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FUZZY LOGIC DESIGN GENERATOR USING A NEURAL NETWORK

TO GENERATE FUZZY LOGIC RULES AND MEMBERSHIP

FUNCTIONS FOR USE IN INTELLIGENT SYSTEMS

BACKGROUND OF THE INVENTION 1. Field of the Invention

The present invention relates to intelligent systems, and in particular, to intelligent controllers using neural network based fuzzy logic.

2. Description of the Related Art Uses of intelligent controllers have become more numerous and varied in keeping pace with the numerous and varied control requirements of complex modern electronics systems. For example, intelligent controllers are being called upon more frequently for use in assisting or use as servomechanism controllers, as discussed in commonly assigned U.S. Patent Applications Serial No. 07/967,992, entitled "Intelligent Servomechanism Controller Using a Neural Network", and Serial No. 07/859,328, entitled "Intelligent Controller With Neural Network and

Reinforcement Learning" (the disclosures of which are each incorporated herein by reference) . Further applications include control systems for robotic mechanisms. One type of intelligent controller seeing increased use and wider application uses "approximate reasoning", and in particular, fuzzy logic. Fuzzy logic, initially developed in the 1960s (see L.A. Zadeh et al., "Fuzzy Sets and Applications", Selected Papers of L.A. Zadeh, by R.R. Yager, S. Ouchinnikov et al. (Eds.), John Wiley & Sons, 1987), has proven to be very successful in solving problems in many control applications where conventional model-based (mathematical modeling of the system to be controlled) approaches are very difficult, inefficient or costly to implement.

An intelligent controller based upon fuzzy logic design has several advantages including simplicity and ease of design. However, fuzzy logic design does have a number of disadvantages as well. As the control system complexity increases, it quickly becomes more difficult to determine the right set of rules and membership functions to accurately describe system behavior. Further, a significant amount of time is needed to properly "tune" the membership functions and adjust the rules before an accurate solution can be obtained. Indeed, for many complex systems, it may even be impossible to arrive at a satisfactory set of rules and membership functions. Moreover, once the rules have been determined, they remain fixed within the fuzzy logic controller, i.e. the controller cannot be modified, e.g. "learn", based upon its experience (except adaptive fuzzy logic systems which do allow some limited adaptation capabilities) .

Furthermore, even once a good set of rules and membership functions has been established and used, the resulting solutions, i.e. control signal sets, must be evaluated in accordance with those rules and then "defuzzified". As is known, rule evaluation and defuzzification are two important steps in fuzzy logic design. Rule evaluation, or fuzzy inferencing as it may be called, combines the output from all of the rules which have "fired". However, the output of the fuzzy inferencing still remains as a fuzzy output. Defuzzification then converts this fuzzy output into numerical, i.e. nonfuzzy, outputs.

Several schemes have been proposed for fuzzy inferencing and defuzzification; however, all are based on some form of heuristics. For example, the most popular conventional fuzzy inferencing method uses the maximum of the outputs from all rules for each universe of discourse. The most popular and effective conventional defuzzification uses the center-of-gravity ("COG") method. For simple systems, these methods generally yield good solutions. However, for more complex systems, the heuristic based algorithms may not yield satisfactory results over a wide range of system control inputs. Indeed, conventional antecedent processing according to a "minimum" operation is often unsatisfactory for not providing consistently accurate outputs over a wide range of inputs and/or applications. A further demanding task is that of determining a good set of rules that will work well with those inferencing and defuzzification techniques.

The application of neural networks to learn system behavior has been suggested to overcome some of the problems associated with fuzzy logic based designs. Using a system's input and output data, a neural network can learn the system behavior and, accordingly, generate fuzzy logic rules. See: B. Kosko, "Neural Nets and Fuzzy Systems", Prentice Hall 1992; J. Nie et al., "Fuzzy Reasoning Implemented By Neural Networks", Proceedings of IJCNN92 (International Joint Conference on Neural Networks, June 1992), pp. 11702-07; and J. Buckley et al., "On the Equivalent of Neural Networks and Fuzzy Logic", Proceedings of IJCNN92, pp. 11691-95.

As is known, a neural network mimics human learning instead of using fixed, preprogrammed approximate reasoning or rules. Also, like fuzzy logic, it uses numerical techniques. In a neural network, many simple processing elements are interconnected by variable connection strengths and the network learns by appropriately varying those connection strengths. It is primarily a data driven system and does not rely heavily upon programming. By proper learning, the neural network can develop good generalization capabilities, and therefore, solve many control problems that would otherwise go unsolved or be inefficiently solved by existing techniques. However, a neural network may not always be the most effective way to implement an intelligent controller, since implementation of a neural network is more costly compared to fuzzy logic implementations. For example, fuzzy logic may be more effective for a particular application and, by proper programming, a conventional embedded controller can be used to implement the fuzzy logic. A neural network implementation by programming of the conventional embedded controller iε also possible, but it will typically be significantly slower.

Furthermore, a dedicated hardware implementation, generally more desirable, is more common for fuzzy logic than for a neural network, particularly when considering the relative costs of each. Another problem with a neural network based solution is its "black box" nature, namely the relationships of the changes in its interlayer weights with the input/output behavior of the system being controlled. Compared to a fuzzy rule based description of the system, a good understanding of the "black box" nature of a neural network is difficult to realize.

Accordingly, it would be desirable to have an improved technique for applying neural network design to the design and implementation of fuzzy logic. Further, it would be desirable to have an improved fuzzy logic design in which antecedent processing, rule evaluation (fuzzy inferencing) and defuzzification can be performed upon control signals generated in accordance with such neural network based fuzzy logic design.

SUMMARY OF THE INVENTION A fuzzy logic system design in accordance with the present invention combines a neural network with fuzzy logic to generate fuzzy logic rules and membership functions, and to provide neural network based fuzzy antecedent processing, rule evaluation and defuzzification.

In accordance with the present invention, a neural network is used as a fuzzy rule generator to generate a set of fuzzy logic rules and membership functions to be used in the controller for the system intended to be controlled. In a preferred embodiment of the present invention, a multilayered feed-forward neural network, based upon a modified version of a back-propagation neural network, is used that learns the system behavior using input and output data and then mapping the acquired knowledge into a new non- heuristic fuzzy logic system. Interlayer weights of the neural network are mapped into fuzzy logic rules and membership functions.

In further accordance with the present invention, antecedent processing is performed according to a weighted product, or multiplication, of the membership value of the antecedents, rather than using their minimum value.

In still further accordance with the present invention, neural network based algorithms are used for performing rule evaluation and defuzzification. In a preferred embodiment of the present invention, rule evaluation and defuzzification are performed in one layer of the neural network, or in one step of the neural network operation. These and other features and advantages of the present invention will be understood upon consideration of the following detailed description of the invention and the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS Figure IA depicts a fuzzy rule generator in accordance with the present invention.

Figure IB illustrates a plant controller driving a plant, wherein the inputs and outputs of the plant controller are used for learning purposes by the fuzzy rule generator of Figure IA.

Figure 2A illustrates a neural network driving a plant, in a feedforward configuration, using the fuzzy logic rules and membership functions generated by the fuzzy rule generator of Figure 1. Figure 2B illustrates a neural network driving a plant, in a feedback configuration, using the fuzzy logic rules and membership functions generated by the fuzzy rule generator of Figure 1.

Figure 3 illustrates a neural network for generating fuzzy logic rules and membership functions in accordance with the present invention.

Figure 4 illustrates an exemplary portion of the fuzzification layer of the neural network illustrated in Figure 3. Figure 5 illustrates in more detail the neural network of Figure 3.

Figure 6 illustrates a flowchart for neural network learning and Recall operations in accordance with the present invention. Figure 7 illustrates an exemplary input membership function generated in accordance with the present invention.

Figure 8 depicts a fuzzy rule verifier and fuzzy system simulator in accordance with the present invention.

DETAILED DESCRIPTION OF THE INVENTION Referring to Figures IA and IB, a fuzzy rule generator 10 for generating fuzzy logic rules and membership functions with a neural network in accordance with the present invention receives input signals 12 and 14a representing the inputs and desired outputs, respectively, of a plant controller 21 used for controlling a plant 22. As is known in the art, the input 12 and desired output 14a data can be generated in a number of ways, such as simulation, measurement and learning of the inverse of the plant model. (See e.g. commonly assigned U.S. Patent Application Serial No. 07/967,992, entitled "Intelligent Servomechanism Controller Using a Neural Network", the disclosure of which is incorporated herein by reference). Based upon this input data 12, 14a, the fuzzy rule generator 10, in accordance with neural network learning techniques, develops fuzzy logic rules and membership functions (discussed further below) , and provides output signals 16 and 18 which represent those fuzzy logic rules and membership functions, respectively.

Referring to Figures 2A and 2B, these fuzzy logic rules 16 and membership functions 18 are used by a fuzzy system controller 20 for generating plant control input(s) 14b in accordance with its inputs 12. (Figure 2A illustrates a system with a feedforward configuration, and Figure 2B illustrates a system with a feedback configuration [with provision for other, optional inputs, such as an error change input].) Ideally, these controller outputs 14b generated by the fuzzy system controller 20 in accordance with the fuzzy logic rules 16 and membership functions 18 are identical to the desired outputs 14a originally used by the fuzzy rule generator 10 for its learning (i.e. for the same controller inputs 12) . In practice, these controller outputs 14a and 14b are quite close, and often identical, when using fuzzy logic rules 16 and membership functions 18 generated with a fuzzy rule generator 10 in accordance with the present invention. Referring to Figure 3, that portion 10a of the fuzzy rule generator 10 which forms the learning mechanism of the neural network includes a comparator 11 and three neural layers: (1) fuzzification; (2) rule base; and (3) rule evaluation and defuzzification. For the sake of simplicity in describing a preferred embodiment of the present invention, the learning mechanism 10a illustrated uses a three-layer neural network for generating the fuzzy logic rules and membership functions of a two- input, one-output system. (However, it should be understood that further embodiments of the present invention include such learning mechanisms using neural networks with more than three layers for generating the fuzzy logic rules and membership functions of systems with other than two inputs or one output.)

The first, or input, layer performs the fuzzification. The values of the input signals X_lf X₂ are matched against the labels used according to the fuzzy control rule. For this example, the fuzzy control rules represent relative input signal amplitudes classified as low ("L") , medium ("M") or high ("H") . The fuzzification layer is used to define the input membership functions. The middle, or rule base, layer (neurons Nl through N9) represents the fuzzy logic rule base. The actual output 14b of the output neuron is compared against the desired output(s) 14a, with the resulting error e being used in the neural network's learning process. In accordance with a preferred embodiment of the present invention, the rule base neurons Nl through N9 have linear functions which perform multiplication, rather than summation, and have slopes of unity. Further, linear neurons whose functions also have slopes of unity are used for the output (rule evaluation and defuzzification) layer neuron. Therefore, the equivalent error e at the input of the output layer can be computed as follows:

€_k = (t_k - o_k)f_k (1) where: e_k = equivalent error at the input of the output _f layer neuron f_k = first derivative of the function of the output layer neuron t_k = desired output 14a of the output layer neuron o_k = actual output 14b of the output layer neuron

Once this equivalent error e_k has been computed, the middle-to-output layer weights W_jk can be modified, or updated as follows:

W_jk(?_ew) ^» W_{jk (old)} + T7e_ko. ( 2 )

= ^Wjk(old) + f7 <- ° where: ^w _jk(n. ₎ = updated weight between middle (hidden) layer neuron j and output layer neuron k W_jkc_oi_d) ⁼ original weight between middle

(hidden) layer neuron j and output layer neuron k η = learning rate e_k = equivalent error at the input of the output layer neuron

O_j = actual output of the hidden layer neuron

(Further discussion of interlayer weight modification can be found in the aforementioned commonly assigned U.S. Patent Application Serial No. 07/859,328, entitled "Intelligent Controller With Neural Network and Reinforcement Learning", the disclosure of which is incorporated herein by reference. )

Next, the equivalent error e_i at the input to the middle, or hidden, layer neurons Nl through N9 can be computed according to back-propagation modeling techniques as follows:

e-_j = f_j∑e_kW_Jk (3) k where: e_? = equivalent error at the input of middle _t (hidden) layer j f_j = first derivative of the function of middle (hidden) layer neuron j

(Further discussion of this back-propagation modeling technique can be found in the aforementioned commonly assigned U.S. Patent Application Serial No. 07/967,992, entitled "Intelligent Servomechanism Controller Using a Neural Network", the disclosure of which is incorporated herein by reference.) Following that, the equivalent error ej at the input to the input, or fuzzification, layer can be computed. However, as noted above, the middle, i.e. rule base, layer neurons Nl through N9 use multiplication instead of summation, as follows:

fj = πwijo, ( 4 )

1 where : f_j = function describing the operation of the middle (hidden) layer neuron

W_ij= weight between input layer neuron i and middle (hidden) layer neuron j o_A = output of input neuron i

Accordingly, the equivalent error e_i at the input to the fuzzification layer is computed as follows:

€i = afi ■∑∑^β€^jjWWn_i3 (5)

_i = equivalent error at the input of input

_f layer i fi = first derivative of the function of input layer neuron i e

_? - equivalent error at the input of middle

(hidden) layer j W_ij= weight between input layer neuron i and middle (hidden) layer neuron j W_mj= weight between input layer neuron m and middle (hidden) layer neuron j o_m = output of input neuron m

The input-to-middle layer weights W^ can then be modified, or updated, as needed using an equation similar to Equation (2) above, with appropriate substitutions for the corresponding equivalent error e_n and output o_n signals. As seen in Figure 3, the inputs to the middle layer neurons Nl through N9 are the preconditions, or antecedents, of the rules, and the outputs are the conclusions, or consequents. Accordingly, rule base layer neuron Nl can be interpreted as representing the rule that "if input X_j is low and input X₂ is low, then the output is N^', wherein ^ can be used to represent the fuzzy conclusion from rule number one. As noted above, and as is evident from Equation (4) above, the antecedent processing of the present invention uses a multiplication operation, rather than a "minimum" operation as is used in conventional fuzzy logic design. As discussed further below, this means that the signals representing the antecedents are multiplied within the rule base layer neurons Nl through N9, rather than summed and compared to a threshold, or minimum, reference value. For a two- input, one-output learning mechanism 10a, as shown in Figure 3, if three membership functions are used, we need a maximum of 3²=9 rules. Therefore, we need nine middle, or rule base, layer neurons as shown. The output layer neuron, as discussed further below, performs the rule evaluation and defuzzification.

To help expedite learning and convergence, the neural network 10a of the fuzzy rule generator 10 is first initialized with some suitable values of interlayer weights W^, W_jk. Following that, a set of input 12 and desired output 14 data is applied for several cycles to allow the neural network 10a to converge. At that point, as discussed further below, the generated fuzzy logic rules and membership functions can be extracted. The inputs 12, 14 should be selected so as to cover the entire potential input range well. This will result in better learning, and therefore, a better set of fuzzy logic rules and membership functions. Also, the learning rates used for modifying the interlayer weights should be selected so that the neural network does not oscillate and so that it converges to a good local minima.

If it is desired to generate membership functions of different shapes, the fuzzification layer of Figure 3 can be constructed of multiple layers of various types of neurons, including combinations of linear and nonlinear function neurons. Accordingly, the weight modification Equations (1) through (5) above will also be modified to correspond to these additional layers of neurons that generate the membership functions. Referring to Figure 4, the fuzzification layer of the learning mechanism neural network 10a of Figure 3 can be constructed of four layers A, B, C, D of neurons. For the sake of simplicity, only one signal path for input X_x is shown. It should be understood that, depending upon how many membership functions are used for each input, additional signal paths will be required.

The input signal X_x is processed by layer A and B neurons whose functions are f_A1 and f_B1, respectively. Their outputs A_x and ₁ are weighted by the interlayer weights W_al and W_bl, respectively, with the result (i.e. the multiplication product _ιf_AιW_alf_BιW_bl) being summed in layer C with a bias signal β₁. The sum output c_{l t} weighted by interlayer weight W_cl, forms the input signal Y for the layer D neuron, whose nonlinear function f_D1 produces the output signal O__ which goes to neuron Nl of the rule base layer (Figure 3) . Mathematically, this signal path can be represented as follows: Y = (Xιf_Aι _Aιf_Bi _bι + 0ι) W_cl ( 6 ) where :

X_: = first input f_A1= first layer neuron function W_A1= weight between the first and second layer neurons f_B1= second layer neuron function W_bl= weight between the second and third layer neurons /3_j = bias no. 1

W_cl= weight between the third and fourth layer neurons

The neural functions f_A1 and f_B1 of layers A and B can be kept constant, e.g. as linear gain functions, with the interlayer weights W_al, W_bl and W_cl available for modification during the learning process. If the nonlinear function f_D1 of layer D is an exponential function (e.g. of the form 1/[1 + e^" ]), then the output D_α can be expressed as follows:

D = f_D = 1/(1 + e^"y) (7)

= 1

1 + exp[-y]

1 + exp [ - (X₁f_A1W_A1f_B1W_bl + /3J W..

During learning, i.e. by modifying weights W_al,

W_bl, and W_cl, an exponential membership function can be established. As will be recognized from Equation (7) above, the size and shape of this membership function is determined by the weights W_al, W_bl and W_cl, and the bias β__ . Accordingly, by using different initial values of weights and biases among the various input signal paths, various exponential membership functions of the same type but with different shapes, sizes and positions, can be generated. For example, by using multiple neurons in layers C and D, and using different weight values for their input weights W_bl and W_cl, any class of exponential type membership functions can be generated. Such membership functions would meet the criteria necessary to back- propagate error signals. However, it should be understood that other suitable mathematical functions could be used as well, such as (l-e^"y)/(l+e^~y) .

After these membership functions have been generated, the weights W_al, W_bl, W_cl remain fixed, and a neural network Recall operation will classify the input X__ in one or more fuzzy logic classes (where each neuron in layer D defines a fuzzy logic class) . (To perform the Recall operation, the input pattern X, or vector, is applied to the input layer and propagated through the network without modifying any interlayer weights.)

The AND operation of the membership function noted above is performed as a multiplication operation. Therefore, the antecedent processing is performed according to:

v_c = v_av_b (8) where: v_c = membership function of the combination of membership functions v_a and v_b v_a = membership function of input A v_b = membership function of input B

This use of multiplication, which is dictated by the use of a neural network, produces significantly improved results over those obtained using a "minimum" operation for processing the antecedents. Since the antecedent processing, and the rule evaluation and defuzzification are all based on neural network learning, as discussed further below, use of a "minimum" operation instead of a multiplication operation produces significant errors. Examples of the errors encountered can be seen by referring to Table 1 below.

TABLE 1

INPUT 1 INPUT 2 MINIMUM MULTIPLICATION

1.0 1.0 1.0 1.0

0.9 0.9 0.9 0.81

0.8 0.8 0.8 0.64

0.7 0.7 0.7 0.49

0.6 0.6 0.6 0.36

0.5 0.5 0.5 0.25

0.4 0.4 0.4 0.16

0.3 0.3 0.3 0.09

0.2 0.2 0.2 0.04

0.1 0.1 0.1 0.01

Referring to Figure 5, a preferred embodiment of a two-input, one-output neural network 10a in accordance with the present invention includes seven layers of neurons, i.e. layers A through G. The input signals X__ and X₂ are received by the layer A neurons Al and A2, which have gains of unity and produce intermediate signals A_x and A₂, respectively. These signals A_{l r} A₂ are multiplied, i.e. weighted, by their respective interlayer weights W_al and W_a2 and received by the layer B neurons Bl and B2, which have gains of unity and produce intermediate signals B_x through B and B₅ through B₈, respectively (where B₁=B₂=B₃=B_<1 and B₅=B₆=B₇=B₈) .

These intermediate signals B_: through B₈ are weighted by their respective interlayer weights W_bl through W_b8 and received by the layer C neurons Cl through C8, which also receive the two bias signals β₁ and β_z . The layer C neurons Cl through C8 perform summation operations on these input signals (C₁=B₁V_bl+β₁ , C₂=B₁W_h2+β_{l l} . . . , C₈=B₈W_b8+0₂) and produce intermediate signals C__ through C₈. These intermediate signals C_: through C₈ are weighted by their respective interlayer weights W_cl through W_c8 and received by the layer D neurons Dl through D8. These intermediate signals C^_cl , C₂W_c2, ..., C₈W_c8 are modified in accordance with the functions f_DI through f_DB of the layer D neurons Dl through D8 (per the discussion regarding Equation (7) above) to produce intermediate signals D_: through D₈. The intermediate signals D₃ through D₈ are weighted by their respective interlayer weights W_dl through W_d8 which, with the exception of weights W_d3 and W_d7, have values of unity. Weights W_d3 and W_d7 have values of negative one (-1) . The resulting product signals D_:W_dl, D₂W_d2, ... , D₈W_d8 are received by the layer E neurons El through E6. Layer E neurons El, E3, E4 and E6 have gains of unity and produce from their respective input signals D^^ , D₄W_d , D₅W_d5 and D₈W_d8 further intermediate signals E__ through E₃, E₇ through E₁₂ and E₁₆ through E₁₈ (where E₁=E₂=-=E₃, E₇=E₈=E₉, E₁₀-=E₁₁=E₁₂ and E₁₆=E₁₇=E₁₈) . Layer E neurons E2 and E5 receive their respective input signals D₂W_d2, D₃W_d3, D₆W_d6 and D₇W_d7, perform summation operations and provide intermediate signals E_< through E₆ and E₁₃ through E₁₅ (where E_<.=E₅=E₆ and E₁₃=Eu=E₁₅) .

The neurons of layers A through E establish the membership functions 18. The membership functions 18 are established by taking the outputs of the layer E neurons 18a (E__ [or E₂ or E₃]) through 18f (E₁₈ [or E₁₆ or E₁₇]) with respect to the network inputs 12a (X__) and 12b (X₂) . (For purposes of simplification, only membership function signals 18a, 18b, 18e and 18f have been labelled.) Thus, the membership functions 18 can be extracted for later use, e.g. by computing and storing the values of the outputs 18 versus inputs 12 functions (see e.g. Figure 7) .

The layer F neurons Fl through F9 serve as the rule base neurons and each perform multiplication, or product, operations on their respective input signals. For example, as shown, layer F neuron Fl receives and multiplies its input signals Ei _βl and ^Eι_o ^wβi_o- Similarly, neuron F2 forms the product of its two input signals E₂W_e2 and E₁₃W_el3, and so on through neuron F9. Resulting product signals F_: through F₉ are multiplied by their respective interlayer weights W_fl through W_f9, with the resulting product signals ^Fι^wfn ^F ₂ ^w _f2 ? ? ? ^F9^w _f9 received by the layer G neuron. The layer G neuron performs a summation of all of these product signals to produce the final output signal Z.

The output layer weights 16a (W_fl) through 16i (W_f9) between the layer F and G neurons form the consequents of the fuzzy logic rules 16. The antecedents for the fuzzy logic rules 16 are formed by FUZZY(E through FUZZY(E₁₈), where the function FUZZY(E_N) represents the value of layer E output E_H (where Ne{l,2,3, ... ,18}) based upon the input XM (where Me{l,2}) in accordance with the membership functions (discussed above) . Accordingly, a typical rule for the configuration of Figure 5 is: "If XI is FUZZY(El) [e.g. "L" per Figure 3], and X2 is FUZZY(EIO) [e.g. "L" per Figure 3], then the output is W ." Thus, the fuzzy logic rules 16 can be extracted for later use by storing the values of the output layer weights 16a (W_fl) through 16i (W_f9) .

From the foregoing it can be seen that a better understanding of the "black box" nature of a neural network can be gained. This allows better initial membership functions and fuzzy logic consequents to be estimated. Accordingly, faster and more effective learning can be achieved.

Referring to Figure 6, the operation of the neural network of Figure 5 can be visualized in accordance with the illustrated flowchart. First, the interlayer weights W_a, V?_b, W_c, V5_d, W_β and \._{f l} and biases β__ and β_z are initialized. Then, the layer A neurons Al, A2 receive the input signals X_λ and X₂, and generate the layer A outputs A. The layer B neurons Bl, B2 receive their respective input signals Sw_a and generate the layer B neuron output signals B. The layer C neurons Cl through C8 receive their respective input signals BS5_b, β__ , β₂, and generate the layer C outputs C The layer D neurons Dl through D8 receive their respective input signals δw_c and produce, in accordance with their respective nonlinear functions f_D1 through f_D8, the layer D outputs DW_d. The layer E neurons El through E6 receive their respective inputs Dw_d and generate the layer E outputs S. The layer F neurons Fl through F9, the rule base neurons, receive their respective input signals Sw_β and generate the layer F outputs FJ The layer G neuron receives signals f ._£ and generates the layer G output Z. In accordance with the discussion above regarding Equation (1) , the output error e_G is then computed and compared against a selected reference, or threshold, error value α. If the output error e_G is greater than or equal to the threshold α, the equivalent errors e_n for each of the hidden layers are computed in accordance with the discussion above regarding Equations (3) and (5) . Following this, the interlayer weights _a, Vl , W_c, _e and _£ are adjusted. (The interlayer weights W_d are not modified, but are kept at their original values of unity or negative one [-1] . )

The foregoing steps are repeated until such time as the output error ε_G is less than the selected threshold error α for all input training patterns X. Once e_G<α for all X, the learning operation is complete.

In accordance with the present invention, rule evaluation (fuzzy inferencing) and defuzzification are combined to form a single operation. As discussed further below, this single operation does not use any division, unlike conventional defuzzification such as COG. Accordingly, defuzzification in accordance with the present invention saves time and is more accurate since it is based on neural network learning, where accuracy can be controlled during learning by selecting the desired error criterion. Defuzzification in accordance with the present invention is performed as follows:

Z = ∑θ_jW_Jk (9) j where:

Z = defuzzified output of neural network °_j ⁼ output of rule base layer neuron j

W_jk= weight between rule base layer neuron j and output layer neuron k

The defuzzified output Z includes the contribution from all of the rules within the rule base layer of neurons and matches the behavior of 'the neural network. Hence, this defuzzification (e.g. "neural defuzzification") is optimal, particularly sines it does not require a division operation. This is to be contrasted with the conventional COG defuzzification operation, which is defined as follows:

Z = (∑υN_i ) / ( ∑υ_i) (10) i i where: υi = membership function i

V__ - universe of discourse i i = index for universe of discourse (dependent upon number of output membership functions, shape of output membership functions and application)

In COG defuzzification, the index "i" for the universe of discourse can be more than the number of rules. On the other hand, in neural defuzzification according to the present invention, the summation index "j" is limited to the number of rules. Therefore, on the average, neural defuzzification has fewer multiplication operations, as well as no division operation. As should be understood from the foregoing discussion, neural defuzzification in accordance with the present invention is actually a form of rule evaluation. Since the output of the rule is a "nonfuzzy" number, actual defuzzification is not required. Therefore, the terminology neural rule evaluation, rather than neural defuzzification, can also be used.

A qualitative example of the foregoing would be as follows. Referring back to Figure 5, if one input 12a (X was "low" (L) and the other input 12b (X₂) was "low" (L) , then the output 14 (Z) would be represented by the product F;-W_fl (of the output F_x and output weight W_fl of the first layer F neuron Fl) . Quantitatively, the neural network implementation and operation of Figures 4 and 5, respectively, have been simulated with very encouraging results. By way of example, one simulation involved the following highly nonlinear analog function:

Y = 2X_j ³ + 3X₂ ² - 1 (11) where X_Z=2X__ has been used to simplify the example.

The corresponding input signal data X_{l f} X₂, the resulting output Y, the learning rate η , learning factor and accuracy factor a used in this simulation are shown below in Table 2.

TABLE 2

Learning Rate ( η) = 0.01 Learning Factor = 0.20 Accuracy Factor (α) = 0.008

INPUT 1 INPUT 2 OUTPUT

XI X2 Y

-2.10 -4.20 33.40

-1.70 -3.40 23.85

-1.30 -2.60 14.89

-0.90 -1.80 7.26

-0.50 -1.00 1.75

-0.10 -0.20 -0.88

0.30 0.60 0.13

0.70 1.40 5.57

1.10 2.20 16.18

1.50 3.00 32.75

1.90 3.80 56.04

2.10 4.20 70.44

In accordance with the foregoing discussion, the learning factor determines the learning rates within the inner layers of neurons, while the learning rate η corresponds to the output layer only. After the neural net has converged with the applied data to an accuracy factor α as indicated in Table 2 using seven input membership functions for each input, the fuzzy logic rules are generated (using the layer F neurons' outputs F__ through F₉ and output weights W_£1 through W_f9) . The resulting fuzzy logic rules for this example are shown below in Table 3.

TABLE 3 X1=\X2= LH MH SH MD SL ML LL_

LH 8.79 14.58 11.36 0.04 0.01 0.01 0.01

MH 14.58 3.62 9.86 0.20 0.01 0.01 0.01

SH 11.36 9.86 22.93 19.68 0.07 0.01 0.01

MD 0.04 0.20 19.68 -1.63 4.46 0.66 0.07

SL 0.01 0.01 0.07 4.46 9.26 1.06 0.89

ML 0.01 0.01 0.01 0.66 1.06 5.89 11.53

LL 0.01 0.01 0.01 0.07 0.89 11.53 13.47 where:

LH = large high MH = medium high

SH = small high

MD = medium

SL = small low

ML = medium low LL = large low and where "large", "medium", "small", "high" and "low" are relative qualitative parameters (e.g. fuzzy variables) .

EXAMPLE: If input XI is SH and input X2 is MH, then output is 9.86.

The data shown in Table 3 are numbers, i.e. singletons, as opposed to fuzzy numbers as used in conventional fuzzy logic. The shape of the input membership functions for input 1 (X is shown in Figure 7, and is generated by the neural network. The shape of the membership function for input 2 (X₂) is similar since input X_: and X2 are related by Equation (11) above. Accordingly, the two-input, one-output system generated 7²=49 rules. The number of rules was reduced, i.e. optimized, to 25 using a "fuzzy rule verifier" which, as discussed further below, is basically a fuzzy logic system using the above-described neural network techniques. A set of Recall inputs was used along with the generated fuzzy logic rules and membership functions for both the non-optimized (49 rules) and optimized (25 rules) cases. The resulting computed outputs are shown below in Table 4.

TABLE 4

Input 1 Input 2 Comp'd Out Comp'd Out

XI X2 (49 Rules) (25 Rules.

-2.100 -4.200 33.321 33.321

-1.900 -3.800 28.594 28.594

-1.500 -3.000 19.245 19.245

-1.100 -2.200 10.805 10.805

-0.700 -1.400 4.212 4.212

-0.300 -0.600 0.016 0.016

0.100 0.200 -0.836 -0.836

0.500 1.000 2.199 2.197

0.900 1.800 10.063 10.059

1.300 2.600 23.697 23.694

1.700 3.400 43.508 43.507

2.100 4.200 70.397 70.397

From the foregoing, a number of observations can be made. First, the result of the antecedent processing, rule evaluation and defuzzification using all of the possible fuzzy logic rules (i.e. all 49 rules) and membership functions generated in accordance with the present invention is substantially equal to the values generated in a neural network Recall operation (column 3) . Further, it can be shown that using even fewer than 49 rules (e.g. 25 rules per column 4) will produce comparable results. This is advantageous in that it takes substantially less time, as well as less memory, to compute the output.

It can be seen by looking at the shape of the membership functions (e.g. Figure 7) that implementing such membership functions on high end processors is not difficult. Moreover, for low end processors, such membership function shapes can be reasonably approximated to convenient geometric shapes, such as triangles or trapezoids. Such approximations can yield reasonably accurate defuzzified outputs. Referring to Figure 8, a fuzzy rule verifier 24 in accordance with the present invention evaluates the fuzzy logic rules 16 and membership functions 18 generated by the fuzzy rule generator 10 of Figure 1. The inputs to the fuzzy rule verifier 24 are the fuzzy logic rules 16, membership functions 18 and inputs 12 for which the verified output 26 needs to be computed. (The fuzzification processor 28, antecedent processor 30, and rule evaluation and defuzzification processor 32 can be constructed in the form of a multilayer neural network and operated in accordance with the discussion above regarding Figures 3, 4 and 5.)

To verify the fuzzy logic rules 16 and membership functions 18, the verified output 26 of the fuzzy rule verifier 24 is compared with the desired outputs 14. Some of the desired outputs 14' , as discussed above, were used during the initial neural network training phase. The other desired outputs 14" can be obtained either by more measurements, or by performing a forward calculation on the learned neural network 10a (Figure 3). (The latter can be done by performing a Recall operation within the fuzzy rule generator 10.)

From the foregoing, a number of observations can be made. The fuzzy rule verifier 24 can verify whether the computed output 26 from the defuzzification process for a set of inputs is the same as that obtained from a Recall operation through the fuzzy rule generator 10. This can check directly the accuracy of the generated fuzzy logic rules 16 and membership functions 18 provided by the fuzzy rule generator 10. Since the fuzzy rule generator 10 has the capability to reduce the number of fuzzy logic rules 16' methodically, a fuzzy rule verifier 24 can be used to verify whether the resulting defuzzified output 26 using fewer fuzzy logic rules 16' is sufficiently close to the value otherwise obtained from the Recall operation. Additionally, if approximated membership functions 18' are used, e.g. with more convenient geometric shapes, the fuzzy rule verifier 24 can be used to compute the resulting approximated defuzzified output 26' and compare it with the output 26 generated from the Recall operation. This way, acceptable results can be ensured while still allowing for the use of approximated membership functions 18' .

After the optimum number of fuzzy logic rules 16" and membership functions 18" has been determined (including the shapes of the membership functions) using the fuzzy rule verifier 24, a fuzzy logic system design can be completed. The completed design can then be implemented on a chosen processor. A conventional automated code converter can be used to take the fuzzy logic rules and the shape of the input membership functions and generate assembly code therefrom that can be executed by the chosen processor.

From the foregoing, it can be seen that the present invention advantageously combines beneficial attributes of neural networks and fuzzy logic, e.g. the adaptability, or learning capability, of the former and the simplicity of the latter. In doing so, the present invention provides an improved technique for applying neural network design to the design and implementation of fuzzy logic. Also, it provides neural network based fuzzy logic design for antecedent processing, rule evaluation (fuzzy inferencing) and defuzzification.

It should be understood that various alternatives to the embodiments of the present invention described herein can be employed in practicing the present invention. It is intended that the following claims define the scope of the present invention, and that structures and methods within the scope of these claims and their equivalents be covered thereby.