(Seborg et al., Ex. 12.1, p291)

A process has the transfer function G(s)=K/(10s+1)(5s+1) where K has a nominal value of one. PID controller settings are to be calculated using the Direct Synthesis approach with τc = 5 min. Suppose that these controller constants are employed and that K changes unexpectedly from 1 to 1+α.

 

a) For what value of α 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 |α|≤0.2. What is the smallest value of τc that can be used?

 

c) What conclusions can be made concerning the effect that the choice of τc has on the stability of the closed-loop system to changes in steady-state gain K?

 

 먼저 다이렉트 신테시스로 Gc를 구하면 (10s+1)(5s+1)/5Ks 가 된다고 생각했는데 이걸 G(s)와 곱해서 캐릭터EQ를 구하고자 하면 K가 사라져 버립니다...아니면 G(s)의 K는 1로 두고 Gc의 K만 살려두면 pole이 -1/5k만 나옴에 1과 1+a에서 무슨 연결고리가 있는지 알 수가 없습니다...어떻게 접근을 해야 할까요..?