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# Proportional-Integral-Derivative Control

Consider the block diagram in Figure Q2.1, describing a process under Proportional-Integral-Derivative control.

1. Is the system open loop stable?

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 Kf = 15. A Bode plot of the open loop transfer function with these parameter values is shown in Figure Q2.2. Sketch a Nyquist plot for this system and use this sketch to determine if the system is closed loop stable. (Only answer this question in the attached document).

#### Solution Preview

1) For the system with poles right at the origin, special attention is required. For system with only one pole at the origin, it is an all pass system, which is still marginally stable.

For the open loop of the system there are three poles 0, -2.6180 and -0.3820. Since there is a pole at origin it will be marginally stable.

2) For Ki= 10, the transfer function will be (Kds^2+ Kps +10)/(s^3+3s^2+s)

Using Routh ...

#### Solution Summary

Solution includes calculations for all three questions and Nyquist diagram for (3).

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