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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.

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