Neuro-fuzzy Control of Integrating Processes

Transcription

1 Nuro-fuzzy Control of Intgrating Procsss Anna Vasičkaninová, Monika Bakošová 1 Fuzzy tchnology is adaptiv and asily applicabl in diffrnt aras.fuzzy logic provids powrful tools to captur th prcption of natural phnomna. Th papr dals with tuning of nuro-fuzzy controllrs for intgrating plant and for intgrating plants with tim dlay. Th dsignd approach is vrifid on thr xampls by simulations and compard plants with classical PID control. Dsignd fuzzy controllrs lad to bttr closd-loop control rsponss thn classical PID controllrs. Kywords: fuzzy logic, intgrating procsss, tim dlay, PID control, fuzzy control Introduction Fuzzy logic is a simpl and powrful tool to modlling systms bcaus it nabls a mathmatical formalization of ill dfind problms, s.g Tanaka and Sano (1991). A usful introduction to fuzzy logic ar for xampl Kosko (1994) and Fang (1997). Fuzzy st thory provids a mathmatical framwork that is appropriat for handling th complxity of systms, s. g. Klir and Yuan (1997). Thr ar a lot of xampls of gological objcts that do not fit wll into traditional classifications: Dmicco and Klir (2004), Fang and Chn (1990), Kosko (1994), Mujumdar and Sasikumar (2002), Aroba t al. (2007). Ratitsch and Schulz purpos applications in th aras of gochmistry and hydrology - Ratitsch (2000), Schulz t al. (1999). Th utility of fuzzy mathmatics in analyzing and undrstanding th compositional htrognity of clay minrals was shown in Varadachari t al. (2003).Th xampls of applications of fuzzy logic to stratigraphy and porosity ar mntiond in Fang (1997). Fuzzy sts wr succssfully incorporatd in th gographic information systms (GIS) supporting gographic problm solving, s. g. Ptry t al. (2005). This book includs xampls of th us of fuzzy sts in issus such as trrain faturs, landscap morphology, spatial xtnts and approachs for spatial intrpolation. A varity of applications using fuzzy sts ar covrd including data mining, spatial dcision making, cological simulation, and rliability in GIS. Th fuzzy logic tchniqu in GIS provids a flxibl tool to tst a concptual xploration modl whn thr is good data covrag within th ara of intrst, s. g. Sawatzky t al. (2009). Spatial modlling using GIS tchniqus for indntifying aras of minral prospctivity hav bcom incrasingly popular bcaus of thir data drivn approach and transparncy, s. g. Bonham-Cartr t al. (1989), Agtrbrg t al. (1993), Harris t al. (2001). Th gnsis of minral dposits is a good xampl for situations whr th high complxity of physical and chmical procsss can lad to an incomplt undrstanding of th dposit information. Application of th fuzzy tchnology in minral procssing procsss can b found in Cirpisz and Hyduk (2002) and Karr and Wck (1996). PID controllrs ar th most widly usd controllrs in industry. Popularity of thm can b attributd partly to thir robust prformanc in a wid rang of oprating conditions and partly to thir functional simplicity. PID controllrs provid robust and rliabl prformanc for most systms if th PID paramtrs ar tund proprly. Various tuning mthods ar dscribd for xampl in Johnson t al. (2005), Ogunnaik and Ray (1994). Intgrating modls appar whil modlling mass or nrgy accumulation, rotation of machinris, tc. Thy contain undsirabl pol which nds to b shiftd by suitabl dsign of a fdback loop. Tim dlay in controlld systms has ngativ and unplasant faturs with rgard to fdback control and its quality, too. Howvr, dlay trms ar inhrnt parts of many procsss, plants and objcts. Th combination of intgrating bhaviour of th systm and tim dlay maks controllr dsign mor difficult, and it rquirs utilization of som advancd procdurs. Nowadays, PID tuning mthods ar proposd to dal with various intgrating procsss, s.g. Åström and Hägglund (2006), Vítčková and Vítčk (2008), Vítčková and Vítčk (2009), Zhang t al. (2004). Chin and Fruhauf proposd an intrnal modl control (IMC) mthod to find th sttings for a PI controllr usd for control of a procss consisting of an intgrator and a tim dlay - Chin and Fruhauf (1990). Tyrus and Luybn proposd an altrnativ approach basd on classical frquncy rspons mthods for PI controllr sttings in Tyrus and Luybn (1992). Wang and Clutt also discussd mntiond control problm and proposd a PID controllr dsign mthod basd on spcification in trms of dmandd control on signal trajctory which is scald with rspct to th magnitud of th cofficint in th scond trm of th Taylor s sris xpansion in Wang and Clutt (1997). In rcnt yars, incrasing attntion has bn paid to th problms of stability analysis and controllr dsign for tim dlay systms, s. g. L t al. (2000), Zhang t al. (1999). Visioli proposd a tuning mthod for intgrating systms 1 Anna Vasičkaninová, Monika Bakošová, PhD., Slovak Univrsity of Tchnology in Bratislava, Faculty of Chmical and Food Tchnology, Institut of Information Enginring, Automation and Mathmatics, Radlinského 9, Bratislava, Slovakia, anna.vasickaninova@stuba.sk, monika.bakosova@stuba.sk (Rviw and rvisd vrsion ) 74

2 in Visioli (2001). Chidambaram and Sr introducd a simpl mthod for th PI, PD and PID controllr sttings for intgrating procsss basd on matching th cofficints of corrsponding powrs of s in th numrator and thos in th dnominator of th closd-loop transfr function for a srvo problm, s.g. Chidambaram and Sr (2003). PID control is widly usd to control stabl procsss; howvr, its application to intgrating procsss is lss common. In this papr, w propos fuzzy PID controllrs for intgrating procsss with tim dlay. Thr numrical xampls ar prsntd to confirm th ffctivnss of th proposd nuro-fuzzy control. Using of fuzzy control can lad in ths cass to mor succssful control than using of classical controllrs. Analytical dominant multipl pol mthod Mthods Th analytical dominant multipl pol mthod (Vítčková (2001), Vítčková and Vítčk (2008), Vítčková and Vítčk (2009)) is basd on th assumption that th dominant pol of th control systm is multipl and ral, which nsurs th stabl non-oscillatory control procss closd to th marginal procss. Simultanously, it is supposd that th influnc of th non-dominant zros and pols can b nglctd. Lt us hav a control systm dpictd in Fig.1. Hr, (s), w(s), u(s) and y(s) ar th Laplac transforms of th control rror, th rfrnc valu, th control input and th controlld output; G C (s) is th controllr transfr function; G P (s) is th plant transfr function. Th transfr function of th standard PID controllr is following G C (s) = K C (1 + 1 T I s + T Ds) (1) whr K C is th proportional trm wight (th controllr gain), T I is th intgral tim, T D is th drivativ tim. Th multipl dominant pol of th control systm is obtaind by th solution of th quation systm d i N(s) ds i = 0, i = 0,1,...,m (2) Fig. 1: Control systm with standard controllr. whr N(s) is th charactristic polynomial of th control systm, and m is th numbr of th controllr adjustabl paramtrs. Th charactristic polynomial N(s) of th control systm Fig. 1 can b dtrmind on th basis of th opn-loop transfr function G 0 (s) (3) in th form (4). Fuzzy PID controllr G 0 (s) = G C (s)g P (s) = M 0(s) N 0 (s) N(s) = M 0 (s) + N 0 (s) (4) Th fuzzy controllrs ar usually basd on th structur of th standard PID controllrs. Fuzzy PID control has following (absolut) form: u(t) = f ((t), d(t) t, (τ)dτ) (5) dt 0 Takagi-Sugno fuzzy infrnc systm is gnratd using subtractiv clustring in th form: I f is A i and d(t) dt is B i and is C i T hn f i = p i + q i d(t) dt + r i (3) + s i, i = 1,...,n (6) 75

3 Anna Vasičkaninová, Monika Bakošová: Nuro-fuzzy Control of Intgrating Procsss whr is th control rror, p i, q i, r i, s i ar consqunt paramtrs, and n is numbr of ruls. Various typs of functions can b usd for fuzzification, and th symmtric Gaussian function (gaussm f in MATLAB) µ is chosn for fuzzification of inputs in this approach. Th symmtric Gaussian function dpnds on two paramtrs σ and c as it is sn in (7), whr x rprsnts, d dt,. µ(x;σ,c) = (x c)2 2σ 2 (7) For obtaining of all ndd paramtrs, it is ncssary to hav th data sts of, d dt, and u at first. Ths data can b obtaind by simulations of control of procsss using classical PID controllrs. Stability of th systm can b studid. In this papr, an Adaptiv Nuro Fuzzy Infrnc Systm (ANFIS) basd PID controllr is applid, s. g. Jang (1993). Th ANFIS structur with first ordr Sugno modl, Gaussian mmbrship functions with product infrnc rul ar usd at th fuzzification lvl. Hybrid larning algorithm that combins last squar mthod with gradint dscnt mthod is usd to adjust th paramtr of mmbrship function. Rsults and Discussion In th papr, th controlld procsss with transfr functions (8)-(10) wr considrd k s(t s + 1) k s T ds (8) (9) k s(t s + 1) T ds whr k is th procss gain, T is th procss tim constant, T d is th procss tim dlay, s is th complx variabl in th Laplac transform. Exampl 1 Lt us hav th first ordr and intgrating plant without tim dlay dscribd by th transfr function (11) (10) 1 s(1.5s + 1) (11) Th fdback PID controllr (1) was tund using abov dscribd dominant multipl pol mthod by Vítčková and Vítčk (2009). Th found controllr paramtrs wr K C = 3.556, T I = 3.375, T D = and th controllr was usd for control of th procss (11). Simulations of control wr usd for obtaining th data sts of, d dt, and u that wr ndd for th fuzzy controllr dsign with th Takagi-Sugno-typ fuzzy infrnc systm (6). Th paramtrs σ and c obtaind for th Gaussian symmtric function (7) ar listd in Tabl 1. Th consqunt paramtrs in th control rul ar listd in Tabl 2. Fig. 2 dmonstrats th graphical rprsntation of th Takagi-Sugno fuzzy infrnc systm, and Fig. 3 shows th structur of Anfis. Tab. 1: Paramtrs of th Gaussian Mmbrship Functions d σ c σ c σ c x Tab. 2: Consqunt Paramtrs p i q i r i s i Nuro-fuzzy and PID controllrs wr compard in th task of st-point tracking and in th task of disturbanc rjction. Fig. 4 prsnts th comparison of th simulation rsults obtaind by dsignd nuro-fuzzy controllr and PID controllr tund using dominant multipl pol mthod whn stp changs of th rfrnc w ar from 1 to 1.2 at 76

4 Fig. 2: Fuzzy infrnc systm for th systm (11). Fig. 3: Structur of Anfis for th systm (11). tim t=20 and from 1.2 to 0.9 at tim t=40. Fig. 5 prsnts th simulation rsults of th nuro-fuzzy and PID control in th cas whn disturbancs affct th controlld procss. Disturbancs ar rprsntd by thir stp changs from 1 to 1.5 at tim t=20 and from 1.5 to 0.5 at tim t=40. Th comparison of th nuro-fuzzy controllr and PID controllr was mad using IAE and ISE intgral prformanc indxs dscribd as follows: Th IAE and ISE valus ar givn in Tabl 3. IAE = dt (12) 0 ISE = 2 dt (13) 0 Tab. 3: Comparison of th Simulation Rsults by IAE and ISE Controllr St-point Tracking Disturbanc rjction IAE ISE IAE ISE Nuro-fuzzy PID Fig. 4: Closd-loop control rsponss in th task of st-point tracking: rfrnc (blu lin), PID controllr (rd lin ), fuzzy controllr (grn lin). 77

5 Anna Vasičkaninová, Monika Bakošová: Nuro-fuzzy Control of Intgrating Procsss Fig. 5: Closd-loop control rsponss in th task of disturbanc rjction: rfrnc (blu lin), PID controllr (rd lin ), fuzzy controllr (grn lin). Exampl 2 Suppos furthr th intgrating plant with tim dlay: s (14) s Th paramtrs of th fdback PID controllr (1) tund using th dominant multipl pol mthod by Vítčková and Vítčk (2009) ar: K C =3.556, T I =3.375, T D = Th paramtrs σ and c obtaind for th Gaussian symmtric function (7) ar listd in Tabl 4. Th consqunt paramtrs ar listd in Tabl 5. Fig. 6 dmonstrats th Takagi- Sugno fuzzy infrnc systm and Fig. 7 shows th structur of Anfis for th systm (14). Tab. 4: Paramtrs of th Gaussian Mmbrship Functions d σ i c i σ i c i σ i c i Tab. 5: Consqunt Paramtrs p i q i r i s i Nuro-fuzzy and PID controllrs wr again compard in th task of st-point tracking and in th task of disturbanc rjction. Fig. 8 prsnts th comparison of th simulation rsults obtaind by dsignd nuro-fuzzy controllr and PID controllr tund using dominant multipl pol mthod whn stp changs of th rfrnc w ar from 1 to 1.2 at tim t=100 and from 1.2 to 0.9 at tim t=200. Fig. 9 prsnts th simulation rsults of th nuro-fuzzy and PID control in th cas whn disturbanc changs from 1 to 1.5 at tim t=70 and from 1.5 to 0.5 at tim t=140. Anfis prforms static non-linar mapping from input to output spac, but it cannot b usd without modification to rprsnt dynamic systm. In ordr to idntifity dynamic systms, a combination of Anfis with som tim dlay units and fdback is rquird. Hnc, nonlinar dynamic systms or varying tim daly can b modlld by Anfis combind with som tim dlay units. Anfis is not availabl for all of th fuzzy infrnc systm options. Spcifically, Anfis only supports Sugno-typ systms. All output mmbrship functions must b th sam typ and ithr b linar or constant. Diffrnt rul cannot shar th sam output mmbrship function, namly th numbr of output mmbrship functions must b qual to th numbr of ruls. Th comparison of th nuro-fuzzy controllr and th PID controllr was mad also using IAE and ISE intgral prformanc indxs and thir calculatd valus ar givn in Tabl 6. 78

6 Tab. 6: Comparison of th Simulation Rsults by IAE and ISE Controllr St-point Tracking Disturbanc rjction IAE ISE IAE ISE Nuro-fuzzy PID Fig. 6: Fuzzy infrnc systm for th systm (14). Fig. 7: Structur of Anfis for th systm (14). Fig. 8: Closd-loop control rsponss in th task of st-point tracking: rfrnc (blu lin), PID controllr (rd lin ), fuzzy controllr (grn lin). Exampl 3 In th 3 rd xampl, lt us hav th first ordr intgrating plant with tim dlay: s + 1 5s (15) 79

7 Anna Vasičkaninová, Monika Bakošová: Nuro-fuzzy Control of Intgrating Procsss Fig. 9: Closd-loop control rsponss in th task of disturbanc rjction: rfrnc (blu lin), PID controllr (rd lin ), fuzzy controllr (grn lin). Th paramtrs of th fdback PID controllr (1) tund using th dominant multipl pol mthod by Vítčková and Vítčk (2009) ar: K C =3.556, T I =3.375, T D = Th paramtrs σ and c obtaind for th Gaussian symmtric function (7) ar listd in th Tabl 7. Th consqunt paramtrs ar listd in Tabl 8. Fig. 10 prsnts th comparison of th simulation rsults obtaind by nuro-fuzzy controllr and PID controllrs tund using dominant multipl pol mthod. Fig. 11 prsnts th simulation rsults of th nuro-fuzzy and PID control in th cas whn disturbancs affct th controlld procss. Disturbancs wr rprsntd by stp changs from 1 to 1.5 at t=50 and from 1.5 to 0.5 at t=100. Tab. 7: Paramtrs of th Gaussian Mmbrship Functions d σ i c i σ i c i σ i c i x Tab. 8: Consqunt Paramtrs p i q i r i s i Comparison of th nuro-fuzzy controllr and th PID controllr don using IAE and ISE critria is givn in Tabl 9. Tab. 9: Comparison of th Simulation Rsults by IAE and ISE Controllr St-point Tracking Disturbanc rjction IAE ISE IAE ISE Nuro-fuzzy PID Conclusion In this papr, simpl nuro-fuzzy PID controllrs ar proposd for intgrating procss, on of thm without and two othr with dad-tim. Dsignd fuzzy controllrs lad to bttr control prformanc in both tasks, th st-point tracking and disturbanc rjction. Comparison of dsignd fuzzy controllrs with classical PID controllr tund by 80

8 Fig. 10: Closd-loop control rsponss in th task of st-point tracking: rfrnc (blu lin), PID controllr (rd lin ), fuzzy controllr (grn lin). Fig. 11: Closd-loop control rsponss in th task of disturbanc rjction: rfrnc (blu lin), PID controllr (rd lin ), fuzzy controllr (grn lin). th dominant multipl pol mthod is basd on simulation xprimnts. Prsntd simulation rsults dmonstrat th supriority of th proposd fuzzy approach. Acknowldgmnt Th authors gratfully acknowldg th contribution of th Scintific Grant Agncy of th Slovak Rpublic undr th grant 1/0537/10. Rfrncs Agtrbrg, F., Bonham-Cartr, G., Chng, Q., Wright, D., Wights of vidnc modlling and wightd logistic rgrssion for minral potntial mapping. Computrs in Gology, 25 Yars of Progrss, Aroba, J., Grand, J., Andújar, J., d la Torr, M., Riqulm, J., Application of fuzzy logic and data mining as tools for qualitativ intrprtation of acid min procsss. Environmntal Gology 53, Åström, K., Hägglund, T., Advancd PID Control. ISA - Th Instrumntation, Systms and Automation Socity, Rsarch Triangl Park, NC. Bonham-Cartr, G., Agtrbrg, F., Wright, D., Wights of vidnc modlling: a nw approach to mapping minral potntial. Statistical Applications in th Earth Scincs, Gological Survy of Canada, Chidambaram, M., Sr, R., A simpl mthod of tuning PID controllrs for intgrator/dad-tim procsss. Computrs and Chmical Enginring 27, Chin, I., Fruhauf, P., Considr IMC tuning to improv prformanc. Chm. Eng. Prog. 10, Cirpisz, S., Hyduk, A., A simulation study of coal blnding control using a fuzzy logic ash monitor. Control Enginring Practic 10,

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