Control Laboratory Experiments in ThermoAcoustics using the Rijke Tube

Jonathan P. Epperlein, Bassam Bamieh and Karl J. Astrom

Abstract— We report on experiments that investigate the

dynamics, identification and control of thermoacoustic phe-

nomena in a Rijke tube apparatus. These experiments are

relatively simple to construct and conduct in a typical, well-

equipped undergraduate controls laboratory, yet allow for

the exploration of rich and coupled acoustic and thermal

dynamics, the associated thermoacoustic instabilities, and the

use of acoustic feedback control for their stabilization. We

describe the apparatus construction, investigation of thermoa-

coustic dynamics and instabilities in both open-loop and closed-

loop configurations, closed-loop identification of the underlying

dynamics, as well as model validation. We also summarize

a transcendental transfer function analysis that explains the

underlying phenomena. These experiments are notable for the

fact that rich thermoacoustic phenomena can be analyzed using

introductory concepts such as the frequency response and root

locus, and thus can be performed and understood by controls

students with relatively little background in acoustics or heat

transfer.

I. INTRODUCTION

The Rijke tube is an experiment that is relatively simpleand inexpensive to build in a typical university laboratory.Despite its construction simplicity, it can serve to illustrate awide variety of mathematical modeling, empirical identifica-tion, verification and feedback control techniques. As such, itis suitable for use in both advanced undergraduate and grad-uate controls laboratory courses. The Rijke tube is perhapsthe simplest illustration of the phenomena of thermoacousticinstabilities. These phenomena typically occur whenever heatis released into a gas in underdamped acoustic cavities. Theheat release can be due to combustion or solid/gas conductiveand convective heat transfer. Under the right conditions, thecoupling between the acoustic and heat release dynamicsin the cavity becomes unstable. This instability manifestsitself as a sustained limit cycle resulting in audible, powerfulpressure oscillations. Thermoacoustic instability phenomenaare most often encountered in combustors [1], [2], [3],where the resulting powerful pressure waves are undesirabledue to the danger of structural damage as well as perfor-mance degradations. In this context, they are often referredto as combustion instabilities, and are notoriously difficultto model due to the additional complexity of combustion

This research is partially supported by NSF grants ECCS-0937539 andCMMI-0626170.

Jonathan Epperlein is with the Department of Electrical and ComputerEngineering, University of California, Santa Barbara, Santa Barbara, Cali-fornia 93117, USA. jpe@engineering.ucsb.edu

Bassam Bamieh is with the Department of Mechanical Engineering,University of California, Santa Barbara, Santa Barbara, California 93117,USA. bamieh@engineering.ucsb.edu

Karl J. Astrom is with the Department of Automatic Control, LundUniversity, Lund, Sweden. kja@control.lth.se

dynamics. The advantage of the Rijke tube experiment isthe ability to produce thermoacoustic instabilities without acombustion process. Many of the identification and feedbackcontrol issues involved in combustion instabilities are presentin the Rijke tube experiment. Thus, this experiment providesan easily accessible platform within which one can explorethe myriad issues relevant to thermoacoustic instabilities andtheir control.

The present paper aims at introducing the Rijke tube as anexperimental platform to explore thermoacoustic dynamicsand their control. We present an empirical investigation —which can be easily reproduced in a controls lab — of thedynamics of the Rijke tube using closed-loop identification,and standard linear techniques such as root locus and Nyquistcriterion. It is remarkable that one can obtain rather usefuland predictive models of the system with this approach.

A Rijke tube can be made out of a vertical, long, narrowand hollow tube, typically made out of glass (Pyrex) in ourcase for ease of visualization. Figure 1(a) shows a basicdiagram. A heating element (typically a resistive coil) isplaced towards the lower end of a vertical, open, hollowtube. If the coil is sufficiently hot, a steady upwards flowof air is achieved. An increase in the power to the coil

heatingcoil

Open glasstube

Controller

z

zo

0

mic

speaker

(a) (b)w

Fig. 1. (a) The Rijke tube shown with a heating element placed towardsthe bottom (suspension mechanism for coil not shown). Upward arrowindicates steady air flow caused by the coil’s heat. (b) The Rijke tubewith microphone, speaker and feedback controller. The external signal w isused for closed-loop identification.

causes an increase in the air flow, and at some criticalvalue, the tube begins to emit a loud steady hum likea pipe organ. Proportional acoustic feedback as shown inFigure 1(b) can make this hum disappear with an appropriatesetting of the gain. It is important to note that this isnot a noise cancellation scenario, but rather a stabilizationproblem, in that the acoustic feedback actually stabilizes

the underlying thermoacoustic instability that generates thesound. Distinguishing between these two settings is one partof the experimental investigation.

This paper is organized as follows. We first describe thebasic aspects of the apparatus construction, which thoughrelatively simple, requires some careful attention to cer-tain parameters so as to obtain an easily humming tube.Section III contains the basic initial observations of thethermoacoustic instabilities in open loop, and with stabilizingas well as further destabilizing feedback gains. Some of theelementary acoustic physics is diagrammatically illustrated.Section IV describes the procedure and typical results offrequency-domain closed-loop empirical identification. Thisproduces an open-loop plant transfer function that can beused for model validation. Though this transfer function hasbeen arrived at without any underlying physical modeling,it is predictive in that it explains the initial thermoacousticinstability, the proportional feedback stabilization, as well asthe high gain instabilities of the controlled system. This isdone in Section V using frequency responses and root loci ofthe identified open-loop model. The open-loop poles of thesystem as well as its right half plane zeros play an importantrole in this analysis. Finally, we include a short analysis sec-tion (Section VI) in which a transcendental transfer functionwith an infinite number of poles (representing acoustics) isanalyzed in feedback with heat release dynamics. A rootlocus analysis shows clearly how the coupling betweenacoustics and convective heat release is the underlying causeof the thermoacoustic instability. Although the analyticalderivations of these transfer functions are beyond the scopeof the present paper, it is included to illustrate the tightcorrespondence between models derived from the underlyingphysics, and those obtained from empirical identification.

II. CONSTRUCTION OF THE RIJKE TUBE APPARATUS

We describe here the particular hardware configurationused in the controls laboratory at the University of Cal-ifornia at Santa Barbara (UCSB). Earlier versions of thisexperiment have appeared elsewhere, namely in [4] where itwas specifically used in a controls laboratory. Details of ourbasic set up can be easily modified according to the user’sparticular laboratory facilities. Our basic Rijke tube apparatusused for this experiment is composed of the following maincomponents:

• Pyrex R� Glass tube, length = 4ft, internal diameter 3in (A very high aspect ratio is necessary to achievethe thermoacoustic instability with only moderate heaterpower.),

• Heater coil made from 24 gauge NiCr wire,• Simple clip-on microphone (with built-in preamplifier),• Audio amplifier,• Speaker,• DC or AC power supply.

We use a DAQ board in connection with Simulink Real-Time Windows Target to realize the variable control gainand collect the requisite data. This is certainly a bit overkill,a simple op-amp circuit, along with some other way of

collecting data, would do. A photograph and a diagram ofthis particular arrangement is shown in Figure 2. The glass

Power Supply

Audio Amplifier

PC

with

DAQ Board

Microphone

Coil

Speaker

Fig. 2. Photograph and diagram of the Rijke tube experimental apparatus.

tube is vertically mounted to a rigid frame, with the heatercoil mounted about 1/4 of the way up from the bottom ofthe tube. The power supply is used to heat the coil. Themicrophone is mounted on top and in the center of the tube.The microphone signal (AC coupled) is fed via the DAQboard to Simulink, where it is recorded and multiplied withthe variable gain. The interrogation signal (see Figure 3) isalso added there. The generated signal is then routed fromthe DAQ board to the audio amplifier and to the speaker.Our working assumption is that the the power and pre-

Rijke Tube

speaker mic pre-amppower amp

w

interrogationsignal

Fig. 3. Equivalent block diagram of the closed-loop identification setup.

amplifier, as well as the microphone and speaker can allbe described by pure proportional gains. In reality, theyeach have their characteristic frequency response which maynot be flat. However, in the frequency range of interest inthis experiment (typically 50-1000 Hz) where the Rijke tubeacoustic dynamics are dominant, we take these componentsto be pure gains and regard the Rijke tube system withthis acoustic feedback as reasonably well modeled by theconceptual diagram shown in Figure 3.

III. EMPIRICAL INVESTIGATION OF THE RIJKE TUBE

The experimental exploration of the Rijke tube beginswith supplying enough power for the initial thermoacousticinstability to appear. The effects of proportional acousticfeedback are then investigated through initial stabilizationand then observing other instabilities at high gains. Oncea stable system is established, the closed-loop identificationcan be performed to obtain a frequency response model. A

root locus analysis can then explain the initial stabilizationand subsequent destabilization with increasing gains.

1) Observing the thermoacoustic instability: The heatercoil power supply is turned on and increased slowly. Duringthis process one can feel the upward flow of hot air byplacing the hand slightly over the tube. Most likely, thewire will also start to glow red. There is a critical heaterpower beyond which the tube will begin to hum loudly. Theincrease in sound level up to saturation occurs slowly enough(a couple of seconds) to be perceptible. If the heater poweris decreased and then increased again, a slight hysteresisphenomena is observed.

(a) (b)

(c) (d)Fig. 4. Time trace of the microphone signal (a) at the onset of instabilityshowing growth, and then saturation of the limit cycle. Linear growth ona semilog plot (b) of the signal’s envelope confirms initial exponentialgrowth of its amplitude. A zoomed-in picture (c) shows the periodic, butnon-symmetric limit cycle behavior. With appropriate proportional feedback,the limit cycle is stabilized as this trace of the speaker’s input signal (d)shows.

The sound frequency f is easily measured with an oscil-loscope (about 140Hz in our setup), its wavelength is foundto be approximately equal to twice the length of the tube.This is consistent with a half-wavelength standing wave inthe tube; it is also the fundamental mode of a tube open atboth ends. The basic physics of that mode is illustrated inFigure 5.

2) Proportional acoustic feedback: With the control loopconnected, we apply proportional feedback to the Rijketube. If the gain is chosen appropriately, i.e. between Kminand Kmax, the humming disappears almost instantly. Thisis usually an impressive demonstration of the power offeedback.

Since many students who perform this experiment are notfamiliar with acoustics, they are often unsure as to what ishappening when the tube’s hum disappears. They often saythat the tube’s noise has been “cancelled”, probably becauseof familiarity with active noise-canceling headsets. However,the process here is fundamentally different. The feedbackhas stabilized the thermoacoustic instability which causedthe limit cycle in the first place. To verify the distinctionbetween stabilization and noise cancellation, it suffices to

vp(a) (b) (c) (d)

Fig. 5. A diagram of the fundamental acoustic mode of the Rijke tubeshowing its “half-wave” nature. The (a) pressure and (b) velocity spatialwaveforms are shown. These oscillate temporally 90o out of phase. Tothe right, diagrams of the corresponding motion of gas particles (ignoringthe mean convective upwards flow) are shown using arrows, and thecorresponding pressure distribution in color. In one half cycle (c) air rushesinto the tube causing a pressure maximum at the middle, while in the secondhalf (d) air rushes out causing a pressure minimum at the tube’s mid point.

observe the control signal into the speakers terminals onan oscilloscope. The oscilloscope will show that the controlsignal decays rapidly and hovers around almost zero if astabilizing feedback gain is used, see also Figure 4(d). Incontrast, a noise canceling system would have a persistentnon-zero control signal canceling the persistent noise.

0 Kfundamental mode unstable stable higher harmonic is unstable

Kmin Kmax

Fig. 6. A depiction of the effects of proportional feedback on the Rijketube. A minimum feedback gain K

min

is necessary to stabilize the otherwiseunstable fundamental mode. There is then a critical higher gain K

max

beyond which a higher harmonic mode (screech) of the tube becomesunstable.

3) High-gain instability: Next, the gain can be increaseduntil it reaches the critical higher gain value K

max

, abovewhich a new instability is triggered and a loud screechappears. A measurement of that screech frequency willreveal it to be an odd harmonic of the initial fundamentalhum frequency. Exactly which harmonic it is will dependon the details of the experimental set up (in ours it istypically the 3rd or 5th harmonic). This phenomenon ishowever repeatable if the experimental set up is unchanged(e.g. microphone, speaker and heater locations). The screechfrequency is recorded as it can be predicted from a root locusanalysis of the identified system model, and therefore can beused to validate that model.

IV. CLOSED-LOOP IDENTIFICATION AND MODELFITTING

Now we perform system identification to obtain someinformation about the dynamics of the Rijke tube. Since itis an unstable system, it will have to be identified whileoperating in a stabilizing closed loop. That comes with a

few pitfalls, in particular simply recording the plant inputand output and applying open-loop identification techniques,ignoring the fact that the input is the result of feedback, mightyield wrong results; for background and more sophisticatedmethods than the one we will employ see e.g. [5], [6] andthe references therein.

Closed-loop Identification: Our method of choice is theso-called indirect method: As shown in the conceptual blockdiagram in Figure 3, an exogenous interrogation signal isadded into the loop. The stable closed-loop transfer functionT (e

jk!

) = Y (e

jk!

)/W (e

jk!

) can then be identified withyour favorite open-loop method, and the unstable open loopcan be recovered from

T =

KG

1�KG

asG =

1

K

T

1 + T

. (1)

Here, K is the gain we set, the gains from the othercomponents such as microphone preamps and speaker areassumed to be independent of frequency and absorbed intoG. The indirect method is the simplest available method todeal with open-loop unstable systems, and as we will see itworks well for the present case.

Open-loop Identification Method: The dynamics under-lying the Rijke tube are a combination of acoustics and heattransfer, and are thus of relatively high order (in fact, theyare infinite dimensional). Nonparametric frequency domainidentification schemes are better suited to those types ofsystems than time domain based ones, since they do notforce us to select a model order a priori. Instead, we willdirectly identify the frequency response T (e

j!

) and thenuse a least-squares based method to fit a model of appro-priate order over the identified frequency domain. So-calledspectral methods (see e.g. [7, Ch. 6]) estimate the frequencyresponse as the ratio of the cross spectrum �

yw

(!) ofoutput and interrogation signal, and the spectrum �

w

(!) ofthe interrogation signal. The MATLAB System IdentificationToolbox [8] offers two implementations of these methods,spa and spafdr.1 Since we are expecting sharp peaks,fine frequency resolution is required, thus spafdr is theright choice. The least-squares fit is then performed usingthe FREQID Toolbox for MATLAB [9], [10].

Interrogation Signal: An interrogation signal shouldhave rich frequency content, while due to actuator andsensor limitations in physical systems, amplitudes shouldbe kept reasonably small. Popular choices include white

1While spa and spafdr both estimate the cross spectrum and inputspectrum by by a smoothing window to what roughly amounts to theDiscrete Fourier Transforms of input and output data, spa performs thewindowing in the time domain, whereas spafdr applies the window inthe frequency domain. Since a narrow frequency domain window, which iswhat is required, corresponds to a wide time domain window, using spafdrallows us to specify a small window, resulting in a drastic decrease incomputation time compared to the large window we would have to specifyto achieve the same resolution using spa. Another important distinctionbetween spa and spafdr is that the latter allows for frequency dependentresolution (hence the name), but we did not make use of this feature.

noise, Schroeder-phased sinusoids [11] and sine sweeps(also known as chirp signals). We experimented with allthree types of signals; sine sweeps, which are of use foridentification of acoustic systems [12], emerged as the mosteffective class, and all shown data was collected using asweep over the shown frequency range.

For the identification experiment, the tube is first broughtto a hum. Then, the feedback with a stabilizing gain is turnedon, and the interrogation signal is added to the feedbacksignal, as shown in Figure 3. The microphone signal isrecorded for the duration of the experiment (in our caseabout 2min), and together with the the applied interrogationsignal forms an input-output pair, which is all the data neededto obtain a spectral estimate. To minimize the effects ofrandom noise, this is done several times, and an average(weighted by the covariance) of the estimated frequencyresponses is formed. Figure 8 shows an averaged closed-loopfrequency response along with the individual experiments.This response has the signature of wave-like dynamics inthe presence of several very lightly damped modes at inte-ger multiples of a fundamental frequency. The fundamentalfrequency corresponds very closely to the frequency of thehum in the unstable Rijke tube.

In order to perform the least-squares fit of a finite di-mensional transfer function model for T to the estimatedfrequency response, a value for the model order needs to beselected. Figure 7 shows a 12th order transfer function fit,which nicely captures the first 6 harmonics in the frequencyresponse.

!60

!40

!20

0

Magnitu

de (

dB

)

100 200 300 400 500 600 700 800 900

!1080

!720

!360

0

Phase

(deg)

Frequency (Hz)

Identified Closed Loop

12th Order Fit

Fig. 7. Closed-loop frequency response obtained by a non-parametricspectral estimation, and a 12th order least-squares fit. Note that due to thelog-scale, the seemingly large deviations in the ranges between the peaksare actually very small.

To obtain the transfer function G as a parametric model ofthe open loop, the fitted model T is then plugged into (1). Ofcourse it is also possible to apply (1) to the nonparametricestimated frequency response at each frequency, therebyobtaining a nonparametric model of the open loop. Both ofthose possibilities are compared in Figure 9, they are in closeagreement, which is encouraging. We note with satisfaction

!60

!40

!20

0M

ag

nitu

de

(d

B)

500 1000 1500 2000 2500

!2880

!2160

!1440

!720

0

Ph

ase

(d

eg

)

Frequency (Hz)

(Weigthed) Average

!60

!40

!20

0

Magnitu

de (

dB

)

100 200 300 400 500 600 700 800 900

!1080

!720

!360

0

Phase

(deg)

Frequency (Hz)

(Weigthed) Average

Fig. 8. Closed-loop frequency response obtained with a sine sweep over therange of 0-2.5 kHz (top) and 0-900 Hz (bottom). The response below 20Hz,which is outside the audible range, and above 1kHz is likely dominated bymicrophone and speaker distortions. The range 0-1 kHz however exhibitstypical wave-like dynamics with resonances occurring at multiples of thefundamental harmonic.

that, while the phase at the first peak of the closed loop T

drops by 180

o indicating a stable pole, it increases by 180

o

in the open-loop response, indicating a pole in the right halfplane (RHP).

Varying the Microphone PositionSo far, we have not stated, where exactly we placed

the microphone during the experiments. It was not placeddeliberately, but rather just somewhere inside, but near thetop of, the tube. The reason is that the microphone positiondoes not influence the position of the poles, hence neitherthe peaks. However, there are special locations that do yieldinteresting results. In Figure 10, we show the identified openloops if we place the microphone a quarter length from thetop of the tube, and in the middle of the tube. Doing soappears to “remove peaks,” in the former case it would beevery fourth, and in the latter case even every even-numberedone.

This is relatively easily explained with the physical modelof the transfer function, which is developed elsewhere [13],but there is also a very intuitive explanation: each peakcorresponds to a mode, a standing pressure wave, in the tube.

!60

!40

!20

0

Magnitu

de (

dB

)

100 200 300 400 500 600 700 800 900

!1080

!720

!360

0

Phase

(deg)

Frequency (Hz)

From Identified Closed LoopFrom Fitted Closed Loop

Fig. 9. Open-loop frequency responses, obtained by applying (1)to the identified closed-loop response at every frequency (green) orto the fitted closed-loop response (black)

So the standing wave corresponding to the first peak wouldbe a half-wave, for the second peak it would be a full wave,for the third one and a half waves and so on. By placing themicrophone at e.g. the center, we placed it where all the evennumbered waveforms have a pressure node, and hence theircontribution is not registered by the microphone, a pressuresensor.

This would not be surprising at all for a tube without aheater; that it still holds true with the heater indicates thatits effect, the thermoacoustic effect, is pulling the first modeinto the right half plane, but besides that, regular acousticsdominate the response.

!60

!40

!20

0

Magnitu

de (

dB

)

100 200 300 400 500 600 700 800 900

!1080

!720

!360

0

Phase

(deg)

Frequency (Hz)

!m = 1/2!m = 3/4

Fig. 10. Bode plots for identified and fitted open-loop responses withdifferent microphone positions. Placing the microphone in the middle ofthe tube (⇠

m

= 1/2) seems to remove every other peak, while placing itat a quarter length from the tube attenuates the fourth peak only. The veryugly identification data at the removed peaks, and especially at the peakaround 660Hz, can be explained by the fact that perfect cancellation of apole by a zero is virtually impossible; instead, one gets a pole and a zerovery close together – notoriously difficult to identify.

V. MODEL VALIDATION: ROOT LOCUS ANALYSIS

We now have a model G of the open loop to validate anduse to explain the experimental observations. We will see thatthe root locus is effective in explaining why proportionalfeedback initially stabilizes the thermoacoustic instability,and why a higher frequency mode becomes unstable athigh gains. It will also give a quantitative prediction of thathigher frequency – a prediction that can be used for modelvalidation.

!2500 !2000 !1500 !1000 !500 0 500 1000

144

288

432

576

720

Real Axis [Hz]

Imagin

ary

Axi

s [H

z]

Fig. 11. A full view of the root locus of the identified open-loop model showing the fundamental frequency at 144Hz and itsharmonics very close to the imaginary axis.

!15 !10 !5 0

144

288

432

576

720

Real Axis [Hz]

Ima

gin

ary

Axi

s [H

z]

(a)

!40 !20 0

144

288

432

576

720

Real Axis [Hz]

Ima

gin

ary

Axi

s [H

z]

(b)

Fig. 12. Root locus of the identified open-loop model showing closed-looppole locations at a gain which (a) just stabilizes the open-loop unstablefundamental mode, and (b) causes a higher mode to become unstable. Inthis particular case, that higher mode is the fifth harmonic, and its frequencymust correspond to the pitch of the screech sound heard at high feedbackgains.

Figures 11 and 12 show the root locus of the identifiedopen-loop dynamics. The pole pattern resembles that of adamped wave equation, with imaginary parts of the polesbeing integer multiples of a fundamental frequency, and thereal parts having successively higher damping as the modefrequency increases. As promised, the fundamental mode is

unstable, having a positive real part. The imaginary part ofthe fundamental mode corresponds to the hum frequencyheard when the tube is initially powered on.

The right half plane zeros explain, why instability reoccursat a higher gain: Figure 12(a) shows the locus and thepole locations at the value of the gain sufficient to initiallystabilize the fundamental mode (denoted K

min

in Figure 6).Note how all poles are in the left half plane. However, dueto the presence of RHP zeros, some poles will eventually beattracted into the right half plane as the gain is increased. Fig-ure 12(b) indicates that for this particular identified model,it is the fifth harmonic mode that becomes unstable at highergain (denoted K

max

in Figure 6). The frequency of thismode must correspond to the frequency of the screech heardin the experiment as the system becomes unstable again athigh feedback gains, which is indeed what we observed.This serves as a useful method of model validation for thisexperiment.

We see that the identification yielded a model that cor-rectly predicted the instability and its frequency, and ex-plained why upon increasing the gain, another instabilityoccurs. It also correctly predicts the frequency of this high-gain instability.

Common problems: Often, the phase of the open-loopfrequency response will also drop, instead of increase, by180

o at the first peak, i.e. the open loop is identified as stable,while we know that the open loop must be unstable. Thestability of the open loop turned out to be very sensitivein particular to the amplitude and phase of T at the firstpeak. This is most easily explained with an argument basedon the Nyquist criterion: Inspecting (1), we notice that G

has the same poles as T in negative unity feedback, so wecan assess stability of G through the Nyquist criterion. Inorder for T to encircle the critical point (�1, 0j), we need|T | > 1 and 6

T = �180

o at the same frequency. InspectingFigure 8 again, we see that for the presented data, the firstpeak reaches only about 2dB, and the range for which itexceeds 0dB is only about 1Hz wide. Hence, if the peak is“cut off,” the identification will result in a stable open loop.Likely culprits are insufficient frequency resolution and toomuch smoothing during the spectral estimation. If increasingthe resolution and decreasing the smoothing do not help, adifferent speaker might be the solution; we found speakersto have quite different frequency responses, and some evenadded considerable phase lag.

It also might happen that the root locus predicts the higherharmonic instability incorrectly. This indicates that the initialclosed-loop identification step was inaccurately performed(insufficient or noisy frequency response data, too low orderselection for the model fit, etc.). A repeat of the identificationstep with more care will typically resolve this issue and themore carefully identified model will then yield the correctprediction of the high gain instability.

Lastly, if experiments are run for a long time, the walls ofthe tube, especially around the heater, absorb a lot of heat.If the identification is stopped and restarted for a new run,there might be no initial humming, due to the tube walls

heating the air around the heater to the point where the heattransfer between air and heater is insufficient to support thehumming. In that case, one can only wait for the tube to cooloff, or, if the setup admits, increase the power to the heaterto increase the coil temperature.

VI. TRANSFER FUNCTIONS FROM PHYSICAL MODELING

In this section we present a very brief description of a first-principles analysis that explains the onset of thermoacousticinstabilities using a root locus argument. The derivationsof system models and underlying transfer functions will bereported elsewhere [13].

The dynamics of the Rijke tube can be succinctly de-scribed by the block diagram in Figure 13. The transferfunction G represents the acoustic dynamics of the tube. It isdriven by both the speaker input as well as the heat releasefrom the heating coil2. There is also a feedback path sinceacoustic velocity oscillations v(t, x

o

) at the heating element(located at x

o

) modulate the convective heat release fromthe coil. This feedback effect is depicted by the transferfunction Q. It is this coupling between acoustic dynamicsand convective heat release from the coil that gives rise tothe thermoacoustic instability under appropriate conditions.

Speaker Signal

Velocity v(t, xo

)at the coil

Heat releasedby the coil

Microphone measurementp(t, x

m

)

G(s)

Q(s)

G

11

(s) G

12

(s)

G

21

(s) G

22

(s)

Fig. 13. A block diagram representation of the Rijke tube dynamicsdepicting the feedback interconnection between the acoustic dynamics G,and coil heat release dynamics Q. Acoustic velocity fluctuations v(t, x

o

) atthe coil location x

o

modulate the convective heat release q(t) from the coil.This heat in turn acts as a driving force for the acoustics. G is a distributedtransfer function which describes the acoustics of the entire tube, while Q

is lumped since it describes local effect of velocity on heat release.

The initial humming is a manifestation of the thermoa-coustic instability, which is caused by the feedback betweenQ and G

22

in Figure 13. The remaining transfer functions inG are not relevant to this analysis and will therefore notbe considered here. The acoustic dynamics are describedby the driven wave equation, and one can derive [13] thetranscendental form for G

22

as

G

22

(s) = � 1

cosh(s/2)

. (2)

This transfer function is irrational and has an infinite set ofpoles lying along the imaginary axis. We note that this formwas derived for acoustics without any damping. The actualsystem’s transfer function would include damping causedprimarily by viscous friction with the tube’s walls. In theRijke tube such damping is relatively small and has the net

2In compressible gas dynamics, heat input acts as a source term in theacoustics equations

!3 !2 !1 0 1 2 3

Im

!7 !

!6 !

!5 !

!4 !

!3 !

!2 !

!!

!

2 !

3 !

4 !

5 !

6 !

7 !

Re

Fig. 14. Root locus plot for the heat release–acoustic velocity feedback loopG

22

(s)Q(s). The analytical model is utilizing dimensionless variables, withthe speed of sound and tube length each scaled to 1. Thus, the frequency ⇡

on the imaginary axis corresponds to a physical frequency f = c/L·⇡2⇡

=c

2L

⇡ 142Hz.

effect of locating the poles of G22

slightly to the left of theimaginary axis rather than being exactly on the imaginaryaxis (typically with higher frequency poles being shiftedfurther into the left half plane than lower frequency modes).

The convective heat release dependence on velocity can bedescribed by a memoryless nonlinearity with a square rootdependence followed by a first order lag. The linearizationof velocity fluctuations around the base velocity thus has atransfer function which acts like a lowpass filter:

Q(s) =

Khr

⌧hrs+ 1

. (3)

Khr is an unknown, positive gain, and ⌧hr is the time constantof the heat release, which for our setup is estimated (see [14]and again [13]) to be of the order of 1.

Since the gain Khr is of known sign but unknown size, aroot locus analysis is the natural choice. The root locus forthe open loop G

22

Q is shown in Figure 14 and we see thatbranches originating from every odd multiple of j⇡ crossinto the right half plane, indicating that for sufficiently largeKhr, the interaction of acoustics and heat release is unstable.This observation is “robust” with respect to the actual valueof ⌧hr.

Since in the actual transfer function with damping terms,the poles at ±j⇡ will be closest to the imaginary axis, theycan be expected to be the first to cross into the right halfplane, at which point the system will exhibit exponentiallygrowing oscillations, with a frequency determined by theimaginary part of the crossing poles, until it settles into alimit cycle. The normalized frequency ⇡ corresponds to aphysical frequency of f ⇡ 142Hz, which corresponds to thefrequency observed in the experiment (see Figure 4).

VII. CONCLUSION

The Rijke tube is an experiment that is simple to con-duct, yet illustrates complex thermoacoustic phenomena, andoffers a platform for the application of basic as well as

advanced methods of identification and control. We believemany undergraduate controls laboratories can use it to illus-trate the power of even simple proportional control, demon-strate high-gain instabilities in non-minimum phase systems,apply closed-loop and frequency domain based identificationtechniques, and demonstrate root locus methods and theNyquist criterion.

We touched briefly on another topic: in addition to theexperimental investigation, a distributed model can be de-rived from first principles, and analyzed with transcendentaltransfer functions and infinite dimensional root loci; thus, theRijke tube allows to contrast experimental identification onthe one hand, and physical modeling on the other. This willbe reported in detail in [13].

REFERENCES

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[7] L. Ljung, System identification: theory for the user. PTR PrenticeHall, Upper Saddle River, NJ, 1999.

[8] ——, System Identification Toolbox for Use with MATLAB, TheMathWorks, Inc., 2007.

[9] R. De Callafon, P. M. Van den Hof, et al., “FREQID – frequencydomain identification toolbox for use with matlab,” Selected Topics inIdentification, Modelling and Control, vol. 9, pp. 129–134, 1996.

[10] R. De Callafon, “http://mechatronics.ucsd.edu/freqid/.”[11] D. Bayard, “Statistical plant set estimation using Schroeder-phased

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[12] J. Burgess, “Chirp design for acoustical system identification.” TheJournal of the Acoustical Society of America, vol. 91, no. 3, p. 1525,1992.

[13] J. P. Epperlein, B. Bamieh, and K. Astom, “ThermoAcoustics andthe Rijke Tube: Experiments, Identification and Modeling,” ControlSystems Magazine, In preparation.

[14] M. Lighthill, “The response of laminar skin friction and heat transferto fluctuations in the stream velocity,” Proceedings of the Royal Societyof London. Series A. Mathematical and Physical Sciences, vol. 224,no. 1156, pp. 1–23, 1954.