A concentration control loop in an industrial chemical process can be represented by a second-order dynamics with a long transmission delay. The two-time constants of the second-order dynamics are T₁ = 4 min and T₂ = 6 min respectively. The steady-state gain is 5 and the dead time is 12 min. The loop is to be controlled to achieve the desired dynamics of first order with time constant T_d = 2 min and zero steady-state offset for a step reference.

Q1
Identify the plant transfer function and the desired closed-loop system transfer function. Then, design a feedback control system with the controller designed using Direct Synthesis method, where the time-delay is approximated using the first-order Taylor expansion. Implement the controller with a standard industrial PID controller.