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A Real Time Supervision System for Adaptive Control
Copyright © IFAC Ada ptive Svstems ill CO lltrol and Signal Processing. Glasgow. U'; . IY/ll1
A REAL TIME SUPERVISION SYSTEM FOR ADAPTIVE CONTROL M. Samaan, M. Duque and M. M'Saad Greco CNRS "Sarta", Laboratoire d'AutoTlUltique de Grenoble (UA 228) , ENSIEG-INPG, E .P. 46, 38402 Saint Martin d'H eres, France
ABSTRACT: What is supervision ?, why do we need to supervise a control loop?, which kind of knowledge does a supervisor integrate ? and how do we build a supervisor ? In this paper, we define the supervision of a control loop using mathematical logic approach, which gives a more precise way to see the problem. Logical resolutions and formal systems approach lead us to a better interpretation of knowledge used by the supervisor. Afterthen knowledge is coded in a declarative way using simple structures. Because general inference can never give satisfaction, we presente some knowledge construction principles to ensure more coherence and self explanatory of the knowledge base. Finally, we present an application of a real time supervised adaptive control system on an aero-thermal plant. KEY WORDS : Supervision, Adaptive control, Recursive functions, Formal system, Expert system, Knowledge base, Logical resolution, Control loop, Predictive control, Skolem functions.
sure, allovs to cOD:l\ll.e on the limitation of this type of approa::hes. The common idea of all supervision architecture comes from the iD:ompletaness of the controllavs vhich are !nore am more complicated. So, ve unierstm:1 the necessity to integrate some knovn logical fun:tionnalities about the plant m:1 the control lav algorithm. These fun:tionnaiities must ensure : _ the validity of hypothese of mathema1ical theory vhich is behiM the controllav. _ the description of the physical plant flm:tionning m1 its limitations. In this paper, ve present IS supervision system based on the ·supervisor" ~ditional loop. Its implementation on an expert system generator allovs the integration of some controlled declaratiYe stN:tures arouni IS library of prOC€ldural fUD:tions used by the supervisor. In the f1l'St three chapters, ve d.ef1D8 in a more formal vay the Supervision Logic. Then ve present briefly the resolution methods used by the "supervisor" . In the forth m1 flth chapters, ve describe the JDBth)d ve used to integrate the supervision knovledge in the system. ArK1 finally, ve present the implement architecture the real time aspects am some experimentation results.
INTRODUCTION The supervision of the control loop during the last years has been the subject of many reflections presented unier several forms. Among the different approa::hes, one fi"'-s those based on the introdldion of an expert system inside the control loop [6]. This choice of "expert control" has been essentially motivated by the declaratiYe aspect that knov1edge can take. Other approa::hes are more direct than those of "expert control". They ~ a nev loop ca11ed "supervisor" onto the control loop [3]. Their motivations are based on the robustness improuvments of the control loop. The implementation of su:h a "supervisor" usually takes the form of a decision tree or graph. The declaratiYe aspect is secorK1ary m:1 pra::tically absent in real time realizatlons . The common care of these tvo types of approa::hes is to giYe the essential role to algorithms issued from the precise mathema1ical mod.elisati.on. A third type of approa::hes considers only declarative knovledge issued from experimentations m:1 logical furnonning of the plant. Therefore it lets the inferm:e strategy in an expert system to make decisions m:1 to calculate the control. The interesting part of this type of approa::hes is the use of simple m1 uniq\.M! stN:ture vhich ignore compleJB mathematical stN:tures. Altboush the idea is attr~tive, it became evident that the simplicity of the representation lea1s to an enormous complexity of the inferea:e level. The non existm::e, until nov, of su:h a cornplm am general1Dterea:e methods vhlch must be reliable am
UNIVERSAL EQUA rION Suppose I-{a,b] vith IS m:1 bEN is the lDluisition interval of a measure instrument Wehave : .cI the c_ algebra of Bool defined on the interval I . .l\-jJ.(X) vith XE Cl am xtE N IS Lebesp measure 397
398
M. Samaan, M. Duque a nd M. M'Saad
mtion defined on C7J. %t is dODe at tbB instant t We call time seqt.m:8 of length (n+ I) tbB ve11-ordered sat of msasures. I
X. - (~-a' 1t-... 1' ... , ~ ) ThI coding of this seqt.m:8 in a Turing ma:bine is [16,17]:
~ ~ $~-"'i
6 n-I
(It-...i
:-
! (1t+1 :-
.say as
It ) - S ( D,,)
a substitution of tbB measure
xt+ I
by the
D
measure xt Sl.l:C8eded by a shift on YJ,n. This formulation by a Turing ma:bine is more precise for tbB impleantation because it mans clear tbB coding stnEtures am paramBtan as so as tbBir limits. Suppose (f) a partial mtion defined on M am takes values in M. We say that (f) is recursive if there exists a procedure (P) (or a program) vritten in a Turing ma:bine 1anguDge so that [ 16, 18 ] : P(n) • .L if n def(f) P(n).f(n) if n e def(f) vith .L IDBIDS that P runs am D8V'81' stops. def(f) is tbB definition domain of (f). Suppose A a sat irdl.l1ed in M. We say that A is recursive if am only if there exists a
fuD:tion f: M~N f(n).1 si ne A f(n).O si A is recursive. We say also that A is recursive1y_enumerab1e if am only if there exists a mtion (f) vith A-c1ef(f). Suppose T(x .. xz, ...,lrsJ a predicate of (n) variables ( x.. xz, ... ,Xn) e F vith F a sat on vhich recursive am recursively_8DUIDBl'ab1e parts all vell defined. We say that T is decidable if tbB sat of solutions : By - {( ~ , Xz, ... ,~) I T ( ~ , ~, ... , ~) is true } is recursive. T is semi_decidab1e if 61 is recursively_enumerable. We define vith the same vay, the recursivity of a time_seqt.m:8 YJ,n as :
n.
suppose
-
1._ signal.
S-(U,,)cM called
3U:
a time_seqt.m:8 in M, called control 3~,hz,
.. ·,haeM
P(
..,a u.;,a ~,~ , ~,hz, .. ·,ha) t~
is decidable. In otbBr terms, for nary sequm:es of msasures of output am refaJm:e, there aista a seqt.m:8 of control am there exists satisfied hypothese so that tbB predicIIa (P) is decidable. We call tbB predicate (P) tbB UDivena1 equation of control. ThI ezistm::e of tbB seqt.m:8 Utn
of control can be ca1cu1at8d by a Sko1em mtion fu.
1t-...it1)
as a seqt.m:8 of instn.£tions of substitution on YJ,n. am tbB up1ate operator by tbB reccurent formula "+1
amreferm:e
called hypothese or perforrnm:es so that :
.define ~ • S ~_I vith (.S) tbB instn.£tion "Sl.IX8SSOr of" . ut could. be a data or an instn.£tion. .defme also tbB substitution instnEtion (:.) on tbB formula ut-I :- ut as tbB substitution of ut-I by ut· This allovs to defme tbB shift operator.
D" -~ .say
vy:,~a cM wo time_sequm:as in M am called rwpawlyoutput
We say that YJ,n is recursive if am only if S is recursive.We also say that S has a ~I everywhere. Nov, let's define tbB folloving predicate :
u: -
f, (
y:, ~a , ~ , hz , ... , ha )
ve call this f'urEtion tbB "control1av" vhich red\E8S tbB number of ezistentia1 variables of U. UDivena1 equation, vtl,~ c • 3 ~I'lz , ... ,~ •• : P( 1l,~,~( 1l,~,~,lz, .. · ,~),~,lz, .. · ,~) • p.(
tl,~,~l'lz, .. · ,~)
We can interprets U. UDi'm'Sal equation by the existen:e of a program vhich implemmts Pu.We suppose that this program can use any control 1av known until nov. It Dl\Bt be able, vith respect to hypothese bt.bz, ... ,hn, to respoIJ1 in a finite time if ")'IS" or "no", it can control U. plant CODDIICted. to it ThI ansver to this question appears to be evident. All U. same, mvertblless, many resaarchm try to inwnt SlI:b a program : - an algorithm able to control a larp class of plants but also to d.etect vhat is happening am vhy, sometimes, it is unable to do U. vork. Suppose that U. only exIerDa1 observations that U. UDiversal equation has, are tbB msasures of output am referm:e. It Dl\Bt be able , using these observations, to estabhsb the truth on hypothese fiJBd on U. control1av. That means, in a formal VI'J, that ve are able to fiD1 Stolem fuD:tions of hypotblse so that [16,18] ,
~-f~ (Y:,~) hz-f~(y:,~ )
(y:,
ha -f.. ~a ) ThI elimination of all aistentia1 variables of thI universal equation giVlS tbB compact folloviDg equation: Vy:,~a cM ,Q(y:,~a) is d.ecidable ThI decidability of Q IDBIDS that the recursive am u. recursively_8DUIDBl'ab1e parts of sequm:es R.~ are vell known. In otbBr terms, ve have to knov perfectly U. plant ve van1 to control. In pra;:tice, this is not frequent aM sometimes, altbousb ve knov all tbB pbBDomenas that affect U. plant, it is
1ft ,
399
A Real Time Supervision System for Adaptive Control
oot easy to f1D1 appropriate stolem t'l.uxtions. On the other hmi. it is m:>re in1aresting to be able to redu::e the problem using stolem f'urEtions established for certain cases BlEb as a class of plants. Let's giw the namp1e of the partial state ganeralized pred.ictiw control vbich is ~ as the controllav of O\D' implaantation [1.2.12.13). Uniwrsal equation : ~ ,036 .. 36 \
P
vith
lit : the initialization horizon. hp : the prediction horizon.
he : the control horizon. ~Pc : veijtrtings.
Pe
: regulation model. AID/Bm : tncting rmde1. p. D : disturbm:es model. A • B • d : plant model.
fh
So. suppose y - f(zt, Xa. x,. ... • z,) a stolem fuD!tion, ve associate a set of veU-ordered. predicates of
length (0+ 1) so that.
",-(PO'P 1 '
.... Pa )
Pi - SP1-1 O~i~n or. in a Turing meclline,
~
",- W Pi This sequm:e defines the coDStrl.l:tion method of the considered Stolem fuD!tion. Many timBs. theIe sequm:es use common sub-seq\llllDS of predicates. So that hierarchical organization becomas necessary.Pro8J'llDll'lin8 a decision grapb can giw satisfa:tion. We eaU qualitatiw model linked to a fuD!tional unit, the logical depeD1eD::y graph vhich ensure the good fuD!tionning of the unit.So that the m:>dularity of the control scbBml introdu::e an equivalent m:>dularity in the supervision graph.
THE LOGICAL RESOLUTION Control1av : De.(t) =(-1 0 0 : . .
Ol[ L:Lr +}..(t)IT' L: E~
vith e,(t) _ p.(q-1) (u(t) _ PA(q-1) r(t+d) )
m1 f
..;
-1,.
t;,(t-1)
-I
r e..(t)
ao(t)"~"'I(q ,~+ ~(q ) ~
p.(q )
P.(q )
m1 ....... etc.
Some simplified stolem f'urEtions : - vb8n there 11 1<*1 disturbm:as. ve use an in18gra1or. - for a plant of order (n). ve baw hp>-2*n+ 1 - if the plant is a secom order then n-2 - for a stable m1 veU damped plant. ve usually use he I - if bp>lS a mem:>ry owrflov vill occurs. so that take hp - 15. - to test the stability of the plant eaU the stability function or search the roots of the A polynomial m1 .... etc. TH! SUPERVISION The iIEomp1eteness of the solution of the uniwrsal equation Q np1ains in a formal vay the necessity of the supaniJion [3."\.5) . The supervision unit contains eaentia11y the set of stolem furdlons of Q ncept the controllav. In tha case of ad.apti91 control. ve consider that pc ameter estimation tim::tion is a unit of the control lav. An lIDalytical description of stolem fuD!tions does DOt UIUIilly aist. but many stolem functions are composed of 18Q\IIIlDS of predicates. This comas from the fact that a sraat number of theIe tim::tions describes the physical reality of the plant m1 the lECumulated expenm:e aroum the controllavs.
Suppose that the set of predicates of the supervision logic is vritten in a first order logic. It is m:>re m1 m:>re difficult to build. a decision graph vb8n the number of predicates becomes important. So ODe can reconsider the problem by putting a part this ~w resolution m1 using a pral resolution methode. This last method. can be the folloving inferm:e rule
[18.19.16] • f.B I-b I'IS ifml only if (f) has the form avfl (g) has the form -, bv gl (b) has the form 0(8 fl v il )
vhere f.g,h,a,b.f1ml il are formulas in a first order logic (clauses). 8 is a substitution, so that et ml i baw DO common variables. ml a is the most sraat unificator of ea ml b. The predicate h is called the resolvant of f ml i vith .., is the Degation ml V is tha disjur¥:tion. This type of resolution is iIEomplet8.so it can enter in an infinite loop trying to fiD1 the solution. The iIEompleteness of this inferm:e leads to introda particular tschniques called inferm:e control strategies to uplore the dedl.l:tion tree. The resolution by inferm:e has the tuMamental property of the coDStrl.l:tion by interpretation of the solution, using the set of predicates. That irEreases the declaratiw aspect of coding ml decreases the impratiw one. It is evident that this dec1aratiw aspect is partial because of the use of control strategies in the search. In our impleznentation, ve use another type of resolution vhich is m:>re used in pndice. It irl:l1l1es vbat ve eaU the forvard, b!dvard ml mm chaining. Usually this infereJxe is based on propositional logic vithgtobal variables (0+)[1"\].
M. Samaan , M. Duque and M. M'Saad
400
KDov1ed.ge has simple str\Eture called rules m1 a rule take the form,
if (cormtions) 1mD (ldions)
Let's see it from a formal system point of viev, the extm:1ed recursivity domain is the highest knov1ed.ge 1e98l axording to the ph}'sical plant. Iba supervisor looks for his axioms ~s inside this domain. The
That easy propositional vay to build knov1ed.ge base imol_ a coDSiderable iwease of the number of rules m1 makes the i.nferm:e control wry difficult A part of this difficulty has been o'mCOmed by using knovledge islm1 sources m1 by introdl.Eing objec1s m1 inlmetm::e i.nferm:e [14]. The pra1 frame of an object is,
hypothese of Q are the theorems 1s of the supervisor. This set of theorems represents the axioms of the controllav scheme. To resume, ve can say that betveen axioms --\ am
OBJECT
CONSTRUCTION METHOD
theorems 1s , the supervisor builds his dedl.£tion graph to search the solution for 8IIcl1 problem.
Name Classes m1 Sub-Classes Subobjacts Attributes. On the other hand, more user effort is requested to str\Eture the knovl8dsa.
In our supervisor implementation, ve use the
N-EXPERT sof'tvare vbich combines these tvo levels of mov1ed.ge representation.
RECURSIVITY AND AXIOUA TIC LEVEL The ~mpleteDess of the lmiwrsal equation Q '-is to split the set of observations into three parts
[16,17,18,191 - a vell know part. It has interpretations m1 solutions for possible problems. Ibis part is a recursiw set D'Q. - a know but badly stW.ied part. We knov the problems related to this part but ve ba98 l81 no solution to them )'It. Por nample, ve knov that the plant is non-liDear for vide referm::e changIs but ve ba98 no-good mstb:ld to o'mCOme possible problems inherent to non-limarities. In this case the operator may stop the plant control. Ibis part is a recursi981~numerable set CQ. - the third part is wry t.:1 know or untnovn. It is a non-recursiw set EQ. Prom this interpretation of the recursivity, ve define the recursiw domain D'Q of the uniwrsal equation as the set of observations for vhich Qhas a solution, D~ - { ~a ) I Q ~ is true} is recursiw
(ll,
(ll, )
In another vords, D'Q is the set of observations for vbich Q is d.ecidable. Ibis domain can be viden by ildllling recursi981y-enumerab1e observations. These observations can start alarms or stop the system to ask external aais1m:e. n~= {(11 ,1t:) I 0(11 ,1t: )islnJe) isrears~-l!IUIlnJe We call D"Q the extm1ed recursivity domain m1 ve can easily vrite,
D~ -D~UCQ Iba supervision logic has to ensure that observations still to be a solution of Q or that the control loop is
alvays inside the extm:1ed recursivity domain.
The preceding discussion on the lmiwrsal equation, on the resolution m:1 on the formal system approa:h leeds us to propose a method to build the knov1ed.ge of the supervisor. Ibis rmthod uses tvo principles,
1) The axiom -+ theorem trams [16,18,19] : Although, rules m1 objects in the supervisor ba98 stm1ard forms, knov1ed.ge coding is not usually evident. Let's take the eDmple of a knov1ed.ge corarning the disturbm:es. We knov that if the output disturbm:es are of 10lld type, one can cm:el them by introdw:ing an integrator in the control loop. Ibis sentm:e can take another use, say that if the control loop has an integrator, 10lld disturbm:es are cm:e11ed. The coding of this information appears to be evident as a rule stru::ture but in reality, that depm1s on vhat ve van! to do vith this rule. In the supervisor the axiom--Mheorem trams, vhich comes from formal system approach, allOYS to formulate the rule. The axiom, in this case, is the observation of 10lld disturbm:es, The theorem tells that tbBre exists a transfer fuixtion called integrator vhich alloys to resolw the problem. Iba 10lld disturbm:es is a physical parameter vbereas the integrator is an abstn£ted stru::ture that only its interpretation can atUdle it to the reality. Anyvay, ve can ba98 another theorem lading to another solution of the same problem. In N:h case, inferm:e control strategies haw to solw possible conflicts. 2) The interpretation base : The axioms of the supervisor are built using a sub-set of observations Ds in:lu1ed in the extemed recursivity
domain. We call Ds the interpretation base. The choice of this base allOYS the supervisor to C098l' a part or all the exteD:1ed recursivity domain. In pnctice, vben the supervision task is supported by the expert or the user, he makes decisions as a tuD=tion of observations saneraUy d.isp1ay9d as graphs by the measure instruments. We call image the time-seqtn:e of measures of lenith (n+l) at instant t. Prom all the imaBes representing the observations, only a finite number reflects a state chan8es vhiche make necessary the supervision action. Usually, ve choose the interpretation base from this finite set of images.
401
A Real Time Supervision System for Adaptive Control
Aftgr
tbBD, tb8 IUpIJ"IiJor DI8dJ to tnda tb8Ie imII8II
at his llliomatic lnal. We use NO ditferent vays to code imqes, -n. shape of tb8 sraph of tb8 imIgIe is recognized by
using ODe of mown form recoption mstbods [5]. In this calli tb8 imIgIe is coded as an object, its attributs are tb8 elements of tb8 recognised shape. - A JDIINI'8 tuD:tion is dafiDBd. on tb8 imIgIe. The computed measure is an 1lli0IDI va1\11 [1,2,11). Por nample, an information measure is DII8ded to decide if tb8 estimation has to be freez8d or DOl This JDIINI'8 is a flmction of tb8 regression wctor or of inputs m1 outputs~ .
Pigure(2) shovs tb8 germal systam architecture Pigure(3) shovs a real tiIDB tum:tioDDiDg mmpII . I
r
IBM/AT
Serial communication
RS232
As a confirmation of this priD:ipla, vhan an apart or a rearcbar vants to analyse a pbmoIDlDl, he tries to project or simply reprod\E8 the interpretation base imqIs. Let's take the namp1e of f'1gUl'8 (I), a lom disturbm:es happeDs at instant 68. n. supervisor detects an output offset vbich vill establish tb8 truth on the disturbm:e llliom. ThIn it introd\1:8S an intaifllOr
temperature
at instant 87. 6,58-
...... i output oltset
5,28 3,98-
AERO-THERMAL PLANT
2,68 -
Figure (2)
1.38 -
IIUIl,f--..,.......-+--+--+--+-+--+-+-I--+---I -1.38 -2,68 -
An intml'ttition ~e elmnt. OutPUt loid distUl'banm. -3.'8 -5.28 -6.5845.811 52,511 6UII 67.58 75.l1li 82.59 99.99 97.59 195.99112.58128.
Pigure (1)
SYSTEW ARCHITECTURE A partial state ~ II1aptive control is implemmted on an IBWAT-386 using EXPERT-AD PIEbia [13]. n. control loop is coDDBCtad to an aar-tbBrmal plant. This plant is DOn-liDBIIr e.:cording to l"efereu::e changes. It is a firt order stable plant vith nc>-de1ay. We vant to control output temperature vbich can wry betveen tb8 surrouMing temperature m1 900 C . n. supervisor is implelDlmted on anotbBr IBWAT-386 using N-EXPER T sottvare PIEbia [14). It communicates vith the control loop via serial port
RS232. Real tiIDB constraintes d\ll to sampling period are supported by tb8 real tiIDB environment of EXPERT-AD ~. On the otbBr hmi, the supervisor intenentiom are asyn:hrODOW. So that the supervisor sm1s orders to the control1av as soon as the resolution has foUll1 the solution. If the solution is DOt reliable, it calls the operator for ezterna1 assistaD::e. n. supervision mov1edga is coded as rules m1 objects. The iDferm:e engin starts vith forvard chaining then it chazIi8s re.oning vbile nploring different mov1edge is1m1s.
3.58 I I I I ~ 811.8'8.8 188.8 278.8 3611.8 458.1 • l'til till! suPtI''1iStd idiPtive conbol of iD mo-thel'tlil pliDt
"'1 10"'
t.,
'r-t. h-
I~
11.58 I 811.8 ,8.8 18U 278.8 36U 458.11 5411.11 639.9 729.9 81U , •• SiJII'ling pniode ( ts : 6 S )
Figure (3)
CONCLUSIONS
In this paper, Ye tried to def'1DI tb8 outliDBs of a supervision system vhich merges tb8 compla matl:mnatica1 modelization vith losica1 m1 1lli0matica1 tua:tioDDiDg of a plant. On the otbBr hmi, tb8re remains a big amr:>lmt of york, coDl'Ding. - Sko1em tua:tiODS m1 recoption mstbods, - the minjmisation of tb8 int8rpl'8tation base. the real tiIDB aspect. A real time
sym:hI'oDOW
supervision affects considerably the stnEture of the mov1edge.
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UDI
Dtbodologie de la mise en oeum des schIDDas de commm1e ~ve',Commm1e Adaptatlve
402
M. Samaan , M. Duque and M. M'Saad
Methodologie et Application, pp 63-91. Ed.. HERMES 1988. (2) M.MSAAD, I.D.LANDAU, M.SAMAAN am M.DUQUE, "PrafOrrDID oriented robust Il1aptiw controUer",papar,ACC, Atlanta, georgia, June 15-17 1988. (3) M.DUQUE, M.SAMAAN, M.M'SAAD am I.D.LANDAU,"Real time supervised Il1aptiw control" Paper,Adaptiw control symposiom, YALE(USA) May 1987. (1) M.SAMAAN,"Station de travail et syst8mBs experts pour la commm1e",Report, DEA in Laboratoire d'AutomaUqua de Grenoble, 1985. (5) R.ISERMANN am K.H. LACHANN, "Patameter Il1aptiw control vith configuration aids am supervision turrioDS",Automatica, Vol. 21,N 3, PP'm-2P1J. 19P1J. (6) K.JASTROM, JANTON m1 K.E.ARZEN,"Expert control",Automatica wl22, N3, PP'm-2P1J. 19P1J. (7) JPITRA T, "Cormaissm:es et mhacormaissm:es" Intelligm:e des mlflcanismes et m!canismes de l'int811ig8D:e", P75-113, Ed.. PAYARD, Po~on Diderol (8) E.RICHE," Artiticia1 Intel1igm:e",McGrav Hill series in artiftcial inte11igm:e. (9) N.J.NILLSON, "Pri.u:ip1es of Artificial Intelligm:e " Spring-Verlag Berlin, Herde1ber"Nev York. (10) WILLEM BLOKLAND m1 JANOS SZTIPANOVITS, "KDov1edgo&-based apprOl£h to reconf'18UI'Ib1e control systems, "paper,ACC, Atlanta, Georgia, June 15-17 1988, pp 1623-1628. (11) ROB ER T L. KOSUT, "Adaptiw robust control via traDSter turrion urartainty estimation",papar, ACC, Atlanta, Georgia, June 15-17 1988,pp 319-351.
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(15) DOUGLAS HOPSTADTER, "Gode1 Escber Ba", Ed..InterEdition, Paris, 1979. (16) S.C. KLEENE, Mathematical Logic, John W11ey & SoDS, Nev York 1967. (17) M.MARGENSTERN," Le ~r8me de Matiyassevitch et resu1tats annnes",Model Theory m1 Aritbmatic Procw1iDjB, Paris 1988,pp 198-211, Springer Verlag. (18) J.P.DELAHA YE, 0uti1s Logiques pour 1'Intelligm:e ArtificieUe, Ed.. EyroUes,Paris, 1987. (19) J.P.DELAHAYE, SYSTEMES EXPERTS: organisation et programmation des bases de cormaiam:e en calcu1 propositiormel, Ed.. EyroUes,Paris, 1987.
A REAL TIME SUPERVISION SYSTEM FOR ADAPTIVE CONTROL M. Samaan, M. Duque and M. M'Saad Greco CNRS "Sarta", Laboratoire d'AutoTlUltique de Grenoble (UA 228) , ENSIEG-INPG, E .P. 46, 38402 Saint Martin d'H eres, France
ABSTRACT: What is supervision ?, why do we need to supervise a control loop?, which kind of knowledge does a supervisor integrate ? and how do we build a supervisor ? In this paper, we define the supervision of a control loop using mathematical logic approach, which gives a more precise way to see the problem. Logical resolutions and formal systems approach lead us to a better interpretation of knowledge used by the supervisor. Afterthen knowledge is coded in a declarative way using simple structures. Because general inference can never give satisfaction, we presente some knowledge construction principles to ensure more coherence and self explanatory of the knowledge base. Finally, we present an application of a real time supervised adaptive control system on an aero-thermal plant. KEY WORDS : Supervision, Adaptive control, Recursive functions, Formal system, Expert system, Knowledge base, Logical resolution, Control loop, Predictive control, Skolem functions.
sure, allovs to cOD:l\ll.e on the limitation of this type of approa::hes. The common idea of all supervision architecture comes from the iD:ompletaness of the controllavs vhich are !nore am more complicated. So, ve unierstm:1 the necessity to integrate some knovn logical fun:tionnalities about the plant m:1 the control lav algorithm. These fun:tionnaiities must ensure : _ the validity of hypothese of mathema1ical theory vhich is behiM the controllav. _ the description of the physical plant flm:tionning m1 its limitations. In this paper, ve present IS supervision system based on the ·supervisor" ~ditional loop. Its implementation on an expert system generator allovs the integration of some controlled declaratiYe stN:tures arouni IS library of prOC€ldural fUD:tions used by the supervisor. In the f1l'St three chapters, ve d.ef1D8 in a more formal vay the Supervision Logic. Then ve present briefly the resolution methods used by the "supervisor" . In the forth m1 flth chapters, ve describe the JDBth)d ve used to integrate the supervision knovledge in the system. ArK1 finally, ve present the implement architecture the real time aspects am some experimentation results.
INTRODUCTION The supervision of the control loop during the last years has been the subject of many reflections presented unier several forms. Among the different approa::hes, one fi"'-s those based on the introdldion of an expert system inside the control loop [6]. This choice of "expert control" has been essentially motivated by the declaratiYe aspect that knov1edge can take. Other approa::hes are more direct than those of "expert control". They ~ a nev loop ca11ed "supervisor" onto the control loop [3]. Their motivations are based on the robustness improuvments of the control loop. The implementation of su:h a "supervisor" usually takes the form of a decision tree or graph. The declaratiYe aspect is secorK1ary m:1 pra::tically absent in real time realizatlons . The common care of these tvo types of approa::hes is to giYe the essential role to algorithms issued from the precise mathema1ical mod.elisati.on. A third type of approa::hes considers only declarative knovledge issued from experimentations m:1 logical furnonning of the plant. Therefore it lets the inferm:e strategy in an expert system to make decisions m:1 to calculate the control. The interesting part of this type of approa::hes is the use of simple m1 uniq\.M! stN:ture vhich ignore compleJB mathematical stN:tures. Altboush the idea is attr~tive, it became evident that the simplicity of the representation lea1s to an enormous complexity of the inferea:e level. The non existm::e, until nov, of su:h a cornplm am general1Dterea:e methods vhlch must be reliable am
UNIVERSAL EQUA rION Suppose I-{a,b] vith IS m:1 bEN is the lDluisition interval of a measure instrument Wehave : .cI the c_ algebra of Bool defined on the interval I . .l\-jJ.(X) vith XE Cl am xtE N IS Lebesp measure 397
398
M. Samaan, M. Duque a nd M. M'Saad
mtion defined on C7J. %t is dODe at tbB instant t We call time seqt.m:8 of length (n+ I) tbB ve11-ordered sat of msasures. I
X. - (~-a' 1t-... 1' ... , ~ ) ThI coding of this seqt.m:8 in a Turing ma:bine is [16,17]:
~ ~ $~-"'i
6 n-I
(It-...i
:-
! (1t+1 :-
.say as
It ) - S ( D,,)
a substitution of tbB measure
xt+ I
by the
D
measure xt Sl.l:C8eded by a shift on YJ,n. This formulation by a Turing ma:bine is more precise for tbB impleantation because it mans clear tbB coding stnEtures am paramBtan as so as tbBir limits. Suppose (f) a partial mtion defined on M am takes values in M. We say that (f) is recursive if there exists a procedure (P) (or a program) vritten in a Turing ma:bine 1anguDge so that [ 16, 18 ] : P(n) • .L if n def(f) P(n).f(n) if n e def(f) vith .L IDBIDS that P runs am D8V'81' stops. def(f) is tbB definition domain of (f). Suppose A a sat irdl.l1ed in M. We say that A is recursive if am only if there exists a
fuD:tion f: M~N f(n).1 si ne A f(n).O si A is recursive. We say also that A is recursive1y_enumerab1e if am only if there exists a mtion (f) vith A-c1ef(f). Suppose T(x .. xz, ...,lrsJ a predicate of (n) variables ( x.. xz, ... ,Xn) e F vith F a sat on vhich recursive am recursively_8DUIDBl'ab1e parts all vell defined. We say that T is decidable if tbB sat of solutions : By - {( ~ , Xz, ... ,~) I T ( ~ , ~, ... , ~) is true } is recursive. T is semi_decidab1e if 61 is recursively_enumerable. We define vith the same vay, the recursivity of a time_seqt.m:8 YJ,n as :
n.
suppose
-
1._ signal.
S-(U,,)cM called
3U:
a time_seqt.m:8 in M, called control 3~,hz,
.. ·,haeM
P(
..,a u.;,a ~,~ , ~,hz, .. ·,ha) t~
is decidable. In otbBr terms, for nary sequm:es of msasures of output am refaJm:e, there aista a seqt.m:8 of control am there exists satisfied hypothese so that tbB predicIIa (P) is decidable. We call tbB predicate (P) tbB UDivena1 equation of control. ThI ezistm::e of tbB seqt.m:8 Utn
of control can be ca1cu1at8d by a Sko1em mtion fu.
1t-...it1)
as a seqt.m:8 of instn.£tions of substitution on YJ,n. am tbB up1ate operator by tbB reccurent formula "+1
amreferm:e
called hypothese or perforrnm:es so that :
.define ~ • S ~_I vith (.S) tbB instn.£tion "Sl.IX8SSOr of" . ut could. be a data or an instn.£tion. .defme also tbB substitution instnEtion (:.) on tbB formula ut-I :- ut as tbB substitution of ut-I by ut· This allovs to defme tbB shift operator.
D" -~ .say
vy:,~a cM wo time_sequm:as in M am called rwpawlyoutput
We say that YJ,n is recursive if am only if S is recursive.We also say that S has a ~I everywhere. Nov, let's define tbB folloving predicate :
u: -
f, (
y:, ~a , ~ , hz , ... , ha )
ve call this f'urEtion tbB "control1av" vhich red\E8S tbB number of ezistentia1 variables of U. UDivena1 equation, vtl,~ c • 3 ~I'lz , ... ,~ •• : P( 1l,~,~( 1l,~,~,lz, .. · ,~),~,lz, .. · ,~) • p.(
tl,~,~l'lz, .. · ,~)
We can interprets U. UDi'm'Sal equation by the existen:e of a program vhich implemmts Pu.We suppose that this program can use any control 1av known until nov. It Dl\Bt be able, vith respect to hypothese bt.bz, ... ,hn, to respoIJ1 in a finite time if ")'IS" or "no", it can control U. plant CODDIICted. to it ThI ansver to this question appears to be evident. All U. same, mvertblless, many resaarchm try to inwnt SlI:b a program : - an algorithm able to control a larp class of plants but also to d.etect vhat is happening am vhy, sometimes, it is unable to do U. vork. Suppose that U. only exIerDa1 observations that U. UDiversal equation has, are tbB msasures of output am referm:e. It Dl\Bt be able , using these observations, to estabhsb the truth on hypothese fiJBd on U. control1av. That means, in a formal VI'J, that ve are able to fiD1 Stolem fuD:tions of hypotblse so that [16,18] ,
~-f~ (Y:,~) hz-f~(y:,~ )
(y:,
ha -f.. ~a ) ThI elimination of all aistentia1 variables of thI universal equation giVlS tbB compact folloviDg equation: Vy:,~a cM ,Q(y:,~a) is d.ecidable ThI decidability of Q IDBIDS that the recursive am u. recursively_8DUIDBl'ab1e parts of sequm:es R.~ are vell known. In otbBr terms, ve have to knov perfectly U. plant ve van1 to control. In pra;:tice, this is not frequent aM sometimes, altbousb ve knov all tbB pbBDomenas that affect U. plant, it is
1ft ,
399
A Real Time Supervision System for Adaptive Control
oot easy to f1D1 appropriate stolem t'l.uxtions. On the other hmi. it is m:>re in1aresting to be able to redu::e the problem using stolem f'urEtions established for certain cases BlEb as a class of plants. Let's giw the namp1e of the partial state ganeralized pred.ictiw control vbich is ~ as the controllav of O\D' implaantation [1.2.12.13). Uniwrsal equation : ~ ,036 .. 36 \
P
vith
lit : the initialization horizon. hp : the prediction horizon.
he : the control horizon. ~Pc : veijtrtings.
Pe
: regulation model. AID/Bm : tncting rmde1. p. D : disturbm:es model. A • B • d : plant model.
fh
So. suppose y - f(zt, Xa. x,. ... • z,) a stolem fuD!tion, ve associate a set of veU-ordered. predicates of
length (0+ 1) so that.
",-(PO'P 1 '
.... Pa )
Pi - SP1-1 O~i~n or. in a Turing meclline,
~
",- W Pi This sequm:e defines the coDStrl.l:tion method of the considered Stolem fuD!tion. Many timBs. theIe sequm:es use common sub-seq\llllDS of predicates. So that hierarchical organization becomas necessary.Pro8J'llDll'lin8 a decision grapb can giw satisfa:tion. We eaU qualitatiw model linked to a fuD!tional unit, the logical depeD1eD::y graph vhich ensure the good fuD!tionning of the unit.So that the m:>dularity of the control scbBml introdu::e an equivalent m:>dularity in the supervision graph.
THE LOGICAL RESOLUTION Control1av : De.(t) =(-1 0 0 : . .
Ol[ L:Lr +}..(t)IT' L: E~
vith e,(t) _ p.(q-1) (u(t) _ PA(q-1) r(t+d) )
m1 f
..;
-1,.
t;,(t-1)
-I
r e..(t)
ao(t)"~"'I(q ,~+ ~(q ) ~
p.(q )
P.(q )
m1 ....... etc.
Some simplified stolem f'urEtions : - vb8n there 11 1<*1 disturbm:as. ve use an in18gra1or. - for a plant of order (n). ve baw hp>-2*n+ 1 - if the plant is a secom order then n-2 - for a stable m1 veU damped plant. ve usually use he I - if bp>lS a mem:>ry owrflov vill occurs. so that take hp - 15. - to test the stability of the plant eaU the stability function or search the roots of the A polynomial m1 .... etc. TH! SUPERVISION The iIEomp1eteness of the solution of the uniwrsal equation Q np1ains in a formal vay the necessity of the supaniJion [3."\.5) . The supervision unit contains eaentia11y the set of stolem furdlons of Q ncept the controllav. In tha case of ad.apti91 control. ve consider that pc ameter estimation tim::tion is a unit of the control lav. An lIDalytical description of stolem fuD!tions does DOt UIUIilly aist. but many stolem functions are composed of 18Q\IIIlDS of predicates. This comas from the fact that a sraat number of theIe tim::tions describes the physical reality of the plant m1 the lECumulated expenm:e aroum the controllavs.
Suppose that the set of predicates of the supervision logic is vritten in a first order logic. It is m:>re m1 m:>re difficult to build. a decision graph vb8n the number of predicates becomes important. So ODe can reconsider the problem by putting a part this ~w resolution m1 using a pral resolution methode. This last method. can be the folloving inferm:e rule
[18.19.16] • f.B I-b I'IS ifml only if (f) has the form avfl (g) has the form -, bv gl (b) has the form 0(8 fl v il )
vhere f.g,h,a,b.f1ml il are formulas in a first order logic (clauses). 8 is a substitution, so that et ml i baw DO common variables. ml a is the most sraat unificator of ea ml b. The predicate h is called the resolvant of f ml i vith .., is the Degation ml V is tha disjur¥:tion. This type of resolution is iIEomplet8.so it can enter in an infinite loop trying to fiD1 the solution. The iIEompleteness of this inferm:e leads to introda particular tschniques called inferm:e control strategies to uplore the dedl.l:tion tree. The resolution by inferm:e has the tuMamental property of the coDStrl.l:tion by interpretation of the solution, using the set of predicates. That irEreases the declaratiw aspect of coding ml decreases the impratiw one. It is evident that this dec1aratiw aspect is partial because of the use of control strategies in the search. In our impleznentation, ve use another type of resolution vhich is m:>re used in pndice. It irl:l1l1es vbat ve eaU the forvard, b!dvard ml mm chaining. Usually this infereJxe is based on propositional logic vithgtobal variables (0+)[1"\].
M. Samaan , M. Duque and M. M'Saad
400
KDov1ed.ge has simple str\Eture called rules m1 a rule take the form,
if (cormtions) 1mD (ldions)
Let's see it from a formal system point of viev, the extm:1ed recursivity domain is the highest knov1ed.ge 1e98l axording to the ph}'sical plant. Iba supervisor looks for his axioms ~s inside this domain. The
That easy propositional vay to build knov1ed.ge base imol_ a coDSiderable iwease of the number of rules m1 makes the i.nferm:e control wry difficult A part of this difficulty has been o'mCOmed by using knovledge islm1 sources m1 by introdl.Eing objec1s m1 inlmetm::e i.nferm:e [14]. The pra1 frame of an object is,
hypothese of Q are the theorems 1s of the supervisor. This set of theorems represents the axioms of the controllav scheme. To resume, ve can say that betveen axioms --\ am
OBJECT
CONSTRUCTION METHOD
theorems 1s , the supervisor builds his dedl.£tion graph to search the solution for 8IIcl1 problem.
Name Classes m1 Sub-Classes Subobjacts Attributes. On the other hand, more user effort is requested to str\Eture the knovl8dsa.
In our supervisor implementation, ve use the
N-EXPERT sof'tvare vbich combines these tvo levels of mov1ed.ge representation.
RECURSIVITY AND AXIOUA TIC LEVEL The ~mpleteDess of the lmiwrsal equation Q '-is to split the set of observations into three parts
[16,17,18,191 - a vell know part. It has interpretations m1 solutions for possible problems. Ibis part is a recursiw set D'Q. - a know but badly stW.ied part. We knov the problems related to this part but ve ba98 l81 no solution to them )'It. Por nample, ve knov that the plant is non-liDear for vide referm::e changIs but ve ba98 no-good mstb:ld to o'mCOme possible problems inherent to non-limarities. In this case the operator may stop the plant control. Ibis part is a recursi981~numerable set CQ. - the third part is wry t.:1 know or untnovn. It is a non-recursiw set EQ. Prom this interpretation of the recursivity, ve define the recursiw domain D'Q of the uniwrsal equation as the set of observations for vhich Qhas a solution, D~ - { ~a ) I Q ~ is true} is recursiw
(ll,
(ll, )
In another vords, D'Q is the set of observations for vbich Q is d.ecidable. Ibis domain can be viden by ildllling recursi981y-enumerab1e observations. These observations can start alarms or stop the system to ask external aais1m:e. n~= {(11 ,1t:) I 0(11 ,1t: )islnJe) isrears~-l!IUIlnJe We call D"Q the extm1ed recursivity domain m1 ve can easily vrite,
D~ -D~UCQ Iba supervision logic has to ensure that observations still to be a solution of Q or that the control loop is
alvays inside the extm:1ed recursivity domain.
The preceding discussion on the lmiwrsal equation, on the resolution m:1 on the formal system approa:h leeds us to propose a method to build the knov1ed.ge of the supervisor. Ibis rmthod uses tvo principles,
1) The axiom -+ theorem trams [16,18,19] : Although, rules m1 objects in the supervisor ba98 stm1ard forms, knov1ed.ge coding is not usually evident. Let's take the eDmple of a knov1ed.ge corarning the disturbm:es. We knov that if the output disturbm:es are of 10lld type, one can cm:el them by introdw:ing an integrator in the control loop. Ibis sentm:e can take another use, say that if the control loop has an integrator, 10lld disturbm:es are cm:e11ed. The coding of this information appears to be evident as a rule stru::ture but in reality, that depm1s on vhat ve van! to do vith this rule. In the supervisor the axiom--Mheorem trams, vhich comes from formal system approach, allOYS to formulate the rule. The axiom, in this case, is the observation of 10lld disturbm:es, The theorem tells that tbBre exists a transfer fuixtion called integrator vhich alloys to resolw the problem. Iba 10lld disturbm:es is a physical parameter vbereas the integrator is an abstn£ted stru::ture that only its interpretation can atUdle it to the reality. Anyvay, ve can ba98 another theorem lading to another solution of the same problem. In N:h case, inferm:e control strategies haw to solw possible conflicts. 2) The interpretation base : The axioms of the supervisor are built using a sub-set of observations Ds in:lu1ed in the extemed recursivity
domain. We call Ds the interpretation base. The choice of this base allOYS the supervisor to C098l' a part or all the exteD:1ed recursivity domain. In pnctice, vben the supervision task is supported by the expert or the user, he makes decisions as a tuD=tion of observations saneraUy d.isp1ay9d as graphs by the measure instruments. We call image the time-seqtn:e of measures of lenith (n+l) at instant t. Prom all the imaBes representing the observations, only a finite number reflects a state chan8es vhiche make necessary the supervision action. Usually, ve choose the interpretation base from this finite set of images.
401
A Real Time Supervision System for Adaptive Control
Aftgr
tbBD, tb8 IUpIJ"IiJor DI8dJ to tnda tb8Ie imII8II
at his llliomatic lnal. We use NO ditferent vays to code imqes, -n. shape of tb8 sraph of tb8 imIgIe is recognized by
using ODe of mown form recoption mstbods [5]. In this calli tb8 imIgIe is coded as an object, its attributs are tb8 elements of tb8 recognised shape. - A JDIINI'8 tuD:tion is dafiDBd. on tb8 imIgIe. The computed measure is an 1lli0IDI va1\11 [1,2,11). Por nample, an information measure is DII8ded to decide if tb8 estimation has to be freez8d or DOl This JDIINI'8 is a flmction of tb8 regression wctor or of inputs m1 outputs~ .
Pigure(2) shovs tb8 germal systam architecture Pigure(3) shovs a real tiIDB tum:tioDDiDg mmpII . I
r
IBM/AT
Serial communication
RS232
As a confirmation of this priD:ipla, vhan an apart or a rearcbar vants to analyse a pbmoIDlDl, he tries to project or simply reprod\E8 the interpretation base imqIs. Let's take the namp1e of f'1gUl'8 (I), a lom disturbm:es happeDs at instant 68. n. supervisor detects an output offset vbich vill establish tb8 truth on the disturbm:e llliom. ThIn it introd\1:8S an intaifllOr
temperature
at instant 87. 6,58-
...... i output oltset
5,28 3,98-
AERO-THERMAL PLANT
2,68 -
Figure (2)
1.38 -
IIUIl,f--..,.......-+--+--+--+-+--+-+-I--+---I -1.38 -2,68 -
An intml'ttition ~e elmnt. OutPUt loid distUl'banm. -3.'8 -5.28 -6.5845.811 52,511 6UII 67.58 75.l1li 82.59 99.99 97.59 195.99112.58128.
Pigure (1)
SYSTEW ARCHITECTURE A partial state ~ II1aptive control is implemmted on an IBWAT-386 using EXPERT-AD PIEbia [13]. n. control loop is coDDBCtad to an aar-tbBrmal plant. This plant is DOn-liDBIIr e.:cording to l"efereu::e changes. It is a firt order stable plant vith nc>-de1ay. We vant to control output temperature vbich can wry betveen tb8 surrouMing temperature m1 900 C . n. supervisor is implelDlmted on anotbBr IBWAT-386 using N-EXPER T sottvare PIEbia [14). It communicates vith the control loop via serial port
RS232. Real tiIDB constraintes d\ll to sampling period are supported by tb8 real tiIDB environment of EXPERT-AD ~. On the otbBr hmi, the supervisor intenentiom are asyn:hrODOW. So that the supervisor sm1s orders to the control1av as soon as the resolution has foUll1 the solution. If the solution is DOt reliable, it calls the operator for ezterna1 assistaD::e. n. supervision mov1edga is coded as rules m1 objects. The iDferm:e engin starts vith forvard chaining then it chazIi8s re.oning vbile nploring different mov1edge is1m1s.
3.58 I I I I ~ 811.8'8.8 188.8 278.8 3611.8 458.1 • l'til till! suPtI''1iStd idiPtive conbol of iD mo-thel'tlil pliDt
"'1 10"'
t.,
'r-t. h-
I~
11.58 I 811.8 ,8.8 18U 278.8 36U 458.11 5411.11 639.9 729.9 81U , •• SiJII'ling pniode ( ts : 6 S )
Figure (3)
CONCLUSIONS
In this paper, Ye tried to def'1DI tb8 outliDBs of a supervision system vhich merges tb8 compla matl:mnatica1 modelization vith losica1 m1 1lli0matica1 tua:tioDDiDg of a plant. On the otbBr hmi, tb8re remains a big amr:>lmt of york, coDl'Ding. - Sko1em tua:tiODS m1 recoption mstbods, - the minjmisation of tb8 int8rpl'8tation base. the real tiIDB aspect. A real time
sym:hI'oDOW
supervision affects considerably the stnEture of the mov1edge.
REP!RENCES (1) M.M'SAAD, M.DUQUE et M.SAMAAN,"Sur
UDI
Dtbodologie de la mise en oeum des schIDDas de commm1e ~ve',Commm1e Adaptatlve
402
M. Samaan , M. Duque and M. M'Saad
Methodologie et Application, pp 63-91. Ed.. HERMES 1988. (2) M.MSAAD, I.D.LANDAU, M.SAMAAN am M.DUQUE, "PrafOrrDID oriented robust Il1aptiw controUer",papar,ACC, Atlanta, georgia, June 15-17 1988. (3) M.DUQUE, M.SAMAAN, M.M'SAAD am I.D.LANDAU,"Real time supervised Il1aptiw control" Paper,Adaptiw control symposiom, YALE(USA) May 1987. (1) M.SAMAAN,"Station de travail et syst8mBs experts pour la commm1e",Report, DEA in Laboratoire d'AutomaUqua de Grenoble, 1985. (5) R.ISERMANN am K.H. LACHANN, "Patameter Il1aptiw control vith configuration aids am supervision turrioDS",Automatica, Vol. 21,N 3, PP'm-2P1J. 19P1J. (6) K.JASTROM, JANTON m1 K.E.ARZEN,"Expert control",Automatica wl22, N3, PP'm-2P1J. 19P1J. (7) JPITRA T, "Cormaissm:es et mhacormaissm:es" Intelligm:e des mlflcanismes et m!canismes de l'int811ig8D:e", P75-113, Ed.. PAYARD, Po~on Diderol (8) E.RICHE," Artiticia1 Intel1igm:e",McGrav Hill series in artiftcial inte11igm:e. (9) N.J.NILLSON, "Pri.u:ip1es of Artificial Intelligm:e " Spring-Verlag Berlin, Herde1ber"Nev York. (10) WILLEM BLOKLAND m1 JANOS SZTIPANOVITS, "KDov1edgo&-based apprOl£h to reconf'18UI'Ib1e control systems, "paper,ACC, Atlanta, Georgia, June 15-17 1988, pp 1623-1628. (11) ROB ER T L. KOSUT, "Adaptiw robust control via traDSter turrion urartainty estimation",papar, ACC, Atlanta, Georgia, June 15-17 1988,pp 319-351.
(12) M.DUQUE, M.SAMAAN m1 M.M'SAAD. "Partial state LQG m1 GPC Il1aptiw control: an nperimenta1 evaluation" papar,INRIA,Analysis m1 Optimization of systems, pp 830-811,1988. (13) EXPERT-AD sof'tvare ~kaga for Il1aptiw control prodlE8d by ADAPTECH (PRANCE GRENOBLE St MARTIN D'HERES). (11) NEXPERT sof'tvare ~. An expert system generator prodlE8d by NEURON DATA INC. CALIP. USA )
(15) DOUGLAS HOPSTADTER, "Gode1 Escber Ba", Ed..InterEdition, Paris, 1979. (16) S.C. KLEENE, Mathematical Logic, John W11ey & SoDS, Nev York 1967. (17) M.MARGENSTERN," Le ~r8me de Matiyassevitch et resu1tats annnes",Model Theory m1 Aritbmatic Procw1iDjB, Paris 1988,pp 198-211, Springer Verlag. (18) J.P.DELAHA YE, 0uti1s Logiques pour 1'Intelligm:e ArtificieUe, Ed.. EyroUes,Paris, 1987. (19) J.P.DELAHAYE, SYSTEMES EXPERTS: organisation et programmation des bases de cormaiam:e en calcu1 propositiormel, Ed.. EyroUes,Paris, 1987.