STABILITY ANALYSIS OF A FUZZY LOGIC CONTROLLER

Poc. of 997 Amican Contol Confnc, pp. 33-335, Albuququ, NM, Jun 4-6, 997. STABILITY ANALYSIS OF A FUZZY LOGIC CONTROLLER Gudal Aslan and Kwang Y. L Dpatmnt of Elctical Engining Th Pnnsylvania Stat Univsity Univsity Pak, PA 68 kylc@ng.psu.du Abstact Th ida of studying th stability of fuzzy logic contolls within th Lu poblm famwok is xtndd to th cas wh th plant modl contains paamt unctainty. A fuzzy logic contoll is dsignd to contol a nucla acto. Thn, th stability and th obustnss of th ovall systm is studid by using th cnt dvlopmnts in th obust contol thoy. To suppot th thotical sults, a sis of simulations is pfomd fo diffnt plant conditions. Whil fuzzy logic contoll povidd good pfomanc fo th nominal plant, it also maintaind th stability und th off-nominal plant conditions.. Intoduction Fuzzy st thoy was intoducd by Lotfi A. Zadh as a mthod fo modling human asoning pocss []. Fuzzy contol has bn a succssful application of this thoy and implmntd in many industial systms. Simplicity and flxibility of fuzzy contolls hav playd an impotant ol in thi accptanc. Moov, it has bn claimd that fuzzy contolls a obust with spct to paamt vaiations in th plant modl. This claim has bn mainly suppotd by vaious applications and simulation sults []. On of th most impotant issus that aiss in th contol thoy is th stability of th systms. Futhmo, mathmatical modls of th systms a typically impcis mostly bcaus of th paamt vaiations in th plant modl and th unmodlld high fquncy dynamics. Thfo, maintaining th systm stability in th fac of unctaintis is a pimay concn fo th dsign ngin. Stability and obustnss of th convntional systms hav bn xtnsivly studid in th past. But, littl pogss has bn mad towads invstigating stability and obustnss of fuzzy contolls [3-9]. Phaps, th difficulty in studying stability and obustnss of th fuzzy contolls is du to th fact that fuzzy logic contol dsign is basd on huistics and xpt knowldg whas convntional contol dsign is basd on mathmatical modl of th plant. Though ths two appoachs a fundamntally diffnt with spct to thi bass, th sulting contolls a almost th sam with spct to thi stuctus, which a nonlina contolls. In this pap, th ida of studying th stability of fuzzy contol systms within th Lu poblm famwok [,] is xtndd to th cas wh th plant modl contains paamt unctainty. In paticula, th conditions of th cicl cition can b vifid by xamining th Nyquist plot of th plant. In addition, cnt dvlopmnts in th paamtic obust contol thoy has mad it asy to min th impotant fquncy domain chaactistics of unctain lina systms. Th stict positiv alnss o th bounday of th Nyquist plots of a tansf function family containing paamtic unctainty can b mind by chcking som pominnt mmbs of th family. Such a study is givn in this pap fo th fuzzy contol of a nucla cto which was pviously tatd by Ramaswamy t al. [] without obustnss analysis.. Fuzzy Logic Contol of a Nucla Racto A fuzzy logic contoll is dsignd to contol th pow lvl of a nucla acto. Th ovall nonlina mapping of th fuzzy logic contoll is obtaind to vify th stability and obustnss poptis of th systm bfo implmntation. Fig.. shows th gnal stuctu of th closd-loop systm, wh n is th acto pow lvl, n d is th pow dmand, z is th od spd, and is th o signal. Th fuzzy logic contoll shown has fou majo pats, a fuzzifi, an infnc mchanism, a ul bas and a dfuzzifi... Fuzzificaton and Mmbship Functions Th basic ol of th mmbship functions is to convt a cisp valu to a fuzzy st. Two sts of mmbship functions hav to b chosn fo both input and output of th infnc mchanism. n d fuzzifi fuzzy logic contoll infnc mchanism ul bas dfuzzifi z acto modl Fig... Gnal stuctu of th closd-loop systm.... Input mmbship functions: Th bll-shapd functions a usd to convt th o signal into th linguistic tms: N (Ngativ), Z (Zo), P (Positiv), as shown in Fig..: tanh[ ( t ( ). )] μ Ngativ [()] t = 5 +, (.) μ Zo [()] t = xp[ ()] t, (.) tanh[ ( t ( ). )] μ Positiv [()] t = + 5, (.3) wh t ( ) dnots th o signal obtaind by subtacting th acto output fom th pow dmand. n

.5 n c i μ i * u ()= t i= n μi i=, (.4) o signal (t) -. -.5.5. Fig... Input mmbship functions.... Output mmbship functions: Th tiangulashapd functions w usd to xpss th od spd in linguistic tms, N (Ngativ), Z (Zo), P (Positiv), as shown in Fig..3. -..5 -.5.5. od spd z () t * wh u ( t ) is th output of th dfuzzification opation, ci is th cnt of th fuzzy output st implid in th i-th ul, and μ i is th lvanc of th i-th ul. By substituting th appopiat vaiabls into (.4), th ovall mapping of this fuzzy logic contoll is computd as follows: [ + + ]. 5 tanh[ 5(. )] tanh[ 5(. )] z( ) = Ψ() = tanh[ 5( +. )] + xp( ) + tanh[ 5(. )] (.5) wh dnots th o signal. Th ovall nonlina mapping of th fuzzy logic contoll is shown in Fig..4, which is a smooth mapping and gos though th oigin. Th fuzzy contoll psntd by such quation smbls a popotional contoll with a satuation nonlinaity..5 od spd z () Fig..3. Output mmbship functions... Rul Bas Th ul bas is simply composd of a collction of th fuzzy if-thn uls. In this wok, th simpl fuzzy if-thn uls a usd to contol th pow lvl of th acto: -. -.5.5. o signal t () if th o is Ngativ, thn th od spd is Ngativ, if th o is Zo, thn th od spd is Zo, if th o is Positiv, thn th od spd is Positiv. -.5.3. Infnc Mchanism Th infnc mchanism simply mins a mapping fom th input fuzzy sts to th output fuzzy sts basd on th fuzzy ifthn uls stod in th ul bas. Typically, th dcision making pocss involvs sval stps xplaind blow. In th fist stp, th fuzzy inputs to th infnc mchanism a compad with th pmiss of all th fuzzy if-thn uls in th ul bas to min th lvanc of th ach ul to th cunt fuzzy inputs. In th scond stp, using th poduct fuzzy implication, a fuzzy dcision is mad by modifying th fuzzy output sts in th consqunt pats of th all cuntly lvant uls. In th final stp, somtims calld agggation, th modifid fuzzy output sts fom all th uls a unifid to hav a singl fuzzy conclusion using th sum mthod..4. Dfuzzification and Ovall Mapping A fuzzy dcision psntd by th agggation of th modifid fuzzy output sts is mad in th infnc mchanism. This fuzzy dcision gading th cunt contol action has to b convtd to a cisp valu to contol th plant. Th pocss of finding th cisp valu that bst psnts th fuzzy dcision is calld dfuzzification. In this pap, th most popula dfuzzification mthod Cnt-Of-Aa (COA) is usd. If th sum mthod is usd fo th agggation, and th output mmbship functions a tiangula-shapd with th sam boundd wihs and a symmtic aound thi cnts, thn Fig..4. Th ovall mapping of th fuzzy logic contoll. 3. Stability and Robustnss Analysis In this sction, th us of nonlina systm and obust contol tchniqus fo th obust stability analysis of th fuzzy contol systm is dmonstatd. As a sult of sval intmdiat stps, th closd loop systm is psntd in th Lu poblm configuation. Thn, using th availabl analytical tools, th obust absolut stability of th fuzzy contol systm with th lina acto modl is shown. 3.. Th Lina Racto Modl To analyz th obust stability of a nonlina systm within th Lu poblm famwok, th systm has to b dcomposd into a lina subsystm and a nonlina fdback tm. An quivalnt psntation of th acto modl in tms of th dviations fom an quilibium is givn by th following quations: dn δ δρ β n ~ n δρ G = δ + + βδ i ci, (3.) Λ Λ Λ i= dc δ i = λδ i n λδ i ci, i = 6, (3.) dt δ f ffp n T T = δ f l μ Ω f μ δ + Ω f μ δ, (3.3) f

dt δ l ( ff ) P Ω M = δ n + T f Tl μc μ δ ( + Ω) δ, c μc (3.4) dδρ = Gδz, (3.5) α δ δρ = δρ + α fδ f + c T T l. (3.6) Claly, th oigin is th quilibium point of th systm (3.)-(3.6). As fa as th closd-loop systm is concnd, th oigin is also th uniqu quilibium point bcaus th fuzzy contoll s output, th od spd dviation δz, bcoms zo only whn th pow lvl dviation δn is zo. Lt x dnot th stat vcto [ δ n δ c δ c δ c δ c δ c δ c δ T f δ T l δρ] T 3 4 5 6, u dnot th contol input δz, and y dnot th output δn. Thn, th fist od appoximation of this modl is givn by: dx = Ax + Bu, and (3.7) y= Cx, (3.8) wh β β β β β β β α fn ~ α cn ~ n ~ 3 4 5 6 Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ λ λ λ λ λ3 λ3 λ4 λ4 A = λ5 λ5, λ6 λ6 ff P Ω Ω μ f μ f μ f ( ff ) P Ω M Ω μc μc μc (3.9) T B= G, (3.) C =. (3.) [ ] [ ] Th matics A and B a th functions of th acto paamts som of which a unctain. Ths matics also dpnd on th quilibium valu of th acto pow lvl which changs accoding to th pow dmand. It will b assumd that th pow dmand may hav valus btwn th % pow lvl and % pow lvl. Th unctainty in th lina acto modl (3.7)-(3.) du to th pow dmand changs a pcisly dfind by. n ~.. Th nominal linaizd modl has an ignvalu at th oigin but is othwis stabl. Th ignvalu at th oigin nts th modl bcaus th od spd is usd as th contol input. If th od position ath than th od spd is usd as th contol input, th al pats of all ignvalus of th lina acto modl bcoms stictly ngativ. 3.. Th Fuzzy Logic Contoll Th o btwn th pow dmand and th acto pow lvl is qual to th pow lvl dviation with th sign invsion, i.., = δ n, sinc th quilibium valu of th acto pow lvl is qual to th pow dmand. Thus, th ovall mapping of th fuzzy logic contoll (.5) in tms of th dviations of th vaiabls taks th following fom: δz = Ψ ( δn), (3.) wh Ψ is dfind in (.5). On popty of th function Ψ is that it psnts an odd function. Scondly, Ψ is tim-invaiant. Moov, Ψ is globally Lipschitz in δn bcaus Ψ / δn is continuous and boundd. Lastly, Ψ is simila to a satuation nonlinaity. It globally blongs to th scto [,4], sinc it passs though th oigin and lis btwn th staight lins of slops and 4 as can b sn in Fig..4. Th closd-loop systm with th lina acto modl, dscibd by (3.7)-(3.) and (3.), has th fuzzy logic contoll psntd by Ψ in th fdback path. Thfo, th fuzzzy logic contol systm xactly fits th Lu poblm configuation []. Sinc th lina acto modl has a zo at th oigin, th lina subsystm block is not Huwitz. It is also known that th nonlina subsystm, th fuzzy logic contoll, blongs to th scto [,4]. In this cas, th cicl cition cannot b usd to asss th stability of this systm. To mak this systm suitabl to th cicl cition, a loop tansfomation is pfomd. Namly, a constant output fdback 5δn is applid aound th lina subsystm to obtain a nw Huwitz lina subsystm. Th ffct of this fdback is movd by subtacting 5δn fom th output of th fuzzy logic contoll. It should b notd that ths fictitious fdbacks do not appa in th al implmntation of th fuzzy contoll and thi m puposs a to mak th oiginal poblm suitabl to th Lu poblm. Nominal tansfomd systm Im{.}.6.4. -. -.4 -.6 ω = Extmal st ω = ω = 5 ω = 5 -.4 -...4.6 R{.} ω = 5 ω = 5 Fig. 3.. Nyquist plots of th xtmal and th nominal systms. Fig. 3. shows th Nyquist plots of th xtmal st and th nominal tansfomd systm as ω changs fom -5 to 5. All th Nyquist plots volv towads th oigin as th magnitud of th fquncy incass bcaus all th tansf functions involvd a stictly pop. Th family of plots xcluding th singl plot makd as Nominal tansfomd systm blongs to th xtmal st associatd with th ovbounding systm. A cicl whos cnt is on th al axis and whos cicumfnc passs though th points

= (-.5+j.) and = (.65+j.) is also shown in Fig. 3.. This cicl cosponds to a nonlinaity that blongs to th scto [ m, m ] wh m = / = 538. and m = / =. Bcaus th Nyquist plots of th xtmal st and th nominal tansfomd systm main insid this cicl, th Nyquist plots of th ovbounding systm must main in this cicl implying that th Nyquist plots of th nti tansfomd systm must also main insid this cicl. Thfo, th obust absolut stability of th tansfomd systm fo th fdback nonlinaitis blonging to th scto [-5.38,] is concludd fom th cicl cition [,]. Sinc th tansfomd nonlinaity blongs to th scto [-5,9] which is containd in th scto [-5.38,], th obust absolut stability of th fuzzy contol systm with th lina acto modl dscibd by (3.7)-(3.) and (3.) is concludd und all paamt vaiations. By obust absolut stability, it is mant that th fuzzy contol systm stabilizs all unctain lina acto modls which a linaizd aound all pow lvls btwn th % pow lvl and th % pow lvl. In addition, th scto bound that can b tolatd fo th nonlina tm without losing th stability is found to b much lag than th scto bound on th tansfomd nonlinaity. This indicats that th fuzzy logic contoll actually can accommodat lag unctainty vaiations. 4. Simulation Rsults In this pap, a spcial attntion is paid to th obust stability of th fuzzy contol systm. It is shown that th quilibium point of th closd loop systm mains asymptotically stabl und all allowabl paamt vaiations and pow lvls. Howv, this sult alon dos not justify th pacticality of th fuzzy contoll bcaus a tuly pactical contoll should also povid a high pfomanc. Poviding a high pfomanc und all paamt vaiations, th obust pfomanc, is a difficult task fo a fixd contoll. It is known that th pow lvl changs hav stong influnc on th nucla cto. To tst th pfomanc of th fuzzy logic contoll in th psnc of pow lvl changs, stp chang in th pow lvl is applid fom % to 9%. This is ptaining to th pow lvl opations in th high o atd pow gion. Fist, th tst is pfomd whn th total od activity G and th coolant tmpatu activity cofficint α c hav thi nominal valus to illustat th nominal pfomanc of th fuzzy logic contoll. Thn, th sam tst is patd fo th cass wh th paamts G and α c hav vaiations in thi valus. It is known that th wost valu of th total od activity G in tms of th systm stability is its maximum valu. Whn th total od activity G has its minimum valu th mildst systm conditions in tms of th stability a obtaind. And, th vaiations in th coolant tmpatu activity cofficint α c usually do not hav a significant ffct on th nucla acto. In th light of this infomation, xtnsiv comput simulations a pfomd to find th most xtm cass in tms of th tim domain spcifications. Mo pcisly, th sttling tim is usd to min th xtm cass bcaus vitually no ovshoot and no stady stat o is obsvd in most cass. Basd on th simulation sults, it is found that th most stssful and th most laxd cass in tms of th systm stability cospond to th fastst and slowst systm sponss, spctivly. Th most distinguishd systm sponss in tms of th sttling tim a obtaind in two cass, wh th two paamts G and α c hav thi maximum valus simultanously and thi minimum valus simultanously. Nith any fast no any slow spons is ctd in any cas. Th sults of th tst fo th nominal cas and th two xtm cass a psntd. Th th cass a pfomd a labld as follows: nominal cas: G =., and α c =.. stssful cas: G =., and α c =.. laxd cas: G =. 5, and α c =. 4. In all tsts and cass, th systm vaiabls a monitod. Ths th impotant vaiabls a th acto pow lvl, th contol od spd, and th xit tmpatu. Th plots of ths vaiabls along with thi analysis a givn in th squl. Th simulation sults a shown in Fig. 4. though Fig. 4.3. In this tst, a stp dcas is applid to th pow st point fom % pow lvl to 9% pow lvl at t=5 sc. Fig. 4. shows th pow lvl sponss fo th th diffnt cass. In th nominal cas, th pow lvl spons sttls to th tagt valu in appoximatly 3 sconds. In th stssful cas, th pow lvl spons is fast than th on in th nominal cas and it sttls to th tagt valu in lss than sconds. Th fastnss of this spons is du to th dastic incas in th valu of th total od activity. In th laxd cas, th pow lvl spons is ath slow bcaus of th big dcass in th valus of th total od activity and th coolant tmpatu activity cofficint. Th sttling tim in this cas is appoximatly 8 sconds. In all cass, th stady stat o is zo. And, no ovshoot occus in any cas. Fig. 4. shows th contol od spd sponss fo th th cass. In all cass, an abupt chang occus in th contol od spd whn th stp dcas is applid to th pow st point at t=5 sc. Thn, th contol od spd mains constant in all cass until th pow lvl o in th cosponding cass bcoms vy small. Whn th pow lvl fist nts th vicinity of its tagt valu, th contol od spd stats incasing towads its stady stat valu in all cass. Th ffcts of th paamt vaiations on th contol od spd a consistant with th cosponding ffcts in th pow lvl. Th contol od spd is fast in th stssful cas and slow in th laxd cas. Fig. 4.3 shows th xit tmpatu spons fo th th cass. Th xit tmpatu has vy simila ovdampd sponss in all cass. Th xit tmpatu sponss sttl in appoximatly 5 sconds in all cass. Th ffcts of th paamt vaiations on ths sponss a also consistnt with th cosponding ffcts in th pvious sponss.

acto pow lvl (n) od spd z xit tmpatu (Tl)..98.96.94.9.9.88 5 5 5 3 35 tim (sc.).5.4.3.. -. -. -.3 -.4 -.5 37.5 37 36.5 36 35.5 35 34.5 solid : nominal cas dashd : stssful cas dottd : laxd cas Fig. 4.. Racto pow lvl sponss. solid : nominal cas dashd : stssful cas dottd : laxd cas 5 5 5 3 35 tim (sc.) Fig. 5.. Contol od spd sponss. solid : nominal cas dashd : stssful cas dottd : laxd cas 6. Conclusions In this pap, a fuzzy contoll is dsignd to contol th pow lvl of a nucla acto. Th dsign of th fuzzy contoll is ssntially basd on huistics and acto opation. Though an itativ pocss, th paamts of th fuzzy contoll a tund to obtain a good pfomanc fo th nominal plant. Thn, obust stability of th fuzzy contoll is shown by using th cicl cition. Finally, a st of simulations is pfomd to tst th pfomanc of th fuzzy contoll und paamt vaiations. Th main conclusion of this wok is that fuzzy contol is not totally ad hoc and that th xist fomal tchniqus fo th analysis of a fuzzy contoll. Fuzzy systms constitut a spcial subst of th nonlina systms. Th ovall mapping of a fuzzy contoll can always b obtaind. Onc this mapping is obtaind, all xisting nonlina tchniqus can b usd to analyz this fuzzy contoll. 5. Rfncs [] L. A. Zadh, Fuzzy sts, Jounal of Infomation and Contol, vol. 8, pp. 338-353, 965. [] P. Ramaswamy, R. M. Edwads, and K. Y. L, An automatic tuning mthod of a fuzzy logic contoll fo nucla actos, IEEE Tansactions on Nucla Scinc, vol. 4, no. 4, pp. 53-6, 993. [3] W. M. J. Kickt, and E. H. Mamdani, Analysis of a fuzzy logic contoll, Fuzzy Sts and Systms, vol., no., pp. 9-44, 978. [4] K. S. Ray, and D. D. Majumd, Application of cicl citia fo stability analysis of lina siso and mimo systms associatd with fuzzy logic contoll, IEEE Tansactions on Systm, Man, and Cybntics, vol. 4, no., pp. 345-349, 984. [5] J. B. Kiszka, M. M. Gupta, and P. K. Nikifouk, Engtistic stability of fuzzy dynamic systms, IEEE Tansactions on Systm, Man, and Cybntics, vol. 5, no. 6, pp. 783-79, 985. [6] G. Langai, and M. Tomizuka, Stability of fuzzy linguistic contol systms, Pocdings of th 9th IEEE Confnc on Dcision and Contol, Honolulu, Hawaii, pp. 85-9, 99. [7] L. X. Wang, Adaptiv fuzzy systms and contol : dsign and stability Analysis. Pntic-Hall, NJ, 994. [8] K. Tanaka, and M. Sugno, Stability analysis and dsign of fuzzy contol systms, Fuzzy Sts and Systms, vol. 45, pp. 35-56, 99. [9] P. Myszkoowski, and R. Longchamp, On th stability of fuzzy contol systms, Pocdings of th 3nd IEEE Confnc on Dcision and Contol, San Antonio, Txas, pp. 75-75, 993. [] H. K. Khalil, Nonlina systms. Macmillan Publishing Company, Nw Yok, 99. [] M. Vidyasaga, Nonlina systms analysis. Pntic-Hall Publishing Co.,Englwood Cliffs, NJ, 993. 34 5 5 5 3 35 tim (sc.) Fig. 4.3. Exit tmpatu sponss.