A process has the following second order transfer function and is controlled by a PID controller, given by: H(s)=(2s+1)(6s+1)3 a) Use the Direct Synthesis approach to determine the controller settings for a time constant c=2min. For a second order plus-time delay model the controller settings are given by: Kc=K1(c+)1+2,I=1+2,D=1+212 [5 marks] b) Explain why the process represented by Eq. (2.1) is stable. [2 marks] c) Suppose the process gain changes from 3 to 3+ in Eq. (2.1). Draw a block diagram of the feedback control system assuming that the transfer function of the control valve GV=1 and the transfer function of the feedback transmitter Gm=1. The transfer function of a PID controller is given by Gc=Kc(1+ls1+Ds). [ 5 marks] d) Write the characteristic equation of the feedback control system. [ 5 marks] e) State the necessary stability condition that must be fulfilled by the characteristic equation. [ 3 marks] f) Determine the values of for which the closed-loop system is stable. [5 marks]