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Application of a measure of proximity to fuzzy control algorithms
330
European Journal of Operational Research 59 (1992) 330-332 North-Holland
Theory and Methodology
Application of a measure of proximity to fuzzy control algorithms C o s t a s P. P a p p i s
Department of Mechanical Engineering, University of Patras, Patras, Greece Received December 1989; revised January 1991
Abstract: A decision procedure is presented concerning the selection of the control action when using fuzzy algorithms. The procedure is based on the notion of approximation of fuzzy values and is intended to include in the decision to be taken considerations about the form and other features of the fuzzy set incurred when applying the inference rule to the fuzzy algorithm.
Keywords: Fuzzy control; fuzzy algorithms; measure of proximity
Introduction Fuzzy control algorithms have been extensively used to model process controllers which are fuzzy in nature (e.g. [1,3,5]). Such algorithms usually consist of a set of fuzzy control statements which are derived from either an operator's experience a n d / o r a control engineer's knowledge, or from the fuzzy model of the process controlled or from an operator's control actions [6]. These statements form a protocol of actions, IF A 1 THEN B 1 ELSE
where A k and B k are fuzzy subsets of U and V, respectively. R is complete if the set A = {A k [k = 1. . . . . n} is an a-cover of U, that is, if for every element u of U there exists an Ak such that I~Ak(U) > 0.5, where lZAk(U) is the grade of membership of u in A k [2]. Alternatively, R may be a fuzzy relation defined on U × V with which each pair (u, v) is assigned a n u m b e r tZR(U, V) which measures the truth value of the statement 'if u, then v'. R is complete in this case if m a x ( / z n ( u , v ) ) _ 0.5
for all u.
v
iv A 2 THEN B 2 ELSE
Execution of fuzzy control algorithms
EESE IF A n THEN
n n
which are usually modelled as a set R of fuzzy conditional statements (FCS),
R={A k ~Bklk=
1 . . . . ,n}
Correspondence to." C.P. Pappis, Department of Mechanical Engineering, University of Patras, 26500 Rio, Patras, Greece.
Let R be a complete set of FCS and tZAk(U) and /ZBk(V)be the grades of membership of u and v in A k and Be, respectively, A k and B k being fuzzy subsets of U and V. If A ' corresponds to a state of inputs, then the control action B ' derived from the fuzzy algorithm is given by
B'=A'oR
0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
C.P. Pappis /Application of proximity measure to fuzzy control algorithms
where ' o ' denotes composition. B' depends on the interpretation of the set R and the operation o. One possible interpretation of the latter is maxmin composition, whereas R can be interpreted as the union of the Cartesian products A k × B k, that is,
R= ~ ( A k × B k ) . k=l
Alternatively, as is noted above, R may be formed directly from the truth values of the statements 'if u, then v' for every pair (u, v). In either case, the fuzzy subset B ' of V which is inferred from the composition A ' o R implies the control action to be undertaken. This is generally a (nonfuzzy) value vk of V which is usually selected to be the one corresponding to the maximum grade of membership in V, i.e.
Uk:IZB,(Uk) =max(IZB,(V))
for all v.
It is noted, however, that by using this selection rule all other information inherent in B', such as dispersion and distribution of membership values, is ignored. One possible way to (partially) avoid this is by using the notion of approximation.
Proximity of fuzzy values
Let B, B' be fuzzy subsets of V and b and b' be the corresponding grades of membership vectors. By 11b - b' II we denote the number max(I bi - b[ I), i.e., the maximum of the absolute values of the differences between all corresponding elements of b and b'. B and B' are said to be approximately equal (and this is denoted by B --- B ' ) iff, given a small nonnegative number e, we have l[ b - b' II -< e. The number e is said to be a proximity measure of B and B'. It can be proven that approximation is preserved under the operations of union, intersection and max-min composition [4].
Selection of the control action
Suppose that upon execution of a fuzzy control algorithm described by R, a fuzzy subset is in-
331
curred corresponding to the control action to be taken. By using the notion of approximation, a procedure for selecting the (nonfuzzy) control input may be defined as follows: Step 1. Identify all possible control actions in terms of fuzzy sets B i, i = 1 . . . . . n. This may be done, e.g. by modelling an operator's experience. Step 2. Infer the fuzzy control action B' using the fuzzy algorithm, the input to the system under control and an appropriate inference rule, e.g. the maxmin compositional rule of inference. Step 3. Find B k such that
II b'
- bk II =
min( II b '
-
bi
II).
Step 4. Apply the (nonfuzzy) control action which corresponds to Bk, e.g. the control input which corresponds to the biggest grade of membership to B k. Clearly, the above procedure takes account of the extent to which the actual control action taken approximates the fuzzy control action inferred from the fuzzy algorithm upon its composition with the fuzzy input. Example. Let a fuzzy algorithm be described by a fuzzy relation R on U × V, with a relation matrix R given by
R=
0 0.3 0.5 0.7
0.3 0.5 0.7 1.0
0.5 0.7 1.0 0.7
0.7 1.0
0.7 0.5
1.0 0.7 0.5 0.3
and a fuzzy set A of U with a membership vector a given by a=(0.3
0.6
0.8
1.0).
Using the maxmin compositional rule of inference, a fuzzy subset B of V is incurred with a membership vector b given by b=(0.7
1.0
0.8
0.7
0.6).
According to the procedure described above suppose that the possible control actions in terms of fuzzy sets of V have the following membership vectors: ba=(1.0
0.6
0.4
0.2
0),
b2=(0.4
0.6
1.0
0.6
0.4),
b3=(0
0.2
0.4
0.6
1.0).
C.P. Pappis / Application of proximity measure to fuzzy control algorithms
332
W e have II b - b I II = 0.6, [2]
II b - b2 II = 0.4, II b - b3 II = 0.8.
[3]
Furthermore, rain( II b - b, II) = 0.4
for i = 2.
T h e r e f o r e , the control action w h i c h corre'sponds to B 2 s h o u l d be applied. T h e e l e m e n t of V with the biggest grade of m e m b e r s h i p in B 2 is v 3. N o t e that the respective e l e m e n t of B is v 2.
References [1] Mamdani, E.H., Ostergaard, J.J., Lembessis, E., "Use of fuzzy logic for implementing rule-based control of indus-
[4] [5] [6]
trial processes", in: H.J. Zimmerman, L.A. Zadeh and B.R. Gaines (eds.), Fuzzy Sets and DecisionAnalysis, North Holland, New-York, 1984, 429-445. Pappis, C.P., "On a fuzzy set-theoretic approach to aspects of decision making in ill-defined systems", Ph.D. Thesis, University of London, 1976. Pappis, C.P., and Mamdani, E.H., "A fuzzy logic controller for a traffic junction", IEEE Transactions on Systems, Man, and Cybernetics 7/10 (1977) 707-717. Pappis, C.P., "Value approximation of fuzzy systems variables", Fuzzy Sets and Systems 39 (1991) 111-115. Sugeno, M. (ed.), Industrial Applications of Fuzzy Control, North Holland, New York, 1985. Takagi, T., and Sugeno, M., "Derivation of fuzzy control rules from human operator's control actions", in: Proceedings of the Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseille, 1983, 5560.
European Journal of Operational Research 59 (1992) 330-332 North-Holland
Theory and Methodology
Application of a measure of proximity to fuzzy control algorithms C o s t a s P. P a p p i s
Department of Mechanical Engineering, University of Patras, Patras, Greece Received December 1989; revised January 1991
Abstract: A decision procedure is presented concerning the selection of the control action when using fuzzy algorithms. The procedure is based on the notion of approximation of fuzzy values and is intended to include in the decision to be taken considerations about the form and other features of the fuzzy set incurred when applying the inference rule to the fuzzy algorithm.
Keywords: Fuzzy control; fuzzy algorithms; measure of proximity
Introduction Fuzzy control algorithms have been extensively used to model process controllers which are fuzzy in nature (e.g. [1,3,5]). Such algorithms usually consist of a set of fuzzy control statements which are derived from either an operator's experience a n d / o r a control engineer's knowledge, or from the fuzzy model of the process controlled or from an operator's control actions [6]. These statements form a protocol of actions, IF A 1 THEN B 1 ELSE
where A k and B k are fuzzy subsets of U and V, respectively. R is complete if the set A = {A k [k = 1. . . . . n} is an a-cover of U, that is, if for every element u of U there exists an Ak such that I~Ak(U) > 0.5, where lZAk(U) is the grade of membership of u in A k [2]. Alternatively, R may be a fuzzy relation defined on U × V with which each pair (u, v) is assigned a n u m b e r tZR(U, V) which measures the truth value of the statement 'if u, then v'. R is complete in this case if m a x ( / z n ( u , v ) ) _ 0.5
for all u.
v
iv A 2 THEN B 2 ELSE
Execution of fuzzy control algorithms
EESE IF A n THEN
n n
which are usually modelled as a set R of fuzzy conditional statements (FCS),
R={A k ~Bklk=
1 . . . . ,n}
Correspondence to." C.P. Pappis, Department of Mechanical Engineering, University of Patras, 26500 Rio, Patras, Greece.
Let R be a complete set of FCS and tZAk(U) and /ZBk(V)be the grades of membership of u and v in A k and Be, respectively, A k and B k being fuzzy subsets of U and V. If A ' corresponds to a state of inputs, then the control action B ' derived from the fuzzy algorithm is given by
B'=A'oR
0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
C.P. Pappis /Application of proximity measure to fuzzy control algorithms
where ' o ' denotes composition. B' depends on the interpretation of the set R and the operation o. One possible interpretation of the latter is maxmin composition, whereas R can be interpreted as the union of the Cartesian products A k × B k, that is,
R= ~ ( A k × B k ) . k=l
Alternatively, as is noted above, R may be formed directly from the truth values of the statements 'if u, then v' for every pair (u, v). In either case, the fuzzy subset B ' of V which is inferred from the composition A ' o R implies the control action to be undertaken. This is generally a (nonfuzzy) value vk of V which is usually selected to be the one corresponding to the maximum grade of membership in V, i.e.
Uk:IZB,(Uk) =max(IZB,(V))
for all v.
It is noted, however, that by using this selection rule all other information inherent in B', such as dispersion and distribution of membership values, is ignored. One possible way to (partially) avoid this is by using the notion of approximation.
Proximity of fuzzy values
Let B, B' be fuzzy subsets of V and b and b' be the corresponding grades of membership vectors. By 11b - b' II we denote the number max(I bi - b[ I), i.e., the maximum of the absolute values of the differences between all corresponding elements of b and b'. B and B' are said to be approximately equal (and this is denoted by B --- B ' ) iff, given a small nonnegative number e, we have l[ b - b' II -< e. The number e is said to be a proximity measure of B and B'. It can be proven that approximation is preserved under the operations of union, intersection and max-min composition [4].
Selection of the control action
Suppose that upon execution of a fuzzy control algorithm described by R, a fuzzy subset is in-
331
curred corresponding to the control action to be taken. By using the notion of approximation, a procedure for selecting the (nonfuzzy) control input may be defined as follows: Step 1. Identify all possible control actions in terms of fuzzy sets B i, i = 1 . . . . . n. This may be done, e.g. by modelling an operator's experience. Step 2. Infer the fuzzy control action B' using the fuzzy algorithm, the input to the system under control and an appropriate inference rule, e.g. the maxmin compositional rule of inference. Step 3. Find B k such that
II b'
- bk II =
min( II b '
-
bi
II).
Step 4. Apply the (nonfuzzy) control action which corresponds to Bk, e.g. the control input which corresponds to the biggest grade of membership to B k. Clearly, the above procedure takes account of the extent to which the actual control action taken approximates the fuzzy control action inferred from the fuzzy algorithm upon its composition with the fuzzy input. Example. Let a fuzzy algorithm be described by a fuzzy relation R on U × V, with a relation matrix R given by
R=
0 0.3 0.5 0.7
0.3 0.5 0.7 1.0
0.5 0.7 1.0 0.7
0.7 1.0
0.7 0.5
1.0 0.7 0.5 0.3
and a fuzzy set A of U with a membership vector a given by a=(0.3
0.6
0.8
1.0).
Using the maxmin compositional rule of inference, a fuzzy subset B of V is incurred with a membership vector b given by b=(0.7
1.0
0.8
0.7
0.6).
According to the procedure described above suppose that the possible control actions in terms of fuzzy sets of V have the following membership vectors: ba=(1.0
0.6
0.4
0.2
0),
b2=(0.4
0.6
1.0
0.6
0.4),
b3=(0
0.2
0.4
0.6
1.0).
C.P. Pappis / Application of proximity measure to fuzzy control algorithms
332
W e have II b - b I II = 0.6, [2]
II b - b2 II = 0.4, II b - b3 II = 0.8.
[3]
Furthermore, rain( II b - b, II) = 0.4
for i = 2.
T h e r e f o r e , the control action w h i c h corre'sponds to B 2 s h o u l d be applied. T h e e l e m e n t of V with the biggest grade of m e m b e r s h i p in B 2 is v 3. N o t e that the respective e l e m e n t of B is v 2.
References [1] Mamdani, E.H., Ostergaard, J.J., Lembessis, E., "Use of fuzzy logic for implementing rule-based control of indus-
[4] [5] [6]
trial processes", in: H.J. Zimmerman, L.A. Zadeh and B.R. Gaines (eds.), Fuzzy Sets and DecisionAnalysis, North Holland, New-York, 1984, 429-445. Pappis, C.P., "On a fuzzy set-theoretic approach to aspects of decision making in ill-defined systems", Ph.D. Thesis, University of London, 1976. Pappis, C.P., and Mamdani, E.H., "A fuzzy logic controller for a traffic junction", IEEE Transactions on Systems, Man, and Cybernetics 7/10 (1977) 707-717. Pappis, C.P., "Value approximation of fuzzy systems variables", Fuzzy Sets and Systems 39 (1991) 111-115. Sugeno, M. (ed.), Industrial Applications of Fuzzy Control, North Holland, New York, 1985. Takagi, T., and Sugeno, M., "Derivation of fuzzy control rules from human operator's control actions", in: Proceedings of the Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, Marseille, 1983, 5560.