DE4303149C1 - Fuzzy logic rule matrix storage for fuzzy controller - Google Patents

Abstract

Approximate rules with n input values and output length of a set of N statements are evaluated, as the premise of each rule is an AND product of statements about the input values, and the conclusion is expressed as output values in that each rule is replaced by a numerical value Zi, then for each sampling instant t output is: Zt=sum of elements from N to i=1 of Wi(t)*Zi/sum of elements from N to i = 1 of Wi(t), where wi(t) is the validity of the ith rule. Rules with the same conclusion are stored together in a matrix with natural numbers. L assorted conclusions Z1....ZL form the L-premise matrix.

Description

The invention relates to a specified in the preamble of claim 1 Method to realize a fuzzy controller. Blurred regulators have become increasingly important in recent years attained, cf. z. B. H.-P. Preuß: "Fuzzy control - heuristic regulation by means of fuzzy logic "(Automation Technology Practice 34, Part I: pp. 176-184, Part II: pp. 239-246), B. Kosko: "Neutral networks and fuzzy systems" (Prentice-Hall, 1992, Proceed., IEEE International Conf. On Fuzzy Systems 1992).

The invention deals with a fuzzy controller having n input quantities and one output. Its actions are represented by a set {R₁,. , ., R _N } linguistic rules controlled. Each of these rules has as a premise either a linguistic statement about an input variable or an AND combination of several linguistic statements about pairwise different input variables, and as a conclusion the rule has a linguistic statement about the output variable Z of the controller. Each linguistic statement is described by a fuzzy set, which is characterized by its membership function. This setup is the most common for a fuzzy controller.

A disadvantage of the usual operation of a fuzzy controller (cf. H.-P. Preuss, a. a. O.) is that at runtime many arithmetic operations with membership functions required are. This can be especially at high Speed requirements, eg. B. in real-time operation, disadvantageous  his. The existing automation device for the technical process, which is supposed to be regulated, often does not allow the blurred regulator to be particularly to dispose of powerful hardware, as this bad in the other components could integrate the automation device and / or would be too expensive. Nevertheless, to meet the existing speed requirements, In recent times, blurred regulators have been introduced at each sampling time, the weighted summary of the previously formed by "defuzzification" of the conclusions obtained N numbers and output as the value of the output quantity. Examples include C.L. McCullough: "An anticipatory fuzzy logic controller using neural net prediction" (Simulation 58 No. 5, pp. 327-332, 1992) and M. Mizumoto: "Realization of PID controls by fuzzy control methods "(Proceed. IEEE International Conf. on Fuzzy Systems 1992, pp. 709-715).

The premise of each rule is either the AND of a linguistic statement about X and one about Y or a single linguistic statement about X or one about Y. FIG. The omission of OR-links is not a real restriction: a rule of the form "IF premise il OR premise ¨ THEN conclusion" can be equivalently replaced by two rules "IF premise İ THEN conclusion" and "IF premise İ THEN conclusion". Let μ _i and ν _{i be} the membership functions for the linguistic statement about X or Y in the premise or R _i and λ _i for the conclusion of R _i . If in the premise of R _i only a linguistic statement about X occurs, but none about Y, then formally ν _i (y) = 1 is set for all y. If only one statement about Y occurs, then μ _i (x) = 1 is set for all x. Let x _t and y _{t be} the current values of the input quantities X and Y. The controller works as follows:

In the first phase, for all i ε {1,. , ., N} the membership function λ _{i is} "defuzzified" and thereby a real number Z _{i is obtained} as an action proposal of the rule R _i (step I).

In the second phase, for all i ε {1,. , ., N} determines the degree of membership of x _t to μ _i and y _t to ν _i , and thereby the values α _i1 (t) and α _i2 (t) are obtained (step II). Thereafter, for i = 1,. , ., N the values α _i1 (t) and α _i2 (t) to the current weighting factor ω _i the rule R _i summarized by the two membership _degrees α _i1 (t) and α _i2 (t) a T-norm ("triangular norm ", see below) (step III). For i = 1,. , ., N, ω _i (t) is multiplied by the precise conclusion Z _{i of} the rule R _i (step IV). The thus weighted N action suggestions are added (step V). For normalization, the sum of all weight factors is formed (step VI). The current value z _{t of} the variable determined by the fuzzy controller is determined by dividing the value formed in step V by the value determined in step VI (step VII), that is

The most frequently mentioned T-standard in the literature so far is the minimum, another T-norm the algebraic product. T-standards are z. In C.C. Lee: "Fuzzy Logic in Control Systems: Fuzzy Logic Controller (IEEE Transactions on Systems, Man, and Cybernetics 20, pp. 404-435, 1990) described.

An advantage of this approach over the classic blurred scheme is that the work of the first phase does not have to take place at runtime, but can be executed in advance when creating the controller. to Running time then the fuzzy controller needs only the work of the second phase to perform. With the exception of step II, these are only arithmetic operations with real numbers, not with membership functions.

The method shown in Fig. 1 does not specify how the rule set is to be stored. An obvious method of storing a set of linguistic rules is e.g. In Abdelnour et al. ("Design of a Fuzzy Controller Using Input and Output Mapping Factors", IEEE Transact. Systems, Man and Cybernetics Vol. 21, No. 5, pp. 952-960):

• - The linguistic statements about the input and output variables are suitable encoded by numbering. For every input and every output the numbering starts at 1.
• - The rule set is stored in an n-dimensional data structure, where n is the number of Input variables of the controller called.

In the case of two input variables X and Y and one output Z, this data structure is a table. Be A₁,. , ., A _nX the statements about X, B₁ _,. , ., B _nY the over Y and C₁ _,. , ., C _nz the over Z. The rule "FALLS 'X = A _i ' AND 'Y = B _j ' THEN 'Z = C _k '" is stored in that in the ith row and the j-th column Number k is entered. This is done for i = 1,. , ., nX, j = 1,. , ., nY and k = 1,. , , nZ. The table therefore has nX rows and nY columns. Each entry is an integer between 1 and nZ.

This obvious method has the following disadvantages:

• - It does not specify how the controller evaluates the data structure at runtime.
• - It does not allow efficient parallelization.

A controller can evaluate the table purely serially at runtime. That can often take too much time in Claim. Remedy creates a suitable parallelization.

Will you in the just described way to save the rule set, achieve that the Regulator performs several actions in parallel, so you have for each rule, d. H. for each Table entry, provide a separate processing unit. The number of required Processing units can be large, as the following calculation example shows: A blurred Regulator has 3 inputs X₁, X₂ and X₃, and about each input 5 formulated linguistic statements. Then there are 5³ = 125 possibilities, one statement each to combine every input. Thus, 125 processing units would be required. If the number of linguistic statements is changed over an input or output variable, so you have to change the number of processing units, so the architecture of the controller. Such a change is common in the design and validation of the controller required.

It has as its object to improve the method mentioned in the introduction and shown in FIG. 1 so that it can be implemented inexpensively and efficiently, ie in such a way that the controller evaluates the set of rules as quickly as possible and uses as little space as possible, and that it is suitable for every possible number of input variables and any number of linguistic statements per input variable is applicable in order to meet the requirements in practice.

This object is achieved by the method described in claim 1 solved: Displayed as a - possibly more extensive - ruleset saved in memory space and quickly evaluated at runtime can be, even if only limited storage and computing capacity be available.  

The idea of the method according to the invention is that all rules are stored together with the same conclusion and evaluated together at runtime. Let {statement ₁,. , ., Statement _L } the set of possible linguistic statements about Z. Let k ε {1,. , ., L} Z ^{(k) is} the real number given by "defuzzification" of the membership function that identifies _k . Formula (1) is rewritten; At each sampling time, the controller works as follows:

• - First, the controller determines k = 1,. , ., L is the real number ζ _k (t), which is the current weighting of the statement _k , according to ζ _k (t) = Σ {ω _i (t) | The conclusion of R _i is "Z = statement _k } (2)
• Then the current output value z _t is calculated as a weighted average of {Z ⁽¹⁾ ,. , ., Z ^(L) }, namely according to

This modification requires only L multiplications instead of N multiplications as formula (1). In many fuzzy regulators described in the literature N is considerably larger than L.

The weighting factors ζ _k (t) determined according to formula (2) depend on the current input values, the membership functions for linguistic statements on the input variables and on the premises of the rules, but not on the conclusions. In formula (3) flow only the weighting factors {ζ₁ (t),. , ., ζ _L (t)} and the "defuzzified" conclusions {Z ⁽¹⁾ ,. , ., Z ^(L) }. This motivates the idea to store the premises of all rules together with the same conclusion. The method according to the invention provides coded storage in a matrix consisting of whole nonnegative numbers. Since there are L different conclusions, one obtains L premise matrices P ⁽¹⁾ ,. , ., P ^(L) . For each k ε {1,. , ., L} contains P ^(k) encodes the premises of all rules of the form "IF premise THEN Z = statement _k ".

To calculate at runtime ζ _k (t) according to formula (2), the fuzzy controller evaluates the matrix P ^(k) . In addition, the numbers Z ⁽¹⁾ ,. , ., Z ^(L) available.

In the following it will first be described how, according to the controller design, the premises of the linguistic rules which control the actions of the fuzzy controller are stored coded in the premises matrices (phase 1), and then it is explained how the fuzzy controller according to claim 1 working at runtime (phase 2). For this purpose, the case of two input variables X and Y is again treated as an example; it is therefore n = 2. Let Z denote the output quantity. X _X linguistic statements are possible via _X, linguistic statements about Y n _Y and Z n _Z. They are numbered from 1 to n _X or to n _Y or to n _Z. Be {μ [1],. , ., μ [n _X ]}, {ν [1],. , ., ν [n _Y ]} and {λ [1],. , ., λ [n _Z ]} the linguistic hedges for X, Y and Z; The linguistic hedge of a quantity is the set of membership functions for the possible linguistic statements about size.

The steps which the method according to the invention provides in phase 1 after the fuzzy controller has been designed and validated are shown in FIG. 2. First, the N linguistic rules {R 1,. , ., R _N } to the quantities T₁,. , , T _L divided (step I). For k ε {1,. , , L} T _k contains all the rules whose conclusions are the k-th linguistic statement about Z. Let N _k be the number of elements of T _k . If N _k = 0, then there is no rule with the k-th linguistic statement about Z as a conclusion, and this linguistic statement is neglected in phase and phase 2. For the sake of clarity, this case is not considered in FIG. 2 and FIG. 3. If N _k <0, then the premises are coded according to the following rule in the premises matrix P ^(k) (step II):

For k ε {1,. , ., L}, P ^{(k) is} a matrix with N _k rows and n = 2 columns. For i ε {1,. , ., N _k } is p _i1 ^(k) , the number in the i-th row and the first column of P ^(k) , equal to 0, if not referenced X in the premise of the ith rule from T _k otherwise, p _i1 ^(k) = r ε {1 _,. , ., n _X }, if in the premise of this rule the rth linguistic statement about X occurs. Similarly, p _i2 ^(k) = 0 if the premise of this rule does not refer to Y, otherwise p _i2 ^(k) = s ε {1 _,. , ., n _Y }.

A simple example illustrates the coding method of the claim. About X and Y are ever 3 linguistic statements possible. After numbering, these are for X:
Statement ₁ over X is "X is small", statement ₂ over X is "X is medium", and statement ₃ over X is "X is large". Accordingly, statement ₁ over Y is "Y is small", statement ₂ over Y is "Y is medium" and statement ₃ over Y is "Y is large".

Statement ₁ over Z is "Z is very large". The conclusion "Z is very big" occurs in the following rules:

IF "X is big" THEN "Z is very big" IF "Y is small" THEN "Z is very big" as well IF "X is small" AND "Y is middle" THEN "Z is very big"

Since these are 3 rules, N₁ = 3 and P ^{(1) is} a 3 × 2 matrix. It has these elements:

p₁₁ ⁽¹⁾ = 3, p₁₂ ⁽¹⁾ = 0, p₂₁ ⁽¹⁾ = 0, p₂₂ ⁽¹⁾ = 1, p₃₁ ⁽¹⁾ = 1, p₃₂ ⁽¹⁾ = 2

In Phase 1, after the controller design, the numbers {Z ⁽¹⁾ ,. , ., Z ^(L) } (step III in FIG. 2). For k ε {1,. , ., L}, Z ^{(k) is} determined by "defuzzification" of λ [k], ie the membership function which characterizes the k-th linguistic statement about Z. One of the methods widely used in literature, such as the centroid method, is used for this.

In order to store the rule set according to the invention, the following storage space is required for n input variables: Let iε {1,. , ., n} n _{i is} the number of linguistic statements about the i-th input, let n _max = max {n₁,. , ., n _n }. Then you need:

• - n₁ +. , , n _n times the space for a membership function
• - N _* n times the space for an integer greater than or equal to 0 and less than or equal to n _max
• - L times the space for a real number
• - L times the space for structure and memory information of a matrix (eg. Memory address of the 1st element), namely for the L premises matrices
• and once the place for structure and memory information of a vector, namely the vector (Z₁, ..., Z _L )

In most fuzzy regulators reported in the literature, both n _max and L are less than 10.

The operation of the fuzzy controller at run time, ie in phase 2, at a sampling time t, Fig. 3 for the case of two input variables X and Y. On the one hand, the numbered membership functions of the linguistic hedges for X and Y, on the other hand, the L premise matrices P ⁽¹⁾ ,. , ., P ^(L) , in addition, the numbers x _t and y _t as values of X and Y at the sampling time t.

First, the L matrices A ⁽¹⁾ (t),. , ., A ^(L) (t) determined (step IV). For k ε {1,. , ., L} has A ^(k) (t) just like P ^(k) N _k rows and n = 2 columns. The elements of A ^(k) (t) are numbers from the interval [0, 1] and depend on the following formula: x _t , y _t , the linguistic hedges of X and Y, and the premise matrix P ^(k) :

The number μ [p _i1 ^(k) ] (x _t ) is the degree of membership of x _t , the value of X at time t, to the membership function for the p _i1 (k) -th linguistic statement about X.

For k ε {1,. , , L}, the current weighting factor ζ _k (t) of the value Z _{k is} determined by applying formula (2) to the matrix A ^(k) (t). Applying formula 2 to A ^(k) (t) means first applying a T-norm to the n numbers of the row for each matrix row and then adding the N _k numbers thus obtained.

Finally, according to formula (3), with ζ (t),. , ., ζ _L (t) weighted summary of values Z ⁽¹⁾ ,. , ., Z ^(L) and thereby computes z _t as the output value at time t.

For n input variables, the procedure shown in FIG. 3 requires the following overall arithmetic operations: Let k ε {1,. , ., L} G _k, the number of those elements of the premise matrix P ^(k) that are greater than 0. It is G _k ≧ N _k , since each rule has at least one linguistic statement in the premise, and the rule subset T _{k has} N _k elements and G _k ≦ N _{k *} n, since P ^{(k) has} N _{k *} n elements , Let G: = G₁ +. , + G _L , it is N ≦ G ≦ N _* n.

At each sampling time, perform:

• - N _* n read accesses to a matrix, namely to a premise matrix, N _* n checks of a number for equality with 0, and G determinations of a membership value for a current input value to the matrices A ⁽¹⁾ (t),. , ., A ^(L) (t) to determine (step IV in Fig. 3)
• - N _* n multiplications and N additions of real numbers between 0 and 1 to obtain from the matrices A ⁽¹⁾ (t),. , ., A ^(L) (t) the weighting factors ζ₁ (t),. , ., ζ _L (t) to calculate (step V)
• - L multiplications, 2 _* L additions and 1 division to determine z _t as a weighted sum according to formula 1 of the claim (steps VI to VIII).

A fuzzy controller which operates according to the inventive method shown in FIG. 2 and FIG. 3, can be realized technically both as a software program as well as by commercially available computer technology hardware modules.

An advantageous choice of occurring in formula 2, but in the claim 1 unspecified T-norm is the algebraic product. It's fast too calculate, but gives away in contrast to the often mentioned in the literature Minimum no information, as the product of two numbers of both Numbers depend on the minimum only of the smaller number.

A further advantageous embodiment of the invention is when each membership function is trapezoidal, that is one of the forms shown in Fig. 4 has. The limitation to trapezoidal membership functions saves memory and runtime or compute hardware. The degree of membership of a current value of an input variable to a trapezoidal membership function, which must be determined in step IV of FIG. 3, is then advantageously realized exactly by linear interpolation over the four interpolation points.

A further advantageous embodiment of the invention is already before the controller design, an upper limit S for the maximum permissible number linguistic statements about each input and output variable. For practical applications S = 7 should be sufficient. The Defining this upper bound has two advantages:

• Since each element p _ÿ ^{(k) of} the premise matrix P ^{(k) is} an integer between 0 and S, one can easily determine the maximum storage space requirement for the premises matrices. For S = 7, a maximum of 4 bits is required for each element of a premise matrix, since the element is a number between 0 and 7. Altogether, the elements of all L premise matrices thus claim 4 _* N _* n bits.
• Since L, the number of linguistic statements about the output quantity Z, is less than or equal to S, one can switch S function or computational hardware components in parallel and thus ensure that each rule set stored according to the invention is processed in parallel as follows: The matrices A ⁽ FIG. ¹⁾ (t),. , ., A ^(L) (t) are determined in parallel (step IV in Fig. 3), also the weighting factors ζ₁ (t),. , ., ζ _L (t) (step V).

A further advantageous embodiment of the invention is to perform the "fuzzification" of the input variables for each linguistic statement at each sampling time t only once for all rules together. Let n be the number of input quantities and for i ε {1,. , ., n} denote n _i the number of linguistic statements about the ith input. Let N be the number of linguistic rules of the fuzzy controller. Then, instead of N _* n provisions of a membership value, as may be required in the procedure of Fig. 3, only n₁ +. , . + n _n determinations of a membership value. Fig. 5 shows the operation of such a fuzzy controller in the case of two input variables X and Y.

Taking all the possibilities presented here to design the invention in an advantageous manner, the procedure shown in FIG. 5 requires the following arithmetic operations at each sampling instant:

• - n₁ +. , . + n _n determinations of a hearing value (step I in Fig. 5) by linear interpolation over 4 nodes and storage of the thus obtained n₁ +. , . + _nn numbers
• - N _* read accesses to a matrix, namely to a premises matrix, N _* n checks of a number for equality with 0 and N _* n read accesses to a previously determined membership value, around the matrices A ⁽¹⁾ (t),. , ., A ^(L) (t) (step II in FIG. 5)
• - N _* n multiplications and N additions of real numbers between 0 and 1 to obtain from the matrices A ⁽¹⁾ (t),. , ., A ^(L) (t) the weighting factors ζ₁ (t),. , ., ζ Calculate _L (t) (Step III)
• - L multiplications, 2 _* L additions and 1 division to determine z _t as a weighted sum according to formula (3) (steps IV to VI),

If maximum linguistic statements were made about each size, so the following estimates are possible:

L ≦ S
n₁ +. , , + n _n ≦ S _* n
N ≦ S _* n,

since there are at most S _* n possibilities to form an n-tuple with a linguistic statement about each input quantity.

Claims (6)

1. Method for storing and evaluating a set of linguistic rules {R₁,. , ., R _N } in a fuzzy controller with n input variables and an output Z, where

• - the premise of each rule consists either of a single linguistic statement about an input variable of the controller or of several linguistic statements about one input variable each,
• - the linguistic statements in each premise are ANDed if the premise consists of more than one linguistic statement,
• - the linguistic statements in each premise apply in pairs different input quantities and
• - the conclusion of each rule is a single linguistic statement about the output quantity,
• - for each linguistic statement that appears in the premise or conclusion of a rule, there is a membership function over the population of the variable involved in the statement, and
• in a first phase for each i ε {1,. , ., N}, where N is the number of linguistic rules, the conclusion of the rule R _{i is} replaced by a numerical value Z _i, which is obtained by "defuzzification" of the membership function characterizing the linguistic statement of the conclusion, and
• in a second phase for each time t, for which the fuzzy controller is to determine a value for the output quantity Z by evaluating the set of rules,
  • - first for each i = 1,. , ., N for each linguistic statement A _ÿ (with j ε {1, _... , N}), which occurs in the premise of the rule R _i , the _degree of membership α _ÿ (t) is determined, which is the current value of the linguistic statement has affected size in the membership function characterizing the linguistic statement,
  • - then for each i ε {1,. , ., N} from these degrees of membership the actual validity ω _i (t) of the premise of R _{i is} determined,
  • - and finally the number z _t according to the formula is formed and used as the current value for the output Z,

characterized in that in the method

• - in the first phase, the linguistic statements about each input and output variable are numbered beginning at 1,
• - The rule set in L disjunkte subsets T₁,. , ., T _L , where L is the number of linguistic statements about Z, for all k ε {1,. , ., L} T _k consists of the set of rules which, as conclusion, have the k-th linguistic statement about Z,
• - for all k ε {1,. , ., L} the rules of the subset T _k in a matrix
  P ^(k) = {p _ÿ ^(k) | 1 ≦ i ≦ N _k and 1 ≦ j ≦ n}
  are stored, where n is the number of input variables, and N _k denotes the number of linguistic rules in the rule-subset T _k,
• - for all k ε {1,. , ., L}, i ε {1,. , ., N _k } and j ε {1,. , ., n} p _ÿ ^{(k) is} an integer greater than or equal to 0, which describes the linguistic statement about the jth input in the premise of the ith rule in the set T _k , where p _ÿ ^(k) = 0 means that this input does not appear in the premise, while p _ÿ ^(k) <0 expresses that the premise is the p _ÿ ^(k) -th linguistic statement about the j-th input, and
• - for all k ε {1,. , . L} the real number Z ^{(k) is} obtained by "defuzzification" of the membership function which characterizes the k-th linguistic statement about Z,
• - in the second phase
• - first for k ε {1,. , ., L}, i ε {1,. , ., N _k } and j ε {1,. , ., n} α _ÿ ^(k) (t) = 1 is set if p _ÿ ^(k) = 0, or α _ÿ ^(k) (t) as a degree of membership of the value of the j-th input quantity at time t the match function, which identifies the p _ÿ ^(k) -th linguistic statement about the j-th input, is determined if p _ÿ ^(k) <0,
• - then for k ε {1,. , ., L} the current weighting ζ _k (t) of the k-th linguistic statement about Z according to the formula where T is a T-norm with n arguments, and
• - last the current output value z _t according to is calculated.

2. The method according to claim 1, characterized

• - that in formula 2 as T-norm the algebraic product is chosen, thus for k ε {1,. , ., L} the current weighting ζ _k (t) of the k-th linguistic statement about Z according to the formula is determined.

3. The method according to claim 1 or 2, characterized, that an upper bound S for the maximum number of linguistic statements over each input variable as well as over the output size and thereby the maximum storage space requirement of the controller are determined can. 4. The method according to claim 1 or 2, characterized, that exclusively trapezoidal membership functions are used. 5. The method according to claim 1 or 2, characterized, that at each sampling time for each input, the membership values their current value to all linguistic statements about them Input size determined in advance, cached, and in the rule set application for this sampling time the stored membership values be used.