Optimal Control Using State-dependent Riccati Equation (SDRE) for a Hydraulic Actuator

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1 Procdings of th World Congrss on Enginring Vol III, WCE, July - 5,, ondon, U.K. Otial Control Using Stat-dndnt Riccati Equation (SDRE) for a Hydraulic Actuator F. iccardo, S. Strano and M. rzo Abstract his ar rsnts a nonlinar control for a hydraulic actuator with significant nonlinaritis. hs ar th ky factors causing dlay and rror in th hydraulic actuation rsons and highly liit th rforancs of th classical linar control. Hnc, a nonlinar control dsign basd on th athatical odl of th hydraulic actuator is loyd. h roosd aroach consists of a nonlinar fdback control basd on th stat-dndnt Riccati quation (SDRE). h control rforanc is donstratd by both siulations and ral-ti xrints. h xrintal rsults alidat th roosd aroach and highlight a good accordanc with siulations. Indx rs Hydraulic actuator, otial control, SDRE, Riccati quation, nonlinar, tracking, ral-ti. I. INRODUCION YDRAUIC actuators loy hydraulic rssur to dri Han outut br. hs ar usd whr high sd and larg forcs ar rquird. h fluid usd in hydraulic actuator is highly incorssibl so that rssur alid can b transittd instantanously to th attachd br. Hydraulic coonnts, bcaus of thir high sd and rssur caabilitis, can roid high owr outut with sall wight and siz in coarison to lctric syst coonnts. Hydraulic actuators can b found in transortations, industrial achinris, sisic alications [], and arth oing quints. Howr, th dynaics of hydraulic systs ar highly nonlinar [] du to th rssur-flow rat rlationshi, th dad band of th control al and th frictions. hs nonlinaritis highly liit th rforanc achid by th classical linar controllr. In th ast, uch of th work in th control of hydraulic systs has usd linar odl [] or local linarization of th nonlinar dynaics about th noinal orating oint [4]. Suitabl adati aroachs ar loyd whn thr is no knowldg of th aratr alus [5], [6]. In ordr to tak syst uncrtaintis into account, robust aroachs can b adotd [7], [8]. In [9], a sliding od control F. iccardo is with th Diartinto di Inggnria Industrial, Unirsità dgli Studi di Naoli Fdrico II, 85 IAY (-ail: flic.liccardo@unina.it) S. Strano is with th Diartinto di Inggnria Industrial, Unirsità dgli Studi di Naoli Fdrico II, 85 IAY (corrsonding author, hon: ; -ail: salator.strano@unina.it). M. rzo is with th Diartinto di Inggnria Industrial, Unirsità dgli Studi di Naoli Fdrico II, 85 IAY, (-ail:.trzo@unina.it). ISBN: ISSN: (Print); ISSN: (Onlin) alid to an asytric singl-rod cylindr was rsntd. In this ar a SDRE-basd control for a hydraulic actuator is roosd and alid to a hydraulic cylindr of a sisic tst bnch. h basic ida of th SDRE tchniqu is to catur th nonlinaritis by bringing th nonlinar syst to a linar structur haing stat-dndnt cofficint (SDC) atrics, and iniizing a nonlinar rforanc indx haing a quadratic-lik structur []. h subotial control action can b obtaind soling onlin an algbraic Riccati quation (ARE) using th SDC atrics []. his ar continus th work don in [] and [] and shows xrintal rsults in ordr to alidat th roosd nonlinar aroach. h rst of th ar is organizd as follows: a dscrition of th roosd control is gin in Sction II. In Sction III th nonlinar odl of th hydraulic actuator is drid and in Sction IV th control has bn articularizd th scific syst. Siulation rsults ar rortd in Sction V and in Sction VI th ain xrintal rsults ar rsntd. II. SDRE FORMUAION Considr a nonlinar obsrabl systs rrsntd in gnral for by quations x f ( x, u) () n whr x and u ar th stat and th control rsctily. Assu that th origin is an quilibriu oint and suos that th dynaic odl of th syst can b lacd in th SDC for x A( x B( u. () Considr th autonoous, infinit-horizon, nonlinar rgulator robl of iniizing th rforanc indx J x Qx u Rudt with rsct to th stat x and th control u, subjct to th nonlinar diffrntial constraints (); whr Q and R ar sytric wighting atrics. h SDRE control thod roids an aroxiat nonlinar fdback solution of th abo robl. h fdback gain quation is gin as () WCE

2 Procdings of th World Congrss on Enginring Vol III, WCE, July - 5,, ondon, U.K. K( R B ( (4) V P β A y Q (9) whr P ( is th sytric, ositi-dfinit solution of th SDRE of th for A ( A( Q B( R B ( h SDRE controllr can b ilntd as a srochanis, siilar to th that of a linar quadratic rgulator [4]. Gin a dsird stat trajctory x d, th SDRE sro control action is thn gin by u SD R B ( ( x x d (5) ). (6) whr V is th olu of ach chabr for th cntrd osition of th iston, Q = (Q A +Q B )/ is th load flow and β is th ffcti Bulk odulus. An orlad four-way al is considrd: tyically, this kind of al is charactrizd by th lands of th sool gratr than th annular arts of th al body. Consquntly, th flow rat is zro (dad band) whn th sool is in th nighbourhood of its cntral osition h load flow dnds on th suly rssur, th load rssur and th al sool osition in accordanc with th following Q Ψ( ) P P () s III. DYNAMICA MODE OF HE HYDRAUIC ACUAION SYSEM In ordr to dscrib th bhaiour of th syst, a st of diffrntial quations is drid in th following. h hydraulic actuator undr considration consists of a doublndd hydraulic cylindr drin by a four-way sool al. h hydraulic cylindr is could to a ass that os on linar guids (Fig. ). For th driation of th athatical odl, th following hyothsis ha bn adotd: a) fluid rortis not dnding on th tratur; b) qual iston aras; c) qual oil olu for ach sid (with th barrl in a cntral osition); d) ngligibl intrnal and xtrnal fluid lakags; ) tank rssur qual to zro. h dynaics of th oabl ass dislacnt (y) is gornd by y σy F ( y ) A P, (7) f whr is th ass of th load, σ is th iscous friction cofficint, F f is th friction forc, A is th iston ara, P = P A -P B is th load rssur, P A and P B ar th rssurs insid th two chabrs of th cylindr. h friction forc is rrsntd by th following quation Fc sgn( y ) μg sgn( y ) y Ff ( y ) Z Z Fc μg y (8) whr F c is th Coulob friction forc in th hydraulic actuator (assud qual to its static alu F c ), μ is th Coulobian friction cofficint of th linar guids (assud qual to its static alu μ ), g is th graitational acclration and Z is th nt tangntial forc that acts on th actuator whn it is not oing [5]. h load rssur dynaics [] is gin by whr is th dislacnt signal of th sool al and Ψ( ) is a ariabl gain functional to dscrib th al dad band. h analytical xrssion of Ψ ( ) can b assud as k Ψ ( ) k q qn n n n () whr th airs ( n, ) and (k qn, k q ) ar th dad band widths and th gains for th ositi and ngati sool dislacnt, rsctily. h dad band nonlinarity is aong th ky factors causing dlay and rror in th hydraulic actuation rsons. h roortional al dynaics can b wll rrsntd by a scond ordr diffrntial quation [6] ω ξ ω k u () whr aratrs ω and ξ ar th natural frquncy and th daing ratio of th al rsctily, is th sool osition bias, k is th inut gain and u is th al coand. ISBN: ISSN: (Print); ISSN: (Onlin) WCE

3 Procdings of th World Congrss on Enginring Vol III, WCE, July - 5,, ondon, U.K. Fig.. Schatic diagra of th hydraulic actuation syst. Finally, th quations gorning th dynaics of th whol syst (oabl ass + hydraulic syst) ar y σy Ff ( y ) AP V P A y Q β Q Ψ ( ) Ps P ω ξω ω ( k u ) () h dlod fifth ordr odl fully dscribs th nonlinar dynaical bhaior of th hydraulic actuation syst and taks th nonlinar friction forcs and th nonlinar flow rat distribution into account. h syst () can b writtn in th following for σ Ff ( y ) AP y y A βψ ( ) P P P β s y, (4) V V ξω ω ωku h syst (4), rrsntd in th stat sac for, is nonlinar in th stat, autonoous and charactrizd by a fully known stat by ans of asurnts. It is ossibl to not in th first and th third quation of (4) a diision by th ariabl, as wll as hans in (). In ordr to rnt diisions by zro, has bn introducd a functional ariabl ε, whr ε is gin by: ε othrwis (5) whr th alu of is chosn lss than th iniu acctabl asurnt rror of th al sool osition. Substituting th ariabl into th fractions of (4) and () follows: σ Ff ( y ) AP y y βa βψ ( ) P P P s y V V ξω ω ωku IV. PARAMEERIZAION OF HE SDRE CONRO (6) his sction xlains th dsign of th SDRE controllr. h SDC aratrization has bn rford using th highly nonlinar quations (6). h atrics in () ha bn chosn as follows: σ βa A( V A ξ ω Ff ( y ) βψ ( ) P V s ω whr th stat ctor is x y y P. P (7) h atrix B dos not dnd on th stat and is gin by: B. (8) ω k For th scific robl, without loss of gnrality, th goal of th tracking control is that th oabl ass follows a dsird dislacnt y d (t); thrfor, th wighting atrics in () bco: Q diag(, qsd,,,) and R r SD, whr q SD and r SD ar two wighting cofficints. V. SIMUAION RESUS In this sction so siulation rsults, concrning th alication of th roosd controllr to a articular hydraulic actuation syst rsntd in Sction VI, ar rortd. h hysical aratr alus of th hydraulic actuator ar: N s 44 kg, σ 555, F C N FC 95, μ μ., N 46 N, A., ISBN: ISSN: (Print); ISSN: (Onlin) WCE

4 Procdings of th World Congrss on Enginring Vol III, WCE, July - 5,, ondon, U.K..4 V, β 9 Pa, k 6.5 7, q s V Pa k ,.4 V,.V, qn s V Pa.V, k. 49, ω 5., P S 66Pa. hs aratrs ha bn obtaind with a aratr idntification tchniqu []. h controllr is usd for tracking sinusoidal otion trajctory charactrizd by an alitud of.4 and a frquncy of Hz. h siulation rsults rortd in Fig. consist of coarisons btwn th targt and th ffcti oabl ass dislacnt, th tracking rror ( y yd ) and th control action. n h isolator undr tst (C) is locatd btwn th sliding tabl and a slid that can o rtically with rsct to th horizontal raction structur (D). A hydraulic jack (E) is ositiond btwn th rtical raction structur (F) and th slid in ordr to ak th isolator undr tst rtically loadd. h suly circuit of th hydraulic actuator is ainly constitutd by an axial iston u, owrd by a 75 kw AC lctric otor, a rssur rlif al and a four waythr osition roortional al. h u is charactrizd by a ariabl dislacnt and it is abl to gnrat a axiu rssur of bar and a axiu flow rat qual to l/in. Fig.. Exrintal tst rig. Fig.. Siulation rsults of th nonlinar SDRE-basd control. h axiu alu of th tracking rror is about.8. h diffrnc btwn th ak to ak alitud of th ffcti dislacnt and th targt on is about., and th actuation syst has lag is of. s. o rify th rdiction caacity of siulation, th rsults rortd in Fig. will b coard with th xrintal ons. VI. REA-IME EXPERIMENS In ordr to tst th ffctinss of th roosd nonlinar tracking controllr, xrintal studis ar conductd on th hydraulic actuation syst of th tst rig shown in Fig.. A. h xrintal tst rig h xrintal tst rig is a achin utilizd to rfor shar tsts on sisic isolators [7], [8]. h actuation syst consists of a doubl-ndd hydraulic actuator lacd btwn a fixd bas and a sliding tabl (.8 x.59 ). h hydraulic actuator is constitutd by a obil barrl, intgral with th sliding tabl (A), and two iston rods linkd to th fixd bas (B) (Fig. ). ISBN: ISSN: (Print); ISSN: (Onlin) h axiu horizontal forc is 9 kn, th axiu sd is. /s and th axiu strok is.4 (±. ). h roal of th raction structurs allows th tsting achin to b usd as a shaking tabl [9]. h full-stat fdback has bn obtaind with th following asurnts: - tabl osition by ans of agntostricti osition snsor (FS =.4 - stiatd uncrtainty = ±.-4 ); - rssur in th two chabrs of th hydraulic cylindr by ans of strain gaug snsor (FS = 4 bar - stiatd uncrtainty = ± bar); - al sool osition by ans of built-in VD snsor; A dspace DS controllr board, quid with a 6-bit A/D and D/A cortr, has bn usd for th ralti xrints. All xrints ha bn conductd with a sal frquncy of khz. o attnuat th influnc of th nois, all asurd signals ar rocssd through a low-ass filtr. h sulid rssur has bn fixd to 66 Pa. B. Exrintal Rsults h xrints on th sisic isolator tsting achin ha bn conductd without th isolator to b charactriz. Hnc, th hydraulic actuation syst has bn utilizd only for th sliding tabl ositioning. For sak of coarison, th xrintal alidation of th roosd control has bn obtaind with th sa targt dislacnt iosd to th oabl ass in th siulations. h robl of couting th SDRE fdback gains WCE

5 Procdings of th World Congrss on Enginring Vol III, WCE, July - 5,, ondon, U.K. rducs to soling (5). h roosd aroach is basd on finding th igalus of th associatd Hailtonian atrix [] A( H( Q B( R A B ( (. (9) h SDRE-basd fdback gains ha bn obtaind with a ol lacnt algorith in trs of th stabl igalus of H(. h SDRE fdback action has bn ilntd as a C- cod function downloadd to th controllr board and ilntd in ral-ti. In Fig. 4 ar rortd th xrintal rsults in trs of th targt and th xrintal ffcti tabl dislacnt, th tracking rror and th control action. Fig. 4. Exrintal rsults of th nonlinar SDRE-basd control. Analyzing th xrintal rsults, it is ossibl to assrt that th axiu alitud rror is qual to -5 and th actuator has lag is qual to.5 s. Concrning th tracking rror, its axiu alu is qual to.9. h xrintal rsults agrs with th rsults rdictd in siulation ironnt. It has bn xrintally donstratd that th roosd nonlinar controllr can achi ry good rforanc in trs of tracking control and stability. VII. CONCUSION In this ar has bn roosd an otial control using th SDRE for a hydraulic actuator. A fifth ordr dynaic odl of th syst has bn drid taking into account th tyical nonlinaritis of th hydraulic actuators. h aratrization of th SDRE fdback control has bn obtaind dirctly fro th fifth ordr odl. h ral-ti ilntation of th SDRE nonlinar otial robl has bn rford with a ol lacnt algorith in trs of th stabl igalus of th Hailtonian atrix. Exrints ha bn conductd on th hydraulic actuation syst quid on a sisic isolator tst rig. h xrintal rsults alidat th roosd aroach and highlight a good accordanc with siulation rsults. REFERENCES [] J.P. Cont,.. robtti, inar dynaic odling of a uniaxial sro-hydraulic shaking tabl syst, Earthquak Enginring and Structural Dynaics, ol. 9, no. 9, ,. [] H.E. Mrritt, Hydraulic control systs, John Wily & Sons, Nw York, USA, 967. [] A. R. Plur and N. D. Vaughan, Robust adati control for hydraulic srosysts, ASME Journal of Dynaic Systs, Masurnt, and Control, ol. 8, no.,. 7 44, 996. [4] Y. Xu, H. Hua, J. Han, Modling and controllr dsign of a shaking tabl in an acti structural control syst, Mchanical Syst and Signal Procssing, ol.,. 97 9, 8. [5] R. Russo, M. rzo, Dsign of an adati control for a agntorhological fluid brak with odl aratrs dnding on tratur and sd, Sart Matrials and Structurs, ol., no., 5 (9),. [6] D.P. Stotn, E.G. Goz, Adati control of shaking tabls using th inial control synthsis algorith, Phil. rans. R. Soc. ond., ol. 59, ,. [7] B. Yao, F. Bu, J. Rdy, G..C. Chiu, Adati robust otion control of singl-rod hydraulic actuators: thory and xrints, IEEE/ASME ransactions on Mchatronics, ol.5, no.,. 79 9,. [8] Y. in, Y. Shi, R. Burton, Modling and Robust Discrt-i Sliding-Mod Control Dsign for a Fluid Powr Elctrohydraulic Actuator (EHA) Syst, IEEE/ASME ransactions on Mchatronics, ol.8, no.,.-,. [9] D. Hissin, Robust tracking control for a hydraulic actuation syst, Procdings of 5 IEEE Confrnc on Control Alications,. 4 47, 5. []. Çin, Sury of Stat-Dndnt Riccati Equation in Nonlinar Otial Fdback Control Synthsis, AIAA Journal of Guidanc, Control, and Dynaics, ol. 5, no. 4,. 5 47,. [] R. Russo, M. rzo, Modlling, aratr idntification, and control of a shar od agntorhological dic, Procdings of th Institution of Mchanical Enginrs, Part I: Journal of Systs and Control Enginring, ol. 5, no. 5, ,. [] S. Pagano, R. Russo, S. Strano, M. rzo, Modlling and control of a hydraulically actuatd shaking tabl loyd for ibration absorbr tsting, (ESDA-88), in Proc. of th ASME th Binnial Confrnc on Enginring Systs Dsign and Analysis (ESDA), ol., ,. [] S. Pagano, R. Russo, S. Strano, M. rzo, Non-linar odlling and otial control of a hydraulically actuatd sisic isolator tst rig, Mchanical Systs and Signal Procssing, ol. 5, no., ,. [4] J.R. Cloutir, D.. Stansbry, Control of a continuously stirrd tank ractor using an asytric solution of th stat-dndnt Riccati quation, in Proc. of th IEEE. Intrnational Confrnc on Control, ol., , 999. [5] A. Bonchis, P. Cork, D. Ry, A rssur-basd, locity indndnt, friction odl for asytric hydraulic cylindrs, in Proc. of th IEEE Intrnational Confrnc on Robotics and Autoation, ol., , 999. [6]. Márton, S. Fodor, N. Shri, A ractical thod for friction idntification in hydraulic actuators, Mchatronics, ol.,. 5 56,. [7] M. Cardon, S. Strano, Fluid-dynaic analysis of arthquak shaking tabl hydraulic circuit, (ESDA-84), in Proc. of th ASME th Binnial Confrnc on Enginring Systs Dsign and Analysis (ESDA), ol.,. 4 5,. [8] S. Pagano, M. Russo, S. Strano, and M. rzo, A ixd aroach for th control of a tsting quint loyd for arthquak isolation systs, Procdings of th Institution of Mchanical Enginrs, Part C: Journal of Mchanical Enginring Scinc, DOI:.77/ ,. [9] G. Di Massa, S. Pagano, E. Rocca, S. Strano, Snsiti quints on WRS-BU isolators, Mccanica, DOI:.7/s ,. [] E.B. Erd, A.G. Allyn, Exrintal ral-ti SDRE control of an undr actuatd robot, in Proc. of th 4th IEEE Confrnc on Dcision and Control, ol., ,. ISBN: ISSN: (Print); ISSN: (Onlin) WCE