Arm Like Example

The questions below are due on Thursday April 06, 2023; 11:59:00 PM.

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An example similar to the Copter Arm.

When we included the thrust-buildup in the model for the copter-arm, the transfer function was not too different from

H(s) = \frac{200}{(s+10)s^2}

where we have assumed a \gamma = 20 and a continuous time thrust build-up pole of -10.

Is there any K_0 for which K_0H(s) has a positive phase margin?

Is there any K_0 for which K_0*H(s) has a positive phase margin?

Suppose we have three possible compensators:

K_A(s) = 10\frac{s+1}{s+10}

K_B(s) = 10\frac{s+5}{s+50}

K_C(s) = 10\frac{s+10}{s+100}

which compensator would produce the best phase margin for the plant described above? Enter a single compensator (such as K_D to indicate K_D(s)) or multiple compensators separated with commas.

Does using K_C(s) give you a positive phase margin?

Is there any \beta > 1 for which 10 \frac{s+ 10 \beta}{s + 100 \beta} gives you a postive phase margin. (Hint: with \beta > 1, the leading zero in the compensator is more negative than the most negative pole in the plant).

Is there a \beta > 1 for which using 10*(s+10\beta)/(s+100\beta) gives a positive phase margin?

In a lead compensator, the zero "leads" the pole. That is, the compensator's zero is less negative than the compensator's pole. Which of the following statements about lead compensators is true?

"If a lead compensator is applied to a stable three real pole system, and the compensator's zero is as negative, or more negative, than the most negative system pole, then the compensator will NOT improve the system's phase margin."

Hint: What about H(s) = \frac{60}{(s+1)*(s+2)*(s+3)}?

No positive phase margin with three stable real poles?

"If a lead compensator is applied to a system with one negative real pole and two poles at 0, and the compensator's zero is as negative, or more negative, than the most negative system pole, then the compensated system will NEVER have a positive phase margin."

No positive phase margin with two poles at 0, the other negative real?