There does not seem to be a consistent mathematical theory about fuzzy logic. Logic inference under uncertainty can not be carried over more than several steps even if it is done mathematically correctly. Moreover we have evidence in the conditional radom variable research to suggest that mathmetically correct inference under uncertainty is not possible). Our view is that fuzzy logic could be viewed as a set of huristics aimed in interpolating existing control laws when precision is not essential. See the writeup below.
Rationale for fuzzy logic control:
Any control law can be implemented with "if-then" rules. But the crisp "if-then" rules are too rigid and too sensitive for practical use. To achieve the robustness one needs to "softern" the edges for the input to the "if-then" rules, and this leads to the fuzzy set representation of the input data. To compute the fuzzyness of the output for the "if- then" rules we need a calculus, which is the fuzzy logic. To understand that rule-based control could tune a PID controller, see the following set of linguistic proctocols for such tuning. This set of protocol is from "Intelligent Control" by Clarence W. De Silva, CRC, 1995.
If the response oscillations are low then slightly decrease the propor- tional gain and slightly increase the derivative time constant; or if the response is moderately oscillatory, then moderately decrease the proportional gain and moderately increase the derivative time constant; or if the response is highly oscillatory, then make a large decrease in the proportional gain and a large increase in the derivative time constant; or if the response is extremely oscillatory, then make a large decrease in the proportional gain and a large increase in the derivative time constant and a slight decrease in the integral rate.
If the response is slow then increase the derivative time constant slightly and increase the proportional gain moderately; or if the speed of response is moderate then increase the derivative time constant slightly and increase the proportional gain slightly; or if the speed of response is high increase the derivative time constant slightly and decrease the proportional gain slightly; or if the speed of response is very high decrease the proportional gain moderately.
If the response diverges slowly then slightly decrease the proportional gain and slightly increase the derivative time constant; or if the response diver- ges moderately then slightly decrease the proportional gain and moderately increase the derivative time constant; or if the response diverges rapidly then slightly decrease the proportional gain and increase the derivative time constant by a large amount and slightly decrease the integral rate; or if the response diverges very rapidly then moderately decrease the propor- tional gain and increase the derivative time constant by a large amount and moderately decrease the integral rate.
If the offset is low then slightly increase the proportional gain, and sli- ghtly increase the integral rate; or if the offset is moderate then moderately increase the integral rate; or if the offset is high then increase the propor- tional gain slightly and increase the integral rate by a large value.; or if the offset is very high then moderately increase the proportional gain and increase the integral rate by a large value. Bla Bla Bla
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