Hi!
I have had this question, ever since I started my research in adaptive fuzzy controllers and that is--- How shall I construct the fuzzy rule to approximate the unknown dynamics- f(x) and g(x) in the dynamical system. If I start off with lets say 3 MFs to approximate the dynamics and if lets say I have some 9 states that completely describe the system dynamics, then, do I need to take all the possible combinations i.e 3^9 as number of fuzzy rules? Or just 3? Or any number between 3 an 3^9? What should be the optimum number of fuzzy rules to approximate the unknown dynamics?
For example, IF there are 3 fuzzy sets A1, A2, A3 on x universe and 3 fuzzy sets B1, B2, B3 on ẋ universe, THEN the adaptive fuzzy controller consists of 32 = 9 rules. The parameter vector is given by:
Θ = [θ1, θ2, θ3, θ4, θ5, θ6, θ7, θ8, θ9].
So if we have 9 such universes, then the fuzzy rules will be 3^9 or we'll need 3^9 parameters?? If so, then don't you think it'll be computationally very expensive?
Yup, this is overambitious and the “curse of dimensionality” has been highlighted previously in your thread about 2 month ago:
https://www.researchgate.net/post/How_can_ANFIS_be_used_for_design_of_adaptive_fuzzy_control_without_training_data
Is your f(x) a vector like f(x) = [f1(x1, x2, x3), f2(x4, x5, x6), f3(x7, x8, x9)]?
If there is no secrecy in the dynamics, can you show the actual f(x) and g(x) here?
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