Consider the transfer function of a PT2 system:
Various technical systems can be approximated by a PT2-System, for example a damped mass-spring oscillator. Here the natural frequency is 20 [rad/s] and the damping factor is 0.05. The plant is stable and minimum-phase.
Your mission, should you decide to accept it, is to design an H∞ controller that renders the closed loop stable and satisfies the control targets below. In a first step, you have only one degree of freedom in choosing the weights and one control objective (the steady-state offset). When that goal is achieved, in a second step two additional degrees of freedom become available, and all control objectives have to be achieved.