A simple unity feedback control system has a process transfer function

The system input is a step function with an amplitude A. The initial condition of the system at time t0 is y(t0) = Q, where y(t) is the output of the system. The performance index is defined as

(a) Show that I = (A - Q)2/(2K).
(b) Determine the gain K that will minimize the performance index I. Is this gain a practical value?
(c) Select a practical value of gain and determine the resulting value of the performance index.

2 E(s)