2nd Order System Example 4

Problem

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.

Solution

Transfer function is developed from the block diagram with below steps (see here for technique):

1. Form equation at summation:
2. Rearrange as output/input:

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.