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A computational causal model for process supervision
Copyright @ IFAC Artificial Intelligence in Real-Time Control, DelftTheNetherlands, 1992
A COMPUTATIONAL CAUSAL MODEL FOR
PROCESS SUPERVISION K. Bousson and L. Trave-Massuyes LAAS-CNRS, 7 avenue du Colonel Roche, 31077 Toulouse Cedex, France
Abstract: This paper proposes a causal modeling framework for the purpose of process supervision. It is based on a qualitative automata (or q-automata) concept that we have devised. A q-automaton captures both the dynamics of a process variable and the expert knowledge necessary for supervising the variable's behavior. We use a two-level representation scheme for the description of the relationships between the q-automata underlying a process: a local constraint level and a global constraint one. The local constraint level describes the qualitative causal relationships between the q-autornata, and the global constraint level states the quantitative constraints among them. The formalism is shown to allow the modeling of deep knowledge as well as compiled knowledge. Furthermore, it is suitable for the modeling of partially-known, hybrid (numeric and symbolic), continuous and discrete processes. A causal engine (CA-EN) using the formalism is under intensive development. It is at the core of the process supervision system.
Keywords: Process Supervision; Qualitative Reasoning; Causal Modeling; Qualitative Automata; Compiled Knowledge; Deep Knowledge; Influence Combination.
1
Introduction
Expert systems are decision support systems; however today, it is widely recognized that their knowledge is too 'shallow' to cope with engineering problems in the sense that they are based on pre-specified relationships between conditions and actions, or causes and effects, (e.g. between system behavior abnormalities and corresponding causes) rather than on a model of the domain of interest (eg. structural description, including physical laws). The weakness of shallowreasoning systems and the lack of mathematical models of incompletely-known engineering systems (e.g. biotechnological processes) have recently led Artificial Intelligencecommunitiesand control engineers to investigate qualitative reasoning about physical systems (or qualitative physics) (Bobrow (1984), Weld and DeKleer (1990»). As a component of qualitative reasoning, the aim of causal reasoning is to derive system behaviors from structural descriptions for the purpose of, among other tasks, prediction, causal explanation and situation assessment. Hence causal reasoning is well suited for process supervision. Causal modeling is the cornerstone of influence representation within devices, more often than not at the basis of causal reasoning. Consider we were asked to supervise a sprinkler (fig. 1) so that the level of the liquid in the tank remains in the interval [Y2, Ya[ and that the color of the liquid stays green during the sprinkling process. To do that, we first have to use a set of conventions to represent the dependences between the tap flow rates and the liquid level, the pipe flow rate and the liquid level, etc .., In other words, we need first to model causally the interrelationships among the system's variables. Here, 'Causality', as we understand it in engineering, stands for something that links the structure (set of numeric or symbolic equations, topological description) of a system to its functions and behavior.
The main tasks of a process supervision system are to assess the process behavior and to act, or to advise a human operator about what occurs and what to do when the process behaves unwantedly. Therefore a supervision system must meet the following requirements: • Real-time: The system must be reliable in response time. For that purpose it must be endowed with fast processing strategies and deal with time. • It must track the behavior of the process over time from state to state. That ensures the detection of early deviations from the nominal behavior.
• In a process, all the variables are not necessarily measured or observed, that is, some of them may be inaccessible. A process supervision system must be provided with means to predict the behavior of those inaccessible variables as well as the behavior of accessible (measured or observed) ones. • It must be supplied with trustworthy structural and behavioral models of the process to enable it to meet the above mentioned requirements. This is all the more achieved as the model captures more relevant available knowledge. Consequently, it must deal with both numeric data (e.g. numerical equations) and symbolic ones (e.g. experential knowledge, physical qualitative constraints). Indeed, supervision appears as the last and skilled level of automation that control theories do not master and that Artificial Intelligence techniques are welcome (Aguilar-Martin (1991)). Meanwhile, knowledge from lower levels (e.g. regulation-oriented numeric models) should be used if need be.
133
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pose of supervision. We use a two-level representation scheme for the description of the relationships between the process variables: a local constraint level and a global constra.int one. The local constraint level represents the qualitative causal relationships between the process variables, and the global constraint level states the quantitative const raints among the variables.
Causal modeling has received considerable attention in qualitative physics. Work has been done in process modeling for supervision tasks (Leyval and Gentil (1991)) using the notion of qualitative transfer function which is the qual. itative counterpart of transfer function in control theory, or propagation function like in (Vescovi (1990)). Meanwhile, the existing approaches suffer from some weakness in that they are generally supported by directed-graph representations:
2
• Multi-variable relationships induce the problem of influence combination so that those systems generally assume linear relationships, which may be a very rough approximation in many cases.
Representing process knowledge
Dealing with the supervision of physical processes requires having a relevant representation of the quantities at hand which are time and the process variables.
• Constraints involving more than two variables (e.g. mass balance relations) are not representable, which makes difficult to model processes whose supervision requires an interminglement of first principles and experential knowledge.
2.1
Time representation
The algebra of temporal intervals (Allen (1983)) is a. temporal reasoning framework which suits most of Artificial Intelligence systems for it has a higher expressive power and is easy to implement. However it is shortcome by the fact that it may accept an inconsistent set of temporal relations as being consistent (incompleteness problem) and it runs in O(n 3 ) , which is a too high computational complexity for coping with real-time applications such as supervision tasks. Vilain and Kautz (1986) have shown that the completeness problem can be solved by representing time in the restricted interval algebra which allows the same expressive power as Allen's temporal algebra. Recently, Ghallab and Alaoui (1989) solved the complexity problem by proposing the IxTeT temporal management system which guarantees linear time complexity for data retrieval and updating, and which has been proven sound and complete ensuring the same expressive power as the restricted interval algebra.
• Although they may allow reasoning in time, they do not allow reasoning on time so that intermittent influences or influences with limited duration are not representable. Time representation is of event type which makes difficult tracking the process behavior. • The formalism used for representing process dynamics is too inflexible so that hybrid processes (i.e. processes involving both numeric and symbolic variables, such as the above mentioned sprinkling system) cannot be handled. A consequence of this is that they do not cope with processes involving both continuous variables and discrete ones. On the other hand, qualitative differential equation based (Kuipers (1986)) or component-based (DeKleer and Brown (1984)) approaches can easily cope with multi-variable relationships and constraints involving more than two variables. However, they also suffer from event-type time representation and from the fact that they do not allow reasoning on time. The last mentioned issue is a problem as well. In addition, causality paths are not explicitely represented. First, that makes it difficult to represent experential knowledge understanding oriented influences which rises the problem of involving "informative" variables, for instance how can you represent in the QSIM formalism that the color of the water in the tank depends on the concentration rate of product but not the converse? Second, it complexifies several tasks like diagnosis and causal explanation. This paper proposes a causal modeling framework which captures most of what is needed to reason about partiallyknown, hybrid, continuous and discrete process for the pur-
Therefore, the temporal representation and management strategies we have chosen for dealing with processes are that of IxTeT system. A time-interval is represented as a pair T = (t l, h) where tl and t2 are time-instants with t l < h. However we regard the temporal axis as regularly sampled with temporal landmarks a.t which the physical process must be checked up. Hence, the temporal unit for the supervision system is the (constant) dista.nce between adjacent landmarks. The temporal unit must be chosen so tha.t the behavior of the process at hand can be assumed to be linear within it.
2.2
Qualitative Automata
When a human operator supervises a process he concerns himself with checking whether the value of each variable equals some (discrete) value, when the variable is discrete, or falls into some interval, when the variable is real-valued.
134
not only at time-instants) when they occur.
When a variable behaves unwantedly, the operator uses knowledge about its dynamics and about the causal relationships between that variable and others to correct the variable's behavior. In other words, on one hand the operator checks the qualitative behaviors of the variables in somewhat regarding each process variable as a kind of automaton moving from an interval or a value to another, and on the other hand he uses a set of knowledge to act on the variable. Therefore, this motivates that we assign a qualitative automaton (or q-automaton for short) to each process variable and to the knowledge-base which is needed to reason about the behavior of that variable.
Definition 2.1: The causal history of a q-automaton is the chronological sequence of its states and the actions that it suffers (if it is a control q-automaton) or that the control q-automata which influence it directly or indirectly suffer. Causal histories are main sources of relevant information to achieve the goals of prediction and explanation in process supervision. From the definition, the causal history of each q-automaton grows as the process runs. However, some of its elements can be discarded when they are no longer necessary for behavior prediction or explanation.
Our perception of a physical process is a world of interacting q-automata influencing one another. The set of q-automata underlying a process is called the q-automata universe. In order to cope with any type of process variables, a q-automaton may be of numeric or symbolic type according to whether the underlying process variable is realvalued or takes discrete (or symbolic) values (e.g. red, green, blue).
2.2.1
Definition 2.2: The variation space of a q-automaton is the set of possible amounts of change it bears within a temporal unit. The variation (amount of change) of a qautomaton X is denoted by AX. The dynamics of a q-automaton is defined by a set of influence values (its inputs), a quality space, a variation space, and mappings on those sets. Definition 2.3: The dynamics of a q-automaton is represented as a 5-tuple D = (I,S,V,f,g) where: I is the input set. It is a set of influence values. S is the q-automaton's quality space, V is the q-auiomaion's variation space, f is a mapping from I into V I g is a mapping from V x S into S .
Quality spaces, states and values
The significant discrete values and intervals according to which the values of a process variable must be evaluated over time by the human operators are pre-specified by process experts. We assume that for real-valued variables, these intervals describe a finite and totally ordered partition of their respective ranges. We call quality space (Hayes (1985) of a numeric q-automaton such a partition. The quality space of a symbolic q-automaton is its value set (which is, in fact, a discrete set), and is assumed to be finite and totally ordered according to the context of use. The state of a q-automaton is an element of the quality space; it corresponds to relevant information for a human operator in the decision-making tasks. Contrarily to conventional automata theory, a q-automaton's state does not necessary summarize the whole history of the q-automaton.
Dynamically, a q-automaton processes as follows (fig. 2) : Mapping f enables to sum up the inputs the q-automaton receives over time. Mapping g is used to combine the result of that computation with the q-automaton's most recent state and its current variation to update the q-automaton's state.
2.2.3
When several q-automata influence the same q-automaton, some of them may influence with a higher degree of sensitivity than others. On the other hand the subset of qautomata influencing a given q-automaton may be timedependent. Therefore, the set of q-automata influencing a given q-automaton is a time-dependent fuzzy subset of the set of all the q-automata.
The value of a q-automaton X at time-instant t is the value of the associated variable at i, It will be denoted by value-of(X, t)j if t==now (i.e. the current time-instant), that is simply written value-of(X). The same shorthand applies to state-of for the description of states. If X is a symbolic q-autornaton, the state of X at t equals its value at tj but if X is numeric, value-of(X, t) is element of state-of(X, t).
2.2.2
Dealing with influences
To reason about influences we make the: Influence Sameness Assumption: The degree of sensitivity of an influence between two q-automata is constant whenever the influence holds.
Dynamics
The assumption states that:
The dynamics of a process is due, on one hand, to the interactions among the q-automata and, on the other hand, to actions coming from the process outside world. An action is an operation made by a human operator or a device in the way to increase or decrease the state of a qautomaton. Actions are made by means of process control q-automata (q-automata underlying the process control variables). The interactions among the variables of a process can be represented by influences from one q-autornaton to another. The influence that a q-autornaton exerts on another q-automaton is due to the variation (change) in the
• The degree of influence (i.e. the degreeof sensibility of the influence) that a q-automaton X exerts on a given q-autornaton Y, denoted by !lY(X), remains steady throughout the time-intervals on which the influence holds (it equals zero elsewhere), • If X influences Y on two distinct time-intervals Tl and T2 , the degree of influence throughout T l equals the degree of influence throughout T2 •
Instead of reasoning by means of the explicit numeric values of influence degrees, we will rather consider relative orders of magnitude at the level of the influences exerted upon the same q-automaton, This is dictated by the way process experts reason about influences. The O(M) formalism (Mavrovouniotis and Stephanopoulos (1988») has been chosen for it has shown to be the closest formalism to the one process experts use. Table 1 presents the primitive relations of the formalism and the explanations of these relations. For instance, assume Xl, X 2 to be two q-automata influencing a q-automaton Y throu.qhout a time interval T.
former. A q-automaton X is said to influence (directly) a q-automaton Y if, assuming no action to be done on Y and all the q-automata to be quiescent except X and Y, a change in X is necessarily followed by a change in Y. A q-automaton X influences indirectly a q-automaton Y if there exists a chain of q-automata X, Xl,. .., X,.,Y which successively influence directly one another. Unless otherwise specified, the word influence, from now on, will refer to direct influence implicitly. We assume influences to have durations, that is, they hold throughout time-intervals (and
135
O(M) relation A«B
Verbal explanation A is much smaller than B A is moderately smaller than E A is slightly smaller than B A is exactly equal to B A is slightly larger than B A is moderately larger Ihan B A is milch larger than B
A --B
A>-B A»B.
In this section, we propose a specification of q-automata and then define the local and global constraint representation levels.
Table I: Primiliverelationsof the OeM) formalism
3.1
==
If /lY(X I ) /lY(X 2) , that is, the degree of influence of Xl on Y is exactly equal to that of X 2 on Y, then, Y is influenced by Xl and X 2 in the same proportion throughout the time-interval T. Now suppose that /ly(Xr) > - /LY(X2 ) , then Y will be influenced by Xl moderately larger than by X 2 , therefore the net influence on Y is nearer to the influence exerted by Xl on Y than the one exerted by X 2 on Y, and so on...
2.3
Monitoring attributes are: Type: numeric or symbolic Accessibility: measured, observed or inaccessible Quality-space: {subrangej,i = l, ...,n} Each subrangej is an interval if the q-autornaton is numeric, or a single value if the q-autornaton is symbolic.
Local and global constraints
A local constraint in the q-autornata universe is a causal relation involving two q-automata, A causal relation describes the interactions between a pair of q-automata, Theft are two kinds of causal relations: one is concerned with cause-effect relations (e.g, if the amount of liquid in the tank increases then we can say that the pressure at the bottom of the tank increases as well) and the other is concerned 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, but the influence rather goes from the concentration rate to the color). We refer to influencebased relations as the former relations and informationbased relations as the latter ones. Influence-based relations are used to predict behaviors, whereas information-based relations allow fast interpretations. The q-automata akin to a physical process can be networked by means of the causal relations linking them. This leads to a directed graph whose nodes are q-autornata and edges are labeled with causal relations. A global constraint in a q-autornata universe is a numerical or qualitative equation involving possibly several q-automata.
3
Q-automata specification
The q-automata specification provides a clear statement of relevant knowledge needed to reason efficiently about qautomata. It includes three parts: monitoring atrt r i.buties , dynamics and control knowledge-base. The monitoring attributes describe the knowledge about the monitoring tasks (e.g. when can we say that a q-automaton is behaving unwantedly ?, or is the q-automaton measured or observed ?). The dynamics consists in a set of information underlying influence combination and state updating. The control knowledge-base contains the compiled knowledge about what action to do to make the q-automaton behave suitably according to situations.
Nominal-range: default nominal range Set of nominal (desired) values of the underlying process variable for the supervision purpose. The nominal range should be one of the above mentioned decisionmaking subranges or a contiguous union of some of them. Since the nominal values may change according to situations, the reasoning system using the model must enable the change to the nominal range in operation when necessary. The dynamics part copes with influence computation tables and methods to calculate net influences and update the q-automaton's states. That requires knowing the variation space, the influence combination function (function f) and the state updating function (function gJ. The dynamics computation is postponed until section 4. The control knowledge-base contains expert knowledge described by means of the following primitive functions: do-action(preconditions, actions): states the actions which should be done when the preconditions are true, set-nominal(preconditions, new-nominal-values): allows us to change nominal value sets to new-nominal-values when preconditions are true,
Modeling with q-automata
Modeling with q-automata starts with the collection of all the relevant process variables and the compiled knowledge as well as the deep knowledge that process experts have about them. Then a q-automaton is assigned to each variable in the way to build the q-automata universe underlying the process. The information needed for a process supervision system consists in :
reduce-history(preconditions, elements-to-be-withdrawnfrom-causal-history): allows us to abstract from some parts of the causal history while predicting behavior in stating the information which is no longer useful for prediction. Example 3.1
• The behavioral and structural relationships between the q-automata. These relationships are phrased by local and global constraints as presented previously.
Q-automaton Y (level of the liquid in the tank, see figure 1) could be specified as follows: Monitoring attributes: Type: numeric Accessibility: observed Quality space: {lO, YI[; [YI, Y2[; [Y2, Ysl; [Ys, Y4[; [Y4' Yma,.]} Nominal-range: [Y2' Ys[.
• The dynamics of each q-automaton, as stated in definition 2.3. • A knowledge-base needed for a reliable monitoring and control of each q-automaton's behavior.
136
the respective nominal ranges).
Dynamics: Tables 2 and 3 contain pairwise combined influence values for q-automata Fa, Fb and Fp which influence Y; they enable to compute the net influence on Y. In these tables, N L, N M, N S, P S, PM and P L mean respectively negative large, negative medium, negative small, positive small, positive medium and positive large.
• The equations are given in the usual interval algebra using the usual equality [i.e, =) or the qualitative equality (i.e. Rl) (Trave-Maseuyes and Piera (1989)). They link the states or the states and variations of q-automata by means of the operators Ell and ® which are respectively the qualitative sum and product (Missier (1989)) defined as the qualitative function associated with the classical sum and product operators on reals. The constraints linking exclusively variations are all considered at the local level, either in causal relations or in the influence combination function f.
Control knowledge·base (an example of rules): do-action{value-oJ(Y) < Ya and value-oJ(col) = green, "turn tap B off") do-action{value-oJ(col) = red and value-oJ(Y) < YI, "turn tap B off and increase flow rate from tap B"), ...
~ 6FIi
-
Example 3.2.1
NL
NS
0
PS
PL
PS
PS
PS
PM
PL
PS
PS
PL
PL
PL
In the sprinkler example, the mass-balance equation linking the tank input and output amounts to the volume variation in the tank during a temporal unit 7' is represented as follows:
(1) 0
where S is the section of the tank given as a real number or interval.
+
PS
PM
PL
PL
PL 3.2.2
The local constraint levelincludes the causal relations among the q-automata, They are influence-based relations which causally link the amounts of change (variations) of two qautomata and information-based relations which constrain two q-automata states.
Table 2: Effects of 6Fa and IlFb on Il Y
~
NL
NS
0
PS
PL
-
PS
PS
PM
PM
PL
0
PS
PS
PS
PS
PM
+
PS
PS
PS
PS
PM
6Fp
Local constraint level
An influence-based relation is of the form: R(X, Y, c, fl, d, p) I where: R is the name of the causal relation. X the influencing q-automaton, Y the influenced q-automaton, c the activation condition of the causal relation. The influence occurs if, and only if c holds. fl is the relative order of magnitude in O(M) formalism (see section 2.2.3) of the influence degrees when X influences Y.
Table 3: Effects of Il Fa and Il Fp on Il Y
d is the delay of the influence. It is a real number.
3.2
p is the influence duration (response time). It is a real number, or equals 00 if the influence holds permanently.
Constraint specification
In this section we present the representation levels germane to the local and global constraints stated in section 2.3. 3.2.1
To model influence-based relations we use predicates It and I;; defined as follows:
Global constraint level
It(X,Y,c,fl,d,p) (resp. I;;(X,Y,c,fl,d,p)) holds if, and only if X influences positively (resp. negatively) Y, where (l( is an integer which describes the rate ofchange,
As mentioned earlier, global constraints are numeric or qualitative equations involving possibly several q-automata in representing deep knowledge about the process. Therefore global constraints are concerned with numeric q-automata. There are two kinds of global constraints which are P+ (resp. P-) and equations:
Let us consider the predicate It. If (l( = 0, Y increases (or decreases) in the same proportion as X does. If (l( = 1, Y increases (or decreases) slightly more than X does. If (l( = -1, Y increases (or decreases) slightly less than X does, and so on... The same example could be dealt with I;; dually. An influence-based relation may be the disjunction of n other relations:
• P+(X,XI, ... ,Xn ) (resp. P-(X,XI, ... ,Xn ) ) holds if, and only if:
- When X behaves normally, so do Xl>..., X n • - When the state of X is below or above the nominal range, so are the states of XI, ... .X; (resp. when the state of X is below (above) the nominal range, those of Xl>..., X n are above (below)
137
n
R(X,Y, C,Jl, d,p) =
VR;(X,Y,C;,Jl,di,pi)
4.2
(2)
i=l
where two different conditions Ci and Ci do not hold throughout the same time-interval. In equation (2), for any pair of different elements i,i, R; = Rj implies that di and dj, or Pi and Pi, are different from one another. Such a relation describes influences whose nature changes over time or according to circumstances, and so, it is termed as composite.
The output of a q-automaton at any instant equals the current global effect of the influences that it suffers.
An information-based relation is not associated with delay, persistence duration, conditions and influence degree because it does not describe influences but informs about what is occurring in some q-automaton, Hereafter is an axiom about the quality spaces of q-automata linked by an information-based relation.
5
An information-based relation is represented by a pair
Applying the algorithm to the causal model of the sprinkler as depicted in fig. 3 will allow us to now, for instance, if the level of the liquid will behave abnormally in mean-term, and to correct the q-automata's behavior if necessary.
I(X, Y) where X is the informing q-autornaton and Y the q-automaton that X informs about. Therefore, X must be an accessible (measured or observed) q-automaton. The information underlying such a causal relation is described by a one-to-one mapping from the quality space of X into that of Y. The idea of mapping quality spaces has originated from the study led by Guerrin (1991) about influence propagation in hydroecological processes.
4.1
Prediction algorithm and sprinkler example
The behavior prediction in the q-autornata universe consists in applying in parallel the behavior updating algorithm (global influence and new state calculi stated in the last section) to each q-automaton, The full causal propagation algorithm which accounts for influence delays and durations will not be presented here because of space.
Quality space axiom: The quality spaces of any pair of q-automaia linked by an information-based relation have the same cardinality.
4
State updating and output
The state of a q-automaton over time is computed by means of mapping g (definition 2.3) using the global effect of the influences it undergoes and its near-past state. But computing the new state that way may bear ambiguities. However, these ambiguities can be filtered out by means of global constraints and the causal history of the q-automaton.
6
Conclusion and perspectives
In this paper, we have presented a causal modeling methodology in the framework of process supervision. The proposed formalism relies on a qualitative automata concept that we have devised for the purpose.
Qualitative automata dynamics
The proposed approach is based on a two-level modeling: a global modeling of the structure (numerical and qualitative constraints involving several q-automata) and a local modeling (the behavioral model of each q-automaton and the interactions with its surroundings). The modeling of a process starts with the representation of every relevant process variable as a qualitative automaton. Then the behavioral influence of qualitative automata variations is represented, at the lower level, by causal relationships among the underlying qualitative automata and a cognitive model of each qualitative automaton. The higher level accounts for global constraints involving sets of qualitative automata. It is used by the reasoning mechanism as a multi-automata coupling level. The main avantages of this formalism are the followings:
Global influence calculus
Let Y be a q-automaton influenced by N other q-automata XI,X2, ...,XN and assume Ct,C2, ...,CN to be the respective variation spaces. Then the input set of Y is defined as the cartesian product 1= Ct x C2 x ... X ON. The mapping f is an operator that combines the influence values composing I to yield an element in a (the variation space of Y). Actually, if e = (Ct, C2, ... , CN) E I, the global effect of the input e is:
(3) where each fit c;) is the marginal effect of the influence value C; (I.e. the effect of C; assuming that it is the sole acting) on the q-automaton Y, and @ an influence combining operator defined on C.
• It deals explicitly with time by using a temporal logic which puts instants and time-intervals on the same footing.
Compiled knowledge about influence combination is described in tables (e.g. tables 2 and 3) for known pairwise combined influences, or given by correspondences. Although in general experiential knowledge is available for the combination of several influences at once, we give hereafter an algorithm to compute the global effect when only a few marginal influences and influence tables are available:
• It allows the representation of both deep knowledge and compiled knowledge. • It copes with hybrid (numeric and symbolic), continuous and discrete processes. A causal engine (eA-EN) using the formalism is under intensive development for the supervision of biotechnological processes. It is meant to be validated as a generic system for the supervision of fed-batch fermentation processes which require tedious tasks. However, our ultimate goal is to make the formalism as robust as possible for most of dynamical processes. Herein, the present limitations are that it assumes relative order of magnitude influence degrees and tables to be given by experts. Some work has already been started to deepen the formalization of influence combination techniques, notably, the investigation of methods to generate automatically influence combination tables
1. Combine the terms in equation (3) relatively to the available tables. (That process yields a number of partial net influences in addition to marginal influences). 2. Calculate the order of magnitude associated with the influence degree of each partial net influence. 3. Compute the net influence using relative orders of magnitude calculus. That may gives rise to several possible net influences some of which may be spurious. 4. Filter out for consistency using global constraints.
138
Globlll constretnt t.euel
Constraints on states or mixed constraints (states & variations)
Constraints on variations
LOCIII constretnt Leuel Figure 3: The causal model of the sprinklerexample
Missier A., Piera N. and 'Irave-Massuyes L. (1989),
given minimum expert information about the interactions among the process variables (Bousson and 'Irave-Massuyes (1992».
Order of magnitude qualitative algebras: a survey, Revue d'intelligence Artificielle, VoI.3,noA, 95-109.
Trave-Massuyee L. and Piera (1989) , The order of
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D'Ambrosio B. (1987), Extending the mathematics in Qualitative Process theory, in Proc, AAAI-87, Seattle, Washington. DeKleer J. and Brown J. S. (1984), A qualitative physics based on confluences. Artificial Intelligence 24, 7-83. Ghallab M. and Alaoui M. A. (1989), Managing efficiently temporal relations through indexed spanning trees, IJCAI-89, 1297-1303. Guerrin F. (1991), Qualitative reasoning about an ecological process: interpretation in hydroecology, Ecological Modelling, 59, 165-20l. Hayes P. J. (1985), The Second Naive Physics Manifesto, Formal Theories of The Commonsense World, eds J. Hobbs and B. Moore, 1-36. Kuipers B. (1986), Qualitative Simulation, Artificial Intelligence 29, 289-338. Leyval L. and Gentil S. (1991), On line event based simulation through a causal graph. Qualitative Reasoning and Decision Support Systems Workshop, Toulouse. Mavrovouniotis M. L. and Stephanopoulos G. (1988), Formal order-of-magnitude reasoning in process engineering, Computer Chemical Engineering 12, 867880, Pergamon Press Ltd.
139
A COMPUTATIONAL CAUSAL MODEL FOR
PROCESS SUPERVISION K. Bousson and L. Trave-Massuyes LAAS-CNRS, 7 avenue du Colonel Roche, 31077 Toulouse Cedex, France
Abstract: This paper proposes a causal modeling framework for the purpose of process supervision. It is based on a qualitative automata (or q-automata) concept that we have devised. A q-automaton captures both the dynamics of a process variable and the expert knowledge necessary for supervising the variable's behavior. We use a two-level representation scheme for the description of the relationships between the q-automata underlying a process: a local constraint level and a global constraint one. The local constraint level describes the qualitative causal relationships between the q-autornata, and the global constraint level states the quantitative constraints among them. The formalism is shown to allow the modeling of deep knowledge as well as compiled knowledge. Furthermore, it is suitable for the modeling of partially-known, hybrid (numeric and symbolic), continuous and discrete processes. A causal engine (CA-EN) using the formalism is under intensive development. It is at the core of the process supervision system.
Keywords: Process Supervision; Qualitative Reasoning; Causal Modeling; Qualitative Automata; Compiled Knowledge; Deep Knowledge; Influence Combination.
1
Introduction
Expert systems are decision support systems; however today, it is widely recognized that their knowledge is too 'shallow' to cope with engineering problems in the sense that they are based on pre-specified relationships between conditions and actions, or causes and effects, (e.g. between system behavior abnormalities and corresponding causes) rather than on a model of the domain of interest (eg. structural description, including physical laws). The weakness of shallowreasoning systems and the lack of mathematical models of incompletely-known engineering systems (e.g. biotechnological processes) have recently led Artificial Intelligencecommunitiesand control engineers to investigate qualitative reasoning about physical systems (or qualitative physics) (Bobrow (1984), Weld and DeKleer (1990»). As a component of qualitative reasoning, the aim of causal reasoning is to derive system behaviors from structural descriptions for the purpose of, among other tasks, prediction, causal explanation and situation assessment. Hence causal reasoning is well suited for process supervision. Causal modeling is the cornerstone of influence representation within devices, more often than not at the basis of causal reasoning. Consider we were asked to supervise a sprinkler (fig. 1) so that the level of the liquid in the tank remains in the interval [Y2, Ya[ and that the color of the liquid stays green during the sprinkling process. To do that, we first have to use a set of conventions to represent the dependences between the tap flow rates and the liquid level, the pipe flow rate and the liquid level, etc .., In other words, we need first to model causally the interrelationships among the system's variables. Here, 'Causality', as we understand it in engineering, stands for something that links the structure (set of numeric or symbolic equations, topological description) of a system to its functions and behavior.
The main tasks of a process supervision system are to assess the process behavior and to act, or to advise a human operator about what occurs and what to do when the process behaves unwantedly. Therefore a supervision system must meet the following requirements: • Real-time: The system must be reliable in response time. For that purpose it must be endowed with fast processing strategies and deal with time. • It must track the behavior of the process over time from state to state. That ensures the detection of early deviations from the nominal behavior.
• In a process, all the variables are not necessarily measured or observed, that is, some of them may be inaccessible. A process supervision system must be provided with means to predict the behavior of those inaccessible variables as well as the behavior of accessible (measured or observed) ones. • It must be supplied with trustworthy structural and behavioral models of the process to enable it to meet the above mentioned requirements. This is all the more achieved as the model captures more relevant available knowledge. Consequently, it must deal with both numeric data (e.g. numerical equations) and symbolic ones (e.g. experential knowledge, physical qualitative constraints). Indeed, supervision appears as the last and skilled level of automation that control theories do not master and that Artificial Intelligence techniques are welcome (Aguilar-Martin (1991)). Meanwhile, knowledge from lower levels (e.g. regulation-oriented numeric models) should be used if need be.
133
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pose of supervision. We use a two-level representation scheme for the description of the relationships between the process variables: a local constraint level and a global constra.int one. The local constraint level represents the qualitative causal relationships between the process variables, and the global constraint level states the quantitative const raints among the variables.
Causal modeling has received considerable attention in qualitative physics. Work has been done in process modeling for supervision tasks (Leyval and Gentil (1991)) using the notion of qualitative transfer function which is the qual. itative counterpart of transfer function in control theory, or propagation function like in (Vescovi (1990)). Meanwhile, the existing approaches suffer from some weakness in that they are generally supported by directed-graph representations:
2
• Multi-variable relationships induce the problem of influence combination so that those systems generally assume linear relationships, which may be a very rough approximation in many cases.
Representing process knowledge
Dealing with the supervision of physical processes requires having a relevant representation of the quantities at hand which are time and the process variables.
• Constraints involving more than two variables (e.g. mass balance relations) are not representable, which makes difficult to model processes whose supervision requires an interminglement of first principles and experential knowledge.
2.1
Time representation
The algebra of temporal intervals (Allen (1983)) is a. temporal reasoning framework which suits most of Artificial Intelligence systems for it has a higher expressive power and is easy to implement. However it is shortcome by the fact that it may accept an inconsistent set of temporal relations as being consistent (incompleteness problem) and it runs in O(n 3 ) , which is a too high computational complexity for coping with real-time applications such as supervision tasks. Vilain and Kautz (1986) have shown that the completeness problem can be solved by representing time in the restricted interval algebra which allows the same expressive power as Allen's temporal algebra. Recently, Ghallab and Alaoui (1989) solved the complexity problem by proposing the IxTeT temporal management system which guarantees linear time complexity for data retrieval and updating, and which has been proven sound and complete ensuring the same expressive power as the restricted interval algebra.
• Although they may allow reasoning in time, they do not allow reasoning on time so that intermittent influences or influences with limited duration are not representable. Time representation is of event type which makes difficult tracking the process behavior. • The formalism used for representing process dynamics is too inflexible so that hybrid processes (i.e. processes involving both numeric and symbolic variables, such as the above mentioned sprinkling system) cannot be handled. A consequence of this is that they do not cope with processes involving both continuous variables and discrete ones. On the other hand, qualitative differential equation based (Kuipers (1986)) or component-based (DeKleer and Brown (1984)) approaches can easily cope with multi-variable relationships and constraints involving more than two variables. However, they also suffer from event-type time representation and from the fact that they do not allow reasoning on time. The last mentioned issue is a problem as well. In addition, causality paths are not explicitely represented. First, that makes it difficult to represent experential knowledge understanding oriented influences which rises the problem of involving "informative" variables, for instance how can you represent in the QSIM formalism that the color of the water in the tank depends on the concentration rate of product but not the converse? Second, it complexifies several tasks like diagnosis and causal explanation. This paper proposes a causal modeling framework which captures most of what is needed to reason about partiallyknown, hybrid, continuous and discrete process for the pur-
Therefore, the temporal representation and management strategies we have chosen for dealing with processes are that of IxTeT system. A time-interval is represented as a pair T = (t l, h) where tl and t2 are time-instants with t l < h. However we regard the temporal axis as regularly sampled with temporal landmarks a.t which the physical process must be checked up. Hence, the temporal unit for the supervision system is the (constant) dista.nce between adjacent landmarks. The temporal unit must be chosen so tha.t the behavior of the process at hand can be assumed to be linear within it.
2.2
Qualitative Automata
When a human operator supervises a process he concerns himself with checking whether the value of each variable equals some (discrete) value, when the variable is discrete, or falls into some interval, when the variable is real-valued.
134
not only at time-instants) when they occur.
When a variable behaves unwantedly, the operator uses knowledge about its dynamics and about the causal relationships between that variable and others to correct the variable's behavior. In other words, on one hand the operator checks the qualitative behaviors of the variables in somewhat regarding each process variable as a kind of automaton moving from an interval or a value to another, and on the other hand he uses a set of knowledge to act on the variable. Therefore, this motivates that we assign a qualitative automaton (or q-automaton for short) to each process variable and to the knowledge-base which is needed to reason about the behavior of that variable.
Definition 2.1: The causal history of a q-automaton is the chronological sequence of its states and the actions that it suffers (if it is a control q-automaton) or that the control q-automata which influence it directly or indirectly suffer. Causal histories are main sources of relevant information to achieve the goals of prediction and explanation in process supervision. From the definition, the causal history of each q-automaton grows as the process runs. However, some of its elements can be discarded when they are no longer necessary for behavior prediction or explanation.
Our perception of a physical process is a world of interacting q-automata influencing one another. The set of q-automata underlying a process is called the q-automata universe. In order to cope with any type of process variables, a q-automaton may be of numeric or symbolic type according to whether the underlying process variable is realvalued or takes discrete (or symbolic) values (e.g. red, green, blue).
2.2.1
Definition 2.2: The variation space of a q-automaton is the set of possible amounts of change it bears within a temporal unit. The variation (amount of change) of a qautomaton X is denoted by AX. The dynamics of a q-automaton is defined by a set of influence values (its inputs), a quality space, a variation space, and mappings on those sets. Definition 2.3: The dynamics of a q-automaton is represented as a 5-tuple D = (I,S,V,f,g) where: I is the input set. It is a set of influence values. S is the q-automaton's quality space, V is the q-auiomaion's variation space, f is a mapping from I into V I g is a mapping from V x S into S .
Quality spaces, states and values
The significant discrete values and intervals according to which the values of a process variable must be evaluated over time by the human operators are pre-specified by process experts. We assume that for real-valued variables, these intervals describe a finite and totally ordered partition of their respective ranges. We call quality space (Hayes (1985) of a numeric q-automaton such a partition. The quality space of a symbolic q-automaton is its value set (which is, in fact, a discrete set), and is assumed to be finite and totally ordered according to the context of use. The state of a q-automaton is an element of the quality space; it corresponds to relevant information for a human operator in the decision-making tasks. Contrarily to conventional automata theory, a q-automaton's state does not necessary summarize the whole history of the q-automaton.
Dynamically, a q-automaton processes as follows (fig. 2) : Mapping f enables to sum up the inputs the q-automaton receives over time. Mapping g is used to combine the result of that computation with the q-automaton's most recent state and its current variation to update the q-automaton's state.
2.2.3
When several q-automata influence the same q-automaton, some of them may influence with a higher degree of sensitivity than others. On the other hand the subset of qautomata influencing a given q-automaton may be timedependent. Therefore, the set of q-automata influencing a given q-automaton is a time-dependent fuzzy subset of the set of all the q-automata.
The value of a q-automaton X at time-instant t is the value of the associated variable at i, It will be denoted by value-of(X, t)j if t==now (i.e. the current time-instant), that is simply written value-of(X). The same shorthand applies to state-of for the description of states. If X is a symbolic q-autornaton, the state of X at t equals its value at tj but if X is numeric, value-of(X, t) is element of state-of(X, t).
2.2.2
Dealing with influences
To reason about influences we make the: Influence Sameness Assumption: The degree of sensitivity of an influence between two q-automata is constant whenever the influence holds.
Dynamics
The assumption states that:
The dynamics of a process is due, on one hand, to the interactions among the q-automata and, on the other hand, to actions coming from the process outside world. An action is an operation made by a human operator or a device in the way to increase or decrease the state of a qautomaton. Actions are made by means of process control q-automata (q-automata underlying the process control variables). The interactions among the variables of a process can be represented by influences from one q-autornaton to another. The influence that a q-autornaton exerts on another q-automaton is due to the variation (change) in the
• The degree of influence (i.e. the degreeof sensibility of the influence) that a q-automaton X exerts on a given q-autornaton Y, denoted by !lY(X), remains steady throughout the time-intervals on which the influence holds (it equals zero elsewhere), • If X influences Y on two distinct time-intervals Tl and T2 , the degree of influence throughout T l equals the degree of influence throughout T2 •
Instead of reasoning by means of the explicit numeric values of influence degrees, we will rather consider relative orders of magnitude at the level of the influences exerted upon the same q-automaton, This is dictated by the way process experts reason about influences. The O(M) formalism (Mavrovouniotis and Stephanopoulos (1988») has been chosen for it has shown to be the closest formalism to the one process experts use. Table 1 presents the primitive relations of the formalism and the explanations of these relations. For instance, assume Xl, X 2 to be two q-automata influencing a q-automaton Y throu.qhout a time interval T.
former. A q-automaton X is said to influence (directly) a q-automaton Y if, assuming no action to be done on Y and all the q-automata to be quiescent except X and Y, a change in X is necessarily followed by a change in Y. A q-automaton X influences indirectly a q-automaton Y if there exists a chain of q-automata X, Xl,. .., X,.,Y which successively influence directly one another. Unless otherwise specified, the word influence, from now on, will refer to direct influence implicitly. We assume influences to have durations, that is, they hold throughout time-intervals (and
135
O(M) relation A«B
Verbal explanation A is much smaller than B A is moderately smaller than E A is slightly smaller than B A is exactly equal to B A is slightly larger than B A is moderately larger Ihan B A is milch larger than B
A --B
A>-B A»B.
In this section, we propose a specification of q-automata and then define the local and global constraint representation levels.
Table I: Primiliverelationsof the OeM) formalism
3.1
==
If /lY(X I ) /lY(X 2) , that is, the degree of influence of Xl on Y is exactly equal to that of X 2 on Y, then, Y is influenced by Xl and X 2 in the same proportion throughout the time-interval T. Now suppose that /ly(Xr) > - /LY(X2 ) , then Y will be influenced by Xl moderately larger than by X 2 , therefore the net influence on Y is nearer to the influence exerted by Xl on Y than the one exerted by X 2 on Y, and so on...
2.3
Monitoring attributes are: Type: numeric or symbolic Accessibility: measured, observed or inaccessible Quality-space: {subrangej,i = l, ...,n} Each subrangej is an interval if the q-autornaton is numeric, or a single value if the q-autornaton is symbolic.
Local and global constraints
A local constraint in the q-autornata universe is a causal relation involving two q-automata, A causal relation describes the interactions between a pair of q-automata, Theft are two kinds of causal relations: one is concerned with cause-effect relations (e.g, if the amount of liquid in the tank increases then we can say that the pressure at the bottom of the tank increases as well) and the other is concerned 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, but the influence rather goes from the concentration rate to the color). We refer to influencebased relations as the former relations and informationbased relations as the latter ones. Influence-based relations are used to predict behaviors, whereas information-based relations allow fast interpretations. The q-automata akin to a physical process can be networked by means of the causal relations linking them. This leads to a directed graph whose nodes are q-autornata and edges are labeled with causal relations. A global constraint in a q-autornata universe is a numerical or qualitative equation involving possibly several q-automata.
3
Q-automata specification
The q-automata specification provides a clear statement of relevant knowledge needed to reason efficiently about qautomata. It includes three parts: monitoring atrt r i.buties , dynamics and control knowledge-base. The monitoring attributes describe the knowledge about the monitoring tasks (e.g. when can we say that a q-automaton is behaving unwantedly ?, or is the q-automaton measured or observed ?). The dynamics consists in a set of information underlying influence combination and state updating. The control knowledge-base contains the compiled knowledge about what action to do to make the q-automaton behave suitably according to situations.
Nominal-range: default nominal range Set of nominal (desired) values of the underlying process variable for the supervision purpose. The nominal range should be one of the above mentioned decisionmaking subranges or a contiguous union of some of them. Since the nominal values may change according to situations, the reasoning system using the model must enable the change to the nominal range in operation when necessary. The dynamics part copes with influence computation tables and methods to calculate net influences and update the q-automaton's states. That requires knowing the variation space, the influence combination function (function f) and the state updating function (function gJ. The dynamics computation is postponed until section 4. The control knowledge-base contains expert knowledge described by means of the following primitive functions: do-action(preconditions, actions): states the actions which should be done when the preconditions are true, set-nominal(preconditions, new-nominal-values): allows us to change nominal value sets to new-nominal-values when preconditions are true,
Modeling with q-automata
Modeling with q-automata starts with the collection of all the relevant process variables and the compiled knowledge as well as the deep knowledge that process experts have about them. Then a q-automaton is assigned to each variable in the way to build the q-automata universe underlying the process. The information needed for a process supervision system consists in :
reduce-history(preconditions, elements-to-be-withdrawnfrom-causal-history): allows us to abstract from some parts of the causal history while predicting behavior in stating the information which is no longer useful for prediction. Example 3.1
• The behavioral and structural relationships between the q-automata. These relationships are phrased by local and global constraints as presented previously.
Q-automaton Y (level of the liquid in the tank, see figure 1) could be specified as follows: Monitoring attributes: Type: numeric Accessibility: observed Quality space: {lO, YI[; [YI, Y2[; [Y2, Ysl; [Ys, Y4[; [Y4' Yma,.]} Nominal-range: [Y2' Ys[.
• The dynamics of each q-automaton, as stated in definition 2.3. • A knowledge-base needed for a reliable monitoring and control of each q-automaton's behavior.
136
the respective nominal ranges).
Dynamics: Tables 2 and 3 contain pairwise combined influence values for q-automata Fa, Fb and Fp which influence Y; they enable to compute the net influence on Y. In these tables, N L, N M, N S, P S, PM and P L mean respectively negative large, negative medium, negative small, positive small, positive medium and positive large.
• The equations are given in the usual interval algebra using the usual equality [i.e, =) or the qualitative equality (i.e. Rl) (Trave-Maseuyes and Piera (1989)). They link the states or the states and variations of q-automata by means of the operators Ell and ® which are respectively the qualitative sum and product (Missier (1989)) defined as the qualitative function associated with the classical sum and product operators on reals. The constraints linking exclusively variations are all considered at the local level, either in causal relations or in the influence combination function f.
Control knowledge·base (an example of rules): do-action{value-oJ(Y) < Ya and value-oJ(col) = green, "turn tap B off") do-action{value-oJ(col) = red and value-oJ(Y) < YI, "turn tap B off and increase flow rate from tap B"), ...
~ 6FIi
-
Example 3.2.1
NL
NS
0
PS
PL
PS
PS
PS
PM
PL
PS
PS
PL
PL
PL
In the sprinkler example, the mass-balance equation linking the tank input and output amounts to the volume variation in the tank during a temporal unit 7' is represented as follows:
(1) 0
where S is the section of the tank given as a real number or interval.
+
PS
PM
PL
PL
PL 3.2.2
The local constraint levelincludes the causal relations among the q-automata, They are influence-based relations which causally link the amounts of change (variations) of two qautomata and information-based relations which constrain two q-automata states.
Table 2: Effects of 6Fa and IlFb on Il Y
~
NL
NS
0
PS
PL
-
PS
PS
PM
PM
PL
0
PS
PS
PS
PS
PM
+
PS
PS
PS
PS
PM
6Fp
Local constraint level
An influence-based relation is of the form: R(X, Y, c, fl, d, p) I where: R is the name of the causal relation. X the influencing q-automaton, Y the influenced q-automaton, c the activation condition of the causal relation. The influence occurs if, and only if c holds. fl is the relative order of magnitude in O(M) formalism (see section 2.2.3) of the influence degrees when X influences Y.
Table 3: Effects of Il Fa and Il Fp on Il Y
d is the delay of the influence. It is a real number.
3.2
p is the influence duration (response time). It is a real number, or equals 00 if the influence holds permanently.
Constraint specification
In this section we present the representation levels germane to the local and global constraints stated in section 2.3. 3.2.1
To model influence-based relations we use predicates It and I;; defined as follows:
Global constraint level
It(X,Y,c,fl,d,p) (resp. I;;(X,Y,c,fl,d,p)) holds if, and only if X influences positively (resp. negatively) Y, where (l( is an integer which describes the rate ofchange,
As mentioned earlier, global constraints are numeric or qualitative equations involving possibly several q-automata in representing deep knowledge about the process. Therefore global constraints are concerned with numeric q-automata. There are two kinds of global constraints which are P+ (resp. P-) and equations:
Let us consider the predicate It. If (l( = 0, Y increases (or decreases) in the same proportion as X does. If (l( = 1, Y increases (or decreases) slightly more than X does. If (l( = -1, Y increases (or decreases) slightly less than X does, and so on... The same example could be dealt with I;; dually. An influence-based relation may be the disjunction of n other relations:
• P+(X,XI, ... ,Xn ) (resp. P-(X,XI, ... ,Xn ) ) holds if, and only if:
- When X behaves normally, so do Xl>..., X n • - When the state of X is below or above the nominal range, so are the states of XI, ... .X; (resp. when the state of X is below (above) the nominal range, those of Xl>..., X n are above (below)
137
n
R(X,Y, C,Jl, d,p) =
VR;(X,Y,C;,Jl,di,pi)
4.2
(2)
i=l
where two different conditions Ci and Ci do not hold throughout the same time-interval. In equation (2), for any pair of different elements i,i, R; = Rj implies that di and dj, or Pi and Pi, are different from one another. Such a relation describes influences whose nature changes over time or according to circumstances, and so, it is termed as composite.
The output of a q-automaton at any instant equals the current global effect of the influences that it suffers.
An information-based relation is not associated with delay, persistence duration, conditions and influence degree because it does not describe influences but informs about what is occurring in some q-automaton, Hereafter is an axiom about the quality spaces of q-automata linked by an information-based relation.
5
An information-based relation is represented by a pair
Applying the algorithm to the causal model of the sprinkler as depicted in fig. 3 will allow us to now, for instance, if the level of the liquid will behave abnormally in mean-term, and to correct the q-automata's behavior if necessary.
I(X, Y) where X is the informing q-autornaton and Y the q-automaton that X informs about. Therefore, X must be an accessible (measured or observed) q-automaton. The information underlying such a causal relation is described by a one-to-one mapping from the quality space of X into that of Y. The idea of mapping quality spaces has originated from the study led by Guerrin (1991) about influence propagation in hydroecological processes.
4.1
Prediction algorithm and sprinkler example
The behavior prediction in the q-autornata universe consists in applying in parallel the behavior updating algorithm (global influence and new state calculi stated in the last section) to each q-automaton, The full causal propagation algorithm which accounts for influence delays and durations will not be presented here because of space.
Quality space axiom: The quality spaces of any pair of q-automaia linked by an information-based relation have the same cardinality.
4
State updating and output
The state of a q-automaton over time is computed by means of mapping g (definition 2.3) using the global effect of the influences it undergoes and its near-past state. But computing the new state that way may bear ambiguities. However, these ambiguities can be filtered out by means of global constraints and the causal history of the q-automaton.
6
Conclusion and perspectives
In this paper, we have presented a causal modeling methodology in the framework of process supervision. The proposed formalism relies on a qualitative automata concept that we have devised for the purpose.
Qualitative automata dynamics
The proposed approach is based on a two-level modeling: a global modeling of the structure (numerical and qualitative constraints involving several q-automata) and a local modeling (the behavioral model of each q-automaton and the interactions with its surroundings). The modeling of a process starts with the representation of every relevant process variable as a qualitative automaton. Then the behavioral influence of qualitative automata variations is represented, at the lower level, by causal relationships among the underlying qualitative automata and a cognitive model of each qualitative automaton. The higher level accounts for global constraints involving sets of qualitative automata. It is used by the reasoning mechanism as a multi-automata coupling level. The main avantages of this formalism are the followings:
Global influence calculus
Let Y be a q-automaton influenced by N other q-automata XI,X2, ...,XN and assume Ct,C2, ...,CN to be the respective variation spaces. Then the input set of Y is defined as the cartesian product 1= Ct x C2 x ... X ON. The mapping f is an operator that combines the influence values composing I to yield an element in a (the variation space of Y). Actually, if e = (Ct, C2, ... , CN) E I, the global effect of the input e is:
(3) where each fit c;) is the marginal effect of the influence value C; (I.e. the effect of C; assuming that it is the sole acting) on the q-automaton Y, and @ an influence combining operator defined on C.
• It deals explicitly with time by using a temporal logic which puts instants and time-intervals on the same footing.
Compiled knowledge about influence combination is described in tables (e.g. tables 2 and 3) for known pairwise combined influences, or given by correspondences. Although in general experiential knowledge is available for the combination of several influences at once, we give hereafter an algorithm to compute the global effect when only a few marginal influences and influence tables are available:
• It allows the representation of both deep knowledge and compiled knowledge. • It copes with hybrid (numeric and symbolic), continuous and discrete processes. A causal engine (eA-EN) using the formalism is under intensive development for the supervision of biotechnological processes. It is meant to be validated as a generic system for the supervision of fed-batch fermentation processes which require tedious tasks. However, our ultimate goal is to make the formalism as robust as possible for most of dynamical processes. Herein, the present limitations are that it assumes relative order of magnitude influence degrees and tables to be given by experts. Some work has already been started to deepen the formalization of influence combination techniques, notably, the investigation of methods to generate automatically influence combination tables
1. Combine the terms in equation (3) relatively to the available tables. (That process yields a number of partial net influences in addition to marginal influences). 2. Calculate the order of magnitude associated with the influence degree of each partial net influence. 3. Compute the net influence using relative orders of magnitude calculus. That may gives rise to several possible net influences some of which may be spurious. 4. Filter out for consistency using global constraints.
138
Globlll constretnt t.euel
Constraints on states or mixed constraints (states & variations)
Constraints on variations
LOCIII constretnt Leuel Figure 3: The causal model of the sprinklerexample
Missier A., Piera N. and 'Irave-Massuyes L. (1989),
given minimum expert information about the interactions among the process variables (Bousson and 'Irave-Massuyes (1992».
Order of magnitude qualitative algebras: a survey, Revue d'intelligence Artificielle, VoI.3,noA, 95-109.
Trave-Massuyee L. and Piera (1989) , The order of
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