Control Systems Final

Control Systems Final - Problem 3

3. In the system below, G(s) is the transfer function for an antenna-motor combination.

Dr. Abner Mallity has a Bode' plot for G(jw). That plot is shown below.

Mallity has to have the answer to the following two questions.

(10 pts) What is the smallest SSE (Step Input) that can be achieved while remaining stable? What is the gain for the smallest SSE?

SSE = ______%

K_p = ______

(10 pts) What is the SSE for a phase margin of f_m = 55^o? What is the gain for that phase margin?

SSE = ______%

K_p = ______

Solution

(10 pts) What is the smallest SSE (Step Input) that can be achieved while remaining stable? What is the gain for the smallest SSE?

To get the smallest SSE you need the largest gain. At 3.3 Hz the phase goes through -180^o. At that point, the gain is -33 db, so you can increase the gain by 33 db. That's a factor of 44.67. Since the DC gain for the system as shown is 1.0 (That's 0 db.), the DC gain can increase to 44.7. That means that the minimum SSE is:

SSE_min = 1/(1 + 44.7) = .022 or 2.2%

We've already computed the gain that will give that result.

K_p for SSE_min = 44.7

(10 pts) What is the SSE for a phase margin of f_m = 30^o? What is the gain for that phase margin?

We can look at the Bode' plot to see where the phase goes through -125^o. That looks soemewher near f = 1.2 Hz. At that point we have a gain of -19 db, so we can raise gain 19 db, i.e. a gain of 8.9. That gives an SSE of:

SSE₃₀ = 1/(1 + 8.9) = .101 or 10%
The gain is 8.9