IFAC Pape sOnLine 50-1 (2017) 4528–4533
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A ailable online a www.sciencedi ec .com
2405-8963 © 2017, IFAC (In e na ional Fede a ion o Au oma ic Con ol) Hos ing by Else ie L d. All igh s ese ed.
Pee e iew unde esponsibili y o In e na ional Fede a ion o Au oma ic Con ol.
10.1016/j.i acol.2017.08.726
©
2017, IFAC (In e na ional Fede a ion o Au oma ic Con ol) Hos ing by Else ie L d. All igh s ese ed.
10.1016/j.i acol.2017.08.726 2405-8963
Bounda y con ol o a Rijke Tube using
i a ional ans e unc ions wi h
expe imen al alida ion
Gus a o A. de And ade ∗Ra ael Vazquez ∗∗
Daniel J. Pagano ∗
∗Depa men o Au oma ion and Sys ems, Uni e sidade Fede al de
San a Ca a ina, 88040-900, Flo ian´opolis, SC, B azil
(e-mail: gus a o.a u @posg ad.u sc.b , daniel.pagano@u sc.b )
∗∗ Depa men o Ae ospace Enginee ing, Uni e sidad de Se illa,
Camino de los Descub imien os, s.n.,41092 Se illa, Spain
(e-mail: [email protected])
Abs ac : This pape is conce ned wi h bounda y s abiliza ion o he moacous ic oscilla ions
in he Rijke ube. This sys em consis s o a e ical ube open in bo h ends and a hea e
placed in he lowe hal o he ube. A speake placed unde he ube is used as ac ua o while
a mic ophone placed nea he op o he ube p o ides he p essu e measu emen . To s udy
his p oblem we conside ha he ma hema ical model akes he o m o wo in e connec ed
compa men s: one o he cold zone and o he o he ho zone. The con ol inpu is applied on
he le bounda y condi ion o he cold zone. F om his model we de i e an i a ional ans e
unc ion o design a s abilizing bounda y con ol in he equency domain. In pa icula , we
de i e necessa y and su icien condi ions o he inpu -ou pu s abili y h ough he use o
Nyquis - ype es . Expe imen al esul s show he e ec i eness and eal-li e applicabili y o he
me hod.
Keywo ds: Pa ial di e en ial equa ions, s abiliza ion, F equency domain, I a ional ans e
unc ions, Nyquis s abili y c i e ion, The moacous ic oscilla ions
1. INTRODUCTION
In his pape , we in es iga e bounda y s abiliza ion o
he moacous ic oscilla ions. Fi s a heo e ical model is
igo ously s udied o ob ain s abilizing con ol laws, which
a e la e es ed in an expe imen al se ing. The phe-
nomenon o he moacous ic oscilla ions is desc ibed by
high le els o sound p oduced due o he eedback be ween
hea elease a e luc ua ions and acous ic p essu e luc u-
a ions in con ined spaces.
The Rijke ube se es as a con enien p o o ype sys em
o s udy he moacous ic phenomena. A Rijke ube is a
e ical ube, ypically made o glass, open on bo h ends
and wi h a hea ing elemen placed owa ds he lowe end.
A speake , placed a a sligh dis ance unde he ube, is
used as ac ua o , and a mic ophone loca ed nea he op
o he ube is used as senso . The ma hema ical model
o he Rijke ube ake a o m o wo in e connec ed com-
pa men s: a cold zone below he hea e and a ho zone
abo e i . These pa s a e gi en by he Eule equa ions o
gas dynamics. The hea e is in oduced as an in e ac ion
be ween he p essu e and eloci y ield o hese compo-
nen s. Fu he mo e, he con ol a iable is conside ed as
a bounda y condi ion o he p essu e in cold zone. De-
This wo k has been pa ially unded by he ollowing p ojec s:
MTM2015-65608-P inanced by Spanish Minis e io de Econom´ıa
y Compe i i idad and by CNPq-BRASIL unde he g an
438387/2016-3.
pending on he s eady-s a e condi ions abou which he
Eule equa ions a e linea ized, he esul ing linea iza ion
is a linea i s -o de pa ial di e en ial equa ion (PDE)
ha beha es like a wa e equa ion and hus desc ibes
acous ic wa e p opaga ion (de And ade e al., 2016). The
linea iza ion o his model leads o he sys em conside ed
h oughou his pape .
Many au ho s con ibu ed o he con ol o he moacous-
ic ins abili ies. The con ibu ions ange om phase shi
con olle s (Heckl, 1988) o LQG con olle s (Mu ugappan
e al., 2003) o H∞ obus con olle s (Campos-Delgado
e al., 2003). Mos o hese wo ks use a ini e dimensional
app oxima ion o he sys em o design con olle s. Recen
app oaches ook in o accoun dis ibu ed ea u e o he
sys em, o example ei he by using a PI con olle (K s ic
e al., 1999), o by a Riemann in a ian app oach (de An-
d ade e al., 2016).
In his wo k we conside a con ol me hodology based on
a equency domain app oach. S a ing om he linea ized
PDE sys em in a s eady-s a e egime, we apply he Laplace
ans o m o conside he linea ized PDE in he equency
domain, and classical equency domain ools a e used
o design he con olle in a simila way as o sys ems
ep esen ed by a ional ans e unc ions. Ou objec i e
in his pape is o conside his app oach wi h a igo ous
pe spec i e, and o show wha can e ec i ely be gua an-
eed by using such a equency domain app oach o he
P oceedings o he 20 h Wo ld Cong ess
The In e na ional Fede a ion o Au oma ic Con ol
Toulouse, F ance, July 9-14, 2017
Copy igh © 2017 IFAC 4624
Bounda y con ol o a Rijke Tube using
i a ional ans e unc ions wi h
expe imen al alida ion
Gus a o A. de And ade ∗Ra ael Vazquez ∗∗
Daniel J. Pagano ∗
∗
Depa men o Au oma ion and Sys ems, Uni e sidade Fede al de
San a Ca a ina, 88040-900, Flo ian´opolis, SC, B azil
(e-mail: gus a o.a u @posg ad.u sc.b , daniel.p[email p o ec ed])
∗∗ Depa men o Ae ospace Enginee ing, Uni e sidad de Se illa,
Camino de los Descub imien os, s.n.,41092 Se illa, Spain
(e-mail: [email p o ec ed])
Abs ac : This pape is conce ned wi h bounda y s abiliza ion o he moacous ic oscilla ions
in he Rijke ube. This sys em consis s o a e ical ube open in bo h ends and a hea e
placed in he lowe hal o he ube. A speake placed unde he ube is used as ac ua o while
a mic ophone placed nea he op o he ube p o ides he p essu e measu emen . To s udy
his p oblem we conside ha he ma hema ical model akes he o m o wo in e connec ed
compa men s: one o he cold zone and o he o he ho zone. The con ol inpu is applied on
he le bounda y condi ion o he cold zone. F om his model we de i e an i a ional ans e
unc ion o design a s abilizing bounda y con ol in he equency domain. In pa icula , we
de i e necessa y and su icien condi ions o he inpu -ou pu s abili y h ough he use o
Nyquis - ype es . Expe imen al esul s show he e ec i eness and eal-li e applicabili y o he
me hod.
Keywo ds: Pa ial di e en ial equa ions, s abiliza ion, F equency domain, I a ional ans e
unc ions, Nyquis s abili y c i e ion, The moacous ic oscilla ions
1. INTRODUCTION
In his pape , we in es iga e bounda y s abiliza ion o
he moacous ic oscilla ions. Fi s a heo e ical model is
igo ously s udied o ob ain s abilizing con ol laws, which
a e la e es ed in an expe imen al se ing. The phe-
nomenon o he moacous ic oscilla ions is desc ibed by
high le els o sound p oduced due o he eedback be ween
hea elease a e luc ua ions and acous ic p essu e luc u-
a ions in con ined spaces.
The Rijke ube se es as a con enien p o o ype sys em
o s udy he moacous ic phenomena. A Rijke ube is a
e ical ube, ypically made o glass, open on bo h ends
and wi h a hea ing elemen placed owa ds he lowe end.
A speake , placed a a sligh dis ance unde he ube, is
used as ac ua o , and a mic ophone loca ed nea he op
o he ube is used as senso . The ma hema ical model
o he Rijke ube ake a o m o wo in e connec ed com-
pa men s: a cold zone below he hea e and a ho zone
abo e i . These pa s a e gi en by he Eule equa ions o
gas dynamics. The hea e is in oduced as an in e ac ion
be ween he p essu e and eloci y ield o hese compo-
nen s. Fu he mo e, he con ol a iable is conside ed as
a bounda y condi ion o he p essu e in cold zone. De-
This wo k has been pa ially unded by he ollowing p ojec s:
MTM2015-65608-P inanced by Spanish Minis e io de Econom´ıa
y Compe i i idad and by CNPq-BRASIL unde he g an
438387/2016-3.
pending on he s eady-s a e condi ions abou which he
Eule equa ions a e linea ized, he esul ing linea iza ion
is a linea i s -o de pa ial di e en ial equa ion (PDE)
ha beha es like a wa e equa ion and hus desc ibes
acous ic wa e p opaga ion (de And ade e al., 2016). The
linea iza ion o his model leads o he sys em conside ed
h oughou his pape .
Many au ho s con ibu ed o he con ol o he moacous-
ic ins abili ies. The con ibu ions ange om phase shi
con olle s (Heckl, 1988) o LQG con olle s (Mu ugappan
e al., 2003) o H∞ obus con olle s (Campos-Delgado
e al., 2003). Mos o hese wo ks use a ini e dimensional
app oxima ion o he sys em o design con olle s. Recen
app oaches ook in o accoun dis ibu ed ea u e o he
sys em, o example ei he by using a PI con olle (K s ic
e al., 1999), o by a Riemann in a ian app oach (de An-
d ade e al., 2016).
In his wo k we conside a con ol me hodology based on
a equency domain app oach. S a ing om he linea ized
PDE sys em in a s eady-s a e egime, we apply he Laplace
ans o m o conside he linea ized PDE in he equency
domain, and classical equency domain ools a e used
o design he con olle in a simila way as o sys ems
ep esen ed by a ional ans e unc ions. Ou objec i e
in his pape is o conside his app oach wi h a igo ous
pe spec i e, and o show wha can e ec i ely be gua an-
eed by using such a equency domain app oach o he
P oceedings o he 20 h Wo ld Cong ess
The In e na ional Fede a ion o Au oma ic Con ol
Toulouse, F ance, July 9-14, 2017
Copy igh © 2017 IFAC 4624
Bounda y con ol o a Rijke Tube using
i a ional ans e unc ions wi h
expe imen al alida ion
Gus a o A. de And ade ∗Ra ael Vazquez ∗∗
Daniel J. Pagano ∗
∗Depa men o Au oma ion and Sys ems, Uni e sidade Fede al de
San a Ca a ina, 88040-900, Flo ian´opolis, SC, B azil
(e-mail: gus a o.a u @posg ad.u sc.b , daniel.pagano@u sc.b )
∗∗ Depa men o Ae ospace Enginee ing, Uni e sidad de Se illa,
Camino de los Descub imien os, s.n.,41092 Se illa, Spain
(e-mail: [email protected])
Abs ac : This pape is conce ned wi h bounda y s abiliza ion o he moacous ic oscilla ions
in he Rijke ube. This sys em consis s o a e ical ube open in bo h ends and a hea e
placed in he lowe hal o he ube. A speake placed unde he ube is used as ac ua o while
a mic ophone placed nea he op o he ube p o ides he p essu e measu emen . To s udy
his p oblem we conside ha he ma hema ical model akes he o m o wo in e connec ed
compa men s: one o he cold zone and o he o he ho zone. The con ol inpu is applied on
he le bounda y condi ion o he cold zone. F om his model we de i e an i a ional ans e
unc ion o design a s abilizing bounda y con ol in he equency domain. In pa icula , we
de i e necessa y and su icien condi ions o he inpu -ou pu s abili y h ough he use o
Nyquis - ype es . Expe imen al esul s show he e ec i eness and eal-li e applicabili y o he
me hod.
Keywo ds: Pa ial di e en ial equa ions, s abiliza ion, F equency domain, I a ional ans e
unc ions, Nyquis s abili y c i e ion, The moacous ic oscilla ions
1. INTRODUCTION
In his pape , we in es iga e bounda y s abiliza ion o
he moacous ic oscilla ions. Fi s a heo e ical model is
igo ously s udied o ob ain s abilizing con ol laws, which
a e la e es ed in an expe imen al se ing. The phe-
nomenon o he moacous ic oscilla ions is desc ibed by
high le els o sound p oduced due o he eedback be ween
hea elease a e luc ua ions and acous ic p essu e luc u-
a ions in con ined spaces.
The Rijke ube se es as a con enien p o o ype sys em
o s udy he moacous ic phenomena. A Rijke ube is a
e ical ube, ypically made o glass, open on bo h ends
and wi h a hea ing elemen placed owa ds he lowe end.
A speake , placed a a sligh dis ance unde he ube, is
used as ac ua o , and a mic ophone loca ed nea he op
o he ube is used as senso . The ma hema ical model
o he Rijke ube ake a o m o wo in e connec ed com-
pa men s: a cold zone below he hea e and a ho zone
abo e i . These pa s a e gi en by he Eule equa ions o
gas dynamics. The hea e is in oduced as an in e ac ion
be ween he p essu e and eloci y ield o hese compo-
nen s. Fu he mo e, he con ol a iable is conside ed as
a bounda y condi ion o he p essu e in cold zone. De-
This wo k has been pa ially unded by he ollowing p ojec s:
MTM2015-65608-P inanced by Spanish Minis e io de Econom´ıa
y Compe i i idad and by CNPq-BRASIL unde he g an
438387/2016-3.
pending on he s eady-s a e condi ions abou which he
Eule equa ions a e linea ized, he esul ing linea iza ion
is a linea i s -o de pa ial di e en ial equa ion (PDE)
ha beha es like a wa e equa ion and hus desc ibes
acous ic wa e p opaga ion (de And ade e al., 2016). The
linea iza ion o his model leads o he sys em conside ed
h oughou his pape .
Many au ho s con ibu ed o he con ol o he moacous-
ic ins abili ies. The con ibu ions ange om phase shi
con olle s (Heckl, 1988) o LQG con olle s (Mu ugappan
e al., 2003) o H∞ obus con olle s (Campos-Delgado
e al., 2003). Mos o hese wo ks use a ini e dimensional
app oxima ion o he sys em o design con olle s. Recen
app oaches ook in o accoun dis ibu ed ea u e o he
sys em, o example ei he by using a PI con olle (K s ic
e al., 1999), o by a Riemann in a ian app oach (de An-
d ade e al., 2016).
In his wo k we conside a con ol me hodology based on
a equency domain app oach. S a ing om he linea ized
PDE sys em in a s eady-s a e egime, we apply he Laplace
ans o m o conside he linea ized PDE in he equency
domain, and classical equency domain ools a e used
o design he con olle in a simila way as o sys ems
ep esen ed by a ional ans e unc ions. Ou objec i e
in his pape is o conside his app oach wi h a igo ous
pe spec i e, and o show wha can e ec i ely be gua an-
eed by using such a equency domain app oach o he
P oceedings o he 20 h Wo ld Cong ess
The In e na ional Fede a ion o Au oma ic Con ol
Toulouse, F ance, July 9-14, 2017
Copy igh © 2017 IFAC 4624
Bounda y con ol o a Rijke Tube using
i a ional ans e unc ions wi h
expe imen al alida ion
Gus a o A. de And ade ∗Ra ael Vazquez ∗∗
Daniel J. Pagano ∗
∗Depa men o Au oma ion and Sys ems, Uni e sidade Fede al de
San a Ca a ina, 88040-900, Flo ian´opolis, SC, B azil
(e-mail: gus a o.a u @posg ad.u sc.b , daniel.pagano@u sc.b )
∗∗ Depa men o Ae ospace Enginee ing, Uni e sidad de Se illa,
Camino de los Descub imien os, s.n.,41092 Se illa, Spain
(e-mail: [email protected])
Abs ac : This pape is conce ned wi h bounda y s abiliza ion o he moacous ic oscilla ions
in he Rijke ube. This sys em consis s o a e ical ube open in bo h ends and a hea e
placed in he lowe hal o he ube. A speake placed unde he ube is used as ac ua o while
a mic ophone placed nea he op o he ube p o ides he p essu e measu emen . To s udy
his p oblem we conside ha he ma hema ical model akes he o m o wo in e connec ed
compa men s: one o he cold zone and o he o he ho zone. The con ol inpu is applied on
he le bounda y condi ion o he cold zone. F om his model we de i e an i a ional ans e
unc ion o design a s abilizing bounda y con ol in he equency domain. In pa icula , we
de i e necessa y and su icien condi ions o he inpu -ou pu s abili y h ough he use o
Nyquis - ype es . Expe imen al esul s show he e ec i eness and eal-li e applicabili y o he
me hod.
Keywo ds: Pa ial di e en ial equa ions, s abiliza ion, F equency domain, I a ional ans e
unc ions, Nyquis s abili y c i e ion, The moacous ic oscilla ions
1. INTRODUCTION
In his pape , we in es iga e bounda y s abiliza ion o
he moacous ic oscilla ions. Fi s a heo e ical model is
igo ously s udied o ob ain s abilizing con ol laws, which
a e la e es ed in an expe imen al se ing. The phe-
nomenon o he moacous ic oscilla ions is desc ibed by
high le els o sound p oduced due o he eedback be ween
hea elease a e luc ua ions and acous ic p essu e luc u-
a ions in con ined spaces.
The Rijke ube se es as a con enien p o o ype sys em
o s udy he moacous ic phenomena. A Rijke ube is a
e ical ube, ypically made o glass, open on bo h ends
and wi h a hea ing elemen placed owa ds he lowe end.
A speake , placed a a sligh dis ance unde he ube, is
used as ac ua o , and a mic ophone loca ed nea he op
o he ube is used as senso . The ma hema ical model
o he Rijke ube ake a o m o wo in e connec ed com-
pa men s: a cold zone below he hea e and a ho zone
abo e i . These pa s a e gi en by he Eule equa ions o
gas dynamics. The hea e is in oduced as an in e ac ion
be ween he p essu e and eloci y ield o hese compo-
nen s. Fu he mo e, he con ol a iable is conside ed as
a bounda y condi ion o he p essu e in cold zone. De-
This wo k has been pa ially unded by he ollowing p ojec s:
MTM2015-65608-P inanced by Spanish Minis e io de Econom´ıa
y Compe i i idad and by CNPq-BRASIL unde he g an
438387/2016-3.
pending on he s eady-s a e condi ions abou which he
Eule equa ions a e linea ized, he esul ing linea iza ion
is a linea i s -o de pa ial di e en ial equa ion (PDE)
ha beha es like a wa e equa ion and hus desc ibes
acous ic wa e p opaga ion (de And ade e al., 2016). The
linea iza ion o his model leads o he sys em conside ed
h oughou his pape .
Many au ho s con ibu ed o he con ol o he moacous-
ic ins abili ies. The con ibu ions ange om phase shi
con olle s (Heckl, 1988) o LQG con olle s (Mu ugappan
e al., 2003) o H∞ obus con olle s (Campos-Delgado
e al., 2003). Mos o hese wo ks use a ini e dimensional
app oxima ion o he sys em o design con olle s. Recen
app oaches ook in o accoun dis ibu ed ea u e o he
sys em, o example ei he by using a PI con olle (K s ic
e al., 1999), o by a Riemann in a ian app oach (de An-
d ade e al., 2016).
In his wo k we conside a con ol me hodology based on
a equency domain app oach. S a ing om he linea ized
PDE sys em in a s eady-s a e egime, we apply he Laplace
ans o m o conside he linea ized PDE in he equency
domain, and classical equency domain ools a e used
o design he con olle in a simila way as o sys ems
ep esen ed by a ional ans e unc ions. Ou objec i e
in his pape is o conside his app oach wi h a igo ous
pe spec i e, and o show wha can e ec i ely be gua an-
eed by using such a equency domain app oach o he
P oceedings o he 20 h Wo ld Cong ess
The In e na ional Fede a ion o Au oma ic Con ol
Toulouse, F ance, July 9-14, 2017
Copy igh © 2017 IFAC 4624
Rijke ube by conside ing an in ini e dimensional model.
Mo eo e , we es he p oposed con ol me hodology in a
expe imen al se up o show i s e ec i eness and eal-li e
applicabili y o he me hod.
The pape is o ganized as ollows. Sec ion 2 p esen s he
desc ip ion o he moacous ic oscilla ions, he Rijke ube,
i s linea ized ma hema ical model and he cha ac e iza ion
o he ans e unc ion. The closed loop sys em and i s
s abili y is s udied in Sec ion 3. The expe imen al esul s
a e shown in Sec ion 4. Finally, he main conclusions a e
p esen ed in Sec ion 5.
No a ion The se o eal numbe s is deno ed by R. The
se o posi i e in ege s is deno ed by N. We deno e Cas
he se o complex numbe s and C+all complex numbe s
wi h eal pa la ge han o equal ze o. By swe deno e
he complex a iable, ha is, s=σ+jω, wi h σand ω
eal numbe s and j2=−1. By H∞, we deno e he Hilbe
space H∞={u:C+→C|uanaly ic and sup
Re(s)>0
|u(s)|<
∞}, wi h no m u∞= sup
Re(s)>0
|u(s)|. The Hilbe space o
measu able and squa e in eg able L2- unc ions is deno ed
by L2(0,∞)=u: [0,∞)→C|∞
0|u( )|2d < ∞, wi h
he L2-no m u2=∞
0|u( )|21/2. The class o con inu-
ously di e en iable unc ions om [a, b] o Rnis deno ed
by C1([a, b]; Rn).
2. SYSTEM DESCRIPTION
2.1 The Rijke ube
In his wo k, all he expe imen s we e pe o med on a
simple 1.3 me e long glass ube wi h an elec ical hea ing
elemen made o nich ome wi e coil. The powe is deli e ed
in o he coil using a DC powe supply wi h powe ou pu
360 W. The loca ion o he elec ical hea ing elemen was
chosen o be a qua e o he ube leng h. The sound
p essu e in he ube is measu ed wi h a clip-on mic ophone
wi h buil -in p eampli ie . This signal is sen o a con ol
compu e h ough a da a acquisi ion de ice. The con ol
sys em is implemen ed as pa o a SCADA p og am
based on a LabVIEW so wa e. In such con igu a ion he
con ol algo i hm is implemen ed as a Ma lab unc ion
execu ed om he SCADA p og am. Fo he closed-loop
expe imen s i was used a 30 W ceiling speake as he
ac ua o oge he wi h a linea ampli ie . A schema ic o
he Rijke ube is depic ed in Figu e 1.
2.2 The he moacous ic phenomenon
In he Rijke ube, he hea sou ce ans e hea o he ai in
he ube, making he ai o ise up and c ea ing an upwa d
low. The ising ho ai becomes dense by coming in
con ac wi h he coole walls o he uppe hal o he ube.
This means ha in he lowe hal o he ube, he ai always
expe iences expansion, while in he uppe pa , he ai
always expe ience comp ession. Mo eo e , acco ding o he
Rayleigh’s c i e ion (Rayleigh, 1945), a s anding p essu e
wa e is sus ained i hea is added du ing condensa ion,
o be aken om i a he momen o a e ac ion. On
he o he hand, i hea is added du ing a e ac ion, o
Fig. 1. Rijke ube schema ic.
abs ac ed a he momen o condensa ion, he p essu e
wa e is discou aged. In ma hema ical e ms, he Rayleigh’s
c i e ion is o mula ed in e ms o he Rayleigh in eg al
o e he con ol olume Vgi en by
I=V
˜
P( )˜
Q( )d , (1)
whe e ˜
Pis he acous ic p essu e luc ua ion, ˜
Qis he
luc ua ion o hea powe eleased in he hea e , and is
ime. Acco ding o he Rayleigh’s c i e ion, i I<0, he
p essu e wa e will be damped. I I>0, hen he p essu e
wa e will g ow. O he wise, i I= 0, he p essu e wa e will
nei he be damped ou no ampli ied.
2.3 Ma hema ical model
The Rijke ube can be modeled as a hea ing sec ion
embedded wi hin a ne wo k o pipes. We assume ha he
luc ua ions o p essu e and eloci y occu only along he
axial di ec ion. The e o e, he sys em can be desc ibed by
he one-dimensional ma hema ical model o comp essible
gas dynamics. Fu he mo e, he hea ing elease zone is
assumed o be loca ed in a e y na ow sec ion.
The Rijke ube is composed o 2 compa men s desc ibed
by he linea ized Eule equa ions o gas dynamics:
∂ ˜ i( , xi)+1
ρ∂xi˜
Pi( , xi)=0,(2)
∂ ˜
Pi( , xi)+γP∂xi˜ i( , xi)=0,(3)
i=1,2,
whe e ∈[0,+∞) is he ime, x1∈(−xu,0), x2∈
(0,x
d), xu,x
d>0, γis he adiaba ic a io, ˜
Pis he
p essu e luc ua ion, and ˜ is he eloci y luc ua ion. The
s eady-s a e densi y and p essu e a e deno ed by ρand
P, espec i ely. I is impo an o emphasize ha in his
wo k he s eady-s a e densi y, p essu e and eloci y a e
assumed o be cons an along he space. Fu he mo e,
he s eady-s a e alues a e conside ed he same o bo h
compa men s o he ube.
The ini ial condi ion is de ined by
˜ i(0,x
i)=˜ i,0(xi),˜
Pi(0,x
i)= ˜
Pi,0(xi),
i=1,2.
We ep esen he in e connec ion be ween he downs eam
and ups eam pa o he sys em by he ollowing algeb aic
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Rijke ube by conside ing an in ini e dimensional model.
Mo eo e , we es he p oposed con ol me hodology in a
expe imen al se up o show i s e ec i eness and eal-li e
applicabili y o he me hod.
The pape is o ganized as ollows. Sec ion 2 p esen s he
desc ip ion o he moacous ic oscilla ions, he Rijke ube,
i s linea ized ma hema ical model and he cha ac e iza ion
o he ans e unc ion. The closed loop sys em and i s
s abili y is s udied in Sec ion 3. The expe imen al esul s
a e shown in Sec ion 4. Finally, he main conclusions a e
p esen ed in Sec ion 5.
No a ion The se o eal numbe s is deno ed by R. The
se o posi i e in ege s is deno ed by N. We deno e Cas
he se o complex numbe s and C+all complex numbe s
wi h eal pa la ge han o equal ze o. By swe deno e
he complex a iable, ha is, s=σ+jω, wi h σand ω
eal numbe s and j2=−1. By H∞, we deno e he Hilbe
space H∞={u:C+→C|uanaly ic and sup
Re(s)>0
|u(s)|<
∞}, wi h no m u∞= sup
Re(s)>0
|u(s)|. The Hilbe space o
measu able and squa e in eg able L2- unc ions is deno ed
by L2(0,∞)=u: [0,∞)→C|∞
0|u( )|2d < ∞, wi h
he L2-no m u2=∞
0|u( )|21/2. The class o con inu-
ously di e en iable unc ions om [a, b] o Rnis deno ed
by C1([a, b]; Rn).
2. SYSTEM DESCRIPTION
2.1 The Rijke ube
In his wo k, all he expe imen s we e pe o med on a
simple 1.3 me e long glass ube wi h an elec ical hea ing
elemen made o nich ome wi e coil. The powe is deli e ed
in o he coil using a DC powe supply wi h powe ou pu
360 W. The loca ion o he elec ical hea ing elemen was
chosen o be a qua e o he ube leng h. The sound
p essu e in he ube is measu ed wi h a clip-on mic ophone
wi h buil -in p eampli ie . This signal is sen o a con ol
compu e h ough a da a acquisi ion de ice. The con ol
sys em is implemen ed as pa o a SCADA p og am
based on a LabVIEW so wa e. In such con igu a ion he
con ol algo i hm is implemen ed as a Ma lab unc ion
execu ed om he SCADA p og am. Fo he closed-loop
expe imen s i was used a 30 W ceiling speake as he
ac ua o oge he wi h a linea ampli ie . A schema ic o
he Rijke ube is depic ed in Figu e 1.
2.2 The he moacous ic phenomenon
In he Rijke ube, he hea sou ce ans e hea o he ai in
he ube, making he ai o ise up and c ea ing an upwa d
low. The ising ho ai becomes dense by coming in
con ac wi h he coole walls o he uppe hal o he ube.
This means ha in he lowe hal o he ube, he ai always
expe iences expansion, while in he uppe pa , he ai
always expe ience comp ession. Mo eo e , acco ding o he
Rayleigh’s c i e ion (Rayleigh, 1945), a s anding p essu e
wa e is sus ained i hea is added du ing condensa ion,
o be aken om i a he momen o a e ac ion. On
he o he hand, i hea is added du ing a e ac ion, o
Fig. 1. Rijke ube schema ic.
abs ac ed a he momen o condensa ion, he p essu e
wa e is discou aged. In ma hema ical e ms, he Rayleigh’s
c i e ion is o mula ed in e ms o he Rayleigh in eg al
o e he con ol olume Vgi en by
I=V
˜
P( )˜
Q( )d , (1)
whe e ˜
Pis he acous ic p essu e luc ua ion, ˜
Qis he
luc ua ion o hea powe eleased in he hea e , and is
ime. Acco ding o he Rayleigh’s c i e ion, i I<0, he
p essu e wa e will be damped. I I>0, hen he p essu e
wa e will g ow. O he wise, i I= 0, he p essu e wa e will
nei he be damped ou no ampli ied.
2.3 Ma hema ical model
The Rijke ube can be modeled as a hea ing sec ion
embedded wi hin a ne wo k o pipes. We assume ha he
luc ua ions o p essu e and eloci y occu only along he
axial di ec ion. The e o e, he sys em can be desc ibed by
he one-dimensional ma hema ical model o comp essible
gas dynamics. Fu he mo e, he hea ing elease zone is
assumed o be loca ed in a e y na ow sec ion.
The Rijke ube is composed o 2 compa men s desc ibed
by he linea ized Eule equa ions o gas dynamics:
∂ ˜ i( , xi)+1
ρ∂xi˜
Pi( , xi)=0,(2)
∂ ˜
Pi( , xi)+γP∂xi˜ i( , xi)=0,(3)
i=1,2,
whe e ∈[0,+∞) is he ime, x1∈(−xu,0), x2∈
(0,x
d), xu,x
d>0, γis he adiaba ic a io, ˜
Pis he
p essu e luc ua ion, and ˜ is he eloci y luc ua ion. The
s eady-s a e densi y and p essu e a e deno ed by ρand
P, espec i ely. I is impo an o emphasize ha in his
wo k he s eady-s a e densi y, p essu e and eloci y a e
assumed o be cons an along he space. Fu he mo e,
he s eady-s a e alues a e conside ed he same o bo h
compa men s o he ube.
The ini ial condi ion is de ined by
˜ i(0,x
i)=˜ i,0(xi),˜
Pi(0,x
i)= ˜
Pi,0(xi),
i=1,2.
We ep esen he in e connec ion be ween he downs eam
and ups eam pa o he sys em by he ollowing algeb aic
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ela ions, which can be di ec ly ob ained by linea izing he
equa ions conse a ion o mass, momen um and ene gy
ac oss he hea zone x= 0,
˜
P2( , 0) + ρ ˜ 2( , 0) −˜
P1( , 0) −ρ ˜ 1( , 0)=0,(4)
γ
γ−1 ˜
P2( , 0) + γ
γ−1P+ρ 2˜ 2( , 0)−
γ
γ−1 ˜
P1( , 0) −γ
γ−1P1+ρ 2˜ 1( , 0) = ˜
Q
A( ),(5)
whe e Ais he c oss-sec ional a ea o he ube and ˜
Q
is he luc ua ion o hea powe eleased in he hea e .
Following Eppe lein e al. (2015), we assume ha he
luc ua ion o hea powe is exp essed by he ollowing
o dina y di e en ial equa ion (ODE)
h ˜
Q( )=−˜
Q( )+h ˜ 1( , 0),(6)
whe e h is he hea elease ime cons an and h is he
eloci y-dependen hea ans e coe icien .
Mo eo e , he sys em (2)-(3) is subjec ed o he ollowing
bounda y condi ions
˜
P1( , −xu)=U( ),˜
P2( , xd)=0,(7)
whe e Uis he con ol inpu .
The ope a ional pa ame e s used in his pape a e shown
in Table 1.
Rema k 1. In Heckl (1988) i was compu ed an explici
es ima ion o he Rayleigh in eg al (1) om he equa ions
(2)-(7). The he moacous ic oscilla ions ampli ude g ows
i he acous ic ene gy s o ed in he ube inc eases in ime
a e age, i.e.,
γ−1
ρc2˜
P(0)Q
A>˜
P(L)˜ (L)−˜
P(0)˜ (0)+
ηLc ˜
P2(0)
ρc2+ρ˜u2(0),(8)
whe e •is he mean alue (in ime) o i s a gumen ,
ηLc ˜
P2(0)
ρc2+ρ˜u2(0)is he S okes laye , and ηis he
a enua ion cons an o he sound wa e a eling along
he ube.
Well-posedness Wi hou loss o gene ali y, i can be
assumed ha , by e-scaling he space a iable, he wo
coun e pa s o PDE (2)-(3) e ol e in he domain om 0
o 1. Fu he mo e, in his amewo k i can be assumed
Table 1. Values o he pa ame e s o he sys-
em.
Symbol Desc ip ion Value
ρDensi y 1.2 kg/m3
PP essu e 105N/m2
Veloci y 0.35 m/s
γAdiaba ic a io 1.4
γ- 0.4
LTube leng h 1.3 m
x0Hea e posi ion 1
4L
dTube diame e 0.0762 m
RuRe lec ion coe icien −0.95
RdRe lec ion coe icien −0.95
h Hea - elease ime cons an 0.002
h Veloci y-dependen hea 200
ans e coe icien
ha he e a e only coupling a he bounda ies be ween
hemsel es. This leads o exp ess he sys em (2)-(7) in o
∂ ξ( , z)+A∂zξ( , z)=0,
ξ(0, )=ξ0(z),
gL(ξ( , z),U( ))=0,g
R(ξ( , 1))=0,(9)
whe e ξ=(
˜
P1,˜ 1,˜
P2,˜ 2), z∈[0,1] is he e-scaled
space a iable, gLand gRa e he le and igh bounda y
condi ions, ξ0is he ini ial condi ion and Ais a ma ix
wi h eal coe icien s. We omi he explici exp ession o
A,gLand gRdue o lack o space.
The exis ence and uniqueness o he solu ion o sys em
(9) can be p o ed by he me hod o cha ac e is ics, which
enables us o es a e he PDE as a se o classical ODEs.
Then, i ξ0and Ua e con inuously di e en iable unc ions
o hei a gumen s and i ξ0and he bounda y condi ions
e i y condi ions o C1compa ibili y, one can show ha
he solu ions o he sys em a e con inuously di e en iable
wi h espec o hei a gumen s, i.e., ξ∈C1([0,1] ×
[0,∞); R4). Mo eo e , based on an ex ension o Li ico
and F omion (2009), he e exis M>0 and ηsuch
ha o any ∈[0,∞), any ξ∈C1([0,1]; R6) and any
U( )∈L2([0, ]; R4)∩C1([0, ],R4), he e exis s K such
ha
ξ(·, )L2([0,1];R4)+
˜
Q( )≤Meη ξ0L2([0,1];R4)+

˜
Q(0)+K 

U( )

2,
whe e U( )deno es he es ic ion o U o [0, ].
2.4 Open-loop ans e unc ion
The well-posedness o he solu ion enables us o use a
equency domain app oach. We s a by applying he
Laplace ans o m o (2)-(7) in o de o ob ain a ans e
unc ion G(s) om he speake o he mic ophone p es-
su e. To de i e his ans e unc ion, no e ha equa ions
(2)-(3) ep esen he wa e equa ion. To see ha , ake
bo h a ime and space de i a i e in (3) and sub ac he
esul ing exp essions. One ob ains
∂ ˜
Pi( , xi)=c2∂xixi˜
Pi( , xi),i=1,2,(10)
whe e c=γP
ρis he speed o sound.
I is well known ha he solu ion o (10) is gi en by he
d’Alambe o mula (E ans, 2010). The e o e, he acous ic
p essu e in he ups eam p opaga es acco ding o
˜
P1( , x)=  −x
c+g +x
c,−xu<x<0,(11)
and simila ly o he downs eam side
˜
P2( , x)=h −x
c+j +x
c,0<x<x
d,(12)
whe e ,g,h,ja e unc ions which sa is y he bounda y
and ini ial condi ions.
Subs i u ing (11)-(12) in o (2) and in eg a ing o e ime,
we ge he exp ession o he eloci y luc ua ions a he
ups eam and downs eam pa o he ube:
P oceedings o he 20 h IFAC Wo ld Cong ess
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˜ 1( , x)= 1
ρc   −x
c−g +x
c,−xu<x<0,
(13)
˜ 2( , x)= 1
ρc h −x
c−j +x
c,0>x>x
d.
(14)
F om he bounda y condi ions (7), we ge ( )=
−g −2xu
c+U( −xu
c) and j( )=−h −2xd
c. These
e lec ions a ube ends a e modeled o ideal condi ion.
A mo e ealis ic model should include acous ic e lec ion
losses a he bounda y. The e o e, we in oduce he con-
s an s Ru,R
d∈(−1,0) o accoun o acous ic e lec ion
losses. I ollows ha hese bounda y condi ions a e ew i -
en o
( )=Rug( −τu)+U −τu
2,(15)
j( )=Rdh( −τd),(16)
whe e τu=2
xu
cand τd=2
xd
c.
The open-loop ans e unc ion equi ed, G(s), is om he
speake o he mic ophone p essu e. This can be ob ained
by conside ing x= 0 and subs i u ing (11)-(16) in o
(4)-(5), and applying he Laplace ans o m in o hese
equa ions. A e some algeb aic manipula ions we ob ain
he inal exp ession o he equi ed ans e - unc ion:
G(s)=Pmic(s)
U(s)=−2e−(xu+xmic
c)s
(γ−1) de (S)×
1+Rde2xmic−xd
cs(1 + (γ−1)φ(s)),(17)
whe e Pmic and Ua e he Laplace ans o m o ˜
P2( , xmic)
and U, espec i ely, xmic ∈(0,x
d) is he loca ion o he
mic ophone, φ(s)= h
h s+1 and
S=

−1−Rue−τus1+Rde−τds
(1 −Rue−τus)
γ−1(1 + (γ−1)φ(s)) 1−Rde−τds
γ−1
.
Poles o he open-loop sys em The poles o G(s) cha -
ac e ize he open-loop dynamic beha io o he linea ized
sys em. They a e gi en by he solu ions o he ollowing
equa ion:
ψ(s)(1 −Rde−τus)(1 −Rde−τds)−
(1 −Rue−τus)(τs+1+(γ−1)h )(1 + Rde−τds)=0.
(18)
In gene al, his equa ion has no explici solu ion. Nume -
ical esolu ion o he alues in Table 1 leads o he poles
depic ed in Figu e 2. As can be seen, he e is a pai o
uns able complex conjuga ed poles a he equency 131
Hz, and in ini e poles on he le hand side o he complex
plane. The ollowing p oposi ion p o ides a closed solu ion
o (18) o explain he poles beha io o high equency, in
which he p oo was omi ed due o lack o space.
P oposi ion 2. When |s|0, he solu ions o (18) end
asymp o ically owa ds
˜p±k=log(RuRd)
τu+τd
±2jπk
τu+τd
,(19)
whe e k∈N, and he app oxima ion e o is a he i s -
o de gi en by:
-6 -4 -2 0 2 4
Real axis (Hz)
0
2000
4000
6000
8000
10000
12000
14000
16000
Imagina y axis (Hz)
131 Hz
Fig. 2. Loca ion o he poles o he open-loop ans e
unc ion (17). They we e calcula ed nume ically by
inding oo s o (17). No e ha he e is an uns able
pole a 131 Hz, and in ini e s able poles ha end
asymp o ically owa ds (19).
p±k≈˜p±k−ψ(˜p±k)
ψ(˜p±k).(20)
Figu e 3 shows he open-loop equency esponse o he
model (17) and he eal expe imen desc ibed in Sec ion
2 o e he equency ange 100 −900 Hz. As can be seen,
by compa ing Figu e 3(a) and Figu e 3(b), he esponse
o he i a ional ans e unc ion (17) and he eal sys em
a e e y simila in he equency ange o in e es .
3. CLOSED-LOOP SYSTEM
3.1 P oposed con ol law
As shown in Equa ion (8), he he moacous ic oscilla ion
in he Rijke ube occu s when acous ic ene gy is g ea e
han he loss. The gain and loss o ene gy depend on he
acous ic ield. Howe e , condi ions can change comple ely
i he ield in he ube is dis u bed by a di e en sound
sou ce, which in his wo k is p oduced by a loudspeake .
The inclusion o a sound sou ce changes he di e ence
be ween ene gy gain and loss, which can become la ge
o smalle han he unpe u bed acous ic ield. The e o e,
i is easonable o conside he con ol law gi en by
u( )=Kc˜
P( −τc,x
mic),(21)
whe e Kcand τca e design pa ame e s. No e ha wi h
his con ol law he loudspeake ep oduces an ampli ied
and delayed p essu e wa e o he ube.
3.2 S abili y analysis
In his wo k, he ollowing de ini ion o s abili y is adop ed.
De ini ion 3. I a sys em maps e e y inpu uin L2(0,∞)
o an ou pu yin L2(0,∞) and
sup
u=0
y2
u2
<∞,
he sys em is s able. A sys em is said o be uns able i i
is no s able.
Rema k 4. S abili y o sys ems desc ibed by hei ans e
unc ions can be checked by Theo em A.2 o Cu ain and
Mo is (2009). In his case, a linea sys em is s able i and
only i i s ans e unc ion Gbelongs o H∞.
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˜ 1( , x)= 1
ρc   −x
c−g +x
c,−xu<x<0,
(13)
˜ 2( , x)= 1
ρc h −x
c−j +x
c,0>x>x
d.
(14)
F om he bounda y condi ions (7), we ge ( )=
−g −2xu
c+U( −xu
c) and j( )=−h −2xd
c. These
e lec ions a ube ends a e modeled o ideal condi ion.
A mo e ealis ic model should include acous ic e lec ion
losses a he bounda y. The e o e, we in oduce he con-
s an s Ru,R
d∈(−1,0) o accoun o acous ic e lec ion
losses. I ollows ha hese bounda y condi ions a e ew i -
en o
( )=Rug( −τu)+U −τu
2,(15)
j( )=Rdh( −τd),(16)
whe e τu=2
xu
cand τd=2
xd
c.
The open-loop ans e unc ion equi ed, G(s), is om he
speake o he mic ophone p essu e. This can be ob ained
by conside ing x= 0 and subs i u ing (11)-(16) in o
(4)-(5), and applying he Laplace ans o m in o hese
equa ions. A e some algeb aic manipula ions we ob ain
he inal exp ession o he equi ed ans e - unc ion:
G(s)=Pmic(s)
U(s)=−2e−(xu+xmic
c)s
(γ−1) de (S)×
1+Rde2xmic−xd
cs(1 + (γ−1)φ(s)),(17)
whe e Pmic and Ua e he Laplace ans o m o ˜
P2( , xmic)
and U, espec i ely, xmic ∈(0,x
d) is he loca ion o he
mic ophone, φ(s)= h
h s+1 and
S=

−1−Rue−τus1+Rde−τds
(1 −Rue−τus)
γ−1(1 + (γ−1)φ(s)) 1−Rde−τds
γ−1
.
Poles o he open-loop sys em The poles o G(s) cha -
ac e ize he open-loop dynamic beha io o he linea ized
sys em. They a e gi en by he solu ions o he ollowing
equa ion:
ψ(s)(1 −Rde−τus)(1 −Rde−τds)−
(1 −Rue−τus)(τs+1+(γ−1)h )(1 + Rde−τds)=0.
(18)
In gene al, his equa ion has no explici solu ion. Nume -
ical esolu ion o he alues in Table 1 leads o he poles
depic ed in Figu e 2. As can be seen, he e is a pai o
uns able complex conjuga ed poles a he equency 131
Hz, and in ini e poles on he le hand side o he complex
plane. The ollowing p oposi ion p o ides a closed solu ion
o (18) o explain he poles beha io o high equency, in
which he p oo was omi ed due o lack o space.
P oposi ion 2. When |s|0, he solu ions o (18) end
asymp o ically owa ds
˜p±k=log(RuRd)
τu+τd
±2jπk
τu+τd
,(19)
whe e k∈N, and he app oxima ion e o is a he i s -
o de gi en by:
-6 -4 -2 0 2 4
Real axis (Hz)
0
2000
4000
6000
8000
10000
12000
14000
16000
Imagina y axis (Hz)
131 Hz
Fig. 2. Loca ion o he poles o he open-loop ans e
unc ion (17). They we e calcula ed nume ically by
inding oo s o (17). No e ha he e is an uns able
pole a 131 Hz, and in ini e s able poles ha end
asymp o ically owa ds (19).
p±k≈˜p±k−ψ(˜p±k)
ψ(˜p±k).(20)
Figu e 3 shows he open-loop equency esponse o he
model (17) and he eal expe imen desc ibed in Sec ion
2 o e he equency ange 100 −900 Hz. As can be seen,
by compa ing Figu e 3(a) and Figu e 3(b), he esponse
o he i a ional ans e unc ion (17) and he eal sys em
a e e y simila in he equency ange o in e es .
3. CLOSED-LOOP SYSTEM
3.1 P oposed con ol law
As shown in Equa ion (8), he he moacous ic oscilla ion
in he Rijke ube occu s when acous ic ene gy is g ea e
han he loss. The gain and loss o ene gy depend on he
acous ic ield. Howe e , condi ions can change comple ely
i he ield in he ube is dis u bed by a di e en sound
sou ce, which in his wo k is p oduced by a loudspeake .
The inclusion o a sound sou ce changes he di e ence
be ween ene gy gain and loss, which can become la ge
o smalle han he unpe u bed acous ic ield. The e o e,
i is easonable o conside he con ol law gi en by
u( )=Kc˜
P( −τc,x
mic),(21)
whe e Kcand τca e design pa ame e s. No e ha wi h
his con ol law he loudspeake ep oduces an ampli ied
and delayed p essu e wa e o he ube.
3.2 S abili y analysis
In his wo k, he ollowing de ini ion o s abili y is adop ed.
De ini ion 3. I a sys em maps e e y inpu uin L2(0,∞)
o an ou pu yin L2(0,∞) and
sup
u=0
y2
u2
<∞,
he sys em is s able. A sys em is said o be uns able i i
is no s able.
Rema k 4. S abili y o sys ems desc ibed by hei ans e
unc ions can be checked by Theo em A.2 o Cu ain and
Mo is (2009). In his case, a linea sys em is s able i and
only i i s ans e unc ion Gbelongs o H∞.
P oceedings o he 20 h IFAC Wo ld Cong ess
Toulouse, F ance, July 9-14, 2017
4627
4532 Gus a o A. de And ade e al. / IFAC Pape sOnLine 50-1 (2017) 4528–4533
Wi h con ol law (21), he closed-loop ans e unc ion is
gi en by
Gcl(s)= G(s)
1−C(s)G(s),(22)
whe e C(s)=Kce−τcs.
Fo he s abili y esul we will use he ollowing necessa y
and su icien condi ion (Desoe and Vidyasaga , 1975).
Theo em 5. The closed-loop sys em is s able i and only i
(1) in
Re(s)>0
|1−C(s)G(s)|>0
(2) C(pi)=0,i=1,...,n
0, whe e pia e he poles
o Gin C+.
Condi ion 1 o Theo em 5 can be checked h ough he
g aphic Nyquis c i e ia. In ou case, he open-loop is non
s ic ly p ope , and he applica ion o he g aphic Nyquis
c i e ia is mo e delica e (Desoe and Vidyasaga , 1975).
Condi ion 2 o he p e ious heo em co esponds o a
condi ion p e en ing an ins abili y due o he cancella ion
o an uns able pole o Gby a ze o o C.
Since he p oposed con ol law (21) is no s ic ly p ope ,
we ha e o ake in o accoun he beha io o he Nyquis
plo a in ini y. We p opose below a way o ci cum en
100 200 300 400 500 600 700 800 900
-40
-20
0
20
40
Magni ude (db)
100 200 300 400 500 600 700 800 900
F equency (Hz)
-1260
-900
-540
-180
Phase (0)
(a) Open-loop equency esponse o model (17).
100 200 300 400 500 600 700 800 900
-40
-20
0
20
40
Magni ude (dB)
100 200 300 400 500 600 700 800 900
F equency (Hz)
-1620
-1260
-900
-540
-180
Phase (
o
)
(b) Open-loop equency esponse ob ained by applying
a sine sweep, o e he ange 100 −900 Hz, in o he eal
expe imen al plan .
Fig. 3. Open-loop equency esponse o model (17) and
he expe imen al plan .
his p oblem by analyzing he closed-loop poles o high
equencies. Fi s , conside ha he sys em ul ills he
ollowing assump ion
Assump ion 6. The se
={xu+xd, cτc+xd+xmic, cτc+3xd−xmic},
is a ionally independen 1.
Then, using Theo em 2.2 and Co olla y 2.4 o Hale and
Lunel (2002) we ha e he ollowing necessa y condi ion o
s abili y:
P oposi ion 7. Le τc,x
mic >0 such ha
=(xu+xd, cτd+xd+xmic, cτc+3xd−xmic),
is a ionally independen . Then, he ollowing inequali y is
a necessa y condi ion o closed-loop s abili y:
|Kc|(1 + |Rd|)<1−RdRu(23)
P oo . Fo |s|0, he closed-loop poles can be app oxi-
ma ed as he solu ion o
1−RuRde−(τu+τd)s−Kce−cτc+xd+xmic
cs
−KcRde−cτc+3xd−xmic
cs=0.(24)
Then, choosing τc,x
mic >0 such ha
=(xu+xd, cτd+xd+xmic, cτc+3xd−xmic),
is a ionally independen we can apply Theo em 2.2 and
Co olla y 2.4 o (Hale and Lunel, 2002) o ob ain he
inequali y (23). This concludes he p oo .
Using his esul we can es ic he es o he Nyquis
c i e ion o a ini e ange o equencies, as s a ed in he
co olla y bellow.
Co olla y 8. I P oposi ion 7 is e i ied, hen he e exis s
s0>0 such ha condi ion 1 o Theo em 5 needs only be
es ed on a ini e ange |s|<s
0.
The e sion o he Nyquis heo em which accomoda es in-
ini e dimensional sys ems can be seen in Theo em A.1.14
o Cu ain and Zwa (1995). In he nex p oposi ion we
show he condi ions ha he Nyquis con ou mus obey
in his ini e ange o equencies in o de o gua an ee he
s abili y o he closed loop sys em.
P oposi ion 9. Le nube he numbe o open-loop poles o
(17) in C+. Deno e he Nyquis con ou o
Ψ(s)1+2Kce−(cτc+xu+xmic
c)s
(γ−1) de (S)(1 + Rde2xmic −xd
cs)×
(1 + (γ−1)φ(s)) (25)
by ΓΨ(s). Then he ans e unc ion Gcl is
(1) Uns able i ΓΨ(s)does no enci cles he o igin nu
imes in he clockwise di ec ion.
(2) S able i ΓΨ(s)enci cles he o igin nu imes in he
clockwise di ec ion.
In he limi ing case ha ΓΨ(s)does no enci cle bu c osses
−1, he s abili y is unde e mined.
1We say ha he eal numbe s a1, ..., a
na e a ionally indepen-
den i he only n- uple o in ege s k1, ..., k
nsuch ha k1a1+···+
knan= 0 is he i ial solu ion in which e e y ki,i=1, ...,n is
ze o.
P oceedings o he 20 h IFAC Wo ld Cong ess
Toulouse, F ance, July 9-14, 2017
4628
-4 -3 -2 -1 0 1 2
Real axis
-3
-2
-1
0
1
2
3
Imagina y axis
ω∈[0,+∞)ω∈(−∞,0]
Fig. 4. Nyquis plo o ΓΨ(s)showing wo clockwise en-
ci clemen s o he o igin as ωdec eases om +∞ o
−∞. This g aphic was ob ained expe imen ally on he
Rijke ube con igu a ion desc ibed in Sec ion 2.
We skip he p oo o P oposi ion 9 since i ollows om
s anda d a gumen s.
In sum, he necessa y and su icien condi ions de eloped
in his sec ion can be checked by he algeb aic equa ion
(23) and by choosing τcand xmic such ha Assump ion 6
and he Nyquis c i e ion a e sa is ied.
4. EXPERIMENTAL RESULTS
In his sec ion, we p esen esul s o expe imen s in he
Rijke ube con igu a ion desc ibed in Sec ion 2 wi h con-
ol law (21). We choose Kc=0.002 and τcwas designed
in o de o ΓΨ(s)enci cles he o igin wice in he clockwise
di ec ion since, as shown in Figu e 3(b), he sys em has a
pai o complex conjuga e poles in C+. The Nyquis plo
o ΓΨ(s) o τc=0.001 is depic ed in Figu e 4. The e a e
wo clockwise enci clemen s o he o igin as ωdec eases
om +∞ o −∞.
Figu e 5 shows he sound p essu e a he mic ophone
loca ion and he con ol signal wi h he con ol law (21).
A he beginning o he expe imen , no con olle is ac i e,
and he sys em is in he limi cycle. A =3.5 s he
con olle is ac i a ed. I can be no ed ha he oscilla ions
a e supp essed and he sys em emains in he ope a ing
poin . A = 12 s he con ol is deac i a ed and as
expec ed, he sys em comes back o he oscilla o y egime.
5. CONCLUSIONS
We ha e add essed he issue o bounda y s abiliza ion o
he moacous ic oscilla ions o he Rijke ube by a e-
quency domain app oach. We ha e used some p ope ies o
he sys em ans e unc ion o de i e necessa y and su i-
cien condi ions o inpu -ou pu s abili y o he bounda y
con olled sys em. Expe imen al esul s o a Rijke ube
p o o ype shows he e ec i eness o he app oach.
REFERENCES
Campos-Delgado, D.U., Schue mans, B.B.H., Zhou, K.M.,
Pasche ei , C.O., Galles ey, E.A., and Ponce , A. (2003).
0 2 4 6 8 10 12 14 16 18 20
-60
-40
-20
0
20
40
60
P essu e (Pa)
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-0.5
-0.25
0
0.25
0.5
Con ol signal (V)
con ol on
con ol o
Fig. 5. P essu e luc ua ions a he mic ophone loca ion
and con ol signal as a unc ion o ime. A =3.5
s he con olle is ac i ed. A = 12 s he con ol is
deac i a ed.
The moacous ic ins abili ies: modeling and con ol.
IEEE T ansac ions on Con ol Sys ems Technology,
11(4), 429–447.
Cu ain, R. and Mo is, K. (2009). T ans e unc ions o
dis ibu ed pa ame e sys ems: A u o ial. Au oma ica,
45, 1101–1116.
Cu ain, R.F. and Zwa , H.J. (1995). An in oduc ion
o in ini e-dimensional linea sys ems heo y. Sp inge
Ve lag, Be lin.
de And ade, G., Vazquez, R., and Pagano, D.J. (2016).
Bounda y eedback con ol o uns able he moacous ic
oscilla ions in Rijke ube. In P oceedings o he 2 h IFAC
Wo kshop on Con ol o Sys ems Go e ned by Pa ial
Di e en ial Equa ions.
Desoe , C.A. and Vidyasaga , M. (1975). Feedback sys-
ems: inpu ou pu p ope ies. Academic P ess, New
Yo k.
Eppe lein, J.P., Bamieh, B., and As om, J. (2015). The -
moacous ics and he Rijke ube: Expe imen s, iden i i-
ca ion and modeling. Con ol Sys ems Magazine, 35(2),
57–77.
E ans, L.C. (2010). Pa ial di e en ial equa ions. Ame i-
can Ma hema ical Socie y.
Hale, J.K. and Lunel, S.M.V. (2002). S ong s abiliza ion
o neu al unc ional di e en ial equa ions. IMA Jou nal
o Ma hema ical Con ol and In o ma ion, 19(1-2), 5–23.
Heckl, M.A. (1988). Ac i e con ol o he noise om a
Rijke ube. Jou nal o Sound and Vib a ion, 124(1),
117–133.
K s ic, M., K upadanam, A., and Jacobson, C. (1999).
Sel - uning con ol o a nonlinea model o combus ion
ins abili ies. IEEE T ansac ions on Con ol Sys ems
Technology, 7(4), 424–436.
Li ico, X. and F omion, V. (2009). Bounda y con ol o
hype bolic conse a ion laws using a equency domain
app oach. Au oma ica, 45, 647–656.
Mu ugappan, S., Acha ya, S., Allgood, D.C., Pa k, S.,
Annaswamy, A.M., and Ghoniem, A.F. (2003). Op i-
mal con ol o a swi l-s abilized sp ay combus o using
sys em iden i ica ion app oach. Combus ion Science and
Technology, 175, 55–81.
Rayleigh, J.W.S. (1945). The heo y o sound. Do e , New
Yo k.
P oceedings o he 20 h IFAC Wo ld Cong ess
Toulouse, F ance, July 9-14, 2017
4629

Gus a o A. de And ade e al. / IFAC Pape sOnLine 50-1 (2017) 4528–4533 4533
-4 -3 -2 -1 0 1 2
Real axis
-3
-2
-1
0
1
2
3
Imagina y axis
ω∈[0,+∞)ω∈(−∞,0]
Fig. 4. Nyquis plo o ΓΨ(s)showing wo clockwise en-
ci clemen s o he o igin as ωdec eases om +∞ o
−∞. This g aphic was ob ained expe imen ally on he
Rijke ube con igu a ion desc ibed in Sec ion 2.
We skip he p oo o P oposi ion 9 since i ollows om
s anda d a gumen s.
In sum, he necessa y and su icien condi ions de eloped
in his sec ion can be checked by he algeb aic equa ion
(23) and by choosing τcand xmic such ha Assump ion 6
and he Nyquis c i e ion a e sa is ied.
4. EXPERIMENTAL RESULTS
In his sec ion, we p esen esul s o expe imen s in he
Rijke ube con igu a ion desc ibed in Sec ion 2 wi h con-
ol law (21). We choose Kc=0.002 and τcwas designed
in o de o ΓΨ(s)enci cles he o igin wice in he clockwise
di ec ion since, as shown in Figu e 3(b), he sys em has a
pai o complex conjuga e poles in C+. The Nyquis plo
o ΓΨ(s) o τc=0.001 is depic ed in Figu e 4. The e a e
wo clockwise enci clemen s o he o igin as ωdec eases
om +∞ o −∞.
Figu e 5 shows he sound p essu e a he mic ophone
loca ion and he con ol signal wi h he con ol law (21).
A he beginning o he expe imen , no con olle is ac i e,
and he sys em is in he limi cycle. A =3.5 s he
con olle is ac i a ed. I can be no ed ha he oscilla ions
a e supp essed and he sys em emains in he ope a ing
poin . A = 12 s he con ol is deac i a ed and as
expec ed, he sys em comes back o he oscilla o y egime.
5. CONCLUSIONS
We ha e add essed he issue o bounda y s abiliza ion o
he moacous ic oscilla ions o he Rijke ube by a e-
quency domain app oach. We ha e used some p ope ies o
he sys em ans e unc ion o de i e necessa y and su i-
cien condi ions o inpu -ou pu s abili y o he bounda y
con olled sys em. Expe imen al esul s o a Rijke ube
p o o ype shows he e ec i eness o he app oach.
REFERENCES
Campos-Delgado, D.U., Schue mans, B.B.H., Zhou, K.M.,
Pasche ei , C.O., Galles ey, E.A., and Ponce , A. (2003).
0 2 4 6 8 10 12 14 16 18 20
-60
-40
-20
0
20
40
60
P essu e (Pa)
0 2 4 6 8 10 12 14 16 18 20
Time (s)
-0.5
-0.25
0
0.25
0.5
Con ol signal (V)
con ol on
con ol o
Fig. 5. P essu e luc ua ions a he mic ophone loca ion
and con ol signal as a unc ion o ime. A =3.5
s he con olle is ac i ed. A = 12 s he con ol is
deac i a ed.
The moacous ic ins abili ies: modeling and con ol.
IEEE T ansac ions on Con ol Sys ems Technology,
11(4), 429–447.
Cu ain, R. and Mo is, K. (2009). T ans e unc ions o
dis ibu ed pa ame e sys ems: A u o ial. Au oma ica,
45, 1101–1116.
Cu ain, R.F. and Zwa , H.J. (1995). An in oduc ion
o in ini e-dimensional linea sys ems heo y. Sp inge
Ve lag, Be lin.
de And ade, G., Vazquez, R., and Pagano, D.J. (2016).
Bounda y eedback con ol o uns able he moacous ic
oscilla ions in Rijke ube. In P oceedings o he 2 h IFAC
Wo kshop on Con ol o Sys ems Go e ned by Pa ial
Di e en ial Equa ions.
Desoe , C.A. and Vidyasaga , M. (1975). Feedback sys-
ems: inpu ou pu p ope ies. Academic P ess, New
Yo k.
Eppe lein, J.P., Bamieh, B., and As om, J. (2015). The -
moacous ics and he Rijke ube: Expe imen s, iden i i-
ca ion and modeling. Con ol Sys ems Magazine, 35(2),
57–77.
E ans, L.C. (2010). Pa ial di e en ial equa ions. Ame i-
can Ma hema ical Socie y.
Hale, J.K. and Lunel, S.M.V. (2002). S ong s abiliza ion
o neu al unc ional di e en ial equa ions. IMA Jou nal
o Ma hema ical Con ol and In o ma ion, 19(1-2), 5–23.
Heckl, M.A. (1988). Ac i e con ol o he noise om a
Rijke ube. Jou nal o Sound and Vib a ion, 124(1),
117–133.
K s ic, M., K upadanam, A., and Jacobson, C. (1999).
Sel - uning con ol o a nonlinea model o combus ion
ins abili ies. IEEE T ansac ions on Con ol Sys ems
Technology, 7(4), 424–436.
Li ico, X. and F omion, V. (2009). Bounda y con ol o
hype bolic conse a ion laws using a equency domain
app oach. Au oma ica, 45, 647–656.
Mu ugappan, S., Acha ya, S., Allgood, D.C., Pa k, S.,
Annaswamy, A.M., and Ghoniem, A.F. (2003). Op i-
mal con ol o a swi l-s abilized sp ay combus o using
sys em iden i ica ion app oach. Combus ion Science and
Technology, 175, 55–81.
Rayleigh, J.W.S. (1945). The heo y o sound. Do e , New
Yo k.
P oceedings o he 20 h IFAC Wo ld Cong ess
Toulouse, F ance, July 9-14, 2017
4629