The feed forward transfer function of a negative feedback system is G(s) = N(s)/D(s), and the feedback transfer function is unity. Let X(s) be the Laplace transform of the input x(t) of the feedback system.

(a) Given that the Laplace transform of the error is E(s) = X(s)[1 − F(s)] where F(s) = G(s)/(1 + G(s)) is the overall transfer function of the feedback system, find an expression for the error in terms of X(s), N(s), and D(s). Use this equation to determine the conditions under which the steady-state error is zero for x(t) = u(t).

(b) If the input is x(t) = u(t), N(s) = 1, and D(s) = (s + 1)(s + 2) find an expression for E(s) and from it determine the initial value e(0) and the final value lim_t→∞e(t)of the error.