An automated design flow from linguistic models to piecewise polynomial digital circuits

An Au oma ed Design Flow om Linguis ic Models o Piecewise Polynomial Digi al Ci cui s Iluminada Ba u one San iago Sánchez-Solano And és A. Ge sno iez, and Ma ía B ox Dp o. Elec ónica y Elec omagne ismo – IMSE IMSE Dp o. A q. Compu ., Elec ón. y Tecn. Elec ónica Uni . o Se ille – CNM (CSIC) CNM (CSIC) Uni . o Co doba Se ille, Spain Se ille, Spain Co doba, Spain [email protected] san [email protected] and [email protected] mb[email p o ec ed] Abs ac —This pape desc ibes how he di e en CAD ools o he en i onmen X uzzy 3, de eloped in Mic oelec onics Ins i u e o Se ille and Uni e si y o Se ille, allow o ansla e exp essi e linguis ic models in o ma hema ical ones, in pa icula , in o a combina ion o piecewise polynomial sys ems ha can be implemen ed e icien ly in ha dwa e. The new syn hesis ool o X uzzy 3 au oma es communica ion wi h Xilinx Sys em Gene a o in Ma lab, hus acili a ing implemen a ion o he linguis ic model in o an FPGA om Xilinx. This is illus a ed wi h he design o a na iga ion con olle o an au onomous obo . I. INTRODUCTION Model-based app oaches a e cu en ly gaining popula i y o ackle he g owing complexi y o embedded sys ems de elopmen since hey allow wo king wi h a high deg ee o abs ac ion, enhance unde s anding and educe he ime o ma ke . Pa icula ly linguis ic models, ha is, desc ip ions o sys ems ob ained om heu is ic knowledge o human expe s exp essed linguis ically, acili a e unde s anding and apid de elopmen . In o de o ansla e linguis ic models in o ma hema ical ones, which can be implemen ed in ha dwa e and/o so wa e, uzzy logic-based sys ems ha e been employed widely in he ecen yea s [1]. Howe e , while exp essi e linguis ic models ha e been employed in many so wa e applica ions, he models implemen ed in ha dwa e and embedded so wa e only con ain a single (plain) ule base wi h simple an eceden s and consequen s, hus educing he applicabili y o ha dwa e app oaches. This is he case o many design en i onmen s o uzzy sys ems, such as FIDE, FLASH, and FuzzyTECH, when gene a ing embedded so wa e o speci ic p ocesso s, and he case o many uzzy digi al ci cui s. This pape desc ibes how he di e en CAD ools o he en i onmen X uzzy 3, de eloped in Mic oelec onics Ins i u e o Se ille and Uni e si y o Se ille [2], ease o ill he gap be ween he desc ip ion o exp essi e linguis ic models and i s e icien ha dwa e implemen a ion. The idea is o ans o m successi ely an ini ial linguis ic model in o a combina ion o plain uzzy ule bases implemen ed e icien ly by piecewise polynomial digi al ci cui s. The pape is o ganized as ollows. Sec ion II summa ies how plain uzzy ule bases a e equi alen o piecewise polynomial sys ems (in pa icula o PWL ones) when imposing ce ain cons ain s. Sec ion III desc ibes b ie ly he di e en CAD ools o X uzzy 3 ha acili a e ans o ming an exp essi e linguis ic model in o he combina ion o piecewise polynomial modules, pa icula ly he new ha dwa e syn hesis ool, x sg, which au oma es he communica ion be ween X uzzy 3 and he Xilinx Sys em Gene a o (SysGen) Simulink oolbox o Ma lab. Sec ion IV illus a es his au oma ed design low wi h he design o a na iga ion con olle o an au onomous obo . Finally, Sec ion V summa izes conclusions. II. RULE BASES AND PIECEWISE POLYNOMIAL SYSTEMS A plain uzzy ule base is a se o IF-THEN ules whose an eceden pa s con ain uzzy e alua ions o he inpu a iables. Fuzzy se s a e ep esen ed by membe ship unc ions ha de ine a pa i ion o he inpu uni e ses o discou se. The se s a e uzzy because hei membe ship unc ions o e lap among hem and ake alues be ween 0 and 1 (null o ull membe ship). Hence, gi en an inpu da a, he e is a piece o ules wi h an ac i a ion deg ee g ea e han ze o ha is esponsible o he ou pu alue. Depending on he an eceden and consequen membe ship unc ions and he uzzy ope a o s employed, a plain ule base can pe o m as a piecewise polynomial sys em. This pe o mance is a e y good ade-o be ween e sa ili y o he ule base (i ea u es uni e sal app oxima ion capabili y) and e iciency o i s ha dwa e implemen a ion. Le us conside B-spline amilies o o de ze o, one and wo o ep esen he inpu membe ship unc ions. Fig. 1 shows how hese unc ions a e no malized, ha is, he sum o he membe ship deg ees o any inpu o hem is always he uni y. Le us conside ha an eceden pa s only in ol e conjunc ions ha a e ep esen ed by wo ope a o s: he ex ension o he mee ope a o [3] and he p oduc . Table I and This wo k has been pa ially suppo ed by Eu opean Communi y unde he MOBY-DIC P ojec FP7-IST-248858 (www.mobydic-p ojec .eu), by Minis e io de Ciencia y Tecnología unde he P ojec TEC2008-04920 and by Jun a de Andalucía unde he P ojec P08-TIC-03674. (a) (b) (c) Figu e 1. B-spline amilies o o de (a) ze o, (b) one and (c) wo, as de ined wi h he ool x edi in X uzzy 3. II summa ize he equi alences be ween piecewise polynomial app oxima o s and ule bases ha employ, espec i ely, a ze o-o de Takagi-Sugeno in e ence me hod ( he ules’ consequen s a e single on o non uzzy alues) and a i s - o de in e ence me hod ( he ules’ consequen s a e linea unc ions in he inpu s). B-splines o deg ee ze o a e no a uzzy solu ion because he e is no o e lapping, and B-splines o deg ee wo ha e low linguis ic meaning since o al membe ship is no possible. Hence, B-splines o o de one a e he mos adequa e o linguis ic models. Using hem, piecewise linea , mul ilinea , quad a ic, and mul iquad a ic sys ems can be gene a ed [3]-[4]. The plain ule bases conside ed he ein (depic ed wi h bold on s in Table I and II) use he p oduc as an eceden connec i e. Se e al au ho s ha e epo ed e icien digi al a chi ec u es o implemen such ule bases. In pa icula , he one conside ed he ein is he ac i e- ule d i en a chi ec u e epo ed in [5]. The cons i uen blocks o his a chi ec u e a e membe ship unc ion ci cui s (MFCs), which can implemen B-spline amilies o o de one wi h uni o m o non uni o m (Fig. 1b) dis ibu ion; ule selec ion block, which selec s he ules ac i a ed by he inpu s; an eceden connec i e, which can be selec ed as he p oduc ; ule memo y, which s o es 1 alue pe ule in he case o ze o-o de Takagi-Sugeno in e ence o n+1 alues (being n he numbe o inpu s) in he case o i s - o de Takagi-Sugeno sys em; de uzzi ie block, which in he case o B-spline amilies and Takagi-Sugeno sys ems, can be selec ed as a weigh ed sum (no di ide is equi ed); and a con ol block, which gene a es he con ol and empo iza ion signals o p ocess sequen ially only he 2n ac i e ules. III. XFUZZY 3 AND EXPRESSIVE LINGUISTIC MODELS Plain piecewise polynomial ule bases a e adequa e o desc ibe a simple linguis ic model wi h 1, 2 o e en 3 inpu s and wi h a ew membe ship unc ions pe inpu , due o he cu se o dimensionali y. Howe e , complex linguis ic models can in ol e many inpu s and membe ship unc ions. The i s s ep o acili a e ha dwa e implemen a ion o complex linguis ic models should be o use hie a chical sys ems as much as possible, which combine modules o 1 and 2 inpu s, p e e ably. An ad an age o linguis ic models is ha hie a chy is usually p esen in he model om he beginning o can be pu sued in subsequen e inemen s. The design en i onmen X uzzy 3 helps in his p ocess because i allows desc ibing hie a chical sys ems consis ing o se e al modules ha can in e change uzzy o c isp in o ma ion among hem. Each module can be a se o uzzy IF-THEN ules o a non uzzy (c isp) module desc ibed by any ma hema ical desc ip ion connec ing i s inpu s and ou pu s. The CAD ool x edi in X uzzy 3 acili a es desc ibing his kind o sys ems. Fig. 2, o example, shows he main window o he ool x edi when desc ibing a sys em wi h h ee modules (one o hem c isp). The uzzy modules included in he hie a chical sys em a e no usually piecewise polynomial ule bases i hey a e ob ained om IF-THEN ules exp essed in na u al language. Such ules in na u al language do no always ollow a Takagi- Sugeno in e ence scheme wi h an eceden s ep esen ed by B- splines o deg ee one and only connec ed by p oduc . X uzzy 3 employs a o mal speci ica ion language, named XFL3, ha acili a es ansla ing IF-THEN ules exp essed in na u al language because o i s exp essi eness. No only inpu a iables can be e alua ed as uzzy ( o example, ‘powe is medium’) bu also he uzzy concep s and he p oposi ions i sel can be e alua ed linguis ically by using linguis ic hedges ( o example, ‘powe is g ea e han medium and mo e o less speed is high o a ea is low’). The las example in TABLE I. ZERO-ORDER TAKAGI-SUGENO RULE BASES An eceden connec i e O de o B-spline amily in an eceden s 0 1 2 Ex ension o mee Piecewise cons an Piecewise linea (PWL) Piecewise quad a ic P oduc Piecewise cons an Piecewise mul ilinea Piecewise mul iquad a ic TABLE II. FIRST-ORDER TAKAGI-SUGENO RULE BASES An eceden connec i e O de o B-spline amily in an eceden s 0 1 Ex ension o mee Piecewise linea (PWL) Piecewise quad a ic P oduc Piecewise linea (PWL) Piecewise mul iquad a ic Figu e 2. Main window o he ool x edi . XFL3 would be: powe > medium & ~ (speed == high | a ea == low) Ano he ea u e o XFL3 exp essi eness is ha con idence weigh s can be assigned o he ules. The CAD ool x edi in X uzzy 3 con ains many g aphical use in e aces so as o allow desc ibing complex ules wi hou a deep knowledge o XFL3. The use o X uzzy 3 can e en de ine new ope a o s (conjunc i e, disjunc i e, implica ion, agg ega ion, e c.) apa om hose al eady de ined in he en i onmen so as o be e ansla e he linguis ic meaning o he ules. The CAD ool x pkg in X uzzy 3 has a g aphical use in e ace o acili a e he in oduc ion o new ope a o s and c isp modules. Once he uzzy modules ha e an ini ial ma hema ical desc ip ion, he ollowing s ep is o ans o m hem in o piecewise polynomial modules. Such ans o ma ion is always possible because piecewise polynomial modules a e uni e sal app oxima o s. Se e al CAD ools o X uzzy 3 a e e y help ul in his p ocess. One o hem is he ool x plo ha allows ep esen ing g aphically he ou pu o he linguis ic module e sus 1 o 2 o i s inpu s and sa e he inpu -ou pu nume ical da a in o a ile. These nume ical da a p o ided by he linguis ic knowledge (and any o he nume ical da a p o ided by ano he kind o knowledge) is employed by he ool x dm in X uzzy 3 o ex ac he ini ial s uc u e o he piecewise polynomial module. This ool con ains se e al g id- based algo i hms so as o iden i y he numbe and ini ial dis ibu ion o B-splines pe inpu . Tuning o his ini ial s uc u e o minimize app oxima ion e o o nume ical da a is done wi h he ool x sl, which includes se e al supe ised lea ning algo i hms [6]. Simpli ica ion o he uned membe ship unc ions and ules o each module is done wi h he ool x sp, which includes p uning, simila i y-based, clus e ing-based and abula ion simpli ica ion algo i hms [7]. A inal uning is again pe o med wi h x sl. The ollowing s ep a e ob aining a hie a chical sys em wi h piecewise polynomial modules is o e i y i s beha io . X uzzy 3 con ains se e al CAD ools o help in his p ocess. One o hem is x sim, which uses a model o he con ex whe e he sys em is in ol ed so as o simula e he sys em dynamically. Ano he ool is x m , which allows moni o ing each o he cons i uen ule bases o unde s and how he ou pu alues a e in e ed om he inpu ones. The ool x plo is he o he e i ica ion ool ha can be again used o compa e he inpu -ou pu beha io (s a ic beha io ) o he ans o med and he ini ial sys ems. S a ic and dynamic e i ica ions in X uzzy 3 do no conside ha dwa e implemen a ion aspec s such as he in luence o pa ame e wo d sizes ( he s a ic beha io analyzed in X uzzy does no conside quan iza ion) and echnology de ails so as o e alua e i he design mee powe , a ea, and speed equi emen s ( he dynamic beha io simula ed in X uzzy is a high le el). Since hese de ails depend on he a ge pla o m, ou app oach has been o exploi he design en i onmen s o he ha dwa e pla o ms and de elop a syn hesis ool in X uzzy whose objec i e is o au oma e communica ion be ween bo h en i onmen s. In his sense, he new syn hesis ool de eloped o X uzzy, named x sg, ac s as an in e ace be ween X uzzy and Xilinx Sys em Gene a o (SysGen) ool in Ma lab. Using he Xilinx Blockse in Simulink, a lib a y o modules, named X uzzy Blockse o X uzzyLib, has been de eloped o implemen piecewise polynomial digi al ci cui s acco dingly o he p e iously commen ed ac i e ule-d i en a chi ec u e. These modules ha e masks ha allow de ining hei pa ame e s by Ma lab command window o a con igu a ion ile. The ool x sg gene a es au oma ically hese con igu a ion iles. Some o he equi ed pa ame e s a e ob ained om he XFL3 desc ip ions (numbe o membe ship unc ions pe inpu , kno s o he B- splines and consequen s o he ules). The o he pa ame e s ela ed o bi size o he a iables in he uzzy and c isp modules should be in oduced by he use h ough a g aphical in e ace. Fig. 3 shows he window o x sg co esponding o he hie a chical sys em al eady shown in Fig. 2. In addi ion, x sg can gene a e he Simulink model o he whole sys em. Hence, simula ion pe o med in X uzzy can now be done in Simulink conside ing ha dwa e de ails. E en he design can be syn hesized and implemen ed in he FPGA and a ha dwa e-in-a-loop simula ion can be ealized. IV. APPLICATION EXAMPLE The design low desc ibed abo e has been applied o de elop an embedded con olle o a ca -like au onomous obo . The p oblem add essed has been o con ol he obo speed, , and he angle o he on wheels ha a e esponsible o he ajec o y cu a u e, , so as o d i e backwa d he obo om any con igu a ion (x, y, , , ) o an objec i e con igu a ion (0, 0, 0, 0, 0) in he e e ence sys em shown in Fig. 4. The con olle , he e o e, has 2 ou pu s and up o 5 inpu s. Ins ead o looking o a single ule base, which would be di icul o design, heu is ic knowledge can be exploi ed o ob ain a hie a chical sys em wi h 1- and 2-inpu modules. The Figu e 3. Main window o he ool x sg. x y  x y  Figu e 4. Na iga ion p oblem add essed. Figu e 7. Ha dwa e/so wa e cosimula ion in Ma lab. (a) (b) Figu e 5. Simula ion esul s in: (a) X uzzy 3, (b) Ma lab. i s s ep is o sepa a e speed and cu a u e con ol. Speed con ol depends on e ical posi ion, y, and p e ious cu a u e. Cu a u e con ol depends on global posi ion, x, y, and o ien a ion, . A second simpli ica ion is o u he decompose cu a u e con ol in o wo modules connec ed in cascade. The i s one e alua es, depending on x and y, which o ien a ion, , should ha e he obo so as o d i e backwa d wi h a ze o cu a u e. The second module e alua es he di e ence be ween and  o decide he cu a u e. The s uc u e o he cu a u e con olle is ha al eady shown in Fig. 2. The 2-inpu module in cha ge o con olling speed is designed by ansla ing linguis ically exp essed ules. Using he ools x plo , x dm, x sl, and x sp, his module is ansla ed in o a piecewise mul ilinea sys em wi h 4 B-spline unc ions co e ing y and 3 unc ions co e ing. Simila ly, he 1-inpu module o he cu a u e con olle is designed om linguis ic ules and hen ansla ed in o a piecewise linea sys em wi h 6 B-spline unc ions co e ing i s inpu . The 2-inpu module o he cu a u e con olle is designed om combining heu is ic knowledge (exp essed as linguis ic IF-THEN ules) and geome ical analysis o he p oblem (exp essed as inpu -ou pu nume ical da a). Again using he ools x plo , x dm, x sl, and x sp, his module is designed as a piecewise mul iquad a ic sys em wi h 5 B-spline membe ship unc ions co e ing x, 3 unc ions co e ing y, and 15 consequen s ha depend linea ly on he inpu s. Beha io o he designed con olle wo king in a closed- loop wi h he obo model is e i ied wi h he ool x sim. Fig. 5a shows one o hese simula ed ajec o ies. Using he ool x sg, con igu a ion iles a e p o ided au oma ically o gene a e a Simulink model ha uses modules o he X uzzy Blockse (which use, in u n, he modules in he Xilinx Blockse ). Fig. 6 shows he Simulink model co esponding o he cu a u e con olle . Ha dwa e/so wa e cosimula ion o SysGen can be employed o implemen he whole con olle in a FPGA (in his case a Spa an3 xc3s700a o a Spa an 3A S a e Ki ) and analyze i s beha iou when wo king in a closed loop wi h a model o he obo (Fig. 7). Visualiza ion u ili ies o Ma lab allow ob aining he ajec o ies o he obo , like ha in Fig. 5b. These esul s ha conside ha dwa e de ails may no mee con ol equi emen s. In hose cases, he use can go back in he design low and i e a e wi h, o ins ance, di e en bi sizes o pa icula wo ds. Since he whole design low is au oma ed, he use can mo e easily bo om o up and up o bo om. V. CONCLUSIONS The di e en CAD ools o X uzzy 3 a e e y help ul o pa e he way be ween exp essi e linguis ic models and e icien ha dwa e implemen a ions. Heu is ic knowledge exp essed linguis ically and any o he knowledge exp essed nume ically can be exploi ed by X uzzy o desc ibe a hie a chical sys em made up o piecewise polynomial modules. Au oma ed communica ion be ween X uzzy and Xilinx Sys em Gene a o in Ma lab allows he use o conside ha dwa e de ails wi hin he whole design low. REFERENCES [1] A. M. Ib ahim, “Fuzzy logic o embedded sys ems applica ions”, Else ie Science, 2004. [2] X uzzy web si e: h p://www.imse-cnm.csic.es/X uzzy. [3] R. Ro a i, “Fuzzy piecewise mul ilinea and piecewise linea sys ems as uni e sal app oxima o s in Sobole no ms”, IEEE T ans. on Fuzzy Sys ems, ol. 6 (2), pp. 235-249, 1998. [4] R. Ro a i, “High speed implemen a ion o piecewise-quad a ic Takagi- Sugeno sys ems”, P oc. IEEE In e na ional Fuzzy Sys ems Con ., pp. 292-297, 1999. [5] S. Sánchez-Solano, A. Ba iga, C. J. Jiménez, and J. L. Hue as, “Design and applica ions o digi al uzzy con olle s,” in P oc. IEEE In . Con . Fuzzy Sys ., ol. 2, pp. 869–874, Jul. 1997. [6] F.J. Mo eno-Velo, I. Ba u one, A. Ba iga, S. Sánchez-Solano, “Au oma ic uning o complex uzzy sys ems wi h X uzzy”, Fuzzy Se s and Sys ems, ol. 158, pp. 2026 – 2038, 2007. [7] I. Ba u one, F. J. Mo eno-Velo, A. A. Ge sno iez, “A CAD app oach o simpli y uzzy sys em desc ip ions”, P oc. FUZZ-IEEE’2006, pp. 2392-2399, Vancou e (Canada), July 2006. Figu e 6. Simulink model o he cu a u e con olle .