$7.$ Consider the I$/$O membership functions and rule base for the inverted pendulum problem discussed in class $($the fuzzy pendulum I$/$O membership functions and corresponding rule base are shown on the next page$).$ Suppose, at time $\backslash ($ t $\backslash ),$ we have the following crisp input values $($also indicated on the next page$)$: $\backslash ($ e$($t$)=\backslash$pi $/8,$ d e$($t$)/$ d t$=-3\backslash$pi $/32\backslash ).$

a$)$ Circle which rules are $"$on$,"$ directly on the rule matrix. Write the $"$on$"$ rules out in English. $[5$ points$]$

b$)$ Suppose we want to use sum$-$min inference with COA defuzzification to compute the crisp control actions. Write the general functional form for the output given this strategy $($i$.$e$.,$ set the problem up so that you can solve for a solution$).[5$ points$]$

c$)$ Now use sum$-$min inference with COA defuzzification to compute the specific crisp control action given the inputs provided in part $($a$),$ above. $[5$ points$]$

The Fuzzy Pendulum I$/$O membership functions and corresponding rule base are shown below $-$ Current inputs $/$ measurements also shown on the plots below $($in red$)$

Rule Base

"oentrol force"