Adaptive fuzzy controller
In this chapter, we have what an adaptive fuzzy controller is and how it works. Adaptive Fuzzy Controller is designed with adjustable parameters as well as a builtin mechanism to adjust them. The adaptive controller has been used to improve the performance of the controller.
Basic steps for implementing the adaptive algorithm
Now let's see the basic steps for implementing the adaptive algorithm.

Collecting observable data  Observable data is collected to calculate controller performance.

Controller Parameter Adjustment  Now with the help of controller performance, controller parameter adjustment calculation would be done.

Improvement in controller performance  In this step, the controller settingsare adjusted to improve controller performance.
Operational Concepts
Design of a controller is based on an assumed mathematical model that looks like a real system. The error between the real system and its mathematical representation is calculated and if it is relatively insignificant, the model is supposed to work efficiently.
A threshold constant which sets a limit for the efficiency of a controller also exists. Command input is fed into both the real system and the mathematical model. Here, suppose that $ x left (t right) $ is the output of the real system and $ y left (t right) $ is the output of the mathematical model. Then the error $ epsilon left (t right) $ can be calculated as follows 
$$ epsilon left (t right) = x left (t right)  y left (t right) $ $
Here, the desired $ x $ is the output we want from the system and $ mu left (t right) $ is theoutput from the controller and going to both the real and math model.
The following diagram shows how the fun ction error is tracked between the output of a real system and the mathematical model 
System settings
A fuzzy controller whose design is based on the fuzzy mathematical model will have the following form of fuzzy rules 
Rule 1  IF $ x_1 left (t_n right) in X_ {11}: AND ... AND: x_i left (t_n right) in X_ {1i} $
THEN $ mu _1 left (t_n right) = K_ {11} x_1 left (t_n right) + K_ {12 } x_2 left (t_n right): + ... +: K_ {1i} x_i left (t_n right) $
Rule 2  IF $ x_1 left (t_n right) in X_ {21}: AND ... AND: x_i left (t_n right) in X_ {2i} $
THEN $ mu _2 left (t_n right) = K_ {21} x_1 left (t_n right) + K_ {22} x_2 left (t_n right): + ... +: K_ {2i} x_i left(t_n right) $
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Rule j  IF $ x_1 left (t_n right) in X_ {k1}: AND ... AND: x_i left (t_n right) in X_ {ki} $
THEN $ mu _j left (t_n right) = K_ {j1} x_1 left (t_n right) + K_ {j2} x_2 left (t_n right): + ... +: K_ {ji} x_i left (t_n right) $
The above set of parameters characterizes the controller.
Mechanism tuning
Controller parameters are adjusted to improve controller performance. The process of calculating the adjustment of parameters is the adjustment mechanism.
Mathematically, let $ theta ^ left (n right) $ be a set of parameters to be adjusted at time $ t = t_n $. The adjustment can be the recalculation of parameters,
$$ theta ^ left (n right) = Theta left (D_0,: D_1,: ...,: D_n right) $$
Here $ D_n$ are the data collected at time $ t = t_n $.
Now this for mulation is reformulated by updating the parameter set to its previous value as,
$$ theta ^ left (n right) = phi (theta ^ {n 1},: D_n) $$
Parameters for selecting an adaptive fuzzy controller
The following parameters must be taken into account to select an adaptive fuzzy controller 

Can the system be approached entirely by a fuzzy model?

If a system can be approached entirely by a fuzzy model, are the parameters of that fuzzy model readily available or do they have to be determined online?

If a system cannot be approximated entirely by a fuzzy model, can it be piecewise approximated by a set of fuzzy models?

If a system can be approached by a set of fuzzy models, do these models have the same format with different settings or do they have different formats?

If a system can be approached by a set of fuzzy models having the same format, each with a different sets of parameters, are these sets of parameters readily available or should they be can they be determined online?