Question: Prove the following statements and give an explanation for each in your own words by referring to a sample physical system.

(a) The phase lag for a gain is zero.

(b) The amplitude ratio for a first-order system goes to zero as the frequency goes to infinity.

(c) The amplitude ratio for a second-order system with a damping coefficient of 0.50 is not monotonic with frequency.

(d) The phase angle decreases without limit as the frequency increases for a dead time.

(e) For an integrator, the amplitude ratio becomes very large for low frequencies and becomes very small for large frequencies

(f) The amplitude ratio for a PI controller becomes very large at low frequencies.