [s]
FIGURE 3.13: Time domain simulation of adaptive control of the n = 0 and n = ±1
unstable modes. The upper two plots show the measured pressure perturbation p′ and the
actuated fractional fuel mass flow rate f∇. The lower two plots show the adaptation of the
modal control vector kn(t) for n = 0 (——), n =+1 (−−−) and n =−1 (—·—).
best approximates this modal voltage. Therefore instead of the discrete Fourier trans-
form pair in (3.12), we simply write
Va(θ , t) =
1
A
N
∑
n=−N
V n(t)Φn2(θ).
For the combustor considered N = 1, so Tmin = Bmin = 3 and therefore we use three
actuators and three transducers. Figure 3.6 shows the location of the three actuators
and three transducers, which are at azimuthal angles of 0◦, 126◦ and 234◦.
Figure 3.14 shows time domain results for this simpler control arrangement, ap-
plied to the combustion geometry with the n =±1 unstable mode only (as in § 3.6.2).
All other control parameters are the same as those used in § 3.6.2. Comparison with
46 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
0 0.2 0.4 0.6 0.8 1 1.2
−5
0
5
x 105
0 0.2 0.4 0.6 0.8 1 1.2
−0.2
0
0.2
0 0.2 0.4 0.6 0.8 1 1.2
0
1
2
x 10−5
0 0.2 0.4 0.6 0.8 1 1.2
−10000
−5000
0
p′
[P
a]
f∇
[-]
kn 1
kn 2
Time [s]
FIGURE 3.14: Time domain simulation of adaptive control using fewer transducers and
actuators. The upper two plots show the measured pressure perturbation p′ and the ac-
tuated fractional fuel mass flow rate f∇. The lower two plots show the adaptation of the
modal control vector kn(t) for n =+1 (——) and n =−1 (−−−).
figure 3.9 reveals the behaviour of this simpler arrangement and the fully-instrumented
arrangement to be very similar. The most obvious difference is the increased magni-
tude of the fuel mass flow rate actuation required for control, which in this simpler case
is 15 % of the mean, compared with 3 % of the mean in figure 3.9. This is explained
by the greater control authority required at each actuation point when the total num-
ber of actuators is reduced. One also observes that the time required for the adaptive
parameters kn to settle to stabilizing values is greater, in particular for the n =−1 cir-
cumferential mode. Finally, the discrepancy between the final control vectors k−1 and
k+1 is greater than that seen in figure 3.9. Nevertheless, both modal controllers arrive
at stabilizing control parameters and the combustion system is stabilized.
3.8. CONCLUDING REMARKS 47
3.8 CONCLUDING REMARKS
In this chapter, two properties of unstable aero-engine combustion systems that com-
plicate their stabilization have been addressed: their annular geometry and therefore
their propensity for circumferential modes; and their potential for large changes in op-
erating conditions. For these reasons the adaptive controller presented is well-suited to
real engine environments.
We finish by remarking that the adaptive feedback controller considered operates
outside of the restrictions placed upon it in its theoretical development. The Lyapunov-
based stability analysis of § 3.3 is valid only for linear plants, and yet control is acti-
vated from within a non-linear limit cycle. The SISO analysis used in § 3.2.1 also
relies on an axisymmetry assumption, and yet the adaptive controller of § 3.7 clearly
neglects this assumption. (Although the controller of § 3.6 respects it.)
48 CHAPTER 3: ADAPTIVE CONTROL OF ANNULAR COMBUSTION SYSTEMS
CHAPTER 4
ADAPTIVE CONTROL FOR UNKNOWN CONTROL DIRECTION
One of the fundamental (and troublesome) requirements of classical, Lyapunov-based
adaptive control is that the sign of the high-frequency gain of the system must be known
(Narendra & Annaswamy, 1989). Since time delays are always present in combustion
systems, they are of infinite dimension. Therefore the sign of the high-frequency gain
is never known a priori and must somehow be determined. For this reason a controller
that does not require this information presents advantages for the adaptive control of
combustion oscillations.
In this chapter, an adaptive feedback controller is applied to a simple unstable com-
bustion system: the Rijke tube. The controller does not require knowledge of the sign
of the high-frequency gain of the system, and this represents the first experimental
demonstration of such a controller. Control is maintained following a large change in
operating conditions, brought about by increasing the length of the Rijke tube after
control of the nominal system has been achieved.
4.1 IMPORTANCE OF THE HIGH-FREQUENCY GAIN
At the heart of Lyapunov-based adaptive control are three assumptions. If the open-
loop system has transfer function P(s), and if we write P(s) as the ratio of two coprime,
monic polynomials
P(s) = go
NP(s)
DP(s)
,
then the three conditions placed on P(s) are (Narendra & Annaswamy, 1989)
(i) The sign of the high-frequency gain go is known.
49
50 CHAPTER 4: ADAPTIVE CONTROL FOR UNKNOWN CONTROL DIRECTION
(ii) The relative degree n∗ of P(s) is known.1 (I)
(iii) The zeros of P(s) are all stable (i.e. the system is minimum phase).
It is the first condition that we concern ourselves with here. For many systems, the
sign of the high-frequency gain (whose physical significance is given in the following)
is obvious from the physics of the problem, and so providing it poses no problem. For
combustion systems, however, which are infinite-dimensional, the sign of the high-
frequency gain of the system is unknown and must be determined. (The phase of such
a system evolves indefinitely with frequency, and so the sign of its high-frequency gain
is not well-defined.)
4.1.1 PHYSICAL SIGNIFICANCE OF THE HIGH-FREQUENCY GAIN
To demonstrate the physical significance of the high-frequency gain, we now look at a
simple first-order unstable system whose transfer function is
P(s) =
y(s)
u(s)
=
go
s−a , (4.1)
where u is the system input, y is the system output, s= iω is the Laplace variable, go is
the high-frequency gain of the system and a (which is positive for an unstable system)
corresponds to the location of the system’s (only) pole.
In the time domain, this corresponds to the first-order differential equation
y˙(t) = ay(t)+gou(t).
Looking at y˙(t) for small time using the initial value theorem, and for a unit step input
u(s) = 1/s:
lim
t→0
y˙(t) = lim
s→∞s
2y(s)
= lim
s→∞s
2 1
s
go
s−a
= go.
1The relative degree is defined as n∗ = deg[DP(s)]−deg[NP(s)].
4.1. IMPORTANCE OF THE HIGH-FREQUENCY GAIN 51
Therefore knowing the sign of the high-frequency gain corresponds to knowing,
in the time domain, the sign of the system’s ‘instantaneous gain’: that is, knowing
whether the system’s response to a positive unit step input is positive or negative for t
sufficiently small. More generally, for an nth order system with a relative degree of n∗,
the same analysis can be applied to the (n∗)th derivative of the output, y(n∗).
4.1.2 IMPORTANCE FOR FEEDBACK CONTROL
We can see the significance of go for feedback control by again considering the simple
first-order unstable system (4.1), this time remaining in the frequency domain through-
out. If we introduce a feedback gain as shown in figure 4.1, the overall closed-loop
transfer function Pcl(s) is (negative feedback convention)
Pcl(s) =
P(s)
1+ kP(s)
=
go
s+(kgo−a) .
Since the controller gain k acts to stabilize the system by moving the closed-loop
system’s pole into the left half plane (i.e. by making the term (kgo− a) positive), we
see that the gain k must be both of the correct sign (so that the product kgo is positive)
and sufficiently large kgo > a. Therefore the sign of go dictates the sign of k required
for closed-loop stability. Again, this result still holds true for an nth order system, as
long as it has a relative degree n∗ of one. (It can be shown, using root locus arguments,
that any minimum phase system with a relative degree of one can be stabilized using a
feedback gain alone.)
P (s)
k
u y
−
Pcl(s)
FIGURE 4.1: Feedback arrangement using a feedback gain k.
52 CHAPTER 4: ADAPTIVE CONTROL FOR UNKNOWN CONTROL DIRECTION
4.1.3 IMPORTANCE FOR LYAPUNOV-BASED ADAPTIVE CONTROL
We now look at how these arguments apply to Lyapunov-based adaptive control. A
system of relative degree one that satisfies the three conditions (I) can be stabilized
using just a feedback gain k (Dorf & Bishop, 2005). We have seen that this feedback
gain needs to be (i) of sufficient magnitude and (ii) of the right sign. For a Lyapunov-
based adaptive controller, the second condition must be provided (see the updating law
in (3.9a), for example). Then the adaptive controller is guaranteed to meet the first
condition and provide closed-loop stability.
Although we have already seen in chapter 3 how sgn(go) is used in the updating
law for Lyapunov-based adaptive control (see equation (3.8), for example), we look
specifically at the feedback gain only case here for ease of exposition. In this case, the
sign of the high-frequency gain of the system is used to form an adaptive controller
with updating rule and control law
k˙(t) = sgn(go)y2(t) (4.2)
u(t) =−k(t)y(t), (4.3)
which guarantees stability of the closed-loop system. It is clear how the sgn(go) is used
in (4.2) to decide in which direction to update the control gain k.
4.2 ADAPTIVE CONTROL USING A NUSSBAUM GAIN
Morse (1983) suggested that knowledge of the sign of the high-frequency gain was in-
trinsic: in particular, that a linear plant cannot be stabilized without knowledge of the
sign of go. Nussbaum (1983) proved this conjecture for first order rational controllers.
More importantly, however, by using a non-rational term he constructed a globally
adaptively stabilizing controller for a first order system which did not rely on knowl-
edge of the sign of go. The work of Willems & Byrnes (1984) extends Nussbaum’s
result to any minimum-phase system with a relative degree of one.
The controllers used by Nussbaum (1983) and by Willems & Byrnes (1984) both
incorporate a Nussbaum gain N(k). Willems & Byrnes (1984) show that for the control
4.2. ADAPTIVE CONTROL USING A NUSSBAUM GAIN 53
of a minimum phase system with a relative degree of one, any function N(k) which
satisfies the conditions
sup
k>0
1
k
∫ k
0
N(σ)dσ = ∞
inf
k>0
1
k
∫ k
0
N(σ)dσ =−∞
(4.4)
is a Nussbaum gain and guarantees closed-loop stability when the sign of the high-
frequency gain is unknown. Such a Nussbaum gain can be used with the updating rule
and control law
k˙(t) = y2(t)
u(t) =−N(k)y(t)
to adaptively stabilize the system. These are very similar to the updating rule (4.2) and
control law (4.3) for the Lyapunov controller. The key difference is that the Nussbaum
gain – and the conditions placed upon it – remove the need for knowledge of sgn(go).
Functions satisfying (4.4) are a type of switching function. The review paper of
Ilchmann (1999) gives some examples of such functions, including:
N1(k) = k2 cosk
N2(k) =
{
k if n2 ≤ |k|< (n+1)2, n even
−k if n2 ≤ |k|< (n+1)2, n odd.
It is now clear how the Nussbaum gain achieves control without knowledge of the
sign of the high-frequency gain. The conditions imposed upon a Nussbaum gain (4.4)
mean that it sweeps across both positive and negative gains, whilst also increasing in
magnitude with increasing k. Provided that this occurs, the system can be stabilized
without knowledge of sgn(go): in contrast to the Lyapunov-based adaptive controller,
a Nussbaum gain automatically meets both requirements for closed-loop stability.
In closed-loop control, unnecessarily high feedback gains are to be avoided, not
least because they can place high demands on the actuator, and because they can lead
54 CHAPTER 4: ADAPTIVE CONTROL FOR UNKNOWN CONTROL DIRECTION
to excitation of other modes of the system. Using a Nussbaum gain which is a smooth
function helps to alleviate any high gain behaviour of the Nussbaum gain. Furthermore,
one can help to ensure that the final stabilizing Nussbaum gain, N∗, is in the vicinity
of the minimum required, and not significantly larger, by observing that the Nussbaum
gain only need switch sign once, if at all. This is explained more fully in appendix D,
where choosing the adaptation rate µ (introduced later in equation (4.7)) is discussed.
4.3 NUSSBAUM GAIN ADAPTIVE CONTROL APPLIED TO A RIJKE TUBE
We now look at applying a Nussbaum-gain adaptive controller to a simple laboratory-
scale experiment: the Rijke tube. We will see that the controller is able to stabilize
the system without knowledge of sgn(go), and that the controller is robust to changing
operating conditions. To the author’s knowledge, it represents the first experimental
results for a Nussbaum-type adaptive controller.
4.3.1 EXPERIMENTAL ARRANGEMENT
The Rijke tube provides a simple means of generating combustion oscillations on a
laboratory scale. It consists of a vertical cylindrical tube open at both ends, with a heat
source inside the tube some distance from the lower end. Large-amplitude acoustic
oscillations are excited when the heat source is in the lower half of the tube (Howe,
1998).
Active control has previously been demonstrated on a Rijke tube (Dines, 1983;
Lang et al., 1987; Heckl, 1988). It has been observed that the system can be stabilized
using a feedback gain alone (i.e. phase compensation is not required), and so the Rijke
tube provides a suitable test-bed for a Nussbaum-gain adaptive controller.
The Rijke tube used consists of a cylindrical quartz tube of length 750 mm and di-
ameter 44 mm inclined vertically as shown in figure 4.2(a). A propane-fuelled Bunsen
burner provides a laminar flame which is stabilized on a grid 210 mm above the bottom
of the tube. In the absence of control, the Rijke tube exhibits oscillations at a frequency
of 244 Hz.
Measurement of the pressure perturbation (denoted p) is provided by a microphone
4.3. NUSSBAUM GAIN ADAPTIVE CONTROL APPLIED TO A RIJKE TUBE 55
filter
DSP board
amplifier
microphone
flame
Microphone
Loudspeaker
Adaptive
controller
pref
Vc
Rijke 
tube
loudspeaker
V
p
(a)
−
N(k)
P (s)V
p
Rijke tube
(b)
FIGURE 4.2: Experimental set-up of the Rijke tube: (a) schematic; and (b) corresponding
block diagram.
fitted to a tube tapping situated 410 mm from the bottom of the tube. The semi-infinite
line technique is used to obtain thermal insulation without distortion from acoustic
reflections. Actuation is achieved using a 50 W low-frequency loudspeaker (with the
control voltage to the loudspeaker before amplification denoted V – see figure 4.2(a))
situated close to the lower end of the tube. The loudspeaker exhibits a flat frequency
response over the frequency range of interest (50–2000 Hz). Outside of this range, its
dynamics becomes more complicated. To eliminate this dynamics, a bandpass filter is
applied to the microphone signal with a pass-band from 50 Hz to 2000 Hz.
4.3.2 NUSSBAUM ADAPTIVE CONTROLLER
The Rijke tube can now be treated like the general plant described in § 4.2, where the
microphone voltage V corresponds to the system input u, and the pressure perturbation
p corresponds to the system output y. This is seen more clearly in figure 4.2(b), where
an arrow crossing a circle indicates an adaptive parameter. The updating rule and
56 CHAPTER 4: ADAPTIVE CONTROL FOR UNKNOWN CONTROL DIRECTION
control law for the adaptive controller are
k˙(t) = γ p2(t) (4.5)
V (t) =−N(k)p(t), (4.6)
and the Nussbaum gain used is
N(k) = k cos(µ|k| 14 ), (4.7)
with γ = 100 and µ = 0.53. Here γ and µ are the adaptation rates of the controller,
explained in more detail in appendix D.
The theoretical development of the Nussbaum gain in § 4.2 ignores the influence of
noise in the system. Since noise will always be present in the pressure measurement p
(even when the system has been stabilized), we see from (4.5) that k(t) will never be
stationary. This problem can be solved by introducing a dead-zone into the updating
rule (Ioannou & Sun, 1996). With this modification, k(t) is only updated if the pressure
p exceeds some pre-defined limit (which is chosen based on the magnitude of the noise
in the system). The updating rule for k(t) then becomes
k˙(t) =
{
γ p2(t) |p(t)| ≥ Pdeadzone
0 |p(t)|< Pdeadzone.
(4.8)
We choose Pdeadzone = 0.04 in this particular case, which is about 5 % of the limit-cycle
amplitude.
This adaptive control algorithm is implemented on a TI C6713DSK DSP board
with ADC and DAC evaluation modules (an ADS8361EVM and a DAC7731EVM).
The pressure measurement p and control voltage V are recorded using a PC-based
data acquisition system. The sampling frequency for both the DSP board and the data
acquisition system is 5000 Hz.
4.4. EXPERIMENTAL RESULTS 57
4.4 EXPERIMENTAL RESULTS
Two sets of experimental results are now presented. First, for fixed operating condi-
tions, the Nussbaum controller achieves control even when it is first updated in the
wrong direction. Second, we study the robustness of the controller by changing the
length of the Rijke tube after control of the nominal system has been achieved. The
controller maintains control following this change.
4.4.1 FIXED OPERATING CONDITIONS
Figure 4.3(a) shows adaptive control results when the Nussbaum gain is initially up-
dated in the right direction. The pressure perturbation has already established a limit
cycle when control is activated at t = 0.2s. Since the Nussbaum gain is initially up-
dated in the right direction, the cosine term in the Nussbaum gain (4.7) does not switch
sign. Hence in this case the controller behaves in a similar way to a Lyapunov-based
adaptive controller, and the growth of the Nussbaum gain N(k) is very similar to that
of the adaptive parameter k.1 Within approximately 0.5 s of control being activated,
the controller finds a stabilizing gain N∗, and the system is stabilized.
Figure 4.3(b) shows adaptive control results when the Nussbaum gain is initially
updated in the wrong direction. The impact of this is clear: when control is first ac-
tivated at t = 0.2 s, the Nussbaum gain N(k) causes the Rijke tube’s limit cycle to
actually increase in amplitude. Soon after, however, the Nussbaum gain changes sign
and finds a stabilizing gain N∗ which is both of sufficient magnitude and of the right
sign. The system is stabilized within approximately 0.4 s of control being activated.
The value of the Nussbaum gain used in this case is 2.10, compared with 1.78 in figure
4.3(a).
4.4.2 VARYING OPERATING CONDITIONS
We now look at the robustness of the adaptive controller by increasing the length of the
Rijke tube after control of the nominal plant has been achieved. The length of the tube
is increased from 770 mm to 870 mm using a variable length attachment, which causes
1k starts from zero here (i.e. k(0) = 0), which is true for all three experiments.
58 CHAPTER 4: ADAPTIVE CONTROL FOR UNKNOWN CONTROL DIRECTION
0 0.2 0.4 0.6 0.8 1
−1
0
1
0 0.2 0.4 0.6 0.8 1
−1
0
1
0 0.2 0.4 0.6 0.8 1
0
1
2
p′
V
N
(k
)
Time [s]
(a)
0 0.2 0.4 0.6 0.8 1
−2
0
2
0 0.2 0.4 0.6 0.8 1
−5
0
5
0 0.2 0.4 0.6 0.8 1
−10
−5
0
p′
V
N
(k
)
Time [s]
(b)
FIGURE 4.3: Experimental results for fixed operating conditions: with (a) N(k) initially
updated in the right direction; and (b) N(k) initially updated in the wrong direction. In both
cases the measured pressure perturbation, control voltage and Nussbaum gain are shown.
4.4. EXPERIMENTAL RESULTS 59
0 5 10 15 20 25 30
−2
0
2
0 5 10 15 20 25 30
−5
0
5
0 5 10 15 20 25 30
−10
−5
0
5
p′
V
N
(k
)
Time [s]
(a)
20 25 30 35 40
−0.2
0
0.2
20 25 30 35 40
−0.5
0
0.5
20 25 30 35 40
3
4
5
p′
V
N
(k
)
Time [s]
(b)
FIGURE 4.4: Experimental results for varying tube length (N(k) initially updated in the
wrong direction): (a) tube length increased at approximately 18 s; and (b) zoomed-in plot
of the controller’s re-adaptation following the change in length. In both cases the measured
pressure perturbation, control voltage and Nussbaum gain are shown.
60 CHAPTER 4: ADAPTIVE CONTROL FOR UNKNOWN CONTROL DIRECTION
a shift in frequency of the unstable mode from 238 Hz to 214 Hz.1 Results are shown in
figure 4.4(a), where control is activated at t = 0.2 s, again from within the limit cycle.
The Nussbaum controller finds a stabilizing gain within approximately 0.4 s.
After approximately 18 s, the length of the Rijke tube is suddenly increased. The
Nussbaum gain is no longer stabilizing for this new length, but the controller re-adapts
and finds a new stabilizing gain. This is seen more clearly in figure 4.4(b) which shows
a zoomed-in section of figure 4.4(a). Here the time axis starts just before the length of
the Rijke tube is increased.
4.5 CONCLUDING REMARKS
This chapter has focused on adaptive control of combustion oscillations in a simple
experiment, and considers the adaptive control problem when the sign of the high-
frequency gain (or ‘control direction’) is unknown.
The proof of stability of the Nussbaum gain is for systems with a relative degree of
one. In the experiments, however, it adaptively stabilizes a gain-stabilizable system:
these two conditions are not equivalent (indeed, based on the observations made in
chapter 2, we expect the actuated Rijke tube to have a relative degree not exceeding
two).2 Furthermore, the proof is valid only for linear systems, while in the experiments,
the controller establishes control from within the Rijke tube’s pressure limit cycle. It
is clear, then, that the Nussbaum controller is capable of adaptive stabilization outside
of the restrictions placed upon it in its theoretical development.
1This first tube length is slightly longer than that for the fixed operating conditions case in § 4.4.1,
owing to the geometry of the variable length attachment.
2For minimum-phase systems, the class of n∗= 1 systems is a subset of the class of gain-stabilizable
systems, and so a n∗ = 1 condition is more restrictive than a gain-stabilizable condition.
CHAPTER 5
DYNAMICS OF CAVITY OSCILLATIONS
In this chapter the compressible flow past a two-dimensional rectangular cavity is con-
sidered. We focus on the dynamics that are important for feedback control purposes.
An historical perspective to the problem is first given. Direct numerical simulations
are then used to find the open-loop transfer function of the cavity flow between a body
force near the cavity’s leading edge and a pressure measurement at some point inside
the cavity. The results are compared to those given by a simple linear model of the
cavity flow, which is intended to be useful for feedback control design. The choice
of parameters for this linear model is based on flow data from the direct numerical
simulations. We finish by deriving some general properties of the cavity flow that are
useful for feedback control, and that will be used for adaptive control in chapter 6.
5.1 INTRODUCTION AND HISTORICAL PERSPECTIVE
Cavity flows are another example of a self-sustained oscillation, and can occur in a
wide variety of guises, including the flow past a sunroof in an automobile, and the flow
past aircraft landing gear wells and weapons bays.
The basic configuration of a cavity flow is shown in figure 5.1. We consider the
compressible flow past shallow cavities. The mechanism leading to self-sustained os-
cillations in such cavities is as follows: the shear layer formed at the cavity’s upstream
corner amplifies disturbances as they convect downstream, which are then scattered
into pressure fluctuations at the cavity’s downstream corner. These pressure fluctua-
tions propagate back upstream, and excite further disturbances in the shear layer via a
61
62 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
shear layer acoustic waves
flow
FIGURE 5.1: Basic configuration of a cavity flow.
receptivity process, creating a feedback loop. For suitable phase change of the distur-
bance, resonance occurs.
Cavity flows were studied as early as the 1950s by Roshko (1955) and Krishna-
murty (1956), but the flow-acoustic resonance phenomenon was first described by
Rossiter (1964). (A similar model had previously been proposed by Powell (1953)
for edge tones.) Rossiter provided a semi-empirical formula for the prediction of the
resonant frequencies, which are often referred to as the Rossiter tones. More details
about the physics of cavity flows can be found in the review articles of Rockwell &
Naudascher (1978), Blake & Powell (1986) and Colonius (2001).
Strategies to eliminate cavity oscillations can be split into two broad categories:
passive control and active control. We define these two broad classes of control as we
did in chapter 2 (figure 2.2). Passive control studies have typically involved geometry
modifications such as spoilers, fences and ramps (Heller & Bliss, 1975). Active, open-
loop control studies date back to Sarohia & Massier (1977), who used steady mass
injection at the base of the cavity, but later studies have typically involved forcing the
shear layer near the cavity’s upstream corner (Sarno & Franke, 1994; Stanek et al.,
2000).
A renewed interest in cavity flows in recent years is in part due to the possibil-
ity of using feedback (or ‘dynamic’) control to suppress oscillations. The idea of
using feedback to suppress cavity oscillations is not new (Gharib, 1987), but many
of the earlier studies used simple (trial-and-error) phase-locked controllers (Shaw &
Northcraft, 1999; Williams et al., 2000), which do not use models of the cavity flow
dynamics. More recently, model-based feedback control has been promoted (Rowley
5.2. CAVITY GEOMETRY AND NUMERICAL METHOD 63
et al., 2006): dynamical models of the cavity flow have typically been found using
either system identification methods (Kook et al., 2002; Kegerise et al., 2007a,b) or
the Proper Orthogonal Decomposition (Samimy et al., 2007), and standard techniques
such as Linear Quadratic control (Cabell et al., 2006) andH∞ loop shaping (Yan et al.,
2006) have been used for controller design. For an excellent recent review of the con-
trol of cavity oscillations, see Cattafesta et al. (2008).
Although many studies on feedback control of cavity oscillations already exist,
many of the system identification procedures have been performed for the open-loop
unstable system, which means that the linear dynamics are masked by non-linearities
which will inevitably appear. Furthermore, the linear modeling approach proposed by
Rowley et al. (2006) has not been fully explored. Since the validity of a linear model is
crucial for the design of an effective model-based feedback controller, we make finding
an accurate linear model our priority in this chapter. We then compare this dynamical
model – found directly from direct numerical simulations using system identification
techniques – to that given by the simple linear model proposed by Rowley et al. (2006).
5.2 CAVITY GEOMETRY AND NUMERICAL METHOD
The two-dimensional, compressible flow past a rectangular cavity is considered. The
flow conditions are for a Mach number M = 0.6, length to depth ratio L/D = 2.0,
a Reynolds number based on momentum thickness θ at the cavity leading edge of
Reθ = 56.8, and L/θ = 52.8.
The flow is solved using direct numerical simulation. The Navier-Stokes equations
are non-dimensionalized according to the usual compressible non-dimensionalization
– see Rowley (2001) for details. The simulations have been carefully validated using
grid resolution and boundary placement studies, together with comparison with exper-
imental data (Rowley et al., 2002). The simulation uses 240× 96 grid points inside
the cavity and 1008× 384 grid points above the cavity, which is sufficient to resolve
all scales at this Reynolds number.
At these conditions, the flow develops into a limit-cycle, with the two most ener-
getic peaks in the spectra occurring at Strouhal numbers of 0.40 and 0.70. These peaks
are in reasonable agreement with the first two modes predicted by the semi-empirical
64 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
formula of Rossiter (1964):
Stn =
fnL
U
=
n− γ
M+1/κ
, n = 1,2, . . . , (5.1)
which predicts the first two modes at Strouhal numbers of 0.32 and 0.74. Here n is
the mode number, and κ and γ are empirical constants corresponding to the average
convection speed of disturbances in the shear layer and a phase delay. γ = 0.25 and
1/κ = 1.75 are typical values, and the values used here.
5.3 SYSTEM IDENTIFICATION OF THE CAVITY FLOW
We now look at finding the open-loop transfer function of the cavity flow. Since the
open-loop system is unstable, the cavity must first be stabilized so that the linear open-
loop transfer function can be found. Stability is achieved by using a dynamic phasor
model, developed for the cavity flow by Rowley & Juttijudata (2005). This dynamic
phasor postulates a low-order model that captures the relevant dynamical features of
the non-linear, limit-cycling flow using
r˙(t) = σr(t)−αr3(t)
θ˙ = ω,
which describes oscillations at a frequency ω , with constants α > 0 and σ > 0. Then
the origin (r = 0) is unstable, and there is a stable periodic orbit given by r =
√
σ/α .
For this model, Rowley & Juttijudata (2005) consider a controller whose control signal
is a sinusoid at the same frequency as the natural flow, with suitably chosen phase, and
slowly varying amplitude. This control signal makes the origin (r = 0) globally attract-
ing, at least for the model. It was found that this controller eliminated oscillations in
direct numerical simulations, and so we use it for the current system identification.
Figure 5.2(a) shows the cavity geometry. Actuation is provided by a body force ρv
– where ρ is the fluid density and v is a velocity component which is perpendicular to
the free-stream – positioned in the shear layer a distance L/20 from the cavity upstream
corner. Three pressure sensors are used in total and measure the pressure half-way up
5.3. SYSTEM IDENTIFICATION OF THE CAVITY FLOW 65
L
D
M
p1
p2
p3
ρvc
(a)
− P (s)
K(s)
cavity
dynamic phasor
ρvf
ρvw ρvc p2
p3
p1
(b)
FIGURE 5.2: System identification: (a) the cavity geometry showing the location of the
body force and the pressure sensors, and (b) the corresponding closed-loop block diagram.
the upstream wall (denoted p1), half-way along the cavity floor (denoted p2) and half-
way up the downstream wall (denoted p3).
Figure 5.2(b) shows the feedback arrangement used for system identification. The
pressure measured on the downstream wall (p3) is used as the feedback signal for
the dynamic phasor. The total body force at the input is given by ρvc = ρvw− ρv f
(negative feedback convention), where ρvw is a broadband forcing signal and ρv f is
the feedback control signal from the dynamic phasor.1 Taking the total body force ρvc
and the pressure vector p = [p1 p2 p3]T as the system input and output, the cavity’s
open-loop transfer function P(s) = [P1(s) P2(s) P3(s)]T can be found, where
[p1 p2 p3]T = [P1(s) P2(s) P3(s)]Tρvc.
(If instead the broadband signal ρvw was used as the system input, the closed-loop,
stabilized transfer function would be recovered.)
Using ρvc and p as the input-output pair as described, spectral analysis can be
used to find the cavity’s open-loop transfer function. Tests were performed to ensure
that the forced cavity was behaving linearly for the forcing amplitudes used. This was
achieved using a sum of two sinusoids at the input, the frequencies of which were
chosen to match the frequencies of the two unstable modes, since it was reasoned that
non-linearities were most likely to first appear when forcing at these frequencies.
The open-loop transfer functions found for the three sensor locations are shown
in figure 5.3. At all three pressure probe locations, the first three Rossiter modes are
1ρvw is band-limited to avoid aliasing – where high frequency components can masquerade as lower
frequencies. Despite this, its bandwidth still extends well above the frequencies of interest.
66 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
−80
−60
−40
−20
0 0.5 1 1.5 2 2.5 3 3.5 4
−1080
−720
−360
0
360
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–](a)
−80
−60
−40
−20
0 0.5 1 1.5 2 2.5 3 3.5 4
−1080
−720
−360
0
360
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–](b)
−80
−60
−40
−20
0 0.5 1 1.5 2 2.5 3 3.5 4
−1080
−720
−360
0
360
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–](c)
FIGURE 5.3: Transfer functions measured from direct numerical simulations: (a) for pres-
sure probe 1, P1(s); (b) for pressure probe 2, P2(s); and (c) for pressure probe 3, P3(s).
5.4. LINEAR MODELS FOR CONTROL OF CAVITY OSCILLATIONS 67
seen at Strouhal numbers of 0.40, 0.70 and 0.98. These values agree with those found
by Rowley et al. (2002) for a Mach number of 0.6, and show reasonable agreement
with Rossiter’s formula (5.1), which predicts Strouhal numbers of 0.32, 0.74 and 1.17.
The phase increase seen across the first and the second Rossiter modes indicates that
these two modes are unstable (Dorf & Bishop, 2005), which explains their dominance
in the open-loop spectra. The third Rossiter mode is stable, since one observes a phase
decrease across it.
We note briefly here the identified system’s time delay. Since the transfer function
of a pure time delay of duration τ is e−sτ , a time delay introduces a negative gradient to
a transfer function’s phase: from this gradient, a system’s time delay can be deduced.
The time delay seen at high frequencies in figure 5.3 will be explained in more detail
in § 5.4.3.
5.4 LINEAR MODELS FOR CONTROL OF CAVITY OSCILLATIONS
Rowley et al. (2006) propose a linear model for cavity oscillations. The model, com-
posed of four constituent transfer functions, is of low-order and is intended for feed-
back control design purposes. Each of the constituent transfer functions corresponds
to one of the four elements in the mechanism first described by Rossiter (1964): the
amplification of disturbances by the free shear layer; the scattering of the disturbances
into pressure fluctuations at the downstream corner; the propagation of these pressure
fluctuations inside the cavity; and the receptivity to pressure perturbations at the up-
stream corner. A block diagram of the linear model is shown in figure 5.4. The ‘o’ and
‘L’ subscripts correspond to the upstream and downstream corners respectively.
The open-loop transfer function given by the linear model – which is between the
body force and the pressure at the upstream corner – is given by
Pˆ1(s) =
po(s)
ρvo(s)
=
G(s)S(s)A(s)
1−G(s)S(s)A(s)R(s) . (5.2a)
If we are interested in a pressure measurement at the downstream corner (pL in figure
68 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
5.4), then we instead have
Pˆ3(s) =
pL(s)
ρvo(s)
=
G(s)S(s)
1−G(s)S(s)A(s)R(s) . (5.2b)
(We use the subscripts 1 & 3, together with a hat, to indicate that these are approxima-
tions of the transfer functions at probe locations 1 & 3.)
Since open-loop transfer functions have been measured from direct numerical sim-
ulations of the cavity, it is possible to compare these measured transfer functions with
those given by the linear model (5.2). (The measured transfer functions P1(s) and P3(s)
involve pressure measurements half-way along the upstream and downstream walls,
whilst the linear models (5.2a) and (5.2b) give the transfer functions at the upstream
and downstream corners, but this should make little difference, especially when the
linear model only considers the longitudinal behaviour of the cavity.)
Furthermore, since flow data are available at regular intervals throughout the com-
putational domain, it is possible to use the same system identification methods to find
approximations to the four constituent transfer functions, and these can be used to
inform the choice of the linear model’s constituent transfer functions in figure 5.4.
5.4.1 CONSTITUENT TRANSFER FUNCTIONS
We now look at each of the four constituent transfer functions in turn. The method for
approximating each transfer function from direct numerical simulations is explained,
R(s)
ρv
L
p0pL
G(s)
shear layer
S(s)
scattering
A(s)
acoustics
receptivity
Pˆ1(s)
+
+ ρv0
FIGURE 5.4: Linear model of the cavity flow proposed by Rowley et al. (2006). The ‘o’
and ‘L’ subscripts used correspond to the upstream and downstream corners respectively.
5.4. LINEAR MODELS FOR CONTROL OF CAVITY OSCILLATIONS 69
and the measured constituent transfer function is used to choose the parameters of the
linear model of Rowley et al. (2006). Finally, the overall open-loop system is treated:
the measured open-loop transfer functions (figures 5.3(a) & 5.3(c)) are compared with
those given by the feedback interconnection of the four constituent transfer functions
G(s), S(s), A(s) and R(s) given by equations (5.2a) & (5.2b).
Each constituent transfer function is measured in direct numerical simulations us-
ing the same spectral method that was used to find the overall open-loop transfer func-
tion P(s) in § 5.3. For the linear model, the parameters stated are all non-dimensionalized
by Strouhal number. For a time delay, for example, the non-dimensional value that is
stated, τ , is related to the dimensional value, τd by τ = τdU/L, where U is the free-
stream velocity.
Shear layer
The shear layer transfer function G(s) describes the amplification of velocity perturba-
tions at the upstream corner as they travel downstream. Therefore the input and output
are taken at the upstream corner and downstream corner respectively. Since the input
used for control is a body force ρv rather than a velocity (figure 5.2(b)), we include the
density in the input and output.
The shear layer transfer function can be found using linear stability theory. As a
simpler alternative for a linear model, Rowley et al. (2006) propose using a second-
order system with a time delay:
G(s) =
ρvL(s)
ρvo(s)
= kg
ω2o
s2+2ζωos+ω2o
e−sτg. (5.3)
Figure 5.5 compares the measured shear layer transfer function to that given by
the linear model (5.3) for kg = 3.35, ωo = 3.44, ζ = 0.07 and τg = 1.43 (which corre-
sponds to κ = 0.70 in Rossiter’s formula (5.1)). The model transfer function captures
the important dynamical features of the shear layer: near-unity gain at low frequen-
cies; amplification of disturbances at mid-frequencies; attenuation of disturbances at
high frequencies; and an appropriate phase. (The phase is given by the time delay τg,
the time taken for a disturbance generated at the upstream corner to propagate to the
70 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
−20
−10
0
10
20
30
0 0.5 1 1.5
−900
−720
−540
−360
−180
0
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–]
FIGURE 5.5: Shear layer transfer function G(s): measured from direct numerical simula-
tion (——); and approximated as a second-order transfer function with time delay (−−−).
downstream corner.) One observes that the maximum gain, cross-over frequency and
phase characteristics are all well-captured by the model.1
Acoustics
The acoustic transfer function A(s) relates pressure perturbations at the downstream
corner, pL , to pressure perturbations at the upstream corner, po, and so the pressures at
these two locations are used as input and output for the measured transfer function.
The linear model accounts for longitudinal acoustic cavity resonances by modeling
acoustic propagation and reflection in the cavity:
+
kae
−sτa p0pL
+
re−sτa
A(s) =
po(s)
pL(s)
=
kae−sτa
1− kare−2sτa . (5.4)
An acoustic wave generated at the downstream corner propagates upstream. Some
of it reflects off the upstream wall, propagates downstream, and arrives back at the
downstream wall, where it is again reflected. The coefficients ka and r measure the
total efficiency of the reflection process, including losses to the far-field. The time
1The cross-over frequency is the frequency at which the gain passes through 0 dB.
5.4. LINEAR MODELS FOR CONTROL OF CAVITY OSCILLATIONS 71
−20
−10
0
10
0 0.5 1 1.5
−360
−270
−180
−90
0
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–]
FIGURE 5.6: Acoustic transfer function A(s): measured from direct numerical simulation
(——); and approximated as a time delay with reflection (−−−).
delay τa represents the time taken for an acoustic wave to propagate the length of the
cavity.
Figure 5.6 compares the measured acoustic transfer function with that given by the
linear model (5.4) for ka = 0.8, r = 0.16 and τa = 0.60. (This acoustic time delay is the
value expected and, in its non-dimensional form, is given simply by the Mach number
M.) One observes reasonable agreement, with the phase characteristics more well-
captured than the gain. It is likely that the discrepancy in the gain at lower frequencies
is caused by the acoustic field generated by the shear layer’s evolving vorticity. The
frequency at which the greatest discrepancy is seen coincides with the frequency at
which the shear layer’s gain is largest, which seems to support this observation.
We briefly note the behaviour of A(s) at high frequencies here. Above Strouhal
numbers of approximately 1.0, the gain of A(s) is poorly captured by the linear model,
and the phase of the measured A(s) has a positive gradient, which suggests a negative
time delay. We will see similar behaviour for the scattering and receptivity transfer
functions: that the agreement between what is measured and the linear model is poor
above Strouhal numbers of approximately 1.0. An explanation for all of these obser-
vations is given in § 5.4.3.
72 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
−20
−15
−10
−5
0
5
0 0.5 1 1.5
−180
−135
−90
−45
0
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–]
FIGURE 5.7: Scattering transfer function S(s): as extracted from direct numerical simula-
tion (——); and approximated as a complex-valued constant (−−−).
Scattering
The scattering transfer function S(s) describes the generation of acoustic waves when
velocity perturbations impinge on the downstream corner. Therefore we use the quan-
tity ρvL and the pressure pL at the downstream corner as the input and output for the
measured transfer function.
The scattering transfer function is modeled as a constant gain and constant phase
lag:
S(s) =
pL(s)
ρvL(s)
= kse−i2piθs. (5.5)
Figure 5.7 compares the measured scattering transfer function with that given by the
linear model (5.5) for ks = 0.40 and θs = 0.22. Whilst the model is crude, its constant
gain and phase lag give reasonable agreement up to Strouhal numbers of about 1.0.
Receptivity
The receptivity transfer function R(s) describes the perturbations in ρv generated by
incident pressure perturbations at the upstream corner. Therefore we use the quantities
po and ρvo as the input and output for the measured transfer function.
The receptivity transfer function is modeled as a constant gain and constant phase
5.4. LINEAR MODELS FOR CONTROL OF CAVITY OSCILLATIONS 73
−20
−10
0
10
0 0.5 1 1.5
−90
−45
0
45
90
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–]
FIGURE 5.8: Receptivity transfer function R(s): as extracted from direct numerical simu-
lation (——); and approximated as a complex-valued constant (−−−).
lag:
R(s) =
ρvo(s)
po(s)
= kre−i2piθr . (5.6)
Figure 5.8 compares the measured receptivity transfer function with that given by the
linear model (5.6) for kr = 0.50 and θr = −0.074. Again, whilst crude, the model’s
constant gain and phase lag give reasonable agreement up to Strouhal numbers of about
1.0.
5.4.2 OVERALL LINEAR MODEL
By using the linear model’s four constituent transfer functions in the feedback arrange-
ment shown in figure 5.4, a low-order model for the overall cavity flow can be formed.
Using (5.2), we can do this for a pressure measurement on the upstream wall, Pˆ1(s)
and on the downstream wall, Pˆ3(s).
We now compare the transfer functions given by the linear model at these two
measurement locations, Pˆ1(s) and Pˆ3(s), to those measured directly, P1(s) and P3(s).
The comparison for both transfer functions is shown in a Bode diagram in figure 5.9,
and in a Nyquist diagram in figure 5.10. It is clear from both figures that, over the
frequency range of the first two Rossiter modes, the agreement is very good for both
transfer functions. This is encouraging, since it is the dynamics over this frequency
74 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
range, where the open-loop gain is the largest, that drives the design of a model-based
controller. The third Rossiter mode, however, is not well-captured by the linear model.
We now look at understanding these findings in terms of the loop gain.
−80
−60
−40
−20
0 0.5 1 1.5 2 2.5 3 3.5 4
−540
−360
−180
0
180
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–]
(a)
−80
−60
−40
−20
0 0.5 1 1.5 2 2.5 3 3.5 4
−1080
−720
−360
0
360
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–]
(b)
FIGURE 5.9: Bode diagram comparison of the open-loop transfer function: (a) for probe 1,
P1(s); and (b) for probe 3, P3(s). In both plots, the transfer function measured from direct
numerical simulation (——) is compared with that given by the linear model (−−−).
5.4. LINEAR MODELS FOR CONTROL OF CAVITY OSCILLATIONS 75
−0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 0.02
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
Re(P1)
Im
(P
1)
(a)
mode 1
mode 2
mode 3
−0.1 −0.05 0 0.05 0.1
−0.1
−0.05
0
0.05
0.1
Re(P3)
Im
(P
3)
(b)
mode 1
mode 2
mode 3
FIGURE 5.10: Nyquist diagram comparison of the open-loop transfer function (positive
frequencies only): (a) for probe 1, P1(s); and (b) for probe 3, P3(s). In both plots, the
transfer function measured from direct numerical simulation (——) is compared with that
given by the linear model (−−−). The three Rossiter modes are also indicated.
Importance of the loop gain
The behaviour of the linear model can be explained in terms of its loop gain L(s),
where
L(s) = G(s)S(s)A(s)R(s).
This appears in the denominator of both transfer functions in (5.2), from which it is
clear that the linear model’s poles occur at those frequencies satisfying L(s) = 1. This
can only be satisfied when the phase of the loop gain satisfies ∠L(s) = 0◦, −360◦, . . . ,
and this condition gives the frequencies of the linear model’s modes. Then the stability
of a mode is determined by the magnitude of the loop gain at that frequency: if the
loop gain is less than one, the mode is stable, whilst if the loop gain is greater than one,
the mode is unstable.
Therefore the success in predicting the first and second Rossiter modes can be
explained – in terms of the linear model at least – by the correct prediction of the loop
gain L(s) at those frequencies.
Similarly, the loop gain can be used to explain the poor prediction of the third
Rossiter mode. The linear model predicts a third Rossiter mode at a Strouhal number
76 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
of 1.15, which is higher than that seen in simulations. The loop gain at this Strouhal
number is very low, so that the third Rossiter mode is highly damped, and is therefore
hardly visible in the linear model’s transfer function. Since the loop gain decreases
with increasing frequency, the over-prediction of the third Rossiter mode’s frequency
has a negative impact on the prediction of the loop gain for the third Rossiter mode,
and so the prediction of both its frequency and its loop gain is poor.
Link to Rossiter’s formula
Setting G(s) = e−sτg , assuming S(s) = e−i2piγ and R(s) = 1, and letting ka = 1 and r= 0
(no reflection) in (5.4) to give A(s) = e−sτa , the predicted open-loop transfer function
for pressure probe 1 becomes
Pˆ1(s) =
e−i2piγe−s(τg+τa)
1− e−i2piγe−s(τg+τa)
which has poles at s = iωn, with
Stn =
fnL
U
=
n− γ
M+1/κ
, n = 1,2, . . . ,
and so we recover Rossiter’s formula (5.1).1
It is now clear why Rossiter’s formula can only predict the cavity’s resonant fre-
quencies and not their growth rates: the loop gain in this case is given by L(s) =
e−i2piγe−s(τg+τa), the magnitude of which is always one.
5.4.3 HIGH-FREQUENCY DYNAMICS
The linear model used shows reasonable success in predicting the gain and phase char-
acteristics of the open-loop cavity flow over the frequency range of the Rossiter tones.
Clearly this linear model is useful for feedback control design purposes. However, the
linear model over-predicts the time delay for pressure probe 1 at higher frequencies
(figure 5.9(a)), and we seek an explanation for this.
1In fact, the constant phase change given by γ can be arbitrarily apportioned between S(s) and R(s)
with no change in the predicted resonant frequencies.
5.4. LINEAR MODELS FOR CONTROL OF CAVITY OSCILLATIONS 77
The linear model neglects the acoustic field generated by the body force ρvc di-
rectly. At low frequencies, this is perfectly valid: the acoustics generated by the
Rossiter mechanism dominate, and the effect of the body force’s direct acoustic field
is negligible. At higher frequencies, though, the shear layer attenuates disturbances
rather than amplifying them, i.e. |G(s)|< 1. This means that the velocity fluctuations
arriving at the downstream corner are very small, and the Rossiter mechanism is cut-
off. Indeed, these considerations lead to a convenient definition of ‘low frequencies’
and ‘high frequencies’: low frequencies are those for which the shear layer amplifies
disturbances, whilst high frequencies are those for which disturbances are attenuated.
(From figure 5.5, the cross-over frequency of the shear layer – where the magnitude of
G(s) passes through 0 dB – is at a Strouhal number of about 1.1.)
This explains the positive gradient of the acoustic transfer function’s phase seen
in figure 5.6. At frequencies above the shear layer’s cross-over frequency, the body
force dominates the acoustic field. Since the body force is much closer to the upstream
corner than to the downstream corner, the effect of the body force is seen at po before it
is seen at pL , and this corresponds to A(s) = po(s)/pL(s) having a negative time delay.
The acoustic transfer function makes clear what is true for all four constituent transfer
functions: it is only valid to seek them over the frequency range at which the Rossiter
modes can occur, i.e. at frequencies where |G(s)| > 1. To do so at higher frequencies
is to seek a mechanism that no longer exists, the open-loop dynamics being instead
dominated by the acoustics directly generated by the body force ρvc. This helps to
explain the poor agreement seen at higher frequencies between measurements and the
linear model for the transfer functions representing the acoustics A(s), scattering S(s)
and the receptivity process R(s).
We now look at a simple modification to the linear model to better capture this
high-frequency behaviour for probe 1. (For brevity’s sake we omit probe 3, although
the same analysis can be performed.) In particular, the modification correctly predicts
the system time delay at high frequencies. The modified linear model is shown in
figure 5.11. A transfer function to model the acoustics generated by the body force
78 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
directly, B(s), is introduced. The transfer function that we choose is
B(s) = kbe−sτb, (5.7)
where kb is a gain and τb is the time taken for an acoustic wave, generated by the body
force, to propagate to the pressure sensor location of interest – in this case probe 1.
The linear model’s open-loop transfer function for pressure probe 1 is now given
by
Pˆ1(s) =
G(s)S(s)A(s)+B(s)
1−G(s)S(s)A(s)R(s) .
At low frequencies, B(s) is negligible and we recover (5.2a). At high frequencies,
|G(s)|  1 and we have simply P(s) = B(s).
The Bode diagram of the modified linear model’s transfer function, together with
that found in direct numerical simulations, is shown in figure 5.12. The parameter
values chosen are τb = 0.15, which is the anticipated acoustic time delay between
the body force and probe location 1, and kb = 0.33. One observes that the model’s
prediction of the Rossiter modes remains intact, and that the high-frequency dynamics
at probe 1 are more well-captured, in particular the time delay (given by the gradient of
the phase plot). Furthermore, this model is able to capture zeros of the transfer function
as well as the poles - this is explored in more detail in § 5.5.1.
R(s)
ρv
L
p0pL
G(s)
shear layer
S(s)
scattering
A(s)
acoustics
receptivity
+
+ ρv0
B(s)
+
+
body force
FIGURE 5.11: Modified linear model to account for the acoustic field generated by the
body force directly.
5.5. PROPERTIES OF THE CAVITY TRANSFER FUNCTION 79
−80
−60
−40
−20
0 0.5 1 1.5 2 2.5 3 3.5 4
−540
−360
−180
0
180
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–]
FIGURE 5.12: Bode diagram comparison of the modified open-loop transfer function
for probe 1: as extracted from direct numerical simulation (——); and approximated by
the modified linear model (−−−). The original linear model is also shown (− ·−) for
comparison.
5.5 PROPERTIES OF THE CAVITY TRANSFER FUNCTION
The results of § 5.4.1 suggest that, for suitable choice of constituent transfer functions,
a linear model of the form
Pˆ1(s) =
G(s)S(s)A(s)+B(s)
1−G(s)S(s)A(s)R(s) (5.8)
is useful for feedback control purposes.
We now develop an explicit expression for this transfer function based on the con-
stituent transfer functions used in § 5.4.1. We focus on the transfer function for probe
location 1, Pˆ1(s), because this corresponds most closely to the pressure signal that we
will use for adaptive feedback control in chapter 6. The same results regarding the
locations of the zeros and the transfer function’s relative degree still hold for probe
location 3, however (which is the probe location that we will use for linear quadratic
control, also in chapter 6).
Since the scattering and receptivity transfer functions are modeled simply as com-
plex constants, we set S(s) = Λs and R(s) = Λr. We write the shear layer transfer
function (5.3) as the product of a rational transfer function Go(s) and a pure time de-
80 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
lay:
G(s) = Go(s)e−sτg.
Substituting these expressions, together with the transfer functions for the acoustics
(5.4) and for the body force (5.7), into (5.8) gives
Pˆ1(s) =
kaΛsGo(s)e−s(τ−τb)+ kb(1− kare−2sτa)
−kaΛsΛrGo(s)e−sτ +(1− kare−2sτa) e
−sτb (5.9)
= Po(s)e−sτb, (5.10)
where in (5.9) we have introduced τ = τg+ τa. The body force’s acoustic delay τb has
been removed from the numerator of Pˆ1(s) in (5.9) and appears as a pure time delay.
(This can be done because the shear layer time delay τg will always be larger than the
body force’s acoustic delay τb.) Thus Pˆ1(s) is written as the product of some transfer
function Po(s) and a pure time delay in (5.10).
We now use this model to look at two important properties of the cavity flow’s
transfer function: the location of its zeros; and its relative degree. These two properties
will be important for the adaptive controller used in chapter 6.
5.5.1 TRANSFER FUNCTION ZEROS
We have already seen in chapters 2 & 3 that the location of the zeros of the open-loop
system are important for adaptive control. Specifically, any zeros must lie in the left
half-plane. We first note that without the body force B(s), the zeros of (5.2a) are given
by
G(s)S(s)A(s) = 0. (5.11)
Since none of the transfer functions G(s), S(s) or A(s) possess any zeros, (5.11) is
never satisfied, and the original linear model does not model any zero dynamics of the
system. With the body force modification made, the zeros of (5.8) are given by
G(s)S(s)A(s)+B(s) = 0,
5.5. PROPERTIES OF THE CAVITY TRANSFER FUNCTION 81
and zeros occur whenever G(s)S(s)A(s) =−B(s). Then from equation (5.9), the zeros
of Pˆ1(s) are given by the roots of
kaΛsGo(s)e−s(τ−τb)+ kb(1− kare−2sτa) = 0. (5.12)
First consider the case where r→ 0, which physically corresponds to neglecting
the reflection of acoustic waves inside the cavity. Then (5.12) simplifies to
kaΛsGo(s)e−s(τ−τb)+ kb = 0. (5.13)
We are interested in whether the (complex-valued) roots λ = σ+ iω of (5.13) lie in
the left half-plane (σ < 0) or in the right half-plane (σ > 0). This amounts to looking
at the magnitude of (5.13), since |e−iω(τ−τb)|= 1:
|e−s(τ−τb)|= e−σ(τ−τb) = |kb|
ka|Λs||Go(s)| , (5.14)
and we see that, in this simplified case, the position of the zeros depends on the shear
layer’s gain, and is therefore frequency-dependent. We expect |kb/kaΛs| ≈ 1. There-
fore at low frequencies where |Go(s)| > 1, we expect e−σ(τ−τb) < 1 at the zeros, and
any zeros are likely to be in the right half-plane (σ > 0). At high frequencies where
|Go(s)| < 1, we expect e−σ(τ−τb) > 1, and any zeros are likely to be in the left half-
plane (σ < 0).
If we now include the effect of the downstream reflection coefficient r, the zeros of
(5.9) are given by
|e−s(τ−τb)|= |kb||1− kare
−2sτa|
ka|Λs||Go(s)| , (5.15)
and the same argument applies: right half-plane zeros are more likely at low frequen-
cies; and left half-plane zeros are more likely at high frequencies.1
These observations are supported by the transfer functions found at probe locations
1 & 3 in figures 5.3(a) & 5.3(c). Both transfer functions have zeros at a Strouhal number
1It is instructive to consider that when |Go(s)| > 1, equation (5.12) is dominated by the first term
kaΛsGo(s)e−s(τ−τb). Whilst this has no zeros, the pure time delay e−s(τ−τb) acts in a similar way to right
half-plane zeros (by making the term kaΛsGo(s)e−s(τ−τb) non-minimum phase).
82 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
of around 1.1, at which point the shear layer has approximately unity gain. The phase
decrease across these zeros indicates that they are in the right half-plane. On the other
hand, probe 3 also has a pair of zeros at a Strouhal number of around St = 3.5, by
which point the shear layer transfer function satisfies |G(s)|  1. The phase increase
across these zeros indicates that they are in the left half-plane.
From (5.15), the shear layer transfer function G(s) is important in determining the
zeros of Pˆ1(s). It is worth pointing out that the result concerning their locations holds
for any shear layer transfer function G(s) – provided that its gain is greater than one at
low frequencies and decreases at high frequencies – and is not exclusive to the second-
order model (5.3) chosen.
5.5.2 RELATIVE DEGREE
Recall that for a rational transfer function Po(s), written as the ratio of two coprime,
monic polynomials
Po(s) = ko
No(s)
Do(s)
,
its relative degree n∗ is defined as the degree of its denominator polynomial Do(s)
minus the degree of its numerator polynomial No(s):
n∗[Po(s)] = deg[Do(s)]−deg[No(s)]. (5.16)
We now see that the relative degree of Pˆ1(s) – when approximated by the product
of a rational transfer function Po(s) and a pure time delay – is zero, and we see it in
two ways. The first involves using the specific linear model of § 5.4, and the second is
based on some more general assumptions about the constituent transfer functions.
Based on the derived linear model
From (5.9), Po(s) for the cavity flow is not a rational transfer function. Therefore we
first form a rational approximation Pro(s). Following Evesque et al. (2003b), we do this
using Padé approximants. For some function f (s), an [L,M]th-order Padé approximant
of f (s) is a rational function whose numerator has order L and denominator has order
5.5. PROPERTIES OF THE CAVITY TRANSFER FUNCTION 83
M. The rational function matches the first (L+M+1) terms of the power series of f (s),
and therefore its range of validity increases as L and M increase. We use the notation
[L/M] f (s) for this Padé approximant. For a pure time delay, which has unity gain at
all frequencies, we choose L = M. Then the poles of the resulting Padé approximant
are complex conjugates of its zeros, and [M/M] f (s) has unity gain at all frequencies as
required. The phase of the approximation will be well-matched over some frequency
range, and this range increases with the order of the Padé approximant, but will deviate
asymptotically.
Replacing each of the delay terms in Po(s) with an [M,M]th-order Padé approxi-
mant [M/M] f (s), and writing Go(s) as Go(s)= kgω2o/DG(s) (with DG(s)= s2+2ζωos+
ω2o ) gives the rational approximation Pro(s):
Pro(s) =
kakgω2oΛs[M/M]e−s(τ−τb)+ kbDG(s)(1− kar[M/M]e−2sτa )
−kakgω2oΛsΛr[M/M]e−sτ +DG(s)(1− kar[M/M]e−2sτa )
. (5.17)
The relative degree of this expression is zero. This can be seen by observing that in the
limit ω → ∞,
lim
ω→∞P
r
o(iω) =
kbDG(iω)(1− kar[M/M]e−2iωτa )
+DG(iω)(1− kar[M/M]e−2iωτa )
= kb,
and a finite gain at infinite frequency implies a relative degree of zero. Here we have
assumed that Go(s) = kgω2o/(s2+2ζωos+ω2o ), but the result holds for any shear layer
transfer function with a relative degree of one or greater.
Based on some more general assumptions
The result above was derived for the specific model of the cavity flow given by the
constituent transfer functions (5.3–5.6), and by using Padé approximants. The ques-
tion now arises: is it possible to arrive at the same result using some more general
assumptions about the constituent transfer functions? We will now see that it is. Con-
sider again equation (5.8). Let G(s)S(s)R(s) =Ψ(s), giving
Pˆ1(s) =
Ψ(s)+B(s)
1−Ψ(s)R(s) .
84 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
Now represent each constituent transfer function as the ratio of two polynomials. Then
Ψ(s) = NΨ(s)/DΨ(s), for example, and Pˆ1(s) can be written as1
Pˆ1 =
NΨDRDB+NBDΨDR
DΨDRDB−NΨNRDB (5.18)
=
DR
DB
(
NΨDB+NBDΨ
DΨDR−NΨNR
)
. (5.19)
We want to find the relative degree of this expression. By (5.16), this is
n∗(Pˆ1) =deg(DΨDR−NΨNR)+deg(DB)
−deg(DΨNB+NΨDB)−deg(DR).
(5.20)
We now make two assumptions on the relative degree of the constituent transfer func-
tions:
(i) the transfer functions Ψ(s) = G(s)S(s)A(s) and R(s) are strictly proper transfer
functions – that is, they have a relative degree of one or greater.
(ii) the body force transfer function B(s) has a relative degree of zero.
These two assumptions are physically motivated. The gain of a strictly proper transfer
function satisfies |P(iω)| → 0 as ω→ ∞. Because of viscosity, we expect the pressure
perturbations generated (indirectly) by velocity perturbations in the shear layer – which
is what the transfer function Ψ(s) = G(s)S(s)A(s) represents – to be small at high
frequencies. Similarly, we expect the response of the shear layer to incoming acoustic
waves to decrease with increasing frequency, and this response is characterized by the
transfer function R(s).
A transfer function with a relative degree of zero, on the other hand, has finite
response at high frequencies. Since viscosity is unimportant in sound waves (Dowling
& Ffowcs Williams, 1983), assumption (ii) is reasonable and, like assumption (i), is
1For clarity’s sake we now drop the (s) notation, and the frequency-dependence is implied.
5.6. SUITABILITY OF THE ADAPTIVE CONTROLLER FOR CAVITY OSCILLATIONS 85
physically motivated.1 Using these two assumptions gives
deg(DΨDR−NΨNR) = deg(DΨDR), and
deg(DΨNB+NΨDB) = deg(DΨNB),
and equation (5.20) becomes
n∗(Pˆ1) = deg(DΨDR)+deg(DB)−deg(DΨNB)−deg(DR) (5.21)
= deg(DΨDRDB)−deg(DΨDRNB) (5.22)
= deg
(
DB
NB
)
= deg
(
1
B(s)
)
, (5.23)
which by assumption is zero.
5.6 SUITABILITY OF THE ADAPTIVE CONTROLLER FOR CAVITY OSCILLATIONS
We now consider applying the adaptive controller of chapter 3 to the cavity flow. We
first summarize the conditions placed on the open-loop plant by the adaptive controller
of chapter 3, and look at to what extent each is met by the model of the cavity that
was developed in § 5.5. The concept of collocated control is then introduced as a way
of dealing with the right-half plane zeros that can occur for the cavity flow, and it is
shown that using this collocated control arrangement, the cavity flow meets all of the
requirements of the adaptive controller.
The open-loop transfer function of interest includes the actuator, PL(s) = L(s)P(s).
The requirements placed on PL(s) by the adaptive controller, and the extent to which
PL(s) meets them, are summarized below.
(i) PL(s) has no right half-plane zeros. Based on the linear model derived in 5.5.1,
P(s) – and therefore PL(s) – may have zeros in the right half-plane, particularly
at low frequencies.
(ii) the sign of the high-frequency gain of PL(s) is known. At high frequencies P(s)
1Indeed, the form chosen for B(s) in (5.7) is ‘all-pass’ and has a relative degree of zero after a Padé
approximant is used for the time delay.
86 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
becomes simply P(s) = B(s) = kbe−sτb , and so the high-frequency gain of PL(s)
is klkb, where kl is the high-frequency gain of the actuator transfer function.
(This assumes that the body force completely characterizes the high-frequency
behaviour of the system, which it may not, and so in general we must assume
that the high-frequency gain must be determined.)
(iii) the relative degree of PL(s) does not exceed two. In § 5.5.2 it was shown that the
relative degree of P(s) is zero. Therefore the relative degree of PL(s) is given
by the relative degree of the actuator transfer function L(s). We expect a typical
actuator to have a relative degree of one or two – see Kegerise et al. (2007a),
for example, where the actuator used is typical of a second-order, under-damped
system. Therefore we assume that the relative degree of PL(s) satisfies n∗ ≤ 2.
Clearly, all of the requirements on PL(s) are met, with the exception of the locations
of its zeros. There is also another, slightly more subtle way in which the cavity flow
does not presently meet the above requirements. For a transfer function to be Strictly
Positive Real, a relative degree of one is necessary but not sufficient. In addition the
pole and zero pairs must ‘interlace’ in frequency: that is, a pair of poles must be
followed by a pair of zeros, who must be followed by another pair of poles, and so on.
Referring to figure 5.3, this condition is not met at any of the three pressure probes,
where in each case the three Rossiter tones neighbour each other with no zeros in
between.
We remedy both of these problems by introducing collocated control. This arrange-
ment is depicted in figure 5.13, where now the pressure sensor and body force region
are collocated. The location of the body force does not change, the pressure sensor
used for control instead moving to join it near the cavity’s upstream corner. Although
this arrangement is not possible in practice, it could be approximated in experiments
by placing a pressure sensor close to the actuator at the upstream corner. It seems rea-
sonable to assume that such an experimental arrangement, although no longer strictly
collocated, could give rise to the same desired properties of only left half-plane zeros
and interlacing poles and zeros.
We introduce this collocation to take advantage of the fact that, for a collocated
transfer function from volume velocity to pressure, the product of the input and out-
5.7. CONCLUDING REMARKS 87
L
D
M
p3
ρvc
pc
FIGURE 5.13: Collocated control arrangement for adaptive control. (Pressure probe 3 is
also shown.)
put has the dimension of power, and is therefore positive real (see Hong & Bernstein
(1998)). Only stable transfer functions can be positive real, and so this positive real-
ness applies to the closed-loop stabilized system (as it did in the development of the
adaptive controller in § 3.3.1).
For a transfer function to be positive real, all of its zeros must lie in the left half-
plane, and its poles and zeros must interlace in frequency (this ensures that its phase
lies between −90◦ and +90◦). Therefore using collocated control, the closed-loop
stabilized system must have only left half-plane zeros, and the pole and zero pairs
must interlace in frequency. These collocation-based arguments do not include the
actuator transfer function L(s), but we have already established in requirement (iii) on
page 86 that the overall system PL(s) can be modeled as having a relative degree not
exceeding two, and now all of the requirements on PL(s) are satisfied.
5.7 CONCLUDING REMARKS
This chapter has focused on linear models of cavity oscillations that are useful for
feedback control purposes. By applying system identification techniques to direct nu-
merical simulations of the cavity flow, its linear dynamics have been very accurately
determined. The transfer function found has been compared to that given by a sim-
ple linear model, which shows reasonable agreement and which is useful for feedback
control design.
By including a simple model of the acoustic field generated directly by the body
force region, two properties of the system are more accurately modeled: its behaviour
88 CHAPTER 5: DYNAMICS OF CAVITY OSCILLATIONS
at high-frequencies (in particular its time delay); and the locations of its zeros.1
Some general properties of cavity oscillations have been derived. These properties
are of general interest for any feedback control scheme, but are particularly useful for
adaptive control. Although the derivation of these properties relies to some extent on
the specific linear model used, many of the underlying assumptions would hold for any
physically-motivated model – for example, that the shear layer amplifies disturbances
only over some limited frequency range, and that the actuator is of low relative degree.
Finally, by comparing these derived properties to the properties required by the adap-
tive controller of chapter 3, it has been shown that such an adaptive controller can be
applied to the cavity flow, provided that collocated control is used.
ACKNOWLEDGEMENT
The work presented in this chapter and to be presented in chapter 6 was done in col-
laboration with Professor Clarence Rowley at Princeton University. As part of the col-
laborative work I spent two months on a research visit to Professor Rowley’s research
group, where we had many discussions on the DNS code and on applying feedback
control to it.
1In fact, the linear model which does not account for the body force’s acoustic field predicts only
the poles of the system and not its zeros, as demonstrated in equation (5.11).
CHAPTER 6
FEEDBACK CONTROL OF CAVITY OSCILLATIONS
Previous attempts at controlling cavity oscillations have met with limited success. One
of the major challenges for the control of cavity flows is dealing with the sensitivity
of the flow’s dynamics to changes in Mach number. See, for example, Rowley &
Juttijudata (2005), where an 8 % change in Mach number from M = 0.60 to M = 0.55
resulted in the closed-loop controller actually increasing the amplitude of oscillations,
or Kegerise et al. (2007b), where closed-loop control maintained suppression of cavity
tones only over a modest range of Mach numbers (0.275≤M ≤ 0.29).
Feedback control is most effective when a reliable model of the system-to-be-
controlled is available. In this chapter, system identification and model reduction tech-
niques are used to form a low order model of the cavity flow. Specifically, the Eigen-
system Realization Algorithm (ERA) is used to find a low order state-space model of
the cavity flow. An optimal (H2) controller is then designed, and this controller main-
tains stability over a wider Mach number range than seen in previous studies. We then
attempt to improve the robustness of the controller by introducing a very simple type
of gain scheduling.
Finally, we look at using a Lyapunov-based adaptive control scheme like that used
in § 3, and see that it maintains stability over an even wider Mach number range.
6.1 REDUCED ORDER MODELING FOR FLUIDS
One of the greatest challenges for feedback control of fluids is their dynamical com-
plexity, the Navier-Stokes equations being both high-dimensional and non-linear. Model
89
90 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
reduction involves finding low-dimensional models that approximate the high-dimensional
dynamics of a system. Since model reduction is so pertinent to the problem of feed-
back control of fluids, a brief review of some of the methods available is given here,
before we use one of these methods – the Eigensystem Realization Algorithm (ERA) –
to form a reduced-order model of the cavity flow which is useful for feedback control
purposes.
Proper Orthogonal Decomposition (POD), also known as principal component anal-
ysis, or the Karhunen-Loève expansion, has been used for some time in developing
low-order models of fluids (Lumley, 1970; Sirovich, 1987). Given a set of data that lie
in a vector space V, proper orthogonal decomposition finds a subspace Vr of dimension
r such that the error in the projection onto the subspace is minimized. The resulting
POD modes are optimal in that they maximize the average energy in the projection of
the data onto the subspace spanned by the modes. However, these POD modes may
not be the best modes for describing the dynamics that generate the data set, since low-
energy features may be critically important to the dynamics. For cavity oscillations,
for example, acoustic disturbances are crucial to the Rossiter mechanism, even though
they have much smaller energy than hydrodynamic pressure fluctuations.
To address this problem, Rowley (2005) introduced a method called balanced POD
which approximates balanced truncation (Moore, 1981) for large systems, and which
has deep connections with POD. The balancing refers to the observability and con-
trollability Gramians of the resulting low order model being equal and diagonal (or
‘balanced’), and physically this means that the dynamics that generate the data set are
properly accounted for. The concepts of observability and controllability of a low order
model will be explained in more detail in the next section.
Because of this balancing procedure, a reduced order model computed by balanced
POD is vastly superior to that given by POD for a given order r, at least when we are
concerned with finding a low order model for feedback control purposes. The main
disadvantage with balanced POD, however, is that solutions of an adjoint system are
required, and so the method can only be used in computational studies and not for
experimental data.
Ma et al. (2009) have shown that the reduced order models generated by the eigen-
6.2. BALANCED REDUCED ORDER MODELING 91
system realization algorithm (Juang & Pappa, 1985) – a method developed for modal
parameter identification and model reduction in the structural dynamics community –
are theoretically identical to those generated by balanced POD. A distinct advantage
of the ERA is that solutions of an adjoint system are not required, and so a balanced
low order model can be found from experimental data as well as from simulations.
6.2 BALANCED REDUCED ORDER MODELING
We now look at some of the important concepts for balanced reduced order modeling,
before looking specifically at the eigensystem realization algorithm.
Consider the linear, time-invariant, discrete-time dynamical system
x(k+1) = Ax(k)+Bu(k)
y(k) =Cx(k)
(6.1)
where u ∈ Rp is a vector of inputs, y ∈ Rq is a vector of outputs and x ∈ Rn is called
the system state: n is the order of the system, and will be important later. k is the time
step index. We consider the discrete-time case, since we are interested in discrete-time
data from simulations or experiments.
We now look at the definitions of two important concepts for balanced reduced
order models: controllability and observability.
6.2.1 CONTROLLABILITY
For zero initial condition x(0) = 0, the state of the system (6.1) evolves according to
x(1) = Bu(0)
x(2) = ABu(0)+Bu(1)
x(3) = A2Bu(0)+ABu(1)+Bu(2)
...
x(s) =
s
∑
k=1
Ak−1Bu(s− k)
92 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
or, in matrix form
x(s) =
[
B AB A2B · · · As−1B
]

u(s−1)
u(s−2)
u(s−3)
...
u(0)

, (6.2)
which relates the state at time step s, x(s) to previous inputs u(k), 0 ≤ k ≤ s− 1. A
system is called controllable if, for every state x¯, there exists an input series u(k) that
takes the state from x(0) = 0 to x(s) = x¯ in a finite time. It can be shown that the system
(6.1) is controllable if and only if the controllability matrix
Q,
[
B AB A2B · · · As−1B
]
, (6.3)
Q ∈ Rn×ps, which appears in (6.2), has rank n. Of course, if the (linear) system can go
from zero state to any x¯, then it can go from any x(0) to any x¯.
6.2.2 OBSERVABILITY
A state-space realization (6.1) is observable if the initial state x(0) can be deduced
from knowledge of u(k) and y(k) for 0≤ k ≤ s for any s > 0.
Without loss of generality, we may take u(k) = 0, ∀k, since the u-dependent part
of the solution can always be subtracted from the full solution. Therefore looking at
the evolution of the output y(k), for some initial state x(0) 6= 0, the system (6.1) with
u(k) = 0 gives
y(0) =Cx(0)
y(1) =CAx(0)
y(2) =CA2x(0)
...
y(s−1) =CAs−1x(0)
6.2. BALANCED REDUCED ORDER MODELING 93
or, in matrix form 
y(0)
y(1)
y(2)
...
y(s−1)

=

C
CA
CA2
...
CAs−1

x(0)
which relates the output at all times to the initial state x(0). A unique solution exists
– and therefore the initial state x(0) can be deduced – if and only if the observability
matrix
P=
[
C CA CA2 · · · CAs−1
]T
, (6.4)
P ∈Rqs×n, has rank n. Having justified our setting u(k) = 0 without loss of generality,
this result holds for any u(k) 6= 0.
6.2.3 EIGENSYSTEM REALIZATION ALGORITHM
In the previous section it was stated that the eigensystem realization algorithm can be
used to find balanced reduced order models. We now look at using the ERA to find
a low order state-space model of the cavity flow. The ERA is valid only for stable
systems, and therefore we use the dynamic phasor – as we did in § 5 – to first stabilize
the flow. Then we use an extension to the ERA called observer/controller identification
(OCID), which allows the open-loop dynamics of a system operating in closed-loop to
be found.
An overview of the ERA is now given, and the ERA for stable systems is described.
Then we look briefly at the problems faced in applying it to an unstable system, and
see how these problems can be addressed using OCID.
The eigensystem realization algorithm involves forming the generalized αq×β p
94 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
Hankel matrix composed of the system Markov parameters:
H(k−1) =

Yk Yk+1 · · · Yk+β−1
Yk+1 Yk+2 · · · Yk+β
...
... . . .
...
Yk+α−1 Yk+α · · · Yk+α+β−2
 (6.5)
where α and β are arbitrary integers. The Markov parameter Yk ∈Rq×p for time index
k is defined as
Yk ,CAk−1B. (6.6)
Setting k= 1 for the Hankel matrix, substituting in (6.6) for the Markov parameters,
and decomposing into two matrices, we find that
H(0) = PαQβ (6.7)
where Pα ∈ Rαq×n is the observability matrix (6.4) and Qβ ∈ Rn×β p is the controlla-
bility matrix (6.3) as defined earlier. Since the Hankel matrix H(0) is composed of the
Markov parameters in (6.5), we see from (6.7) that the Markov parameters are closely
related to the observability and controllability matrices.
Now, factorizing the Hankel matrix H(0), which has rank n, using the singular
value decomposition, and truncating to order r < n, we have
H(0) = RΣST ∼= RrΣrSTr . (6.8)
Rr and Sr are made up of the first r (orthonormal) columns of R and S respectively. Σr
is a rectangular matrix, Σr = diag = [σ1 σ2 · · · σr], with σ1 ≥ σ2 ≥ ·· · ≥ σr ≥ 0,
the first r singular values of H(0). Examining (6.7) and (6.8) as a whole, we can write
H(0) = PαQβ ∼= (RrΣ1/2r )(Σ1/2r STr ). (6.9)
One possible choice for the decomposition of H(0) is then Pα = RrΣ
1/2
r and Qβ =
6.2. BALANCED REDUCED ORDER MODELING 95
Σ1/2r STr , and this choice makes both Pα and Qβ balanced in the sense that
PTαPα = Σ
1/2
r RTr RrΣ
1/2
r = Σr
QβQ
T
β = Σ
1/2
r STr SrΣ
1/2
r = Σr,
(6.10)
where PTαPα and QβQ
T
β are the observability and controllability Gramians.
The system matrices B and C can then be found from (6.3) and (6.4) respectively.
To find the A matrix, let k = 2 in (6.5) and use (6.9) to give
H(1) = PαAQβ ∼= (RrΣ1/2r )A(Σ1/2r STr ),
from which A can be found. For a more rigorous treatment of the ERA, see Juang &
Pappa (1985) or Juang (1994).
The state-space model (6.1) given by the ERA is balanced: the controllability and
observability Gramians are diagonal and equal by (6.10), and this means that the dy-
namics of the system are properly taken into account. Furthermore, an upper bound
for the error in the reduced order model can be derived. Let Gr denote the transfer
function of the reduced order model of order r, and let G denote the transfer function
of the full system. Then the H∞-norm of the error satisfies
||G−Gr||∞ ≤ 2
n
∑
j=r+1
σ j
(‘twice the sum of the tails’).
Finally, we briefly note a connection to POD here: the modes found by POD are the
most controllable modes, and the observability of the modes is not taken into consid-
eration. This helps to explain some of the inadequacies of POD for feedback control,
since the most controllable modes may be only weakly observable.
96 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
Observer/controller identification (OCID)
The ERA is only applicable to stable systems. We can see the problem with applying
the ERA to an unstable system by again considering equation (6.6):1
Yk ,CAk−1B.
For a stable system subjected to an impulse, the term Ak−1 becomes progressively
smaller, and the Markov parameter (or ‘impulse response’) Yk ' 0 for sufficiently large
k. For an unstable system, however, Ak−1 becomes progressively larger, and no such
approximation can be made.2
For an unstable system made stable by feedback, OCID remedies this problem
by first computing the Markov parameters of the closed-loop, stable system. Using
these Markov parameters, and the fact that both the forcing signal and the feedback
signal can be measured (ρvw and ρv f in figure 5.2(b)), OCID then finds the Markov
parameters of the open-loop, unstable system and of the controller. These Markov
parameters can be used to find a state-space realization of the open-loop system (and,
if desired the closed-loop controller).
For more details of OCID, see Juang (1994), chap. 8.
6.3 APPLICATION OF THE ERA TO THE CAVITY FLOW
We now look at applying the ERA to the cavity flow. To validate the reduced order
models obtained, we compare the frequency response of the reduced order model to
that found using spectral analysis in § 5.3
We find that by using a model with 140 states, the frequency responses of the
reduced order model and the spectral estimate are almost indistinguishable: the com-
parison is shown in figure 6.1. We seek a low order model of the cavity flow, and so
the question now arises: How many states are required for feedback control purposes?
1This is without considering the non-linearities that the growing response will give rise to.
2A similar problem can occur for lightly damped systems, where the k required to satisfy Yk ' 0 can
become prohibitively large.
3The reduced order model is in state-space form (6.1) and is therefore in the (discrete) time domain.
Its frequency response can be found by using z-transforms.
6.4. LQG CONTROL 97
−80
−60
−40
−20
0 0.5 1 1.5 2 2.5 3 3.5 4
−1080
−720
−360
0
360
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–]
FIGURE 6.1: Open-loop Bode diagram comparison: transfer function found using spectral
analysis (—–), and using the ERA with 140 states (−−) and with 8 states (–·–). All for
probe 3.
By using just 8 states in the ERA-derived reduced order model, one observes (figure
6.1) that the first three Rossiter modes are very well-captured. (Mathematically, this
means that the H∞-norm of the error ||G−Gr||∞ is sufficiently small using 8 states.)
This 8-state reduced order model is used in the next section for the design of a
linear quadratic Gaussian regulator.
6.4 LQG CONTROL
The measured open-loop transfer function, as well as giving valuable insight into some
of the physical features of the cavity flow, can be used for feedback control design
purposes. Using the state-space model found using the ERA, it is possible to use
standard robust control techniques such as Linear Quadratic Gaussian (LQG) control
and H∞ control (Zhou & Doyle, 1998). Here we focus on LQG control, and use it to
find a stabilizing regulator for the cavity flow at a Mach number of 0.6.
98 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
− P (s)
L(s)
V
ρvc pc
K(s)
d
n
FIGURE 6.2: Closed-loop cavity arrangement. The effect of noise (n) and disturbances
(d) has been included. The subscript c is used to denote the control parameters ρvc and
pc, which could be situated anywhere in the cavity’s domain, and V is the control voltage.
6.4.1 CONTROLLER DESIGN
The cavity flow’s transfer function P(s) found in § 5 and § 6.3 is from the control body
force ρvc to the pressure measurement pc. In an experiment, an actuator would be
used to provide the body force, and so we include an actuator transfer function here,
which we denote L(s). Then the regulator is designed for the overall plant PL(s) =
L(s)P(s), which is the transfer function between a control voltage V and the pressure
measurement pc. The overall arrangement is shown in figure 6.2.
For the discrete-time state space model defined by (6.1), a Linear Quadratic (LQ)
regulator uses the state feedback law
u =−Kx (6.11)
to minimize the quadratic cost function
J =
∞
∑
i=0
(xT Qxx+uT Quu). (6.12)
Qx ∈Rn×n and Qu ∈Rp×p are weighting matrices used to penalize large system states
and large control inputs respectively.1
1Note the difference between the scalar transfer function K(s) in figure 6.2 and the matrix of (con-
stant) control gains K in the state feedback law (6.11).
6.4. LQG CONTROL 99
Substituting (6.11) into (6.1), we obtain the closed-loop system
x(k+1) = (A−BK)x(k)
y(k) =Cx(k)
(6.13)
and we see that, by suitable choice of the feedback gain matrix K, the eigenvalues of
(A−BK) – and therefore the dynamics of the closed-loop system – can be made stable.
Linear Quadratic (LQ) control assumes that the full state x is available to the regu-
lator – this state is used in the feedback law (6.11). In many cases, though, the full state
is not available for measurement, the regulator having direct access only to the inputs
u and the outputs y. Linear Quadratic Gaussian (LQG) control can be employed to
remedy this by using an observer to form an estimate xˆ of the real state x. This estimate
of the state is used in the LQG control feedback law
u =−Kxˆ. (6.14)
It remains to see how the estimate of the state is formed: the observer determines
the estimate of the state by
xˆ(k+1) = Axˆ(k)+Bu(k)+L[y(k)− yˆ(k)]
yˆ(k) =Cxˆ(k),
(6.15)
a system that mimics (6.1), but is forced by the output error y− yˆ. It is straightforward
to show that, if the eigenvalues of (A−LC) are stable, the estimation error xˆ− x tends
to zero. The Gaussian term in Linear Quadratic Gaussian control refers to the specific
type of observer used, a Kalman filter. A Kalman filter amounts to a specific choice
of the observer gain matrix L which is optimal, in the sense that the error converges in
the presence of stochastic disturbances d and noise n (see figure 6.2), assumed to be
zero-mean, Gaussian, white-noise processes.
The LQG and optimal estimation problems can be solved using standard routines
in Matlab. The resulting LQG controller and observer can then be combined to give
an overall regulator K(s), which is of the same order as the plant model used PL(s)
(including actuator).
100 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
The actuator transfer function used L(s) is a second-order, under-damped system:
L(s) = kL
ω2
L
s2+2ζLωLs+ω2L
,
with kL = 1.0, ωL = 1.9 (which corresponds to a non-dimensional frequency of fL =
0.3) and ζL = 0.3. See Kegerise et al. (2007a) for example, where the frequency
response of the piezoelectric actuator used was typical of this type.
Since we have a single input system, Qu in (6.12) is simply a scalar. We choose the
output measurement y =Cx as our output cost (giving Qx =CTC), and the quadratic
cost function has a simple form
J =
∞
∑
i=0
(y2+quu2),
where the scalar qu is used to specify the relative importance of maintaining a small
input signal and maintaining a small output measurement. Very little noise is encoun-
tered in the direct numerical simulations, but we expect more in a real experiment, so
Gaussian white noise is artificially added to the pressure measurement. The covariance
of this noise is used in the design of the observer.
The Bode diagram of the overall regulator K(s), given by the combination of the
state estimator (6.15) and the feedback law (6.14), is shown in figure 6.3. The Nyquist
diagram of the overall open-loop plant PL(s) is plotted in figure 6.4(a): this shows the
(complex-valued) frequency response of the system in the complex plain for both pos-
itive and negative frequencies. In figure 6.4(b), the Nyquist diagram of the loop-gain
K(s)PL(s) is plotted. Since the open-loop system has two pairs of unstable poles, we
require 4 counter-clockwise encirclements of the -1 point. The regulator K(s) provides
those 4 encirclements, and it does so with a gain margin of 4.5 dB and a phase margin
of 36◦.
6.4.2 RESULTS
We now look at how the LQG regulator performs in direct numerical simulations of
the cavity flow. The LQG regulator is first applied at the design point: a free stream
6.4. LQG CONTROL 101
−20
−10
0
10
20
30
0.5 1 1.5 2 2.5 3 3.5 4
−360
−270
−180
−90
G
ai
n
[d
B
]
Ph
as
e
[d
eg
.
]
St = f L/U∞ [–]
FIGURE 6.3: Bode diagram of the LQG regulator (—–); and of the gain-scheduled regu-
lator for a Mach number of M = 0.8 (−−).
Mach number of M = 0.6. Since we are confident that the state-space model of the
system is accurate, and since the optimal control techniques used should provide rea-
sonable robustness to changes in the plant, the regulator should exhibit good robustness
to changes in Mach number. We test this by applying the controller at off-design con-
ditions, for Mach numbers in the range 0.4≤M ≤ 0.8.
Closed-loop control results for the design case of M = 0.6 are shown in figure 6.5:
control is activated at approximately 76 non-dimensional time units. The introduction
of the LQG regulator stabilizes the flow, and the pressure measurement is reduced to
the background noise level.
Figure 6.6 summarizes the results of the LQG regulator applied at off-design Mach
numbers. Results for Mach numbers of 0.4, 0.5, 0.7 and 0.8 are shown – this range
of Mach numbers is based on the range over which the Rossiter mechanism (or ‘shear
layer mode’) is observed (Rowley et al., 2002). The LQG regulator performs well at
the lower Mach numbers, providing closed-loop stability for M = 0.4 and M = 0.5. At
the higher Mach numbers, though, it performs less well: for M = 0.7 and for M = 0.8,
the limit cycle, although reduced in amplitude by the regulator, persists. This poorer
performance at higher Mach numbers is a result of the cavity’s modified dynamics, and
102 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
−0.1 −0.05 0 0.05 0.1
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Re(PL)
Im
(P
L)
(a)
mode 1
mode 2
mode 3
−2 −1.5 −1 −0.5 0
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Re(KPL)
Im
(K
P L
)
(b)
mode 1
mode 2
mode 3
FIGURE 6.4: Nyquist diagrams for LQG control: (a) the open-loop plant PL(s); and (b)
the loop-gain K(s)PL(s). Positive and negative frequencies are shown, and in each plot the
three Rossiter modes are indicated.
is not a result of the 8 state reduced order model used: the same limitations would be
encountered by the controller if a higher order model of the cavity were used.
These same observations can be made by looking at the time delay phase portraits
of the pressure signals, which can help to distinguish between a stable, noise-driven
regime and a limit-cycling regime. The phase portrait of a limit-cycling system will
be a closed curve (and, with noise, a fuzzy curve). The phase portrait of a stable
system, forced by noise, will be concentrated about a point. Figure 6.7 shows the time
delay phase portraits for each of the four off-design Mach numbers: both open-loop
and closed-loop data are shown. Each of the plots helps clarify the observations made
based on figure 6.6. For Mach numbers of 0.4 and 0.5, control transforms the pressure
phase portrait from a closed-curve to being concentrated about a point, suggesting
closed-loop stability. For Mach numbers of 0.7 and 0.8, the closed-curve nature of the
phase portrait remains intact with feedback control activated.
6.4. LQG CONTROL 103
30 40 50 60 70 80 90 100 110 120
0.68
0.69
0.7
0.71
0.72
0.73
30 40 50 60 70 80 90 100 110 120
−0.2
−0.1
0
0.1
0.2
p c
[–
]
ρv
c
[–
]
t = tdU∞/L [–]
FIGURE 6.5: Closed-loop control using the LQG regulator at the design Mach number of
M = 0.60. The pressure measurement (top) and the control body force (bottom) are shown.
For the pressure measurement, the closed-loop case (—–) is compared to the open-loop
case (−−).
104 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
0
5
10
15
20
25
30
35
40
45
50
0.
70
8
0.
71
0.
71
2
0.
71
4
0.
71
6 0
5
10
15
20
25
30
35
40
45
50
−
0.
04
−
0.
020
0.
02
0.
04
p
c
[–]
ρvc[–]
t
=
td
U
∞
/L
[–
]
(a)
0
10
20
30
40
50
60
0.
690.
7
0.
71
0.
72
0.
73
0
10
20
30
40
50
60
−
0.
1
−
0.
050
0.
050.
1
p
c
[–] ρvc[–]
t
=
td
U
∞
/L
[–
]
(b)
0
10
20
30
40
50
60
70
0.
66
0.
680.
7
0.
72
0.
74
0
10
20
30
40
50
60
70
−
0.
20
0.
2
p
c
[–]
ρvc[–]
t
=
td
U
∞
/L
[–
]
(c)
0
10
20
30
40
50
60
70
0.
650.
7
0.
75
0
10
20
30
40
50
60
70
−
0.
50
0.
5
p
c
[–]
ρvc[–]
t
=
td
U
∞
/L
[–
]
(d)
F
IG
U
R
E
6.
6:
Pe
rf
or
m
an
ce
of
th
e
L
Q
G
re
gu
la
to
r
at
of
f-
de
si
gn
M
ac
h
nu
m
be
rs
:
fo
r
a
fr
ee
st
re
am
M
ac
h
nu
m
be
r
of
(a
)
M
=
0.
4;
(b
)
M
=
0.
5;
(c
)M
=
0.
7;
an
d
(d
)M
=
0.
8.
L
eg
en
d
sa
m
e
as
fig
ur
e
6.
5.
6.4. LQG CONTROL 105
FIGURE 6.7: Time delay phase portraits of the LQG regulator for free stream Mach num-
bers of (a) M = 0.4; (b) M = 0.5; (c) M = 0.7; and (d) M = 0.8. Pressure measurements
are compared for the closed-loop (•) and the open-loop case (•).
6.4.3 GAIN SCHEDULING
In this section we ask a simple question: can Rossiter’s formula, used for predicting
the frequencies of cavity oscillations, be used to increase the robustness of a feedback
controller?
Rossiter’s formula (Rossiter, 1964) has proved to be a reliable way of predicting the
frequencies of cavity oscillations. Whilst modifications have since been made (Heller
et al., 1971), these modifications retain the same underlying mechanism: a disturbance
in the free shear layer spanning the cavity is amplified via Kelvin-Helmholtz instabil-
ity; acoustic scattering of the disturbance at the cavity trailing edge gives rise to further
disturbances to the shear layer; for suitable phase change of the disturbance, resonance
106 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
occurs. Rossiter’s formula predicts resonant modes at Strouhal numbers given by
Stn =
n− γ
M+1/κ
, n = 1,2, . . .
where γ is a constant phase delay, and κ is given by the phase speed of disturbances in
the shear layer.
Gain-scheduling is a technique commonly employed in feedback control systems to
increase robustness to changing operating conditions. The method involves scheduling
the parameters of the feedback controller based on some global system property that
can be measured. In our case, this amounts to modifying the LQG regulator in response
to the free stream Mach number.
The exact form of the scheduling that we consider is very crude: simply ‘stretch-
ing’ or ‘shrinking’ the regulator’s frequency response along the frequency axis: the
stretching factor used is given by Rossiter’s formula. Using Rossiter’s formula, it is
easy to show that, for a given resonant mode number n, the relationship between the
Strouhal number of that resonant mode at two different Mach numbers M1 and M2 is
given by
Stn,M2
Stn,M1
=
M1+1/κ
M2+1/κ
, (6.16)
which gives us the stretching factor to use in the gain scheduling: this expression holds
for any n. The result of (6.16) for a Mach number of M = 0.8, which gives a stretching
factor of 0.922, is shown in figure 6.3. The regulator’s frequency response is squeezed
along the frequency axis to anticipate the reduction in Strouhal number that this new
Mach number brings.
We now look at applying this gain-scheduling to the regulator at those Mach num-
bers where closed-loop stability was not achieved (M = 0.7, M = 0.8 – see figure 6.6).
The results are shown in figure 6.8. For clarity, the open-loop pressure data are omitted,
and only the original regulator and gain-scheduled regulator data are compared. The
same scale as that in figure 6.6 is used for the pressure data, however, so comparison
with the open-loop data can easily be made.
We see that the gain-scheduled regulator actually performs more poorly at both
Mach numbers, with the amplitude of oscillations increased relative to the original
6.4. LQG CONTROL 107
0 10 20 30 40 50 60 70
0.66
0.68
0.7
0.72
0.74
0 10 20 30 40 50 60 70
−0.2
0
0.2
p c
[–
]
ρv
c
[–
]
t = tdU∞/L [–]
(a)
0 10 20 30 40 50 60 70
0.65
0.7
0.75
0 10 20 30 40 50 60 70
−0.5
0
0.5
p c
[–
]
ρv
c
[–
]
t = tdU∞/L [–]
(b)
FIGURE 6.8: Performance of the gain-scheduled LQG regulator at off-design Mach num-
bers: for a free stream Mach number of (a) M = 0.7; and (b) M = 0.8. For both the
pressure plots and the body force plots, the original regulator (—–) is compared to the
gain-scheduled regulator (−−).
108 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
regulator, despite a larger control body force being provided.
It appears that such a crude form of gain-scheduling is, in fact, too crude. This
is perhaps not surprising, since its foundations in Rossiter’s formula means that it ac-
counts solely for changes in frequency, and makes no allowance for the changes in
growth rates of the modes which will inevitably occur.
These observations provide clear motivation for an adaptive feedback controller,
which we look at next.
6.5 ADAPTIVE CONTROL
We now look at applying the adaptive controller – developed for combustion oscil-
lations in chapter 3 – to the cavity flow for free stream Mach numbers in the range
0.40 ≤M ≤ 0.80. Based on the observations made in § 5.6, we use a collocated con-
trol arrangement, so-called because the actuator and pressure sensor are now collo-
cated (see figure 5.13), using which the adaptive controller provides stability at all
Mach numbers tested.
With the requirements on PL(s) all met using collocated control as shown in § 5.6,
we apply the adaptive controller for n∗ ≤ 2 of § 3.3.1 (equation (3.9)):
k˙(t) = Γsgn(go)p(t)dλ (t)
V (t) =−kT (t)d(t)− k˙T (t)dλ (t),
(with, as before, k= [k1 k2]T , d= [p Vz]T , Vz = Vs+zc and dλ =
1
s+λ d) to the cavity flow
with collocated input and output. The parameter values used for the adaptive controller
are zc = 0.3, γ1 = 3.0×105, γ2 = 3.0×101, and sgn(g0) = +1.
Results of applying the updating rule and control law given by (3.9) to the cavity
flow for Mach numbers of 0.40, 0.50, 0.60, 0.70 and 0.80 are shown in figures 6.9–
6.13.
To make the adaptive controller’s behaviour clear, noise has not been artificially
added, although this could be accounted for in the updating rule with a dead-zone like
that used in chapter 4 (equation (4.8)). For each Mach number, two pressure signals
are plotted: the (collocated) pressure used for feedback control pc (see figure 5.13),
6.6. CONCLUDING REMARKS 109
and the pressure at probe 3, p3, which is included to make comparison with the LQG
control results easier, since p3 was the pressure signal used in that case.
The time scales used are the same as those used in figures 6.5 & 6.6 for LQG con-
trol, except for M = 0.70 and M = 0.80: for these two cases, the adaptation takes
longer and therefore the pressure fluctuations take longer to eliminate. Nevertheless,
the adaptive controller provides stability at all five Mach numbers. The fixed controller
that each of these controllers corresponds to once adaptation has stopped are given in
table 6.1.
Although the collocated control arrangement makes the adaptive controller less
practical than the LQG regulator, the advantage of adaptive control is clear, with sta-
bility provided over the entire Mach number range.
TABLE 6.1: Corresponding fixed controller K(s) at each of the five Mach numbers.
M = 0.4 M = 0.5 M = 0.6 M = 0.7 M = 0.8
25.8
s+0.30
s+0.36
39.7
s+0.30
s+0.47
47.4
s+0.30
s+0.59
34.7
s+0.30
s+0.47
49.7
s+0.30
s+0.68
6.6 CONCLUDING REMARKS
This chapter has built on the linear modeling approach used in chapter 5, and has done
so in two ways. First, model reduction using the eigensystem realization algorithm
has been used to form a low-order state space model of the cavity flow. Validation
of the reduced-order model has been performed in the frequency domain, where its
frequency response was compared to that found in chapter 5 using spectral methods.
This reduced-order model has been used to design a robust (H2) feedback controller,
which provides stability over a wider Mach number range than seen in previous studies.
Second, by taking advantage of the properties of the cavity flow derived in § 5.5,
together with the collocation-based arguments of § 5.6, the adaptive controller of chap-
ter 3 has been applied to the cavity flow. The adaptive controller provides stability over
the entire Mach number range, albeit at the cost of a less practical control arrangement.
110 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
0 5 10 15 20 25 30 35 40 45 50
0.706
0.71
0.714
0.718
0 5 10 15 20 25 30 35 40 45 50
0.708
0.712
0.716
0 5 10 15 20 25 30 35 40 45 50
−0.02
0
0.02
0 5 10 15 20 25 30 35 40 45 50
0
10
20
30
0 5 10 15 20 25 30 35 40 45 50
0
0.04
0.08
p 3
[–
]
p c
[–
]
ρv
c
[–
]
k 1
[–
]
k 2
[–
]
t = tdU∞/L [–]
FIGURE 6.9: Adaptive control for M = 0.40. The top two plots show the pressure mea-
surement at probe 3 and the feedback control pressure measurement pc for both the open-
loop (−−) and closed-loop (—–) case; the middle plot shows the control body force; and
the bottom two plots show the control parameters k1 and k2.
6.6. CONCLUDING REMARKS 111
0 10 20 30 40 50 60
0.69
0.7
0.71
0.72
0.73
0 10 20 30 40 50 60
0.7
0.71
0.72
0 10 20 30 40 50 60
−0.1
0
0.1
0 10 20 30 40 50 60
0
25
50
0 10 20 30 40 50 60
0
0.1
0.2
p 3
[–
]
p c
[–
]
ρv
c
[–
]
k 1
[–
]
k 2
[–
]
t = tdU∞/L [–]
FIGURE 6.10: Adaptive control for M = 0.50. Legend same as in figure 6.9.
112 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
30 40 50 60 70 80 90 100 110 120
0.68
0.69
0.7
0.71
0.72
0.73
30 40 50 60 70 80 90 100 110 120
0.69
0.7
0.71
0.72
0.73
30 40 50 60 70 80 90 100 110 120
−0.4
−0.2
0
0.2
0.4
30 40 50 60 70 80 90 100 110 120
0
25
50
30 40 50 60 70 80 90 100 110 120
0
0.2
0.4
p 3
[–
]
p c
[–
]
ρv
c
[–
]
k 1
[–
]
k 2
[–
]
t = tdU∞/L [–]
FIGURE 6.11: Adaptive control for M = 0.60. Legend same as in figure 6.9.
6.6. CONCLUDING REMARKS 113
0 10 20 30 40 50 60 70 80 90 100
0.66
0.68
0.7
0.72
0.74
0 10 20 30 40 50 60 70 80 90 100
0.68
0.7
0.72
0.74
0 10 20 30 40 50 60 70 80 90 100
−0.4
−0.2
0
0.2
0.4
0 10 20 30 40 50 60 70 80 90 100
0
20
40
0 10 20 30 40 50 60 70 80 90 100
0
0.1
0.2
p 3
[–
]
p c
[–
]
ρv
c
[–
]
k 1
[–
]
k 2
[–
]
t = tdU∞/L [–]
FIGURE 6.12: Adaptive control for M = 0.70. Legend same as in figure 6.9.
114 CHAPTER 6: FEEDBACK CONTROL OF CAVITY OSCILLATIONS
0 50 100 150
0.65
0.7
0.75
0 50 100 150
0.65
0.7
0.75
0 50 100 150
−0.5
0
0.5
0 50 100 150
0
25
50
0 50 100 150
0
0.25
0.5
p 3
[–
]
p c
[–
]
ρv
c
[–
]
k 1
[–
]
k 2
[–
]
t = tdU∞/L [–]
FIGURE 6.13: Adaptive control for M = 0.80. Legend same as in figure 6.9.
CHAPTER 7
CONCLUSIONS
This chapter contains a summary of the contributions of this thesis, together with some
suggestions for future work.
7.1 SUMMARY
The elimination of combustion oscillations using adaptive feedback control has been
considered. An adaptive feedback controller for annular combustor geometries has
been developed and tested in a low-order thermoacoustic network model called LOTAN.
The adaptive controller uses a modal decomposition of the pressure field from a num-
ber of azimuthally-spaced pressure transducers for sensing and a number of azimuthally-
spaced fuel valves for actuation. The controller stabilizes circumferential modes and
longitudinal modes, including the simultaneous control of multiple modes when those
modes are uncoupled. Control is maintained following a large change in operating
conditions. Finally, the controller still achieves control of a circumferential instability
using a reduced number of pressure transducers and fuel valves, which represents a
more practicable control arrangement.
One of the requirements of Lyapunov-based adaptive control has then been consid-
ered in more detail: that the sign of the high-frequency gain of the open-loop system
is known. By using a Nussbaum gain, an adaptive controller that does not require
this information has been developed and tested experimentally on a Rijke tube. The
Nussbaum gain adaptive controller stabilizes the system, and maintains control fol-
lowing a 13 % change in length of the Rijke tube (which leads to a 10 % change in the
115
116 CHAPTER 7: CONCLUSIONS
instability’s frequency).
Feedback control of a different type of fluid-acoustic resonance has also been con-
sidered: the compressible flow past a shallow cavity. First, system identification tech-
niques were used to very accurately determine the cavity’s linear dynamics or ‘transfer
function’. This dynamical model of the cavity – found directly from direct numerical
simulations – was then compared to that given by a simple linear model, which models
individual components of the cavity physics separately, and which is intended to be
useful for feedback control purposes.
Then we looked at using feedback control to eliminate these cavity resonances, and
two control strategies were considered. First, standard H2 robust control methods –
specifically a Linear Quadratic Gaussian regulator – were applied to the cavity flow,
the state-space model for which being provided by the Eigensystem Realization Algo-
rithm (ERA). The LQG regulator provides control over a much wider Mach number
range than seen previously in the literature. Second, we looked at using the adaptive
controller, earlier developed for combustion oscillations, to the cavity flow using col-
located control. The adaptive controller achieves closed-loop stability over an even
wider Mach number range.
This thesis has considered both robust (H2) control and adaptive control for sta-
bilizing thermo-acoustic and fluid-acoustic resonances. These two control approaches
are distinct, and bring with them their own sets of advantages and inconveniences.
Robust control makes no restrictions on the dynamics of the open-loop system, such
as the locations of its zeros or its relative degree, and the open-loop system’s being
observable and controllable is sufficient. This flexibility comes at a price, since an
accurate dynamical model of the system must be available, and this can be difficult to
obtain, especially for unstable systems. The adaptive controller considered requires no
such model of the system, and control can be achieved over a wide range of operating
conditions, which has been observed both for combustion systems and for cavity reso-
nances. However, the open-loop system must meet some general properties, and these
in themselves can be restrictive: for combustion oscillations, they place demands on
the nature of acoustic and entropy disturbances; and for cavity resonances, they could
only be met using collocated control.
7.2. SUGGESTIONS FOR FUTURE WORK 117
7.2 SUGGESTIONS FOR FUTURE WORK
Although feedback control of combustion oscillations has already been demonstrated
at large- and full-scale, most studies involve simple phase-locked loops. It seems, then,
that there is scope for more sophisticated controllers – which could include robust and
adaptive control – at large- and full-scale, and in particular for annular geometries,
allowing the adaptive controller for circumferential modes developed in this thesis to
be applied experimentally.
In terms of adaptive control, it seems that it is possible to adaptively stabilize a sys-
tem without knowledge of the sign of its high-frequency gain, although the controller
required to achieve it becomes more complex. The adaptive controller used, though,
is slightly at odds with the assumption made earlier in this thesis regarding the rela-
tive degree of a general combustion system. For a system with a relative degree not
exceeding two, a feedback controller must provide both gain and phase compensation,
and yet the Nussbaum gain controller provides only a feedback gain. The development
of an adaptive controller which utilizes a Nussbaum-type function and which provides
gain and phase compensation would resolve this.
It is encouraging that a simple linear model of cavity resonances can be used to
predict the system’s transfer function with a reasonable level of success, and which is
certainly useful for feedback control purposes. What remains to be seen is whether
a feedback controller designed using this simple linear model would be successful
in stabilizing the system in direct numerical simulations, and then whether the same
strategy could be used to model the system – and therefore to design a model-based
controller – at other Mach numbers. The emphasis, as it has been in this thesis, would
need to be on providing the information that is important for feedback control. This
seems to lend itself toH∞ loop-shaping techniques and to ν-gap metric analysis, which
indicates in what sense a model should be accurate if it is to be useful for feedback
design (Vinnicombe, 2000).
Finally, it seems that there is still scope for an adaptive controller that is more
well-suited to the control of cavity resonances, and which does not rely on collocated
control. This could be achieved by further development of the Lyapunov-based adap-
tive controller, although presumably this will result in a higher-order and therefore
118 CHAPTER 7: CONCLUSIONS
more complex controller. An alternative approach would be to develop a different type
of feedback controller which did not rely on any general properties of the open-loop
system, and Model Predictive Control (MPC) seems particularly attractive, since it can
be applied to systems with many inputs and many outputs; to both linear and non-
linear systems; and to systems with constraints (Maciejowski, 2002). MPC has found
diverse applications in systems with slow dynamics, in particular in the petrochemical
industry. The main barrier to its application to faster processes is the computational
burden of the method and therefore its restriction to low sampling frequencies. This
constraint is being overcome with more powerful computing capabilities and more ef-
ficient optimization algorithms. Of course, such techniques could equally be applied
to combustion oscillations, and this seems a promising area for future research.
APPENDIX A
GENERAL PROPERTIES OF COMBUSTION SYSTEMS
Here we repeat part of the analysis of Evesque (2000), but consider both longitudinal
and circumferential modes. Referring to figure 3.2, for the single-input single-output
(B = T = 1), longitudinal geometry case, Evesque made some general assumptions
about the open-loop transfer function P(s). She assumed that the actuator transfer
function L(s) had a relative degree not exceeding two, and that the flame transfer func-
tion F(s) was stable and had limited bandwidth, i.e.
lim
ω→∞F(iω) = 0.
For the modal open-loop transfer function given by equation (3.2), the actuator transfer
function Lb(s) and the flame transfer function Fb(s) are both for an individual burner,
and so the assumptions on the actuator and the flame are still valid in this case.
Another key part of Evesque’s analysis, however, was to use a wave expansion
of the acoustic field to characterize the acoustic transfer function G(s) between the
unsteady heat release and the pressure perturbation. By doing so Evesque was able to
show that the relative degree of P(s) was given by the relative degree of the actuator
L(s), and that all of the zeros of G(s) (and therefore of P(s)) would be in the left half-
plane. These results do not necessarily hold for the modal case Gn(s), and so we repeat
this part of the analysis here, but use a wave expansion that includes circumferential
dependence, which is important for the annular geometries of chapter 3.
Figure A.1 shows a general combustion chamber, which could represent a longi-
tudinal geometry, or one half of a circumferential slice through an annular geometry.
119
120 APPENDIX A: GENERAL PROPERTIES OF COMBUSTION SYSTEMS
Chapter 3. Adaptive control of circumferential modes 27
3.1 Theory
3.1.1 Plant transfer function
Before implementing an adaptive controller for circumferential modes, we
must first reconsider the three characteristics of the combustion system pre-
sented in §2.1.1. For the modal transfer functions, two of these conditions
are automatically met: that the relative degree of the open-loop system is
the same as the relative degree of the actuator; and that the sign of the high
frequency gain is positive 1. The remaining assumption - that the zeros of the
open-loop plant all lie in the left half-plane - is not necessarily met, however,
and it is this condition that we concern ourselves with here.
We will derive an analytical expression for the modal transfer functionW n(s)
for a circumferential mode in an annular combustor of the form shown in
figure 3.1. Here, n denotes the circumferential mode number. For a thin
annulus, we can set the radial velocity v′ to zero and ignore radial dependence
of the flow parameters. We also assume zero swirl velocity w′ = 0 2. Figure
3.1 shows a one half of a circumferential slice through the annulus of the
combustor. Zone 1, upstream of the flame, has downstream propagating
wave f and upstream propagating wave g. Zone 2, downstream of the flame,
has downstream propagating wave h and upstream propagating wave j.
xu x = 0 xd
Ru Rd
f
g
h
j
1 2
Q
Flame
Figure 3.1: One half of a circumferential slice through the combustor annulus
We will neglect the conversion of combustion generated entropy waves into
1this condition is looked at in more detail in §4
2including swirl would not change the equations of conservation of mass, momentum
and energy across the flame. It would, however, change the expression for the wave number
FIGURE A.1: Disturbances in a comb stion chamber.
Evesque (2000) lo ked at the longitudinal case, but we concentrate on the annular case
here. Upstream of the flame (zone 1) is a downstream propagating wave f and an
upstream propagating wave g, and downstream of the flame (zone 2) is a downstream
propagating wave h and an upstream propagating wave j. For a thin annulus, we can
set the radial velocity v′ to zero and ignore radial dependence of the flow parameters.
We also assume zero swirl velocity w′ = 0.1
We will neglect the conversion of combustion-generated entropy waves into acous-
tic waves at the downstream nozzle. This will be a good approximation when the time
taken for the entropy waves to convect through the duct exceeds their diffusion time.
We assume that the reflection coefficients at the upstream and downstream boundaries
satisfy |Ru(s)| < 1 and |Rd(s)| < 1. For linear disturbances, the perturbation in the
pressure p′, density ρ ′ and axial velocity u′ are given by (Stow & Dowling, 2001)
p′ = A±eiωt+ik±x+inθ
ρ ′ =
1
c2
A±eiωt+ik±x+inθ
u′ =− k±
ρα±
A±eiωt+ik±x+inθ ,
(A.1)
where c is the mean speed of sound and
ck± =
Mω∓ [ω2− (cnR )2(1−M
2)]
1
2
1−M2
(A.2a)
α± = ω+ cMk±. (A.2b)
1Including swirl would not change the equations of conservation of mass, momentum and energy
across the flame, but it would change the expression for the wave number.
121
The + and the − subscripts refer to the f ,h and g, j waves respectively, depending on
which zone the equations are applied to. M is the mean Mach number (assumed to be
less than unity) and R is the mean radius of the annulus. The cut-off frequency ωc is
given by
ωc =
cn
R
(1−M2) 12 ,
and R is the combustor radius. For ω < ωc, the waves decay and mode n is ‘cut-off’.
The equations of conservation of mass, momentum and energy across the combus-
tion zone can be written in a form independent of downstream density and temperature
(Dowling, 1997):
p2− p1+ρ1u1(u2−u1) = 0
γ
γ−1(p2u2− p1u1)+
1
2
ρ1u1(u22−u21) =
Q
Acomb
,
where Acomb is the cross-sectional area of the combustor. We now set each flow vari-
able to a mean value plus a perturbation (u1 = u1+u′1 for example, with u1 the mean
velocity and u′1 the perturbation). Linearizing these equations, and using the boundary
conditions at the upstream and downstream boundaries
f eik f xu = Rugeikgxu
jeik jxd = Rdheikhxd ,
we arrive at a matrix equation in the frequency domain:[
X11 X12
X21 X22
](
g
h
)
=
[
Y11 Y12
Y21 Y22
](
Rugei(kg−k f )xu
Rdhei(kh−k j)xd
)
+
(
0
Q′
Acombc1
)
. (A.3)
The full expressions for the X and Y matrices are given in appendix B. Expressed
in the time domain, (A.3) would give the time evolution of the outgoing waves g(t) and
h(t) in terms of the unsteady heat release Q′(t). Using (A.3) we can find a frequency-
domain expression relating the modal pressure pnref (at some axial location xref) to the
modal unsteady heat release at the flame Q′n. When measuring pnref downstream of the
122 APPENDIX A: GENERAL PROPERTIES OF COMBUSTION SYSTEMS
flame, we find
pnref(s)
Q′n(s)
=
[X11(s)−RuY11(s)ei(kg−k f )xu][1+Rdei(kh−k j)(xd−xref)]
Acombc1
eikhxref
= Gn0(s)e
ikhxref
= Gn(s).
(A.4)
Giving an expression for Gn(s) from equation (3.2) as the product of a transfer function
Gn0(s) and a pure time delay. If the pressure is instead measured upstream of the flame,
this becomes
pnref(s)
Q′n(s)
=
[Y12(s)Rdei(kh−k j)xd −X12(s)][1+Ruei(k f−kg)(xref−xu)]
Acombc1
eikgxref
= Gn0(s)e
ikgxref
= Gn(s).
(A.5)
We now concentrate on the upstream-sensing case (A.5). For the longitudinal case (n=
0), (A.5) simplifies and the X and Y matrices become independent of frequency (see
appendix B). By forming a rational approximation to G00(s) using Padé approximants
of its time delays, Evesque showed that, for longitudinal modes
(i) the relative degree of the overall open-loop transfer function P0(s) is equal to the
relative degree of the actuator L(s)
(ii) G00(s) (and therefore P
0(s)) has only left half-plane zeros.
With the additional circumferential dependence of the flow variables, the same
Padé approximant technique can be used to find a rational approximation to Gn0(s), and
assumption (i) regarding the relative degree of the modal open-loop transfer function
Pn(s) can still be made. Assumption (ii) regarding the position of the zeros of Gn0(s) is
not necessarily met, though, and now we look at this condition in more detail.
We want to show that any zeros of Gn0(s) lie in the left half-plane of the Laplace
variable s= iω , which is equivalent to their lying in the upper half-plane of the complex
123
frequency ω . For upstream-sensing (A.5), zeros may occur when either
|ei(kh−k j)xd |= |X12(s)||Rd(s)Y12(s)| , (A.6a)
or
|ei(k f−kg)(xref−xu)|= 1|Ru(s)| . (A.6b)
(These conditions are necessary but not sufficient, since the phase of the left hand-
side and right hand-side of (A.6a) or (A.6b) must also be equal for a zero to occur.) We
will now see that any zeros that lie well above the cut-off frequency ωc (i.e. frequencies
for which ω2 ω2c ) must lie in the left half-plane, Re(s)< 0.
For frequencies well above cut-off, the X and Y matrices simplify to expressions
that are independent of frequency (see appendix B), and then |X12| > |Y12|. There-
fore, since |Rd(s)|< 1 and |Ru(s)|< 1 by assumption, a zero can only occur when the
magnitude of the exponential term in (A.6a) or (A.6b) is greater than one. Well above
cut-off, these two equations become
e−sτd =
|X12|
|Rd(s)Y12| (A.7a)
e−sτu =
1
|Ru(s)| , (A.7b)
where τd and τu are acoustic propagation times given by τd = 2xd/(c2(1−M22)) and
τu = 2(xref−xu)/(c1(1−M22)). Therefore the magnitude of the exponential terms can
only be greater than one – and zeros can only occur – in the upper half-plane of ω ,
which corresponds to the left half-plane for the Laplace variable s. A similar proof can
be shown for downstream pressure sensing, if one makes the further assumption that
u2
u1
≥ 2−M1
1−M1
.
124 APPENDIX A: GENERAL PROPERTIES OF COMBUSTION SYSTEMS
APPENDIX B
X AND Y MATRICES
Here we collect together the full expressions for the X and Y matrices introduced in
chapter 2 for two cases: (i) the general case; and (ii) for frequencies well above the
cut-off frequency. For simplicity, we drop the over-bar notation for a mean value in the
flow parameters c, M and u, the mean value now being implied.
GENERAL CASE
The four terms in the X matrix introduced in appendix A are
X11(s) =−1+ kgc1αg M1
(
2− u2
u1
)
+M21
(
u2
u1
−1
)
,
X12(s) = 1−M2 khc2αh ,
X21(s) =
kgc1
αg − γM1
γ−1 +
kgc1
αg
M21 +
1
2
M21
(
M1− kgc1αg
)(
u22
u21
−1
)
,
X22(s) =
c2
c1
−khc2
αh + γM2
γ−1 −
khc2
αh
M1M2
ρ1
ρ2
,
125
126 APPENDIX B: X AND Y MATRICES
and the four terms in the Y matrix are
Y11(s) = 1− k f c1α f M1
(
2− u2
u1
)
−M21
(
u2
u1
−1
)
,
Y12(s) =−1+M2 k jc2α j ,
Y21(s) =
−k f c1
α f + γM1
γ−1 −
k f c1
α f
M21 −
1
2
M21
(
M1− k f c1α f
)(
u22
u21
−1
)
,
Y22(s) =
c2
c1
k jc2
α j + γM2
γ−1 +
k jc2
α j
M1M2
ρ1
ρ2
.
WELL ABOVE CUT-OFF
For frequencies well above the cut-off frequency ω2  ω2c , the X and Y matrices
can be approximated by expressions independent of frequency. These are the same
expressions that one finds for the longitudinal case, since ω2  ω2c is equivalent to
saying that the term cnR (1−M
2)1/2 in (A.2a) is negligible. For X we have
X11 =−1+M1
(
2− u2
u1
)
+M21
(
u2
u1
−1
)
,
X12 = 1+M2,
X21 =
1− γM1
γ−1 +M
2
1 +
1
2
M21 (M1−1)
(
u22
u21
−1
)
,
X22 =
c2
c1
1+ γM2
γ−1 +M1M2
ρ1
ρ2
,
and for Y
Y11 = 1+M1
(
2− u2
u1
)
−M21
(
u2
u1
−1
)
,
Y12 =−1+M2,
Y21 =
1+ γM1
γ−1 +M
2
1 −
1
2
M21 (M1+1)
(
u22
u21
−1
)
,
Y22 =
c2
c1
1+ γM2
γ−1 +M1M2
ρ1
ρ2
.
APPENDIX C
PROOF OF CLOSED-LOOP STABILITY FOR A NUSSBAUM GAIN
We consider the first order system (4.1). For the more general case of minimum phase,
n∗ = 1 systems, see Willems & Byrnes (1984). The first order system (4.1) can be
written in the time domain as
y˙(t) = ay(t)+g0u(t). (C.1)
Suppose the adaptive control law1
k˙(t) = y2(t)
u(t) = N(k)y(t)
is used, where N(k) is the Nussbaum gain. The resulting closed-loop system is given
by
y˙(t) = ay(t)+g0N(k)y(t) (C.2)
k˙(t) = y2(t). (C.3)
We want to find a Nussbaum gain that will guarantee closed-loop stability without
knowledge of the sign of g0. To study the closed-loop stability of the system, we
1Note the positive feedback convention used here, which is the opposite of that used in chapter 4
(equation (4.3)): the Nussbaum gain is ultimately insensitive to the sign adopted, and we adopt a positive
convention for simplicity.
127
128 APPENDIX C: PROOF OF CLOSED-LOOP STABILITY FOR A NUSSBAUM GAIN
introduce an indicator function,
Λ(t) =
1
2
y2(t). (C.4)
Evaluating Λ˙(t) along solutions to (C.2) gives
Λ˙(t) = y(t)y˙(t)
= {a+g0N(k)}y2(t)
= {a+g0N(k)}k˙(t)
Integrating over time t (using k(0) = k0 and introducing σ = k(τ)), we have∫ t
0
Λ˙(τ)dτ = a
∫ t
0
k˙(τ)dτ+g0
∫ t
0
N(σ)k˙(τ)dτ (C.5)
= a[k(t)− k0]+g0
∫ k(t)
k0
N(σ)dσ (C.6)
and so
Λ(t)−Λ(0) = [k(t)− k0]
{
a+
g0
k(t)− k0
∫ k(t)
k0
N(σ)dσ
}
. (C.7)
Suppose that the Nussbaum gain N(σ) satisfies the two conditions
sup
k>k0
1
k− k0
∫ k
k0
N(σ)dσ = ∞
inf
k>k0
1
k− k0
∫ k
k0
N(σ)dσ =−∞.
(C.8)
Seeking a contradiction, suppose that k(t)→∞ as t→∞ (from (C.3), k(t) is monotone
non-decreasing). Since the Nussbaum gain (C.8) takes arbitrarily large positive and
negative values as k(t)→ ∞, we derive a contradiction at (C.7). Therefore k(t) must
be bounded. From (C.3), this is equivalent to y(t) ∈ L2(0,∞]. Then from (C.2) y˙(t) ∈
L2(0,∞]. Finally, lemma 2.12, Narendra & Annaswamy (1989) shows that y(t) is
guaranteed to converge asymptotically to zero.
APPENDIX D
ADAPTATION RATES FOR THE NUSSBAUM GAIN
The adaptation rates γ and µ , introduced into the adaptive controller’s updating rule
(4.5) and control law (4.6–4.7) in § 4.3.2 are now explained in more detail, together
with details of how they are chosen.
CHOOSING γ
Looking at the updating rule (4.5), one can see that γ sets the rate at which k adapts.
Therefore γ can be set by specifying the desired speed of adaptation of k (Yildiz et al.,
2007). If k starts at zero, this corresponds to 3τn for a 5% band around the set point
(where τn is the period of the unstable mode):
k˙(t) = γ p2(t) =
|k∗|
3τn
,
and therefore
γ =
|k∗|
3τn pˆ2
,
where k∗ is the value of k for which closed-loop stability is achieved, and pˆ is a char-
acteristic value of p. The value of k∗ will depend on whether the Nussbaum gain is
initially updated in the right direction. To get around this we assume that the Nuss-
baum gain is initially updated in the right direction – then k∗ will be similar in size to
N∗, i.e.
γ =
|N∗|
3τn pˆ2
.
129
130 APPENDIX D: ADAPTATION RATES FOR THE NUSSBAUM GAIN
If the Nussbaum gain is instead initially updated in the wrong direction, then this sim-
ply means that the controller will take longer to reach k∗ (and therefore N∗) and achieve
closed-loop stability.
CHOOSING µ
Looking at the Nussbaum gain used (4.7), one can see that µ dictates, via the cosine
term, exactly when N(k) will switch sign. The Nussbaum gain N(k) provides closed-
loop stability by providing a gain that is (i) large enough and (ii) of the right sign.
Furthermore, N(k) only needs to switch sign once (if at all). Therefore it suffices to
ensure, via µ , that N(k) sweeps over a sufficiently large gain before it switches sign.
This will allow the controller to ‘try’ a big enough gain before N(k) switches sign.
We can ensure that this happens by specifying the turning point of N(k), that is by
specifying the value of N(k) at which dNdt = 0:
dN
dt
=
dN
dk
dk
dt
= 0.
We already have an expression for dkdt (4.5): this is zero when p is zero. Therefore we
are interested in dNdk = 0. From (4.7),
dN
dk
= cos(µ|kt | 14 )− 14µ|kt |
1
4 sin(µ|kt | 14 ) = 0,
where kt is the k for which dNdk = 0. Letting µ|kt |
1
4 = X , we have
X tanX = 4,
which we can solve for X . We need two equations for the two unknowns µ and kt : the
second equation comes from specifying Nt , the value of N(k) for which dNdk = 0, i.e.
Nt = N(k)| dN
dk =0
= kt cos(µ|kt | 14 )
Nt = kt cosX .
131
Then kt and µ can be found using
kt =
Nt
cosX
µ = X |kt |− 14 .
Therefore to choose the two adaption rates γ and µ , it suffices to have an estimate (to
order of magnitude accuracy only) of N∗, τn and pˆ. In addition, Nt needs to be chosen
so that Nt > |N∗| is guaranteed.
132 APPENDIX D: ADAPTATION RATES FOR THE NUSSBAUM GAIN
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