Modular and Multi-view Representation of Complex Hybrid Systems

Transcription

1 Modular and Muli-view Represenaion of Complex Hybrid Sysems Luc THEVENON, Jean-Marie FLAUS Laboraoire d Auomaique de Grenoble UMR 5528 CNRS-INPG-UJF ENSIEG, BP 46, F-38402, Sain Marin d Hères Cedex FRANCE Absrac: - A modular and muli-view approach for he modelling of complex hybrid sysems is presened. The aim is o define a global represenaion of his class of sysem, i.e. a represenaion which groups ogeher in a same srucure several ypes of informaion coming from differen origins. Such a model, used for differen purposes like he simulaion of he process or he diagnosis, consiues a firs sep oward an objecive of supervision of he plan. For ha a decomposiion of he process based on he noion of inpu-oupu blocks is examined, and a hybrid generic objec is hen deduced. The modular building of he process model from hese objecs is shown, as he way o projec he model on differen views. Key-Words: - Hybrid sysems, complex sysems, bach plan, muli-models approach, modular modelling, supervision IMACS/IEEE CSCC'99 Proceedings, Pages: Inroducion Our aim is o define a modelling approach for hybrid complex sysems which is based on he idea ha any complex sysem can be analysed wih a sysemic poin of view and seen as a se of basic elemenary sysems ineracing ogeher. Our challenge is o find he righ sandard aomic sysem, o define is srucure in order o be able o describe any behaviour, and hen o define a semanic for he ineracions which is rich enough in order o allow he descripion of complex sysems. As a by-produc of our approach, we propose a formal definiion of a muli-view modelling approach in which he main idea is o projec (or resric he complee sysem on a se of a relaion of a specific ype. We will illusrae our idea on a class of such sysems which are bach processes. Indeed, hese processes are becoming more and more imporan in indusries dealing wih maerial ransformaion (chemical, food bioechnological or pharmaceuical indusries hanks o he high level of flexibiliy of hese processes which allows o manufacured several ypes of producs in small quaniies wih he same equipmen. A model of such complex sysems mus hold enough informaion for differen purposes, like he simulaion of he process or he diagnosis, ec. I consiues a firs sep oward an objecive of supervision of he plan by offering a mehod o represen wih a coheren way informaion coming from differen origins. The idea is o represen he plan according differen poins of view (physical, behavioural, funcional, ec., and o mix hese views o ge a full model of he plan. We mus hen define a srucure in order o have an inernal unified way of represening an objec wih heir differen views and he rules o combine hese views. The srucure mus allow o decompose he whole process in smaller pars easier o manipulae in order o reduce he complexiy of he modelling. The main characerisics of he represenaion are he following: - global in order o ake ino accoun all he various pars and he main aspecs of he plan (opology, physical ransformaion, conrol, ec. - modular in order o build he whole model of he plan (ofen complex by inerconnecion and aggregaion of basic objecs, - flow ransformaion oriened: an objec is a physical equipmen like a valve or a reacor which acs on he energy or maerial flow, in order o build he plan model by following is opology, and o reuse objecs already defined. For ha, we have defined a represenaion based on he noion of inpu-oupu blocks inerconneced. Block diagrams used in Simulink for he modelling of coninuous sysems are a classic example of such an approach, whose he ineres lies in he modulariy of he model building. We have exended his approach o he hybrid dynamical sysems and

2 defined a generic inpu-oupu objec (secion 2. Such an aemp has been made in [] and [2] where he coninuous dynamic is approximaed by a discree model, and in [3] and [4]. Then, we show he building rules of complex objecs in secion 3, and he projecion of he full model of he process on differen views in secion 4. 2 Generic Granular Objec We have defined a generic objec wih an inernal dynamic which allows o represen any physical equipmen belonging o a complex hybrid process. This dynamic is described by a coninuous subsysem and a discree one which are inerconneced by heir inpus-oupus hrough appropriae inerfaces. Bu, in order o describe he behaviour of a hybrid process, coninuous signals are no sufficien, and oher inpu-oupu ypes mus be defined. We presen hem in his secion, and hen we describe he inernal dynamic of he generic objec. 2. Inpu-oupu signals and flows Firs, le us define he noion of variable and vecor. Definiion : Variable There are four ypes of variables. - Coninuous variables are described by a real or ineger number and can ake an infinie number of values. - Boolean variables have wo possible values: TRUE or FALSE. - Symbolic variables (noed beween are described by a characer sring ( open, sar and can ake a finie number of values defined in a lis. Definiion 2: Vecor A vecor is a n-uple composed of variables of he same ypes (simple vecor or of differen ypes (composie vecor. The inpus-oupus of a generic objec can be filed in wo caegories: signals which convey informaion, and flows which describe a physical flow. Conrary o signals for which he informaion is always conveyed in he same direcion (acuaor owards process, sensor owards conroller, ec., he direcion of a flow can change during he evoluion of he sysem, according o he direcion of he physical flows of he process. Signals and flows can be uniary or muliple (composed of several signals or flows. 2.. Coninuous signals ( They convey coninuous informaion (acuaor or sensor values. A coninuous signal is represened by a coninuous variable Even signals ( They convey discree sae ransiion informaion of a sysem. An even signal is a poinwise non-zero signal represened by a Boolean variable Condiion signals ( They convey informaion abou he discree sae of a sysem. A condiion signal is a piecewise consan signal represened by a symbolic variable Coninuous flows ( They describe he physical flows which circulae ino he sysem a a given poin. They are specific o an applicaion domain. For example, in he case of chemical engineering, hey describe he maerial and energy flow. A coninuous flow is represened by a coninuous vecor and conains he following informaion abou he physical flow: - Rae of flow, i.e. a coninuous variable. - Sae, i.e. exensive and inensive coninuous variables (pressure, concenraions, ec.. - Characerisics, i.e. a se of consan parameers (densiy, ec.. - Direcion, i.e. an informaion concerning he curren direcion of he physical flow Discree flows ( They describe he flows of physical objecs which circulae ino he sysem a a given poin (boles or barrels. Conrary o he coninuous flow for which several componens can be presen a a given poin of he flow (mixure of wo liquids, only one objec is presen a a given poin of a discree flow (a bole. A discree flow is represened by a composie vecor and conains he following informaion abou he flow of physical objecs: - Rae of flow, i.e. a Boolean variable which is TRUE when an objec is presen as regards of he flow, and FALSE he res of he ime. - Characerisics, i.e. a se of consan parameers which describe he curren objec (size, color, composiion... - Direcion, i.e. an informaion concerning he curren direcion of he flow of physical objecs. 2.2 Coninuous subsysem I represens he coninuous dynamic of he generic objec. Many formalisms exis o describe such a dynamic: ransfer funcions, bond-graphs, ordinary

3 differenial equaions, ec. The more general represenaion is given by differenial and algebraic equaions (DAEs of he form: 0 = f y( = ( x( x( h( x( ( where x( is he coninuous sae, is he inpu vecor (se of coninuous signals, and y( is he oupu vecor (se of coninuous signals. We add also a se of coninuous flows ϕ( as inpu and/or oupu of he sysem. Bu, such a represenaion does no allow o represen discree swichings of he sysem s dynamics. For ha, an oher inpu is required o know he discree sae and o selec he appropriae se of equaions. As we saw, he discree sae of a sysem is conveyed by a condiion signal. We obain hen he following subsysem: ϕ( 0 = f y = h ( x( x( ϕ( ( x( ϕ( Figure : Coninuous subsysem y( ϕ( 2.3 Discree subsysem I represens he discree dynamic of he generic objec. As for he coninuous one, many formalisms exis o describe such a dynamic: auomaon, Peri nes, Saechar, Grafce. The weakness of mos of hem is ha he communicaion beween subsysems is no well suied, wha no allows o build easily modular models. An oher formalism, called condiion/even (C/E sysems and inroduced by Sreenivas and Krogh [5], is well suied for modular modelling of discree even sysems, and is similar o coninuous block diagram. I is based on wo ypes of inpus-oupus: condiion and even signals. As we saw, condiion signals allow o convey informaion abou he sae of a discree sysem. This is very convenien by comparison o auomaon which communicae only wih even signals, wha impose he addiion of discree saes in order o obain he same resul. The behaviour of his sysem is described by hree equaions, as shown Figure 2. c( e( δ d( = γ s( = β (, c( e( ( c( (, e( Figure 2: Discree subsysem I remains he reamen of discree flows, for which an oher discree formalism needs o be used o describe heir ineracion. d( s( 2.4 Subsysems inerconnecion To inerconnec discree and coninuous subsysems, we have defined appropriae signal/flow (S/F converers Coninuous S/F o C/E signals converer Also called even deecion, i convers coninuous signals and/or flows ino condiion and even signals. When he coninuous inpus cross a hreshold described by an even funcion g of he form: ( w( 0 = g (2 an oupu even signal is generaed, and an oupu condiion signal is se o a specific value. Here, he coninuous vecor w( is composed of he oupus coninuous S/F y( and ϕ(. As inpu, here are also a condiion signal c( and an even signal r( whose he meaning depends on he size of w(. If he size of w( is non-zero, he ime of he even depends on he evoluion of he coninuous inpus (sae even deecion. The inpu r( is no used, and c( conveys he curren value of he discree sae in order o selec he appropriae se of even funcions and remove he ones no ye or no longer useful. Two ses of oupus are considered, according wheher he hreshold is crossed from a negaive value (posiive direcion, or a posiive one (negaive direcion as shown Figure 3. w 2 c l c h 2 2 w( r( c( ( w( r( 0 = g, e l e h < > 2 2 Figure 3: Sae even deecion c l ( e l ( c h ( e h ( In a simulaion objecive, he exac deerminaion of he ime of he sae even is very imporan, bu poses many problems, because i can occur during an inegraion sep. This is no he purpose of his paper o explain hese problems, bu hey mus be aken in accoun for such an objecive. If he size of w( is zero, he ime of he even is known, because i depends only of he ime (ime even deecion. The converer is hen a simple imer wih one se of oupus, and whose he sar

4 ime is indicaed by he inpu r( under he condiion c(. When he waned ime is reached, he imer is auomaically deacivaed unil he occurrence of a new inpu even Even signals o coninuous S/F converer I convers even signals ino coninuous signals and/or flows. When an inpu even ev( occurs a ev, values of a se of coninuous variables z( are modified according o a funcion f x (Figure 4. The vecor z( is composed of he inpu coninuous S/F and ϕ( wha corresponds o a modificaion of he coninuous evoluion, and also of he coninuous sae x( wha corresponds o a sae jump. ev( z( ev f ( z( = x ev z( ev z( ev Figure 4: Even o coninuous converer 2.5 The generic objec From he discree and coninuous subsysems inerconneced hrough signals and flows converers, a generic hybrid objec wih he five ypes of inpusoupus seen previously can be defined (Figure 5. c( a( ϕ( e( ϕ d ( c( C/D Coninuous subsysem: x( y( ϕ( x( ϕ( e( d( s( y( - s( Discree subsysem: D/C ϕ( - y( ϕ( s( ϕ d ( Figure 5: Generic hybrid objec More formally, a generic objec is defined by: - a coninuous sae vecor x( R n, - a discree sae vecor Q a se of symbols, - a composie inpu vecor (signals and flows, - a composie oupu vecor (signals and flows, - a composie inpu/oupu vecor (coninuous and discree flows, - a se of coninuous equaions, - a se of equaion defining he c/e sysem, - a super-objec and several sub-objecs. We see ha flows can be eiher inpu or oupu or inpu/oupu of he objec, according wheher he direcion of he corresponding physical flow can change or if i is se by he opology of he process. We show Figure 6 a simple example of he modelling of an on/off valve. d( Valve on/off (F, P (F, P 2 F = αcv P P 2 e (=close valve e 2 (=open valve Converer α {0,} s ( s 2 ( {open, closed} d( {open, closed} s (=valve opening s 2 (=valve closing Figure 6: On/off valve objec In a firs ime, his objec can be used in order o represen any basic physical equipmen, like for example a valve, a reacor, or a coninuous conroller. Then, elemenary objecs can be inerconneced by heir inpus-oupus in order o build more complex equipmen (disillaion column, or a sequenial conrol (manufacuring recipe, or he whole model of he plan. An advanage of such an approach, in addiion o he simplificaion of he model building, is ha he inernal dynamic of objecs can be described wih oher formalisms (ransfer funcions, black box model, ec., or several accuracy levels (saic relaions. Only he inpu-oupu ypes of he objec mus be idenical o he ones defined previously. 3 Complex objecs Complex objecs are buil by aggregaion of oher objecs, i.e. he new objec (called super-objec is composed of oher objecs (called sub-objecs which can be inerconneced or no as seen Figure 7. For ha, specific rules allow o check he validiy of he model and creae i. Type I Sub-obje Sub-obje 2 Sub-obje 3 I Super-obje Figure 7: Aggregaion of hree objecs 3. Connecions beween objecs Figure 7 shows he hree ypes of connecion beween objecs, which are realised hrough variables forming signals and flows. Rule : The ype and he dimension of he wo conneced signals or flows mus be he same. Type I is a connecion beween wo elemenary or complex sub-objecs. I consiss in replacing in he equaions of each sub-objec he inpu and oupu

5 variables by he corresponding flow or signal variables which become sae variables added o he coninuous sae vecor of he super-objec. is a connecion beween a sub-objec and a super-objec. I defines he inpus and oupus of he super objec from hose of he sub-objecs. I is a connecion beween he inpus and oupus of he super-objec. The corresponding variables are no modified by he objec. The validiy of connecions can be checked by looking a he causaliy of he variables involving in he connecion (Inpu, Oupu or Inpu/Oupu: Rule 2: The validiy of a connecion beween wo variables A (sub- or super-obje and B (sub-objec is defined by he following able, according o he ype of he connecion. 9 means a valid connecion, whereas means a wrong one. A B Type I I I 9 I O 9 9 I/O I 9 9 O O 9 I/O I I/O O I/O Objec aggregaion A complex objec has he same srucure ha a elemenary one. Is aribues are defined a he ime of he sub-objecs aggregaion by he following rules: - The inpus-oupus are defined by hose of he sub-objecs hrough ype II or III connecions. Non conneced sub-objecs inpus-oupus become non accessible from he exerior. - Coninuous sae vecor is defined by aggregaing hose of sub-objecs. The inpus and oupus variables inerconneced of sub-objecs become new saes added o he sae vecor. - Discree sae vecor is defined by aggregaing he vecors of sub-objecs. - Coninuous sysem is defined by grouping ogeher he ses of equaions of each subobjecs modified by he connecions. - C/e sysem is defined by he c/e sysems of each sub-objecs which communicae hrough inpu and oupu condiion and even signals which become inernal signals for he new objec. 3.3 Srucural analysis According o he goal of he modelling, ha can need a numerical simulaion of he coninuous equaion sysem. Bu, he raw se of equaions can no be used wihou a srucural analysis whose he purpose is o check he validiy of he coninuous sysem, and o arrange he order of equaions Srucural singulariies During he aggregaion of objecs, specific rules are applied, and connecions are checked. Bu, nohing assures ha he se of coninuous equaions obained for he whole sysem is correc, i.e. ha he equaion sysem is resolvable in a mahemaical or simulaion sense. An equaion sysem is resolvable if he number of equaion and coninuous variables are he same, and if each variable can be paired wih an equaion in which i appears. If no, he sysem is said srucurally singular and can no be resolved. Because he se of equaions picked changes according o he discree sae, such a verificaion mus be performed for all he possible coninuous models. One soluion o deec such anomalies, is o permue he equaions o make all diagonal elemens of he incidence marix non-zero [7] Causal ses We define a causal se S, as a se of coninuous equaions (and by exension a se of objecs whose all he inpu variables are se by he exernal environmen and do no depend on he evoluion of S, and all he oupus variables are se by S and do no depend on he exernal environmen. The decomposiion of a model in several causal subses or blocks is useful because each block can be simulaed independenly from he ohers by using a sequenial resoluion mehod, wha reduce he size and he complexiy of he problem in a numerical poin of view. The deerminaion of causal ses can be performed by making a Block Lower Triangular pariion of he incidence marix of he se of equaions [7]. 4 Noions of muli-view model A muli-view model is a model which groups ogeher in a same srucure, differen descripions or aspecs of a process [6]. For example, a bach process can be regards from: - he equipmens which compose i, - he maerial and energy ransformaions ha occur ino hese equipmens, - he auomaic sysems which conrol he plan,

6 - he operaing mode which describes how make a produc, - he safey acions realised in case of an abnormal behaviour. These aspecs are described by differen ypes of informaion (PID diagram, process diagram, auomaa language, ec.. In he represenaion we have defined in he previous secions, each inpu-oupu ype of an objec represens a paricular view of his objec. So, he full model of a process, represened by all he inpus-oupus, can be projeced on differen views by considering only he corresponding inpusoupus. For examples, in a bach process, he ransformaion view is given by he coninuous and discree flows, he conrol view by he coninuous and C/E signals, and he recipe view by only C/E signals. Sub-views can also be considered like for example he componen view which is given by only he coninuous flow wih he paricular componen whose he follow-up hrough he process is waned. Such a decomposiion reduces he complexiy of he model which can be regarded from differen views separaely, and allows o consider and sudy only a paricular view according o he objecives. Examples of he ransformaion and conrol view of a small bach process are shown Figure 8 and 9 where he objecs R i and V i are he same objecs, bu projeced on differen views. R R 2 V V 2 R R 2 R 3 h h 2 V 4 R R 2 (F,P (F,P 2 V (F 3,P 3 (F R 3 V 2,P 2 3 V 2 (F 2,P 22 Figure 8: Transformaion view Transfer R R 3 0 = h -L 0 = h 2 -L 2 close close 3 h = L h 2 = L 2 Close V 2 4 Close V 2 = mn V V 2 5 Hea R 3 valve sae & Transfer R 2 R 3 6 Sop heaing Open V 3 valve sae sar coun Figure 9: Conrol view sar heaing 0 = - sop heaing open (F 3,P 32 R 3 V 3 5 Conclusion In his paper, we have presened a modular approach for he modelling of complex hybrid sysems. The main ineress of his approach are: - o separae he represenaion (or modelling phase from he analysis and simulaion phase, - o propose a generic granular objec which can be used for modelling complex sysems, - o propose a consisen way of describing relaions beween he objecs, - o formalise he muli-view noion of a sysem. Alhough we have illusraed i wih examples coming from he chemical engineering area, i can be used for he modelling of any coninuous, discree or hybrid complex process. Then, he model obained holds enough informaion for differen purposes (simulaion, ec., and can be used in an objecive of supervision of he plan. References: [] S. Kowalewski, S. Engell, M. Friz, R. Geshuisen, G. Regner and M. Sobbe, Modular Discree Modelling of Bach Process by Means of Condiion/Even Sysems, Workshop Analysis and Design of Even-Driven Operaions in process Sysems, 995. [2] O. Sursberg, S. Kowalewski, J. Preuβig and H. Treseler, Block-diagram based Modelling and Analysis of Hybrid Processes under Discree Conrol, 3 rd Inernaional Conference on Auomaion of Mixed Processes: Hybrid Dynamical Sysems, 998, pp [3] B.H. Krogh, Condiion/Even Signal Inerfaces for Block Diagram Modeling and Analysis of Hybrid Sysems, Proceedings of he 8 h Inernaional Symposium on Inelligen Conrol Sysems, 993, pp [4] S. Engell, Modelling and Analysis of Hybrid Sysems, Proceedings of he IMACS Symposium on Mahemaical Modelling, 997, pp [5] R.S. Sreenivas and B.H. Krogh, On Condiion/Even Sysems wih Discree Sae Realizaions, Discree Even Dynamic Sysems: Theory and Applicaions, Vol., 99, pp [6] K.E. Arzen, Using Muli-view Objecs for Srucuring Plan Daabases, Inelligen Sysems Engineering, 993, pp [7] S.E. Mason, Simulaion of Objec-Oriened Coninuous Time Models, Proceedings of he IMACS Symposium on Mahemaical Modelling, 994, pp