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Effect of phase-shift feedback on thermoacoustic coupling
13th IFAC IFAC Workshop Workshop on on Time Time Delay Delay Systems Systems 13th 13th Workshop on Delay June 22-24, 2016. Turkey 13th IFAC Workshop on Time Time Delay Systems Systems JuneIFAC 22-24, 2016. Istanbul, Istanbul, Turkey 13th IFAC Workshop on Time Delay Systems June June 22-24, 22-24, 2016. 2016. Istanbul, Istanbul, Turkey Turkey Available online at www.sciencedirect.com June 22-24, 2016. Istanbul, Turkey
ScienceDirect IFAC-PapersOnLine 49-10 (2016) 065–070
Effect of phase-shift feedback Effect of feedback Effect of phase-shift phase-shiftcoupling feedback thermoacoustic thermoacoustic coupling thermoacoustic coupling
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Beatriz M. M. Mor´ Mor´ an, n, A. A. Saurabh, Saurabh, C. C. O. O. Paschereit, Paschereit, L. L. Kabiraj Kabiraj Beatriz a Beatriz M. Mor´ a n, A. Saurabh, C. O. Paschereit, L. Kabiraj Beatriz a Beatriz M. M. Mor´ Mor´ an, n, A. A. Saurabh, Saurabh, C. C. O. O. Paschereit, Paschereit, L. L. Kabiraj Kabiraj Chair o Chair of of Fluid Fluid Dynamics, Dynamics, Hermann-F¨ Hermann-F¨ ottinger-Institut ttinger-Institut Chair of Fluid Universit¨ Dynamics, Hermann-F¨ ottinger-Institut Chair of Dynamics, o Technische a Berlin, 10623 Technische att Hermann-F¨ Berlin, Germany Germany 10623 Chair of Fluid Fluid Universit¨ Dynamics, Hermann-F¨ ottinger-Institut ttinger-Institut Technische Universit¨ a t Berlin, Germany 10623 Technische Universit¨ a t Berlin, Germany 10623 Technische Universit¨ a t Berlin, Germany 10623 (corresponding author:[email protected]) (corresponding author:[email protected]) (corresponding author:[email protected]) (corresponding (corresponding author:[email protected]) author:[email protected]) Abstract: Abstract: In In the the present present investigation investigation we we report report our our experiments experiments dealing dealing with with the the effect effect of of Abstract: In the the present present investigation we report report our experiments experiments dealing with theburner—a effect of of Abstract: In investigation we our dealing with the effect phase-shift feedback on chaotic thermoacoustic oscillations in a premixed flame Rijke phase-shift feedback on chaotic thermoacoustic oscillations in a premixed flame Rijke burner—a Abstract: In the present investigation we report our experiments dealing with the effect of phase-shift feedback on chaotic thermoacoustic oscillations in of premixed flame Rijke burner—a phase-shift on thermoacoustic in aaa premixed flame burner—a fundamentalfeedback configuration employed frequentlyoscillations in the the study study thermoacoustic coupling, which fundamental configuration employed frequently in of thermoacoustic coupling, which phase-shift feedback on chaotic chaotic thermoacoustic oscillations in of premixed flame Rijke Rijke burner—a fundamental configuration employed frequently in the the such study thermoacoustic coupling, which fundamental configuration employed frequently in study of thermoacoustic coupling, which is currently a pressing problem in engineering systems as gas turbine combustors, furnaces is currently a configuration pressing problem in engineering systems such asofgas turbine combustors, furnaces fundamental employed frequently in the study thermoacoustic coupling, which is currently a pressing problemThe in engineering systems such as gas gas turbine combustors, furnaces is currently a problem in systems such combustors, and rocket combustors. phenomenon of coupling itself another and rocket engine engine combustors. The phenomenon of thermoacoustic thermoacoustic coupling itself is is yet yet furnaces another is currently a pressing pressing problemThe in engineering engineering systems such as as gas turbine turbine combustors, furnaces and rocket engine combustors. phenomenon of thermoacoustic coupling itself is yet another and rocket engine combustors. The phenomenon of thermoacoustic coupling itself is yet another nonlinear feedback coupling phenomenon, which has only very recently been identified exhibit nonlinear feedback coupling phenomenon, which has only very recently been identified exhibit and rocket engine combustors. The phenomenon of thermoacoustic coupling itself is yet another nonlinear feedbackofcoupling coupling phenomenon, which has has onlyThe veryultimate recently aim beenofidentified identified exhibit nonlinear feedback which only very recently been a wide wide spectrum spectrum complexphenomenon, dynamics, including including chaos. research exhibit in this this a ofcoupling complex dynamics, chaos. The ultimate aim ofidentified research in nonlinear feedback phenomenon, which has only very recently been exhibit afield wide spectrum of complex dynamics, dynamics, including chaos. The ultimate ultimate aim of of research inchaos this a wide spectrum complex including The aim this is control thermoacoustic oscillations. That thermoacoustic coupling can exist field is to to control of thermoacoustic oscillations. Thatchaos. thermoacoustic coupling canresearch exist as asin chaos afield wide spectrum of complex dynamics, including chaos. The ultimate aim of research in this is to control control thermoacoustic oscillations.on That thermoacoustic couplingWe canshow existhere as chaos field is thermoacoustic oscillations. That thermoacoustic coupling can exist as and nonlinear states existing control that and other other nonlinear states has has implications implications on existing control strategies. strategies. We show here that field is to to control thermoacoustic oscillations.on That thermoacoustic couplingWe canshow existhere as chaos chaos and other nonlinear states has implications existing control strategies. that and other nonlinear states has implications on existing control strategies. We show here that the commonly employed phase-shift control results in an additional complexity, and that the the commonly employed phase-shift control results in an additional complexity, and that the and other nonlinear states has implications on existing control strategies. We show here that the commonly employed phase-shift control nonlinear results in inoscillations an additional additional complexity, and that the the the commonly employed phase-shift control results an complexity, and that resulting state could could correspond to different different depending on the the phase-shift resulting state correspond to nonlinear oscillations depending on phase-shift the commonly employed phase-shift control results in an additional complexity, and that the resulting state could correspond correspond to different different nonlinear oscillations oscillations depending on on the phase-shift phase-shift resulting state feedback feedback parameters. parameters. resulting state could could correspond to to different nonlinear nonlinear oscillations depending depending on the the phase-shift feedback parameters. parameters. feedback feedback parameters. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Thermoacoustic Thermoacoustic instabilities, instabilities, Nonlinear Nonlinear time time series series analysis, analysis, Chaos, Chaos, Control, Control, Keywords: Keywords: Thermoacoustic instabilities, Nonlinear time series analysis, Chaos, Control, Keywords: Thermoacoustic instabilities, Nonlinear Nonlinear time time series series analysis, analysis, Chaos, Chaos, Control, Control, Phase-shift feedback. Phase-shiftThermoacoustic feedback. Keywords: instabilities, Phase-shift feedback. Phase-shift feedback. Phase-shift feedback. 1. In 1. INTRODUCTION INTRODUCTION In aa general general fashion, fashion, the the term term chaos chaos describes describes complex complex 1. INTRODUCTION In aa general fashion, the term chaos describes complex 1. INTRODUCTION In general fashion, the term chaos describes complex behavior of a dynamical system which is aperiodic aperiodic but behavior of a dynamical system which is but 1. INTRODUCTION In a general fashion, the term chaos describes complex behavior of dynamical system which is aperiodic but behavior of aaanearby dynamical system whichrapidly is aperiodic aperiodic but bounded and and nearby solutions separate rapidly in time. time.but It bounded solutions separate in It behavior of dynamical system which is bounded and nearby solutions separate rapidly in time. It bounded andlong nearby solutions separate rapidly in time. time. It implies that term predictions of the system are almost implies that long term predictions of the system are almost bounded and nearby solutions separate rapidly in It Phase-shift feedback feedback control control have have been been used used before before as as aa implies that long term predictions of the system are almost Phase-shift implies thatdespite long term term predictions of the system system are are almost impossible the system deterministic (Glendinimpossible despite thepredictions system is is of deterministic (GlendinPhase-shift feedback control have been used before as that long the almost Phase-shift feedbacksuppress control have have been used used oscillations. before as as aaa implies strategy to to actively actively suppress thermoacoustic oscillations. impossible despite the system is deterministic (Glendinstrategy thermoacoustic Phase-shift feedback control been before impossible despite the system is deterministic (Glendinning (1994)). Even though no definition of chaos has ning (1994)). Even though no definition of chaos has strategy to actively suppress thermoacoustic oscillations. despite the systemnois definition deterministic (Glendinstrategy to actively suppress thermoacoustic oscillations. However, these strategies have not for ning (1994)). Even though of chaos has However,to these strategies havethermoacoustic not been been evaluated evaluated for the the impossible strategy actively suppress oscillations. ning (1994)). Even though nothe definition of agrees chaos that has universally been accepted yet, the literature agrees that universally been accepted yet, literature However, these strategies have not been evaluated for the ning (1994)). Even though no definition of chaos has However, these strategies have not not oscillations been evaluated evaluated for the the control chaotic thermoacoustic because it been accepted yet, the literature agrees that control of ofthese chaotic thermoacoustic oscillations because it universally However, strategies have been for universally beenpresent accepted yet, the literature literature agrees Strothat chaotic some certain characteristics. chaotic systems systems present some certain characteristics. Strocontrol of chaotic thermoacoustic oscillations because it universally been accepted yet, the agrees that control of chaotic thermoacoustic oscillations because it assumed that combustion instabilities exist only as limit chaotic systems present some certain characteristics. Stroassumed that combustion instabilities exist only as limit control of chaotic thermoacoustic oscillations because it chaotic systems present some certain characteristics. Strogatz (1994) gives a more elaborated definition of chaos gatz (1994) gives a more elaborated definition of chaos assumed that combustion instabilities exist only as limit systems present some certain characteristics. Stroassumed that combustion combustion instabilities exist of only as limit limit chaotic cycle Therefore, the phase-shift gatz (1994) gives aa more elaborated definition of chaos cycle oscillations. oscillations. Therefore, the application application of phase-shift assumed that instabilities exist only as gatz (1994)these givescharacteristics: more elaborated elaborated definition of chaos chaos embracing these characteristics: chaosdefinition is aperiodic aperiodic longembracing chaos is longcycle oscillations. Therefore, the application of phase-shift gatz (1994) gives a more of cycle oscillations. Therefore, the application of phase-shift control to chaotic oscillations, which has been investigated these characteristics: chaos is aperiodic longcontrol to chaotic Therefore, oscillations,the which has beenofinvestigated cycle oscillations. application phase-shift embracing embracing these characteristics: chaos is isthat aperiodic longterm behavior in a deterministic system that exhibits senterm behavior in a deterministic system exhibits sencontrol to chaotic oscillations, which has been investigated embracing these characteristics: chaos aperiodic longcontrol to chaotic chaotic oscillations, which in hasthe been investigated in is important development of behavior in deterministic system that exhibits senin this this paper, paper, is an anoscillations, important study study in the development of term control to which has been investigated term behavior in aaa on deterministic system that that exhibits sensitive dependence initial conditions. In this definition sitive dependence on initial conditions. In this definition in this paper, is an important study in the development of term behavior in deterministic system exhibits senin this paper, is an important study in the development of combustor (thermoacoustic systems) technology. dependence on initial conditions. In this definition combustor (thermoacoustic systems) technology. in this paper, is an important study intechnology. the development of sitive sitive dependence on that initialhave conditions. In this this definition definition there some to The there are are some terms terms that have to be be explained. explained. The first first combustor (thermoacoustic systems) sitive dependence on initial conditions. In combustor (thermoacoustic (thermoacoustic systems) systems) technology. technology. there are some terms that have to be explained. The first combustor there are some terms that have to be explained. The first one is isare aperiodic long-term behavior which means thatfirst in Thermoacoustic instability instability refers refers to to self-excited, self-excited, coupled, coupled, there one aperiodic long-term behavior which means that in Thermoacoustic some terms that have to be explained. The one is aperiodic long-term behavior which means that in Thermoacoustic instability refers to self-excited, coupled, one is aperiodic aperiodic dynamics long-term the behavior which means that in ina Thermoacoustic instability in refers to self-excited, self-excited, coupled, one the system does not high amplitude amplitude oscillations pressure (i.e. the asymptotic asymptotic dynamics the systemwhich does means not present present a high oscillations in pressure (i.e. in in the the acoustics acoustics is long-term behavior that Thermoacoustic instability refers to coupled, the asymptotic dynamics the system does not present aa high amplitude oscillations in pressure (i.e. in the acoustics the asymptotic dynamics the system does does behavior. not present present high amplitude oscillations oscillations in pressure pressure (i.e. in in the the acoustics acoustics fixed-point, periodic orbits or The of and rate (in fixed-point, periodic orbits the or quasi-periodic quasi-periodic behavior. The of combustors) combustors) and heat heat release release rate fluctuations fluctuations (in the the the asymptotic dynamics system not a high amplitude in (i.e. fixed-point, periodic orbits or quasi-periodic behavior. The of combustors) and heat release rate fluctuations (in the fixed-point, periodic orbits or or quasi-periodic quasi-periodic behavior. The of combustors) and heat release rate fluctuations fluctuations (in the the fixed-point, second term termperiodic is deterministic deterministic which states states behavior. the idea idea that that flames), which can appear at when second is which the flames), which and can heat appear at conditions conditions when density density orbits The of combustors) release rate (in second term is deterministic which states the idea that flames), which can appear at conditions when density second term is deterministic which states the idea that flames), which can appear at conditions when density the chaotic behavior is not due to noisy or random forces, fluctuations that result from the unsteady flame(s) add the chaotic behavior is not due to noisy or random forces, fluctuations that result from the unsteady flame(s) add second term is deterministic which states the idea that flames), which can appear atthe conditions when density the chaotic behavior is not due to noisy or random forces, fluctuations that result from unsteady flame(s) add the chaotic behavior isnonlinearity not due due to to noisy noisy or random random forces, fluctuations that result from from the unsteady unsteady flame(s) add the but rather to present in of energy to to the the combustor acoustic field at at aaflame(s) rate higher higher but chaotic rather due due to the theis nonlinearity present in this this kind kind of energy combustor acoustic field rate behavior not or forces, fluctuations that result the add but rather due to the nonlinearity present in this kind of energy to the combustor acoustic field at aa rate higher but rather due to the nonlinearity present in this kind of energy to the combustor acoustic field at rate higher systems. The last term is sensitive dependence on initial than the rate of acoustic energy dissipation present in the systems. The last termnonlinearity is sensitivepresent dependence onkind initial than thetorate ofcombustor acoustic energy dissipation present in the but rather due to the in this of energy the acoustic field at a rate higher systems. The last term is sensitive dependence on initial than the rate of acoustic energy dissipation present in the systems. The last term term isnearby sensitive dependence on initial than the In rate of feedback acoustic energy energy dissipation present in the the systems. conditions, it means means thatis nearby initial conditionson separate system. the process that generated, conditions, it that initial conditions separate system. In theof feedback processdissipation that is is thus thus generated, The last sensitive dependence initial than the rate acoustic present in conditions, it means that nearby initial conditions separate system. In the feedback process that is thus generated, conditions, it means that nearby initial conditions separate system. In the feedback process that is thus generated, exponentially fast while the system evolves in time. acoustic oscillations perturb the flame, which in turn exponentially fast while the system evolves in time. acoustic oscillations perturb the flame, which in turn conditions, it means that nearby initial conditions separate system. In the feedback process that is thus generated, exponentially fast while the system evolves in acoustic oscillations perturb the flame, which in turn exponentially fast fast while while the the system system evolves evolves in in time. time. acoustic oscillations perturb the the flame, which in turn turn exponentially generates oscillations acoustic oscillations. oscillations. Some examples of systems systems generates acoustic Some examples of acoustic perturb flame, which in While, clearly clearly it it is is of of academic academic and practical practical time. interest to to While, and interest generates acoustic oscillations. Some examples of systems generates acoustic oscillations. Some examples of systems vulnerable to this instability are gas turbines combustors, clearly it is of academic and practical interest to vulnerableacoustic to this instability areSome gas turbines combustors, generates oscillations. examplescombustors, of systems While, While, clearly it is is of of and academic and practical practical interest to to consider the existence and characterization of bifurcations, consider the existence characterization of bifurcations, vulnerable to this instability are gas turbines While, clearly it academic and interest vulnerable to this this instability instability are gasindustrial turbines boilers, combustors, industrial domestic and and the existence and characterization of bifurcations, industrial furnaces, furnaces, domesticare andgas industrial boilers, and consider vulnerable to turbines combustors, consider theother existence and characterization characterization of bifurcations, bifurcations, chaos nonlinear states syschaos and and other nonlinear states in in thermoacoustic thermoacoustic sysindustrial furnaces, domestic and industrial boilers, and consider the existence and of industrial furnaces, domestic and industrial boilers, and aero-engines. Combustion oscillations are highly nonlinear chaos and other nonlinear states in thermoacoustic sysaero-engines. Combustion oscillations are highly nonlinear industrial furnaces, domestic and industrial boilers, and chaos and other nonlinear states in thermoacoustic systems, ultimately the most practically relevant question is tems, ultimately the most practically relevant question is aero-engines. Combustion oscillations are highly nonlinear chaos and other nonlinear states in thermoacoustic sysaero-engines. Combustion oscillations are highly nonlinear in nature nature and andCombustion exhibit variety dynamical states ultimately the most practically relevant question is in exhibit aa rich rich variety of of dynamical states tems, aero-engines. oscillations are highly nonlinear tems, ultimately the most practically relevant question is to study the implications on control strategies. Phase-shift to study the implications on control strategies. Phase-shift in nature and exhibit aa rich variety of dynamical states tems, ultimately the most practically relevant question is in nature and exhibit rich variety of dynamical states such as chaos via well-defined scenarios (Kabiraj et al. study the implications on control strategies. Phase-shift such as chaos via well-defined scenarios (Kabiraj states et al. to in nature and exhibit a rich variety of dynamical to study the the implications on control strategies. Phase-shift feedback control, which on is the the focus of this this report, report, has feedback control, which is focus of has such as chaos via well-defined scenarios (Kabiraj et al. study implications control strategies. Phase-shift such as Kabiraj chaos via via well-defined scenarios (Kabiraj etThis al. to (2012), et (2015), et (2011)).et This feedback control, which is the focus of this report, has (2012), Kabiraj et al. al. (2015), Gotoda Gotoda et al. al. (2011)). such as chaos well-defined scenarios (Kabiraj al. feedback control, which is the focus of this report, has been applied previously to control instabilities in rocket been applied previously to control instabilities in rocket (2012), Kabiraj et al. (2015), Gotoda et al. (2011)). This feedback control, which is the focus of this report, has (2012), Kabiraj et al. (2015), Gotoda eton al.the (2011)). This work is is Kabiraj a part part et of al. our(2015), ongoing studyet onal. the control of been applied previously to control instabilities in rocket work a of our ongoing study control of (2012), Gotoda (2011)). This been applied previously to who control instabilities in rocket rocket engines by (1952), used this theoengines by Tsien Tsien (1952),to who used this method method theowork is part of our ongoing study on the control of applied previously control instabilities in work is aaa part part instability, of our our ongoing ongoing study onpresently the control control of been thermoacoustic instability, and we we focuson presently on the the engines by Tsien (1952), who used this method theothermoacoustic and focus on work is of study the of engines by Tsien (1952), who used this method theoretically to low-frequency oscillations in a liquid monoretically to low-frequency oscillations in a liquid monothermoacoustic instability, and we focus presently on the engines by Tsien (1952), who used this method theothermoacoustic instability, and we focus presently on the analysis of the application of phase-shift feedback control to low-frequency oscillations in a work liquid monoanalysis of the application of phase-shift feedback control thermoacoustic instability, of and we focus presently on the retically retically torocket low-frequency oscillations in liquidwas monopropellantto rocket motor. Shortly Shortly after, in this was expropellant motor. after, this work exanalysis of the application phase-shift feedback control low-frequency oscillations aa work liquid monoanalysis of the the behavior. application of of phase-shift phase-shift feedback feedback control control retically during propellant rocket motor. Shortly after, this was exduring chaotic chaotic behavior. analysis of application propellant rocket motor. Shortly after, this work was exduring chaotic behavior. propellant rocket motor. Shortly after, this work was exduring chaotic behavior. during chaotic behavior. Copyright © 2016, 2016 65 Copyright 2016 IFAC IFAC 65 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 65 Copyright ©under 2016 responsibility IFAC 65 Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 65 10.1016/j.ifacol.2016.07.474
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tended to the liquid bi-propellant rocket motors in Marble (1953). Heckl (1988) proved the reliability of feedback control implementing experimentally a phase-shift closed-loop in a Rijke tube. Paschereit et al. (1998) controlled thermoacosutic instabilities in an experimental low-emission swirl-stabilized combustor. More recently, Moeck et al. (2007) applied phase-shift control to an atmospheric swirlstabilized premixed combustor using two different actuators: an on-off valve to modulate secondary pilot fuel to modulate the flame and a loudspeaker located upstream of the burner, which provided excitation to the air mass flow, and thus modifies acoustics. The control scheme studied in this last study has been followed-up in the present work with the exception that only one actuator was used: the loudspeaker. These and several other studies that deal with control of thermoacoustic oscillations using feedback control only considered limit cycle thermoacoustic oscillations. Following the developments of chaos and chaos control in various fields, the study of chaos control strategies applied to thermoacoustic oscillations will lead to a huge step in the improvement of combustion systems.
Fig. 1. Schematic representation of the experimental setup.
As mentioned, in this paper, we investigate the effect on the oscillations dynamics after applying a phase-shift feedback to chaotic thermoacoustic oscillations in a laminar premixed combustor configuration. To fulfill this task, first we carry out a experimental bifurcations analysis which allows us to identify nonlinear states (incl. the chaotic state) in the combustor model. The investigation is executed through the application of nonlinear time series analysis techniques. Subsequently, the phase-shift feedback case is discussed. Fig. 2. Perforated plate geometry (left) and an amplification of section A (right). The flame is anchored on the plate on ignition.
2. EXPERIMENTAL SETUP Figure 1 shows a schematic representation of the model which consists of three vertical ducts: a steel upstream duct with four positions for microphones, a quartz glass tube which allows the visibility of the flame and a steel downstream duct with another four microphone positions. The total length of the duct is 1300 mm and it has an inner diameter of 105mm.
characterize the combustion oscillations in the system and find the good conditions of fuel and air for obtaining chaotic oscillations inside the prototypical model. This experiment consisted of subjecting the system to bifurcations by changing the control parameter which in this case was the equivalence ratio φ (controls the intensity of combustion—the flame). The equivalence ratio was varied keeping constant the air mass flowrate at 2250 g/h and increasing the fuel mass flowrate from φ = 0.722 to φ = 0.897 in equal steps of ≈ 0.008 which corresponds to 1 g/h of fuel. In each step of the bifurcation experiment, pressure and chemiluminescence data were acquired taking samples of 8 seconds length with a sampling frequency of 8192 Hz. Measurements were recorded only after waiting enough time to ensure that only asymptotic dynamics is captured.
The premixed reactants of the combustion process are injected at the bottom of the first duct. After passing the upstream tube, the flow meets a perforated plate employed as a holder to stabilize the flame. Considering a one-dimensional configuration in longitudinal direction, the flame remains stationary if the flame speed is equal to the speed of the unburnt flow at each location. By placing perforated plates as burners in a cross section of the gas flow, heat is lost from the flame and the burning velocity decreases until it equals the unburnt mixture velocity. Therefore, a stable laminar flat-flame is produced over a range of conditions. Fig. 2 shows the geometry of the perforated plate and an amplification of it in area A. It can be seen that the plate diameter is 50mm and its thickness 2mm. The hexagonal pattern of the holes can also be observed.
2.1 Phase-shift control Phase-shift control is a process of control which belongs to the group of closed-loop active control. The general scheme of the phase-shift control process applied in this work is shown in Fig. 3. The process starts with the measurement of the unsteady pressure in the combustor by a sensor, in this case a microphone. Then, this information is sent to the controller which for this process has three
Before applying phase-shift control to the instabilities in the combustor, we executed a bifurcation experiment in order to perform a bifurcation analysis which allows to 66
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3. RESULTS AND DISCUSSIONS 3.1 Bifurcation Analysis The results of the bifurcation experiment showing the transition of the investigated system between different nonlinear states are shown in a bifurcation diagram (Fig. 5). In this diagram, the equivalence ratio—the bifurcation parameter—is represented on the x-axis while, for each different equivalence ratio, on the y-axis are plotted the pressure wave amplitudes of the local maxima obtained from a section of the time series of the acquired pressure data. Distinction between different oscillatory states can be identified by the spread of points: For purely sinusoidal limit cycle oscillations, all the local maxima from the time series section will correspond to one amplitude, and will hence coincide. For more complex oscillations, amplitudes of maxima can assume non-identical values, hence points will be scattered for the corresponding parameter. The diagram is divided into three different regions, each of which is designated with a roman number. The lines which separate the regions roughly represent bifurcations in the thermoacoustic oscillations. These bifurcations were detected and characterized using different techniques from nonlinear time series analysis.
Fig. 3. Schematic of the phase-shift control process used in the present work. Detailed information can be found in B.M.Moran (2015).
important functions: shifting the phase of the pressure wave, reducing/amplifying proportionally the amplitude of the wave and, only if required, saturating the wave in such a manner that the driving signal sent to the actuator does not exceed the operating limits of the loudspeaker. Finally, the actuator generates the feedback signal which modulates the air mass flow into the burner and produces a sound wave. This wave interacts with the unsteady heat release and acoustic field to suppress the instabilities.
The sequence of bifurcations observed in the oscillations, while the control parameter was increased, is the one shown below. A subcritical Hopf bifurcation gave rise to limit cycle oscillations. Beyond this region, increasing φ brought the system to a state dominated by chaos, appearing indicatively via torus-breaking. And finally, a third bifurcation produced a state of period 2 oscillations in the system. Note that increasing the parameter does not cause monotonous changes in the flame properties. Hence, with this parameter, well-defined scenarios are not expected.
The control parameters of the controller are the time delay and the gain of the acoustic wave generated. The first one, the time delay, modifies the phase of the input signal in the controller while the gain modifies the amplitude of this signal.
Limit cycle −→ Chaos −→ P eriod 2
The bifurcation experiment led to conclude that the best flowrates for the phase-shift control experiment are an air- and fuel-flowrates respectively of m ˙ a = 2250 g/h and m ˙ f uel = 100 g/h which correspond to an equivalence ratio (φ) equal to 0.76. At these conditions the oscillations presented a chaotic behavior. Furthermore, Fig. 4 shows the control circuit employed. The pressure signal was used as the input to the circuit. This signal was phase-shifted and amplified according to the values of the driver/control parameters: time delay (τ ) and gain (K). Next, the signal was filtered by a low-pass filter with a cut-off frequency of 600 Hz. Finally, to prevent damages to the loudspeaker the signal was saturated at a value of ±0.2 volts. The data were recorded for 8 seconds at a sampling frequency of 8192 Hz.
(1)
Therefore, the suitable region for applying the control process is region II. We present the nonlinear analysis of a representative condition from this region which cor-
Fig. 4. Control circuit layout used for phase-shift control during thermoacoustic instability.
Fig. 5. Bifurcation plot summarizing the experiment at m ˙ air =2250 g/h. The Roman numerals (I, II and III) indicate different regions. 67
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Fig. 7. Analysis for data acquired by the PMT at φ = 0.745 with a moving average filter of 15 points: (a)time series, (b) power spectrum (c) phase space portrait using 8192 points from the time series and (d) Poincar´e section at the plane p(t-2τ )=0.
Fig. 6. Analysis for data acquired by the microphone at φ = 0.745: (a)time series, (b) power spectrum (c) phase space portrait using 8192 points from the time series and (d) Poincar´e section at the plane p(t-2τ )=0.
terms of datapoints and as data have been acquired with a sampling rate of 8192Hz, the value of λm for this condition is equal to 50.8. The positive value of λm indicates the presence of chaotic behavior in the system.
responds to φ=0.745. Figure 6 depicts the oscillations analysis of the data acquired by the microphone. In the phase space portrait, a toroidal attractor of a large crosssection thickness—evident in the Poincar´e section—can be seen. The presence of two independent frequencies in the power spectrum for this condition, f ∼ 223 Hz and f2 ∼ 32 Hz, suggests the existence of quasi-periodic behavior as a precursor to the chaotic state. The chaotic state, on other hand is clear from the dispersed (broken torus) Poincar´e section and the absence of a clear torus shape in the portrait.
3.2 Phase-shift control As mentioned before, the control parameters of the phaseshift feedback experiment are the time delay and the gain of the controller. We modified both parameters and studied the changes in the oscillation dynamics inside the combustor. The first parameter changed was τ while K was kept constant at 0.5. It was shifted from τ = 0 sec to τ = 0.005 sec in steps of 0.0002 sec. Fig. 9 displays the bifurcation diagram of the combustion oscillations behavior with the time delay, τ as the parameter. In this diagram τ is represented in the x-axis while in the y-axis the local maxima of the pressure amplitude from a section of the acquired time series are depicted. The diagram is divided into six different regions, each of which corresponds to a
Moreover, the analysis for the data acquired by the PMT (fluctuations in the flame intensity) at this condition is shown in Fig. 7. The flame intensity fluctuation data is in general quite noisy owing to the inherent noise of the optical measurement and incoherent fluctuations in the combustion process. Despite the moving average filter used (the strong dip in the spectra at ∼ 520 Hz is an effect of the filter), there is a large amount of noise which is quite evident in the phase space portrait and in its Poincar´e section. The power spectrum exhibits the same independent frequencies observed in the microphone spectrum, f ∼ 223 Hz and f2 ∼ 32 Hz and also linear combinations of them. To verify the existence of chaotic behavior we calculated the maximal Lyapunov exponents (Kantz (1994)) for the acquired data in this region. The Lyapunov exponents are a basic indicator of chaotic oscillations (Holzfuss and Lauterborn (1989)) such that a positive maximal exponent is a signature of chaos (Strogatz (1994); Kantz (1994)). The algorithm employed for this aim computes the average separation between neighboring trajectories in the phase space portrait as the system evolves in time. Fig. 8 shows the representation of the average separation S(∆n) respect to the temporal separation ∆n for the data acquired by the microphone at φ = 0.745. The black line in the graphic indicates a linear fit of the data whose slope represents the maximal Lyapunov exponent per step of time. The slope of the line, as indicated, is ∼ 0.0062 when calculated in
Fig. 8. Estimation of the maximal Lyapunov exponent at φ = 0.745. The black line indicates a linear fit and the slope of this line is equal to λm per time step. 68
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Fig. 10. Bifurcation diagram of the thermoacoustic oscillations shifting the gain of the input signal to controller. K was changed from K=0.05 to K=0.65 in steps of 0.05 while the time delay was kept constant at τ = 0.0018 sec.
Fig. 9. Bifurcation diagram of the thermoacoustic oscillations shifting the time delay of the input signal in the controller. τ was changed from τ = 0 sec to τ = 0.005 sec in steps of 0.0002 sec while the gain was kept constant at K=0.5.
one behavior to the other. Region III includes K=0.35 to K=0.65 and presented a more complex behavior, chaos. specific dynamical state, and is designated with a roman number. The bifurcations, represented by the vertical lines, were detected and characterized using different techniques from nonlinear time series analysis.
In the last two experiments, τ shifting and K shifting, we observed characteristic indications of the Ruelle-Takens transition to chaos scenario, specifically in transition between region III and region IV in Fig. 9 and in Fig. 10 between regions II and III. In both cases the route to chaos started from a limit cycle state whose power spectrum exhibit only one dominant frequency. The limit cycle oscillations evolved into a quasi-periodic state where the oscillations orbits are influenced by two irrational related frequencies. Eventually, the two torus structure of the quasi-periodic attractor breaks down leading to chaos due to the birth of a third incommensurate frequency. The scenario was particularly distinct for variation in K.
The first region of Fig. 9, region I, extends from τ = 0 to τ = 0.0002 sec, this region showed a periodic behavior of period 2. The next region, region II, extends from τ = 0.0004 to τ = 0.0008 sec and presented oscillations with a quasi-periodic behavior. Region III, τ = 0.0010 to τ = 0.0012 sec, showed again a periodic behavior. This time, oscillations belonged to a limit cycle. The following region, region IV, occupies the largest region in the bifurcation diagram. It ranges from τ = 0.0014−0.0032 sec and exhibited chaotic behavior. At the beginning of this region it was observed that quasi-periodic behavior existed in between the chaotic oscillations within the same time series. Furthermore, for some values of the time delay, the attractor could be distinguished as one that corresponds to a frequency-locked state. After this region, the oscillations changed back to a periodic limit cycle, and period-2 oscillations in region V, which extends from τ = 0.0034 to τ = 0.0046. Finally, a short region came out, region VI, which comprises τ = 0.0048 − 0.0050 sec. In this region the oscillations had limit cycle behavior.
In Fig. 11 we summarize the behaviors observed in the oscillations while applying the control process for the different values of the control parameters. In the x-axis is represented the time delay while in the y-axis is depicted the gain. The values of τ and K which correspond to a same behavior of the oscillations are grouped in an ellipse. Without phase-shift feedback, the dynamical state of the system corresponds to chaos. For a significant feedback input (K = 0.5), dynamical state transitions could be introduced by changing the delay. In addition, varying the strength of the feedback for the choice of a single delay also causes dynamical changes. However, in this latter case, the behavior was limited to chaos and quasi-periodicity, most likely because, as seen in the map, in the area that corresponds to the time delay, the system shows a strong inclination to chaos. Through the experiment, only a small subset of the entire K-τ region was traversed. In a full investigation, a more complete picture containing patches in the map corresponding to different dynamical states can be expected. The main message, however, that comes of the investigations is that phase-shift control, which was considered previously as a candidate for instability control leads an increase in the complexity of the system; The inherent nonlinear dynamics of the system together with the additional complexity of phase-shift feedback would need
The second control parameter of the controller was the feedback signal gain. This parameter was shifted from K=0.05 to K=0.65 in steps of 0.05 while τ was kept constant at 0.0018 sec. In the same manner as in the previous case, the results of the bifurcations are exhibited in a bifurcation diagram as showed in Fig. 10. In this case, we divided the diagram in 3 different regions designated with a roman number. The first region of the diagram, region I, comprises only K=0.05. In this region the oscillations presented a quasi-periodic behavior. The following region, region II, extends from K=0.1 to K=0.3 and exhibited two different behaviors: limit cycle and quasi-periodic. Both states were contained within the same time series data and the system underwent constant transition from 69
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TU Berlin and her master thesis. The financial support received by L.K from the DFG to conduct research at the TU Berlin is gratefully acknowledged. The authors also thank Jonas Moeck for providing the phase-shift routine for the experiments.
to be considered to identify the impact of feedback control. Progress in this direction has recently been made by Olgac and colleagues (Olgac et al. (2015)). The authors propose a novel methodology to identify phase-shift control parameters, K and τ . In contrast to previously existing methods, the proposed methodology is based on a formulation that treats the system as infinite-dimensional, and hence, does not require the assumption of finite number of acoustic modes. The approach is promising, however, essentially linear. It will be of interest to identify whether nonlinear dynamics that arise from complex flame dynamics and resulting complex oscillatory states affect the performance of phase shift control. Specialized techniques, such as OGY control (Ott et al. (1990)), might be required for control.
REFERENCES B.M.Moran (2015). Active Control to Suppress Nonlinear Thermoacoustic Oscillations. Master’s thesis, Technische Universitt Berlin. Glendinning, P. (1994). Stability, instability and chaos: an introduction of the theory of nonlinear differential equations. Cambridge University Press. Gotoda, H., Nikimoto, H., Miyano, T., and Tachibana, S. (2011). Dynamic properties of combustion instability in a lean premixed gas turbine combustor. AIP Publishing. Heckl, M. (1988). Active control of the noise from a rijke tube. Journal of Sound and Vibration, 117–133. Holzfuss, J. and Lauterborn, W. (1989). Lyapunov exponents from a time series of acoustic chaos. Physical Review A. Kabiraj, L., Saurabh, A., Karimi, N., Sailor, A., Mastorakos, E., Dowling, A., and Paschereit, C.O. (2015). Chaos in an imperfectly premixed model combustor. University of Glasgow. Kabiraj, L., Saurabh, A., Wahi, P., and Sujith, R. (2012). Route to chaos for combustion instability in ducted laminar premixed flames. AIP Publishing. Kantz, H. (1994). A robust method to estimate the maximal Lyapunov exponent of a time series. Physics Letters A, 185(1), 77– 87. doi:10.1016/0375-9601(94)90991-1. URL http://dx.doi.org/10.1016/0375-9601(94)90991-1. Marble, F. (1953). Servo-stabilization of low-frequency oscillations in a liquid bipropellant rocket motor. Journal of the American Rocket Society, 63–81. Moeck, J., Bothien, M., Guyot, D., and Paschereit, C.O. (2007). Phase-shift control of combustion instability using (combined) secondary fuel injection and acoustic forcing. Institute for Fluid Dynamics and Technical Acoustic. Olgac, N., Zalluhoglu, U., and Kammer, A.S. (2015). A new perspective in designing delayed feedback control for thermo-acoustic instabilities (TAI). Combustion Science and Technology, 187(5), 697–720. doi: 10.1080/00102202.2014.960924. Ott, E., Grebogi, C., and Yorke, J.A. (1990). Controlling chaos. Phys. Rev. Lett., 64(11), 1196–1199. doi:10.1103/physrevlett.64.1196. URL http://dx.doi.org/10.1103/PhysRevLett.64.1196. Paschereit, C.O., Gutmark, E., and Weisenstein, W. (1998). Structure and control of thermoacoustic instabilities in a gas-turbine combustor. Combsution Science and Thechnology, 138. Strogatz, S.H. (1994). Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry and Engineering. Westview Press., Massachusetts. Tsien, H. (1952). Servo-stabilization of combustion in rocket motors. Journal of the American Rocket Society, 256–263.
Fig. 11. States observed in the oscillations for the different values of the control parameters K and τ . 4. CONCLUSIONS The main objective of this work was to demonstrate the effect that phase-shift feedback has on chaotic thermoacoustic oscillations. We have first presented the bifurcation analysis of the thermoacoustic oscillations without feedback to illustrate bifurcations and different dynamics states appearing in the system as a result of changes in the equivalence ratio of the combustion process. Subsequently the response of the chaotic state to phase-shift feedback was studied while varying the feedback parameters: time delay and the gain of the phase-shifted signal. The experiments illustrated that the behavior of the instabilities underwent transitions among chaotic state, periodic and quasi-periodic states. Previously, phase-shift control was proposed as an instability control strategy. We demonstrated experimentally that phase-shift control of thermoacoustic coupling can results in additional complexity, particularly when chaos is present. The results reached in this work are important from a practical perspective because it is the first investigation where chaos is taken into account in the implementation of active control of thermoacoustic instabilities. ACKNOWLEDGEMENTS B.M.M. thanks the ERASMUS programme and Universidad Polit´ecnica de Madrid to support her stay at the 70
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Effect of phase-shift feedback Effect of feedback Effect of phase-shift phase-shiftcoupling feedback thermoacoustic thermoacoustic coupling thermoacoustic coupling
on on on
Beatriz M. M. Mor´ Mor´ an, n, A. A. Saurabh, Saurabh, C. C. O. O. Paschereit, Paschereit, L. L. Kabiraj Kabiraj Beatriz a Beatriz M. Mor´ a n, A. Saurabh, C. O. Paschereit, L. Kabiraj Beatriz a Beatriz M. M. Mor´ Mor´ an, n, A. A. Saurabh, Saurabh, C. C. O. O. Paschereit, Paschereit, L. L. Kabiraj Kabiraj Chair o Chair of of Fluid Fluid Dynamics, Dynamics, Hermann-F¨ Hermann-F¨ ottinger-Institut ttinger-Institut Chair of Fluid Universit¨ Dynamics, Hermann-F¨ ottinger-Institut Chair of Dynamics, o Technische a Berlin, 10623 Technische att Hermann-F¨ Berlin, Germany Germany 10623 Chair of Fluid Fluid Universit¨ Dynamics, Hermann-F¨ ottinger-Institut ttinger-Institut Technische Universit¨ a t Berlin, Germany 10623 Technische Universit¨ a t Berlin, Germany 10623 Technische Universit¨ a t Berlin, Germany 10623 (corresponding author:[email protected]) (corresponding author:[email protected]) (corresponding author:[email protected]) (corresponding (corresponding author:[email protected]) author:[email protected]) Abstract: Abstract: In In the the present present investigation investigation we we report report our our experiments experiments dealing dealing with with the the effect effect of of Abstract: In the the present present investigation we report report our experiments experiments dealing with theburner—a effect of of Abstract: In investigation we our dealing with the effect phase-shift feedback on chaotic thermoacoustic oscillations in a premixed flame Rijke phase-shift feedback on chaotic thermoacoustic oscillations in a premixed flame Rijke burner—a Abstract: In the present investigation we report our experiments dealing with the effect of phase-shift feedback on chaotic thermoacoustic oscillations in of premixed flame Rijke burner—a phase-shift on thermoacoustic in aaa premixed flame burner—a fundamentalfeedback configuration employed frequentlyoscillations in the the study study thermoacoustic coupling, which fundamental configuration employed frequently in of thermoacoustic coupling, which phase-shift feedback on chaotic chaotic thermoacoustic oscillations in of premixed flame Rijke Rijke burner—a fundamental configuration employed frequently in the the such study thermoacoustic coupling, which fundamental configuration employed frequently in study of thermoacoustic coupling, which is currently a pressing problem in engineering systems as gas turbine combustors, furnaces is currently a configuration pressing problem in engineering systems such asofgas turbine combustors, furnaces fundamental employed frequently in the study thermoacoustic coupling, which is currently a pressing problemThe in engineering systems such as gas gas turbine combustors, furnaces is currently a problem in systems such combustors, and rocket combustors. phenomenon of coupling itself another and rocket engine engine combustors. The phenomenon of thermoacoustic thermoacoustic coupling itself is is yet yet furnaces another is currently a pressing pressing problemThe in engineering engineering systems such as as gas turbine turbine combustors, furnaces and rocket engine combustors. phenomenon of thermoacoustic coupling itself is yet another and rocket engine combustors. The phenomenon of thermoacoustic coupling itself is yet another nonlinear feedback coupling phenomenon, which has only very recently been identified exhibit nonlinear feedback coupling phenomenon, which has only very recently been identified exhibit and rocket engine combustors. The phenomenon of thermoacoustic coupling itself is yet another nonlinear feedbackofcoupling coupling phenomenon, which has has onlyThe veryultimate recently aim beenofidentified identified exhibit nonlinear feedback which only very recently been a wide wide spectrum spectrum complexphenomenon, dynamics, including including chaos. research exhibit in this this a ofcoupling complex dynamics, chaos. The ultimate aim ofidentified research in nonlinear feedback phenomenon, which has only very recently been exhibit afield wide spectrum of complex dynamics, dynamics, including chaos. The ultimate ultimate aim of of research inchaos this a wide spectrum complex including The aim this is control thermoacoustic oscillations. That thermoacoustic coupling can exist field is to to control of thermoacoustic oscillations. Thatchaos. thermoacoustic coupling canresearch exist as asin chaos afield wide spectrum of complex dynamics, including chaos. The ultimate aim of research in this is to control control thermoacoustic oscillations.on That thermoacoustic couplingWe canshow existhere as chaos field is thermoacoustic oscillations. That thermoacoustic coupling can exist as and nonlinear states existing control that and other other nonlinear states has has implications implications on existing control strategies. strategies. We show here that field is to to control thermoacoustic oscillations.on That thermoacoustic couplingWe canshow existhere as chaos chaos and other nonlinear states has implications existing control strategies. that and other nonlinear states has implications on existing control strategies. We show here that the commonly employed phase-shift control results in an additional complexity, and that the the commonly employed phase-shift control results in an additional complexity, and that the and other nonlinear states has implications on existing control strategies. We show here that the commonly employed phase-shift control nonlinear results in inoscillations an additional additional complexity, and that the the the commonly employed phase-shift control results an complexity, and that resulting state could could correspond to different different depending on the the phase-shift resulting state correspond to nonlinear oscillations depending on phase-shift the commonly employed phase-shift control results in an additional complexity, and that the resulting state could correspond correspond to different different nonlinear oscillations oscillations depending on on the phase-shift phase-shift resulting state feedback feedback parameters. parameters. resulting state could could correspond to to different nonlinear nonlinear oscillations depending depending on the the phase-shift feedback parameters. parameters. feedback feedback parameters. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Thermoacoustic Thermoacoustic instabilities, instabilities, Nonlinear Nonlinear time time series series analysis, analysis, Chaos, Chaos, Control, Control, Keywords: Keywords: Thermoacoustic instabilities, Nonlinear time series analysis, Chaos, Control, Keywords: Thermoacoustic instabilities, Nonlinear Nonlinear time time series series analysis, analysis, Chaos, Chaos, Control, Control, Phase-shift feedback. Phase-shiftThermoacoustic feedback. Keywords: instabilities, Phase-shift feedback. Phase-shift feedback. Phase-shift feedback. 1. In 1. INTRODUCTION INTRODUCTION In aa general general fashion, fashion, the the term term chaos chaos describes describes complex complex 1. INTRODUCTION In aa general fashion, the term chaos describes complex 1. INTRODUCTION In general fashion, the term chaos describes complex behavior of a dynamical system which is aperiodic aperiodic but behavior of a dynamical system which is but 1. INTRODUCTION In a general fashion, the term chaos describes complex behavior of dynamical system which is aperiodic but behavior of aaanearby dynamical system whichrapidly is aperiodic aperiodic but bounded and and nearby solutions separate rapidly in time. time.but It bounded solutions separate in It behavior of dynamical system which is bounded and nearby solutions separate rapidly in time. It bounded andlong nearby solutions separate rapidly in time. time. It implies that term predictions of the system are almost implies that long term predictions of the system are almost bounded and nearby solutions separate rapidly in It Phase-shift feedback feedback control control have have been been used used before before as as aa implies that long term predictions of the system are almost Phase-shift implies thatdespite long term term predictions of the system system are are almost impossible the system deterministic (Glendinimpossible despite thepredictions system is is of deterministic (GlendinPhase-shift feedback control have been used before as that long the almost Phase-shift feedbacksuppress control have have been used used oscillations. before as as aaa implies strategy to to actively actively suppress thermoacoustic oscillations. impossible despite the system is deterministic (Glendinstrategy thermoacoustic Phase-shift feedback control been before impossible despite the system is deterministic (Glendinning (1994)). Even though no definition of chaos has ning (1994)). Even though no definition of chaos has strategy to actively suppress thermoacoustic oscillations. despite the systemnois definition deterministic (Glendinstrategy to actively suppress thermoacoustic oscillations. However, these strategies have not for ning (1994)). Even though of chaos has However,to these strategies havethermoacoustic not been been evaluated evaluated for the the impossible strategy actively suppress oscillations. ning (1994)). Even though nothe definition of agrees chaos that has universally been accepted yet, the literature agrees that universally been accepted yet, literature However, these strategies have not been evaluated for the ning (1994)). Even though no definition of chaos has However, these strategies have not not oscillations been evaluated evaluated for the the control chaotic thermoacoustic because it been accepted yet, the literature agrees that control of ofthese chaotic thermoacoustic oscillations because it universally However, strategies have been for universally beenpresent accepted yet, the literature literature agrees Strothat chaotic some certain characteristics. chaotic systems systems present some certain characteristics. Strocontrol of chaotic thermoacoustic oscillations because it universally been accepted yet, the agrees that control of chaotic thermoacoustic oscillations because it assumed that combustion instabilities exist only as limit chaotic systems present some certain characteristics. Stroassumed that combustion instabilities exist only as limit control of chaotic thermoacoustic oscillations because it chaotic systems present some certain characteristics. Strogatz (1994) gives a more elaborated definition of chaos gatz (1994) gives a more elaborated definition of chaos assumed that combustion instabilities exist only as limit systems present some certain characteristics. Stroassumed that combustion combustion instabilities exist of only as limit limit chaotic cycle Therefore, the phase-shift gatz (1994) gives aa more elaborated definition of chaos cycle oscillations. oscillations. Therefore, the application application of phase-shift assumed that instabilities exist only as gatz (1994)these givescharacteristics: more elaborated elaborated definition of chaos chaos embracing these characteristics: chaosdefinition is aperiodic aperiodic longembracing chaos is longcycle oscillations. Therefore, the application of phase-shift gatz (1994) gives a more of cycle oscillations. Therefore, the application of phase-shift control to chaotic oscillations, which has been investigated these characteristics: chaos is aperiodic longcontrol to chaotic Therefore, oscillations,the which has beenofinvestigated cycle oscillations. application phase-shift embracing embracing these characteristics: chaos is isthat aperiodic longterm behavior in a deterministic system that exhibits senterm behavior in a deterministic system exhibits sencontrol to chaotic oscillations, which has been investigated embracing these characteristics: chaos aperiodic longcontrol to chaotic chaotic oscillations, which in hasthe been investigated in is important development of behavior in deterministic system that exhibits senin this this paper, paper, is an anoscillations, important study study in the development of term control to which has been investigated term behavior in aaa on deterministic system that that exhibits sensitive dependence initial conditions. In this definition sitive dependence on initial conditions. In this definition in this paper, is an important study in the development of term behavior in deterministic system exhibits senin this paper, is an important study in the development of combustor (thermoacoustic systems) technology. dependence on initial conditions. In this definition combustor (thermoacoustic systems) technology. in this paper, is an important study intechnology. the development of sitive sitive dependence on that initialhave conditions. In this this definition definition there some to The there are are some terms terms that have to be be explained. explained. The first first combustor (thermoacoustic systems) sitive dependence on initial conditions. In combustor (thermoacoustic (thermoacoustic systems) systems) technology. technology. there are some terms that have to be explained. The first combustor there are some terms that have to be explained. The first one is isare aperiodic long-term behavior which means thatfirst in Thermoacoustic instability instability refers refers to to self-excited, self-excited, coupled, coupled, there one aperiodic long-term behavior which means that in Thermoacoustic some terms that have to be explained. The one is aperiodic long-term behavior which means that in Thermoacoustic instability refers to self-excited, coupled, one is aperiodic aperiodic dynamics long-term the behavior which means that in ina Thermoacoustic instability in refers to self-excited, self-excited, coupled, one the system does not high amplitude amplitude oscillations pressure (i.e. the asymptotic asymptotic dynamics the systemwhich does means not present present a high oscillations in pressure (i.e. in in the the acoustics acoustics is long-term behavior that Thermoacoustic instability refers to coupled, the asymptotic dynamics the system does not present aa high amplitude oscillations in pressure (i.e. in the acoustics the asymptotic dynamics the system does does behavior. not present present high amplitude oscillations oscillations in pressure pressure (i.e. in in the the acoustics acoustics fixed-point, periodic orbits or The of and rate (in fixed-point, periodic orbits the or quasi-periodic quasi-periodic behavior. The of combustors) combustors) and heat heat release release rate fluctuations fluctuations (in the the the asymptotic dynamics system not a high amplitude in (i.e. fixed-point, periodic orbits or quasi-periodic behavior. The of combustors) and heat release rate fluctuations (in the fixed-point, periodic orbits or or quasi-periodic quasi-periodic behavior. The of combustors) and heat release rate fluctuations fluctuations (in the the fixed-point, second term termperiodic is deterministic deterministic which states states behavior. the idea idea that that flames), which can appear at when second is which the flames), which and can heat appear at conditions conditions when density density orbits The of combustors) release rate (in second term is deterministic which states the idea that flames), which can appear at conditions when density second term is deterministic which states the idea that flames), which can appear at conditions when density the chaotic behavior is not due to noisy or random forces, fluctuations that result from the unsteady flame(s) add the chaotic behavior is not due to noisy or random forces, fluctuations that result from the unsteady flame(s) add second term is deterministic which states the idea that flames), which can appear atthe conditions when density the chaotic behavior is not due to noisy or random forces, fluctuations that result from unsteady flame(s) add the chaotic behavior isnonlinearity not due due to to noisy noisy or random random forces, fluctuations that result from from the unsteady unsteady flame(s) add the but rather to present in of energy to to the the combustor acoustic field at at aaflame(s) rate higher higher but chaotic rather due due to the theis nonlinearity present in this this kind kind of energy combustor acoustic field rate behavior not or forces, fluctuations that result the add but rather due to the nonlinearity present in this kind of energy to the combustor acoustic field at aa rate higher but rather due to the nonlinearity present in this kind of energy to the combustor acoustic field at rate higher systems. The last term is sensitive dependence on initial than the rate of acoustic energy dissipation present in the systems. The last termnonlinearity is sensitivepresent dependence onkind initial than thetorate ofcombustor acoustic energy dissipation present in the but rather due to the in this of energy the acoustic field at a rate higher systems. The last term is sensitive dependence on initial than the rate of acoustic energy dissipation present in the systems. The last term term isnearby sensitive dependence on initial than the In rate of feedback acoustic energy energy dissipation present in the the systems. conditions, it means means thatis nearby initial conditionson separate system. the process that generated, conditions, it that initial conditions separate system. In theof feedback processdissipation that is is thus thus generated, The last sensitive dependence initial than the rate acoustic present in conditions, it means that nearby initial conditions separate system. In the feedback process that is thus generated, conditions, it means that nearby initial conditions separate system. In the feedback process that is thus generated, exponentially fast while the system evolves in time. acoustic oscillations perturb the flame, which in turn exponentially fast while the system evolves in time. acoustic oscillations perturb the flame, which in turn conditions, it means that nearby initial conditions separate system. In the feedback process that is thus generated, exponentially fast while the system evolves in acoustic oscillations perturb the flame, which in turn exponentially fast fast while while the the system system evolves evolves in in time. time. acoustic oscillations perturb the the flame, which in turn turn exponentially generates oscillations acoustic oscillations. oscillations. Some examples of systems systems generates acoustic Some examples of acoustic perturb flame, which in While, clearly clearly it it is is of of academic academic and practical practical time. interest to to While, and interest generates acoustic oscillations. Some examples of systems generates acoustic oscillations. Some examples of systems vulnerable to this instability are gas turbines combustors, clearly it is of academic and practical interest to vulnerableacoustic to this instability areSome gas turbines combustors, generates oscillations. examplescombustors, of systems While, While, clearly it is is of of and academic and practical practical interest to to consider the existence and characterization of bifurcations, consider the existence characterization of bifurcations, vulnerable to this instability are gas turbines While, clearly it academic and interest vulnerable to this this instability instability are gasindustrial turbines boilers, combustors, industrial domestic and and the existence and characterization of bifurcations, industrial furnaces, furnaces, domesticare andgas industrial boilers, and consider vulnerable to turbines combustors, consider theother existence and characterization characterization of bifurcations, bifurcations, chaos nonlinear states syschaos and and other nonlinear states in in thermoacoustic thermoacoustic sysindustrial furnaces, domestic and industrial boilers, and consider the existence and of industrial furnaces, domestic and industrial boilers, and aero-engines. Combustion oscillations are highly nonlinear chaos and other nonlinear states in thermoacoustic sysaero-engines. Combustion oscillations are highly nonlinear industrial furnaces, domestic and industrial boilers, and chaos and other nonlinear states in thermoacoustic systems, ultimately the most practically relevant question is tems, ultimately the most practically relevant question is aero-engines. Combustion oscillations are highly nonlinear chaos and other nonlinear states in thermoacoustic sysaero-engines. Combustion oscillations are highly nonlinear in nature nature and andCombustion exhibit variety dynamical states ultimately the most practically relevant question is in exhibit aa rich rich variety of of dynamical states tems, aero-engines. oscillations are highly nonlinear tems, ultimately the most practically relevant question is to study the implications on control strategies. Phase-shift to study the implications on control strategies. Phase-shift in nature and exhibit aa rich variety of dynamical states tems, ultimately the most practically relevant question is in nature and exhibit rich variety of dynamical states such as chaos via well-defined scenarios (Kabiraj et al. study the implications on control strategies. Phase-shift such as chaos via well-defined scenarios (Kabiraj states et al. to in nature and exhibit a rich variety of dynamical to study the the implications on control strategies. Phase-shift feedback control, which on is the the focus of this this report, report, has feedback control, which is focus of has such as chaos via well-defined scenarios (Kabiraj et al. study implications control strategies. Phase-shift such as Kabiraj chaos via via well-defined scenarios (Kabiraj etThis al. to (2012), et (2015), et (2011)).et This feedback control, which is the focus of this report, has (2012), Kabiraj et al. al. (2015), Gotoda Gotoda et al. al. (2011)). such as chaos well-defined scenarios (Kabiraj al. feedback control, which is the focus of this report, has been applied previously to control instabilities in rocket been applied previously to control instabilities in rocket (2012), Kabiraj et al. (2015), Gotoda et al. (2011)). This feedback control, which is the focus of this report, has (2012), Kabiraj et al. (2015), Gotoda eton al.the (2011)). This work is is Kabiraj a part part et of al. our(2015), ongoing studyet onal. the control of been applied previously to control instabilities in rocket work a of our ongoing study control of (2012), Gotoda (2011)). This been applied previously to who control instabilities in rocket rocket engines by (1952), used this theoengines by Tsien Tsien (1952),to who used this method method theowork is part of our ongoing study on the control of applied previously control instabilities in work is aaa part part instability, of our our ongoing ongoing study onpresently the control control of been thermoacoustic instability, and we we focuson presently on the the engines by Tsien (1952), who used this method theothermoacoustic and focus on work is of study the of engines by Tsien (1952), who used this method theoretically to low-frequency oscillations in a liquid monoretically to low-frequency oscillations in a liquid monothermoacoustic instability, and we focus presently on the engines by Tsien (1952), who used this method theothermoacoustic instability, and we focus presently on the analysis of the application of phase-shift feedback control to low-frequency oscillations in a work liquid monoanalysis of the application of phase-shift feedback control thermoacoustic instability, of and we focus presently on the retically retically torocket low-frequency oscillations in liquidwas monopropellantto rocket motor. Shortly Shortly after, in this was expropellant motor. after, this work exanalysis of the application phase-shift feedback control low-frequency oscillations aa work liquid monoanalysis of the the behavior. application of of phase-shift phase-shift feedback feedback control control retically during propellant rocket motor. Shortly after, this was exduring chaotic chaotic behavior. analysis of application propellant rocket motor. Shortly after, this work was exduring chaotic behavior. propellant rocket motor. Shortly after, this work was exduring chaotic behavior. during chaotic behavior. Copyright © 2016, 2016 65 Copyright 2016 IFAC IFAC 65 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016 IFAC 65 Copyright ©under 2016 responsibility IFAC 65 Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 65 10.1016/j.ifacol.2016.07.474
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tended to the liquid bi-propellant rocket motors in Marble (1953). Heckl (1988) proved the reliability of feedback control implementing experimentally a phase-shift closed-loop in a Rijke tube. Paschereit et al. (1998) controlled thermoacosutic instabilities in an experimental low-emission swirl-stabilized combustor. More recently, Moeck et al. (2007) applied phase-shift control to an atmospheric swirlstabilized premixed combustor using two different actuators: an on-off valve to modulate secondary pilot fuel to modulate the flame and a loudspeaker located upstream of the burner, which provided excitation to the air mass flow, and thus modifies acoustics. The control scheme studied in this last study has been followed-up in the present work with the exception that only one actuator was used: the loudspeaker. These and several other studies that deal with control of thermoacoustic oscillations using feedback control only considered limit cycle thermoacoustic oscillations. Following the developments of chaos and chaos control in various fields, the study of chaos control strategies applied to thermoacoustic oscillations will lead to a huge step in the improvement of combustion systems.
Fig. 1. Schematic representation of the experimental setup.
As mentioned, in this paper, we investigate the effect on the oscillations dynamics after applying a phase-shift feedback to chaotic thermoacoustic oscillations in a laminar premixed combustor configuration. To fulfill this task, first we carry out a experimental bifurcations analysis which allows us to identify nonlinear states (incl. the chaotic state) in the combustor model. The investigation is executed through the application of nonlinear time series analysis techniques. Subsequently, the phase-shift feedback case is discussed. Fig. 2. Perforated plate geometry (left) and an amplification of section A (right). The flame is anchored on the plate on ignition.
2. EXPERIMENTAL SETUP Figure 1 shows a schematic representation of the model which consists of three vertical ducts: a steel upstream duct with four positions for microphones, a quartz glass tube which allows the visibility of the flame and a steel downstream duct with another four microphone positions. The total length of the duct is 1300 mm and it has an inner diameter of 105mm.
characterize the combustion oscillations in the system and find the good conditions of fuel and air for obtaining chaotic oscillations inside the prototypical model. This experiment consisted of subjecting the system to bifurcations by changing the control parameter which in this case was the equivalence ratio φ (controls the intensity of combustion—the flame). The equivalence ratio was varied keeping constant the air mass flowrate at 2250 g/h and increasing the fuel mass flowrate from φ = 0.722 to φ = 0.897 in equal steps of ≈ 0.008 which corresponds to 1 g/h of fuel. In each step of the bifurcation experiment, pressure and chemiluminescence data were acquired taking samples of 8 seconds length with a sampling frequency of 8192 Hz. Measurements were recorded only after waiting enough time to ensure that only asymptotic dynamics is captured.
The premixed reactants of the combustion process are injected at the bottom of the first duct. After passing the upstream tube, the flow meets a perforated plate employed as a holder to stabilize the flame. Considering a one-dimensional configuration in longitudinal direction, the flame remains stationary if the flame speed is equal to the speed of the unburnt flow at each location. By placing perforated plates as burners in a cross section of the gas flow, heat is lost from the flame and the burning velocity decreases until it equals the unburnt mixture velocity. Therefore, a stable laminar flat-flame is produced over a range of conditions. Fig. 2 shows the geometry of the perforated plate and an amplification of it in area A. It can be seen that the plate diameter is 50mm and its thickness 2mm. The hexagonal pattern of the holes can also be observed.
2.1 Phase-shift control Phase-shift control is a process of control which belongs to the group of closed-loop active control. The general scheme of the phase-shift control process applied in this work is shown in Fig. 3. The process starts with the measurement of the unsteady pressure in the combustor by a sensor, in this case a microphone. Then, this information is sent to the controller which for this process has three
Before applying phase-shift control to the instabilities in the combustor, we executed a bifurcation experiment in order to perform a bifurcation analysis which allows to 66
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3. RESULTS AND DISCUSSIONS 3.1 Bifurcation Analysis The results of the bifurcation experiment showing the transition of the investigated system between different nonlinear states are shown in a bifurcation diagram (Fig. 5). In this diagram, the equivalence ratio—the bifurcation parameter—is represented on the x-axis while, for each different equivalence ratio, on the y-axis are plotted the pressure wave amplitudes of the local maxima obtained from a section of the time series of the acquired pressure data. Distinction between different oscillatory states can be identified by the spread of points: For purely sinusoidal limit cycle oscillations, all the local maxima from the time series section will correspond to one amplitude, and will hence coincide. For more complex oscillations, amplitudes of maxima can assume non-identical values, hence points will be scattered for the corresponding parameter. The diagram is divided into three different regions, each of which is designated with a roman number. The lines which separate the regions roughly represent bifurcations in the thermoacoustic oscillations. These bifurcations were detected and characterized using different techniques from nonlinear time series analysis.
Fig. 3. Schematic of the phase-shift control process used in the present work. Detailed information can be found in B.M.Moran (2015).
important functions: shifting the phase of the pressure wave, reducing/amplifying proportionally the amplitude of the wave and, only if required, saturating the wave in such a manner that the driving signal sent to the actuator does not exceed the operating limits of the loudspeaker. Finally, the actuator generates the feedback signal which modulates the air mass flow into the burner and produces a sound wave. This wave interacts with the unsteady heat release and acoustic field to suppress the instabilities.
The sequence of bifurcations observed in the oscillations, while the control parameter was increased, is the one shown below. A subcritical Hopf bifurcation gave rise to limit cycle oscillations. Beyond this region, increasing φ brought the system to a state dominated by chaos, appearing indicatively via torus-breaking. And finally, a third bifurcation produced a state of period 2 oscillations in the system. Note that increasing the parameter does not cause monotonous changes in the flame properties. Hence, with this parameter, well-defined scenarios are not expected.
The control parameters of the controller are the time delay and the gain of the acoustic wave generated. The first one, the time delay, modifies the phase of the input signal in the controller while the gain modifies the amplitude of this signal.
Limit cycle −→ Chaos −→ P eriod 2
The bifurcation experiment led to conclude that the best flowrates for the phase-shift control experiment are an air- and fuel-flowrates respectively of m ˙ a = 2250 g/h and m ˙ f uel = 100 g/h which correspond to an equivalence ratio (φ) equal to 0.76. At these conditions the oscillations presented a chaotic behavior. Furthermore, Fig. 4 shows the control circuit employed. The pressure signal was used as the input to the circuit. This signal was phase-shifted and amplified according to the values of the driver/control parameters: time delay (τ ) and gain (K). Next, the signal was filtered by a low-pass filter with a cut-off frequency of 600 Hz. Finally, to prevent damages to the loudspeaker the signal was saturated at a value of ±0.2 volts. The data were recorded for 8 seconds at a sampling frequency of 8192 Hz.
(1)
Therefore, the suitable region for applying the control process is region II. We present the nonlinear analysis of a representative condition from this region which cor-
Fig. 4. Control circuit layout used for phase-shift control during thermoacoustic instability.
Fig. 5. Bifurcation plot summarizing the experiment at m ˙ air =2250 g/h. The Roman numerals (I, II and III) indicate different regions. 67
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Fig. 7. Analysis for data acquired by the PMT at φ = 0.745 with a moving average filter of 15 points: (a)time series, (b) power spectrum (c) phase space portrait using 8192 points from the time series and (d) Poincar´e section at the plane p(t-2τ )=0.
Fig. 6. Analysis for data acquired by the microphone at φ = 0.745: (a)time series, (b) power spectrum (c) phase space portrait using 8192 points from the time series and (d) Poincar´e section at the plane p(t-2τ )=0.
terms of datapoints and as data have been acquired with a sampling rate of 8192Hz, the value of λm for this condition is equal to 50.8. The positive value of λm indicates the presence of chaotic behavior in the system.
responds to φ=0.745. Figure 6 depicts the oscillations analysis of the data acquired by the microphone. In the phase space portrait, a toroidal attractor of a large crosssection thickness—evident in the Poincar´e section—can be seen. The presence of two independent frequencies in the power spectrum for this condition, f ∼ 223 Hz and f2 ∼ 32 Hz, suggests the existence of quasi-periodic behavior as a precursor to the chaotic state. The chaotic state, on other hand is clear from the dispersed (broken torus) Poincar´e section and the absence of a clear torus shape in the portrait.
3.2 Phase-shift control As mentioned before, the control parameters of the phaseshift feedback experiment are the time delay and the gain of the controller. We modified both parameters and studied the changes in the oscillation dynamics inside the combustor. The first parameter changed was τ while K was kept constant at 0.5. It was shifted from τ = 0 sec to τ = 0.005 sec in steps of 0.0002 sec. Fig. 9 displays the bifurcation diagram of the combustion oscillations behavior with the time delay, τ as the parameter. In this diagram τ is represented in the x-axis while in the y-axis the local maxima of the pressure amplitude from a section of the acquired time series are depicted. The diagram is divided into six different regions, each of which corresponds to a
Moreover, the analysis for the data acquired by the PMT (fluctuations in the flame intensity) at this condition is shown in Fig. 7. The flame intensity fluctuation data is in general quite noisy owing to the inherent noise of the optical measurement and incoherent fluctuations in the combustion process. Despite the moving average filter used (the strong dip in the spectra at ∼ 520 Hz is an effect of the filter), there is a large amount of noise which is quite evident in the phase space portrait and in its Poincar´e section. The power spectrum exhibits the same independent frequencies observed in the microphone spectrum, f ∼ 223 Hz and f2 ∼ 32 Hz and also linear combinations of them. To verify the existence of chaotic behavior we calculated the maximal Lyapunov exponents (Kantz (1994)) for the acquired data in this region. The Lyapunov exponents are a basic indicator of chaotic oscillations (Holzfuss and Lauterborn (1989)) such that a positive maximal exponent is a signature of chaos (Strogatz (1994); Kantz (1994)). The algorithm employed for this aim computes the average separation between neighboring trajectories in the phase space portrait as the system evolves in time. Fig. 8 shows the representation of the average separation S(∆n) respect to the temporal separation ∆n for the data acquired by the microphone at φ = 0.745. The black line in the graphic indicates a linear fit of the data whose slope represents the maximal Lyapunov exponent per step of time. The slope of the line, as indicated, is ∼ 0.0062 when calculated in
Fig. 8. Estimation of the maximal Lyapunov exponent at φ = 0.745. The black line indicates a linear fit and the slope of this line is equal to λm per time step. 68
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Fig. 10. Bifurcation diagram of the thermoacoustic oscillations shifting the gain of the input signal to controller. K was changed from K=0.05 to K=0.65 in steps of 0.05 while the time delay was kept constant at τ = 0.0018 sec.
Fig. 9. Bifurcation diagram of the thermoacoustic oscillations shifting the time delay of the input signal in the controller. τ was changed from τ = 0 sec to τ = 0.005 sec in steps of 0.0002 sec while the gain was kept constant at K=0.5.
one behavior to the other. Region III includes K=0.35 to K=0.65 and presented a more complex behavior, chaos. specific dynamical state, and is designated with a roman number. The bifurcations, represented by the vertical lines, were detected and characterized using different techniques from nonlinear time series analysis.
In the last two experiments, τ shifting and K shifting, we observed characteristic indications of the Ruelle-Takens transition to chaos scenario, specifically in transition between region III and region IV in Fig. 9 and in Fig. 10 between regions II and III. In both cases the route to chaos started from a limit cycle state whose power spectrum exhibit only one dominant frequency. The limit cycle oscillations evolved into a quasi-periodic state where the oscillations orbits are influenced by two irrational related frequencies. Eventually, the two torus structure of the quasi-periodic attractor breaks down leading to chaos due to the birth of a third incommensurate frequency. The scenario was particularly distinct for variation in K.
The first region of Fig. 9, region I, extends from τ = 0 to τ = 0.0002 sec, this region showed a periodic behavior of period 2. The next region, region II, extends from τ = 0.0004 to τ = 0.0008 sec and presented oscillations with a quasi-periodic behavior. Region III, τ = 0.0010 to τ = 0.0012 sec, showed again a periodic behavior. This time, oscillations belonged to a limit cycle. The following region, region IV, occupies the largest region in the bifurcation diagram. It ranges from τ = 0.0014−0.0032 sec and exhibited chaotic behavior. At the beginning of this region it was observed that quasi-periodic behavior existed in between the chaotic oscillations within the same time series. Furthermore, for some values of the time delay, the attractor could be distinguished as one that corresponds to a frequency-locked state. After this region, the oscillations changed back to a periodic limit cycle, and period-2 oscillations in region V, which extends from τ = 0.0034 to τ = 0.0046. Finally, a short region came out, region VI, which comprises τ = 0.0048 − 0.0050 sec. In this region the oscillations had limit cycle behavior.
In Fig. 11 we summarize the behaviors observed in the oscillations while applying the control process for the different values of the control parameters. In the x-axis is represented the time delay while in the y-axis is depicted the gain. The values of τ and K which correspond to a same behavior of the oscillations are grouped in an ellipse. Without phase-shift feedback, the dynamical state of the system corresponds to chaos. For a significant feedback input (K = 0.5), dynamical state transitions could be introduced by changing the delay. In addition, varying the strength of the feedback for the choice of a single delay also causes dynamical changes. However, in this latter case, the behavior was limited to chaos and quasi-periodicity, most likely because, as seen in the map, in the area that corresponds to the time delay, the system shows a strong inclination to chaos. Through the experiment, only a small subset of the entire K-τ region was traversed. In a full investigation, a more complete picture containing patches in the map corresponding to different dynamical states can be expected. The main message, however, that comes of the investigations is that phase-shift control, which was considered previously as a candidate for instability control leads an increase in the complexity of the system; The inherent nonlinear dynamics of the system together with the additional complexity of phase-shift feedback would need
The second control parameter of the controller was the feedback signal gain. This parameter was shifted from K=0.05 to K=0.65 in steps of 0.05 while τ was kept constant at 0.0018 sec. In the same manner as in the previous case, the results of the bifurcations are exhibited in a bifurcation diagram as showed in Fig. 10. In this case, we divided the diagram in 3 different regions designated with a roman number. The first region of the diagram, region I, comprises only K=0.05. In this region the oscillations presented a quasi-periodic behavior. The following region, region II, extends from K=0.1 to K=0.3 and exhibited two different behaviors: limit cycle and quasi-periodic. Both states were contained within the same time series data and the system underwent constant transition from 69
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TU Berlin and her master thesis. The financial support received by L.K from the DFG to conduct research at the TU Berlin is gratefully acknowledged. The authors also thank Jonas Moeck for providing the phase-shift routine for the experiments.
to be considered to identify the impact of feedback control. Progress in this direction has recently been made by Olgac and colleagues (Olgac et al. (2015)). The authors propose a novel methodology to identify phase-shift control parameters, K and τ . In contrast to previously existing methods, the proposed methodology is based on a formulation that treats the system as infinite-dimensional, and hence, does not require the assumption of finite number of acoustic modes. The approach is promising, however, essentially linear. It will be of interest to identify whether nonlinear dynamics that arise from complex flame dynamics and resulting complex oscillatory states affect the performance of phase shift control. Specialized techniques, such as OGY control (Ott et al. (1990)), might be required for control.
REFERENCES B.M.Moran (2015). Active Control to Suppress Nonlinear Thermoacoustic Oscillations. Master’s thesis, Technische Universitt Berlin. Glendinning, P. (1994). Stability, instability and chaos: an introduction of the theory of nonlinear differential equations. Cambridge University Press. Gotoda, H., Nikimoto, H., Miyano, T., and Tachibana, S. (2011). Dynamic properties of combustion instability in a lean premixed gas turbine combustor. AIP Publishing. Heckl, M. (1988). Active control of the noise from a rijke tube. Journal of Sound and Vibration, 117–133. Holzfuss, J. and Lauterborn, W. (1989). Lyapunov exponents from a time series of acoustic chaos. Physical Review A. Kabiraj, L., Saurabh, A., Karimi, N., Sailor, A., Mastorakos, E., Dowling, A., and Paschereit, C.O. (2015). Chaos in an imperfectly premixed model combustor. University of Glasgow. Kabiraj, L., Saurabh, A., Wahi, P., and Sujith, R. (2012). Route to chaos for combustion instability in ducted laminar premixed flames. AIP Publishing. Kantz, H. (1994). A robust method to estimate the maximal Lyapunov exponent of a time series. Physics Letters A, 185(1), 77– 87. doi:10.1016/0375-9601(94)90991-1. URL http://dx.doi.org/10.1016/0375-9601(94)90991-1. Marble, F. (1953). Servo-stabilization of low-frequency oscillations in a liquid bipropellant rocket motor. Journal of the American Rocket Society, 63–81. Moeck, J., Bothien, M., Guyot, D., and Paschereit, C.O. (2007). Phase-shift control of combustion instability using (combined) secondary fuel injection and acoustic forcing. Institute for Fluid Dynamics and Technical Acoustic. Olgac, N., Zalluhoglu, U., and Kammer, A.S. (2015). A new perspective in designing delayed feedback control for thermo-acoustic instabilities (TAI). Combustion Science and Technology, 187(5), 697–720. doi: 10.1080/00102202.2014.960924. Ott, E., Grebogi, C., and Yorke, J.A. (1990). Controlling chaos. Phys. Rev. Lett., 64(11), 1196–1199. doi:10.1103/physrevlett.64.1196. URL http://dx.doi.org/10.1103/PhysRevLett.64.1196. Paschereit, C.O., Gutmark, E., and Weisenstein, W. (1998). Structure and control of thermoacoustic instabilities in a gas-turbine combustor. Combsution Science and Thechnology, 138. Strogatz, S.H. (1994). Nonlinear Dynamics and Chaos: With Applications To Physics, Biology, Chemistry and Engineering. Westview Press., Massachusetts. Tsien, H. (1952). Servo-stabilization of combustion in rocket motors. Journal of the American Rocket Society, 256–263.
Fig. 11. States observed in the oscillations for the different values of the control parameters K and τ . 4. CONCLUSIONS The main objective of this work was to demonstrate the effect that phase-shift feedback has on chaotic thermoacoustic oscillations. We have first presented the bifurcation analysis of the thermoacoustic oscillations without feedback to illustrate bifurcations and different dynamics states appearing in the system as a result of changes in the equivalence ratio of the combustion process. Subsequently the response of the chaotic state to phase-shift feedback was studied while varying the feedback parameters: time delay and the gain of the phase-shifted signal. The experiments illustrated that the behavior of the instabilities underwent transitions among chaotic state, periodic and quasi-periodic states. Previously, phase-shift control was proposed as an instability control strategy. We demonstrated experimentally that phase-shift control of thermoacoustic coupling can results in additional complexity, particularly when chaos is present. The results reached in this work are important from a practical perspective because it is the first investigation where chaos is taken into account in the implementation of active control of thermoacoustic instabilities. ACKNOWLEDGEMENTS B.M.M. thanks the ERASMUS programme and Universidad Polit´ecnica de Madrid to support her stay at the 70
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