Fuzzy Logic Controller Output Equation Analysis

Transcription

1 Fuzzy Logic Controllr Output Equation Analysis Kuldip S. Rattan and Thomas Brhm Dpartmnt of Elctrical Enginring Wright Stat Univrsity Dayton, Ohio Introduction Th prvious chaptr discussd th basic componnts of th fuzzy logic controllr (FLC and how thos componnts oprat. This chaptr focuss on how th individual componnts function togthr to driv an output. Th goal is to prsnt a drivation of an output quation of a proportional FLC (PFLC and a proportional-plus-drivativ FLC (PDFLC. Analysis of th FLC output quation shows that th PDFLC is a picwis linar controllr with many similaritis to th classical proportional-plus-drivativ (PD controllr. This chaptr vrifis this hypothsis and also shows why th FLC is considrd a picwis linar controllr. Classical controllrs, as dscribd in this chaptr, us ithr proportional, intgral and drivativ (PID gains, proportional and intgral (PI gains, proportional gains (P or as

2 Introduction 2 mntiond arlir, PD gains. Ths gains ar adustd to achiv th dsird output. Th FLC control action is dpndnt on input and output gains as wll th fuzzification procss, nowldg bas, and th dfuzzification procss. As w now, thr ar svral typ of fuzzification and dfuzzification schms. Analysis of all fuzzification and dfuzzification schms is byond th scop of this thsis. Thrfor, th fuzzification and dfuzzification procsss, and th nowldg bas ar constraind as follows. CONSTRAINT 4.: Th input valus from th snsors ar considrd crisp valus. Thrfor, fuzzification consists of matching th valus to th input fuzzy sts ovr th domain of th input variabl. Mmbrship is dtrmind by applying th function that dscribs th matchd fuzzy st. CONSTRAINT 4.2: Th fuzzification procss uss th triangular mmbrship function. Sinc th goal is to provid a picwis linar li PD controllr, th linar natur of th function is rquird. CONSTRAINT 4.3: Th width of a fuzzy st xtnds to th pa valu of ach adacnt fuzzy st and vic vrsa as shown in Figur. Th sum of th mmbrship valus ovr th intrval btwn two adacnt sts will b on. Thrfor, th sum of all mmbrship ovr th univrs of discours at any instant for a control variabl will always b qual to on. This constraint is also commonly rfrrd to a fuzzy partitioning. CONSTRAINT 4.4: Th dfuzzification mthod usd is th modifid cntr of ara mthod. This mthod is similar to obtaining a wightd avrag of all possibl output valus. Thrfor, this componnt is also linar.

3 Introduction 3 µ Blif µ (a A J µ (a A J A A J J A a A Input a A, A -Pa Valus A - a A - A a - A A - A µ (a A J µ (a A J µ (a A J - Input µ (a A J Figur Two mmbrship functions in th univrs of discours for th variabl a. CONSTRAINT 4.5: Th ruls in th nowldg bas will covr all possibl mmbrships of th input variabls. For xampl, for a two variabl systm with fiv fuzzy sts ach, thr ar 25 ruls. As shown in Tabl, ach lmnt of th control matrix will hav a valu. Tabl. PD control rul matrix. Error NB NS ZO PS PB NB NB NB NB NS ZO Chang in NS NB NB NS ZO PS Error ZO NB NS ZO PS PB PS NS ZO PS PB PB PB ZO PS PB PB PB PB - Positiv Big PS - Positiv Small ZO - Zro NS - Ngativ Small NB - Ngativ Big ErrorInput-Output Chang in ErrorPrvious Error- Currnt Error

4 FLC Output Equation Drivation 4 CONSTRAINT 4.6: Th controllrs in this chaptr ar for a normalizd input. A nonnormalizd stp input rquirs a gain to normaliz th input and a gain to scal th output. FLC Output Equation Drivation FLC is basd on linguistic xprssion of th dsird control action. Thrfor, to driv an output quation for th FLC, th numrical xprssions of th fuzzification and dfuzzification procsss ar usd to translat th "English" li trms to a mathmatical form. Th fuzzification procss uss functions to rturn mmbrship valus for th crisp input. Ths mmbrship functions ar substitutd into th quation for dfuzzification to giv th output xprssion of th FLC. PFLC Output Equation For th PFLC, thr is on control variabl which has mmbrship in xactly two fuzzy sts as shown in Figur 2. This figur also shows that if th rror valu (dsird valu minus actual valu, is btwn and, fuzzy sts E J and E J ar activ. Th mmbrship for E J is µ E ( ( and E J is: ( (2 µ E J

5 PFLC Output Equation 5 µ Blif µ ( Ε J µ ( Ε J Ε J Ε J µ ( Ε J µ ( Ε J E - E - E - E E - E E E Error Figur 2 Exampl mmbrship functions for rror input. As rquird, th sum of th mmbrships givn by ( and (2 is on. For th control variabl, rror, that has mmbrship in two fuzzy sts, thr will b two applicabl ruls xprssd as R : if is E J thn u is U J R : if is E J thn u is U J whr and u ar th rror and output. Ths two ruls form a 2-lmnt sub vctor from th fuzzy rul vctor shown in Tabl 2. Tabl 2. P control rul vctor. Error NB E J E J PS PB Output NB U J U J PS PB Th crisp output control action is dtrmind by applying th modifid cntroid of ara dfuzzification schm to th two control ruls and is givn by

6 PDFLC Output Equation 6 u µ ( U µ ( U EJ J EJ J µ ( µ ( EJ EJ (3 Th xprssions for µ and th output valus U ar substitutd into quation (3 giving u [ ] UJ [ ] [ ] [ ] Rmoving th common dnominator ( - givs u U J ( ( ( ( U U J J Expanding th trms in th numrator and dnominator and grouping li trms yilds th final xprssion for th PFLC output ( u U U U U J J J ( J ( ( (4 PDFLC Output Equation Drivation for th PDFLC output quation follows th sam procdurs as th PFLC. Howvr, th PDFLC has two input control variabls; rror (dsird valu minus actual valu and chang in rror (currnt rror minus prvious rror dividd by th tim intrval. Li th PFLC, ach input variabl of th PDFLC has mmbrship in xactly two fuzzy sts. Th mmbrship functions for rror ar th sam as th PFLC and ar xprssd in quations (( and (2. Figur shows th scond input control variabl,

7 PDFLC Output Equation 7 chang in rror. For th chang in rror input, if th valu is btwn and, thn th mmbrship for E K is µ E ( K (5 and th mmbrship for E K is µ E K ( (6 For th two control variabls, rror and chang in rror, with two activ sts, thr will b four applicabl ruls xprssd as µ Blif µ ( Ε K µ ( Ε K Ε K Ε K µ ( Ε K µ ( Ε K E E - E E E E E E Chang in Error Figur 3. Exampl mmbrship functions for chang in rror.

8 PDFLC Output Equation 8 R, : R, : R, : R, : if is E J and is E K thn u is U J,K if is E J and is E K thn u is U (J,K if is E J and is E K thn u is U J,(K if is E J and is E K thn u is U (J,(K whr,, and u ar th rror, chang in rror and output, rspctivly. Ths four ruls form a 2 x 2 sub matrix from th fuzzy rul matrix shown in Tabl 3. Tabl 3 PD control rul matrix for FLC. Error NB E J E J PS PB NB NB NB NB NS ZO Chang in E K NB U J,K U (J,K ZO PB Error E K NB U J,(K U (J,(K PS PB PS NS ZO PS PB PB PB ZO PS PB PB PB Th crisp output control action is dtrmind by applying th modifid cntroid of ara dfuzzification schm to th four control ruls and is givn by u 4 i i µ U i i i 4 (7 µ

9 PDFLC Output Equation 9 whr µ i is calculatd by th product rul applid to th antcdnt of th fuzzy rul and U i is th output st for th ith rul. Th product rul is dfind as th product of ach mmbrship valu. Thrfor, for a givn rul, µ i is calculatd by multiplying th valu of mmbrship of th rror input for th givn rror fuzzy subst and th valu of mmbrship of th chang in rror input for th chang in rror fuzzy subst as givn in (8. [mmbrship of in E]x[mmbrship of in E] (8 Th product rul is ncssary to obtain an xprssion for th PDFLC output. For th four applicabl ruls, mmbrship functions (, (2, (5 and (6 ar substitutd for mmbrship valus in th product rul to obtain th xprssion for ach µ i. Th xprssions for µ i and th corrsponding valus of U i ar Rul R, : µ, U U J,K Rul R (, : µ 2, U 2 U (J,K Rul R,( : µ 3, U 3 U J,(K Rul R (,( : µ 4, U 4 U (J,(K Th xprssions for µ i and th output valus U i ar substitutd into quation (7 giving [ ] [ ] [ ] [ ] [ ] u U U U U JK J K E JK J K,,,, [ ] [ ] [ ] Rmoving th common dnominator [( - ( - ] givs

10 FLC As A Picwis Controllr 0 u ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( U U U U,,,, Expanding th trms in th numrator and dnominator and grouping li trms yilds th final xprssion for th PDFLC output [ U U U U UJ K UJ K UJ K,, ] [,, ] [,, ] [, UJ, K ] ( ( ( ( J K J K J K J K u [ U U ] [ U U ] ( ( J, K J, K J, K J, K ( * [ UJ, K UJ, K ] [ UJ, K UJ, K] ( ( (9 FLC As A Picwis Controllr Th output quation for th PFLC is a function of th input rror and for th PDFLC, th output is a function of rror and chang in rror. Howvr, for both FLC typs, th quations ar dpndnt on th fuzzy sts for th currnt rang of oprations. Thrfor, as th valus of th input control variabls chang, th controllr output quation changs. PFLC As A Picwis Classical P Controllr As dmonstratd in quation (4, th output of th PFLC is similar to th classical proportional controllr. Li th quation for a classical proportional controllr, quation (4 has rror multiplid by a gain trm but thr is an additional constant trm. Equation (4 can b writtn as u K Const pff whr th ffctiv proportional gain is givn by

11 PFLC As A Picwis Classical P Controllr K pff ( UJ UJ ( (0 and th constant controllr output is givn by ( U J UJ Const ( ( If th ffcts of th constant trm ar ngligibl, thn th form of th PFLC is idntical to th classical proportional controllr. K p-ff and Const givn by quations (0 and ( hav a common dnominator; th width btwn th adacnt fuzzy sts. Th numrator of th ffctiv gain is th diffrnc btwn th adacnt output valus. Th numrator of th constant trm is dpndnt on th valu of th pa valus of th rror fuzzy sts and th valu of th outputs. Thrfor, a chang to ithr th rror fuzzy sts or th output valus will chang th ffctiv gain, K p-ff and th constant trm. Th valus of quations (0 and ( ar valid for rror in th rang to. If th valu of rror wr to fall in anothr rang (i.. and 2, thn th valu of th ffctiv gain and th constant trm would b ( U U ' J 2 J K pff ( 2 (2 ( U U ' 2 J J 2 Const ( 2 (3

12 PDFLC As A Picwis Classical PD Controllr 2 Th ffctiv gain and th constant trm ar dpndnt on th diffrnc btwn th pa valus of th rror fuzzy sts and th output valus. Thrfor, for th nw rangs givn in quations (2 and (3, if th diffrnc ( 2 - is not th sam as ( -, th ffctiv gain and th constant trm will chang. Th sam is tru for th output valus (i.. (U J2 -U J is not th sam as (U J -U J. For th constant trm, unlss ( 2 U J - U J2 and ( U J - U J ar both zro, a chang in ithr th output or rror pa valus changs th constant for that rang. PDFLC As A Picwis Classical PD Controllr Equation (9 dmonstrats that th output of th PDFLC is similar to a classical PD controllr output. Th controllr quation consists of four trms; rror multiplid by a gain, chang in rror multiplid by a gain, a nonlinar trm (* multiplid by a gain, and a constant trm. Th contribution from th nonlinar trm is vry small. If th nonlinar trm is ignord, thn quation (9 can b writtn in th form whr uk p-ff K d-ff Const [ UJ, K UJ, K ] [ UJ, K UJ, K] K p-ff ( ( (4 [ UJ, K UJ, K] [ UJ, K UJ, K ] K d-ff ( ( (5

13 PDFLC As A Picwis Classical PD Controllr 3 [ U, U, ] [ U, U, ] Const ( ( (6 K p-ff, K d-ff and Const givn by quations (4-(6 hav a common dnominator whos valu is dtrmind by th product of th width of th rror and th width of th chang in rror sts. Th numrator for ach trm uss th output valus from th 2x2 rul sub matrix. Th rror gain, K p-ff uss th diffrnc btwn th valus in th rows (i.. U J,(K -U (J,(K and U (J,K -U J,K tims th chang in rror sts pa valus. Th chang in rror gain, K d-ff uss th diffrncs btwn th columns (i.. U J,(K -U J,K and U (J,K - U (J,(K tims th rror sts pa valus. Th constant trm uss th pa valus for both input fuzzy sts and th output valus. Thrfor, sinc both ffctiv gain trms and th constant ar mad up of rror and chang in rror pa valus, changs in any on of th pa valus will affct all trms. Th sam is also tru with th output valus. Changs in any on of th four output valus affcts both gain trms and th constant trm. Th trms in (4 through (6 only apply to th rang of opration btwn th pa valus and for rror and btwn and for chang in rror. Th width btwn th pa valus for th nxt fuzzy sts may not b th sam as th prvious sts. Also, th diffrnc in output valus U may not b th sam for th nxt fuzzy st. Thrfor, th ffctiv gain valus and th constant trm may b diffrnt. As an xampl, th trms for th nxt rang of opration could b ' K p-ff [ UJ, K 2 UJ 2, K 2] 2[ UJ 2, K UJ, K ] ( ( 2 2 (7

14 PDFLC As A Picwis Classical PD Controllr 4 [ U U ] [ U U ] K ' 2 J, K 2 J, K J 2, K J 2, K 2 d-ff ( 2 ( 2 (8 Const 2[ 2UJ K UJ K 2] [ UJ 2 K 2 2UJ 2 K ] ( ( ',,,, 2 2 (9 As dmonstratd in (7-(9, if th diffrnc btwn th st pa valus is diffrnt as compard to th prvious rang, th product trm [( 2 - ( 2 - ] for th gains and constant trm ar diffrnt. Th diffrnc btwn th output trms along th rows (i.. U (J,(K2 -U (J2,(K2 and U (J2,(K -U (J,(K and along th columns (i.. U (J,(K2 - U (J,(K and U (J2,(K -U (J2,(K2 for th 2x2 rul sub matrix may also b diffrnt. Thus, K p-ff, K d-ff and Const for this rang of opration ar not th sam as K p-ff, K d-ff and Const for th prvious rang. This xampl dmonstrats th picwis linar natur of th PDFLC. Equations (4 and (5 indicat that th ffctiv gains of ach input control variabl ar mutually dpndnt. Changing a fuzzy st of ithr input control variabl will affct th magnitud of both ffctiv gains. Howvr, th picwis linarity of ithr ffctiv gain is still dpndnt on th corrsponding input control variabl. For xampl, to calculat th picwis linar rror gain K p, th valu of chang in rror is hld constant which givs K p C0[ UJ, K UJ, K ] C[ UJ, KUJ, K] ( C2, whr C 0, C and C 2 ar fixd valus dtrmind by th chang in rror sts. As th rror input changs for th fixd valu of chang in rror, th ffctiv K p will dpnd on th rror and output fuzzy sts. Th sam

15 Summary 5 analysis can b applid to dtrmin th ffctiv picwis linar K d for a fixd valu of rror. Summary This chaptr uss mathmatical xprssions of fuzzification and dfuzzification to driv an input-output quation for a spcific PFLC and PDFLC. Ths quations dmonstrat that th PFLC and PDFLC hav similar form as thir classical countrparts. Howvr, unli thir classical countrparts, th FLC is picwis linar. As th input control variabls chang valu, th ffctiv gain of th FLC changs. Th nxt chaptr xplors th FLC picwis linarity using graphical tchniqus. Th FLC also has a constant valu that is addd to th product of th ffctiv gains and th input. Th ffcts of th constant trms will b studid in th chaptr on tim domain analysis.