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One can also anticipate harmonic models governed by the trigonometric functions

The generality of the mapping can be increased even further by utilizing neural networks in the condition part of the rule

- if condition is A then y = NN(x, connections)

(the right hand side of the rule is practically a highly nonlinear mapping furnished by the neural network).

Instead of static dependences we can admit differential or difference equations in the conclusion part of the rule, say

if condition is A then = f (x, param)

or

if condition is A then x (k+1) =B x (k) +C u (k)

(here “k” stands for a discrete time moment and u(k) is a sequence of control actions).

4.4.2. A design of fuzzy rule - based systems

Assuming that a collection of if-then statements is provided (we can relax this request afterwards by considering their learning mechanisms), arises a fundamental question of the design of the rule - based systems. The literature in this area is abundant; the reader can find plenty of useful hints and could be easily mislead by the meaningless technical variations that make some methods look new even though they are nothing but slight variations of some fundamental approaches. Our intent is to get a better grasp on the design of rule - based systems, underline main design problems and eventually study some algorithms. Let us emphasize first (that is nothing new and astonishing yet it puts the design in a proper perspective) that fuzzy sets are abstract mathematical concepts useful in capturing linguistic domain knowledge about a problem. Do fuzzy sets exist ? The same applies to probabilities -all of these are conceptual and helpful artifacts. This invokes two fundamentally distinct development framework within which fuzzy rule - based systems are to be constructed and tested:

(i)  conceptual level of fuzzy sets. Here we anticipate that the system is built and tested vis a vis linguistic information. This design mode is often present in expert systems and decision support systems
(ii)  physical level of systems. Here we anticipate that the conclusion needs to be provided in a numeric way; the antecedents can be either numeric or linguistic. This mode of design is prevalent in the design of fuzzy controllers. At the end of the design the resulting constructs have to interact with the numeric rather than linguistic world. As such all such rule - based systems are essentially nonlinear numeric mappings between the input variables and the outputs. It is worth underlining that the factor of fuzziness resides within the designer and his domain knowledge about the problem and the way he perceives the control protocol as well as expresses specific control objectives.

Depending upon the type of the design and application framework, we exercise two relevant testing strategies, Fig. 4.4.


Figure 4.4  Fuzzy rule-based systems and their interaction with an environment

At the linguistic level the goal is to design a rule - based system such that it maps all rules

  if condition is Ak then conclusion is Bk

to the highest extent. This means that if A = Ak then the result of inference supported by the system should return B that is the same as Bk (or follows Bk as close as possible). In other words, we require that

Bk = rule_based_system(Ak)

for all the rules in the protocol, k=1, 2, ..., N. Note that the testing is fully fuzzy set oriented. The testing at the level of numeric quantities is somewhat more convoluted as the performance of the inference processing going on within the rule-based system is somewhat masked by the encoding and decoding procedures, Fig. 4.4 (these two schemes are usually referred to as a fuzzification and defuzzification). The best mapping criterion is applied against the numeric outputs of the rule-based system, meaning that if x =xk then the output y

should follow yk as close as possible, k=1, 2, ..., N. Where (xk, yk) are the elements of the numeric testing set against which the system has to be verified. Here and denote the decoding and encoding algorithms; for more details refer also to Chapter 3.

The implementation of the rule-based mapping is done in many possible ways. The main criterion uses a notion of transparency of the mapping. On one hand, there are pure logic-inclined constructs. As driven by logic features, these are highly transparent as far as interpretation aspects are concerned. They are far less flexible and may not well approximate (represent) all the rules in the system. The examples of this implementation track include fuzzy logic and fuzzy relational equations. On the other hand, we have neural network realizations that become quite dominant. By their nature, those are highly flexible yet difficult to interpret constructs. Obviously, the logical spirit of rule-based computing is not supported at all. There are also some realizations in-between. For instance, fuzzy Hebbian learning maintains a relatively high modularity of the implementation (that helps interpret the structure). It can be flexible once equipped with some extra parameters. Symbolically, these two features of transparency and flexibility allow us to locate any implementation in the flexibility -transparency plane, Fig. 4.5. When going for any new realization, it is worthwhile to identify its place in this plane to realize better its eventual advantages and disadvantages.


Figure 4.5  Realizations of rule-based systems in the transparency - flexibility plane

To gain a better insight into the algorithmic details of the implementation of rule-based systems, we discuss a Hebbian-like scheme of fuzzy associative memories.


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