Q1. Consider the liquid-level control system shown in Figure 1, where R1 and R2 are water resistances, C1 and C2 are the capacitance of tanks 1 and 2, respectively. A proportional controller is added to regulate the inflow water rate (q; (t) = Kha ()), assuming the changes in the variables are small. Consider disturbance water inflow as input and water height in tank 2 as output. List all system equations and the Laplace transform of system equations; Draw the block diagram of the control system. (Please construct the block diagram from system equations directly, do not derive or combine the equations. Do not need to reduce it) Proportional controller Q2 Reduce the following block diagram to get its transfer function. Y + + G, G, E G3 G R + H H3Question 2 0.5 pts Thermal systems are those that involve the transfer of heat from one substance to another. Thermal systems may be analyzed in terms of resistance and capacitance. Consider a thermal system as shown in the following figure, cold liquid flows through the thermal system, and is heated up to the desired temperature by the heater. @, () is the temperature of the outflowing liquid, h (t) is the heat input rate (by Heater); R is the thermal resistance and C is the thermal capacitance. Heater Cold liquid (a) (b) If considering the heat input rate as the system input and the outflowing liquid temperature as the system output, the transfer function of the system can be presented by: H(s) ata In this system the thermal resistance R=12.7 C sec/kcal, and the thermal capacitance C=1.9 kcal/ C. obtain the expression of the system response to a step input x,=8.2 calculate the settling time of this system to a unit step input, and fill the value to the following blank (please keep one digit after the decimal point). It is known that the thermal resistance R is related to the liquid flow rate (G) kg/sec and the specific heat of liquid (c) kcal/kg C by the following relationship. If the liquid flow rate increases in the system, will the system's response time be faster or slower? Please explain your judgment with the necessary calculation. 4 R= &, A spring-mass-damper system has transfer function as follow: _ k TF = (3s?-++4s-+k) Spring constant is found to be k= 0.6 If step input H= 2.6 is applied to this system, find out the expression of system response, and fill the following blank with the state-state response of this system under the given step input. (please keep two digits after decimal point)