PHYSICAL REVIEW E 89, 022910 (2014)
Detection and control of combustion instability based on the concept of dynamical system theory
Hiroshi Gotoda,* Yuta Shinoda, Masaki Kobayashi, and Yuta OkunoDepartment of Mechanical Engineering, Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu-shi, Shiga 525-8577, Japan
Shigeru Tachibana†
Propulsion Systems Research Group, Institute of Aeronautical Technology, Japan Aerospace Exploration Agency, 7-44-1 Jindaiji-Higashii,Chofu, Tokyo 182-8522, Japan(Received 25 July 2013; published 11 February 2014; corrected 5 March 2014)
We propose an online method of detecting combustion instability based on the concept of dynamical systemtheory, including the characterization of the dynamic behavior of combustion instability. As an important casestudy relevant to combustion instability encountered in fundamental and practical combustion systems, wedeal with the combustion dynamics close to lean blowout (LBO) in a premixed gas-turbine model combustor.The relatively regular pressure ﬂuctuations generated by thermoacoustic oscillations transit to low-dimensionalintermittent chaos owing to the intermittent appearance of burst with decreasing equivalence ratio. The translationerror, which is characterized by quantifying the degree of parallelism of trajectories in the phase space, can beused as a control variable to prevent LBO.
DOI: 10.1103/PhysRevE.89.022910PACS number(s): 05.45.Tp, 47.70.Pq
I. INTRODUCTION
Self-excited oscillations caused by closed-loop couplingbetween unsteady heat-release ﬂuctuations, acoustic velocity,and/or the pressure ﬁeld are an important thermoacoustic insta-bility of much interest in wide a spectrum of combustion, ther-mal ﬂuid, and physics ﬁelds [1–10]. The development of newengine systems using the concept of thermoacoustic instabilityhas recently been attempted by Biwa and coauthors in the ﬁeldof physics [7–10]. On the contrary, the complex interactionbetween the combustion process and the acoustic ﬁeld insidecombustors generates thermoacoustic combustion oscillationswith nonlinear dynamic behavior, leading to serious damagein practical combustors such as propulsion and land-basedgas-turbine engines. There have been many experimental andnumerical investigations exploring the physical mechanismsresponsible for the excitation of thermoacoustic combustionoscillations regardless of the existence or nonexistence ofa swirling ﬂow [11–17]. As a representative study on theﬂame-acoustic interaction mechanisms in a swirl-stabilizedpremixed combustor, Polifke and Lawn [15] have proposedthree essential mechanisms by which acoustic velocity ﬂuctua-tions give rise to perturbations in equivalence ratio, ﬂame frontkinematics, turbulence intensity, and length scale, possiblyincluding large-scale coherent vortex structures. All theseperturbations result in heat-release ﬂuctuations. More recently,the physical mechanisms identiﬁed to date in the ﬁeld ofcombustion physics have been summarized in a review paper[18] and book [19].Lean premixed combustion is a highly promising methodfor effectively reducing nitrogen oxide (NOx) emission fromgas-turbine engines without sacriﬁcing combustion efﬁciency.However, it is highly susceptible to combustion instabilitiessuch as thermoacoustic combustion oscillations, lean blowout(LBO), and ﬂame ﬂashback. Recently, we showed that with
*gotoda@se.ritsumei.ac.jp†tachibana.shigeru@jaxa.jp
increasing equivalence ratio, the dynamic behavior of combus-tion instabilities in a premixed gas-turbine model combustorundergoes a signiﬁcant transition from fractional Brownianmotion in close proximity to LBO to periodic oscillationsgenerated by thermoacoustic combustion oscillations throughchaos [20,21]. In addition to thermoacoustic combustionoscillations, LBO is another problem that must be avoidedwhen operating gas-turbine engines, since it leads to severesafety problems and high operational costs. Current gas-turbine combustor engines are operated within a wide safetymargin far from actual blowout limits due to the difﬁcultyin detecting LBO precursors in engines. Most approaches todetecting or correlating LBO limits are based on the idea thatLBO is caused by the competition between characteristic timescales of ﬂow residence and chemical kinetics. For instance,Zhang et al. [22] concluded that the global Damk¨ohler number(given by the ratio of the ﬂow residence and chemical kinetictime scales) describes the onset condition of local ﬂameextinction near LBO, which should have a direct relationwith global LBO. The phenomenological approach in theirstudy provides an in-depth understanding of the complex LBOphenomenon and should ultimately lead to the establishmentof a design parameter to determine actual LBO conditions.Meanwhile, a novel method of detecting LBO precursors bymonitoring signals in engines is of importance for controllingthe complex combustion state under low equivalence ratios.Nonlinear time series analysis based on dynamical systemtheory conveys striking and revealing information for clar-ifying the dynamic properties of irregular time series dataobtained in a wide range of experiments in the ﬁelds ofmedicine, biology, electronics, mechanical engineering, andmany other sciences [23]. It widely covers areas ranging fromthe quantiﬁcation of many important invariants such as fractaldimension, Lyapunov exponents, and entropies, which yielda physical description of the dynamical structure, to practicalapplications such as nonlinear prediction and noise reduction.In fact, it has recently been used to investigate the temporalbehavior of combustion and ﬂame front instabilities observed
1539-3755/2014/89(2)/022910(8)022910-1©2014 American Physical Society
GOTODA, SHINODA, KOBAYASHI, OKUNO, AND TACHIBANAPHYSICAL REVIEW E 89, 022910 (2014)
in experiments (e.g., a thermal pulse combustor [24], a ductedpremixed combustor [3–5], a spark ignition engine [25,26], andswirling premixed ﬂames [27]). These previous works havebeen devoted to the characterization of complex combustiondynamics from the viewpoint of the self-similar structure andorbital instability in a phase space using the correlation dimen-sion obtained by the Grassberger-Procaccia (GP) algorithm[28], multifractal analysis, and the Lyapunov exponent. Someof the authors have also used more sophisticated nonlinear timeseries analysis to extract the short-term predictability and long-term unpredictability characteristic of chaos in combustion[20] and ﬂame front instabilities [29]. The importance of ourapproach has also been discussed from a practical point of view[20,29]. The relevance of chaos to combustion instabilitieshas thus far been shown by nonlinear time series analysis[4,5,20,24,29,30], but there have been no attempts to proposean online method of detecting an unstable combustion stateto appropriately control a combustion system, focusing on theconcept of orbital instability in the phase space. Developing asophisticated method of detecting a precursor of LBO basedon the extracted nonlinear nature of the complex dynamicsunderlying combustion instability is a major challenge incontemporary time series analysis, which will be of interest innonlinear physics and engineering disciplines.The main purpose of this study is to propose an onlinemethod of detecting combustion instability based on theconcept of dynamical system theory. As an important casestudy relevant to combustion instability encountered in afundamental and practical combustion system, we deal withcombustion dynamics in a laboratory-scale lean premixedgas-turbine model combustor with an axial swirler ﬂameholder. A simple and useful nonlinear time series analysisbased on orbital instability in the phase space, referred to asthe translation error method [31], represents the degree ofparallelism of trajectories in the phase space which has beenused to successfully characterize the dynamic properties ofdifferent types of ﬂame front instability [27]. In this study, wediscuss the applicability of the translation error as an onlinedetector. The translation error is estimated for trajectories inthe phase space constructed from short time series data ofcombustion-chamber pressure ﬂuctuations. On the basis ofthe extracted nonlinear nature of the unstable combustionstate close to LBO, we attempt to control the combustionstate so as to prevent LBO using the translation error as acontrol variable. The intermittent phenomenon of irregularalternation between regular ﬂuctuations and burst is of muchinterest in the ﬁelds of modern nonlinear science and physics.As will be presented in this work, in a narrow range of theequivalence ratio, an interesting intermittent phenomenon isformed, with its pressure ﬂuctuations switching back and forthbetween regular ﬂuctuations generated by thermoacousticinstability and irregular ﬂuctuations with small amplitude. Wetherefore investigate the dynamic behavior of this intermittentphenomenon from the viewpoint of dynamical system theory,in addition to proposing an online method of detectingcombustion instability. This paper is organized as follows.In Sec. II, we brieﬂy describe the experimental apparatus andthe detection method of combustion instability. In Sec. III,we present the results and discussion, and our conclusions aregiven in Sec. IV.
Main fuel
Pressure transducer
Amplifier
Mass flow controller
Mass flow controller
Inlet of Cooling water
Outlet of Cooling water
Nozzle
plifier
Combustion chamber 
Lean premixed flame 
Secondary fuel
FIG. 1. (Color online) Experimental system.
II. EXPERIMENTAL APPARATUS AND DETECTIONMETHOD OF COMBUSTION INSTABILITY
The conﬁguration of our experimental system is shownin Fig. 1. The combustion test rig is composed of massﬂow controllers, a nozzle, an axial swirler, and a combustionchamber. The chamber has a length of 620 mm with a 100 ×100 mm2 cross section. The rest of the chamber is composedof a water-cooled stainless-steel duct. Methane gas is used asthe main fuel. There is a secondary fuel line for modifyingthe total fuel ﬂow rate near the LBO. An axial swirler,which is normally adopted for holding ﬂame in most practicalcombustion devices, is installed at the inlet of the combustionchamber. The inlet air temperature and volume ﬂow rate (meanaxial ﬂow velocity at the inlet of the combustion chamber)are 300 K and 148 L/min (5.8 m/s), respectively. The equiv-alence ratio φ is decreased from 0.74 to 0.46, because it isunder these conditions that the combustion instability changesfrom thermoacoustic combustion oscillations to LBO. Thepressure ﬂuctuations p′ are measured by a pressure transducer(JTEKT Products, PD104K-10 kPa) as a physical quantity thatrepresents the nonlinear dynamics of the combustion state.In practice, a pressure transducer is commonly preferablefor monitoring the combustor status in practical combustionsystems from the viewpoint of robustness. The pressuretransducer is located on the wall of the combustion chamber,20 mm from its inlet. Signals from the pressure transducerare acquired through a data acquisition system (LabVIEWsystem, National Instruments). The sampling frequency of p′ is5 kHz. The noise level of the pressure ﬂuctuation measurementis within ±0.004 kPa in the absence of a ﬂow, which issmall compared with the pressure ﬂuctuations in combustioninstabilities we observed in this work.If the determinism is clearly visible in a time series obtainedin experiments, the neighboring trajectories in the phase spacewill have similar directions. However, as the determinismis lost, the trajectories diverge with time due to the orbitalinstability in the phase space. On the basis of this concept, weintroduce a useful algorithm [31] to extract the determinismfrom complex time series data obtained in experiments. Aphase space based on Takens’ embedding theorem [32] is ﬁrstconstructed from p′. The time-delayed coordinates used for
022910-2
DETECTION AND CONTROL OF COMBUSTION . . .PHYSICAL REVIEW E 89, 022910 (2014)
the construction of the phase space are expressed as
p′(ti) = [p′(ti),p′(ti + τ),p′(ti + 2τ),...,p′(ti + (D − 1)τ)],(1)where i = 0,1,...,n (n is the data number of the time series),p′(ti) are the phase space vectors, p′(ti) are the pressureﬂuctuations at time ti, D is the embedding dimension, thatis, the dimension of the phase space, and τ is the time lag.In this work, D is set to 3. As reported in previous studies[20,29], an appropriate choice for the time lag is made usingmutual information. After randomly selecting a vector p′(ti)from the phase space, we ﬁnd its K nearest neighbors p′(tk)with k from 1 to K. Here, K is the number of neighboringpoints to p′(ti). By setting the images of all the vectors top′(tk + T �t) at a suitable time interval T , the diversity in thedirections of neighboring trajectories is measured in terms ofthe translation error Etrans, deﬁned by
Etrans =1
∥v(tk) − ¯v∥2
∥¯v∥2,(2)
k=0v(tk).(3)
Here, v(tk) [=p′(tk + T �t) − p′(tk)] approximates the tan-gential vectors of trajectories at time tk. T and K are setto 8 and 5, respectively, for the estimation of Etrans. Themore parallel the trajectories are to each other, that is, themore visible determinism there is in the time series, the closerEtrans will be to zero. In the estimation of the translation error,the stochastic error can be reduced by obtaining the mediansof Etrans for Q sets of M randomly chosen p′(ti) and thentaking the mean of the Q medians. In a preliminary test, wesystematically varied M and Q from 50 to 500 and from 1 to10, respectively. The values of M and Q within these rangeswere found to have little effect on the translation error. On thebasis of the preliminary test, M and Q are set to 100 and 1 inthis work, respectively. Etrans is calculated for the trajectoriesof the phase space constructed from short time series data ofp′. In this work, Etrans is obtained every 100 ms, that is, thelibrary data of the phase space consists of 100 ms time seriesdata of p′.
III. RESULTS AND DISCUSSION
Time variations of the pressure ﬂuctuations p′, three-dimensional phase space [p′(t), p′(t + τ), p′(t + 2τ)], andtranslation error Etrans for φ = 0.62 are shown in Fig. 2.Irregular pressure ﬂuctuations with small amplitudes suddenlyappear in the regular ﬂuctuations with large amplitudesgenerated by thermoacoustic combustion oscillations. The tra-jectories at t = 5.0–5.2 s ﬁll the core of the phase space and donot appear to exhibit a strong deterministic nature, while theybecome limit cycles with a large width at 15.0–15.2 s. Kabirajand Sujith [33] recently found that intermittent bursts withlarge amplitudes irregularly appear in periodic oscillationswith small amplitudes in a ducted laminar premixed conicalﬂame. They showed the presence of type-II intermittencyusing a recurrence analysis based on dynamical system theory.On the basis of our previous studies [20,21], the entire ﬂow
FIG. 2. (Color online) Time variations of pressure ﬂuctuationsp′, three-dimensional phase space (p′(t), p′(t + τ), p′(t + 2τ)), andtranslation error Etrans for equivalence ratio φ = 0.62. The tempera-ture of the inlet air is 300 K. The volume ﬂow rate of the inlet air is148 L/min. The amplitude of p′ drastically changes with the onsetof intermittency. Etrans responds to the drastic change in p′.
ﬁeld inside our current combustion chamber predominantlyconsists of two large-scale recirculation regions: one is avortex breakdown bubble in the wake of the center body dueto swirling ﬂow, and the other is an outer recirculation regionlocated in the dump plate due to the backward-facing step ﬂow.In the absence of thermoacoustic combustion oscillations, theentire ﬂame is sustained by the vortex breakdown bubbleand stands off from the centerbody. When thermoacousticcombustion oscillations are induced, the ﬂame base propagatesinto the upstream ﬂow and becomes attached to the axialswirl vane. The entire ﬂame exposed to the thermoacousticcombustion oscillations is sustained by both recirculationregions, with the roll up of the ﬂame caused by vorticalstructures probably stemming from convective interaction inthe shear layers. Although the overall dynamic behavior of thepressure ﬂuctuations with intermittency observed in this workdiffers from that in a ducted laminar premixed conical ﬂame[33], on the basis of our observation, it is conceivable that theabrupt switching between these states results in the formationof the intermittent phenomena. The implementation of laserdiagnostics, such as OH-PLIF and PIV measurements [34],is required in a future work to obtain a more comprehensiveunderstanding of the dynamics of the thermoacoustic combus-tion oscillations observed in our current experimental system.As shown in Fig. 2, when the amplitude of p′ drasticallychanges with the onset of intermittency, Etrans also changessimultaneously. The response of the translation error to thedrastic change in pressure ﬂuctuations indicates that it isfeasible to use the translation error to detect a drastic change in
022910-3
GOTODA, SHINODA, KOBAYASHI, OKUNO, AND TACHIBANAPHYSICAL REVIEW E 89, 022910 (2014)
the combustion state in the lean premixed gas-turbine modelcombustor. In this study, we investigate the dynamic behaviorof the intermittent pressure ﬂuctuations using the recurrenceplots and a sophisticated nonlinear forecasting method basedon orbital instability in the phase space. Note that therehave been no attempts in previous studies on combustionto discuss the possible existence of chaos in the intermit-tent pressure ﬂuctuations from the viewpoint of nonlinearforecasting.Recurrence plots (RPs), introduced by Eckmann et al. [35],are two-dimensional representations suitable for qualitativelyextracting hidden order and disorder pattern structures ina phase space, and have been applied to a wide rangeof experimental data sets in chemistry and physics. InRPs, black and white dots are commonly used to expressvalues of one and zero, respectively, from the binary matrixRi,j [=�(ε − ∥p′(ti) − p′(tj)∥),i,j = 1,...,N, where is �the Heaviside function and ε is a threshold parameter]. In thiswork, we use the color-coded version of RPs consisting of256 gradations without the need to consider the threshold ofε, following Eq. (4):
Ri,j =∥p′(ti) − p′(tj)∥ − ∥p′(ti) − p′(tj)∥min
∥p′(ti) − p′(tj)∥max − ∥p′(ti) − p′(tj)∥min.(4)
Here, a color code at location (i,j) represents the distancebetween the pair of p′(ti) and p′(tj) in the phase space, andvarious color codes are assigned to different distances. As thedeterministic nature appears in the time series data, regularlyarrayed structures come to be contained in the RPs. RPscomposed of 50,000 × 50,000 points, corresponding to p′ for0 ⩽ t ⩽ 10 s at φ = 0.62, are depicted in Fig. 3. The patterncorresponding to a burst with small amplitudes is shown inthe lower-left corner and center regions. Regularly arrayedgeometrical structures do not seem to be contained in theseregions. In contrast, they appear in the upper-right corner,corresponding to regularly ﬂuctuating pressures with largeoscillation amplitudes, but interestingly, the local structureslose their regularity. Note that the kite-like structure, which isas an important indicator of the presence of type-II intermit-tency and was found in a ducted laminar premixed combustor[33], is not observed in our experimental system. It is also notobserved in our preliminary test using usual RPs consistingof a binary matrix. The results obtained by the RPs indicatethe coexistence of large square, rectangular, regular andlocally irregular patterns, suggesting the possible existence ofchaos.One of the authors has recently reported that a nonlinearforecasting method taking into account the updating of librarydata is feasible for predicting the short-term complex temporalbehavior of ﬂame front instability [29] and the intermittentchaos in Rayleigh-B´enard convection subjected to a magneticﬁeld [36] from the viewpoint of practical application. Theauthor has also proposed a nonlinear forecasting methodologyto capture the chaotic dynamics in spatially extended systems[37]. An important point to note for this methodology isthat the dynamic behavior has the short-term predictabilityand long-term unpredictability characteristic of chaos if thecorrelation coefﬁcient between the actual and predicted timeseries data, including the case of the predicted increments
4.05 4.1 4.15 4.2 4.25 4.3 
054321
FIG. 3. (Color online) Recurrence plots (RPs) composed of50,000 × 50,000 points for equivalence ratio φ = 0.62. The experi-mental conditions of the inlet air ﬂow are the same as those in Fig. 2.RPs consist of large square, rectangular, regular, and locally irregularpatterns, indicating the possible existence of chaos.
of the time series data, gradually decays upon increasing theduration of the actual time series data added to library data. Inthis methodology, the time series of p′ at φ = 0.62 is initiallydivided into two parts: the library data and the referencedata for comparison with the time series data predicted usingthe library data. The neighboring vectors in the phase space,denoted by p′(tk) (k = 1,2,...,K), to vector p′(tp) are searchedfor among all the points in the phase space constructed fromthe library data, where p′(tp) is the ﬁnal point of the trajectoryin the phase space. Note that p′(ti) = [p′(ti),p′(ti − τ),p′(ti −2τ),...,p′(ti − (D − 1)τ)] when applying our proposed non-linear forecasting method. p′(tk + Ts�t) represents the futurevalue corresponding to p′(tk) after Ts time steps. Then, thepredicted value ˆp′(tp + Ts�t) corresponding to p′(tp) after Tstime steps is obtained by the nonlinearity weighted sum of thelibrary data p′(tk + Ts�t) shown in Eq. (5):
ˆp′(tp + Ts�t) =�Kk=1 exp(−dk) · p′(tk + Ts�t)
�Kk=1 exp(−dk),(5)
where dk = ∥p′(tp) − p′(tk)∥, and K = 1000 for the non-linear forecasting. When predicting the time evolution ofp′, the library data is updated so as to continue capturing
022910-4
DETECTION AND CONTROL OF COMBUSTION . . .PHYSICAL REVIEW E 89, 022910 (2014)
the determinism governing the current temporal behavior,keeping the size of the updated library data constant. Inthis work, values of p′ of up to t = 25 s are used asthe library data and those after t = 20 s are used as thereference.The correlation coefﬁcient C between the actual andpredicted dynamic behaviors in the pressure ﬂuctuations p′
at φ = 0.62 is shown in Fig. 4 as a function of the durationtP of the actual pressure ﬂuctuations added to the librarydata. C at tP = 2 × 10−4 s is nearly unity, which meansthat one-step-ahead prediction of the intermittent pressureﬂuctuations is feasible with high accuracy. C remains nearlyunchanged up to tP = 3.6 × 10−3 s. The appearance of theconstant region up to tP = 3.6 × 10−3 s is the manifestationof the short-term predictability characteristic of chaos. As tPexceeds this critical value, C gradually decreases due to thelong-term unpredictability characteristic of chaos associatedwith intermittently occurring burst regions in the pressureﬂuctuations. As reported in recent study on a spatially extendedsystem [37], a similar distribution of C in terms of tPfor the incremental process should be observed when thedynamical behavior of time series data is chaos. In fact, asshown in Fig. 4, a similar trend is observed for increments
FIG. 4. (Color online) Changes in the correlation coefﬁcient Cbetween the actual and predicted dynamic behaviors in the inter-mittent pressure ﬂuctuations p′ at φ = 0.62 as a function of theduration tP of the actual pressure ﬂuctuations added to the librarydata. Changes in C for increments of �p′(=p′(ti+1) − p′(ti)) arealso shown as a function of tP . The experimental conditions ofthe inlet air ﬂow are the same as those in Fig. 2. C for p′ attP = 2 × 10−4 s is nearly unity, and remains nearly unchanged upto tP = 3.6 × 10−3 s, which is the manifestation of the short-termpredictability characteristic of chaos. The long-term unpredictabilitycharacteristic of chaos appears as tP exceeds this critical value.A similar trend is observed for increments of �p′, indicating thepresence of chaotic dynamics.
of �p′(=p′(ti+1) − p′(ti)), indicating the presence of chaoticdynamics. In our preliminary test, we obtained a similar trendfor the changes in C in terms of tP for periodic ﬂuctuations withan intermittently formed small burst [36], i.e., low-dimensionalintermittent chaos produced by Lorenz equations, includingthe case of incremental process of Lorenz equations solution.This means at least that the dynamic behavior of the inter-mittent pressure ﬂuctuations at φ = 0.62 has the same degreeof short-term predictability and long-term unpredictability asthe low-dimensional intermittent chaos. As mentioned in ourprevious work [20], the Lorenz equations in themselves arenot a suitable model for dealing with combustion instability,but they enable us to discuss the degree of complexityin the dynamic behavior. On the basis of the results inFigs. 2–4, it appears that the irregular components of thepressure ﬂuctuations with small amplitude affect the rela-tively regular ﬂuctuations due to thermoacoustic combustionoscillations, resulting in low-dimensional chaotic behavior.Our experimental system differs from the ducted laminarpremixed combustor, but the appearance of low-dimensionalchaos provides us with signiﬁcant understanding of thetransient process from thermoacoustic combustion oscilla-tions to lean blowout in a laboratory-scale, lean, premixedgas-turbine model combustor with an axial swirler ﬂameholder. Revealing the intermittent characteristics by statisticalanalysis [26,38] may also be of importance to obtain abetter understanding of combustion dynamics in our nextstudy.The standard deviation of pressure ﬂuctuations p′rms andthe translation error Etrans are shown in Fig. 5 as functionsof equivalence ratio φ. Note that, similarly to in our previouswork [20], in which a different type of gas-turbine model
0.75 0.7 0.65 0.55 0.6 0.5 0.45 0 
LBO Thermoacoustic combustion oscillations 
Onset of intermittent oscillations 
0.2 0.4 0.6 0.8 1 
0.3 0 
Etransp'rms (kPa) 
FIG. 5. Standard deviation of pressure ﬂuctuations p′rms andtranslation error Etrans as functions of equivalence ratio φ. Theexperimental conditions of the inlet air ﬂow are the same as thosein Fig. 2. p′rms remains nearly unchanged for φ ⩽ 0.61. In contrast,Etrans increases as φ decreases from 0.61 to 0.49, close to LBO, whichshows that the translation error clearly captures the signiﬁcant changein combustion close to LBO.
022910-5
GOTODA, SHINODA, KOBAYASHI, OKUNO, AND TACHIBANAPHYSICAL REVIEW E 89, 022910 (2014)
combustor was considered, the dynamic behavior of thepressure ﬂuctuations observed in this work becomes complexas the combustion state approaches LBO. p′rms graduallydecreases with decreasing φ below φ = 0.70. It drasticallydecreases when the intermittent pressure ﬂuctuations occurbut remains nearly unchanged for φ ⩽ 0.61. This means thatthe standard deviation of the pressure ﬂuctuations cannotcapture the change in the complex combustion state closeto LBO, which indicates that it is difﬁcult to determinean appropriate threshold of p′rms for detecting LBO. Etransdrastically increases with the appearance of the intermittencyin p′, and monotonically increases with decreasing φ downto 0.49 in close proximity to LBO. The important pointto note in Fig. 5 is that the translation error captures thesigniﬁcant change in the complex combustion state evenunder low equivalence ratio conditions close to LBO. Thisindicates that the translation error can be used to detect theprecursor of LBO by setting a threshold. It is therefore ofconsiderable interest to examine the possibility of controllingthe combustion state using the translation error as an onlinedetector. For this purpose, we control the combustion state soas to prevent LBO by continuing to capture the determinismgoverning the current temporal behavior of the pressureﬂuctuations.An active control system using the power spectrum oftemperature ﬂuctuations has been adopted for preventingLBO during (I) power reduction and (II) transient processes[39]. In this work, we conduct closed-loop feedback controlby adjusting the secondary fuel supply with the translationerror used as the control variable to consider the above twoprocesses. The time variation of the equivalence ratio φ withdecreasing volume ﬂow rate of the main fuel QCH4,main isshown in Fig. 6 as an example of power reduction. As shownby the dashed line in Fig. 6, only QCH4,main is decreased untilLBO takes place. We attempt to control the combustion stateby correlating the volume ﬂow rate of the secondary fuelQCH4,second with the difference between Etrans,thre and Etrans
QCH4,main (L/min) 
Decrease in QCH4,main
t (s)120
20 40 60 80 100 
FIG. 6. (Color online) Time variation of equivalence ratio φ withdecreasing volume ﬂow rate of main fuel QCH4,main. Lean blowout(LBO) is prevented even when QCH4,main is decreased to 6 L/min.
ΦQCH4,main (L/min) 
Decrease and increase in QCH4,main
t (s)20 40 60 80 100 120 0
FIG. 7. (Color online) Time variation of equivalence ratio φ withdecreasing and then increasing volume ﬂow rate of main fuelQCH4,main. The prevention of LBO is achieved for the transient processas well as for the case of a power reduction.
[QCH4,second = α(Etrans − Etrans,thre)]. Here, α is a coefﬁcientand Etrans,thre is the threshold of Etrans used for the controlregulator. In this work, we set Etrans,thre and α to 0.15 and 1.0,respectively. Our results clearly show that LBO is preventedeven when QCH4,main is decreased to 6 L/min. The timevariation of the equivalence ratio φ with decreasing and thenincreasing the volume ﬂow rate of the main fuel QCH4,mainis shown in Fig. 7 as an example of a transient process.QCH4,main is decreased, leading to LBO, but the addition offuel from the secondary fuel supply system prevents LBO.After preventing LBO by increasing QCH4,main, the secondfuel supply system returns to the initial setting. These resultsindicate that without knowing the actual LBO limit, the onlinedetection method using the translation error has considerablepotential for preventing LBO in a premixed gas-turbine modelcombustor.In reality, the noise inherently included in experimentalsystems inevitably affects the deterministic dynamics govern-ing the temporal behavior of combustion instability. Althoughthe translation error method in itself may be consideredconventional in the ﬁelds of nonlinear science and physics,it can provide reliable estimates from a short time series andis robust to the noise contamination of obtained experimentaldata [31]. In this sense, the translation error involving theonline update of a small amount of library data proposedin this work is applicable to combustion control, which willbe worthwhile from the practical viewpoint of combustionsystems. It is well recognized in the ﬁeld of applied physicsthat plasma processing is an indispensable technology for themanufacture of semiconductors. A major drawback of plasmaprocessing, however, is that the onset of anomalous dischargesleads to serious problems such as damage to microdevices.With the aim of detecting the onset of anomalous discharges,the dynamical properties of acoustic emission by anomalousdischarges in a plasma processing system has been investigated
022910-6
DETECTION AND CONTROL OF COMBUSTION . . .PHYSICAL REVIEW E 89, 022910 (2014)
relatively recently [40]. It is considered that the onlinedetection method proposed in this work has potential usefor detecting anomalous discharges in the plasma processingsystem in Ref. [32]. In this sense, our work may provide usefulguidelines for detecting and controlling dynamic behavior ina wide spectrum of complex nonlinear phenomena, includingcombustion phenomena. In this work, we have investigated theapplicability of dynamical system theory as a useful detectorto control the combustion state from a practical viewpoint.We need to examine whether and to what extent the onlinedetection method we proposed is feasible for appropriatelycontrolling the combustion state without the occurrence ofLBO under a broad range of experimental conditions, as wellas to reduce the response time of the current control system sothat it can be applied to combustion instability with quite highfrequencies. Our system enables us to control the combustionstate up to Etrans = 0.20, but the appropriate control of thecombustion state close to the lean blow-out limit is difﬁcultat present owing to the limitation of the current fuel ﬂowcontrol system used as an actuator. We demonstrated in thiswork that our method based on dynamical system theory iscapable of preventing LBO in both (I) power reduction and (II)transient processes [39]. A possible application of our methodin engineering is the operation control of gas-turbine engines,including the control of fuel ﬂow-rate schedules. Typically,the frequency responses required for such control are lessthan 10 Hz. The method of detecting instability proposedin this study should fulﬁll a key role in such control asit continuously provides updates of the status of monitoredtargets. The operation can be made more ﬂexible by virtueof the updated information, eventually leading to stable andlow-emission combustion. These issues will be focused on inour future work to enable the practical application of the onlinedetection method based on orbital instability in the phasespace for controlling the dynamic behavior of combustioninstability.
IV. CONCLUSION
We have proposed an online method of detecting combus-tion instability based on the concept of dynamical system the-ory, including investigation of the dynamic behavior of com-bustion instability. As an important case study relevant to com-bustion instability encountered in fundamental and practicalcombustion systems, we deal with combustion dynamics closeto lean blowout (LBO) in a laboratory-scale premixed gas-turbine model combustor. With decreasing equivalence ratio,irregular pressure ﬂuctuations with small amplitudes suddenlyappear in the relatively regular pressure ﬂuctuations generatedby thermoacoustic combustion oscillations, resulting in low-dimensional intermittent chaos. This is clearly shown usingrecurrence plots and the nonlinear forecasting method recentlyproposed by Gotoda and co-workers [29,36]. The translationerror, which can be characterized by quantifying the degree ofparallelism of trajectories in the phase space constructed fromthe short time series data of combustion-chamber pressureﬂuctuations, is used to detect the signiﬁcant change in thecombustion state under low equivalence ratio conditions. Ourresults show that the translation error has potential use as acontrol variable to prevent LBO, which is worthwhile fromthe practical viewpoint of combustion systems.
ACKNOWLEDGMENTS
One of the authors (H.G.) was partially supported by aresearch grant from the Paloma Environmental TechnologyFoundation, the Adaptable & Seamless Technology TransferProgram through Target-Driven R&D from Japan Science andTechnology Agency (JST), and a Grant-in-Aid for YoungScientists (A) from the Ministry of Education, Culture, Sports,Science, and Technology of Japan (MEXT). This work isbased on many useful ﬁndings obtained by collaborative workswith Prof. Takaya Miyano [20,29]. The authors would like toexpress their sincere thanks for his enormous contributions.
[1] S. Candel, Proc. Combust. Inst. 29, 1 (2002).[2] Combustion Instabilities in Gas Turbine Engines: OperationalExperience, Fundamental Mechanisms and Modeling, editedby T. C. Lieuwen and V. Yang, Progress in Astronautics andAeronautics Vol. 210 (American Institute of Aeronautics andAstronautics, Reston, VA, 2005).[3] K. Balasubramanian and R. I. Sujith, Phys. Fluids 20, 044103
(2008).[4] L. Kabiraj, A. Saurabh, P. Wahi, and R. I. Sujith, Chaos 22,
023129 (2012).[5] K.Kashinath,Ph.Dthesis,UniversityofCambridge,2013.[6] T. Biwa, Y. Tashiro, U. Mizutani, M. Kozuka, and T. Yazaki,
Phys. Rev. E 69, 066304 (2004).[7] T. Biwa, Y. Tashiro, M. Ishigaki, Y. Ueda, and T. Yazaki, J. Appl.Phys. 101, 064914 (2007).[8] T. Yoshida, T. Yazaki, H. Futaki, K. Hamaguchi, and T. Biwa,
Appl. Phys. Lett. 95, 044101 (2009).[9] T. Biwa, D. Hasegawa, and T. Yazaki, Appl. Phys. Lett. 97,
034102 (2010).
[10] H. Hatori, T. Biwa, and T. Yazaki, J. Appl. Phys. 111, 074905
(2012).[11] G. Searby and D. Rochwerger, J. Fluid Mech. 231, 529
(1991).[12] V. Bychkov, Phys. Fluids 11, 3168 (1998).[13] X. Wu, M. Wang, P. Moin, and N. Peters, J. Fluid Mech. 497,
23 (2003).[14] C. J. Lawn and W. Polifke, Combust. Sci. Technol. 176, 1359
(2004).[15] W. Polifke and C. J. Lawn, Combust. Flame 151, 437 (2007).[16] V′. Akkerman and C. K. Law, Phys. Fluids 25, 013602
(2013).[17] J. Y´a˜nez, M. Kuznetsov, and R. Redlinger, Combust. Flame 160,
2009 (2013).[18] Y. Huang and V. Yang, Prog. Energy Combust. Sci 35, 293
(2009).[19] T. C. Lieuwen, Unsteady Combustor Physics (CambridgeUniversity Press, Cambridge, UK, 2012).[20] H. Gotoda, H. Nikimoto, T. Miyano, and S. Tachibana, Chaos21, 013124 (2011).
022910-7
GOTODA, SHINODA, KOBAYASHI, OKUNO, AND TACHIBANAPHYSICAL REVIEW E 89, 022910 (2014)
[21] H. Gotoda, M. Amano, T. Miyano, T. Ikawa, K. Maki, andS. Tachibana, Chaos 22, 043128 (2012).[22] Q. Zhang, S. J. Shanbhogue, and T. Lieuwen, J. Eng. GasTurbines Power 132, 111502 (2010).[23] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis(Cambridge University Press, Cambridge, UK, 1997).[24] C. S. Daw, J. F. Thomas, G. A. Richards, and L. L.Narayanaswami, Chaos 5, 662 (1995).[25] T. Kaminski, M. Wendeker, K. Urbanowicz, and G. Litak, Chaos14, 461 (2004).[26] A. K. Sen, G. Litak, T. Kaminski, and M. Wendeker, Chaos 18,
033115 (2008).[27] H. Gotoda, T. Miyano, and I. G. Shepherd, Phys. Rev. E 81,
026211 (2010).[28] P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).[29] H. Gotoda, T. Ikawa, K. Maki, and T. Miyano, Chaos 22, 033106
(2012).[30] A. C. Noble, G. B. King, N. M. Laurendeau, J. R. Gord, andS. Roy, Combust. Sci. Technol. 184, 293 (2012).
[31] R. Wayland, D. Bromley, D. Pickett, and A. Passamante, Phys.Rev. Lett. 70, 580 (1993).[32] F. Takens, Dynamical Systems of Turbulence, Lecture Notes inMathematics Vol. 898 (Springer, Berlin, 1981).[33] L. Kabiraj and R. Sujith, J. Fluid Mech. 713, 376 (2012).[34] S. Tachibana, J. Yamashita, L. Zimmer, K. Suzuki, and A. K.Hayashi, Proc. Combust. Inst. 32, 1795 (2009).[35] J. Eckmann, S. O. Kamphorst, and D. Ruelle, Europhys. Lett. 4,
973 (1987).[36] H. Gotoda, R. Takeuchi, Y. Okuno, and T. Miyano, J. Appl.Phys. 113, 124902 (2013).[37] H. Gotoda, M. Pradas, and S. Kalliadasis (unpublished).[38] M. Pradas, D. Tseluiko, S. Kalliadasis, D. T. Papageor-giou, and G. A. Pavliotis, Phys. Rev. Lett. 106, 060602(2011).[39] H. Li, X. Zhou, J. B. Jeffries, and R. K. Hanson, Proc. Combust.Inst. 31, 3215 (2007).[40] T. Miyano, T. Moriya, H. Nagaike, N. Ikeuchi, and T.Matsumoto, J. Phys. D 41, 035209 (2008).
022910-8