Is a positive PID (proportional, integral and approximate derivative) controller (plus unit negative feedback) able to control your plant?

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No, the controller C(s) would be non-causal

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Partially, the complete control system T(s) is stable and precise to step, but the system is both too slow and too oscillatory (or with too much overshoot)

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Partially, the complete control system T(s) is stable and precise to step, but the system is too oscillatory (or with too much overshoot)

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No, the controller C(s) would be unstable

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Partially, the complete control system T(s) is stable but precision (and oscillations) are bad

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No, the stability range for the complete control system T(s) is too narrow, unattainable in real life

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The automatic control system T(s) is stable and not great but the best that I could find, so I have to take it

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Yes, for the adequate placement and gain, this controller is able to obtain an automatic control system T(s) stable, precise enough and with both acceptable response speed and small enough (or zero) oscillations

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No, the controller C(s) would be absolutely unstable

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Partially, the complete control system T(s) is stable and precise to step, but the system is too slow

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No, the controller C(s) would need to supply outputs physically impossible for this system

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No, the controller C(s) would be too expensive

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No, the resulting control system T(s) would be non-causal

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No, the resulting control system T(s) would be unstable for every gain

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Is a positive PID (proportional, integral and approximate derivative) controller...

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