Consider a negative feedback system used to control a plant with transfer function

G(s) = 1/(s(s + 1)(s + 2)).

The output y(t)of the feedback system is connected via a sensor with transfer function H(s) = 1 to a differentiator where the reference signal x(t) is also connected. The output of the differentiator is the feedback error e(t) = x(t) − v(t) where v(t) is the output of the feedback sensor

(a) Carefully draw the feedback system, and find an expression for E(s), the Laplace transform of the feedback error e(t).

(b) Two possible reference test signals for the given plant are x(t) = u(t) and x(t) = r(t), choose the one that would give a zero steady-state feedback error.

(c) Use numeric MATLAB do the partial fraction expansions for the two error functions E₁(s), corresponding to when x(t) = u(t) and E₂(s) when x(t) = r(t). Use these partial fraction expansions to find e₁(t) and e₂(t), and thus verify your results obtained before.