Sliding Mode Flow Rate Observer Design

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Alexandrina Wade
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1 Sliding Mod Flow Rat Obsrvr Dsign Song Liu and Bin Yao School of Mchanical Enginring, Purdu Univrsity, Wst Lafaytt, IN797, USA Abstract Dynamic flow rat information is ndd in a lot of applications; howvr it is normally not masurabl Calculation of th flow rat from fluid prssur dynamic quations usually rsults in poor stimats du to th vry noisy prssur masurmnts and unavoidabl modlling uncrtaintis This papr proposs a sliding mod dynamic flow rat obsrvr to stimat th mtr-in and mtr-out flow rats of a hydraulic cylindr Thortical convrgnc can b guarantd and simulation and xprimntal rsults show th ffctivnss of th proposd flow rat obsrvr Kywords Dynamic flow rat, sliding mod obsrvr Introduction Dynamic flow rat information is ndd in a lot of applications, such as automatd modlling of valv flow mapping, fault dtction, diagnostics and prognostics Howvr, with a convntional flow mtr which is only capabl of masuring static flow rat, th dynamic flow rat is not masurabl In ordr to obtain th flow rat information, on must sk othr approachs to avoid th dmand of dirct masurmnt In hydraulic systms activatd by hydraulic cylindrs, th flow rat is dirctly rlatd to th cylindr prssur dynamics and th motion of th cylindr rod Th cylindr prssur dynamics can b writtn as [] V P = Ax + Q + β () V P = A x Q + β whr x is th cylindr rod displacmnt, A and A ar th ram aras of th cylindr, P and P rprsnt prssurs at th two chambrs, Q and Q stand for th mtr-in and mtr-out flow rats rspctivly, β is th ffctiv bulk modulus, and V (x) and V (x) ar th total cylindr volums of th had and rod nds including conncting hos volums, and rprsnt th lumpd disturbanc flows that ar not associatd with th control valvs (i, th cylindr intrnal and xtrnal lakag flows, and so on) It is assumd that: a th cylindr rod position x and th vlocity x as wll as th prssurs P and P ar availabl, b th ffctiv bulk modulus β dos not chang or changs slowly, th actual valu of β may not b known but th bounds β min and β max ar known c th lumpd disturbanc flows and ar boundd by som constant known bounds δ and δ, rspctivly From () on may think to calculat flow rats from th prssur dynamic quations, g, V Q = Ax + P () β Though th prssur P is masurabl, it is vry noisy It is not practical to numrically diffrntiat P unlss a low pass filtr is usd, which would introduc phas lag Th accuracy of th calculation basd on () also dpnds on th accuracy of th ffctiv bulk modulus β, which is usually of larg varianc and vry hard to b dtrmind xactly Th apparanc of th disturbanc flow furthr complicats th problm A flow rat obsrvr may b a solution to this problm Howvr, bcaus th flow rat Q i, i= or, is not a stat of th systm dynamics, Lunbrgr obsrvr dos not work in this cas vn whn all paramtrs ar known and th disturbanc flow is ngligibl In ordr to dsign th flow rat obsrvr, on must ovrcom th following difficultis: a Th systm dynamics is not linar du to th chang in total cylindr volums V (x) and V (x) b Som systm paramtrs, such as β may not b xactly known c Th flow rat Q and Q ar inputs to th systm instad of bing systm stats This papr proposs a sliding mod flow rat obsrvr to ovrcom th nonlinarity and modl uncrtaintis

2 [,3,] Th proposd sliding mod flow rat obsrvr has strong thortical prformanc and robustnss It guarants convrgnc in finit tim whn thr is no modl uncrtaintis and boundd stimation rror in th prsnc of modlling rror Sliding Mod Flow Rat Obsrvr Sinc th obsrvrs for Q and Q ar basically sam, only th on for Q would b dsignd hr, th othr can b workd out in th sam way Sliding mod flow rat obsrvr dsign Lt θ =, th prssur dynamics () can b β rwrittn as V θ P = Ax + Q + (3) Th flow rat obsrvr, basd on th cylindr prssur dynamics (3), is givn by ˆ V θ Pˆ = Ax + Qˆ K( Pˆ P) () whr ˆP, ˆQ and ˆ θ rprsnt th stimats of P, Q and θ, rspctivly, K is a positiv obsrvr gain Subtract (3) from (), on can obtain: ( Pˆ P ) ( ˆ ) V ˆ θ + V θ θ P = Qˆ Q K( Pˆ P) (5) Dfin = Pˆ P as th prssur stimation rror ˆ as th paramtr stimation rror, and θ = θ θ rwrit (5) in trm of, V ˆ θ + K= Qˆ Q V θ P (6) Whn paramtrs lik th bulk modulus β is known and th disturbanc flow is ngligibl (i, assum = ), Eq (6) can b simplifid into: V ˆ θ + K= Qˆ Q (7) Sinc th mtrd flow rat Q must go through th valv orific, Q is always limitd by th valv or pump capability Mathmatically, Q is boundd by a known bound, i, Q Qmax In (7), ˆQ can b considrd as an input to th prssur stimation rror dynamics Thrfor, on may want to choos a propr ˆQ to mak th stimation rror (t) convrg to zro A discontinuous ˆQ is chosn as Qˆ = Q sign() (8) max Not that during th sliding mod whn =, mathmatically, (8) is dfind by th so-calld quivalnt control [,3] and can b approximatd obtaind by applying a low pass filtr to th discontinuous ˆQ : ˆ ˆ Q + ξω Q + ω Qˆ = Qˆ q n q n q whr ξ and ω n ar th damping ratio and natural frquncy of th filtr Thorm: Assuming paramtrs ar known and th disturbanc flow is ngligibl, th flow rat obsrvr () and (8) guarants that th prssur stimation rror (t) rachs zro in finit tim and, subsquntly, th quivalnt flow stimat Q ˆ q convrgs to th actual valv flow Q Proof: Dfin a positiv dfinit function V = V ˆ θ, whr V = V + Ax, V is th cylindr volum including hos and fitting volums whn th cylindr rod is fully rtractd, i, x = Diffrntiating V and noting (7), on can obtain: V = V ˆ θ + V ˆ θ ˆ = Ax θ K + ( Qˆ Q) ˆ = ( K Ax θ) ( Qmaxsign( ) + Q) (9) () Sinc th cylindr rod vlocity is always boundd du to finit flow capacity of any pump, it is asy to choos a K gratr than ˆ Axθ Thrfor th drivativ of V satisfis:

3 V < ( Q sign( ) + Q ) < η () max for som positiv η Thrfor, (t) rachs zro in finit tim and stays at zro Onc (t) is qual to zro, ˆP P as wll as ˆP P would b zro Hnc Eq (7) indicats th quivalnt flow stimat Q ˆ q would convrg to Q This nds th proof Robust prformanc analysis Whn thr is paramtric uncrtainty θ or th disturbanc flow in Eq (6), th drivativ of V would chang corrspondingly from () to th following: V = ( K Ax ˆ θ) ( Qmaxsign( ) + Q+ V θp + ) as () Sinc both V θ P and ar boundd, as long Q max is chosn larg nough to dominat Q as wll as V P θ and, th drivativ of V still satisfis: V < ( Qmax sign( ) + Q + V θp + ) < µ (3) for som positiv µ It is obvious that th prssur stimation rror still rachs zro in finit tim Aftr that, during th sliding mod whn = and =, from (6), th quivalnt flow stimat is Qˆ = Q + V θ P + () q Thus, thr will b a boundd valv flow stimation rror ˆ ( ) that dpnds on th lvl of of Q q Q = V x θp+ paramtric uncrtaintis and th disturbanc flow Howvr, th convrging rat of th valv flow stimation is similar to th prvious cas whn no modl uncrtaintis xist as sn from (3) whn compard to () 3 Rmarks Th thortically xcllnt robustnss of sliding mod control is prsrvd in th sliding mod obsrvr Th two main drawbacks of sliding mod control, i, larg control authority and control chattring, do not limit th sliding obsrvr s practical application This is du to th fact that th chattring and larg control authority issus in th sliding mod obsrvr dsign ar only linkd to numrical implmntation rathr than hard mchanical limitations [3] Th quivalnt flow stimat Q ˆ q is obtaind by snding ˆQ through a low pass filtr (LPF) to rmov th high frquncy chattring componnts and kp th ffctiv low frquncy componnts Th LPF usd in th sliding mod obsrvr has a diffrnt ffct from th on usd to filtr prssur masurmnts in () Th LPF usd in flow rat calculation is to smooth th masurmnt nois, which is almost whit and covrs th ntir frquncy rang from DC to Nyquist frquncy Th cut off frquncy of that LPF must b chosn vry low to mak th prssur diffrntiation possibl On th othr hand, th LPF usd to obtain quivalnt Q ˆ q is to rmov th chattring in ˆQ, whos frquncy mainly dpnds on th obsrvr gain K, th magnitud of th discontinuous trm Q max and th frquncy of digital implmntation Th largr K, Q max and digital implmntation frquncy ar, th highr th chattring frquncy is Practically, th obsrvr can run at a highr sampling frquncy than th control systm sampling frquncy, which would nabl th chatting at a frquncy vn highr than th control systm sampling frquncy Hnc LPF in th obsrvr dsign can hav a much highr cutoff frquncy and dos not introduc svr phas lag to th flow rat stimation 3 Simulations and Exprimnts Simulations ar don in Simulink A simpl PI controllr is usd to control a singl-rod doubl-acting hydraulic cylindr with a srvo valv, whos dynamics is nglctd Th cylindr rod is controlld to track a point-to-point rfrnc trajctory Whit nois is addd to prssur masurmnts

4 Th cylindr ram aras A and A ar m and m, rspctivly Th supply and tank prssurs ar st as 69KPa (PSI) and KPa, rspctivly Th ffctiv bulk modulus β is qual to 8 Th cylindr intrnal lakag is addd as th disturbanc flow: = K P P sign( P P ) (5) lak whr K lak is chosn to b -8 Th flow obsrvr paramtrs K and Q max ar st as K=- and Q max =-m 3 /s A scond ordr LPF with cut off frquncy qual to Hz is usd to rmov th chattring in ˆQ convrg to th actual valus vry wll without any phas lag, as prdictd by th thorm Th flow rat stimats also convrg vry wll xcpt for two spiks whr actual flow rats hav discontinuous changs Howvr, this may not b a problm bcaus flow rat would not chang discontinuously in practic du to th valv and flow dynamics P, Pa x 6 3 Simulation prssur with modlling rror Estimatd prssur Actual Prssur Estimation rror P, Pa 6 x 6 Simulation prssur w/o modlling rror Estimatd prssur Actual Prssur Estimation rror P, Pa 8 x 6 6 Estimatd prssur Actual Prssur Estimation rror P, Pa Mtr-in flow, m 3 /sc Mtr-out flow, m 3 /sc x 6 6 Estimatd prssur Actual Prssur Estimation rror x - - Tim, scond Fig Simulation prssurs w/o modlling rror Simulation flow rat, w/o modlling rror Estimatd Flow Rat Actual Flow Rat Estimation rror x Tim, scond Estimatd Flow Rat Actual Flow Rat Estimation rror Fig Simulation flow rats w/o modlling rror Figurs and show th prssur and flow rat stimats in th absnc of disturbanc flows It is obvious that th prssur stimats for both chambrs Mtr-in flow, m 3 /sc Mtr-out flow, m 3 /sc Tim, scond Fig 3 Simulation prssurs with modlling rror x - Simulation flow rat, with modlling rror Estimatd Flow Rat Actual Flow Rat Estimation rror x Tim, scond Estimatd Flow Rat Actual Flow Rat Estimation rror Fig : Simulation flow rats with modlling rror Figurs 3 and illustrat th prssur and flow rat stimats in th prsnc of disturbanc flows Th prssur stimats, as prdictd by th thorm, ar not affctd by th disturbanc flows and convrg to th actual valus vry wll Th flow rat stimats, though affctd by th cylindr intrnal lakag, still show vry good convrgnc Exprimnts ar don with th cylindr usd to activat th swing motion of a thr dgr-of-frdom

5 lctro-hydraulic robot arm A nonlinar controllr is usd to control th swing motion to track an angular trajctory Sinc th cylindr rod motion in th xprimnt is diffrnt from th on in simulation, th xprimnt flow rats look diffrnt from th simulation Sinc thr is no way to know th actual flow rat, w can not compar th stimats with thir actual valu Th flow rat stimats ar shown in Fig 5 Mtr-in flow, m 3 /sc Mtr-out flow, m 3 /sc x - Exprimnt stimatd flow rat x Tim, scond Fig 5: Exprimnt flow rat stimats Conclusions and Futur Works Th proposd sliding mod dynamic flow rat obsrvr taks advantag of th us of th quivalnt control for th stimation of flow rats with discontinuous trm It succssfully ovrcoms th nonlinar dynamics and uncrtain modlling rror, and thortically guarants that th valv flow rat stimats convrg to thir tru valus in th absnc of th disturbanc flow and th paramtr stimation rror, or a boundd stimation rror in th prsnc of th disturbanc flow and paramtr stimation rror Simulation and xprimntal rsults ar obtaind to illustrat th prformanc of th proposd flow rat obsrvr Th flow rat obsrvr is th first stp toward an automatd modlling mchanism of cartridg valv flow mapping Futur works includ adopting adaptiv robust obsrvr (ARO) tchniqu to dal with th changing paramtrs in th prssur dynamics, such as ffctiv bulk modulus β, and to rduc ffcts of paramtr variation and th disturbanc flow stimation rrors Acknowldgmnt Th work is supportd in part through th National Scinc Foundation undr th grant CMS-79 Rfrncs [] H E Mrritt, Hydraulic control systms, John Wily & Sons, 967 [] V I Utkin, Sliding Mods and Thir Application in Variabl Structur Systms Mir Publishrs, 978 [3] J J E Slotin, J K Hdrick and E A Misawa, On Sliding Obsrvrs for Nonlinar Systms ASME Journal of Dynamics Systms, Masurmnt, and Control, Vol 9, pp 5-5, 987 [] R A McCann, M S Islam, and I Husain, Application of a Sliding-Mod Obsrvr for Position and Spd Estimation in Switchd Rluctanc Motor Drivs IEEE Trans On Industry Applications, Vol 37, No, pp 5-58,

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