B-4-10. Consider the liquid-level control system shown in Figure 4-52. The inlet valve is controlled by a hydraulic integral controller. Assume that the steady-state inflow rate is and steady-state outflow rate is also Q, the steady-state head is H, steady-state pilot valve displacement is t = = 0, and steady-state valve position is Y. We assume that the set point R corresponds to the steady-state head H. The set point is fixed. Assume also that the disturbance inflow rate qd, which is a small quantity, is applied to the water tank at 0. This disturbance causes the head to change from A to H+h. This change results in a change in the outflow rate by qo. Through the hydraulic controller, the change in head causes a change in the inflow rate from 0 to + q. (The integral controller tends to keep the head constant as much as possible in the presence of disturbances.) We assume that all changes are of small quantities. We assume that the velocity of the power piston (valve) is proportional to pilot-valve displacement x, or dy dt = K₁x where K₁ is a positive constant. We also assume that the change in the inflow rate q; is negatively proportional to the change in the valve opening y, or qi = -K₁y where K,, is a positive constant. Assuming the following numerical values for the system, C = 2m², R = 0.5 sec/m², a = 0.25 m, b = 0.75 m, obtain the transfer function H(s)/Qd(s). K₁ = 1 m²/sec K₁ = 4 sec1 Problem 3 (20 points) Solve problem B-4-10. Then answer the following: 1. If the head h is the input to the hydraulic controller and the output from the controller is derive the transfer function of the controller? 2. What is the control action of this controller? b- Y+y +qi 龍 9d H+h C (Capacitance) Figure 4-52 Liquid-level control system. R (Resistance) +40