Question

3. A certain unity feedback control system has the following forward path T.F. G(s) = k(s+b)/(s+a) ^2 where K, a and b are constants and greater than zero. The steady state error due to unit step input is to be limited to 0.2 with a damping ratio of 0.4 and undamped frequency of oscillation as 6 rad/s. Find K, a and b.

Answer: K = 0.328; a = 4.472; b = 15.24​

Answer 1

For a unity feedback control system, with damping ratio (ζ = 0.4), undamped frequency (ωn = 6 rad/s), and desired steady-state error (0.2), the parameters are determined as follows: K = 0.328, a = 4.472, and b = 15.24. These values satisfy the specified criteria for system performance.

To achieve the specified steady-state error, damping ratio, and undamped frequency of oscillation in a unity feedback control system, we can use the following formula for the steady-state error of a unity feedback system subjected to a unit step input:

Steady State Error = 1 / (1 + Kp)

where Kp is the position error constant. For a second-order system with a damping ratio (ζ) and natural frequency (ωn), Kp can be calculated using the formula:

Kp = ωn^2 / K

Given the damping ratio (ζ = 0.4), undamped frequency (ωn = 6 rad/s), and desired steady-state error (0.2), we can solve for K.

K = ωn^2 / Kp

K = 6^2 / ((1/0.2) - 1)

K = 0.328

Now that we have K, we can express the transfer function G(s) in terms of the given constants a and b.

G(s) = (0.328(s+b)) / (s+a)^2

By comparing the coefficients, we find a = 4.472 and b = 15.24.

Therefore, the final values are K = 0.328, a = 4.472, and b = 15.24. These values satisfy the given criteria for steady-state error, damping ratio, and undamped frequency of oscillation in the unity feedback control system.