Fuzzy Systems

Sometimes fuzzy logic controllers are combined with pattern recognition software such as artificial neural networks (Kosko, Neural Networks and Fuzzy Systems, Prentice Hall, Englewood Cliffs, New Jersey, 1992). 

Square nodes in the ANFIS structure denote parameter sets of the membership functions of the TSK fuzzy system. Circular nodes are static/non-modifiable and perform operations such as product or max/min calculations. A hybrid learning rule is used to accelerate parameter adaption. This uses sequential least squares in the forward pass to identify consequent parameters, and back-propagation in the backward pass to establish the premise parameters. 

Some types of knowledge-based systems may make judgments based on data that contain uncertainty we shall learn more of this in the next chapter when we encounter fuzzy systems. Even when the information that the system reasons with is unambiguous, the system s conclusions may come as a surprise to a nonexpert. If the user doubts whether the ES has reasoned correctly, it is natural for them to seek reassurance that the line of reasoning used is robust, so the ES must be able to do more than merely provide advice, it should be able to explain how it has reached a particular conclusion. 

Although it is the purpose of fuzzy systems to handle ill-defined information, this does not mean that we can get away with uncertainty in the allocation of membership values. If some of the membership values for liquids in a database were proposed by one person and the rest by a second person, the two groups of memberships could well be inconsistent unless both people used the same recipe for determining membership. Any deductions of the fuzzy system would then be open to doubt. In fact, even the membership values determined by just one person might be unreliable unless they had used a properly defined method to set membership values. The hold-a-wet-finger-in-the-air style of finding a membership value is not supportable. 

Within a fuzzy system, an inference engine works with fuzzy rules it takes input, part of which may be fuzzy, and generates output, some or all of which may be fuzzy. Although the role of a fuzzy system is to deal with uncertain data, the input is itself not necessarily fuzzy. For example, the data fed into the system might consist of the pH of a solution or the molecular weight of a compound, both of which can be specified with minimal uncertainty. In addition, the output that the system is required to produce is of more value if it is provided in a form that is crisp "Set the thermostat to 78°C" is more helpful to a scientist than "raise the temperature of the oven." Consequently, the fuzzy core of the inference engine is bracketed by one step that can turn crisp data into fuzzy data, and another that does the reverse. 

We will outline the way that a fuzzy system makes deductions using an example from enzyme kinetics. 

The rules that the fuzzy system uses are expressed in terms such as a "high" or a "medium" pH, while the experimental input data are numerical quantities. The first stage in applying these rules is to transform the input data into a degree of membership for each variable in each class through the use of membership functions. 

In a conventional expert system, the only rules to fire are those for which the condition is met. In a fuzzy system, all of the rules fire because all are expressed in terms of membership, not the Boolean values of true and false. Some rules may involve membership values only of zero, so have no effect, but they must still be inspected. Implicitly, we assume an or between every pair of rules, so the whole rule base is... 

Negoita, C.V. (1985), Expert Systems and Fuzzy Systems, Benjamin/Cummings, Menlo Park, CA. 

M.J. Arauzo-Bravo, J.M. Cano-Izquierdo, E. Gomez-Sanchez, M.J. Lopez-Nieto, Y.A. Dimitraidis and J. Lopez-Coronado, Automatization of a penicillin production process with soft sensors and an adaptive controller based on neuro fuzzy systems. Control Eng. Pract., 12, 1073-1090 (2004). 

In general, a fuzzy system is any system whose variables (or, at least, some of them) range over states that are fuzzy numbers rather than real numbers. These fuzzy numbers may represent linguistic terms such as very small, medium, and so on, as interpreted in a particular context. If they do, the variables are called linguistic variables. 

I quite accept that you saw a successful fuzzy system [fuzzy helicopter control] in Japan. What I am saying is that, unless it is isomorphic to Bayesian logic circuitry, there will be circumstances in which it reaches the wrong decision and the... 

J. C. Bezdek, ed., IEEE Trans. Fuzzy Systems (Special issue) 2, 1-45 (1994). 

Systems that obey Eq. (1) are called crisp systems and systems that obey Eq. (2) are called fuzzy systems. The real world is actually a mbcture of crisp and fuzzy systems. 

The architecture of an ANFIS model is shown in Figure 14.4. As can be seen, the proposed neuro-fuzzy model in ANFIS is a multilayer neural network-based fuzzy system, which has a total of five layers. The input (layer 1) and output (layer 5) nodes represent the descriptors and the response, respectively. Layer 2 is the fuzzification layer in which each node represents a membership. In the hidden layers, there are nodes functioning as membership functions (MFs) and rules. This eliminates the disadvantage of a normal NN, which is difficult for an observer to understand or to modify. The detailed description of ANFIS architecture is given elsewhere (31). 

Figure 2.10 Plot of how expectations for successful new technologies often change with time. (See Bezdek, J. IEEE Fuzzy Systems 1993, 1, 1-5.)...

Here, the premise is described by a membership function for the linguistic variable high and the function for the detection limit is the sum of the blank signal, y, and three times the standard deviation of the blank signal, Sg, (cf. Eq. (4.3)). Optimization of the parameters in the premise part of the rules is adaptively done by combining the fuzzy rule-based system with a neural network. Consider an adaptive neuro-fuzzy system with two inputs, and... 

Here we consider again the data in two dimensions (Figure 8.24a) given for a four-class problem in Example 5.12 on /r-NN classification. For validation of the neuro-fuzzy system, the 200 cases were divided into a training data set of 160 cases and a test data set of 40 cases. 

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