Inference error minimisation: fuzzy modelling of ambiguous functions.

Transcription

1 Inference error minimisation: fuzzy modelling of ambiguous functions. Antonio Sala and Pedro Albertos Dept. de Ingeniería de Sistemas y Automática. Universidad Politécnica de Valencia Aptdo , E Valencia, Spain, asala@isa.upv.es, pedro@isa.upv.es Abstract. In this paper, the problem of modelling ambiguous (multi-valued) mappings from noisy experimental data and the use of these models in control strategies are considered. An inference method based on an equation image of a rulebase and minimisation of a performance index (inference error) is described so that the ideal model is the one that minimizes contradiction. Its functional interpretation is used in consistent fuzzy modelling of ambiguous functions. Some function approximation properties are discussed. A feedback linearization control application based on a consistent inversion of a fuzzy model representing a multivalued mapping is finally considered. Keywords: Fuzzy set theory, inference methods, linguistic modelling, fuzzy control, function approximators. 1. INTRODUCTION Fuzzy logic [21] has been a key element in the application of rule-based systems to engineering areas such as process control and diagnosis. It is based on a relatively seamless integration of human knowledge and a parallelism with human deduction processes. Nevertheless, in industrial control applications [2] the choice of fuzzy operators, defuzzification methods and rulebase construction, are based on ease of implementation, user understandability and experience, without any further optimisation or validation.

2 2 Many current practical fuzzy systems carry out mere interpolation under a reasoning mask: these are similar to human solutions to some problems. But fuzzy logic possibilities of representing vague information are thus unnecessarily restricted. There are several reasons for using fuzzy logic in control in just an interpolative fashion, being one of them the existence of multiple possibilities of implication [8], rule interpretations [6] and defuzzification methods [2][17], some of them with fat shape dominance, or erroneous output if nonconvex sets are present. Some more involved algorithms use entropy or heuristics to avoid this [20]. Another reason explaining the interpolative bias of fuzzy control research is the easy generalisation of results from the neural network field (neuro-fuzzy systems) and nonlinear control (specially for affine systems), while actual performance of pure logic-based controllers is dependent on an expert by manual tuning. These are some of the reasons motivating the research in the field of logic-based formal fuzzy control. Thus, this line of research is different and complementary of that of using formal stability proofs of linearin-parameter systems (to which many practical fuzzy systems can be proved to belong). In this paper the conception of fuzzy systems is biased towards control applications, by stressing the consequences of their capability of being function approximators. A functional interpretation of rulebases is based on the extension principle[22], presenting rules as examples over fuzzy sets of the function to be modelled. An alternative functional interpretation of fuzzy logic from vague environments and extensional mappings is detailed in [9]. Formal logic-based control must manage system uncertainty in the framework of fuzzy logic. A simple list of parameter values y i may be a poor description of what knowledge means for human beings. A truly intelligent agent must include both a system model and an evaluation of its validity range. To summarize, in actual applications and many research trends, some facts can be stated: - some of the initial spirit of fuzzy logic (computing with words, computing with vague concepts) has been lost. - Fuzzy logic in control has mostly become just a question of man-machine interface for linearly parameterized universal approximators.

3 3 - Many actually working fuzzy algorithms are disguised versions of interpolation ("local models, local controllers"). Most of these algorithms reduce consequent flexibility to one or at most two parameters (centroid, area). Therefore, something is missing when uncertainty and vagueness are represented by fuzzy systems i i. i described by unambiguous averaging expressions such as ϕ y ϕ The previous ideas point out the need of research in applications of formal logic-based methods to fuzzy control. This research line must investigate issues related to knowledge validation: consistency (knowledge quality) - completeness (knowledge quantity) and used resources (redundancy, simplicity and preservation of readability). The results must lead to an improvement of learning capabilities and representational power. Validation issues obviously arise in other fields apart from fuzzy control [10]. The inference error methods [13][14] can be used to define ideal inference as a theoretical reference point for algorithm comparisons and rulebase verification: for a coherent rulebase, inference is analogous to solving a set of equations (one for each rule). This concept can be generalised so that an ideal conclusion is the one that minimises contradiction, to allow extracting conclusions from incoherent rulebases. Apart from providing a way to develop validation methods of fuzzy inference algorithms, generalising those from binary logic [10][15], the approach also provides ways to easily incorporate different IF and IIF rules and to allow conclusions such as x is A OR y is B that do not easily fit in conventional defuzzification paradigms. The issue of fuzzy system validations has been also dealt with in [19] [7]. An interpretation based on the extension principle formalises the use of fuzzy systems as function approximators. Antecedent and consequent membership functions (MF s) are interpreted as fuzzy set examples of a function the rulebase models. Ambiguous functions can be considered as multivalued mappings (i.e., those whose output is a set) [16]. The sources of this "ambiguousness" may be noise or insuficient expresivity of chosen antecedents to model the process nonlinearities, hysteresis, etc. These ideas provide interesting insights to be applied into fuzzy controller design and learning algorithms, as shown in the following sections. The paper is structured as follows: fuzzy logic systems and rule equivalent equations are presented in the second section, defining the concepts of contradiction and completeness as equivalent to the existence

4 4 and uniqueness of solutions. The use of rulebases as function approximators is also defined, including the multivalued mapping case. The general formulation of the coherence performance index (inference error) will be presented in the third section. Algorithms to learn ambiguous models are described in section 4. A feedback linearisation control application example is described in section 5. A conclusion section will stress the most important results. 2. PRELIMINARY CONCEPTS AND DEFINITIONS Concepts about fuzzy rule-based systems and inference methods are widely reported in the literature (see, for instance [1], [2], [17]). As it is well known, Implication ( ) and Double Implication ( ) are the basic operators attached to rules. The generalisation of their binary logic equivalence in terms of other operators (for example, -A+B and -(A B) ) to the fuzzy case is not obvious, giving rise to various interpretations [5]. In boolean logic, stating the truth of each of the premises and rules may be conceived as one equation. Inference is the way of solving them, either by chaining or combinatorial search, for example, applying allowed boolean algebra operations. In order to establish an equivalent equation image of fuzzy rulebases, an implication equation will be also sought. One of the problems is that there are several interpretations of fuzzy rules [5][7]. If the semantics of A B is true is translated into Not A Or B is true (i.e. -A+B=1) then, depending on the conorm used in the disjunction, some intuitive requirements, related to the referred interpretations, cannot be fulfilled. The simplest of those requirements in control applications are here stated: 1. The equation for A B is true must have a solution (logic value of B) for all logic values of A in [0,1]. 2. If A {0,1} there should exist a solution B {0,1} (to keep fuzziness in chained inferences, according to the unformalized intuition of fuzzy logic as interpolation). 3. If A=0 any B [0,1] should be a solution. 4. If A=1, only B=1 should be a solution.

5 5 For example, if -A+B=1, is translated into max(1-a,b)=1 (Kleene-Dienst operator), the equation does not meet the second requirement. Notwithstanding, the use of the bounded sum (Lukasiewicz) implication operator yields the equation: min(1,1-a+b)=1 (1) whose solution B verifies the requirements. This equation is equivalent to a simpler inequality that will be used as a basic definition for implication and its associated double implication fuzzy operators. Inferring will thus be transformed into solving a set of equations and inequations that represent the available knowledge. The third requirement involves a Lukasiewicz-type interpretation, instead of an evidential interpretation associated to Zadeh-Mamdani operators [5]. The Lukasiewicz class of implications is not the usual choice in the fuzzy control literature, although it is initially best suited for validation purposes [7]. Definition 1. The equivalent inequality that replaces fuzzy implication A B is true (IF rule) is defined as: A B (2) Hence, the double implication A B is true, represented as A B AND B A (IIF rule) will be equivalent to the equation: A=B (3) The presented choice of the implication operator is also considered by some authors: the Gödel- Brouwer implication operator and other R-implications [8] give rise to the same equation. If implication equations are applied over a fuzzifier s result they present the form: µ ( x) µ ( y) µ ( x) = µ ( y) (4) A B A B for the rule IF x is A Then y is B and its IIF version, respectively. Thus, the equivalent equations are closely related to the ``gradual rule [5] interpretation with a Rescher-Gaines implication (i.e., used assuming a crisp relation between x and y [7], as it is usually done in function approximation for control). Definition 2. Given a premise x, each rule has an associated conclusion set defined as the set of y that verify its equivalent equation or inequality.

6 6 Example. The following fuzzy rulebase: 1-If x is High Then y is Not Low AND z is High 2-If and Only if x is Medium Then y is High 3-If x is High Then y is Low OR z is High 4-If x is High AND m is Not Zero Then y is Low is equivalent to the set of equations: High(x) min(1-low(y), High(z) ) Medium(x) = High(y) High(x) max(low(y), High(z) ) min( High(x), 1-Zero(m) ) Low(y) So given x, the range of values of y and z verifying these equations will be the conclusions of the inference. If some rules are contradictory then conflicts arise and no solution exists (individual conclusion sets are disjoint). The concept of inference error, to be presented later on, will evaluate a conceptual distance between conclusion sets in case of conflict, to distinguish absolutely contradictory from nearly coherent rulebases. As another example, note that the equivalent equation representing mutually exclusive concepts M and N (being this exclusivity defined by the rules If x is M then x is Not N and If x is N then x is Not M ) is: µ ( x) + µ ( x) 1 x M N This expression points out the conceptual importance of the membership sum of 1 as the boundary between overlapping (non exclusive) concepts and exclusive (disjoint) ones. Remark 1. By using equations, defuzzification is not needed in inference: ideally, only fuzzification exists. Defuzzifiers are just part of some of the algorithms devised to actually carry out the equation solving or a fast approximation of it, but not a basic theoretical entity. Remark 2. Fuzziness is only existent in premises: the rules are assumed to be stated as A B is true. Rule truth in the interval (0,1) may lead to presumption, prejudice and paradoxes [18]. Intermediate cases will be dealt with by confidence functions (section 3).

7 Rulebase validation. The previous approach provides a tool to determine the validity of a rulebase based on algebraic properties of the equivalent equation set, according to the following all-or-nothing definitions: A rulebase is coherent (solvable) for some premises if, for them, there exists at least one solution to the equivalent equations. A rulebase is complete (uniquely determined) for some premises if, for them, there exists one and only one solution to the equations. A rulebase is contradictory (unsolvable) for some premises if, for them, there exists no solution to the equations, i.e., there exist nonoverlapping solutions for simultaneously active rules. A rule is redundant if the elimination of the corresponding equation does not change the solution. A candidate conclusion is said to be coherent with a rulebase and its premises if it is a solution of the related equation set (maybe not the only one). In [7], Rescher-Gaines implications and reflection on the input [19] are used to derive an equivalent approach to coherence definitions. The coherence and contradiction definitions will be generalised in section 3, grading the contradiction via the definition of inference error. As an example, consider the following rulebase for a temperature conclusion: IF A then T<10 ; If A then T>12 It is a contradictory rulebase for premises such that A is true, and incomplete for those such that A is false. The contradiction can be explained because there are two non-overlapping solutions of individual rules (there is simultaneous evidence for T<10 and for T>12), so there is no solution to the doublerestriction problem (for A True, no value of the temperature exists so that both rules are fulfilled). The incompleteness arises due to the fact that a non-fired rule translates to equation 0 µ(t), fulfilled for any temperature and any set defined over that universe (so, as an alternate interpretation, there is evidence of nothing). In rulebase design or in learning algorithms, to take into account validation concepts can contribute to preserve readability of the results, in such a way that fuzzy systems remain readable while training. Certain structural properties and constraints on the rulebases and MFs can ensure consistency and completeness. Preservation of readability has beneficial effects in the user acceptance and even in the

8 8 performance of gradient learning paradigms, in the sense of best directing the search [11]. The importance of the supervision of knowledge acquisition has to be realised Function Approximation. In the fuzzy modelling and control field, rulebases are mostly used to model functions f:u V by means of fuzzy sets defined over U R n and V R, and rules relating them (one layer, one conclusion rulebases) in the form If x is A Then y is B. Thus, fuzzy logic methods and formal validation tools need to be thought of in terms of its adequacy to function approximation purposes to emulate controllers or plant models. A function y=f(u) is coherent with a rulebase if y is a solution of the equivalent equation set for premise u. The extension principle[22] is the key concept to extend the rule subsethood interpretation ( x is A x is B is equivalent to µ ( x) µ ( x), i.e. A B) to sets belonging to different domains. A B Given a function f:u V it can be extended to f:π(u) Π(V) where Π denotes the set of all fuzzy sets over a universe: µ ( y) = sup µ ( x) f( M) f( x) = y M (5) µ ( x) µ ( f( x)) (6) = f 1 ( N) N If f is bijective the expression is simpler: µ ( y ) = ( ) µ ( 1 f ( y )) f M M (7) Some properties of functions of sets are listed in [13]. As µ ( x) µ ( f ( x)), the following lemmas can be proved: M f ( M) Lemma 1. Given a function f:u V and a fuzzy set N V, the function is coherent with the rule IF AND ONLY IF x is M THEN y is N if and only ifm = f 1 (N). Proof. (Sufficiency) If M=f -1 (N) then: µ ( x) µ ( f( x)), hence y=f(x) satisfies the equation = f 1 ( N) N

9 9 (4), µ ( x) = µ ( y). M N (Necessity) If f is coherent with the rule, f(x) satisfies eq. (4) µ ( x) = µ ( f( x)) for all x, so it defines the set M given N, as in (6), hence M f 1 (N). Corollary. If f is coherent with the referred IIF rule, then f(m)=n. The reciprocal is not true. M N Lemma 2. Given a function f:u V and a fuzzy set N V, the function is coherent with the rule IF x is M THEN y is N if and only if M f 1 (N) (equivalently f(m) N ). Proof. (Sufficiency) If M f 1 (N) then µ ( x) µ 1 ( x) = µ ( f( x)) so, f(x) satisfies (4). M f ( N) N (Necessity) If f is coherent with the rule µ ( x) µ ( f( x)), i.e. it is less or equal to that given by (6), being thus a subset of f -1 (N). M N By application of lemma 1, the rule IF and only IF x is M Then y is f(m) is coherent with f only when the expression f -1 (f(m))=m holds for M. This motivates the following definition: Definition 3. Given the function f, if the set M verifies f -1 (f(m))=m it is said to be a proper antecedent for f. Equivalently, M is a proper antecedent if it exists a fuzzy set N in the consequent domain such that M = f 1 (N). Note that if f is bijective all antecedents are proper; that includes monotonic functions. Rules with antecedent M and consequent N in which equality f(m)=n holds are called proper rules with respect to the function f. All coherent IIF rules are proper (directly from lemmas 1 and 2), but some IF rules are not: given a function f, proper rules have the minimum consequent N (i.e., the most specific) for a given antecedent M. All supersets of N form coherent (not proper) IF rules with antecedent M (coherent stands for coherent with f ). IF improperness is present in many of the rules, it may result in incompleteness of the overall rulebase. Notwithstanding, that will allow to define fuzzy systems representing ambiguous functions in the same framework (see next section). The intuitive meaning of the inverse function of a set N may be such as all the input conditions producing an output in N. Stemming from that fact, the lemmas agree with intuition: An IF rule has in its antecedent some of the inputs in which its consequent is produced, An IIF rule has in its antecedent

10 10 all of them. It can be proved that the union and intersection of proper antecedents give as a result proper antecedents, too. In the particular function approximation application, the connection between the algebraic approach to inference and the conventional interpolative defuzzification schemes is drawn by stating that a valid defuzzifier algorithm must be a coherent function with respect to that rulebase from which it extracts the numeric conclusions. Depending on the actual algorithm, some conditions on the fuzzy sets must be fulfilled. Some common algorithms and MF arrangements are validated in [14], in particular centroid defuzzification: n n y*( u) = i( u) yc() i i ( u) µ µ i= 1 i= 1 1 with an add-1 partition of triangular consequents and an arbitrary add-1 partition in antecedents, and rules such that overlapping antecedents correspond to overlapping consequents Multivalued mappings. In control applications where uncertainty is present some mathematical representation of that uncertainty is needed. One option is the probability approach, i.e. the use of a system model that produces, given an input u, a conditional density function for the output y, i.e. f(y u). From the knowledge engineering point of view, an alternative representation is a vague plant model where the output for a given input u is the set of output values that do not contradict the present knowledge about the plant. With this approach, two extremes in the knowledge quantity can be thought of: the first is the absolute lack of knowledge (for an input u, all outputs in the output domain are possible as no restriction is known); the other extreme is the total knowledge of a deterministic system (i.e. the actual function f(u)=y is known). The proposed mathematical representation of these "vague" or "ambiguous" models is a multivalued mapping, defined as follows: Definition 4. Given two universes U and Y, being P(Y) the sets of all subsets of Y, a correspondence F: U P(Y) is named a multivalued mapping, i.e. a correspondence in which a subset F(u) of Y is associated to each element of u [3][16]. If F(u) is a convex set for all u, the mapping F is called a convex multivalued mapping. There are several particular representations of multivalued mappings, even unrelated with fuzzy logic. For example,

11 11 one of them, quite common in robust and sliding control applications, is the parametric uncertainty so that for a certain input (system state) a different output (state derivatives) is obtained depending on the parameter. Notwithstanding, the representation to be analysed here is a rule-based representation of multivalued mappings. In control applications it is often required to find the inverse of a model (for example to apply cancellation or feedback linearisation techniques). Definition 5. Given an element y Y, the set: F ( y) = { u U y F( u)} is named the inverse image of y, i.e., the inverse image of a point y Y is the set of input values that may produce y as the system output (but not necessarily). Incomplete rulebases can be intentionally used to model multivalued mappings representing partly unknown systems. A straightforward method of obtaining those models from noisy data will be presented in section 4. A functional interpretation of rules can also be given in the multivalued case, similar to the one presented for ordinary non-multivalued functions. In order to give this functional interpretation to rule equations, a generalisation of the extension principle is needed. Definition 6. The image of a fuzzy set A under a multivalued mapping is defined as: µ ( y) = sup µ ( u) F( A) u y F( u) Inverse images of sets under multivalued mappings need two definitions arising from vagueness. The idea stems from modal logic [4] approaches: the lower inverse of a nonfuzzy set V Y is the set of input values that possibly produce an output into V. The upper inverse of V is the set of inputs that necessarily produce an output belonging to V according to present knowledge. If the knowledge is very scarce, the upper inverse image may be the empty set. Definition 7. The inverse images of a fuzzy set B, defined over the output universe set, can be written as: A

12 12 µ ( y) = sup µ ( y) F ( B) y F( u) µ + ( y) = inf µ ( y) F ( B) y F( u) where F and F + stand for the lower and upper images, respectively. Definition 8. A rulebase is consistent with a multivalued mapping F if the solution to the equivalent equation set includes the output of F for all u U. Thus, the former definition states that a consistent rulebase is at least as vague as the mapping it tries to model. With these definitions, it can be proved that IF rules are consistent with the ambiguous mapping if the image of the antecedent is a subset of the consequent, thus generalising Lemma 2 Note: In the multivalued case, in general, If and Only If rules will usually be inconsistent (the only proper antecedent is the whole input universe) due to vagueness: the inverse image of the image of a set A is greater (vaguer) than the original A, so Lemma 1 cannot be generalised. B B Inversion of multivalued mappings represented by rules (inequalities). If a multivalued mapping is defined by a consistent rulebase, the set of equivalent equations can be solved for the input u if the output y is assumed known (a target reference to be reached, in some control situations). The solution of the rule equivalent equations is the lower inverse image of a given point y. The lower inverse image of an interval is the union of the lower inverse image of all its points. If the multivalued mapping fulfils certain monotonicity conditions, the calculation needs to be done only at the extreme points of the interval. The upper inverse images are obtained by complementation (the set of values that necessarily produce an output inside a set O is the one that does not possibly produce an output outside O). Note: The proofs for all these cases, as well as further details, are omitted for brevity and to keep clarity at the main points of the exposition. Only some of the most important and intuitive concepts for control purposes have been presented here (see [12] for details and proofs). For a comprehensive review of multivalued mapping theory the reader is referred to [16].

13 13 3. INFERENCE ERROR In the previous section, inference over coherent rulebases and coherent rule transformations have been established. In some cases an approximate solution has to be found despite the inconsistency due to modelling or measuring errors. This section undertakes the problem of defining approximate inference with incoherent rulebases in an optimum way, i.e. keeping as close as possible to the previous results and intuitive interpretations. To allow inference with incoherent rulebases, a cost index is defined. It assesses how many rules are in contradiction when a certain candidate conclusion is assumed. The ideal inference conclusion will be the minimum-cost one, in a similar way to, for example, least-squares solutions of equation systems. The cost index is a positive inference error function ε:[0,1] [0,1] [0,1]. By definition: ε ( x ) = 0 {x is a solution of the equation}. Definition 9. The inference error (IE) functions for the IIF rule (3) and the IF rule (2) may be defined as: p IIF Rule: ε IIF ( AB, )= B A (8) 0 A B IF Rule: ε IF ( AB, ) = p ( A B) A> B (9) where p is an arbitrary positive parameter, whose reference value is defined as p=1. Higher values of p accentuate big errors and lower ones do the opposite. These error functions generalise both the binary contradiction measure shown in table 1 and the rule equation definitions (def. 1), so that if and only if the inference error here defined is zero, the equivalent rule equations are fulfilled. If the error is not zero, it provides a "distance" to the equation fulfilment. The particular form of that generalisation satisfies some interesting properties, over the whole product set [0,1] [0,1], not just over the coherent subset (either A=B or A B): - ε IIF (A,B) = ε IIF (NOT(A),NOT(B)) = ε IIF (B, A) - ε IF (A,B)+ε IF (NOT(A), NOT(B) )=ε IIF (A,B) - ε IF (A,B)=ε IF (NOT(B),NOT(A)) - ε IIF (1,B)=ε IF (1,B), ε IF (0,B)=0, ε IF (A,1)=0

14 14 The presented error functions are a graded measurement of contradiction, in the same way that fuzzy sets are a gradation of membership: a candidate conclusion B can be either coherent with premise A and a rule (ε=0), fully contradictory (ε=1) or somewhere in between. If the inference error equations are combined with fuzzification on antecedent and consequent, the following fuzzified error functions ε:u V [0,1] are obtained (superscripts a and c refer to antecedent and consequent MF, u U, y Y): ε (, uy) = µ () u µ () y IIF a c p ε IF a c 0 µ ( u) µ ( y) ( uy, ) = a c p a c ( µ ( u) µ ( y) ) µ ( u) > µ ( y) (10) Equations (rules) are thus substituted by an equivalent inference error function. A rulebase is represented by a global error function aggregating the individual rule error functions: N ε( x) = φε ( x) i= 1 i i (11) where N is the total number of rules, ε i are the individual rule error functions, φ i correspond to rule confidence levels, and x U 1 U 2... V 1 V 2... contains all the premises and conclusions. The constant confidence levels could be generalised to confidence functions φ i (x) by asserting different validity to rules depending on measured or previously concluded variables. Definition 10. Given a rulebase R and an inference error function ε(x) defined on it, for some given premises, the ideal inference is the set of conclusions minimising ε(x). If the rulebase is coherent, this minimum is zero. In the general case, minimising inference error is equivalent to minimising the number of rules in contradiction (weighted by their confidence level). The ideal inference finds a "central" conclusion given each rule s conclusion set. It is calculated on the basis of the conceptual distance (IE contradiction measure), not on the geometric one. Although, in a general case, the actual implementation of ideal inference might require involved search operations, piecewise linear consequent MF produce piecewise linear inference error curves, so successful implementation of ideal inference is feasible with low computer overhead. A careful arrangement of MF also enables common inference-defuzzifying algorithms (as centroid ones) to provide zero IE conclusions (see [14]).

15 15 In order to see how this works, let us consider a simple example. Define the MF s (for input and output) as shown in figure 1, labelled as Negative, Zero, Positive. Assume that there is only the rule 1: IIF u is Zero Then y is Zero in the rulebase. First, consider a binary case (i.e., u,y {-1,0,1}), expressing the inference errors as a truth table (table 2). For example, for premise u = -1, y = {-1,1} is the conclusion set. Conversely, the conclusion y = 1 might have been produced due to the hypothesis set u={-1,1}. In the general fuzzy case, the rule IIF u is Zero Then y is Zero produces an inference error given by ε(u,y)= Zero(u)-Zero(y) as depicted in Fig.2(a). For the premise u=-0.5, the error curve for y is: ε(y)= 0.33 Zero(y), i.e., an u constant cut on the represented surface, fig 2.b. Error equations prove the intuition that if u is Not Zero it may be either positive or negative, so incompleteness (ambiguity) arises. Note that, as expected, the plot is piecewise linear. Another example may be the following contradictory rulebase (if u is Zero): IIF u is Positive Then y is Positive IIF u is Negative Then y is Negative IF u is Zero Then y is Not Zero The error plot (Fig. 3) detects this contradiction. No contradiction arises if u is not significantly Zero, but error 1 is present in that case. These shapes would have changed accordingly if different confidence levels were introduced Validation of fuzzy systems. A fuzzy system can be considered as a black-box with a set of input and output physical variables. Internally, some operations are carried out. These are usually summarised in the following modules: fuzzification, inference over a set of rules (individual rule inference and aggregation of conclusions), and defuzzification. Extension of fuzzy systems allow either inputs or outputs to be fuzzy (linguistic) via extended fuzzy reasoning. Some details on the various options to implement conventional and extended fuzzy systems can be found in, e.g. [2], [17].

16 16 The coherence mismatch index defined as inference error (def. 10) may be used to generalise the definitions in section 2.1. It expresses how many rules are in contradiction when testing a rulebase as a function approximator. It may be also used in the opposite way: given a certain fuzzy system, the knowledge encoded in it may be recovered by finding which function f makes its coherence mismatch as low as possible. If f is generic, the sought function is the result of ideal inference. Notwithstanding, if f is restricted to be the result of a classical fuzzy inference technique, a validation method is thus defined: If f(x) is the output of a certain fuzzification-inference-defuzzification process over a coherent rulebase, the coherence mismatch index is a quality measure of the algorithm valitidy. Thus, the ideal inference (error minimisation) may not need to be actually implemented in a particular fuzzy system. If the conclusions of the actually programmed algorithm (for example, centroid defuzzification) have a low inference error, its validity can be asserted. Nevertheless, this validity depends on particular rules and MF arrangements [14]. For alternative views on fuzzy systems validation, see, for instance [19], [7]. 4. MODELLING AMBIGUOUS FUNCTIONS The resulting inference error curves can quantify the "ambiguousness" of the knowledge represented by a rule base, if the set of consistent solutions of the equations (zero error zone) is thought of as a multivalued mapping. The smaller the conclusion set, the more precise the knowledge is. As an example of function modelling, various learning algorithms based on the concepts above are presented Algorithm for determining shapes of antecedents or consequents. This algorithm is based on the expression of the image of a fuzzy set under a multivalued mapping (def. 6), i.e. the image is the maximum of the MF of inputs that may produce a certain y. If descriptive enough data are available, the following algorithm produces an approximation of the image of a fixed antecedent. If that image is used as the consequent of a rule, the consistency with the data is guaranteed. Given a point (u, y) from a training set, and an antecedent µ A, if the rule equivalent equation is to be fulfilled at point y, then the consequent membership value has to be greater than µ A (u).

17 17 After setting a mark in the output space at (y, µ A (u) ) for each point in the training set, any fuzzy set that encompasses all marks can be a coherent conclusion for the antecedent A, given the actual training set, in the sense of the previously presented definitions. The application of the algorithm to fuzzy modelling of imperfectly known or noisy functions is straightforward. Example: Characteristic curve of a valve. A target function such as q=k height 1/2 +a, in the interval 0.1<q<0.9, is going to be learnt, from a data set obtained from three different situations (Fig 4). A triangular add-1 fuzzy partition is defined in the height input domain for all cases, as shown in fig. 5. Antecedents are named (Ah Bh Ch Dh Eh). The problem is to determine which are the most coherent consequent fuzzy sets for them. Interpreting consistency via the extension principle and the previously defined inference error, the following consequent plots are obtained: In the case (1), as data are not noisy (nonambiguous function), and the function is monotonous the fuzzy sets have an add-1 fuzzy partition shape (slightly non-triangular due to the nonlinearity of the function) depicted in figure 6. These are obtained by direct application of the extension principle to a bijective function (i.e., µ ( )( y) = µ ( u) ). f A A In the second case, a slightly imprecise situation is present (Fig. 7) giving the approximately minimum consistent consequent fuzzy sets (i.e. proper consequents, in this case approximated to trapezoidal ones for simplicity) shown in Fig 8. With the noisiest data, the output fuzzy sets are bigger and overlapping to a greater extent (they have been represented separately, in figure 9, as the points from different sets get too mixed up if grouped in just one plot). The application of the ideal inference to the consequents presented in fig. 9 leads to conclusions with zero inference error over an interval, thus having learnt both the function and its ambiguousness. The algorithm produces wider and more overlapping sets according to the amount of vagueness in the data set.

18 18 Note: the proposed algorithm produces a model which is consistent with all input/output data. But some of them may be experimentally not valid (for example, outliers), and others may be partially valid (subject to noise, etc.). So the model might be in some cases too general (the extreme, always coherent, would be concluding the whole output universe). Excessive vagueness has a cost in terms of less prediction power and more inadequate control actions in model-based controllers. In particular a greater closed-loop error bounds in fuzzy-inverse feedback linearisation and greater control discontinuity in fuzzy-inverse sliding control will result. So, in some cases, the loss of expressivity of too ambiguous models must be balanced against their contradiction with experimental data. The kind of decisions to be made in balancing contradiction, excessive vagueness, outliers and points possibly obtained by selecting an antecedent without the adequate geometry ( improper) to represent the mapping, are illustrated in fig. 10. Experimental data should be validated, for example by attaching a filtering coefficient v j to each pair (u j, y j ) so that outliers have v j =0. c Thus, in a general case, given a set of parameterized consequent MFs µ ( y θ), and a set of validated experimental data ( uj, y j, v j) a compromise should be reached by optimising a cost index such as: i a c J ( θ ) = v C( µ ( u ) µ ( y θ)) i j i j i j 2 (12) where wc x x Cx ( ) = wa x x is a cost function with maybe different weights for contradiction and ambiguousness, The contradiction weight should be greater for good data matching. A dual algorithm can be easily thought of by fixing the consequents and setting the appropriate marks on the antecedents universe so that any consistent antecedent must encompass no mark at all. > Algorithms for fixed sets. Let us now consider the case of choosing the best pairings given predefined fuzzy antecedents and consequents. Besides the ideas from previous sections in this paper, some other possibilities can be used

19 19 for that problem as, for example, heuristic approaches based on frequency histogram. It can be outlined as follows: for each experimental data (u k,y k ) k=1...n, given an antecedent MF µ a i (), the evidence for each k a c consequent j=1,...n c is calculated as: h = T( µ ( u ), µ ( y )), where T is a T-norm, and a n c n c evidence correlation matrix is formed: ij i k k k k F = F = min( h, h ) jl jl k= 1... N k= 1... N so that the quantity (relative frequency) and the reliability indexes can be defined as: j k ij il q ij = k ki,, j h h k ij k ij r ij = F is s= 1... nc s j F jj F ss Negative reliability would indicate that the rule has more counterexamples (greater evidence for another rule) than positive examples. Both indexes can be used in determining heuristics for rulebase reduction or supervision. Correlation is used in r ij to distinguish the evidence of several rules on the same experimental point (overlapping consequents, redundancy) from different points producing evidence for different consequents ( contradiction). Contradiction-based approach: The algorithm to be presented is based on the contradiction minimisation concept presented in section 3. If antecedents and consequents are fixed (perhaps set up by either an expert or a previous clustering algorithm) then, by calculation of the inference errors over the data set, the best pairings can be obtained in the sense that for a given antecedent the consequent to be chosen is the one that minimises the contradiction. The data points induce a cumulative inference error for each pairing given by: * a c Eij = max( µ i ( uk) µ j( uk ), 0) k= 1,... N p so that for a fixed i, the lower error E * will pinpoint the less contradictory consequent. The inference error formula assumes that rules are thought of in their IF version so that the expression used is the second one in (10). Error values of consecutive antecedents and consequents can show design tips for changing the shape of the MF and improving the accuracy of representation.

20 20 Example. Let us consider the previous application. A trapezoid (non triangular, as suggested by the ambiguous function to be modelled) partition is defined on the output domain and a one-pass coherence test will determine which is the best grouping for the rules. The user-generated consequents to be tested are those depicted in Fig 11. The Lukasiewicz union - µ A B = max(1, µ A +µ B ) - of each set and its neighbour is also tested for consistency with the function, so that a total of 9 sets are used to compute the errors. The result of the Lukasiewicz unions is also a (wider) trapezoidal MF. The names of those sets will be: A, B, C, D, E, AB, BC, CD, DE, where sets AB, BC, CD, DE denote A OR B, B OR C, C OR D and D OR E, respectively, with the aforementioned conorm. The test is done for both IF and IIF rules, to see whether a fuzzy antecedent is a sufficient and necessary condition or not, for any of the 9 consequents. The necessary software to implement the algorithm produces the data output presented in Table 3 (for noiseless-data). From the inference error values, the most perfect pairings are: Height is Ah Flow is A OR B(in most cases only B) Height is Bh Flow is B OR C Height is Ch Flow is C OR D Height is Dh Flow is D Height is Eh Flow is E In the example, total activation is used, i.e., the sum of all firing strengths of the antecedents of IF rules. This differentiates the fact that from the data set, low cumulative inference errors can be originated either due to good consistency of the rule or due to the fact that an antecedent only fires in very few situations (if input data are not uniformly distributed). Note that IIF rules fire even when their antecedent is false. Notwithstanding, the IIF error figures have been also divided by the total activation to make easier the comparisons with IF rules. Nevertheless, total activation is not essential as it is antecedent dependent, hence it does not help in choosing particular consequents. Its use just makes the tables a bit more readable. Due to the inaccuracy of the set design, the more precise IIF rules have a much higher cumulative inference error. Also the equally-spaced candidate consequents are not well designed, as for low u s, they should be more spaced. Note that error is greater for low y s. Moreover, the set A should be removed and

21 21 B should be made a bit larger at its left side, considering the low error of B for antecedent 1. These facts stem from the nonlinearity of the target function to be modelled. If the noisiest data are used, the results for the same sets are presented in Table 4. From this data the results are similar, but the higher uncertainty makes necessary to change the two last rules to account for the less precise knowledge: Height is Ah Flow is A OR B(in most cases, B ) Height is Bh Flow is B OR C Height is Ch Flow is C OR D Height is Dh Flow is D OR E (in most cases, D) Height is Eh Then Flow is D OR E (in most cases, E) Now, a slight redesign of sets B,D,E is advised by observing the error values. Note the difference between average and worst-case behaviour (maximum error). 5. FUZZY FEEDBACK LINEARISATION CONTROL. The previous ideas about multivalued function modelling and inversion can be applied to control calculation via inverse models. Let us have a process described by a pair of equations : x = f( x, u) y = h( x) with stable zero dynamics, where the state is measurable, so that by successive derivation of output an expression can be found in the form: n d y n = f * ( x, u) dt A vague multivalued rule-based fuzzy model F is assumed available, so that f *( x, u) F( x, u) = [f -, f + ], and an uncertainty bound ν is known, so that f + f 2ν. Let us express F(x,u)=F x (u) to stress the fact that u is the only unknown variable to invert the model for. Let us assume that the output and its derivatives up to the degree n-1 can be exactly calculated from the state measurements. Under these assumptions, the following lemma can be proved: Lemma 3. Any of the feedback linearisation controllers given by:

22 22 + ux ( ) F ([ wt ( ) ν, wt ( ) + ν]) x n d r wt ()= + n dt n 1 i= 0 k de i i i dt where k i are design parameters such that G(s)=1/(s n +Σk i s i ) is a stable system (target linearised dynamics), r(t) is the reference signal and e(t) is the loop error, produces a stable closed loop response with loop error bounded by ν g(t) 1 i.e. ν times the integral of the absolute value of the impulse response of G. Invertibility of the system model is required, in the sense that the inverse image is assumed to be inside the physical range of the input actuator. Proof. The signal w(t) can be calculated from measurements. The uncertainty bound ensures the existence of the upper inverse image of the interval. Being u * one of these inputs in F +, it ensures that necessarily w(t)-f * (x,u * ) <ν, i.e. f * (x,u * )=w(t)+ε(t), ε(t) <ν. By substitution in the process equations, the final closed loop error dynamics is e (n) +Σk i e (i) =ε(t) and the boundedness of ε ensures the boundedness of the error in the terms stated in the lemma. The invertibility assumption, in limited power situations (bounded input) usually implies a bound on the n-th derivative of the reference signal. Additional restrictions appear in unstable systems, related to the concept of recoverable states. Example: for a nonlinear mechanical system described by an equation y =f(u-y), the function f is modelled via a fuzzy system as described in the previous algorithms. The antecedents are chosen as the cartesian product of a 11 set partition in both u and y. A much lower number of rules would have been necessary if the chosen antecedents were proper (i.e., defined directly over u-y), but this geometric fact has been assumed as unknown. For example, the function and the ambiguous model for y=0 are plotted in figure 12, where the near-zero inference error zone and the medium-high error one are depicted in black and gray, respectively. If sudden changes appear at the reference, the initial condition effects (due to impulse terms in w(t)) make this lemma only valid from some time after the last reference jump. An approach based on calculating the center point of the lower inverse image will be actually programmed because although it is not accurately representing the assumptions of the lemma the

23 23 programming effort is much lower and the existence of the uncertainty bound can be observed in figure 12. Fortunately, the uncertainty around the equilibrium point y=u is small so a low steady state error can be expected. The results (output and control action) are shown in figure CONCLUSIONS The use of consistent fuzzy rulebases as function approximators in modelling and control is discussed via an equation representation of the rulebases and a general subsethood interpretation, in which rules are understood as examples of a function to be modelled, obtained by applying that function to some fuzzy sets. The concepts of proper rules and antecedents formalise some intuitive notions. These concepts can be generalised to multivalued mappings. By defining an error function related to the rule equations, ideal inference is defined in terms of the minimisation of the global rulebase contradiction. Rule confidence levels stand as weights on the referred error function. Different learning algorithms based on the extension principle and error functions have been presented to fit noisy experimental data to a fuzzy system with prescribed antecedents. Algorithms to select the best antecedent-consequent pairings for given sets are also discussed. A feedback linearisation control application based on consistent fuzzy model inversion of a multivalued mapping has been reported. The methods here presented pave the way to automatic knowledge acquisition supervision: detection of contradictions, rulebase simplification (elimination of redundant rules) and preservation of readability, providing tools for fusion of rulebases from different experts and learning approaches. 7. REFERENCES [1] P. Albertos, Fuzzy Controllers, in : L. Boullart, A. Krijsman, R. A. Vingerhoeds., Ed., Applications of Artificial Intelligence in Process Control. (Pergamon Press, 1992) [2] P. Albertos, M. Martinez, J.L. Navarro, F. Morant, Fuzzy Controller Design: A methodology. Proc. of IEEE Conference on Control Applications, Vancouver (1993) Vol. 1, [3] C. Berge, Espaces topologiques et functions multivoques. (Dunod, Paris, 1966).

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