Problem $3(40$ points$)$

Consider a fuzzy logic controller with two inputs and an output designed for an inverted pendulum system. In this controller, the inputs are the error, $($e$),$ and the derivative of the error, $($de$),$ and the output is the force $(f)$ to be applied to the cart. For each input, three membership functions have been used, which are negative $(N),$ zero $($Z$),$ and positive $($P$).$ The output has also $3$ membership functions: negative $($N$),$ zero $($Z$),$ and positive $($P$)$

The membership functions for error, derivative of error, and output force are given as follows: $($Note that $$

The rule base of the Sugeno$-$type fuzzy logic controller is given below:

$\backslash$table$[[,$Derivative of Error$],[,N,Z,P$