Figure P9.5 shows a two-tank system. The liquid inflow to the upper tank can be controlled using a valve and is represented by F₀. The upper tank’s outflow equals the lower tank’s inflow and is represented by F₁. The outflow of the lower tank is F₂. The objective of the design is to control the liquid level, y(t), in the lower tank. The open-loop transmission for this system is Y(s)/F_o(s) = a₂a₃/s² + (a₁ + a₄)s + a₁a₄ (Romagnoli, 2006). The system will be controlled in a loop analogous to that of Figure P9.1, where the lower liquid level will be measured and compared to a set point. The resulting error will be fed to a controller, which in turn will open or close the valve feeding the upper tank.

a. Assuming a₁ = 0.04; a₂ = 0.0187; a₃ = 1; and a₄ = 0.227, design a lag compensator to obtain a step response steady-state error of 10% without affecting the system’s transient response appreciably.

b. Verify your design through MATLAB simulations.

Fo F1 F2 FIGURE P9,56