Figure 7.43 shows a liquid-level system in which two tanks have hydraulic capacitances

C₁ and C₂, respectively. The volume flow rate into tank 1 is q_i. The liquid flows from tank 1 to tank 2 through a valve of linear resistance R₁ and leaves tank 2 through a valve of linear resistance R₂. The density ρ of the liquid is constant.

a. Derive the differential equations in terms of the liquid heights h₁ and h₂. Write the equations in matrix form.

b. Assume that the volume flow rate q_i is the input and the liquid heights h₁ and h₂ are the outputs. Determine the state-space form of the system.

c. Construct a Simulink block diagram to find the outputs h₁(t) and h₂(t) of the liquid-level system. Assume that ρ = 1000 kg/m³, g = 9.81 m/s², C₁ = 0.15 kg・m²/N, C₂ = 0.25 kg・m²/N,

R₁ = R₂ = 100 N・s/(kg・m²), and initial liquid heights h₁(0) = 1 m and h₂(0) = 0 m. The volume flow rate q_i is a step function with a magnitude of 0 before t = 0 s and a magnitude of

0.25 m³/s after t = 0 s.