Consider a single tank of water lled by a pump and draining through a hole at the base. The inow to the tank is proportional to the voltage V applied to the pump: F,,,(t) == KmV(t) cma/sec. From Bernouilli's Law, the velocity of the ow out of the tank is unt) = \\/ 29W) CID/sec, where g is the gravitational acceleration in cm/secz, and L(t) is the height of water in the tank in cm. The volume ow rate out is thus Fault) = avzgw) 0313/ sec, Where a is the area of the outow orice in cm2. Since the density of the water doesn't change, conservation of mass can be replaced by conservation of volume. Letting A be the cross-sectional area (m2 ) of the tank, AL(t) = KmV(t) a,/2gL(t). (1) 1. Determine the equilibrium level L0 as a function of a constant applied voltage Va. 2. Linearize (1) about the equilibrium Lo determined in Question 1 with voltage V as the control input. 3. Dene a new state variable a:(t) = L(t) L0, input 11(3) = V(t) - V0 and write the linearized system. The measured output is the water height $(t). 4. Compute the transfer function, P(s), of the linearized system determined in Question 2. \f