Exercise A liquid-level process control system is shown in Figure 1 below. The system parameters are: A=2m, RF = 15 s/m, H = 1 V/m (unity gain), Kv=0.1 m/sV, K = 1 (controller gain). Rf 1+ ARfs a) Model the system and obtain the transfer function b) What are the values of T, and when the undamped natural frequency wn is 0.1 rad/s? (Hint: Use the block diagram to obtain the forward path transfer function G(S), then use G(s) to obtain the closed loop transfer function for the system [Ha(s)/Hd(s)]). c) Find an expression for the time response of the system when there is a step change of ha(t) from 0 to 4m. Assume zero initial conditions. d) Use MATLAB to plot the time response for the system. h0 he(0) He(s) PI Controller H_(s) E(s) LI(1) Control Valve K Pressure transducer H Ha -(s) U(s) /HIN Tank Area A Figure 1: Liquid level control system. Note: V(t) is the inflow to the tank, K, is the valve gain, v(t) is the tank outflow, R, is the outlet valve resistance, Ha(t) is the height of liquid in the tank. PI Controller H Control valve K VI(1) Pressure tranducer Outlet valve Resistance R V(s) /1\ V (1) Tank and valve Figure 2: System block diagram h(1) R 1+ARS H(s)