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    Control Systems - Proportional-Integral-Derivative Controlled Process and Nyquist Plots

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    Consider the block diagram attached, describing a process under Proportional-Integral-Derivative control.

    1) Is the system open loop stable? Justify your answer

    2) Let Ki = 10. Use the Routh-Hurwitz criterion to find the range of Kd, and Kp in terms of Kd, so that closed loop stability is achieved.

    3) Suppose that Ki = 10, Kd = 1, and Kp = 15. A Bode plot of the open loop transfer function with these parameter values is shown in the attachment. Sketch a Nyquist plot for this system and use this sketch to determine if the system is closed loop stable.

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    Solution Preview

    1) At this point we assume that the feedback does not exist.
    Note that the PID part can be shown as a single block in the form of (Kds+ Kp+ Ki/s)= (Kds^2+ Kps+ Ki)/s
    Therefore, the open-loop path from R(s) to Y(s) has this transfer function:

    Y(s)/R(s)= [(Kds^2+ Kps+ Ki)/s]*[ 1/(s^2+ 3s+ 1)] =Gs(s)
    Y(s)/R(s)= (Kds^2+ Kps+ Ki)/(s^3+ 3s^2+ s)= G(s)

    If we assume that Ki is a nonzero constant (which makes sense otherwise we ...

    Solution Summary

    This solution is provided in 346 words. It discusses the calculation for the transfer function of the path and uses the Routh-Hurwitz method to find system stability.