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Control Systems Test 6

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Control Systems Test 6
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  • Question 1
    1 / -0
    The loop transfer function of a system is given by G(s)=e0.1ss. The phase crossover frequency is given by
    Solution

    At phase cross over frequency (ωpc),

    ∠G(jωpc) = -180°

    ⇒ -90° - 0.1 ωpc = -180°

    As ωpc is in radians, by converting in to degrees

    90(0.1)(ωpc)180π=180

    90+18ωpcπ=180

    ωpc=5π=π0.2

  • Question 2
    1 / -0
    It is known that the Nyquist plot of a control system does not intersect the negative real axis. Then the gain margin of the system is
    Solution
    • Phase cross-over frequency is the frequency at which the phase angle of the system is -180o.
    • The gain margin is the reciprocal value of the transfer function at the phase cross-over frequency.
    • If any plot is not intersecting the negative real axis, then the phase cross-over frequency will not exist and the gain margin will be ∞ or -∞ depending upon the absolute stability
  • Question 3
    1 / -0

    The open loop transfer function of a unity negative feedback control system is given by

    G(s)=400(1+0.2s)2(s+4)(1+0.5s)2(s2+6s+9)

    The slope of the asymptote between the corner frequency 3 rad/sec and 4 rad/sec in the bode plot of the system is ______ (in dB/dec)
    Solution

    G(s)=400(s+5)2(s+4)(0.2)2(0.5)2(s+2)2(s+3)3

    =64(s+4)(s+5)2(s+2)2(s+3)2

    corner frequencies are: ω = 2, 3, 4, 5

    At ω = 2 rad/sec, slope added = -40 dB/dec

    At ω = 3 rad/sec, slope added = -40 dB/dec

    system slope = -80 dB/dec

    At ω = 4 rad/sec, slope added = 20 dB/dec

    system slope = -60 dB/dec

    At ω = 5 rad/sec, slope added = 40 dB/dec

    system slope = -20 dB/dec

    Slope of the bode plot between ω = 3 and ω = 4 is -80 dB/dec

  • Question 4
    1 / -0

    A unit feedback system has a loop transfer function G(s)=πe0.5ss 

    Which of the following statements is/are true?

    Solution

    Concept:

    Gain margin (GM): The gain margin of the system defines by how much the system gain can be increased so that the system moves on the edge of stability.

    It is determined from the gain at the phase cross over frequency.

    GM=1|G(jω)H(jω)|ω=ωpc

    Phase crossover frequency (ωpc): It is the frequency at which phase angle of G(s) H(s) is -180°.

    G(jω)H(jω)|ω=ωpc=180

    Phase margin (PM): The phase margin of the system defines by how much the phase of the system can increase to make the system unstable.

    PM=180+G(jω)H(jω)|ω=ωgc=180

    It is determined from the phase at the gain cross over frequency.

    Gain crossover frequency (ωgc): It is the frequency at which the magnitude of G(s) H(s) is unity.

    |G(jω)H(jω)|ω=ωgc=1

    Important Points:

    • If both GM and PM are positive, the system is stable (ωgc < ωpc)
    • If both GM and PM are negative, the system is unstable (ωgc > ωpc)
    • If both GM and PM are zero, the system is just stable (ωgc = ωpc)

     

    Calculation:

    G(s)H(s)=πe0.5ss

    |G(jω)H(jω)|=πω

    ⇒ ωc = π rad / sec

    G(jω)H(jω)=π20.5ω=180

    900.5(ω)(180π)=180

    ⇒ ωpc = π rad / sec

    Gain margin =GM=1|G(jω)H(jω)|=ωpcπ=1 

    = 0 dB

    Phase margin = 180 + ∠G(jω)H(jω)

    = 180 + (-180) = 0°

    The system is marginally stable.

  • Question 5
    1 / -0
    A unity feedback system has the open loop transfer function G(s)=1(s1)(s+2)(s+3). How many times the Nyquist plot of G(s) encircles the origin?
    Solution

    G(s)=1(s1)(s+2)(s+3)

    For Nyquist plot of G(s), enrichment is

    N = P – Z

    Where P = right side poles

    Z = right side zeros

    Z = 0, P = 1.

    ⇒ N = 1 – 0 = 1

    Therefore, Nyquist plot encircles only once.
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