Controller Design and Actuator Dynamics Identification for a Hybrid Simulation Testing System

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1 Intrnational Journal of Applid Enginring Rsarch ISSN Volum 12, Numbr 14 (2017) pp Rsarch India Publications. Controllr Dsign and Actuator Dynamics Idntification for a Hybrid Simulation Tsting Systm M. Russo Dpartmnt of Elctrical Enginring and Information Tchnology, Univrsity of Napls Fdrico II, Via Claudio 21, 80125, Napls, Italy. Abstract This papr dscribs th dvlopmnt of a fdforward controllr and an actuator dynamics idntification procdur for a hybrid simulation tsting systm. Th fdforward control is intgratd with a linar fdbac control that has th function to compnsat for tracing rrors du to modlling uncrtainty and unnown raction forc of th dvic undr tst. During hybrid simulations it is crucial th compnsation of actuation systm dynamics in ordr to avoid instability. Th compnsation can b providd idntifying th systm dynamics. To this aim, a nonlinar paramtr stimator, basd on th xtndd Kalman filtr, has bn dvlopd. Exprimntal rsults ar prsntd in ordr to highlight th prformanc of th proposd nonlinar stimator Kywords: fdforward control, hybrid simulation, nonlinar dynamics, xtndd Kalman filtr, nonlinar paramtr stimation INTRODUCTION Hydraulic actuation systms xhibit significant nonlinar bhaviour du to th prssur-flow rat rlationship, th dad zon of th control valv [1] and frictions [2]; ths nonlinaritis ma th mathmatical modl mor complx and, at th sam tim, highly limit th prformanc achivd by th classical linar controllr [3]. Th traditional and widly usd approach to th control of hydraulic systms is basd on a linar modl or on th local linarization of th nonlinar dynamics about th nominal oprating point [4]. Suitabl adaptiv approachs ar also mployd whn thr is no nowldg of th paramtr valus [5] and whn initial adaptation stags ar accptabl. In this papr a modl-basd control systm, abl to follow th dsird displacmnt law, is prsntd and its robustnss proprtis ar chcd. To this nd, a fifth ordr nonlinar modl (5 th OM) of th tst rig is drivd and its validation illustratd. Th 5 th OM fully dscribs th isolator tst rig in trms of typical dynamics, soft and hard nonlinaritis and consquntly is assumd as compltly faithful to th plant. Succssivly a fdforward control basd on th nonlinar invrs modl of th systm is obtaind. Th fdforward loop is dsignd to compnsat for th nonlinarity of th lctrohydraulic systm, including th dad zon of th systm and th nonlinar flow gain of th control valv. Th fdforward control is intgratd with a fdbac control in ordr to compnsat for th tracing rrors du to modlling uncrtainty and unnown raction forc of th dvic undr tst, Th controlld actuation systm is mployd for hybrid simulations (HSs). This mthod, also nown as substructur tsting, consists in dividing a structural systm into physical substructurs that ar xprimntally tstd and numrical substructurs that contain th rst of th structur, which ar numrically simulatd [6], [7]. In ths actuation systms, invitabl amplitud and phas rrors xist btwn th actuator command displacmnt and th ffctiv on. This phas rror (which can also b viwd as a tim dlay) introducs additional nrgy into th systm and th xprimnt could bcom unstabl [8]. Consquntly, suitabl actuator dynamics compnsation mthods ar gnrally adoptd in HSs. Th prformanc of a compnsation algorithm is strictly rlatd to th nowldg of th actuation systm dynamics that varis in accordanc with svral tsting conditions. This rsult implis mthods that can b usful to idntify in raltim th actual actuation systm dynamics. In this papr, a nonlinar paramtr stimator, basd on th Extndd Kalm Filtr (EKF), is proposd for th ral-tim idntification of th systm modl paramtrs. Th robustnss of th EKF with rspct masurmnt nois mas this mthod particularly suitabl for HSs. Th papr is organizd as follows: in Sction II, th tst rig is dscribd; in Sctions III and IV, th tst rig modlling and th controllr dsign ar illustratd, rspctivly. Th HS procdur, th EKF mthodology and th proposd modl paramtr stimation ar prsntd in Sctions V, VI and VII, rspctivly; in Sction VIII, th xprimntal validation of th nonlinar stimator is dscribd. 4624

2 Intrnational Journal of Applid Enginring Rsarch ISSN Volum 12, Numbr 14 (2017) pp Rsarch India Publications. TEST RIG DESCRIPTION Th tst rig (Fig. 1, 2) consists of a fixd bas, a hydraulic actuator and a sliding tabl (1.8 m x 1.59 m). Th xprimntal stup usd has bn dsignd to prform static and dynamic tsts on bas isolation systms. Th tabl motion is constraind to a singl horizontal axis by mans of linar guids. Th isolator undr tst is placd btwn th sliding tabl (A) and th vrtical slid (B) (Fig. 1). Figur 1: Tst rig Th tst rig is instrumntd to dtct th following masurmnts: - tabl position y by mans of magntostrictiv position snsor (FS = 0.4 m - stimatd uncrtainty = ± m); - actuation forc by mans of strain gaug load cll (FS = N - stimatd uncrtainty = ± 500 N); - prssur in th two chambrs of th hydraulic cylindr (P A and P B in Fig. 3) by mans of strain gaug snsor (FS = 400 bar - stimatd uncrtainty = ± 1 bar); - supply prssur Ps by mans of strain gaug snsor (FS = 400 bar - stimatd uncrtainty = ± 1 bar); - valv spool position x v by mans of VDT snsor intgratd in th valv body; - vrtical load prssing on th spcimn undr tst by mans of strain gaug load cll (FS = 16 N - stimatd uncrtainty = ± 230 N). Th rmoval of th raction structurs allows th tsting machin to b usd as a shaing tabl and consquntly as sismic simulator for tst structurs [12]-[14]. TEST RIG MODEING In th following sction, th tst rig mathmatical modl is drivd. Th modlling rfrs to th tsting machin in which no isolator is installd: th hydraulic cylindr has to mov only th sliding tabl (Fig. 3). Th modlling procdur is basd on th following hypothsis: a) fluid proprtis not dpnding on th tmpratur; b) qual piston aras; c) qual oil volum for ach sid (with th barrl in a cntral position); d) ngligibl intrnal and xtrnal fluid laags. Figur 2: Tst rig componnts Th hydraulic jac (C) allows th isolator undr tst (Fig. 2) to b vrtically loadd (max 850 N). Th jac load and th forc acting on th tabl ar balancd by th vrtical (D) and horizontal (E) raction structurs rspctivly. Th hydraulic circuit consists of a four way-thr position proportional valv and a hydraulic cylindr. Th cylindr is constitutd by two qual parts sparatd by a diaphragm and contains two pistons whos rod is connctd to th bas: so, th actuator is charactrizd by a mobil barrl and fixd pistons [9]-[11]. Th maximum horizontal forc is 190 N, th maximum spd is 2.2 m/s and th maximum stro is 0.4 m (± 0.2 m). Th tst rig is modlld as a singl DOF systm subjctd to both actuation and friction forc. In particular, th actuator can b modlld as a doubl-ndd hydraulic cylindr drivn by a four-way spool valv. PA Q A y P B Q B P =cost S P =0 T Figur 3: Schm of th hydraulic actuator adoptd for th mathmatical modl x v m 4625

3 Intrnational Journal of Applid Enginring Rsarch ISSN Volum 12, Numbr 14 (2017) pp Rsarch India Publications. In th following, th diffrntial quations govrning th hydraulic actuation systm dynamics ar givn [15]. Th prssur dynamics is givn by: and rror in th hydraulic actuation rspons. As rgards th valv dynamics, th following scond ordr diffrntial quation is adoptd: V0 P 2β A y Q, (1) p v ω 2 nv 2ξ ω v nv v v v, (6) c whr P = P A-P B is th load prssur, P A and P B th prssurs in th cylindr chambrs, V 0=V A=V B th oil volum btwn th piston and th valv in ach sid for th cntrd barrl position, A p th piston ara, Q = (Q A+Q B)/2, commonly calld load flow, β th ffctiv Bul modulus and y th tabl displacmnt. Th motion quation of th sliding tabl is: m y μn sgn( y ) F A P, (2) f p whr m is th movabl mass, N th vrtical load on th linar guids, μ th Coulombian friction cofficint and F f th hydraulic friction forc. Th hydraulic friction is modlld taing th viscous and th Coulombian trm into account [16]-[19]: F f σy F sgn( y ), (3) c whr σ is th viscous friction cofficint and F c th Coulomb friction forc. An ovrlappd four-way valv is considrd: this ind of valv is typically charactrizd by th lands of th spool gratr than th annular parts of th valv body. Consquntly, th flow rat is zro (dad zon) whn th spool is in th nighbourhood of its cntral position. Undr th assumption of a tan prssur P T qual to zro, th load flow dpnds on th supply prssur, th load prssur and valv spool position in accordanc with th following: Q DZ ( v ) P sgn( v ) P, (4) s whr v is th voltag signal proportional to th valv spool position x v and DZ(v ) is th dad zon function. Mor gnrically, nithr th bra-points nor th slops of th dad zon ar qual and, as consqunc, th analytical xprssion of DZ(v ) is: qn ( v vn) if v vn DZ ( v ) 0 if vn v vp, (5) qp ( v vp) if v vp whr v n and v p ar th limits of th dad zon, qn and qp ar th adoptd gains if v is ngativ or positiv rspctivly. Th dad zon nonlinarity is among th y factors causing dlay whr paramtrs ω nv and ξ v ar th natural frquncy and th damping ratio of th valv rspctivly and v c is th valv command voltag du to its lctronic drivr. Th rlationship btwn v c and th valv input voltag u is: pu v0 if u 0 vc v0 if u 0, (7) nu v0 if u 0 whr v 0 is th bias, and p, n th gains. Finally, th quations govrning th dynamics of th whol systm (sliding tabl + hydraulic systm) ar: my (μn Fc )sgn( y ) σy Ap P V0 P Ap y Q 2β. (8) Q DZ ( v ) Ps sgn( v ) P 2 2 v ω v 2ξ vωnvv ω v nv nv c Th dvlopd 5 th OM fully dscribs th nonlinar dynamical bhaviour of th hydraulic actuation systm and tas th nonlinar friction forcs and th nonlinar flow rat distribution into account. CONTROER DESIGN This sction dscribs th dvlopmnt of th control systm. Th chosn controllr schm consists of a fdforward control, obtaind by th validatd 5 th OM, intgratd with a linar fdbac controllr with th function to compnsat for tracing rrors du to modlling uncrtainty and unnown raction forc of th dvic undr tst. In this way, th fdbac controllr is not rquird to compnsat for th larg hydraulic proportional valv nonlinaritis [20]-[23]. Th fdforward procss can b dscribd as follows: dnoting th nonlinar mapping G as th actual plant to b controlld, G is bron down into th nown part, G n (nominal), and th uncrtain part, ΔG. That is, G= G n+δg. Th trm G n rprsnts a nonlinar mapping of all th nown charactristics of th "plant" to b controlld. Th trm ΔG rprsnts additiv modlling uncrtainty (.g., th raction forc of th spcimn undr tst) of G. Sinc th mapping G, is nown, its invrs,g -1, can b dtrmind. Fig. 4 shows a control bloc diagram intgrating th fdforward controllr 4626

4 Intrnational Journal of Applid Enginring Rsarch ISSN Volum 12, Numbr 14 (2017) pp Rsarch India Publications. with th fdbac control loop. Th fdforward path uss th nowldg of th controlld plant, G n, and th fdbac path with loop gain factor K compnsating for imprfctions in th nowldg of th plant, ΔG. Figur 6: Sismic isolator. Figur 4: Control Bloc Diagram; Intgrating Fdforward with Fdbac control If th paramtr variation ΔG is zro, y prfctly follows r. Whn ΔG is non-zro, a highr achivabl stabl gain (K) rducs th ffct of ΔG on th ability of y to trac r (i.., improvs tracing prformanc). Howvr, thr ar practical stability limits to th achivabl gain. HYBRID SIMUATION SYSTEM In this sction, th hybrid tsting procdur is dscribd. For HSs, th physical substructur intracts with a computational modl (numrical substructur) by mans of a fdbac loop, xchanging information in ral tim with minimum rror btwn thm. Th HS tst st-up is shown in Fig. 5. Th figur dpicts a dvic, charactrizd by hystrtic nonlinarity [24], placd btwn th horizontal sliding tabl and th vrtical actuator. EKF METHODOOGY Th EKF is a mathmatical tool for an optimal stat stimation of nonlinar systms [27]. This algorithm can b usd to stimat unnown paramtrs by taing th paramtrs as additional stats and augmnting stat quations [28]-[29]. Th systm and th masurmnt quations ar: x f( x( t), u( t)) ψ( t) ( t), u( t) g( ) z( t) h x t bing x th stat vctor, u th input vctor, f a nonlinar function, ψ th procss nois with covarianc Q, z th masurmnt vctor, h a nonlinar function and g (t) th Gaussian whit masurmnt nois with covarianc R. Th stimator has bn implmntd in a discrt tim form with th stat stimats and an stimation of th rror covarianc givn by: ˆ 1 1 x f(ˆ x, u ) T T A 1 P 1 A 1 1 Q 1 1 P rspctivly. In th prvious formula th following paramtrs ar dfind Figur 5: Schm of th HS tst st-up. In HS of a bas isolation systm, a horizontal displacmnt is imposd on th baring by mans of th hydraulic actuator. Thn, th masurd raction forc is fd bac to th numrical modl for th solution of th quation of motion. Th adoptd isolation dvic is commonly mployd in sismic isolation [25]-[26], it consists of altrnat las of stl and lastomr connctd by curing (Fig. 6). 1 f A 1 f ψ x ˆ 1 1 Th filtr gain can b computd as in (15) to valuat th masurmnt rsidual. Th updats of stat stimats can b computd as in (16), whil th stimation of th rror 4627

5 Intrnational Journal of Applid Enginring Rsarch ISSN Volum 12, Numbr 14 (2017) pp Rsarch India Publications. covarianc can b dtrmind as in (17): T T T 1 K P H ( H P H M R M ) K z, u ) h(ˆ x P ( I K H ) P whr H M h h g EKF-BASED CONTROED SYSTEM MODE PARAMETER IDENTIFICATION Th dynamics of th controlld systm dpnds on th actual plant (hydraulic actuation systm, controllr, spcimn). Th dynamics of th controlld srvo-systm can b rasonably wll approximatd by a lowr-ordr modl with variabl paramtrs that includ th ffcts of th hydraulic actuation systm, th controllr and th spcimn. Th approximation can b considrd accurat whn highfrquncy dynamics can b nglctd (.g., dynamics associatd with oil-column rsonanc). Th systm transfr function can b writtn as: G, (20) s 1 whr τ is a tim constant and is an amplitud gain. Th paramtr valus of Eq. (20) vary in function of th spcimn and th actuator controllr algorithm; morovr, th paramtrs chang with rspct to th amplitud of th rfrnc signal bcaus of invitabl nonlinaritis. Th basic ida of this papr consists in using th EKF to idntifid th modl paramtrs in ral-tim. With rfrnc to th cas study, a suitabl nlargd stat vctor has to b dfind in ordr to stimat both th stat variabls and th dsird paramtrs. Th diffrntial quation associatd to th transfr function (20) is y y r, (21) y t t 1 y 1 r 1 whr Δt is a discrtization tim. Considring th stat vctor x x x y,, 1, 2, 3,, (22) x, (23) Eq. (22) can b formulatd in th following stat spac form: x x x 2, 3, t x x2, 1 x w 1, 1 x 2, 1 3, 1 w 2, 1 3, 1 1, 1 x3, x 1 2, t r 1 1 w 1, 1 (24) whr w 1 is a Gaussian nois associatd to th actuator displacmnt dynamics, w 2 and w 3 ar fictitious Gaussian nois rlatd to th unnown paramtrs. In addition to Eq. (24), th masurmnt quation has to b introducd: z x 1, v (25) whr v is th masurmnt nois. EXPERIMENTA RESUTS Th prformancs of th proposd stimator hav bn xprimntally valuatd considring shar tsts on th spcimn shown in Fig. 6. Considring th stup dscribd in Sction V, a sinusoidal targt displacmnt charactrizd by a frquncy f=1 Hz and an amplitud A=0.05 m has bn imposd to th sismic isolator. Fig. 7 shows th diagrams of stimatd paramtrs. that can b writtn in discrt tim domain using th forward diffrnc approximation: 4628

6 Intrnational Journal of Applid Enginring Rsarch ISSN Volum 12, Numbr 14 (2017) pp Rsarch India Publications. Applications, Vol. 38(Issu 10): , [2]. Márton, S. Fodor, N. Sphri, A practical mthod for friction idntification in hydraulic actuators. Mchatronics, Vol. 21: , 2011 [3] S. E. Salcudan, M. R. Sirouspour, On th Nonlinar Control of Hydraulic Srvosystms, Proc. ICRA IEEE Intrnational Confrnc on Robotics and Automation, Vol. 2, pp , San Francisco, CA, Figur 7: Exprimntal rsults: EKF-basd paramtr stimation; a) tim constant diagram, b) amplitud gain diagram. Th diagrams of th stimatd paramtrs, rportd in Figs. 7a and 7b, show that th EKF convrgs to almost constant valus for both paramtrs with a physical consistncy concrning th actual srvo-systm dynamics. CONCUSION An xprimntal/thortical study has bn carrid out on th lctrohydraulic actuation systm of a hybrid simulation tsting systm. Th plant consists of a hydraulically actuatd unidirctional moving platform. A fifth ordr dynamic modl of th systm has bn drivd. Th validatd modl has bn considrd for a position control dsign. Th proposd approach consists in a fdforward control intgratd with a fdbac control. Th fdforward control has bn dvlopd on th nonlinar invrs modl of th systm. Th fdbac controllr has bn adoptd to compnsat for th tracing rrors du to modlling uncrtainty and unnown raction forc of th dvic undr tst. Succssivly, a nonlinar paramtr stimator, basd on th xtndd Kalman filtr, has bn proposd for th dynamics idntification of th controlld tst rig achivd for hybrid simulations of sismic isolators. Th srvo-systm dynamics has bn modlld with a first ordr modl with paramtrs that vary in function of th actuation systm configuration, th control algorithm and th spcimn undr tst. Exprimntal rsults highlightd that th EKF-basd stimator is abl to rproduc th actual plant paramtrs in ral-tim. REFERENCES [1]. d. S. Colho, M. A. B. Cunha, Adaptiv cascad control of a hydraulic actuator with an adaptiv dadzon compnsation and optimization basd on volutionary algorithms, Exprt Systms with [4] R. Gunthr, M. A. B. Cunha, E. R. D Piri, V. J. D Ngri. VS-ACC applid to a hydraulic actuator. In. Proc. Amrican Control Confrnc, pp , [5] G. Tao, P. V. Kootovic, Adaptiv control of systms with actuator and snsor nonlinaritis. Nw Yor, USA. John Wily & Sons, [6] A. Calabrs, S. Strano, M. Trzo, Ral-Tim Hybrid Simulations vs. Shaing Tabl Tsts: cas study of a fibr-rinforcd barings isolatd building undr sismic loading, Structural Control and Halth Monitoring, Vol. 22(Issu 3): , [7] A. Calabrs, G. Srino, S. Strano and M. Trzo, Invstigation of th Sismic Prformancs of an FRBs Bas Isolatd Stl Fram through Hybrid Tsting, Proc. of th World Congrss on Enginring WCE 2013, Vol. III, pp ondon, U.K., July 3 5, [8] S. Strano, M. Trzo, Actuator dynamics compnsation for ral-tim hybrid simulation: an adaptiv approach by mans of a nonlinar stimator, Nonlinar Dynamics, 1-16, 2016, Articl in Prss. [9] M. Cardon, S. Strano, Fluid-Dynamic Analysis of Earthqua Shaing Tabl Hydraulic Circuit, (ESDA ), Proc. of th ASME 11th Binnial Confrnc on Enginring Systms Dsign and Analysis (ESDA2012), Vol. 2, pp , [10] S. Pagano, R. Russo, S. Strano, M. Trzo, Modlling and Control of a Hydraulically Actuatd Shaing Tabl Employd for Vibration Absorbr Tsting, (ESDA ), Proc. of th ASME 11th Binnial Confrnc on Enginring Systms Dsign and Analysis (ESDA2012), Vol. 1, pp , [11] M. Cardon, S. Strano, M. Trzo, G. Vorraro, A Numrical and Exprimntal Fluid-dynamic Analysis of a Hydraulic Actuator by mans of Closd oop Tsts, Proc. of th World Congrss on Enginring WCE 2013, Vol. III, pp , ondon, U.K., 4629

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