Fuzzy AntiWindup Schemes for PID Controllers


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1 Intrnational Journal of Applid Enginring Rsarch ISSN Volum Numbr 3 (26) pp Rsarch India Publications Fuzzy AntiWindup Schms for PID Controllrs E. Chakir El Alaoui, H. Ayad and S. Doubabi Laboratory of Systm Elctric & Tlcommunication (LSET), Faculty of Scincs & Tchniqus Guliz Av. Abdlkrim Elkhattabi B.P 549 Marrakch MOROCCO, ayad@fstgmarrakch.ac.ma Abstract In this papr, w propos fuzzy compnsator basd on Anti Windup (AW) approachs for linar systms subjct to control constraints which ar in turn controlld by a standard PID controllr and a fuzzy PID controllr. Th fuzzy compnsation block provids fdback corrction trms to improv th systm bhaviour. Th fuzzy PID controllr is dsignd to improv th control prformanc. Simulation rsults show that th proposd fuzzy mthod givs gnrally bttr tracking prformancs than th conditioning tchniqu. Kywords: Linar Systms, Control constraints, Compnsator, AntiWindup, Fuzzy. Introduction PID controllr is on of th arlist industrial controllrs. It has many advantags: it is conomic, simpl, asily tund and robust. It has provd it ffctivnss in rgulating a wid rang of procsss. It dos not rquir an xact modl, and hnc it can b usd for procsss whos modls ar considrably difficult to driv (s []). PID controllr consists of thr trms: Proportional action, Drivativ action (to spd up th rspons) and Intgral action (to liminat th stady stat rror). In practic, th constraints ar inhrnt to all kinds of physical, chmical or ral procss, may aris from physical limitations on th procss or may dpnd on its natur. In industry, for xampl, on can impos or hav th maximum flow for valvs, limitations on voltags and currnts in lctrical procsss, limitations on ph and concntrations in chmical procsss, tc. Hncforth, constraind systms ar connctd to a wid sprad of applications and thir study is of continuing intrst in control community.
2 296 H. Ayad t al Whn limitation happns, if th controllr is initially dsignd to oprat in a linar rang, th closd loop prformanc will significantly dtriorat with rspct to th xpctd linar prformanc. This prformanc dtrioration is rfrrd to as windup. Morovr, th nonlinarity of th actuator is not always known in advanc. A mor common approach in practic to diminish th ffct of windup is to add xtra fdback compnsation at th stag of control implmntation. As this compnsation aims, it is rfrrd to as antiwindup (AW). Th topic of antiwindup has bn studid ovr a long priod by many authors, and th most popular tchniqus ar dscribd (s for xampl [2][6] without bing xhaustiv). Most ral systms cannot b rprsntd by linar dynamics. But, undr som assumptions it is oftn possibl to modl th dynamical bhaviour of practical systms with a linar modl having som uncrtaintis. Ths ar gnrally inducd from th diffrnc, soms considrabl, btwn th ral bhaviour of th systm and th modl usd to dsign th controllr. Fuzzy controllrs ar rulbasd nonlinar controllrs; thrfor, thir main application should b th control of nonlinar ons. Howvr, sinc linar systms ar good approximations of nonlinar systms around th oprating points, it is of intrst to study fuzzy control of linar systms. Additionally, fuzzy controllrs du to thir nonlinar natur may b mor robust than linar controllrs vn if th plant is linar [7]. Our contribution consists of proposing two anti windup schms basd on fuzzy logic dsign. Th first on uss a classical PID controllr with a fuzzy AW compnsation (PID_FAW). Th scond uss a fuzzy PID controllr with th sam fuzzy AW compnsation (FPID_FAW). Th papr is organizd as follows: sction 2 is dvotd to prsnt classical PID controllr, two AW approachs: Conditioning tchniqu in subsction 2. and th proposd PID_FAW in subsction 2.2. In sction 3, th scond approach FPID_FAW is prsntd. Sction 4 prsnts simulation rsults on a st of bnchmark systms: in subsction 4., tracking prformancs of th conditioning tchniqu and PID_FAW ar compard, whil in subsction 4.2, th potntial of FPID_FAW is tstd. AW approachs for Classical PID Controllr Conditioning Tchniqu Th windup problm can b xplaind by th fact that whn th control signal saturats th actuator, a furthr incras of th control signal will not lad to a fastr rspons of th systm. If intgration rror continus in this cas, th intgrator valu bcoms vry larg, without having any ffct on th procss output. Th control rror thn has to b of th opposit sign for a long to bring th intgrator back to its stady stat valu. Windup problms ar ncountrd in PI / PID controllrs for controlling linar systms with control input nonlinaritis. To illustrat this phnomnon, considr a closdloop control systm (Fig.) containing a PID controllr and a magnitud saturation givn by th following quation:
3 Fuzzy AntiWindup Schms for PID Controllrs 297 ur umax if u = (t) > min < umax (t) u(t) if u u(t) umax (2.) umin if u(t) umin w PID Controllr u(t) ur(t) Procss y(t) P (s) saturation Figur. Closd loop control systm Th PID controllr can b dscribd as follows: = E(s) st Y(s) st s T d U(s) K E(s) (2.2) i d N whr u is th controllr output, y is th procss output, w is th rfrnc signal, =wy is th procss tracking rror and th capital lttrs U, E and Y dnot th Laplac transforms of u, and y rspctivly, K is th proportional gain, T i is th intgral constant, and T d is th drivativ constant, th high frquncy gain N is usually st btwn 7 and 5. Lt s suppos that th controllr and th procss ar in stady stat. A larg positiv stp chang in w causs a jump in u, so that th actuator saturats at high limit (if K>). Thus, ur bcoms smallr than u, and y is slowr than in th unlimitd cas. Th intgral trm incrass much mor than on in th unlimitd cas and it bcoms larg. Whn y approachs w, ur still rmains saturatd or clos to saturation du to th larg intgral trm; u dcrass aftr th rror has bn ngativ for a sufficintly long. This lads to a larg ovrshoot and a larg sttling of th procss output. Th conditioning tchniqu (Fig.2) consists to fdback uur to th intgral trm of th PID controllr through a compnsator with transfr function F(s) = K ( K is th proportional gain).
4 298 H. Ayad t al w sktd s T d N u K ur P(s) y K st i F(s) Figur 2: Conditioning Tchniqu PID Controllr with Fuzzy AW Compnsation Actually, fuzzy logic is applid in much application ara from hom appliancs to industrial control systms. In gnral mploying fuzzy logic should achiv th following objctivs: oprat without human intrvntion, b abl to cop with highly variabl, complx, nonlinar systms, satisfy oprational spcifications and prformanc critria... In th cas of varying procsss, fuzzy logic can b mployd to adapt th paramtrs of PID controllr. In our cas, w intrst to act on signals rathr than on paramtrs to improv th PID bhavior using fuzzy compnsation. Th fuzzy AW compnsation is dsignd whos aim is to rduc th intgral action whn th actuator saturats. It has two inputs v, dv/dt and on output ua (s Fig.3). Paramtrs K, K2 and K3 ar scaling gains. If th rror is positiv for a substantial, th control signal gts saturatd at th high limit u max. If th rror rmains positiv for som subsqunt to saturation, th intgrator continus to accumulat th rror causing th control signal to bcom mor saturatd. Th control signal rmains saturatd at this point bcaus of th larg valu of th intgral. It dos not lav th saturation limit until th rror bcoms ngativ and rmains ngativ for a long to allow th intgral part to com down to a smallr valu. Th advrs ffct of this intgral windup is in th form of larg ovrshoots in th output y and soms vn instability. To avoid windup, an xtra fdback path is providd in th controllr by masuring th actuator output u r and forming an rror signal as th diffrnc btwn th output u of th controllr and th actuator output u r. This rror signal v is th input of th intgrator through th fuzzy AW compnsator. Whn th actuator saturats, th fdback signal ua attmpts to driv v to zro. This nabls us to stablish th rul bas for fuzzy AW compnsation (s tabl ).
5 Fuzzy AntiWindup Schms for PID Controllrs 299 Th following figur shows th structur of th proposd control schm. w sktd s T d N u K ur K st i u I P(s) y ua v y K3 Fuzzy AW Compnsator K d/dt K2 Figur 3: PID controllr with fuzzy AW compnsation To dsign th fuzzy AW compnsation w follow th standard procdur of fuzzy controllr s dsign which consists of: fuzzification, rul bas stablishmnt and dfuzzification ([9]). All Mmbr Functions of inputs and output ar chosn triangular in form and normalizd in th rang [,]. Tabl : Rul bas for fuzzy AW compnsation v N Z P N N Z P dv/dt Z N Z P P N Z P Whr N, Z and P ar Ngativ, Zro and Positiv fuzzy sts rspctivly. Th usd fuzzy infrnc mthod is th sumprod mthod which intrprts th and and thn connctivs as th Cartsian product and th or connctor by th man valu. Th dffuzification mthod applid is th cntr of gravity mthod. AW Approach for Fuzzy PID Controllr In this sction w dscrib th basic structur of th control schm, and motivat th rational for FuzzyProportionalIntgralDrivativ.
6 3 H. Ayad t al Th proposd structur of th control schm is th sam as prsntd in Fig.3 but instad of th convntional PID controllr w us a fuzzy PID controllr, kping th sam fuzzy AW compnsation. In th nxt, w prsnt th dsign of th fuzzy PID controllr. A convntional PID control algorithm can b xprssd as: u = K ( T d ) dt Ti (3.) whr is th procss tracking rror, and ar th first and th scond drivativ of th rror rspctivly. If, and ar fuzzy variabls, quation (3.) will bcom fuzzy PID control algorithm. As thr ar thr fuzzy inputs, th rul bas is a thr dimnsional on and should hav n 3 ruls (n is th numbr of mmbr functions) for a complt rul bas. This will xpand th rul bas gratly and mak th dsign mor difficult. In ordr to rduc th complxity of th rul bas dsign and incras fficincy, a simplifid fuzzy PID controllr ([9]) can b prsntd as shown in Fig.4 by sharing a common rulbas for both FuzzyProportionalIntgral and FuzzyProportional Drivativ parts. This simplifid fuzzy PID can achiv th similar prformanc as th Fuzzy PID. Th simplifid fuzzy PID controllr is simpl in structur, asy in implmntation and fast in computation. Th global structur of th control schm is shown in th following figur whr K, Kd, Ku and Ku 2 ar scaling gains. w Simplifid Fuzzy PID controllr d/dt K Kd Two Dim. Rul Bas ud Ku 2 /s Ku u ur Procss P (s) K3 Fuzzy AW K d/dt v K2 Figur 4: Fuzzy PID controllr with fuzzy AW compnsation
7 Fuzzy AntiWindup Schms for PID Controllrs 3 Th mmbr functions associatd to fuzzy variabls, d/dt and ud ar chosn triangular in form and normalizd in th rang [,]. Th following complt rul bas tabl is applid. Tabl 2: Rul bas for simplifid fuzzy PID controllr N Z P N GN MN Z d/dt Z MN Z MP P Z MP GP whr N, Z, P, GN, MN, GP, MP ar Ngativ, Zro, Positiv, Grat Ngativ, Mdium Ngativ, Grat Positiv, Mdium Positiv fuzzy sts rspctivly. Th dfuzzification and infrnc mthods ar th sam as thos adoptd in subsction 2.2. Simulations and Prformanc Analysis Th proposd fuzzy AW mthods ar tstd through simulation on th following bnchmark systms ([9]): G (s) = 3 (, G s s) 2 (s) = 7 ( s), G 3 (s) = 2s 3. ( s ) Th simulation is carrid out using Simulink in Matlab. PID_FAW and Conditioning Tchniqu In this subsction, tracking prformancs of PID_FAW and conditioning tchniqu ar compard. Th PID paramtrs ar tund using th mthod basd on constraind optimization dvlopd in []. For systms G, G 2 and G 3 th PID paramtrs obtaind in [] ar summarizd in th following tabl. Tabl 3: PID paramtrs K Ti Td G G G Th filtr factor N is takn qual to.
8 32 H. Ayad t al In this study, th scaling gains K, K2 and K3 of th fuzzy AW compnsation ar tund by hand. In nxt rsarch papr a mthod for choosing ths gains will b studid. For ach bnchmark systm, th simulation rsults will corrspond to th following cass: a) unlimitd cas : th systm is controlld by only th PID with no saturation nor AW compnsation b) Limitd cas: th systm is controlld by th PID with saturation and no AW compnsation. This cas will hnc illustrat th windup phnomnon. c) Conditioning Tchniqu d) PID_FAW First xampl: G (s) = P(s) = s( s ) 3 Th following paramtr valus ar chosn: K= ; K2= ; K3=; umin=.; umax=.6. For a stp chang of w from to, w obtain th following rsults: As w can s, th limitd cas xhibits clarly th windup problm which is rsolvd by both AW approachs and (d). Th proposd PID_FAW mthod givs good tracking prformancs than th conditioning tchniqu. Scond xampl: G 2 (s) = P(s) = 7 ( s) Th following paramtr valus ar chosn: K = ; K2 = ; K3 = ; umin = ; umax = (d)..8 output y.8.6 control ur.6.4 (d) Figur 5: Stp rsponss (lft) and control signals (right) of G (s) Unlimitd cas, Limitd cas, Conditioning Tchniqu, (d) PID_FAW
9 Fuzzy AntiWindup Schms for PID Controllrs (d) (d) output y.8.6 control ur Figur 6: Stp rsponss (lft) and control signals (right) of G 2 (s) Unlimitd cas, Limitd cas, Conditioning Tchniqu, (d) PID_FAW As shown in Fig.6, w can driv th sam conclusions as in th first xampl. Third xampl : G 3 (s) = P(s) = 2s ( s ) 3 Th following paramtr valus ar chosn: K = ; K2 = ; K3 = ; umin = ; umax =.5. Also in this cas, w can driv th sam conclusions as in th first xampl..2.4 (d).2.8 (d) output y.6.4 control ur Figur 7: Stp rsponss (lft) and control signals (right) of G 3 (s) Unlimitd cas, Limitd cas, Conditioning Tchniqu, (d) PID_FAW Fuzzy PID with fuzzy AW compnsation (FPID_FAW) Also in this cas, for ach bnchmark systm, th simulation rsults will corrspond to th following cass: a) unlimitd cas : th systm is controlld by only th fuzzy PID with no saturation nor AW compnsation
10 34 H. Ayad t al b) Limitd cas: th systm is controlld by th fuzzy PID with saturation and no AW compnsation. c) FPID_FAW First xampl: G (s) = P(s) = s( s ) 3 Th following paramtr valus ar chosn: K = ; K2 = ; K3 = ; umin =.; umax =.8; K =.99; Kd = 6; Ku =.5; Ku2 =.8. For a stp chang of w from to, w obtain th following rsults: output y.8.6 control ur Figur 8: Stp rsponss (lft) and control signals (right) of G (s) Unlimitd cas, Limitd cas, FPID_FAW As w can s, th Limitd cas xhibits always th windup problm which is rsolvd by our proposd AW approach. Also, FPID_FAW givs good tracking prformancs. Scond xampl: G 2 (s) = P(s) = 7 ( s) Th following paramtr valus ar chosn: K = ; K2 = ; K3 = ; umin =.94; umax =.5; K =.7; Kd = 2.; Ku =.29; Ku2 =.9. For a stp chang of w from to, w obtain th following rsults: As shown in Fig.9, w can driv th sam conclusions as in th first xampl. Third xampl : G 3 (s) = P(s) = 2s 3 ( s ) Th following paramtr valus ar chosn: K = ; K2 = ; K3 = ; umin =.94; umax =.5; K =.7; Kd = 2.; Ku =.29; Ku2 =.9.
11 Fuzzy AntiWindup Schms for PID Controllrs output y control ur Figur 9: Stp rsponss (lft) and control signals (right) of G 2 (s) Unlimitd cas, Limitd cas, FPID_FAW For a stp chang of w from to, w obtain th following rsults.2.2 output y control ur Figur. Stp rsponss (lft) and control signals (right) of G 3 (s) Unlimitd cas, Limitd cas, FPID_FAW Also in this cas, w can driv th sam conclusions as in th first xampl. Conclusion In this papr, w hav tratd th windup problm in linar systms with input saturation and controlld by a classical PID or by a fuzzy PID controllr. Two AW mthods hav bn dsignd: th first on consists of th classical PID controllr plus a fdback fuzzy compnsation block, whil th scond consists of a fuzzy PID controllr plus th sam fdback fuzzy compnsation block. Th proposd mthods ar tstd through simulation on a st of bnchmark systms. Th obtaind rsults affirmd th potntial of both mthods to trat th windup problm. Morovr, th first mthod givs gnrally bttr tracking prformancs than th conditioning tchniqu.
12 36 H. Ayad t al Rfrncs [] K. J. Astrom and T. Häggland, Automatic tuning of PID Controllrs, Instrumnt Socity of Amrica, 988. [2] K. J. Astrom and B.Wittnmark, ComputrControlld Systms Thory and Dsign. Englwood Cliffs, NJ: PrnticHall, 984. [3] M.V. Khothar, P.J. Campo, M. Morari and C.N. Ntt, A unifid Framwork for th Study of AntiWindup Dsigns, Automatica, Vol.3, No. 2, pp , 994. [4] C. Bohn and D.P. Athrton, An analysis packag comparing PID antiwindup stratgis, IEEE Control Systms, 5 (2), 344, 995. [5] Y. Png, D. Vrancic, R. Hanus, A rviw of antiwindup, bumplss and conditiond transfr, IFAC, 3 th trinnial world congrss, San Francisco, 79 84, 996. [6] R. Hanus and Ph. Bogarts, Fdback and Fd forward Conditioning Tchniqus Europan Journal of Control, Vol. 6, pp , 2. [7] W. Gharib and G. Nagib, Fuzzy intrvntion in PID controllr dsign, IEEE procdings, , 2. [8] D. Driankov, H. Hllndoorn and M. Rinfrank, An introduction to fuzzy control, SpringVrlag Brlin Hidlbrg, 993. [9] Li HangXiong and H.B. Gatland, Convntional Fuzzy Control and its Enhancmnts, IEEE transactions on systms, Man and Cybrntics, 26(5), 79797, 996. [] H. Panagopoulos, K. J. Astrom and T. Häggland, Dsign of PID controllr basd on constraind optimization, Procdings of th Amrican Control Confrnc, , 999.
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