Fuzzy inference rules

Define the set of fuzzy inference rules (Fuzzy Rule Base, FRB), which relate the linguistic terms (fuzzy sets) of the input physical variables to those of the output level of the IF considered. 

ABSTRACT Smart structures usually incorporate some control schemes that allow them to react against disturbances. In mechanics we have in mind suppression of mechanical vibrations with possible applications on noise and vibration isolation. A model problem of a smart beam with embedded piezoelectric sensors and actuators is used in this paper. Vibration suppression is realized by using active control. Classical mathematical control usually gives good results for linear feedback laws under given assumptions. The design of nonlinear controllers based on fuzzy inference rules is proposed and tested in this chapter. 

Fuzzy inference rules systematize existing experience, available in terms of linguistic rules, and can be used for the realization of nonlinear controllers. The feedback is based on fuzzy inference and may be nonlinear and complicated. Knowledge or experience on the controlled system is required for the application of this technique. Since the linguistic rules are difficult to be explained and formulated for multi-input, multi-output systems, most applications are based on multi-input, single-output controllers. 

The knowledge to be used by the system is formulated in terms of fuzzy inference rules. There are two distinct ways in which these fuzzy rules can be determined (a) by using the experience of human operators and (b) by obtaining them from empirical data found through suitable learning, for example, using neural networks. As stated before, the canonical form of the fuzzy inference rules is... 

For each fuzzy inference rule, the fuzzy sets associated with each input variable are applied to their actual values, to determine the degree of truth for each rule premise. The degree of truth for a rule premise, sometimes denoted by a, is computed as a = Ti(A(ei), 6( 2)), where Ti is a Tnorm. If ot for a certain rule is nonzero, then the rule is said to fire. 

Using these operators, fuzzy inference mechanisms are then developed to manipulate rules that include fuzzy values. The largest difference between fuzzy inference and ordinary inference is that fuzzy inference aUows partial match of input and produces an interpolated output. This technology is useml in control also. See Ref. 94. 

Fuzzy Logic Control The apphcation of fuzzy logic to process control requires the concepts of Fuzzy rules and fuzzy inference. A fuzzy rule, also known as a fuzzy IF-THEN statement, has the form ... 

In order to combine the conclusions of several such Fuzzy Inference Systems, they will be translated into belief structures, according to the method proposed in [22]. Each Fuzzy Inference System presented Figure 12 is the association of fuzzification functions (providing a numerical evaluation of the membership of a variable to fuzzy sets) and of rules linking these observations to different classes which can be states or disjunctions of states of the process. 

The rules of a Fuzzy Inference System link observations on the space of observed variables to singletons or disjunctions of singletons from the frame of discernment 17. It means that a rule can be expressed by ... 

Reasoning based on fuzzy propositions of the four types, possibly quantified by various fuzzy quantifiers, is usually referred to as approximate reasoning. Although approximate reasoning is currently a subject of intensive research, its basic principles are already well established. For example, the most common inference rules of classical logic, such as modus ponens,... 

Fuzzy inference systems are also known as fuzzy associative memories, fuzzy models, fuzzy-rule-based systems, or fuzzy controllers. 

The results of this rule-base operation are then post-processed using a fuzzy inference engine to obtain the final prediction. 

In Fig. 2, the fuzzy inference flow from linguistic variable fuzzification to defuzzification of the aggregate output is shown it proceeds up from the inputs in the lower left, then across each row, or rule, and then down the rule outputs to end in the lower right. 

Figure 9 shows the complete process of developing a fuzzy inference system (FIS) for sorting and grading process using MATLAB [25]. The process consist of three main steps deflning the input and output in Membership Function Editor, set the fuzzy rule in Rule Editor, and obtaining the output for each rule in Rule and Surface Viewer. 

Based on the distance between the point under test and the reference, a weight for that reference will be chosen. This is done after calculating the grades of membership of the point of the FM under test using the fuzzy inference. In this stage the following simple IF THEN rules are used ... 

Figure 4 Control Force Membership Function (output). Table I Fuzzy Inference System Rules.

Zadeh (1965) introduced fuzzy set theory to treat imprecision and uncertainty that is often present in implementation of problems in real world. Mamadani (1974), by applying Zadeh s theories of linguistic approach and fuzzy inference, successfully used the if-then rule on the automatic operating control of steam generator. Since then fuzzy control theory has been applied to anumber of linear and nonlinear systems. 

Radial basis functions networks are good function approximation and classification as backpropagation networks but require much less time to train and don t have as critical local minima or connection weight freezing (sometimes called network paralysis) problems. Radial basis fimction CNNs are also known to be universal approximators and provide a convenient measure of the reliability and confidence of its output (based on the density of training data). In addition, the functional equivalence of these networks with fuzzy inference systems have shown that the membership functions within a rule are equivalent to Gaussian functions with the same variance (o ) and the munber of receptive field nodes is equivalent to the number of fuzzy if-then rules. 

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