Please explain it step by step in detail as much as much as possible.

1 Exercise 12.1 in the textbook A process has the transfer function: G(s)=(10s+1)(5s+1)K where K has a nominal value of K=1. PID controller settings are to be calculated using the Direct Synthesis approach with c=5min. Suppose that these controller constants are employed and that K changes unexpectedly from 1 to 1+. (a) For what values of will the closed-loop system be stable? (b) Suppose that the PID controller constants are calculated using the nominal value of K=1 but it is desired that the resulting closed-loop system be stable for 0.2. What is the smallest value of c that can be used? (c) What conclusions can be made concerning the effect that the choice of c has on the robustness of the closed-loop system to changes in steady-state gain K