Linear identification of a servo-pneumatic system

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. 2 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. 3 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. 4 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. DOI: 10.17667/ iim.2015.1-2/7. 5 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. 6 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. 7 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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