Q1 [50pt] For the process shown in figurc 2, a Proportional controller is used, the measurement device (level transmitter) and the control valve liave no dynamics but constant gains of Km and Kv respectively. The process open-loop transfer functions are given below. (a) Define The state variable, the controlled variable, the manipulated variable and the disturbance. (b) Draw the corresponding block diagram for the process. Derive the closed-loop transfer function relating the controlled variable to its set point. Determine the closed-loop static gain and time constant. (c) If A=0.5m2,R=6min/m2,Km=Kv=10, find the values of closed-loop static gain, time corstant and offset for Kc=200. Assume the set point is stepped by unity. (d) Assume now a PI controller is used and the rest are the same. Derive the closed-loop transfer function relating the controlled variable to its set point. Determine the closed-loop damping ratio and natural period of oscillation. Determine the value of K that makes the closed-loop system has a natural period of oscillation of 4 and under-damped response, i.e. =0.8. q2(s)h(s)=Gp(s)=s+1Kpq1(s)h(s)=Gd(s)=s+1KpKp=R,=AR