Recen Inno a ions in Mecha onics (RIiM) Vol. 2. (2015). No. 1-2.
DOI: 10.17667/ iim.2015.1-2/7.
1
Linea iden i ica ion
o a se o-pneuma ic sys em
Széll Ká oly
Depa men o Mecha onics, Op ics and Enginee ing
In o ma ics
Budapes Uni e si y o Technology and Economics
Budapes , Hunga y
[email p o ec ed]me.hu
Czme k And ás
Depa men o Mecha onics, Op ics and Enginee ing
In o ma ics
Budapes Uni e si y o Technology and Economics
Budapes , Hunga y
Abs ac —The iden i ica ion o a nonlinea sys em is qui e
challenging o enginee s. This pape p esen s he au oma ic
iden i ica ion o a se o-pneuma ic cylinde based on a
amewo k implemen ed in MATLAB. The in oduced
applica ion sho ens he p ocess leng h o iden i ica ion and gi es
a e e ence model, impo an o con olling.
Keywo ds—se o-pneuma iccylinde ; iden i ica ion; s a e space
I. INTRODUCTION
I is a common ask in enginee ing p ac ice, o iden i y he
pa ame e s o an exis ing de ice wi hou o wi h only li le
in o ma ion abou he sys em. The mo e p ecise model we
need, he mo e ime his p ocess equi es. Ne e heless
nowadays he e is always a p essu e onde elope s o be he
i s on he ma ke , o ha e esul s wi h sho e deadlines. The
lack o ime claims such as solu ions like he sys em
iden i ica ion whe e he complex sys em models a e no
needed.
The de i a ion o an abs ac model using con en ional
analysis has he ollowing s eps:
1. Choosing he pa ame e s which bes desc ibe he
sys em.
2. Building an abs ac model ep esen ing he eal sys em.
3. Analysis o he esul s and e inemen o he model.
4. Gene aliza ion o he esul s and de e mining
co ela ions.
The mos p oblema ic equi emen o he s eps abo e is o
de ine he needed accu acy. The mo e accu a e model we need,
he mo e pa ame e s and a iables a e needed o he building
o he model and he less physical phenomenon ha we can
neglec . A pe ec model can be bene icial, bu i also has o be
aken in o accoun whe he an app op ia e ha dwa e- esou ce is
a ailable. The iden i ica ion and he calcula ions a e wa ds
may ake longe ime han easonable.
The e a e h ee di e en g oups o he sys em-models:
1. Homolog model: scaled-down e sion o he eal
sys em using a ini y laws.
2. Analog model: he applied physical phenomenon di e s
om he modelled phenomenon bu he ou pu s o bo h
sys ems a e he same o he same inpu s (e.g.
subs i u ion o a pneuma ic sys em wi h an elec ical
ci cui )
3. Ma hema ical model: desc ip ion o he physical
beha io o he eal sys em by equa ions.
Due o he de elopmen s in compu ing he applica ion o
ma hema ical models became also widesp ead. I is a as and
ela i ely cheap solu ion o pe o m analy ical asks. The main
poin o his me hod is o de ine a ma hema ical model which
desc ibes he beha io o he eal sys em and which is as and
accu a e enough om con olling poin o iew. One
possibili y is o desc ibe he physical phenomena di ec ly by
equa ions. In his case deepe knowledge is needed o ind he
mos app op ia e o mula and pa ame e s.
Ano he possibili y o de ine he ma hema ical model is he
sys em iden i ica ion, when he model is buil up based on an
inpu signal sequence and he belonging ou pu signal
sequence. The esul is a linea ized empi ical model. This
p ocess does no equi e so deep knowledge o he physical
backg ound o he sys em. The mos impo an poin o be
de ined is he numbe o physical elemen s ha can s o e
ene gy and hus he cha ac e is ic o he expec ed unc ions.
This me hod gi es as esul s bu i has o be handled a bi
scep ic. This model loses he con ac o he eal sys em and can
only be applied in he linea ized domain. Ou side o he
linea ized domain he esponse o he eal sys em and he
esponse o he abs ac model de ined by sys em
iden i ica ionmigh be di e en . The main disad an age o he
me hod is ha we ha e no in o ma ion in he abs ac model
abou he physical backg ound. I is no possible o change a
componen o he eal sys em, and implemen his modi ica ion
in ou ma hema ical model by eplacing some pa ame e s, bu
he whole sys em iden i ica ion p ocess has o be epea ed o
any change in he eal sys em. This kind o abs ac ion canno
highligh he physical ela ionship o he elemen s o ou
expe imen al se up.
Recen Inno a ions in Mecha onics (RIiM) Vol. 2. (2015). No. 1-2.
DOI: 10.17667/ iim.2015.1-2/7.
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II. EXPERIMENTAL SETUP
The expe imen al se up is a se o-pneuma ic sys em (see
Fig. 1.). The ha dwa e elemen s used du ing he iden i ica ion
a e lis ed in Table I. The aim o he expe imen is o se he
posi ion o he ca iage o he odless pneuma ic cylinde wi h
he help o wo 3/3 p opo ional al es (which in his case a e
used as one 5/3 p opo ional al e since hey a e exci ed
simul aneously and in e sely). The posi ion is sensed by an
op ical inc emen al encode wi h 5 [µm] accu acy. P ocessing
o he measu ed da a, calcula ion and ealiza ion o he con ol
signals a e done by a 16-bi DSP mic ocon olle . Due o he
di e en ol age le els an in e ace ci cui is also needed. This
in e ace ci cui handles he sinusoidal signals o he
inc emen al encode and supplies he quad a u e module o he
DSP wi h digi al signals ha can be p ocessed wi h he needed
equency. The in e ace ci cui is also esponsible o he
ol age le el shi be ween he ou pu o he mic ocon olle
(3.3 [V]) and he con ol signals o he p opo ional al es (10
[V]). The p ocessed da a is sen by UART communica ion
om he mic ocon olle o he PC whe e aMATLAB based
g aphical use in e ace (Fig. 2.) logs he da a o pos -
p ocessing.
This expe imen al se up allows he au oma ic iden i ica ion
o he odless pneuma ic cylinde which is discussed in chap e
IV, bu be o e ha i is p ac ical o analyze he elemen s o he
sys em. Exac pa ame e s and unc ions a e no needed, only
he main cha ac e is ics o he sys em o ha e an idea abou he
o de o he sys em model. The necessa y linea ized model is
desc ibed in he nex chap e .
Fig. 1.Expe imen al se up
TABLE I. ELEMENTS OF THE EXPERIMENTAL SETUP
N .
Elemen
Type
1
P essu e egula o
HOERBIGER SFRL-1/4
2
Inc emen al linea encode
Mi u oyo AT112
3
Pneuma ic cylinde
HOERBIGER P210-20 Ø32 700
4
P essu e senso
FESTO SDE-10-10V/20mA
5
3/3 p opo ional al e
HOERBIGER 94701
6
Mic ocon olle de elopmen boa d
DM240001 (Explo e 16)
7
PIC ail Plus
AC164126
8
USB o se ial con e e
PL-2303 USB o RS232
9
In e ace ci cui
10
Powe supply
VOLTCRAFT TNG35
III. LINEARIZED STRUCTURE
A se o-pneuma ic sys em is nonlinea and ime- a ian [1-
5] while ou iden i ica ion p ocess will esul in a linea
ma hema ical model, which has only li le ela ionship o he
physical beha io o he o iginal eal sys em. Ou goal in his
chap e is o ind he main physical phenomena which de ine
he minimal o de o he sys em model o he needed
accu acy.
Recen Inno a ions in Mecha onics (RIiM) Vol. 2. (2015). No. 1-2.
DOI: 10.17667/ iim.2015.1-2/7.
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The posi ioning o he sys em depends on he p essu es in
he le and igh chambe s. The ca iage is pushed by he
esul an o ce which is he di e ence o he p essu es ac ing
on he le and igh su aces o he pis on. The dynamic o he
sys em is depending u he mo e on he mass o he ca iage
and he ic ion be ween he pis on and he wall o he cylinde
[6-8]. The la e is a complex nonlinea phenomenon [9-10].
The S ibeck ic ion model is applied in he analyzed model
bu i is no discussed in de ails as he aim o he pape is a
linea model.Based on hese conside a ions he mo emen o
he ca iage can be desc ibed by a second o de di e en ial
equa ion (1).
m
- mass
A
- su ace o he pis on
x
- pis on posi ion
F
- ic ion o ce
a
p
- p essu e o he le chambe
b
p
- p essu e o he igh chambe
C
F
- Coulomb ic ion coe icien
V
F
- iscous ic ion coe icien
S
F
- S ibeck ic ion o ce

- hea coe icien a io
i
m

- mass low
R
- speci ic gas cons an o ai
T
- chambe empe a u e
i
g
- coe icien o he linea ized a iable
x
- posi ion o he al e body

- damping coe icien o he al e

- na u al equency
A
- opening c oss-sec ion
u
- inpu ol age o he al e
0 ba FApApxm 
(1)
, whe e
   
xxFxFxsignFF S VC  ,
(2)
The di e en ial equa ion can be in e p e ed as a balancing
o he p essu es in he chambe s. Fo example i a mo emen o
he igh di ec ion is needed hen he p essu e in he le
chambe has o o e come he p essu e o he igh chambe and
also he ic ion o ce. The su plus o ce accele a es he
ca iage making he posi ioning possible. The linea ized
ela ionship based on hese conside a ions can be seen in
equa ion (3).
xFppAxm Vba   )(
(3)
As hese phenomena de ine he mos impo an beha io o
ou expe imen al se up, he oo s o his equa ion a e expec ed
o be he dominan oo s o ou model. Depending on he
damping, hese oo s a e eal oo s o complex conjuga e
pai swhich o m is non i ial acco ding o he nonlinea sys em
model.
The chambe p essu e i sel is also a unc ion o se e al
a iables. The p essu e migh be changed in h ee ways. By
changing he amoun o ai in he chambe wi h he help o he
p opo ional al e. By changing he olume o he chambe
when he pis on is mo ed. By changing he empe a u e o he
chambe . Du ing his expe imen he la e one is neglec ed,
due o he di e ence o ime cons an s o mechanical and
he mal p ocesses. Thus he chambe p essu es can be
desc ibed by he di e en ial equa ions (4) and (5).
 
xA
pxAmTR
paaa
a





(4)
 
xA
pxAmTR
pbbb
b





(5)
The linea ized o m o he di e en ial equa ions as a
unc ion o he amoun o subs ance, he p essu e and he
posi ion o he pis on can be seen in equa ions (6) and (7).
xgpgxgp xaapa maa~~~ 

(6)
xgpgxgp xbapb mbb~~~ 

(7)
The mass low o he ai depends on he opening c oss-
sec ion and he low ac o o he p opo ional al e, which is
also a complex nonlinea unc ion. Since his co ela ion is no
a di e en ial equa ion, om he poin o iew o ou deduc ion,
i is i ele an .
The mass lows a e con olled by wo independen
p opo ional al es. In he case o ou expe imen al se up he
al es a e handled as one 5/3 p opo ional al e as hey a e
ac ua ed he same ime bu in e sely. I sdynamic beha io can
be desc ibed by a mass-sp ing model. This elemen is no likely
o gi e he dominan conjuga e pai s and hus we do no go in o
u he de ails. The linea ized o m can be seen in equa ion (8).
uAxxx  22
2


(8)
As summa y o he conside a ions abo e, he se o-
pneuma ic cylinde can be desc ibed by a second o de
di e en ial equa ion o he ca iage mo emen (3), wo i s
o de di e en ial equa ions o he chambe p essu es (6),(7)
and a second o de di e en ial equa ion o he p opo ional
al es (8). I can be concluded ha o he iden i ica ion a
leas a six h o de s a e space model is necessa y.
Recen Inno a ions in Mecha onics (RIiM) Vol. 2. (2015). No. 1-2.
DOI: 10.17667/ iim.2015.1-2/7.
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Fig. 2. MATLAB use in e ace
IV. PREPROCESSING OF THE MEASUREMENT DATA
Du ing iden i ica ion he shape and ype o he exci a ion
signal, applied on he sys em‟s inpu has
de e mina i ein luences. The ideal exci a ion signal is he so-
called pe sis en signal, which can be any o he con en ional
es signals, o example:
 impulse
 s ep signal
 whi e noise
 sum o sin signals
The measu emen accu acy can be imp o ed, i he numbe
o samples pe ime uni is inc eased; also bydec easing he
ime delay be ween he samples.Di e en signal ypes can
achie e he same measu emen accu acy,achie ed wi h
di e en sampling ime. In ou case he da a was sampled wi h
100 [Hz], also a sample was aken in e e y 10 h [ms], so based
on he Shannon- heo em, he use ul equency ange is 50
[Hz]. Signal“da a1” was used o he iden i ica ion, while
da a2 o he e i ica ion(Fig. 3.).
The squa e signal was chosen, because i s Fou ie
ans o m is ich in ha monics, and p o ides he exci a ion o
he mos sys em na u al equencies. The esul , based on he
ou pu signal, gi es a p elimina y o eknowledge i he e a e
u he dominan pole/pole pai s nea o he sys em‟s dominan
poles. As nex s ep i we neglec he poles ha a e a leas
h ee imes as e han he sys em‟s dominan pole/ pole pai s,
we do no make a big mis ake because he e ec s o hese
poles die down as e han he sys em‟s se ling ime.
Fo he pe iodic signals o e he window leng h, he
disc e e Fou ie ans o m gene a es he complex Fou ie
coe icien s. Thesqua es o absolu e alues o hese
coe icien s gi e he signal‟s powe densi y. The squa e o he
disc e e Fou ie ans o m is called he pe iodog am. A
pe iodog am also can be used o es ima e he spec um o a
s ochas ic signal.
Fig. 3. Val e con ol signal and pis on posi ion
Be o e iden i ica ion, du ing he p epa a ion o he
measu ed da a, he ollowing s eps a enecessa y:
 il e ing highe equency noise
 emo ing measu emen e o s
 emo ing DC componen om he signal
As i can be seen on he measu emen esul , he abo e
men ioned p epa a ion s eps can be omi ed, because he
measu ed signals a e clean om noise, u he mo e he
s a ing posi ion was 0 [m] hanks o he ini ializa ion s eps,
implemen ed on he mic ocon olle .
Fo da a eco ding and iden i ica ion, a dedica ed MATLAB
g aphical use in e ace was c ea ed and used (Fig. 2.). The
mul i- unc ion in e ace is implemen edwi h he pu pose o
easing he u he wo k wi h he di e en con ol me hods.
Wi h he “Re e ence posi ion” slide on he “Manual con ol”
ab, we can se he PWM du y a io o he p opo ional al e‟s
con ol signal.Choosing “Sys em iden i ica ion” adio bu on
on he men ioned in e ace ab, a 6 h o de s a e space model
iden i ica ion uns au oma ically a e collec ing iden i ica ion
da a.The accu acy o he iden i ied model is highligh ed on he
same panel igh a e he p ocess is done. Wi h he “LTI iew”
bu on, he model can be analyzed in he ime and equency
domain.
V. IDENTIFICATION
Du ing iden i ica ion a cos unc ion is de ined, which is
he di e ence be ween he eal sys em and he ou pu o he
de e mined model, and i is used o op imize he model
pa ame e s o each he minimum o his unc ion.Wi h o he
wo ds, we wan om he model‟s ou pu o app oxima e he
eal sys em‟s beha io accu a ely. This can be eached by he
op imiza ion o he model‟s polynomials, which is possible
wi h he leas squa e me hod, nume ic op imum sea ching
algo i hms o wi h hese combina ions.
Recen Inno a ions in Mecha onics (RIiM) Vol. 2. (2015). No. 1-2.
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Fig. 4. Es ima ed spec ums o he exci a ion signal and pis on posi ion
The s uc u e o he modeldoes no ha e o be he same
as hes uc u eo he es sys em, bu has o show he same
beha io ; o he wise, i canno eplace he sys em du ing
analysis.
Iden i ica ion esul s ob ained om disc e e- ime linea
models a e expec ed o belong o linea , sampled con inuous-
ime sys ems.
I 𝑠𝑖is he pole o he con inuous- ime sys em, hen i has o
be mapped in he 𝑧𝑖=𝑒𝑠𝑖𝑇 pole in he sampled sys em‟s
disc e e- ime ans e unc ion. This means, i 𝑧𝑖 on he
nega i e eal axes is pole o he iden i ied model and he
mul iplici y is e en, han i canno belong o a con inuous- ime
linea sys em, because i s 𝑠𝑖=ln 𝑧𝑖
𝑇 complex poles can appea
only oge he wi h i s 𝑠 𝑖 complex conjuga ed pai s.
Fig. 5.Simula ion esul s o he iden i ied models
Fig. 6.E o s o he iden i ied models
Model accu acy based on simula ion:
 4 h o de s a e space (n4s4): 95.83%
 2 h o de s a e space (n4s2): 95.06%
 6 h o de s a e space (n4s6): 93.98%

Recen Inno a ions in Mecha onics (RIiM) Vol. 2. (2015). No. 1-2.
DOI: 10.17667/ iim.2015.1-2/7.
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Fig. 7. 20 s eps p edic ion o he iden i ied models
Fig. 8. E o o he 20 s eps p edic ion o he iden i ied models
Modell accu acy based on 20 s eps p edic ion:
 4 h o de s a e space (n4s4): 94.41%
 6 h o de s a e space (n4s6): 94.57%
 2 h o de s a e space (n4s2): 93.19%
F om he iden i ied sys em models, we go he bes esul s
wi h he linea ime in a ian s a e space models, which
s uc u e can be w i en in he o m(9)-(10).
𝒙
𝑡 =𝑨𝒙 𝑡 +𝑩𝒖 𝑡 +𝑲𝒆 𝑡 (9)
𝒚 𝑡 =𝑪𝒙 𝑡 +𝑫𝒖 𝑡 +𝒆 𝑡 , (10)
whe e 𝑨.𝑩.𝑪.𝑫a e he s a e ma ices, Kis he noise
ma ix, 𝒖 𝑡 is he inpu , 𝒚 𝑡 is he ou pu , 𝒙 𝑡 is hen ho de
s a e ec o and 𝒆 𝑡 is he noise ec o .
The ee unable pa ame e s o he sys em a e he noise and
he s a e ma ices excep he D ma ix, because i is assumed,
ha he e is no eed o wa d in he sys em, so can be neglec ed.
Based on he esul s he sys em can be bes desc ibed by a
4 h o de s a e space model. The de e mined model is
accep able, i he emaining e o does no con ain any
s uc u e, also i is ee o any pa e n, and he e is no any
co ela ion wi h he inpu o ou pu :
 he a e age o whi e noise is null
 he p ocess has Gaussian-dis ibu ion
 he elemen s o he e o se ies and he p e ious inpu
alues do no co ela e
The au oco ela ion and he c oss co ela ion es (Fig. 9.)
can be used o he examina ion o he esidual e o , in which
he au oco ela ion o he esidual e o and he c oss
co ela ion o he inpu signal and he e o signal is es ima ed.
The mo e simila i y he emaining e o has o he whi e noise,
he mo e he simila i y be ween he au oco ela ion and he
impulse unc ion is.The c oss co ela ion unc ion is used o
analyze he ela ionship be ween he inpu signal and he
esidual e o . Wi h a su icien ly accu a e model he e is no
any co ela ion be ween he inpu and he e o unc ion, also
he alue o he c oss-co ela ion unc ion is nea o null by any
delay.
Fig. 9.Au o and c oss co ela ion be ween he signals
Recen Inno a ions in Mecha onics (RIiM) Vol. 2. (2015). No. 1-2.
DOI: 10.17667/ iim.2015.1-2/7.
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The ze os and he poles belonging o he iden i ied models
can be seen in Fig. 10. No e ha he e a e no any nega i e eal
poles wi h he mul iplici y o one, so he models can belong o
a eal sys em.No e ha all he models con ain an in eg a o , as
i was expec ed a e he eal sys em‟s s uc u e (zi=1).
Fig. 10.Ze o, pole map o he iden i iac ed models
Fig. 11.Bode diag ams o he iden i ica ed models
VI. CONCLUSION
The pape discussed he de ini ion o he ans e unc ion
o a se o-pneuma ic cylinde using wo di e en me hods.
Fi s he analy ical way and hen he au oma ic sys em
iden i ica ion based on measu emen da a. Based on he esul s
he ou h o de s a e space model has p o en o be he mos
app op ia e o desc ibe he physical beha io o he eal
sys em.
Due o he sys em iden i ica ion me hod a as esul could
be ob ained ha is accu a e enough o apply in simula ions,
u ilize in con olle de elopmen o as a e e ence model.
Applying he model ob ained by sys em iden i ica ion, a obus
s a e space con olle wi h load es ima ion had been
implemen ed ha can p ope ly posi ion he eal sys em.
ACKNOWLEDGMENT
The au ho s wish o hank he suppo o he Hunga ian
Au omo i e Technicians Educa ion Founda ion, o he
Hunga ian Resea ch Fund (OTKA K100951), and he Con ol
Resea ch G oup o HAS. The esul s discussed abo e a e
suppo ed by he g an TÁMOP-4.2.2.B-10/1-- 2010-0009.
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