To explore the performance of a proportional-plus-derivative controller on a second-order system, let G_p(s) = 1/(s(s + 1)) be the transfer function of the plant and Gc(s) = K₁ + K₂s be the controller.

(a) Find the transfer function H(s) = Y(s)/X(s) of a negative feedback system with G_c(s) and G_p(s) in the feed forward path and unity in the feedback. The input and the output of the feedback system are x(t) and y(t) with Laplace transforms X(s) and Y(s).

(b) Suppose (i) K₁ = K₂ = 1; (ii) K₂ = 0, K₁ = 1, and (iii) K₁ = 1, K₂ = 0. For each of these cases indicate the new location of the poles and zeros, and the corresponding steady-state responses due to x(t) = u(t). Comparing the absolute value of the steady-state error |ε(t)|= |y_ss(t)−1|for the three cases, which of (i)–(iii) gives the largest error?