MCQs on Control Systems

01․ Find the relation of damped natural frequency (ω_d)

Damped natural frequency (ω_d) = (ω_n) √(1-ζ²)

02․ A second order control system has F(jω) = 100/(100 - ω² + 10√2 jω). Find the resonant peak?

Resonant peak is the maximum value of magnitude occurring at resonant frequency ω_r. Resonant peak µ_r = 1/(2ζ√(1-ζ²)) Given transfer function F(jω) = 100/(100 - ω² + 10√2 jω) Compare to standard second order equation, Damping ratio ξ = 1/√2 Resonant peak = 1

03․ For a second order under damped system subjected to unit step input the time response has first peak value to be 4 times the second over shoot. Determine the damping ratio ξ ?

First peak value = 1 + M_p1 Second peak overshoot = M_p2 Where, Mp = Peak over shoot = e^{-nξπ/√(1-ζ²)} M_p1 = e^{-ξπ/√(1-ζ²)} M_p2 = e^{-3ξπ/√(1-ζ²)} From give data, 1 + M_p1 = M_p2 By solving above equation, Damping ratio ξ = 0.087

04․ Find the bandwidth of the system, when rise time of the system is give as 1msec?

Bandwidth is the range of frequencies over which the magnitude has a value of 1/√2. Band width indicates the speed of response of the system. Wider bandwidth indicates that, faster response. Bandwidth = 0.35/Rise time = 0.35/(1*10^-3) = 350 Hz

05․ If the bandwidth of the system is very large, the system response is

Bandwidth is the range of frequencies over which the magnitude has a value of 1/√2. Band width indicates the speed of response of the system. Wider bandwidth indicates that, faster response. Bandwidth = 0.35/Rise time

06․ The magnitude of a transfer function G(jω)H(jω) at gain cross over frequency is

Stability from frequency response plots 1 + G(s)H(s) = 0 G(s)H(s) = -1 + j0 Put s = jω G(jω)H(jω) = -1 + j0 |G(jω)H(jω)| at gain cross over frequency = 1 ∠G(jω)H(jω) at phase cross over frequency = -180°

07․ Find the conditions for the system to be stable?

Stable system: Gain margin and phase margins are positive and Gain cross over frequency is less than phase cross over frequency. Unstable system: Gain margin and phase margins are negative and Gain cross over frequency is greater than phase cross over frequency. Marginally stable system: Gain margin and phase margins are zero and Gain cross over frequency is equal to phase cross over frequency.

08․ A second order under damped system has damping ratio of 0.3. Find the phase margin of the system?

Phase margin is the allowable phase lag. ∠G(jω)H(jω) at gain cross over frequency = φ Phase margin = 180° + φ And also Phase margin ≈ 100*ξ Therefore, for given system Phase margin = 30°

09․ If the magnitude of G(jω)H(jω) at phase crossover frequency is 0.5. Find the gain margin of the given system?

Gain margin is the allowable gain. |G(jω)H(jω)| at phase crossover frequency = X Gain margin = 1/X Gain margin in db = 20 log(1/X) For given system gain margin = 1/0.5 = 2

10․ A second order system has peak over shoot = 50% and period of oscillations 0.2 seconds. Find the resonant frequency?

Peak over shoot = 0.5 Peak over shoot M_p = e^{-nξπ/√(1-ζ²)} = 0.5 Damping ratio ξ = 0.215 Time = 0.2 f_d = 1/0.2 = 5 Hz Damped frequency ω_d = 2πf_d = 31.41 rad/s Damped frequency ω_d = (ω_n) √(1-ζ²) Natural frequency (ω_n) = 32.16 rad/s Resonant frequency (ω_r) = (ω_n) √(1-2ζ²) = 30.63 rad/s