MATLAB Real-Time Two-Level Fuzzy Control of Nonlinear Plant

Transcription

1 Procdings of th th WSEAS Intrnational Confrnc on SYSTEMS, Agios Nikolaos, Crt Island, Grc, July 3-5, 7 83 MATLAB Ral-Tim Two-Lvl Fuzzy Control of Nonlinar Plant SNEJANA YORDANOVA, RUSANKA TSEKOVA, BILYANA TABAKOVA, VALERI MLADENOV Faculty of Automatics Tchnical Univrsity of Sofia 8 Klimnt Ohridski st., Sofia BULGARIA sty@tu-sofia.bg, rusanka_ptrova@yahoo.com, btabakova@yahoo.com, valrim@tu-sofia.bg Abstract: - Th aim of th rcnt papr is to dvlop a ral-tim systm for a fuzzy two-lvl control of an lctrical modl of a nonlinar plant. Th ral world plant is th anarobic digstion of organic wast in wastwatr tratmnt, whr th biogas proction rat is to b controlld via th input dilution rat undr variabl organic loading and plant paramtrs. Th plant charactristics in diffrnt oprating points ar modlld on an lctronic FESTO trainr, basd on oprational amplifirs. Th fuzzy controllr is dsignd using Fuzzy Logic Toolbox of MATLAB and compltd in Simulink. Th lctrical plant modl is controlld in MATLAB ral-tim nvironmnt, using lctronic board PCI- 64 to connct th plant modl output to th Simulink fuzzy controllr via analogu input block, and th controllr s output - to th plant modl input via analogu output block. Th main contributions ar: a dvlopd two-lvl Mamdani ral-tim controllr for a nonlinar plant, a ral-tim control of lctrical modl of th plant and study of th fuzzy systm prformanc improvmnts by comparison with th prformanc of a systm with PI controllr. Ky-Words: - Ral-tim fuzzy suprvisory control, anarobic organic wast dgradation, MATLAB, stability Introction Most instrial procsss ar complx nonlinar timvarying plants, difficult to modl and control. Thrfor advancd control approachs ar usd that combin knowldg of plant xprts, masurmnts and oprational xprinc to satisfy th high prformanc dmands to th contmporary control systms accounting for plant nonlinarity, uncrtainty and complxity and saving ffort for building of accurat and simpl plant classical mathmatical modl. Th fuzzy sts thory offrs a mthodology for rprsnting of huristic xprt knowldg in a computabl way by linguistic labls implmntd in linguistic ruls thus daling with uncrtaintis and avoiding complx mathmatical rlationships [, ]. Th fuzzy infrnc procss involvs mmbrship functions (MFs), fuzzy logic oprators mainly and and or and knowldg in if-thn ruls. It has bn succssfully applid to modlling, prdiction and control of various complx procsss [-7]. Th tuning problm in th fuzzy control is succssfully ovrcom by a scond lvl suprvisory Mamdani-typ controllr for autotuning of th scaling cofficints [3]. Th systm prformanc improvmnts hav bn confirmd via simulations. Howvr, th ral world application is still on trial. A stp towards it is to apply th two-lvl fuzzy approach in ral-tim control of a physical modl of a nonlinar plant using a Simulink fuzzy controllr and MATLAB ral-tim facilitis [8, 9]. Th aim of th rcnt invstigation is to dsign and study in MATLAB nvironmnt a ral-tim fuzzy twolvl systm for th control of an lctrical plant modl for th biogas proction rat via th dilution rat in th anarobic digstion of organic wast in wastwatr tratmnt and to compar th fuzzy systm prformanc with th prformanc of a convntional PI controllr systm. Problm Statmnt Th anarobic digstion (mthan frmntation) is considrd on of th most promising mthods for trating organic wasts, providing clan tchnology and nrgy rcovry. It includs convrsion of organic littr in absnc of oxygn into safly xposabl in th nvironmnt substancs through a numbr of complx bactrial ractions. Thr ar thr principal byprocts of anarobic digstion: ) biogas mostly of mthan and carbon dioxid, which can b usd to gnrat lctricity; ) rich in nutrints liquid, utilizd in frtilizrs; 3) organic matrial, comprisd largly of lignin and chitin, but also of a varity of plastics and minral componnts in a matrix of dad bactrial clls, usd as compost or in low grad building procts such as fibrboard. Th anarobic digstion is oftn rgardd as a thr phas convrsion of organic wast into biogas [, ] hydrolysis, acidognsis and mthanognsis. Th modls usd ar nonlinar both in trms of paramtrs and variabls making th classical idntification and control thory inapplicabl. In rcnt yars mor and mor complx mathmatical

2 Procdings of th th WSEAS Intrnational Confrnc on SYSTEMS, Agios Nikolaos, Crt Island, Grc, July 3-5, 7 84 modls ar bing introcd in ordr to bttr prsnt th biodgradabl procsss [, ]. Hr for th purpos of control of th biogas proction rat th fifth ordr Barth-Hill nonlinar modl is usd: dso = DSo β So + DYp Soi d = ( μ k D) ds μ = DS + β So Y d = ( μ k D), () ds μ = DS + Ybμ Y Q = Ygμ μmaxs μ maxs μ =, μ = k + S k + S s whr: S is th concntration of solubl organics (mg/l); of acidognic bactria, mg/l; S - of substrat for acidognic bactria, mg/l; of mthanognic bactria, mg/l; S of substrat for mthanognic bactria, mg/l. Th output is th spcific biogas proction rat y=[q], l/l.d, Q()=.94. Th inputs ar th control variabl - th dilution rat D [D (,.3)], d, D()=.4, and th disturbanc th influnt organic concntration Soi, [S oi, (3, 7)], g/l. Th initial conditions for th stat spac vctor ar: т ( ) = [ So (), (), S(), (), S ()] =. = [ ] Th vctor of th paramtrs T q = [ βypμmax ks k Y μ max k s k Y Yb Yg ] consists of th cofficints β, d, Y, mg/l, Y, mg/g p b and, l/mg, th maximal spcific growth rat of Y g acidognic μ max, d and mthanognic μ, d max bactria rspctivly, th yild cofficints Y, mg/mg and Y, mg/mg, th saturation, mg/l and th k, s k s dcay k, k, d cofficints for th corrsponding bactria. Th nominal paramtr valus ar: q ot =[ ]. Th plant charactristics from () ar physically rprocd by mans of oprational amplifirs from a FESTO trainr by a sris connction of thr tim lags with potntiomtr-adjustabl gains and tim-constants. Thus th ral world plant stp rsponss at diffrnt oprating points ar modlld by diffrnt cofficints of th lctrical plant modl. Th plant gain varis from K=. Th lctrical plant modl is studid in ral- s Fig.. Exprimntal stup tim simulations by th hlp of th input and output channls of th NI board PCI 64 and th Ral Tim Workshop Toolbox of MATLAB, which convrts xtrnal signals (voltag, tc.) from th FESTO trainr into MATLAB radabl C-cod and vic vrsa []. Th xprimntal stup is rprsntd in Fig.. Th signals ar monitord by virtual scop and digital multimtr. Biogas flowrat Q Dilution rat D Plant output Q Plant input D Tim, d Fig.. Plant stp rsponss at diffrnt oprating points Th plant nonlinarity is sn in th diffrnt magnitud of th sttld rsponss to stp inputs for D with qual magnitud, applid from diffrnt initial stady stat points as shown in Fig.. Th aim of th invstigation is to dsign a ral-tim two-lvl fuzzy control systm, which stabilizs th trainr output voltag that modls th biogas proction rat Q at dsird rfrncs, corrsponding to Q r, at changs in th oprating points and to stimat th systm prformanc. 3 Dsign of a Ral-Tim Two-Lvl Fuzzy Control Systm In [3] a two-lvl Mamdani controllr with autotuning for a complx nonlinar plant is suggstd. Simulations show that it nsurs bttr control systm prformanc in trms of shortr sttling tim and smallr ovrshoot than th prformanc of systms with ordinary fuzzy or PI controllr. Th two-lvl controllr has, howvr, a

3 Procdings of th th WSEAS Intrnational Confrnc on SYSTEMS, Agios Nikolaos, Crt Island, Grc, July 3-5, y r y d/ d Diffrntiator Norma lisation K K d Fuzzification Fuzzy Controllr - Infrnc ngin Dfuzzi fication Fuzzy Tuning Controllr - o Dnormalisa tion K Post Procssing K a Intgrating actuator with timconstant /K a u σ or Max y y r t s ( t) o or Max o Fig.3. A two-lvl Mamdani fuzzy controllr complx structur. In Fig.3 is shown its principal functional block diagram. Th main fuzzy controllr- has two inputs th rror - = y r y as th diffrnc btwn th rfrnc yr and th masurd systm output y, and th drivativ of th rror d, obtaind at th output of a diffrntiator. Th two inputs ar normalizd by mans of th scaling cofficints K d. Th normalizd controllr output o aftr dnormalization with K yilds th incrmntal control action, which is passd onto an intgrating actuator to control th opning of th valv for th plant input flow. Th Mamdani fuzzy controllr- for th suprvisory tuning of K, K d is composd of thr fuzzy units. Th input of th first unit in [3] is th ovrshoot σ, calculatd for a tst stp rspons, and its output is K - th rtuning is onc aftr th tst stp rspons. Th input of th scond unit in [3] is th intgral squard t s rror I =, calculatd till th sttling tim t ( t) s, and th output is K d. During a tst stp rspons K d taks minimal possibl valu and at th nd it is rtund and rmains unchangd. Th input of th third unit in [3] is o and th output is K and th tuning is continuous. Th ruls for th thr fuzzy units ar basd on th principl th biggr th input to th corrsponding fuzzy unit th smallr th scaling cofficint, so all scaling cofficints ar smallr at highr input thus nsuring stability. Concrning th pculiaritis of th ral-tim control modifications of th input variabls for th fuzzy controllr- ar suggstd. First, th ral world nois infrnc may caus continuous rtuning of th scaling cofficints at ach tim stp, which may rsult in oscillations in th control and in th plant output. Bsids, th systm may hav no ovrshoot. Thrfor, th controllr s inputs for ovrshoot and o ar substitutd with y max and max o rspctivly - y r th maximum is rtaind till a nw maximum at disturbanc or rfrnc and paramtr chang ariss. Thus though K ar continuously adjustd, this lads to slow and smooth changs. Scond, th diffrntiator paramtrs should conform to th charactristics of th ral world nois, th sampl priod and th rquirmnt to nsur clos to idal diffrntiation. Accounting for th lctrical plant modl dynamic proprtis a sampl priod Δt=., s is slctd. For th spcific plant modl an actuator with Ka=. is nvisagd and th diffrntiator is tund to yild a transfr function. s + Th rangs and th mmbrship functions for е, d and ar spcifid out of mpirical ruls for th lctrical plant modl in Fig.4, considring th magnitud of rfrnc changs and disturbancs and th limitation to ral-world signals magnitud.

4 Procdings of th th WSEAS Intrnational Confrnc on SYSTEMS, Agios Nikolaos, Crt Island, Grc, July 3-5, 7 86 μ gn n z p gp μ gn n z p gp -5 μ d n z p d 5 Fig.4. Mmbrship functions of fuzzy controllr- μ z m b I, max(y/y r ) max o Fig.5. Mmbrship functions of fuzzy controllr- Th MFs dscrib th following trms: - grat ngativ (gn), ngativ (n), zro (z), positiv (p), grat positiv (gp) for and ; - ngativ (n), zro (z), positiv (p) for dе. Th MFs of th thr units of th fuzzy controllr - in Fig.5 dscrib th following linguistic trms for th input: z for zro; m for mdium and b for big. Th cofficints K, K d tak th sam linguistic valus, dnotd with zk, mk and bk with th sam MFs. In Fig.6 is givn th Simulink part of th dsignd ral-tim fuzzy two-lvl control systm. From plant Q r Analog input - Q d/ d Fuzzy Two-Lvl Controllr D Analog output To plant Fig.6. Simulink part of th fuzzy two-lvl control systm 4 Ral-Tim Invtigations Two ral-tim control systms ar invstigatd with th dsignd Simulink two-lvl Mamdani controllr and with a Simulink PI controllr with a gain K p =.8 and intgral action tim T i =5, s, mpirically tund to nsur an ovrdampd transint rspons for nominal plant paramtrs gain K=8, tim-constant T=.5,s and tim dlay τ=.5, s. Th following xprimnts wr carrid out: ) Initially th rfrnc Q r is kpt to.94 for initial conditions to sttl. ) At t=,s Q r changs from.94 to.74. 3) At t=4,s Q r changs from.74 to.94. 4) Aftr t=8,s th plant is subjctd to paramtr changs, simulating chang of oprating point. Th plant gain K is changd in th following ordr: from 8 to 9.4; from 9.4 to 5.6; from 5.6 to 8 and from 8 to. Th purpos of ths changs is to obsrv how fast and fficintly th fuzzy or PI control systm compnsats th paramtr disturbancs. Th transint rsponss ar shown in Fig.7 and Fig.8. Th stability of th two systms is invstigatd using th phas-plan mthod. Th phas trajctoris of th systm with Mamdani two-lvl controllr and with th PI controllr for diffrnt stp rsponss (tim sctions) ar shown in Fig.9 and Fig. and prov systm stability for th applid inputs. 5 Conclusion Th main contributions of this papr can b systmizd in th following way: - an lctrical modl of a nonlinar tim-varying plant is dvlopd and studid by th hlp of a FESTO trainr and lctronic NI board - PCI-64 to simulat th charactristics of an anarobic digstion of organic wast in wastwatr tratmnt undr various organic loading and microorganisms stat in MATLAB ral-tim nvironmnt; - a ral-tim fuzzy two-lvl systm for th control of th lctrical plant modl of th biogas proction rat in th anarobic digstion via th dilution rat is dsignd and studid using Ral-Tim Workshop and Simulink of MATLAB; - a ral-tim systm with a convntional PI controllr for th control of th lctrica plant modl of th biogas proction rat in MATLAB nvironmnt is dsignd and studid to provid a basis for comparison; - th fuzzy control systm prformanc is compard with th prformanc of a convntional PI controllr systm in ordr to stimat th advantags of ral-tim fuzzy two-lvl control; - th stability of th two control systms is studid using th phas portrait approach; Th dsignd fuzzy two-lvl controllr nsurs fastr transint rsponss at diffrnt oprating points, lss maximal dynamic dviation, gratr robustnss to paramtr changs and disturbancs as wll as mor smooth, conomic and ffctiv control. Th phas trajctoris of th fuzzy control systm kp closr to th origin, which is an vidnc of highr facilitis to kp stability, gratr robustnss as wll as ovrdampd and fastr transint rsponss. Acknowldgmnt This work is supportd by a rsarch projct Computr Intgratd Systm for Dvlopmnt and Ral-Tim Exprimntation of Instrial Prototyps of Fuzzy Control Systms, fundd by th Tchnical Univrsity of Sofia Rsarch Cntr, Bulgaria.

5 Procdings of th th WSEAS Intrnational Confrnc on SYSTEMS, Agios Nikolaos, Crt Island, Grc, July 3-5, Qr=.94 Qr= Qr= Qr= K=8 K=8 K=8 K=8 K=9 K=6 K=8 K= Qr Qf.5 Kd.5 K K D Tim, s Fig.7. Stp rsponss of th closd loop systm with Mamdani two-lvl controllr Qr=,94 K=8 Qr=,94-,74 Qr=,94-,74 K=8 Qr=,74-,94 K=8 K=8 K=9 K=6 K=8 K= Qpi Qr.5 D Tim, d Fig.8. Stp rsponss of th closd loop systm with PI controllr s

6 Procdings of th th WSEAS Intrnational Confrnc on SYSTEMS, Agios Nikolaos, Crt Island, Grc, July 3-5, d/ 4 d/ Fig.9 Phas trajctoris of th systm with Mamdani two-lvl controllr -4 - Fig. Phas trajctoris of th systm with PI controllr Rfrncs: [] T. J. Ross, Fuzzy Logic with Enginring Applications, McGraw Hill, Inc., 995. [] J.-S.R. Jang, C.-T. Sun, and E. Mizutani, Nuro- Fuzzy and Soft Computing. A Computational Approach to Larning and Machin Intllignc. MATLAB Curriculum Sris, Prntic-Hall, Inc., N.J., 997. ISBN [3] S. Yordanova, Fuzzy Two-Lvl Control for Anarobic Wastwatr Tratmnt, Proc. of th -nd Int. IEEE Conf. Intllignt Systms, Varna, Jun -4, 4, Sofia, Bulgaria 4, pp [4] S. Yordanova, R. Ptrova and V. Mladnov, Nuro- Fuzzy Control for Anarobic Wastwatr Tratmnt, WSEAS Trans. on Systms, 3, 4, pp. 74-9, ISSN [5] S. Yordanova, R. Ptrova, N. Noykova and P. Tzvtkov. Nuro-Fuzzy Modlling in Anarobic Wastwatr Tratmnt for Prdiction and Control, Int. Sc. J. of Computing, Issu, Vol.5, 6, pp.5-56, ISSN [6] Gotsva, D. Dvlopmnt of Fuzzy Ruls using FUZZY C# Extnsion, Intrnational Scintific Confrnc Computr Scinc 4, Sofia, pp [7] Plamnka Borovska, Subhi Bahudaila, Prformanc Estimation of th Hat Distribution Solution Basd on Mont Carlo Mthod on Clustr Platform, Bulgarian-Grk Confrnc, Computr Scinc 5, Chalkidiki, 3 th Spt.- nd Octobr, 5, Procdings, Part, pp.3-37, ISBN [8] J.-S. R Jang, ANFIS: Adaptiv-Ntwork-Basd Fuzzy Infrnc Systms, IEEE Trans. on Systms, Man, And Cybrntics, Vol. 3, No. 3, May 993. [9] MATLAB Fuzzy Logic Toolbox. Usr s Guid. Mathworks, Inc., 99. [] DT. Hill and CL. Barth, A Dynamical Modl for Simulation of Animal Digstion, J. Watr Pollution Contr. Fd, 977, pp [] D. Dochain, and P. Vanrollghm, Dynamical Modlling and Estimation in Wastwatr Tratmnt Procsss, IWA Publishing, UK,, p. 3 [] MATLAB Ral-Tim Workshop. Usr s Guid. Mathworks, Inc., 99.