Fuzzy Causal Simulation in Process Engineering

FUZZY CAUSAL SHvlULATION IN PROCESS ENGINEERING

Kouamana Bousson and Louise Trave-Massuyes

LAAS-CNI1S. 7. <wenuc du Colonel Hoche, :no77 Toulouse Cedex- FRANCE £-mail: bousson ·C}/aas.Jr or louise@laas.fr

Abstract

This paper presents a causal· simulation method for in­ completely known dynamic systems in process engineer­ ing. The causal model of a process is represented as both a causal network of interacting elementary dynamic systems, called qualitative aut.omata. influencing one an­ other, and a set of qualitative constraints linking possibly several of such automata. Associated with each influence is a weight which expresses its sensitivity. A procedure is presented which allows us to generate fuzzy weights from the relative order of magnitude relations between them. Formulae preserving non-linearities and fulfilling some relevant requirements are provided to compute net influences on extensive variables as well as on intensive ones. Finally an algorithm is given for simulating the causal model, and an example application to a fed-batch fermentation process is presented.

[. [!'lTRODUCTION

One of the most difficult tasks to deal with while con­ trolling and monitoring a dynamic process is to predict its behavior. The problem is that some of the process variables may be difficult to measure on-line. or are not measured at all (e.g. simply because appropriate sen­ sors are not available) . Control theories provide t.ools for state prediction and estimation when precise quantitative models in terms of equations and funct.ions are available. Meanwhile in process engineering. that type of models are not often available because of the complexity of the sys­ tems to cope with. Therefore. to reason with incompletely known models, one should recur to qualitative reasoning.

To endow computer systems with abilities _to reason about qualitative models. work in Qualitative Physics (QP) [Weld and De Kleer , 1990] has focused on qualita­ tive simulation [Forbus. 1984; l~uipers. 1986] of dynamic systems. that is, qualitat ive differential equation (QDE) based reasoning. The advantage of QDE-based simulators is the ability to yield all the possible qualitative behaviors of any incompletely known. or described. dynamic system. However, they suffer from some limitations mainly related to the weakness of their expressive power . the time-scale abstraction they bear in themselves. and to the fact that they are only concerned with real-valued variables and remain silent about pure symbolic variables . Causal rea­ soning, that is reasoning based on the causal interactions between the variables and parameters of a system. may be seen as a way to overcome some of t l1e short.com i ng;s of QDE-based syst.ems in 1 hat I) it does not. require having differential equations, 2) it deals with directed rt>lation­ ships. :3 ) it. may cope with time expliritly which makes it possible to take into arrount. inlluenre delays and dura­ tions. -1) it can providf' a l'ranll'work for knowkrl.~·· r·li(itation since experts and engineers use concepts of causality for the interpretation, prediction and explanation of dy­ namic systems.

An advantage of this causal approach is the possibil­ ity to use the same support (i.e. the causal network) for prediction (going forward) as well as for explanation (go­ ing backward) , following the influence chart. Meanwhile, attention has only been paid to it episodically. Guer­ rin [Guerrin, 1991] proposes ways to combine states of variables using qualitative operators. Leyva! and Gen­ til (1991), Vescovi (1990), and Djerroud et al. (1992) combine the trends of variables using classicnl summa­ tion operator thus neglecting the use of influence weights and the capability to approximate non-linear relation­ ships. Schmoldt (1991) has built a causal simulation for­ malism which seems to be too specific for ecophysiologi­ cal processes; furthermore, although he associates weights to influences and takes into account non-linearities, the computational complexity of his method makes it cum­ bersome for real-time applications in process engineer­ ing. D'Ambrosio ( 1987) has proposed an algorithm for reducing the undecidability during influence combination in qualitative process theory [Forbus, 1984) using linguis­ tic descriptions of influence sensitivities; but his method requires having the necessary problem specific informa­ tion to combine influences properly, and like Schmoldt, he neither provides any formal means to determine in. fluencc weights of variables, nor to check the consistency of a set of influences weights when they are given by ex­ perts. Another aspect which has been neglected in pre­ vious approaches is that they do not distinguish exten­ sive variables from intensive ones. Assume the marginal influences of two variables on the length (an extensive variable) of a metal bar to be respective increases of its value by 1cm and 2cm with the same influence weight, then the net (total) influence of those variables on the length is 3cm. On the contrary, if the marginal influences of the two variables on the temperature (which is an in­ lensil·e variable) of the metal bar are increases by 1° F and 2° F, then the net influence will never be 3° F, but less, whatever the influence weights are.

This paper proposes a causal simulation method which addresses most oft he limitations of the existing ones. The method describes a part of the CA-EN (CAusal ENgine) system which aims to supervise incompletely known dy­ namic engineering processes. Section 2 presents an in­ sight into the modeling formalism which is completely dt:'scribed in [Bousson and Trave-Massuyes, 1992a); Sec­ tion :3 proposes a method for computing fuzzy influence weights from linguistic description of their relative orders of magnitude: sertion ·I is concerned with weighted in­ fluence combination . and nn algorithm for propagating intluences through the causal net of the process variables,

II. THE 1\IODELING FORJ\.IALIS!\1

The process variables are either numeric (real-valued) or symbolic-valued. The value set of a numeric variable is a real interval or the whole real line partitioned into fi­ nite numbers of intervals according to the context of use; that of a symbolic variable is assumed to contain a fi­ nite number of elements. \Ve refer to quality space for a numeric variable, the partition of its value set; the qual­ ity space of a symbolic.variable equals its value set . The variation space of a numeric variable is the partition of its change value set (i.e. the set of all the possible change values during a temporal unit) into a finite number of intervals. Symbolic variables have no variation spaces. Qualitatively, the behavior of a process variable is a se­ quence of discrete states which are either real intervals (if the variable is real-valued) or symbolic values. Therefore, the commonsense perception of a . process variable is on one hand that of an automaton moving from an interval or a discrete value to another, and on the other hand the knowledge that is necessary to reason about its behavior with regard to its interactions with other variables. That suggests that we assign an elementary dynamic system, a qualitative automaton (or q-autom11ton for short), to each process variable and to the knowledge which is needed to reason about the behavior of the variable.

Conceptually, a q-automaton consists of a quality space, a variation space and an influence combining function if the q-automaton is numeric. a transition function that represents the dynamics of the q-aut.omaton in the ab­ sence of any external effect, an output funct.ion. and a state updating function.

A process is represented as two levels of constraints: local constraints and global constraints. Local constraints are causal relations involving two q-aut.omata. There are two kinds of causal relations: iuflueuce-based relations are concerned with cause-effect interactions among numeric q-automata only (e.g. if the amount of liquid in the tank increases then the pressure at the bottom of the tank in­ creases as well) and information-ba.sed relations are con­ cerned with information about q-automata (e.g. the color of the liquid informs us about the concentration rate of a certain chemical product in it) .

An influence-based relation is described by means of predicate It(X,Y, c, w,d,p). or I;;(X, Y,c. w.d,p), where: X is the influencing q-automaton: Y the influ­ enced q-automaton; c the activation precondition of the causal relation ( the influence from .\ to Y is said to be activated whenever c holds); w is the weight. of t.he in­ fluence of X on Y; d is a real number representing the delay of the influence; p is a real number representing the influence duration . or equals 00 if the influence holds permanently. Actually, weight w is a fuzzy \'alue au­ tomatically computed, as presented in the next section. from its relative order of magnitude with respect to other weights of influences on }' using the fuzzy counterpart of the 0(1\l) formalism (fig. I and table I) (~lavrovouniotis and Stephanopoulos. 1988] .

Predicate!;!'(.\. Y.c . w.d. p) (resp . !;;(.\. L c. u·.d.p))

holds if. and only if.\ influences positively (resp. nega­ tively) Y , where a E {0, 1,2,3} is the relative variation rate; a = 0 if Y increases (resp. decreases) in the same proportion as X; a = l if Y increases (resp. decreases) slightly more than X; a = 2 if Y increases (resp. de­ creases) moderately more than X; a = 3 if Y increases (resp. decreases) much more than X. The relative order of magnitude of the variation of Y with respect to that of X are phrased in the O(M) formalism as well.

An information-based relation is represented as /(.\, Y) where X is the informing q-automaton and Y a numeric q-automaton that X informs about. The description of such a relation is expressed by means of a mapping from the quality space of X to that of Y . Whilst influence-based relation only link numeric q­ automata, information-based relations concern any type of q-automata. Information-based relations take prece­ dence over influence-based relations.

A global constraint is a functional numeric constraint relating possibly more than two process variables associ­ ated with interval domains. In other words, a global con­ straint is any numeric equation (in the classical sense) in which each unkown is supposed to take on interval values.

Ill. GENERATING FUZZY WEIGHTS OF INFLUENCES

Experience acquired through collaboration with process experts indicates that these experts estimate net influ­ ences on variables using the relative orders of magnitude between individual influence weights rather than their pure numeric values; furthermore, they hesitate to pro­ vide numeric weights to influences when they are asked to. However t.o automate the weighted influence computation, it is convenient that weights be numeric or numeric-like. . The hesitation of experts stems from the fact that weights may have hazy values. Hence, we propose a method to attribute fuzzy values to weights from the linguistic de­ scription of their relative orders of magnitude.

Let Y be a q-automaton, and X1, ... , Xn be n numeric q-automata influencing Y with respective influ­ ence weights w1 , .. . , Wn through influence-based relations. The linguistic description of orders of magnitude of the weights is a set of pairwise link between these weights (e.g. if the influence weight w; of X; on Y is much smaller than the influence wi of Xi on Y, we write w; < Wj ) . That gives rise to a graph whose nodes are weights, and edges are labeled by fuzzy O(M) relations (because of the symmetry of O(l\1) relations, we only use relations «. - <. -<.and ==). To compute the absolute values of the weights we assume the graph to be connex and that there exists a weight Wmar such that for any weight w;, there exists a path in the graph from w; to Wmaz · There­ fore the graph must be an upper semi-lattice for which Wma.r is the greatest element . We term such a lattice as the weight lattice associated with q-automaton Y.

£3efore we describe the influence weights generation pro­ cess we set an axiom:

Mat·ginal influence axiom: The marginal influence of a q-aulomalon .\ 011 another q-automaton Y is the highest influence that .\ can exert on Y tchen combined with other influwces .

In other words. when q-automata X 1 ..... .\,. impart in­ fluences on a given one, say Y . the contribution of any Xi remains weaker than , or equal to, the influen('e that it individually exerts on Y assuming all the other influences non-existing. Therefore, the maximum influence weight equals 1 (or the 4-uple ( 1 1 1 1) as fuzzy number) 1 .

The algorithm for generating the fuzzy weights of the influences impinging on a given q-automaton consists in exploring the weight lattice associated with that q­ automaton from the top (i.e. the node having the maxi­ mum weight) to the bottom of the lattice in a breadth-first manner:

• ( 1) Set Wma~ = ( 1 1 1 1)
• {2) For each node:
• (2.1) Intersect the supports of its fuzzy values corre­ sponding to its parents' values and the underly­ ing fuzzy 0( M) relations. Stop the generation process if the intersection is empty.
• (2.2) Intersect the cores of its fuzzy values corre­ sponding to its parents ' values and the underly­ ing fuzzy 0( l\I) relations. Stop the generation process if the intersection is empty.

I* If one of the abot·e intersections is empty, that means the weigltt lattice is (partially) Ill­ consistent. *I

• (2.3) If (a, b] and (c. d] are the respect.ive non-empty intersections comput.ed in steps 2.1 and 2.2. then the fuzzy weight corresponding to the node is (a c db) .

Since the weight of an influence cannot be equal to zero, when the weight lattice is consistent , the algorithm yields n fuzzy weights w1 , w2 · . . . , Wn in ]0, 1] .

Example: During a fed-batch fermentation process, the antifoam (AA.f), the pressure in the ferment.or ( P), the agitation speed (AG) of the fermentation liquid, the oxygen inflow rate ( 0 2 ) and the gaz inflow rate ( Q ~) in­ fluence the dissolved oxygen (DO) in the fermentor . Let us denote the weight of .\ by W x . The weight. lattice associated with the dissolved oxygen is (fig.2) : (W,.w -< WAM; Wq.- < w . .tJ\1 ; Wq. -< I·V.w; Wp -< Wq,; Wp- < WAa; Wo, -< IVp ; Wo,- < !V,w) . Ap­ plying the procedure above to that weight lattice where WAM is the maximum weight , the following fuzzy weights are generated: w.-~M = 1. W.w = (0.59 0.77 1 1), Wq, = (0.35 0.59 0.59 0.77) , IVp = (0.21 0.·15 0.5!) U.ii) and W0 , = (0.14 0.35 0.5!) 0.77) .

1 We represent a fuzz~· number as i\ ·1-upl" (·~ 1 ··2 'LJ ···) wh.,re ClJ ~ a2 ~ a 3 ~ 114; the intC'rvals (tt( . ·t·J and [11 2 , ·J_,J arr rt"spr.c­ tively the Jttpport anclthe ··on uf clu~ !'\0 desi)l;ll i\tt~d fu1. :t.~· 1\tllllhel'.

We start this section with a definition that we shall use next. Let S = (a, b) be a real interval. We call fu::::y lrtwcature relative to S the mapping denoted by T 5 defined from the set of fuzzy numbers to the set of fuzzy numbers in S such that, for any fuzzy number 1' = (7't r2 r3 r4) : T 5 (r) = (rJ. r2 r3 r4), where for each i, 1·: =max( a, min(r;, b)), i.e. r: =a if r; <a, r: = r; if r; E 5, or 1·: = b if ri > b.

A. Combining influences

Consider numeric q-automata X 1, X 2, ... , Xn influenc­ ing another numeric q-automaton Y through influence­ based relations. We denote the variation space of Y by t. Y, and the variation (change) of X; on a relatively small time-interval T by 6x;, for each i. Let 6y; be the marginal influence of xi on y (i .e. the influence of x, on y as­ suming that there is no other influence on Y) when the variation of .\ is 6x;. The problem of combming influ­ ences of .\1, X2, ... , Xn on Y is to describe the influence combining function of q-automaton Y which yields the weighted effects of the variations of X1 , X2 , . ·. , Xn on Y.

The influence combining function of q-automaton Y is a mapping© : ~yn - t.Y, which should meet the following conditions:

• ( ii) n-symmetry:

Consistency ensures no contradiction with regard to the concept of marginal influence; one of its consequences is @(0, 0, ... , 0) = 0, that is, no change occurs in a q­ automaton as long as no influence exerts on it. The n­ symmet.ry condition ensures the unicity of the net influ­ ence underlying a set of marginal influences.

To compute the net influence, we must only consider activated influences. If I~ {1,2, ... ,n} is the subscript set of all the activated influences during the time-interval T, then for j fl. I, 6yi = 0. In such a case, combining the effects of Xt. X2, ... , Xn on Y reduces to the combination of the effects of (Xi )iet on Y . That suggests that we tune the weight of each activated influence to-the overall set of activated influences while computing the net influence. The general form of the combining function we consider IS:

where w: is the weight of the influence of X; on Y tuned to t.lw activated influence set . Thus, if all the influences iute1Hhxl to impinge on Y are activated, then w: = w;. ( Eq . l) fulfils the condition of n-symmetry. To meet the condition of consistency, we impose that in the case where only one influence is activated. the net influence equals that one . Therefore. weight tuning is only required when more than one influence are activated whereas all the in­ fluences are not . The wis depend on the w;s. i E I, and on whether the process variable underlying q-automaton Y is an extensive variable or an intensive one. We call extensive {resp. intensive) influences, influences on a q­ automaton assigned to an extensive (resp. intensive) vari­ able.

Case of extensive influences: When the influences of (X;);eJ on Yare extensive, we set, for each i E I:

where max is the fuzzy maximum operation (Dubois and Prade, 1987), U denotes the unit interval (0 , 1], and ru the fuzzy truncature relative to U. The truncature operation guarantees that w;· takes on values from ]0, 1].

Proposition 4.1.1: If the influence hat·ing the maxi­ mum weight (wma~ = ( 1 1 1 1 )) in the weight lattice is activated, then for each i E l, wi = w; .

Case of intensive influences: That corresponds to averaging the influences of (X; );er, hence for each i E I:

and we impose the wis to be linked by equation ( 4) since equation ( 1) should be a mean:

Computing (1) under constraint (4) is termed as fuzzy interactive summation. which is thoroughly studied by Dubois and Prade ( 1987) .

Proposition 4.1.1 holds for 'the intensive case too. Fur­ thermore, a property mimicking idempot.ency is met:

Proposition 4.1.2: Let the 111j1uences 011 Y be inten­ sive. If all the influence weights a1·e identical. and so arc the marginal influences. then the net influence on Y equals the common value of the margmal influences.

B. Simulation Algorithm

The input data are the causal model - including initial conditions ·- and the evolution of the control variables of the process over time which may be data acquired on­ line, or recorded dat. The output. of the system is the behavior of each process variable which is displayed as the process runs. The simulation starts with the com­ pilation of the causal model. The compilation consists in generating the fuzzy weights of influences. 111arginal influence tables according to the rt'lat.ive \'ariat.ion rates in the influence-based relat.ions and partial dt>scription of these tables (13ousson and Trav·<\lassuyi·s. 19!)2b] . tables containing influence delays and durations. The temporal unit of the simulation is that of a logical clock that is set accot·ding to the swiftness of the process.

The simulation is based on the propagation of influences through the causal network describing the interactions among the process q-automata. At each clock tick t, the following steps are performed for each q-automaton fol­ lowing a breadth-first exploration of the causal network:

( 1) Update exogenous, and measured q-automata

I* That consists in reading a data file or acquir­ ing on-line data and computing the qualitative val­ ues (states) and t·ariations for each exogenous, and measured q-automaton accordingly. *I

(2) Fm· each unmeasured q-automaton y do

• ( 2.1) If (the q-au tomaton appears as the first argu­ ment of an information-based relation) Then deduce its new state from the mappiag describ­ ing that relation.

(2.2) Else

• (2.2.1) Let 6y(t) be the net influence at clock tick t. Compute the likeliest variation, that is, the element of the variation space which is the closest to by( t) in the sense of the Hamming distance. Denote the likeliest variation by 6y" ( t) .

I* The likeliset variation is taken as the output influence of the q-automaton (i.e. the influence that the q-automaton exerts on other q-nutomata). *I

• (2.2.2) Set y 1 (t) = y"(t - 1) + by"(t), where y· ( t - 1) is the most recent state of the q-automaton, + the sum operator among intervals, by"(t) the likeliest variation at t. The quantity y 1 (t) is an interval which may encompasses several subintervals of the q­ automaton's quality space. (See hereafter for more details) .
• ( 3) Determine the new state of each q-automaton y by fil­ tering t.he ranges of possible states y 1 (t) (computed in step 2.2.2) out for consistency using the global (interval-based) const.raints. The result of the fil­ tering gives rise to a new state y·(t) for each q­ automaton.

Determi11ing new states. Since the ranges of possible states may contain several possibles states, there is a need to refine them. This is done in two steps. First, the global constraints (interval equations) are used to restrict the ranges in doing a tolerance propagation [Hyvonen, 1991] on t.hem (t.he new ranges may encompass more than one state too). Then. the new state is computed for each non­ measured q-automaton as the likeliest state in the sense of t.hc Hamming distance (similarly to the computation of likeliest variations). However the lower and the upper st.at<'s may he optionally calculated. The above mentioned three states are computed for their relevancy. Indeed. the upper ( resp. lower) state corresponds to the most germane overestimation ( resp. underestimat-ion) of the state of the q-automaton. the likeliest st.ate is interpreted as its most probable state. The simulation process retains (displays and records) all the three states but the likeliest state is taken as the state for the current time-point.

V. EXAMPLE AP.PLICATION TO FED-BATCH FERMENTATION

Cultures in the biotechnological industry are based on the degradation of a substrate (S) by a population of micro­ organisms, the biomass (X), into metabolites. The type offermentation in biomass production depends on the way that fresh nutrients are fed into the fermentor. Fed-batch fermentation is characterized by the fact that it starts with an initial volume of nutrients; · t.hen t.he action vari­ ables (antifoam (.4.M), gas inflow rate (Q 9 ), oxygen inflow rate (02), agitation (AG). pressure (P), etc .. . ) are acted upon, and additional nutrients are brought during the fer­ mentation process if need be. A fermentation process aims to increase the concentration rate of biomass so as to ob­ tain the maximum concentration of product through the use of a suitable substrate and oxygen feeding.

Fig.3 depicts the network of causal influences among the relevant variables of the process. In our process plant, dis­ solved oxygen (DO) is the witness variable on which com­ monsense interpretations of the process behavior rely as studied by Steyer [Steyer, Queinnec and Simoes. 1992] . To show what kind of information we can get from the simulation based on the method presented above, the be­ haviors of dissolved oxygen and substrate resulting from the simulation of the causal network are presented in fig.4, for r = 0 to 20, i.e during 60 seconds. The quality spaces of DO and S are respectively {[0, 20[; [20, 80[: [80. 100]} and {[0, 2[; [2, 4[; [4, 5]} as provided by the process ex­ perts; interval [20, 80[ is the nominal range of DO. The initial conditions for these variables have been taken equal to 40 and 4.5 respectively, and we assume t.hat no more substrate is fed then. The temporal unit considered is three seconds.

Fig.4 shows that the likeliest state of DO is normal till r = 20, but the lower state remains in imerval [0. 20[ till r = 17 then it goes up to interval [20, 80[. The increase of the lower state is interpreted as a possible increase of the dissolved oxygen although its actual value remains qualitatively unchanged . That increase in DO can be ex­ plained by the fact t.hat the bacteria consume less oxygen than fed into the fermentor because of the metabolites slowness at the beginning of the fermentation process. In fact, ten to fifteen minutes after the start of the process. the dissolved oxygen begins to decrease. The lower state. likeliest and upper state of substrate are t.he interval [4. :>] from r = 0 to T = 13. then the likeliest and lower ~tales decrease to interval [2, 4[. That means the substrate con­ centration rate decreases, which is quite normal. since no more substrate is fed into the ferment or whilst the bacte­ ria grow and consume t.he init.ial substrate.

The simulation of t.he causal network allows to contin­ uously t>st.imate the state of non-sensored variables (such as suhst.rat.e in our process plant) during the process life­ time.

VI. DISCUSSION AND PERSPECTIVES

In the proposed method the causal model of a process is represented as both a causal network of interacting quali­ tative automata, and a set of qualitative constraints link­ ing possibly several of such automata. Associated with each influence is a weight which expresses its sensitiv­ ity. A procedure is presented for generating fuzzy weights from the relative order of magnitude relations between them. An algorithm is given for simulating the causal model.

The advantages of the proposed approach over existing ones are : 1) the consistency checking of relative weights provided by experts and the generation of fuzzy weights from their relative description, 2) the computation of net influences which takes into account non-linearities and the fact. that the influences can be extensive or intensive, 3) the simulation algorithm has been devised so as it can be used for real-time applications in process engineering.

The present work has been basically motivated by the development of the CA-EN system which is intended to be used as a supervisory control system in biotechnology. Therefore, ongoing work is the extension of the simula­ tion module to a monitoring and diagnosis module (Milne, 198i] .