A GENERAL PURPOSE FUZZY CONTROLLER FOR MONOTONE FUNCTIONS

C. J. Wu and A.H. Sung

Abstract: In this work, a general purpose fuzzy controller is proposed to handle the class of monotone functions. A set of rules on the selection of fuzzy subsets and decision tables based on the mean-of-inversion (MOI) defuzzification method for guaranteed convergence and accuracy is given and proved. Unlike the mean-of-maximum (MOM) and the center-of-area (COA) methods, the MOI method defuzzifies each fired rule separately instead of superimposing fired rules before defuzzification.

I. INTRODUCTION

Fuzzy logic was introduced by Zadeh in [1-2] and used by Mamdani and Assilian to control dynamic systems in [3-4]. Since then, many successful applications of fuzzy controllers have been reported [5-16]. Although the design of fuzzy controllers is well known for its simplicity as noted in [17], the effect of fuzzy controller components (e.g., fuzzy sets, decision tables, and defuzzification methods) on the overall performance of controllers is not quite clear from the mathematical point of view. Thus, most of the time, the verification of accuracy of fuzzy controllers is done via simulations instead of theoretical analysis.

As mentioned in [17], the lack of rigorous stability analysis was the primary reason for the slow acceptance of fuzzy controllers outside Japan. In the literature, some models for stability analysis had been proposed [18-26]; however, they either imposed many restrictions on the design of controllers or focused on the analysis after design is completed.

In [27-28], a fuzzy controller based on the mean-of-inversion (MOI) defuzzification method for JPEG [29] in the application of image compression was proposed. Although simulations indicate that the designed controller worked very well, a theoretical analysis for accuracy is preferred.

In this work, a set of rules on the selection of fuzzy subsets and decision tables based on the MOI method for guaranteed convergence and accuracy is given and proved. Section II describes the mechanism of the designed fuzzy controllers. The proof of accuracy is given in Section III. Conclusions and discussions are given in Section IV.

II. THE MECHANISM OF THE FUZZY CONTROLLER

We assume a generic system as depicted in Figure 1. In Fig. 1, the plant can be characterized as

(1)

where c and a are the input and output of the plant, respectively, and the function f is assumed to be a monotone function.

In order to measure and analyze the controller's performance in terms of accuracy, we define successful trials and unsuccessful trials.

Definition 1. Successful Trials: For a desired output value , the fuzzy controller succeeds at a trial if one of the following three conditions is met:

Case 1: If for a non-increasing monotone function (or for a non-decreasing monotone function), where and are the minimum and maximum values in the input domain C, then exactly one of the two following conditions is met:

(i) The final output is where is the final value of c after convergence.

(ii) If there exists no such that , then

[1] where is the smallest (greatest) possible value in the case of non-increasing (non-decreasing) monotone functions such that and is the greatest value smaller than in ;

[2] where is the greatest (smallest) possible value in the case of non-increasing (non-decreasing) monotone functions such that and is the smallest value greater than in .

Case 2:

For a non-increasing monotone function, if , then .

(For a non-decreasing monotone function, if , then . )

Case 3:

For a non-increasing monotone function, if , then .

(For a non-decreasing monotone function, if , then .)

Note that only one of Case 1, Case 2, or Case 3 can be satisfied.

Definition 2. Unsuccessful Trials: Any trial that is not a successful trial.

Thus, to guarantee that the controller always produces a successful trial, the feedback must not be 0 until one of the conditions of successful trials is met.

The internal structure of the fuzzy controller is depicted in Figure 2.


The two inputs, p and e, to the fuzzy controller at time t are defined as

(2)

(3)

where is the desired output of the plant and . E and P are the universal sets of e and p, respectively. C, E, and P are finite sets of real numbers.

In general, , the membership function of a fuzzy subset A, in the universal set X is defined [30, 31] as

(4)

where is the closed interval of real numbers with 0 and 1 being the minimum and maximum membership values, respectively.

Without loss of generality, we assume there are N () fuzzy subsets for the fuzzy variable e, that is,

(5)

In addition, the following four restrictions are imposed on .

(1) A special triangular fuzzy subset ZE

(6)

is the only fuzzy subset across 0 in the input distribution E and

(7)

(2) is on the negative side of E if ; and is on the positive side of E if .

(3) For a crisp input value and , there exists at least one i, , such that .

(4) (8)

As for the fuzzy variable p, there are D () fuzzy subsets, that is,

(9)

Additionally, the following four restrictions are imposed on .

(1) A special triangular fuzzy subset ZE

(10)

is the only fuzzy subset across 0 in the distribution P and

(11)

(2) is on the negative side of P if ; and is on the positive side of P if .

(3) For a crisp input value and , there exists at least one j, , such that .

(4) (12)

The decision tables used by the fuzzy controller are Table 1 (for non-increasing monotone functions) and Table 2 (for non-decreasing monotone functions). Since the output of the fuzzy controller is a crisp value of P, we assume, for simplicity, that the entries of the decision table also use the same group of fuzzy subsets as p does. Thus, given the definition of e and p in Eqs. 2 and 3, it is quite natural to assign ZE's to all the entries of the decision table in row k and in column l, as shown in Tables 1 and 2.

In Tables 1 and 2, the membership value of the decision table entry ij, , is calculated by using the fuzzy min (or intersection) operator as

(13)

where and . Table 3 below is an example of Table 1.




As for defuzzification, we use the MOI method to calculate (for ), the feedback from the controller to JPEG,

(14)

In order to guarantee that is no larger than , is decided by the following algorithm:

Step 1. Calculate the corresponding crisp values of fuzzy membership value .

(15)

where A is the corresponding fuzzy subset of and is the inverse function of , the membership function of fuzzy subset A. Note that there might be more than one value generated by .

Step 2. Adjust the inverse values for accuracy.

Case 1. Non-increasing monotone functions:

(16)

Case 2. Non-decreasing monotone functions:

(17)

Step 3. Adjust each inverse value for guaranteed convergence.

(18)

Step 4. Pick the largest absolute value of , , and set .

In addition, after the calculation of Eq. 14 is adjusted as follows:

If overshooting occurs and ,

then (19)

else (20)

where m is a positive integer, u is the measure unit of c , and .

Finally, the new input is calculated as

(21)

When , a special signal STOP is generated to stop the adjusting process.

III. GUARANTEED CONVERGENCE AND ACCURACY

A. Guaranteed convergence

The convergence of this general purpose fuzzy controller for discrete monotone functions is guaranteed by the following three conditions (as will be proved in Theorem 3):

(i) The absolute value of the current adjustment is no larger than that of the previous adjustment , that is, in the case of no overshooting.

(ii) The absolute value of the current adjustment is smaller than that of the previous adjustment , that is, in the case of overshooting where u is the measure unit of input c and .

(iii) , that is, the discrete input domain is bounded.

Lemma 1. At step t+1, .

Proof: According to Eqs. 21 and 3, there are three different cases (exclusive of the first three conditions of Eq. 21).

Case 1. : Since and , . Thus, .

Case 2. : Since , and , . Thus, .

Case 3. : .

Lemma 2. At step t, .

Proof: According to Eq. 18, the absolute values of all inverse values are no larger than , that is, for and . According to Eq. 14,

Thus, after the calculation of Eq. 14, . In the process of adjusting (Eqs. 19 and 20), we have two cases.

Case 1. Overshooting and : According to Eq. 19, will be adjusted such that because .

Case 2. In all other situations: Since u is the measure unit of c, all (for ) are adjusted to nu where n is an integer according to Eqs. 19 and 20; so does according to Eqs. 21 and 3. Thus,

(i) if , then since before the adjustment;

(ii) if , then .

Therefore, according to Eqs. 19 and 20, after the adjustment if no overshooting occurred; otherwise, .

Theorem 3. The absolute values of the feedback produced by the fuzzy controller using the MOI method form a non-increasing monotone series. More specifically, if and have the same signs; otherwise, .

Proof: Based on Lemmas 1 and 2, if no overshooting occurs; otherwise, .

In summary, the produced by the fuzzy controller using the MOI method are a non-increasing monotone series. Therefore, since the input domain is discrete and bounded, after a finite steps of adjustment, the adjusting process will terminate, that is, or for .

B. Guaranteed accuracy for non-increasing monotone functions

Lemma 4. At step t, all non-zero in Eq. 14 have same signs, that is, they are all either positive or negative values.

Proof:

Case 1. Non-increasing monotone functions and : According to definition (Eq. 2),

.

Since (the fuzzy subset ZE) is the only fuzzy subset across 0,

Thus, according to Eq. 13,

In addition, the positive are adjusted to 0 according to Eq. 16. Therefore, there are no positive values in Eq. 14, that is, all according to Table 1.

Case 2. Non-increasing monotone functions and :

.

Thus, as seen in Table 1 and Eq. 13,

In addition, the negative are adjusted to 0 according to Eq. 16. Therefore, all .

Case 3. :

.

Thus, according to Eqs. 16 and 17, all .

Case 4. Non-decreasing monotone functions and : According to definition (Eq. 2),

.

Thus, according to Table 2 and Eq. 13,

In addition, the negative are adjusted to 0 according to Eq. 17. Therefore, there are no negative values in Eq. 14, that is, all according to Table 2.

Case 5. Non-decreasing monotone functions and :

.

Thus, as seen in Table 2 and Eq. 13,

In addition, the positive are adjusted to 0 according to Eq. 17. Therefore, all .

Theorem 5. At step t, the feedback if and only if one of the following three conditions is met:

(1) All the fired fuzzy subsets in the decision table are fuzzy subset ZE's with membership values 1's.

(2) Overshooting occurs and where u is the minimum measure unit of input c.

(3) .

Proof: According to Eqs. 19 and 20, is equal to 0 if and only if either (i) and overshooting occurred. (Thus, the second condition of Theorem 5 is proved.) or (ii) in Eq. 14. Since there exists at least one in Eq. 14,

if and only if

(according to Lemma 4)

where and . Since the fuzzy subset ZE is the only fuzzy subset across 0 in the domain P and (according to Eq. 11.), the inverse value if and only if either (i) (according to Eqs. 11 and 12) or (ii) as defined in Eqs. 16 and 17. Thus, the first and third conditions of Theorem 5 are proved as well.

Lemma 6. For a non-increasing monotone function , if (i) Table 1 is used and (ii) either overshooting does not occur or , then when , and when .

Proof: As seen in Cases 1 and 2 of Lemma 4, all when and all when . However, none of the three conditions of Theorem 5 is met here. (Since , the first condition of Theorem 5 is not true according to Eqs. 7 and 13.) Hence, and in the cases of and , respectively.

Theorem 7. For a non-increasing monotone function and the fuzzy controller using Table 1, if where is the desired output, m is an integer, and s is either a positive integer or 0, then the final input is in the interval .

Proof: For serial inputs

Case 1. : Since , the adjusting process will stop at step t according to Theorem 5 and Eq. 21.

Case 2. :

(i) Overshooting (): Since , . Therefore, according to Lemma 6, and because , , and .

(ii) No overshooting (): According to Lemma 6, and .

Case 3. :

(i) Overshooting (): Since , according to Lemma 6, . Thus, according to Eq. 21, because , , and .

(ii) No overshooting (): According to Lemma 6, since no overshooting occurred. Thus, according to Eq. 21, .

Theorem 8. For a non-increasing monotone function and the fuzzy controller using Table 1, if , then either or where is the final value of c after convergence.

Proof: For serial inputs ,

Case 1. : As seen in Case 2 of Theorem 7, .

Case 2. : ()

(i) Overshooting (): Since , . Therefore, according to Lemma 6, and because , , and .

(ii) No overshooting (): According to Lemma 6, and .

Case 3. :

(i) Either overshooting with or no overshooting (): According to Lemma 6, . Thus, according to Eq. 21, because , , and .

(ii) Overshooting with : (according to Theorem 5) and (according to Eq. 21).

Case 4. : ()

(i) Either overshooting with or no overshooting (): According to Lemma 6, . Thus, according to Eq. 21, because , , and .

(ii) Overshooting with : (according to Theorem 5) and (according to Eq. 21).

Lemma 9. For a non-increasing monotone function and the fuzzy controller using Table 1, if , then where is the final value of c after convergence.

Proof: Since for , according to Lemma 6, because of no overshooting. The value of c will continue to decrease until , and according to Eq. 21. Since , (according to Lemma 6) and , the adjusting process will stop at step t+1 according to Eq. 21.

Lemma 10. For a non-increasing monotone function and the fuzzy controller using Table 1, if , where is the final value of c after convergence.

Proof: Since for , according to Lemma 6, because of no overshooting. The value of c will continue to increase until , and according to Eq. 21. Since , (according to Lemma 6) and , the adjusting process will stop at step t+1 according to Eq. 21.

Theorem 11. For a non-increasing monotone function , the fuzzy controller using Table 1 as the decision table will always yield successful trials as defined in Definition 1.

Proof: Based on the conclusions on Theorems 7 and 8, Lemmas 9 and 10, and Definition 1, for a non-increasing function, the fuzzy controller using Table 1 as the decision table will always yield successful trials.

In summary, if the fuzzy controller is configured as described in Section II, successful trials as defined in Definition 1 can be always obtained for discrete monotone functions. (For non-decreasing monotone functions using Table 2, the proof of accuracy can be obtained in a similar way.) However, the given configuration may not be the only configuration which can always yield successful trials for discrete monotone functions.

IV. CONCLUSIONS AND DISCUSSIONS

In this work, a set of configuration on fuzzy subsets and decision tables based on the MOI method [27-28] for guaranteed accuracy in terms of successful trials defined in Definition 1 is stated and proved. Unlike the COA method, the MOI method defuzzifies each fired rule individually instead of superimposing fired rules before defuzzification. The restrictions imposed on the selection of fuzzy subsets for fuzzy variable e as defined in Eq. 2 are

(1) A special triangular fuzzy subset ZE is the only fuzzy subset across 0 in the distribution E and

.

(2) For a crisp input value and , there exists at least one for .

(3)

Similar restrictions are also imposed on another variable p as defined in Eq. 3.

Strictly speaking, these restrictions are quite common in the design of fuzzy controllers. Thus, the proposed fuzzy controller can be very flexible in terms of the selection of fuzzy subsets for various applications. In addition, the optimization of convergence speed is possible via simulations.

Although the fuzzy controller is well known for its ease of design, the analysis and testing (verification) of accuracy is very difficult in general. Therefore, the set of rules for guaranteed accuracy proposed in this work is very useful and can relieve users' burden on testing via possible thousands of simulations. The further relaxation of the restrictions on fuzzy subsets and decision tables is the ongoing research direction.

ACKNOWLEDGMENTS

The authors would like to thank Drs. S. Mazumdar, S. Schaffer, H. Soliman, and D. Carlson for their helpful suggestions.

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