SOME APPROACHES TO FUZZY MODELING AND CONTROL OF DYNAMIC PROCESSES Eliezer Colina Morles

1. SOME APPROACHES TO FUZZY MODELING AND CONTROL OF DYNAMIC PROCESSESEliezer Colina Morles WSEAS INTERNATIONAL CONFERENCES MÉRIDA, DECEMBER 2010

2. Agenda • Some aspects on fuzzy modeling • Fuzzy modeling of dynamic systems • Fuzzy clustering algorithms • Fuzzy model based control

3. Some aspects on fuzzy modeling Commonly used modeling approaches • Phenomenological or mechanistic models, based on physical considerations • Empirical data-driven models • Qualitative models White-box models Grey-box models Black-box models Fuzzy models Knowledge-based models

4. Some aspects on fuzzy modeling • Rule-based fuzzy models types • Linguistic fuzzy models (Mandani):This type of model is useful to capture qualitative information available in the form of “if-then” rules expressed as • Rk : If x is Ak them y is Bk, k = 1, 2, ...,K • Rkdenotes the k-th rule.Kis the total number of rules • x E X represents the fuzzy system input, (antecedent variables). • y EY represents the fuzzy system output, (consequent variables). • Akand Bk are fuzzy sets, described in terms of their membership functions • μAk(x) →[0, 1] and μBk(x) →[0, 1], respectively. R. Babuska. Fuzzy modeling and identication, phd dissertation. Deft University of Technology, The Netherland, 1996.

5. Some aspects on fuzzy modeling • Rule-based fuzzy models types • Linguistic fuzzy models (Mandani) • The rule base R = {Rk| k = 1, 2, ...,K}and the fuzzy setsAandBare the • knowledge base of the linguistic model. • A fuzzy inference method for obtaining the fuzzy system output may be • summarized as follows: • Each rule Rk : X ∗ Y → [0, 1] specifies a fuzzy relation defined as • μRk(x, y) = I(μAk(x), μBk(x)) • where the operator “I” is a fuzzy implication, interpreted in terms of a • t-norm conjunction operator.

6. Some aspects on fuzzy modeling • Rule-based fuzzy models types • Linguistic fuzzy models (Mandani) • The K rules are combined to form a global relation “R”, (i.e conjunction of • individual Rk rules). • Given an input x is A’ and the relation R, the corresponding fuzzy output • is expressed as • B′ = A′ ◦ R, • where “°” denotes the “sup-t” composition. • The most common used “t-norn” composition is the minimum, and • therefore the above fuzzy output is interpreted as • μB′ (y) = maxxminx;y(μA′ (x), μR′ (x, y))

7. Some aspects on fuzzy modeling • Rule-based fuzzy models types • Linguistic fuzzy models (Mandani) • If the implication “I” is selected as the minimum conjunction operator, the • compositional inference rule is the classical Mandani MaxMin Inference • Rule: • For each rule k, the level of compliance of the antecedents is evaluated as • βk = Mink (μA1k(x1), μA2k(x2) ,…..,μAnk(xn)) k = 1, ...,K • For each rule k, the fuzzy set output membership value is calculated using the t-norn as • μB′k (y) = Mink (βk ,μBk (y)) • The total fuzzy set membership value is obtained by aggregation, using the Max operation, as • μB′ (y) = Maxk=1;:::;K(μB′ k (y)) Difuzzyfication by COG or MOM

8. Some aspects on fuzzy modeling • Rule-based fuzzy models types • Relational fuzzy model: It is considered an extension of the fuzzy linguistic model of Mandani. • If Aj and B denotes the linguistic terms of the antecedents and consequent • variables, respectively • Aj = {Ajk| k= 1, 2, ...,Nj}, j = 1, 2, ..., p, • B= {Bk| k= 1, 2, ...,M}, • the rule base may be represented as an ordinary relation R between the • linguistic terms in the antecedent and consequent as • R : A1 × A2 × ... × An × B → 0, 1 • If A = A1 × A2 × ... × Anis the cartesian product space of the • linguistic terms of the antecedent, then • R : A×B → {0, 1} W. Pedrycz. Fuzzy Control and Fuzzy System. 2 edition, 1993

9. Some aspects on fuzzy modeling • Rule-based fuzzy models types • Relational fuzzy model • A fuzzy relational model is obtained by generalizing R as fuzzy relationship • R : A × B → [0, 1]

10. Some aspects on fuzzy modeling Rule-based fuzzy models types 3. Linguistic singleton fuzzy model: A special case of the linguistic model is obtained when the consequent terms Bk are reduced to singletons. The k-th rule sintax is as follows: Rk : If x is Ak them y = ck, k = 1, 2, ...,K The defuzzification COG method reduces to a mean calculation: The consequent parameters may be easily estimated by least square tecniques using measurement data.

11. Some aspects on fuzzy modeling Rule-based fuzzy models types 3. Linguistic singleton fuzzy model: The singleton fuzzy model belongs to a class of approximators of functions, called basis function expansion, which can be expressed as follows: Under certain conditions it is posible to obtain a multilinear interpolation among the rules consequents, and the fuzzy singleton model may be exactly inverted.

12. Some aspects on fuzzy modeling Rule-based fuzzy models types 3. Linguistic singleton fuzzy model: The input-output multilinear mapping capabilities of the fuzzy singleton model may be illustrated as follows Clearly, a fuzzy singleton model may represent any linear mapping of the form

13. Some aspects on fuzzy modeling Rule-based fuzzy models types 4. Takagi-Sugeno fuzzy model: Consequents are real-valued functions. The i-th rule sintax is expressed as: Ri : If x1 is Ai1 and ... and xs is Ais them yi = fi(x1, x2, ..., xs); i = 1, 2, ..., c M. Sugeno and T. Yasukawa. A fuzzy logic based approach to qualitative modeling. IEEE Transactions on Fuzzy Systems, pages 710, 1993.

14. Fuzzy modeling of dynamic systems Different types of fuzzy models can be used to approximate the state transition functionof a dynamic process. Since the state of a process is usually not measurable, input-output modeling is applied as a viable alternative solution. A common method is the NARX model, which may be represented as y(k+1) = f(y(k), y(k−1), ..., y(k−ny+1), u(k), u(k−1), ..., u(k−nu+1)) A singleton fuzzy model can consist of rules as follows: Ri: If y(k) is Ai1 and y(k − 1) is Ai2 and...and y(k − n + 1) is Ain and u(k) is Bi1 and u(k − 1) is Bi2 and...and u(k − m + 1) is Bin them ˆy(k + 1) is ci I. Leonaritis and S. Billings. Input-output parametric models for non-linear systems. International Journal of Control, pages 303-324.

15. Fuzzy modeling of dynamic systems The system dynamic behavior is carried out by dynamic filters added to the fuzzy system Fuzzy models can approximate any continuous function with any degree of accuracy.  L. Wang. A course in Fuzzy Systems and Control. 1 edition, 1997.

16. Fuzzy modeling of dynamic systems • In designing fuzzy models, must distinguish two main parts: the structure and parameters. • Structure determines the flexibility of the model.Once selected, we estimate the values of the parameters for the model to reproduce the behavior described by the measured data. • In fuzzy models, structure selection involves the choice of : • Input-output variables. (process order) • Rules structure. (among the 4 types mentioned before) • Number and type of membership fuctions for each variable. (granularity of the model) • Inference mechanism, connective operators and difuzzification method. • The adjustable parameters are the rules and the antecedent and consequent membership fuctions.

17. Fuzzy modeling of dynamic systems Singleton fuzzy modeling example. Oil production separator example Selected input variables: Current oil level in the separator y(τ) Valve position at the production output u(τ). Output variable: Oil level at the next sampling instant y(τ+1).

18. Fuzzy modeling of dynamic systems Singleton fuzzy modeling example. Oil production separator example Four fuzzy sets were assigned for the two inputs. The table shows the resultant singleton fuzzy model. Membership fuctions for y(t) Membership fuction for u(t)

19. Fuzzy modeling of dynamic systems Oil production separator example

20. DATA CLUSTERING Fuzzy clustering algorithms • Fuzzy clustering techniques are mostly unsupervised algorithms that are used to decompose a given set of objects into subgroups or clusters based on similarity. • The goal is to separate the data set in such a way that objects (or example cases) belonging to the same clusters are as similar as possible, whereas objects belonging to different clusters are as dissimilar as possible. J.C. Bezdek. Patter recognition with fuzzy objective function algorithms. Plenum Press, New York, 1987

21. Fuzzy clustering algorithms • In ordinary clustering algorithms each input must be assigned to a single class. • The fuzzy clustering analysis relaxes this requirement by allowing gradual memberships and offering the possibility of a data set may belong to several classes simultaneously. • Most of fuzzy clustering analytical techniques are based upon the optimization of an c-mean objective function or some modification of it. M. Delgado, F. Gomez, and Martín F. A fuzzy clustering-based rapid prototyping for fuzzy rule-based modelling. IEEE Transaction on Fuzzy System, pages 223233, 1997

22. Fuzzy clustering algorithms Cluster types depend on the selection of the norm: For B=I Spherical classes For B a diagonal NxN matrix with different variances wrt axis Hyper ellipsoidal classes Hyper ellipsoidal classes with arbitrary orientation For B a different norm for each class

23. Fuzzy clustering algorithms • Steps for fuzzy models identification • Design of experiments ------------------white noise corrupted sine or step • functions with variable amplitude and • frequency. 2.Structure selection -----------------------which are the input and output variables relevant to the modeling problem. 3.Process dynamics representation -------This step transforms the dynamic system identification problem in a static regression problem.

24. Fuzzy clustering algorithms Steps for fuzzy models identification 4.Fuzzy model granularity --------------------This parameter is related to both the number of rules as to the number of linguistic terms for each variable. Clustering is used. 5.Data clustering --------------------------------With the purpose to identify possible operating regimes in the chosen data space. Clustering is used.

25. Fuzzy clustering algorithms Steps for fuzzy models identification Start with a large enough number and make connections among compatible classes with the Compatible Classes Matching (CCM) algorithm. Either 6. Selecting the number of classes ------------- Perform data clustering for different values of “c” and use the varianza criterion or dispersion criterion to validate the obtained partitions. Or 7. Initial model

26. Fuzzy clustering algorithms Oil production separator example 745 data were collected for the construction of the fuzzy model

27. Fuzzy clustering algorithms Oil production separator example The input data u (t) y (t) were grouped using Gustafson-Kessel algorithm. Initially c=4 D. Gustafson and W. Kessel. Fuzzy clustering with a fuzzy covariance matrix. Proceedings of IEEE Conference on Decision and Control, San Diego USA,1979.

28. Fuzzy clustering algorithms Oil production separator example The CCM algorithm was used to check on the compability among classes. There were two compatible classes.

29. Fuzzy clustering algorithms Oil production separator example The Gustafson-Kessel algorithm with c=3 produces the following results

30. Fuzzy clustering algorithms Oil production separator example To determine the fuzzy model, the projection method of the fuzzy partition matrix was used on each of the antecedent variables.

31. Fuzzy clustering algorithms Oil production separator example • The consequent parameters were obtained using the recursive least square algorithm. The resulting rule base is as follows: • If u(τ ) is closed and y(τ ) is low them ˆy(τ + 1) = −0,0012u(τ ) + 0,9629y(τ ) + 0,2845 • If u(τ ) is half open and y(τ ) is normal them ˆy(τ+1) = −0,0023u(τ)+1,0181y(τ ) + 0,072 • If u(τ ) is open and y(τ ) is high them ˆy(τ + 1) = −0,0019u(τ ) +0,9847y(τ ) + 0,1554

32. Fuzzy clustering algorithms Oil production separator example The TS fuzzy model obtained was validated with 170 different input / output data

33. r u y M-1 M x1 Fuzzy Model x2 y xn x1 y Inverse Model Inverse Model y x2 x2 x1 xn xn A B Fuzzy model based control When an ideal model M mapping the control actions u to the system outputs y is considered, the control actions may be simply given by , where r is the reference to be followed Some fuzzy models structures can be exactly inverted and these inversions may be used for control purposes.

34. Fuzzy model based control Let us assume that a singleton fuzzy model of a certain process is available.The k-th rule of the model is given by Theorem: Let the process be represented by the singleton fuzzy model given above, with the weighted-mean defuzzification method. Further, let ∑μXi(x)=1, and ∑ μUj(u)=1. At a certain time the process is at the state x(), and the inverse of the singleton model is given by the fuzzy rules:

35. ... CM-1 CM C1 C2 C3 ... ... c1() c2() c3() cM-1() cM() Fuzzy model based control Where Cj are fuzzy sets that form a partition as indicated in the following figure The cores cj of the fuzzy sets Cj are given by The inference and defuzzification of the rules in equation is accomplished by the fuzzy-mean method

36. Fuzzy model based control The following figure shows the adaptive IMC architecture.

37. SOME APPROACHES TO FUZZY MODELING AND CONTROL OF DYNAMIC PROCESSES Thank you