Guaranteed accurate fuzzy controllers for monotone functions

C. J. Wu

Department of Library and Information Science, Fu-Jen University, Hsin-Chuang, Taiwan, R.O.C. 242

Abstract

In this work, a general purpose fuzzy controller which allows the fuzzy set ZE to be used anywhere in decision tables is proposed to handle the class of monotone functions. For guaranteed convergence and accuracy, the rules on the selection of fuzzy sets based on the defuzzification method mean-of-inversion (MOI) are given and proved. Such a guideline can relieve the users' burden on testing accuracy after design. In addition, these imposed restrictions on the selection of fuzzy sets are not unusual in the design of fuzzy controllers. Thus, the optimization on convergence speed is possible for the proposed fuzzy controller in various applications.

Keywords: Control theory; Fuzzy sets; Monotone functions; Guaranteed accuracy; Decision tables

1. Introduction

Since the introduction of fuzzy logic [22-23] and the fuzzy controllers [1, 9], as mentioned in [10], 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 [2, 5-8, 11-14]; however, those models either imposed many restrictions on the design of controllers or else focused on the analysis after the design was completed. Though the use of fuzzy controllers as universal approximators has been studies in the literature [3-4], they either only proved what can be achieved instead of how to achieve it or else required a complex analysis to acquire proper design details in real applications. Sometimes, the price of those analyses could be very high. Thus, most of the time, the verification of accuracy of fuzzy controllers is done via simulations instead of theoretical analysis.

This work was partially inspired by the author's experiments on image compression [16-19]. Although the lossy JPEG model is one of the most widely used digital image compression techniques for both grayscale and color still images [15], the compression results are not predictable in advance due to a wide variety of image data. For a picture of MxN pixels with P grayscale levels, there are up to possible combinations. With this extremely large number of combinations, it is very difficult for any lossy compression model to predict the relation of its parameters to the compression results, e.g., mean square error. Therefore, in a sense, the compression and distortion ratios depend heavily on the source image data.

Fig. 1 shows the relation of the JPEG input parameter Quality and the distortion ratio AGE (average grayscale error) on 15 USC pictures using the JPEG model developed by Independent JPEG Group. Those 15 USC pictures include eight 256x256 pictures (Girl-1, Couple, Lady, Girl-II, House, Tree, Ball-I, and Ball-II) and seven 512x512 pictures (Splash, Girl-III, Baboon, Lena, F16, Park, and Pepper). The ratio AGE is defined by

(1)

where G is the maximum grayscale levels of pictures and MAE (mean-absolute-error) is defined as

(2)

where and are the corresponding frames in the original and reconstructed pictures, respectively, N is the dimension of the input vector I (or frame M), and F is the total number of frames. As seen in Fig. 1, each image has its own unique curve which can only be known via simulations. Therefore, our aim is to design a fuzzy controller that can deal with a large collection of monotone functions so it is not necessary to change the controller's configuration from function (or image in the application of JPEG) to function.

For achieving better trade-off between distortion and compression ratios, incorporating a fuzzy controller with the JPEG model to achieve the automatic control during compression process on the source side was proposed in [16]. The relation of JPEG (including coder and decoder) and the fuzzy controller on source side is depicted in Fig. 2 where the plant was the JPEG model in those experiments.

As shown in Fig. 2, the plant can be characterized as

(3)

where c and a are the input and the output of the plant, respectively, the function f is a monotone function, and e is one of the two inputs to the fuzzy controller and is defined as

(4)

where is the value of e at step t, is the desired output for the plant, and is the output value of the plant at step t. The other input p to the fuzzy controller at step t is

(5)

where and are the maximum and the minimum values in the input domain C, respectively, and . E and P are the universal sets of e and p, respectively. C, E, and P are FINITE sets of real numbers. Without loss of generality, we assume there are N () fuzzy subsets for the fuzzy variable e as

(6)

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

(7)

To help readers get the better picture of the designed fuzzy controller, the author briefly describes its mechanism used in one of simulations in [20] and summarizes the simulation results as follows:

Experiments were performed on 15 USC images which have been widely used in image compression experiments and include eight 256x256 images and seven 512x512 images with 256 grayscales. Ten random initial values (38, 58, 13, 15, 51, 27, 10, 19, 12, and 86) of q generated by the random number generation function rand() in the standard C programming environment are used to test the accuracy of the fuzzy controller. The function used to describe the behavior of JPEG is

where g (or a as in Eq. 3) is the generated AGE, and q (or c as in Eq. 3) is the value of parameter Quality. Various components of the designed fuzzy controller are introduced as follows:

(1) The fuzzy sets: Seven fuzzy subsets are used--PB (positive big), PM (positive medium), PS (positive small), ZE (zero), NS (negative small), NM (negative medium), and NB (negative big).

For the input in Fig. 2 and the entries of decision tables, the following fuzzy subsets are used.

NB(-100, -50, -20), NM(-45, -25, -5), NS(-20, -10, 0),

ZE(-5, 0, 5),

PS(0, 10, 20), PM(5, 25, 45), PB(20, 50, 100)

For the input in Fig. 2, the following fuzzy subsets are used.

NB (-100, -6, -3) NM (-5, -3, -1) NS (-2, -1, 0)

ZE (-0.5, 0, 0.5)

PS (0, 1, 2) PM (1, 3, 5) PB (3, 6, 100)

The fuzzy subsets PM, PS, ZE, NS, NM are the triangular type of fuzzy sets which can be characterized by three values: right boundary R, center C, and left boundary L, as shown in Fig. 3.

Thus, for an input crisp value x, the membership value of x on a triangular type of fuzzy subset A is calculated as

(8)

The fuzzy subset NB is the trapezoidal type of fuzzy set as depicted in Fig. 4 and can be characterized by two values, right boundary R and center C.

Thus, for an input crisp value x, the membership value of x on the fuzzy subset NB is calculated as

(9)

The fuzzy subset PB is also the trapezoidal type of fuzzy set as depicted in Fig. 5 and can be characterized by two values, left boundary L and center C.

Thus, for an input crisp value x, the membership value of x on the fuzzy subset PB is calculated as

(10)

(2) The decision table:

As shown in Table 1, when we deal with non-increasing monotone functions, the upper part of decision table should use those fuzzy subsets on the positive side of the universal Q. For example, if the generated AGE is bigger than the desired output, we should increase the value of c, i.e., , to decrease the AGE. Additionally, when overshooting occurs, that is, those entries in either upper-left or lower-right parts are fired, we decrease the magnitude of adjustment; for example, we use fuzzy subset PM instead of PB for the entry in the upper-left corner.

The membership grade of the decision table entry ij, , is calculated by the fuzzy-min operator as

where is the membership function defined in Eqs. 8-10, , and . The example given will illustrate how the membership values of entries are calculated. For example, the entry at row 3 and column 2 can be translated into the following if-then rule:

If is NS and is NM, then is PS.

where a is a fuzzy variable which uses the same universe and fuzzy subsets as p in this work. For example, if (1) and , (2) and , then .

(3) The defuzzification method: MOI is introduced in Section 2, the mechanism of the fuzzy controller.

For the performance of controllers, guaranteed accuracy is used and measured in terms of successful trials defined as follows:

Definition 1. Successful Trials: For a desired output , we say that the fuzzy controller succeeds at a trial if any of the following 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 two following conditions is met:

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

(ii) If there exists no such that and , then either or where is the biggest value(s) smaller than , and is the smallest value(s) bigger than .

Case 2:

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

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

Case 3:

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

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

The simulation results of three test points are summarized as follows:

As shown in Fig. 1, despite a few images that exhibit slight non-monotonicity toward one end of the plot, the designed fuzzy controller still produced all successful trials as seen in Table 2.

Although simulations indicate that the designed controller worked very well, a theoretical analysis for accuracy is preferred. In [21], for non-increasing discrete monotone functions, a fuzzy controller based on the defuzzification method mean-of-inversion (MOI) and the following decision table Table 3 for guaranteed convergence and accuracy was proposed and proved. In Table 3, (for ) are fuzzy subsets on the negative side of E, (for ) are fuzzy subsets on the positive side of E, and is the fuzzy subset ZE on E. Similarly, (for ) are fuzzy subsets on the negative side of P, (for ) are fuzzy subsets on the positive side of P, and is the fuzzy subset ZE on P.

Although fuzzy controllers are well known for their 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 [21] is very useful; otherwise, thousands of simulations may be needed to verify the accuracy of design. In spite of the usefulness of guaranteed accuracy, one of the restrictions could affect the system convergence speed is the restriction in which fuzzy subset ZE can only be used in the column l and the row k as shown in Table 3. In this work, a new set of rules on the selection of fuzzy subsets was proposed to eliminate this restriction, that is, we allow the fuzzy subset ZE to be used anywhere in the decision table, and still keep the guaranteed accuracy intact. Thus, to guarantee the controller always produces the successful trials, the feedback from the controller to the plant must not be 0 until one of the conditions of Definition 1 is met. The rest of the paper is organized as follows: Section 2 describes the mechanism of the proposed fuzzy controller. The proofs of convergence and accuracy are given in Sections 3 and 4, respectively. Finally, the conclusions and discussions are given in Section 5.

2. The mechanism of the fuzzy controller

In this work, the decision tables used by the fuzzy controller in Fig. 2 are depicted in Tables 4 (for non-increasing monotone functions) and 5 (for non-decreasing monotone functions). As seen in Table 4, the restriction where the fuzzy subset ZE can only be used in the row k and the column l has been eliminated. Since the output of the fuzzy controller is a crisp value on P, for simplicity, we assume that the entries of the decision table also use the same group of fuzzy subsets as the input p does.

(1) The decision tables:


In Tables 4 and 5, the meanings of , , , , , and are as defined in Table 3. Table 6 is an example of Table 4. As seen in Tables 4-6, the fuzzy subset ZE is allowed to be used at any position of decision tables.

The membership grade of the decision table entry ij, , is calculated by the fuzzy-min operator as

(11)

where is the membership function, , and .

(2) The fuzzy sets:

In order to obtain guaranteed accuracy, the following general restrictions are imposed on .

(i) A special triangular fuzzy subset ZE

(12)

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

(13)

(ii) For a crisp input value and , there exists at least one fuzzy subset i, such that for .

(iii) (14)

Similarly, the following restrictions are imposed on .

(i) A special triangular fuzzy subset ZE

(15)

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

(16)

(ii) For a crisp input value and , there exists at least one fuzzy subset j, such that for .

(iii) (17)

In addition, the following restriction is needed for guaranteed accuracy as well.

Restriction S. If the fuzzy subset ZE is assigned to the entry ij (row i and column j) where and , then the fuzzy subsets for the two input variables e and p must be chosen as follows:

For those crisp values and , there exists at least one entry in the decision tables such that

(i) min(, ) > 0 if the fuzzy subset of entry is not ZE, or

(ii) 1 > min(, ) > 0 if the fuzzy subset of entry is ZE.

Example 1: A group of fuzzy subsets meets the criteria above.

(1) The fuzzy subsets on P

NB(-100, -50, -20), NM(-45, -25, -5), NS(-20, -10, 0),

ZE(-5, 0, 5),

PS(0, 10, 20), PM(5, 25, 45), PB(20, 50, 100).

(2) The fuzzy subsets on E

NB(-100, -6, -2.5), NM(-5.25, -3, -0.75), NS(-2, -1, 0),

ZE(-0.5, 0, 0.5),

PS(0, 1, 2), PM(0.75, 3, 5.25), PB(2.5, 6, 100).

(3) The defuzzification method:

As for defuzzification, we use the mean-of-inversion (MOI) method to calculate (for ), the feedback from the controller to JPEG, [16, 21]

(18)

where N and D are the number of rows and columns in the decision table, respectively.

In order to guarantee that is no larger than in terms of crisp value, is decided by the following algorithm:

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

(19)

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

Step 2. Adjust these inverse values for accuracy.

Case 1. Non-increasing monotone functions:

(20)

Case 2. Non-decreasing monotone functions:

(21)

Step 3. Adjust each inverse value for guaranteed convergence.

(22)

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

In addition, after the calculation of Eq. 18 is adjusted as follows:

Step 1. If overshooting occurs and ,

; (23)

otherwise,

(24)

where m is a positive integer and u is the measure unit in the domain C ().

Step 2. The new input is calculated as

(25)

where and are the maximum and minimum values in the input domain C. Whenever , a special signal STOP is generated to stop the adjusting process.

3. 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. 25 and 5, there are three different cases (exclusive of the first three conditions of Eq. 25).

Case 1. : Since and , . Thus, .

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

Case 3. : .

Lemma 2. At step t, .

Proof: According to Eq. 22, the absolute values of all inverse values are no larger than , that is, for and . According to Eq. 18,

Thus, after the calculation of Eq. 18, . In the process of adjusting (Eqs. 23 and 24), we have two cases.

Case 1. Overshooting and : According to Eq. 23, 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. 23 and 24; so does according to Eqs. 25 and 5. Thus,

(i) if , then since before the adjustment;

(ii) if , then .

Therefore, according to Eqs. 23 and 24, 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, for .

4. Guaranteed Accuracy

We now prove the guaranteed accuracy.

Lemma 4. At step t, all non-zero in Eq. 18 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. 4),

.

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

Thus, according to Eq. 11 (the fuzzy min-operator),

In addition, the positive are adjusted to 0 according to Eq. 20. Therefore, there are no positive values in Eq. 18, that is, all according to Table 4.

Case 2. Non-increasing monotone functions and :

.

Since

,

In addition, the negative are adjusted to 0 according to Eq. 20. Therefore, there are no negative values in Eq. 18, that is, all according to Table 4.

Case 3. :

.

Thus, according to Eqs. 20 and 21, all .

Case 4. Non-decreasing monotone functions and :

.

Since

In addition, the negative are adjusted to 0 according to Eq. 21. Therefore, there are no negative values in Eq. 18, that is, all according to Table 5.

Case 5. Non-decreasing monotone functions and :

.

Since

In addition, the positive are adjusted to 0 according to Eq. 21. Therefore, there are no positive values in Eq. 18, that is, all according to Table 5.

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

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

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

(3) .

Proof: According to Eqs. 23 and 24, is equal to 0 if and only if either

(i) and overshooting occurs. (Thus, the second condition of Theorem 5 is proved.), or

(ii) in Eq. 18.

Since there exists at least one in Eq. 18, 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. 16), the inverse value if and only if either

[i] (according to Eqs. 16 and 17) (Note: For simplicity, we assume that use the same group of fuzzy subsets as the input p.), or

[ii] as defined in Eqs. 20 and 21.

Thus, the first and third conditions of Theorem 5 are proved as well.

Lemma 6. The adjusting process is stop if and only if one of the three following conditions is met.

(i)

(ii)

(iii)

Proof: The special signal STOP is the only way to stop the adjusting process. According to Eq. 25, the three conditions above will make ; thus, a special signal STOP is generated.

If there exists no fuzzy subset ZE assigned to entries ij (row i and column j) of Table 4 where and , then Table 4 is reduced to Table 3. Since there is one additional restriction (Restriction S) on the selection of fuzzy subsets for the proposed fuzzy controllers in this work, the fuzzy subsets in this work are the subset of those in [21]. As proved in [21], the fuzzy controllers will always yield successful trials.

For Tables 4 and 5 with fuzzy subset ZE at entry ij where and , the guaranteed accuracy will be proved as follows:

Lemma 7. For a non-increasing monotone function and the fuzzy subset ZE is assigned to entries ij (row i and column j) of Table 4 where and , if (i) either overshooting does not occur or , and (ii) one of the following three conditions is true: (1) , , and Restriction S, (2) , (3) , then when and, respectively, when .

Proof: As can be seen in the Case 1 of Lemma 4, all when according to Table 4. However, none of the three conditions of Theorem 5 is met here. (According to Restriction S, Eqs. 11, 13, and 16, the first condition of Theorem 5 is not true.) Hence, . Similarly, as seen in the Case 2 of Lemma 4, when .

Theorem 8. For a non-increasing monotone function and the fuzzy subset ZE is assigned to the entries ij of Table 4 where and , if (i) and (ii) , , and Restriction S 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 Lemma 6.

Case 2. :

(i) Overshooting (): Since , . Therefore, according to Lemma 7, and because , , and .

(ii) No overshooting (): According to Lemma 7, and .

Case 3. :

(i) Overshooting (): Since , . Therefore, according to Lemma 7, and because , , and .

(ii) No overshooting (): According to Lemma 7, and .

Theorem 9. For a non-increasing monotone function and the fuzzy subset ZE is assigned to entries ij of Table 4 where and , if (i) and (ii) , , and Restriction S, then either or where is the final value of c after convergence.

Proof: For serial inputs ,

Case 1. : As seen in the Case 2 of Theorem 8, .

Case 2. : ()

(i) Overshooting (): Since , . Therefore, according to Lemma 7, and because , , and .

(ii) No overshooting (): According to Lemma 7, and .

Case 3. :

(i) Either overshooting with or no overshooting (): According to Lemma 7, and because , , and .

(ii) Overshooting with : (according to Theorem 5) and (according to Eq. 25).

Case 4. : ()

(i) Either overshooting with or no overshooting (): According to Lemma 7, and because , , and .

(ii) Overshooting with : (according to Theorem 5) and (according to Eq. 25).

Lemma 10. For a non-increasing monotone function and the fuzzy subset ZE is assigned to entries ij of Table 4 where and , if (i) and (ii) , , and Restriction S, then where is the final value of c after convergence.

Proof: Since , according to Lemma 7, because of no overshooting. The value of c will continue to decrease until , and according to Eq. 25. Since , (according to Lemma 7 and ); thus, the adjusting process will stop at step t+1 according to Lemma 6.

Lemma 11. For a non-increasing monotone function and the fuzzy subset ZE is assigned to the entries ij of Table 4 where and , if (i) and (ii) , , and Restriction S, then where is the final value of c after convergence.

Proof: Since for , according to Lemma 7, because of no overshooting. The value of c will continue to increase until , and according to Eq. 25. Since , (according to Lemma 7 and ); thus, the adjusting process will stop at step t+1 according to Lemma 6.

Theorem 12. For a non-increasing monotone function and the fuzzy subset ZE is assigned to entries ij of Table 4 where and , if (i) , (ii) , and (iii) Restriction S, then the controller will always produce the successful trial as defined in Definition 1

Proof: Based on the conclusions on Theorems 8 and 9, Lemmas 10 and 11, and Definition 1, the fuzzy controller using Table 4 as the decision table will always yield the successful trial.

For non-decreasing monotone functions, guaranteed accuracy can be proved in a similar way. In summary, if the fuzzy controller is configured as described in Fig. 2 and Section 2, the successful trial as defined in Definition 1 can be always obtained for discrete monotone functions.

5. Conclusions and Discussions

In this work, the fuzzy controller which utilizes the defuzzification method mean-of-inversion (MOI) [16, 21] and the decision tables Tables 4 and 5 is proposed. In this proposed fuzzy controller, the fuzzy subset ZE is allowed to use at any place of decision tables to optimize the convergence speed in various applications. We prove that the accuracy in terms of successful trials defined in Definition 1 for discrete monotone functions can be obtained as long as the following rules on the selection of fuzzy subsets are used.

(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)

(4) If the fuzzy subset ZE is assigned to the entry ij (row i and column j) of decision tables where and , then (as seen in Example 1) there exists at least one entry such that either

[1] min(, ) > 0 if the fuzzy subset of entry is not ZE

or

[2] 1 > min(, ) > 0 if the fuzzy subset of entry is ZE.

Similar restrictions are also imposed on another variable p as defined in Eq. 5. 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, and the optimization of convergence speed is possible via simulations.

Although fuzzy controllers are well known for their ease of design, the analysis and testing (verification) of accuracy is very difficult in general. Thus, the set of rules for guaranteed accuracy proposed in this work is very useful since the selection of fuzzy sets according to the proposed rules is the only requirement to obtain accuracy for monotone functions. In addition, the proposed fuzzy controller is quite robust and can work in a less than perfect environment, e.g. neural nets, as shown in [17-18]. The possible application of the fuzzy controller utilized MOI on other mathematical functions is the ongoing research direction.

Acknowledgments

The author would like to thank Dr. A. H. Sung for his helpful suggestions.

References

[1] S. Assilian, Artificial Intelligence in the control of Real Dynamic Systems, Ph.D. Thesis, Queen Mary College (London, 1974).

[2] M. Braae and D.A. Rutherford, Theoretical and Linguistic Aspects of the Fuzzy Logic Controller, Automatic 15 (1979) 553-577.

[3] J.J. Buckley, "Sugeno Type Controllers are Universal Controllers," Fuzzy Sets and Systems 53 (1993) 299-304.

[4] J.L. Castro, "Fuzzy Logic Controllers are Universal Approximators," IEEE Trans. Syst. Man. Cybern. 25 (4) (1995) 629-635.

[5] S.S. Farinwata and G. Vachtsevanos, Stability Analysis of the Fuzzy Logic Controller, in IEEE International Conf. on Fuzzy Systems '93 1377-1382.

[6] G.-C. Hwang and S.-C. Lin, A stability approach to fuzzy control design for non-linear systems, Fuzzy Sets and Systems 48 (1992) 279-287.

[7] C. Jianqin and C. Laijiu, Study on stability of fuzzy closed-loop control systems, Fuzzy Sets and Systems 57 (1993) 159-168.

[8] J.B. Kiszka, M.M. Gupta, and P.N. Nikiforuk, Energetistic Stability of Fuzzy Dynamic Systems, IEEE Trans. Syst. Man Cybern. 15 (5) (1985) 783-792.

[9] E.M. Mamdani, Applications of Fuzzy Algorithms for Simple Dynamic Plants, Proc. IEEE 21 (12) (1974) 1585-1588.

[10] E.M. Mamdani, Twenty Years of Fuzzy Control: Experiences Gained and Lessons Learnt, in Proc. of IEEE International Conf. on Fuzzy Systems '93, 339-344.

[11] A. Pagni, R. Poluzzi, and G. Rizzotto, Automatic Synthesis, Analysis and Implementation of a Fuzzy Controller, in Proc. od IEEE International Conf. on Fuzzy Systems '93 105-110.

[12] K.S. Ray and D.D. Majumder, Application of Circle Criteria for Stability Analysis of Linear SISO and MIMO Systems Associated with Fuzzy Logic Controller, IEEE Trans. Syst. Man, Cybern. 14 (2) (1984) 345-349.

[13] K. Tanaka and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems 45 (1992) 135-156.

[14] K. Tanaka and M. Sano, Fuzzy Stability Criterion of a class of Nonlinear Systems, Information Sciences 71 (1993) 3-26.

[15] G. K. Wallace, "The JPEG still Picture Compression Standard," Communications of the ACM 34 (4) (1991) 30-44.

[16] C.J. Wu and A.H. Sung, The Application of Fuzzy Logic to JPEG, IEEE Trans. Consumer Electronics 40 (4) (1994) 976-984.

[17] C.J. Wu, A.H. Sung, and H.S. Soliman, "A Fuzzy ART Network with Fuzzy Control for Image Data Compression," in Proc. of IASTED Int. Conf. on Modeling, Simulation and Control in the Process Industry (Otiwa, Canada, 1994) 95-98.

[18] C.J. Wu, A.H. Sung, and H.S. Soliman, "An ART Network with Fuzzy Controller for Image Data Compression," in Proc. of IEEE Int. Conf. on Fuzzy Systems (Orlando, FL, 1994) 1743-1748.

[19] C.J. Wu and A.H. Sung, "Comparison of the MOI, the COA, and the MOM Methods in the application of JPEG," in Proc. of IEEE Int. Conference on Systems, Man and Cybernetics (Vancouver, British Columbia, Canada, 1995) 2019-2024.

[20] C.J. Wu, "A General Purpose Fuzzy Controller for JPEG and Monotone functions," Ph.D. Thesis, New Mexico Tech (Socorro, New Mexico, 1995).

[21] C.J. Wu and A.H. Sung, A General Purpose Fuzzy Controller for Monotone Function, IEEE Trans. Syst. Man. Cybern., to appear.

[22] L.A. Zadeh, Fuzzy Sets, Inform. Contr. 8 (1965) 338-353.

[23] L.A. Zadeh, Outline of a New Approach to the Analysis of Complex Systems and Decision Processes, IEEE Trans. Syst. Man. Cybern. 3 (1) (1973) 28-44.