System model block diagram:
Determine A which results in fastest response with no overshoot. Then, plot the response when the above system is commanded to rotate from 0 to 60 degrees (step input).
Finally, gyro failed and A = 0. Describe what happens.
Transfer function is developed from the block diagram with below steps (see here for technique):
From the previous equation,
Fastest response with no overshoot is achieved when ζ = 1 so,
Block diagram without gyro is:
With the same approach as taken before, transfer function is developed:
which has properties,
ζ = 0, so the response is undamped (sine wave with period 2π*0.5, where 0.5 is ωn). Plots for the original system and the system after gyro failure are shown below:
Response as a function of time are as follows:
Finally, for the critically damped function, y(4τ) = y(8) ≈ 54.505 which is about 9.158% from the steady state value; therefore, it is not true to say that second order response reaches within 2% of steady state value in 4 time constants.