Transfer function of controller
In fact, what you define as G(s) in your solution is just F(s):    

$$F(t)= 100 * (1-0,3e^{-6t}-0,7e^{-10t})u(t)$$ and $$u(t) = 1 $$    

u(t) is the step function
When you apply the laplace transform you will get:`

$$F(s)=\frac{100}{s}-\frac{30}{s+6}-\frac{70}{s+10}=G(s)\frac{1}{s}$$ 1/s is the transformation of the step function.
So, to find G(s) you have to multiply by s the result of F(s).