Analysis of Chattering in Sliding Mode Control Systems with Continuous Boundary Layer Approximation of Discontinuous Control

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1 011 mrican Control Confrnc on O'Farrll Strt San Francisco C US Jun 9 - July nalysis of Chattring in Sliding Mod Control Systms with Continuous Boundary Layr pproximation of Discontinuous Control Igor M. Boiko Snior Mmbr IEEE bstract It has bn a widly accptd notion that approximation of discontinuous control by crtain continuous function in a boundary layr rsults in chattring limination in sliding mod (SM) control systms. It is shown through thr diffrnt typs of analysis that in th prsnc of parasitic dynamics this approach to chattring limination would work only if th slop of th continuous nonlinar function within th boundary layr is low nough which may rsult in th dtrioration of prformanc of th systm. fw xampls ar providd. S I. INTRODUCTION LIDING mod (SM) control faturs such valuabl proprtis as low snsitivity to xtrnal disturbancs (thortically insnsitiv) and robustnss with rspct to plant paramtr uncrtaintis and variations. Howvr th pric bing paid for that is an undsirabl phnomnon calld chattring which is rvald as high-frquncy vibrations (oscillations) of th control which in turns rsults in th oscillations of th stat trajctory around th sliding surfac [1]. This drawback of th SM control is wll known and many fforts of th rsarchd wr aimd at limination of chattring or mitigation of its ffcts. On of th proposd rmdis is th rplacmnt of th discontinuous control with its continuous approximation in th vicinity of th sliding surfac which is also known as boundary layr (around th sliding surfac) introduction []- [4]. Howvr it is wll known that chattring happns in a SM control systm whn parasitic dynamics (that wr not accountd for in th dsign of th sliding surfac) ar prsnt. It should b notd that th prsnc of som kind of parasitic dynamics is invitabl [5] [6]. This is why chattring is invitabl whn th control is discontinuous. Yt dspit th popularity of th boundary layr approximation th conditions of th occurrnc of chattring wr not analyzd if parasitic dynamics ar prsnt. Th ida of chattring supprssion basically stms from th logical conclusion that chattring is a fatur of discontinuous control and rplacmnt of discontinuous control with a continuous on would rsult in chattring limination. Howvr it was shown in [7] that chattring (as high-frquncy vibrations of continuous natur) may xist in SM systms with continuous control too. This fact motivats I.M. Boiko is with IMB Controls and Univrsity of Calgary Calgary B TN 1N4 (-mail: i.boiko@i.org). th prsnt rsarch aimd at finding out if and whn chattring may xist in SM systms with continuous approximation of discontinuous control in th boundary layr. Th papr is organizd as follows. t first th modls of th continuous approximations ar considrd. Thn analysis with th usd of th dsibing function (DF) mthod is undrtakn. ftr that th sufficint conditions ar drivd through th us of th Popov s itrion of absolut stability [8]. ftr that xact analysis of chattring is proposd with application of th Poincar map to th problm. Finally an xampl of analysis is givn. II. CONTINUOUS PPROXIMION OF DISCONTINUOUS CONTROL IN BOUNDRY LYER t first w considr a linar plant and a discontinuous SM controllr. Lt th linar part of th systm which includs th plant th sliding surfac and som kind of parasitic dynamics (for xampl th dynamics of snsors and actuators not accountd for in th modl that is usd for dtrmination of th sliding surfac) b dsibd by th following quations: x& x Bu (1) σ Cx () n whr x R is th stat vctor (that includs th stats of both th principal and th parasitic dynamics) σ is th n n n 1 1 n sliding variabl R B R C R ar matrics is assumd nonsingular (for singular matrix th approach of [9] can b usd). Th sign - bfor Bu is attributd to th ngativ fdback so that convntional shaps of nonlinaritis with location in 1 st and 3 rd quadrants can b considrd. It is worth noting that th modl (1) () includs parasitic dynamics in th form of th actuator dynamics but not of th snsor dynamics bcaus it is assumd blow that th sliding variabl σ is th dirct input to th controllr. In ral lif applications thr would b som additional dynamics (of snsors) btwn th sliding variabl that combins th plants stats (and thrfor involvs masurmnts of ths stats) and th controllr. Howvr parasitic dynamics of an actuator and of a snsor possss th duality proprty bcaus both of thm ar connctd in sris with th plant which rsults in th sam ffct rgardlss of whr thy ar connctd: at th plant input or output [6]. Thrfor th ffct of parasitic /11/$ CC 757

2 dynamics in th modl (1) () is containd in th rlativ dgr of th linar part from th control u to th output σ: th rlativ dgr is qual to on if thr ar no parasitic dynamics and it is gratr than two if thr ar parasitic dynamics [6]. lso w do not us th popular trm unmodld dynamics bcaus w ar including ths dynamics in th modl; howvr w raliz that thr still a mismatch btwn th actual actuator (and snsor) dynamics and thir modls so that thr still may b unaccountd for or rmaining unmodld dynamics in this cas too [10]. Th discontinuous control is givn as follows: h if σ > 0 u (3) h if σ < 0 It is wll known that if som kind of parasitic dynamics ar prsnt (which always happns and rsults in th rlativ dgr from th control u to th output y highr than two [6]) thn chattring occurs in th SM control systm. It was proposd that chattring could b liminatd through a continuous approximation of th discontinuous control (3) in a boundary layr around th sliding surfac. In particular it was proposd in [] that a crtain boundary layr should b atd in th vicinity of th sliding surfac: B { x σ ( x) Φ} (4) whr Φ 0 is th boundary layr thicknss. ccording to [] outsid B th control is givn by (3) as bfor (which would guarant that th boundary layr is attractiv and thrfor invariant: all trajctoris stay insid B onc gt thr). It is supposd to provid a guarantd prcision rathr thn idal prcision. Yt insid B it was proposd that th linar function should b usd as th following linar control: u hσ / Φ x B (5) nd thrfor in ovrall th discontinuous control (3) is rplacd with th saturation function: u h sat( σ / Φ) (6) Othr typs of continuous approximation of discontinuous control in th boundary layr ar possibl too. Th following xampl givs anothr continuous control function [3] [4]: σ u h (7) σ ε whr ε > 0 is th boundary layr width (a small numbr). Th ida of th boundary layr from th point of viw of its authors and subsqunt contributors was supposd to totally liminat chattring. Howvr th aspct of growth of th rlativ dgr du to th introduction of parasitic dynamics was not considrd and th conclusion about chattring limination was mad for th systm containing only principal dynamics. W aim to analyz prformanc of SM systms with boundary layr continuous control in which parasitic dynamics ar prsnt. Th analysis is prsntd in th following thr sctions with th us of thr diffrnt mthods: th dsibing function th Popov s mthod and th Poincar map mthod. III. DESCRIBING FUNCTION NLYSIS OF SM SYSTEMS WITH BOUNDRY LYER CONTINUOUS CONTROL W apply th dsibing function (DF) mthod [11] to th analysis of motions in th systm (1) () having th control (6) (7) or anothr singl-valud monoton odd-symmtric continuous function f(σ) fully locatd in th 1st and 3rd quadrants and satisfying th following conditions: f (σ ) h (8) d f ( σ ) h dσ σ 0 Φ d f ( σ ) < 0 (10) dσ Th first condition spcifis that th function must b saturating and th scond condition sts th limit on th drivativ of this function matching it with th drivativ of th linar approximation (5). Th nonlinarity satisfying conditions (8)-(10) is fully locatd in th ara dfind by th saturation function (6) (Fig. 1). Fig. 1. Boundary layr nonlinaritis (9) W find th DF of th nonlinarity f (σ ) as follows [11]: π N ( a) f ( a sinψ ) dψ (11) πa 0 Bcaus th function is odd-symmtric w rwrit (11): π / 4 N ( a) f ( a sinψ ) dψ (1) πa 0 It follows from (8)-(10) that th DF of th nonlinarity f (σ ) is a monoton function of amplitud a satisfying h Φ Φ Φ 0 < N( a) sin 1 Φ a a π a (13) N ( 0) h / Φ (14) For th nonlinarity givn by (7) d f ( σ ) N(0) h ε that falls into th dfinition (8)- dσ σ 0 (10) whn ε Φ. Th condition of th absnc of chattring as slf-xcitd oscillations in th closd-loop 758

3 systm that includs th plant (with possibl parasitic dynamics) th sliding surfac and th considrd nonlinar controllr can b formulatd as th absnc of solution of th harmonic balanc quation: ω 0 a 0 Wl / N( a) (15) whr W l is th frquncy rspons of th linar part (plant and sliding surfac). Condition (15) can b dpictd in th complx plan as in Fig. as th condition of th absnc of intrsction btwn th Nyquist plot of th linar part and th ngativ rciprocal of th DF of th nonlinarity. Fig.. Condition of chattring xistnc Th ngativ rciprocal DF of th nonlinarity dfind by (8)-(10) is always locatd on th ral axis lft of th point ( Φ h j0) ( 1 km j0). From Fig. on can s that th systm with continuous approximation of control in th boundary layr may or may not hav chattring dpnding on th contribution of parasitic dynamics. If parasitic dynamics ar not prsnt and th Nyquist plot approachs th origin from blow along th imaginary axis so that thr is no point of intrsction (plot W 1 (jω)) chattring dos not occur. If parasitic dynamics ar prsnt and thr is a point of intrsction of W l (jω) with th ral axis but this point of intrsction is locatd right of th point ( 1 km j0) (plot W (jω)) thn chattring dos not occur ithr. Yt if parasitic dynamics ar prsnt and thr is a point of intrsction of W l (jω) with th ngativ rciprocal DF of th nonlinarity (plot W 3 (jω)) thn chattring occurs. Th frquncy and th amplitud of chattring (whn it occurs) can b asily found from th harmonic balanc quation. Bcaus ngativ rciprocal of th DF for any f(σ) from th considrd class is locatd on th ral axis th o frquncy of chattring Ω will corrspond to 180 phas o lag of th linar part: argw l ( jω) 80. Th amplitud of chattring is found from th sam harmonic balanc quation but will dpnd on th nonlinarity f(σ). It is also worth noting that th prsntd DF analysis fully agrs with analysis of this systm as of a linar on subjct to th considration of only saturation nonlinarity and th assumption that th motion occur in th linar zon of th nonlinarity. Indd th Nyquist stability itrion would stat that th frquncy rspons plot of th opn-loop systm should not ncircl th point (-1j0) or altrnativly th plot W l should not ncircl th point ( 1 km j0) which coincids with th DF analysis rsults for th sam nonlinarity. Yt th DF analysis is valid for th whol class of nonlinaritis. Thrfor chattring supprssion by th continuous approximation of control in th boundary layr is possibl only if th valu of k m is sufficintly small. This howvr would rduc th systm prformanc which rvals a trad-off btwn th possibility of chattring supprssion and systm prformanc. IV. NLYSIS OF SM SYSTEMS WITH BOUNDRY LYER CONTINUOUS CONTROL BSED ON POPOV S STBILITY CRITERION It is known that V.M. Popov s itrion of absolut stability [8] allows on to stablish sufficint conditions of stability of an quilibrium point for som class of nonlinar functions. This proprty bcoms vry convnint with rspct to th analysis of th control approximation in th boundary layr. gain w considr th class of singl-valud oddsymmtric nonlinar functions that dfin control u givn by (8)-(10). Ths formulas as was mntiond arlir dfin th class of nonlinaritis satisfying th so-calld sctor condition that fully falls in th scop of th conditions of th Popov s thorm [8]. It says that for th systm comprising th nonlinarity satisfying th sctor condition [0;k m ] (s Fig. ) and th stabl linar part with controllabl pair (;B) th point of origin will b globally asymptotically stabl if thr xists a strictly positiv numbr α such that 1 ω 0 R[ ( 1 jαω) Wl ] (16) km for an arbitrarily small >0. Considring th constraints on th class of nonlinaritis (8)-(10) w can s that for this class k m is k h m for Φ vry nonlinarity and thrfor th stability analysis again arrivs at chcking th location of th point ( 1 k m j0) ( Φ h j0) rlativ to a crtain frquncydomain charactristic in th complx plan. This frquncydomain charactristic is th Popov s curv (modifid frquncy rspons) which is formd from th frquncy rspons W l by multiplying th imaginary part by frquncy ω: Wm ( ω) RWl jω ImWl. Th sufficint condition of absolut stability of th origin and thrfor th condition of th limination of chattring is th possibility of drawing a straight lin through th point ( 1 km j0) that would not intrsct th plot W m (ω) (Fig. 3). This is possibl only if th valu of k m is sufficintly small which in turn would rduc th systm 759

4 prformanc. Thrfor as it was notd arlir thr is a trad-off btwn th possibility of chattring supprssion and th systm prformanc. Fig. 3. Condition of chattring limination (pr Popov s itrion) V. NLYSIS OF CHTERING IN SM SYSTEMS WITH BOUNDRY LYER CONTINUOUS CONTROL BSED ON POINCRE MP W now approach th problm bing studid assuming that chattring xists and driv Poincar map for th systm bing analyzd. W find analytical solution for vry part of th pic-wis linar control and join th solutions. Considr solutions for th following controls: h if σ b u Kσ if b < σ < b (17) h if σ b ssum th xistnc of a symmtric limit cycl and find paramtrs of this limit cycl. Lt θ 1 b th duration of motion undr control uh and θ b th duration of motion undr control ukσ so that th priod of th oscillation is T ( θ 1 θ ). W assum that tim t0 corrsponds to th starting point of control uh so that th following rlationships hold: y ( 0) b h K y& ( 0) > 0 y ( θ 1 ) b h K y& ( θ 1 ) < 0 y( θ 1 θ ) b h K y& ( θ1 θ ) < 0 (s Fig. 4). Considring th following rspons of th linar part to th constant control u: t t x ( t ) x (0) ( I) Bu (18) w can find th following mappings: ρ x( 0) η x( θ1) and η x( θ 1) ρ x( θ1 θ ) th fixd point of which will b dfind by th idntity ρ ρ (considring th condition of a symmtric motion). Th mapping ρ x( 0) η x( θ1) is givn by th following formula (on th tim intrval t [ 0; θ1] th control uh): θ η ρ ( I) 1 θ1 Bh. (19) Th mapping η x( θ 1) ρ x( θ1 θ ) can b drivd as follows considring that on th tim intrval t [ θ 1; θ1 θ ] th control ukσ: Th quations of th systm can b considrd as quations of th fr (unforcd) motion in th closd-loop systm (follows from (1) and (17): x & x BKσ x KBC x (0) whr KBC. Thrfor ρ θ η. (1) Now solv th quation ρ ρ that dfins th fixd point. θ θ1 θ1 ρ ( ρ ( I) Bh) From th last quation w find th following solution: / BCKθ / ( I θ ) ( ) ρ () BCKθ Bh Considring th following rlationship btwn ρ and η θ η ρ w can find η as follows: / BCKθ ( I / θ 1 ) ( I ) Bh η. (3) From formula () on can s that if intrvals duration θ 1 and θ ar tratd as indpndnt variabls th priodic motion of priod T is stablishd in th systm with th valu of th stat vctor at t0 givn by (). Howvr durations θ 1 and θ ar not indpndnt variabls but th rsult of th switching conditions (from on control to th othr on). Thrfor θ 1 and θ ar found from th following two quations: Cρ b and Cη b which ar complmntd by th inqualitis y& ( 0) > 0 and y& ( θ 1 ) < 0 that nd to b chckd. Considring also that b h K w writ th following two quations for θ 1 and θ. / BCKθ T / K 1 1 ( θ ) ( BC θ ) B K C I (4) / BCKθ T / 1 1 ( θ ) ( I ) B K C I. (5) Solution of (4) (5) can b bttr prsntd as th solution for th frquncy Ω π ( θ 1 θ ) (or priod) of oscillations and for th rlativ intrval duration γ θ θ. ( ) 1 1 θ It is worth noting that (4) and (5) do not dpnd on such paramtr of th control as h but dpnd only on K which rsults in th invarianc of chattring frquncy and rlativ intrval duration with rspct to th control amplitud h (as far as th valu of K is constant). nalysis of th drivd rlationships is also intrsting from th point of viw of th dpndnc of th solution on th valu of gain K. W can lgitimatly assum that th inas of gain K will rsult in th inas of θ 1 and in th das of θ so that whn K th control transforms into th rlay control with an idal rlay that corrsponds to θ 0 and θ 1 T /. This limiting cas rsults in th following 760

5 rlationship: / / ( I ) ( I) Bh ρ r whr th subsipt r rfrs to th rlay systm. For th practical solution of (4) and (5) it is rasonabl to assum that th frquncy Ω π ( θ 1 θ ) of chattring blongs to th intrval [ Ω r ; Ω l ] whr Ω r is th frquncy of chattring in th rlay systm with th sam plant which corrsponds to th cas of θ / 1 T θ 0 and Ω l is th frquncy of oscillations (that ar only thortically possibl) in th marginally stabl linar systm with th sam plant which corrsponds to th cas of θ 1 0 θ T /. Ths two frquncis ar clos and from th point of viw of th dsibing function analysis ar vn th sam. This fact significantly simplifis th solution of (4) (5) as solution of (4) (5) is complx and involvs sorting out th solutions that do not provid a fixd point of Poincar map (quation (4) or (5) is th condition of th quality of th plant output to th valu of b which may b providd by a non-priodic signal). Th proposd solution schma involvs variation of paramtrs Ω and γ within th intrvals: Ω [ ; ] [ 0;1 ] Ω r Ω l γ (6) Th practical aspcts of finding θ 1 and θ ar illustratd by th xampl in th following sction. VI. RELION TO STBILITY OF LINER SYSTEMS If th valu of K is dasd to th lvl of K K whr K is th gain of th proportional control that maks th systm marginally stabl thn th paramtrs of th oscillations approach th following limits: θ 1 0 θ T /. In fact th valu of K can b found as th limiting cas of (4) (5) whn θ 1 0 θ T /. In this cas th nonlinar systm with saturation transforms into th marginally stabl linar systm rvaling an undampd oscillation. Th valu of th saturation can srv as a natural masur of th siz of th domain in th dfinition of stability. Bcaus w ar considring a linar systm th bhavior of th systm output σ can charactriz th stability of th systm. Thrfor considring th dfinition of stability that says that th quilibrium point x 0 is stabl if for any R > 0 thr xists r > 0 such that if x (0) < r thn x ( t ) < R for all t 0 and analyzing th bhavior of σ w can conclud that if σ (0) < r and σ ( t ) < R b h / K for all t 0 th systm will b stabl. This in turn happns if K < K. W thrfor can formulat th following statmnt. Thorm (robust stability). Linar systm with plant givn by (1) () and control u Kσ is stabl for all K ( BCK ) T / 0 < K < if matrix H I is singular K whr T is th priod of oscillations in th marginally stabl systm with stat matrix bing BCK and K bing th smallst ral positiv numbr that maks H singular. Proof. Considr th limiting cas of (4) for θ 0 θ T / : BCK T ( / ) C I / / θ lim θ T / BCKT / 1 B K / lim θ θ T / 1 On can s that 0 which rquirs that ( BCK ) T / H I should b singular (for K to b finit) which in turn mans that th following should hold: dt H 0. Bcaus th matrix BCK is th stat matrix of th closd-loop systm and this systm is marginally stabl this matrix should hav a pair of ignvalus with zro ral parts. Th matrix ( BCK ) T / as bing th stat transition matrix with th tim bing th half-priod has at last two ignvalus qual to 1 and othr ignvalus within th unit disk. (Not: ( BCK ) T / can b considrd a Poincar map in th marginally stabl systm having a priodic procss.) This lads to dt H 0. VII. EXMPLE Considr th systm givn by th following transfr function W p ( s) 1/ ( s s 1) with th sliding surfac givn by W ss ( s) s 1 and th parasitic dynamics is givn by W pd ( s) 1/ ( s 0.01s 1). Ths modls giv th following transfr function of th linar part: s 1 W l ( s) s s s 1.01s 1 and th stat spac modl with th following matrics: B C [ ] Th task of analysis is to find th itical gain K of th saturation nonlinarity that nsurs stability of th systm K 0; and find paramtrs of priodic motions for [ K ] for a fw diffrnt points of [ ; ) K K. Th analysis of th linar systm with th fdback gain K givs th itical gain valu of K with th frquncy of thortically possibl oscillations in th marginally stabl systm Ω s. Th ignvalus l 761

6 of th matrix BCK j and ignvalus of th matrix ( BCK ) T / ar ( ). s a rsult matrix H is singular bcaus ( BCK dt ) T / I. ar ( j100.01) ( ) 0 Th rsults of th solution of quations (4) (5) for K with trating paramtrs Ω and γ as indpndnt variabls satisfying (6) whn solving quations (4) and (5) ar prsntd in Tabl 1 (for a fw diffrnt valus of γ ). Tabl 1. Computd gain and frquncy of chattring γ K Ω match to th prdictd valus of frquncy Ω and rlativ intrval durationγ. s shown by th analysis prsntd abov Ω and γ do not dpnd on th control amplitud (saturation valu) h. If w considr h4 in our xampl (as in Fig. 4) w will conclud that chattring will b supprssd in th sliding mod systm if th width of th boundary layr is lss than This valu may b accptabl to nsur insignificant dtrioration of th systm prformanc. VIII. CONCLUSION fw approachs to analysis of th xistnc of chattring in sliding mod control systms having a continuous approximation of discontinuous control in th boundary layr is proposd in th papr. Ths approachs ar basd on th dsibing function mthod th Popov s approach and th Poincar map analysis. In th last cas analysis is providd only for th boundary layr approximation givn by th saturation function. nalysis of paramtrs of chattring (frquncy and shap of oscillations) is givn for th boundary layr approximation givn by th saturation function. (a) For K (b) For K11.06 REFERENCES [1] V. Utkin Sliding Mods in Control and Optimization Brlin: Springr-Vrlag 199. [] J.J. Slotin and S. S. Sastry Tracking control of nonlinar systms using sliding surfacs with application to robot manipulator Int. J. Contr. vol. 38 no. pp [3] G. mbrosino G. Clntano and F. Garofalo Variabl structur mod-rfrnc adaptiv-control systms Int. J. Control vol. 39 no. 6 pp [4] J.. Burton and. S. I. Zinobr Continuous approximation of variabl structur control Int. J. Systm Scinc vol. 17 no. 6 pp [5] K.D. Young V.I. Utkin and U. Ozgunr control nginr s guid to sliding mod control IEEE Trans. Control Systm Tchnology vol. 7 no. 3 pp [6] I. Boiko nalysis of Closd-Loop Prformanc and Frquncy- Domain Dsign of Compnsating Filtrs for Sliding Mod Control Systms IEEE Trans. utomatic Control vol. 5 no. 10 pp [7] I. Boiko and L. Fridman nalysis of chattring in continuous slidingmod controllrs IEEE Trans. utomatic Control vol. 50 no. 9 pp [8] V.M. Popov bsolut stability of nonlinar control systms of automatic control. utomation and Rmot Control vol. no.8 pp [9] I. Boiko Oscillations and transfr proprtis of rlay srvo systms with intgrating plants IEEE Trans. utomatic Control vol. 53 no. 11 pp [10] I. Boiko Discontinuous Control Systms: Frquncy-Domain nalysis and Dsign Boston: Birkhausr 009. [11] D.P. thrton Nonlinar Control Enginring Dsibing Function nalysis and Dsign Workingham Brkshir: Van Nostrand Rinhold Company Limitd (c) For K Fig. 4. Priodic motions in systm with saturation Th rsults of simulations of th systm with saturation for th 3 diffrnt gains from Tabl 1 ar prsntd in Fig. 4. In ovrall th rsults of simulations provid a vry good 76