Constrained hopfield neural network for real-time predictive control (IECON'94)

Cons ained Hop ield Neu al Ne wo k o Real-Time P edic i e Con ol J.M. Que o, C.L. Jane , and L.G. F anquelo,Membe , IEEE Dp o de Ingenie ia de Sis emas y Au omi ica, Uni . de Se illa, A da Reina Me cedes s/n, 41012 Se illa - Spain phone: +34-5-4556873 ; ax: +34-5-4556849 ; e-mail: que oQg ex02.us.es Absl cscd-The ha dwa e implemen a ion o an op imiza ion ne wo k wi h es ic ions o pe o m eal- ime Gene alized P edic i e Con ol (GPC) is desc ibed. The use o space-e icien s ochas- ic a chi ec u e allows a a ealiza ion on a p o- g ammable logic de ice. As a esul a p o- g ammable neu al chip cop ocesso ha sol es op imiza ion p oblems subjec o es ic ions has been de elopped. Exp essions o ne wo k pa am- e e s a e p o ided o implemen GPC. An adap- i e con olle is achie ed using RAM memo ies o s o e he ne wo k pa ame e s. Expe imen al esul s om a simple implemen a ion o he con- olle a e included. I. INTRODUCTION Robus , eal- ime con ol is essen ial o mos p ocesses in indus y. Mode n con ol me hods ha e p o ed o be e y e icien [l]. Howe e hey a e high- ime consuming specially when conside ing cons ain sa is ac ion, he e- o e mo e powe ul p ocesso s a e needed. The GPC al- go i hm consis s o op imizing a cos unc ion in o de o achie e he bes expec ed con ol sequence [2]. New app oaches [3] conside cons ain s in con ol sequences leading o cons ained op imiza ion p oblems. We sugges he use o Hop ield ne wo ks o o e ide he compu a ional e o o implemen such con olle . Ap- plica ions o his class o neu al ne wo ks can be ound elsewhe e [4]. A i icial neu al ne wo ks consis o a se o e y simple compu ing elemen s highly in e connec ed ha pe o m an o e all du y. Due o he simplici y o he compu a ional elemen s i is wo h ying o make spe- ci ic elec onic implemen a ions a he han using a gen- e al pu pose digi al compu e o simula e i s beha iou . Elec onic ealiza ion o neu al ne wo ks can be aced in di e en ways. On one hand analog ap oaches a e e y simple in e ms o ci cui y and ha e as con e gence imes, specially when hey a e compa ed wi h digi al im- plemen a ions, hu on he o he hand hei p og amming lexibili y is e y low. Digi al implemen a ions pe o m high p og amming lexibili y and easy in e ace wi h gen- e al pu pose compu e s bu hei e iciency in e ms o consumed silicon a ea is e y low, as a loa ing poin mul- iplie is needed in e e y neu on o calcula e he p esy- nap ic ac i i y. One way o ci cum en his p oblem is employing s ochas icism o e alua e p esynap ic p oduc s Ou aim is o design a neu al cop ocesso which eleases he con ol compu e in he ac o y om his algo i hmic compu a ional bu den. The simplici y o s ochas ic neu al ne wo ks a chi ec u es [?I minimizes he amoun o ha d- wa e needed, he e o e allowing he use o p og ammable logic de ices o i s implemen a ion. This pape is o ganized as ollows. In Sec ion I1 i is discussed how Hop ield’s Neu al Ne wo k can be u ilized o sol e cons ained linea quad a ic op imiza ion p ob- lems and how such p oblems can be posed in e ms o adimen ional quan i ies, yielding a o mula ion ha can be implemen ed in s ochas ic a chi ec u e. In Sec ion I11 i is shown ha gene alized p edic i e con olle s apply a con ol sequence o he sys em ha is being egula ed in o de o minimize a quad a ic cos unc ion. Due o he ac ha all ac ua o s ha e a limi ed ange o ac ua ions and slew a es, linea es ic ions mus be imposed o his op imiza ion p oblem. In Sec ion IV he s ochas ic a chi- ec u e ha has been used o implemen he Hop ield neu- al ne wo k and why his a chi ec u e is sui able o GPC is desc ibed. Sec ion V is de o ed o he ac ual ha dwa e ha has been used o ealize he neu al ne wo k and why a p og ammable logic de ice has been used. In Sec ion VI we conside speci ic examples o demons a e he neu- al ne cop ocesso pe o mance. Finally, conclusions a e d awn in sec ion VII. [51, PI. 11. CONSTRAINED HOPFIELD NEURAL NETWORK. The way in which neu al ne wo ks could be applied o sol e linea p og amming ne wo ks was i s desc ibed by Tank and Hop ield [4]. They sugges ed an op imiza ion 0-7803-1328-3/94$03.000 1994 IEEE 1727 ne wo k, ha is able o minimize a cos unc ion The Gene al P edic i e Con ol algo i hm consis s o ap plying a con ol sequence ha minimizes a mul is age cos --I T=A*V (1) unc ion o he o m Na whe e A' is an N-dimensional ec o o coe @ien s o he J(N1, Nz) = E{ Y( + j I ) - w( + j)12 N a iables which a e he componen s o V. This mini- s ain s among he a iables: j=N1 miza ion is accomplished subjec o a se o M linea con- Na-d + A[Au( +j - (9) j=l (2) whe e E {.} is he expec a ionbpe a o and y( +j I ) is an op imum j-s ep ahead p edic ion o he sys em ou pu on da a up o ime . u( ) and y( ) a e he con ol and Gj = [Zjl, 6j2,. . ., 3jnlT (3) whe e he 6,, o each j, con ain he N a iable coe icien s in a cons ain equa ion and he Bj a e he bounds. Fo he case o a linea quad a ic op imiza ion p oblem, i can be conside ed an ene gy unc ion o he o m M-1 whe e F(z) is a p imi i e o (z) gi en by kz o z > 0 0 o z <= 0 (5) Conside equa ion 4. We shall deno e he ypical al- ues o Gj;, Ij and j by Go, IO and VO. These numbe s a e such ha he adimen ional quan i ies xji, pj and qj de ined in 6 ake absolu e alues anging om 0 o 1. ou pu sequence o he plan . NI and Nz a e he minimum and maximum cos ing ho izons. A is weigh ing coe icien and w( +j) is a u u e se -poin o e e ence sequence. The objec i e o p edic i e con ol is o compu e he u u e con ol sequence u( ), u( +l), ... in such a way ha he u u e plan ou pu y( +j) is d i en close o w( +j). This is accomplished by minimizing J(N1, Nz). Howe e , i can be compu a ionally p ohibi i e o eal ime applica ions. In p ac ice, he no mal way o using GPC is o compu e u( ) and apply i o he p ocess. I ~( ) iola es he con- s ain i is sa u a ed o he bounds, ei he by he con ol p og am o by he ac ua o . The case o u( + l), --e, u( + N) iola ing he cons ain s is no e en conside ed as in mos cases he signals a e no e en compu ed. This way o ope a ing es ic s he op imalli yo he GPC when cons ain s a e iola ed. Fu he mo e, he main pu pose o he GPC, which is o op imize equa ion (9), is no longe alid and he bes expec ed con ol is no achie ed. Mos p ocesses in indus y can be desc ibed by he ol- lowing ans e unc ion: Subs i ui ing in 4 Xij = Xji (8) I VO is chosen so ha = 1 we ob ain he same linea -quad a ic op imiza ion p oblem in e ms o adimen- ional quan i ies. This o mula ion is sui able o be im- plemen ed in a s ochas ic a chi ec u e as i will be seen la e . 111 APPLICATION TO CONSTRAINED GENERAL PREDICTIVE CONTROL. I an op imal p edic o is used, as shown in [8], he dead ime can be igno ed o designing pu poses. I we compa e he ene gy unc ion (4) o a Hop ield ne o he cos unc ion (9) o he GPC, conside ing ha he ou pu o he neu al ne V;. co esponds o he con ol se quence uj, we ob ain he exp essions o he pa ame e s o he ne wo k as un ions o he sys em pa ame e s, con ol pa ame e s and he sys em inpu and ou pu sequences o a gi en con ol ho izon [9]. As an example, esul s o his p ocedu e a e shown in Appendix I o sys ems desc ibed in (10). I is ema kable ha equa ions which de ine he conduc- ances Zj only depend on he sys ems pa ame e s, whils equa ions de ining he ne inpu s also depend on he sys- em inpu and ou pu sequences and he e e ence. The alues o Ci and R, a e chosen a bi a ily. We+can_de ine he cons ain subne iden i ying he e ms o 0, 2 Bj o each es ic ion. 1728 ...................................... !PI--; PREDICTOR J P(Z‘ ) P-l 71’ I Figu e 1: Con ol Scheme The esul ing con ol scheme is shown in Fig. 1. The neu al con olle p o ides he inpu signal sequence o he plan . The iden i ie es ima es he plan model pa ame- e s om i s inpu and ou pu signals.These alues a e ed in o block T ha ep esen s he calculus o he polynomi- als o ob ain he conduc ances o he neu al con olle . In block I in ensi y dependen sou ces a e calcula ed acco d- ing o he cu en e e ence. As i will be explained la e , block I compu a ions a e also pe o med by he neu al ne wo k while iden i ica ion and Tij e alua ions a e ca - ied ou by he con ol compu e . IV ARCHITECTURE. A cons ained Hop ield neu al ne wo k will be consid- e ed in his a icle as a se o sa u a ing linea in eg a o s whose inpu s a e linea combina ions o o he in eg a o s’ alues and ime a ying e ms. The dynamics o hese e ms a e supposed o be much slowe han ne dynamics so ha hey may be ega ded as cons an s du ing in e- g a ion. Analog e sions o his con olle can be ound elsewhe e [lo], bu hey lack o p og ammabili y. S ochas ic sys ems use s ochas ic signals whose alues andomly ake he alue 0 o 1. The a e age o hese I . F - mm A Ii l==l mm :...................................a Figu e 2: S ochas ic Neu al Ne P ocesso alues can be iewed as an analog alue in he ange [- l,l]. S ochas ic signals can be mul iplied using only a simple AND ga e. The e o e, p oduc e ms in he new al ne wo ks can be cacula ed by a s ochas ic mul iplie . The op imiza- ion p oblem should be posed in e ms o adimen ional qua i ies as cons an s and a iables a e o be ansla ed in o s ochas ic s eams o pulses. This has been done in sec ion 11. The high-le el a chi ec u e is shown in Fig. 2. I con- sis s o a s ochas ic neu al p ocesso , which is basically composed o a s ochas ic p oduc e alua o and a s a e machine con olle , and wo andom access ead and w i e memo ies. No ice ha neu on ac i i y alues e ol e due o ei he o he neu ons’ ac i i y alues o ne inpu s. A any a e hese e ms may be conside ed as p oduc s ha can be compu ed by he s ochas ic p oduc e alua o . Cons ained Hop ield neu al ne wo k’s in e connec ion weigh s and ne inpu s’ cons an e ms a e s o ed in one o he memo ies and neu on ac i i ies and a iable e ms o he ne inpu s a e s o ed in he emaining one. All hese magni udes may ei he ake posi i e o nega i e alues. Two kinds o neu ons a e in ol ed in he ne wo k’s e o- lu ion: sys em neu ons and es ic ion neu ons. Sys em neu ons a e sa u a ing linea in eg a o s ha can ei he ake posi i e o nega i e alues. Res ic ion neu ons a e sa u a ing nonlinea in eg a o s. They ake nega i e al- ues when he p oblem es ic ions a e iola ed and o ce he sys em neu ons o e ol e in such a way ha cons ic- ions a e ul illed. Neu ons’ in o ma ion (ac i i y alue, sign and ype o neu on) a e se ially ad essed by he con olle and loaded in he up/down coun e N and la ches S and T. The 1729 0 aooo 1000 0 -1000 -2000 ----.-.-.-. .-.- .-.- -.-. -.-.- --. I I I I I I T o zoo00 ~oooo 60000 eoow io0000 iaoooo uoooo 160000 iaoooo ZOQOOO -3000 I o I . . ioP. Figu e 4: Con olle ’s chip ansien esponse. Figu e 3: Tes boa d pho og aph. s ochas ic p oduc e alua o calcula es he inhibi o y o exci a o y in luence ha neu ons (uj, j = l..n) (and in- pu ime a ying e ms) ha e o e he one which has been loaded (ui). To achie e his goal hese alues and hei ela ed cons an s a e se ially ad essed by he con olle and ead. Random numbe s a e hen gene a ed and com- pa ed wi h hem, p oducing wo s ochas ically indepen- den s eams o pulses ha a e ANDed and ed o logic block B. This logic block ei he inc emen s o dec emen s he up/dowm coun e acco ding o he signs o he h ee numbe s in ol ed and he kind o neu on ha is being in- eg a ed. I also changes he sign bi (S la ch) whene e a ze o c ossing akes place. When he p ocess has inished he new compu ed alue o o i is w i en in he ex e nal memo y and ano he neu on is loaded. PHYSICAL REALIZATION. A neu al op imiza ion p oblem accele a o has been de el- opped (Fig. 3) and es ed ollowing he desc ibed s uc- u e. The s ochas ic neu al p ocesso has been imple- men ed in an E asable P og amable Logic De ice. EPLDs can be designed and p og amed in-house, elimina ing he long enginee ing lead imes and high ooling e o s and cos s o ull cus om o semicus om de ices. They a e also e y app op ia e o sho se ies ab ica ion whe e an Applica ion-Speci ic In eg a ed Ci cui would be un- economical. An Al e a EPM5130 de ice has been chosen o de elop his applica ion [lo]. The numbe o sys em es ic ion neu ons and ne - wo k inpu signals a e ully p og ammable. Two ex e nal RAMS ha e been used o da a s o age, one o neu ons’ alues and o he o synap ic weigh alues. Ne inpu s’ ime- a ying e ms a e sampled by he hos compu e and w i en wi hin he neu on RAM. Thei associa ed con- s an s a e plan model pa ame e s which we e s o ed by he hos compu e in he weigh RAM when he ne was con igu ed. Ne con igu a ion is accomplished by simply loading he ne ’s in e connec ion weigh s and ne inpu s’ cons an e ms in one o he ex e nal memo ies and by loading he ini ial neu ons’ ac i i y alues and ne inpu s’ ime a ying ini ial alues in he o he one. The sys em’s clock may be ei he ex e nal, in case moni- o ing he ne ’s e olu ion is desi ed, o in e nal, o no mal ope a ion. Mos indus ial p ocesses ha e slow dynamics. Fo his eason high speed memo im ha e no been used, yielding a sys em’s clock speed o 5MHz. VI APPLICATION. As an example he sys em o be con olled has a pu e delay o one sampling pe iod and is desc ibed by he ol- lowing polynomials: A(.-’) = 1 - 0.4~-I - 0.32.~-~ B(z-1) = 0.1 Fo he i s se o es s he u u e e e ence was consid- e ed o be a p e iously known squa e wa e. The weighing ac o A was made equal o 0.0001. The maximun con ol signals we e cons ained by 10, he slew a es we e con- side ed o be limi ed by 5 and a con ol ho izon o 5 has been used. The ansien esponse o he neu al ne wo k is shown in Fig. 4 when calcula ing he i s con ol signal se quence. When he s able s a e is eached, V(5) ep esen s he con ol signal o be applied o he sys em. R(l) and R(2) a e he ou pu s o he es icc ions V(5) < 10 and 1730 -- -10 1 1 0 5 15 20 a5 30 35 10 I o s ep* Figu e 5: Sys em's closed loop beha iou . V(5) - V(4) < 5 espec i ely. This s able s a e ma ches heo e ical esul and i s compu a ion ime has been 0.3 seconds. In o de o check he pe o mance o he neu al con- olle , he esponse o he closed loop sys em has been s udied when changing he e e ence. In Fig. 5 ' e ' ep- esen s he applied e e nce signal''V(5)' he s eady s a e o he i s sys em neu on in he con olle and 'Y' he ou pu o he plan as unc ions o he numbe o he con- olle 's ac ua ions. VI1 CONCLUSIONS. Digi al s ochas ic a chi ec u es a e a e y e icien way o ealize neu al ne wo ks as hey signi ican ly educe he needs o silicon a ea. We ha e implemen ed a gene al pu - pose e sion o a cons ained Hop ield ne wo k on a p o- g ammable logic de ice. This cop ocesso sol es any con- s ained quad a ic op imiza ion p oblem. I has been used o implemen he Gene al P edic i e Con ol p oblem in o de o elease he con ol compu e om he hea y com- pu a ional bu den o his algo i hm. Exp essions o ne pa ame e s a e p o ided o a common plan model which pe mi an adap i e con ol. The chip has been inco po- a ed in o a pe sonal compu e . Resul s ob ained wi h his expe imen al chip ha e been epo ed. APPENDIX I NETWORK PARAMETERS Fo his sys em, he ne pa ame e s a e gi en by he ol- lowing equa ions: i=l + 2Au -1 i j odd i j e en 2n--i .. 1=J n-j+l whe e odd(i) = emainde (i/2) and e en(i) = 1 - odd(i). REFERENCES [l] M. Gopal. Model Con ol Sys em Theo y. Wiley Eas e n Limi ed 1985. [2] D. W. Cla ke C. Moh adi and P.S. Tu s. Gene al- ized P edic i e Con ol. Pa I The Basic Algo i hm. Au oma ica ~01.23, pp. 137-148. 1988 [3] E. F. Camacho. Cons ained Gene alized P edic i e Con ol IEEE T ans. on Au oma ic Con ol, ol 38, pp. 327-332, 1993. [4] D. W. Tank and J. J. Hop ield. Simple Neu al Op i- miza ion Ne wo ks: An A/D Con e e , Signal De- cision Ci cui , and a Linea P og amming Ci cui . IEEE T ans. on Ci cui and Sys ems, ~01.35, pp. 554562, 1988 1731 [SI D.E. Van den Bou and T.K. 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