In order to design an underwater vehicle that has the characteristics of both a long-range transit vehicle (torpedo-like) and a highly maneuverable low-speed vehicle (boxlike), researchers have developed a thruster that mimics that of squid jet locomotion (Krieg, 2008). It has been demonstrated there that the average normalized thrust due to a command step input, U(s) = T_ref/s is given by:

T(t) = T_ref(1 - e_-λt) + a sin(2πf t)

where T_ref is the reference or desired thrust, λ is the system’s damping constant, a is the amplitude of the oscillation caused by the pumping action of the actuator, f is the actuator frequency, and T(t) is the average resulting normalized thrust. Find the thruster’s transfer function

T(s)/U(s). Show all steps.