Frequency Domain Methods for Controller Design

Question 1

Gain Margin

Answer 1

Change in open-loop gain required to make the closed-loop system unstable

Question 2

Phase Margin

Answer 2

Change in open-loop phase shift required to make the closed-loop system unstable. Also measures system’s tolerance to delay time.

Question 3

Gain Crossover Frequency

Answer 3

Additional amount of phase lag that is required for the open-loop system’s phase to reach -180 degrees at the frequency where the open loops system’s magnitude is 0 dB

Question 4

Phase Crossover Frequency

Answer 4

Additional gain required for open-loop systems magnitude to reach 0 dB at the frequency where the open-loop system’s phase equals -180 degrees

Question 5

Bandwidth Frequency

Answer 5

Frequency at which the closed-loop magnitude drops 3 dB below its DC magnitude

Question 6

2nd Order Approximation of Bandwidth Frequency

Answer 6

Frequency at which the open-loop magnitude response is between -6 and -7.5 dB assuming open-loop response is between -135 and -225 degrees

Question 7

Bode Design Method

Answer 7

Evaluates how a sinusoidal input to a system is scaled and shifted by the system.Provides information about system’s time response. Used to determine robustness or how close system is to becoming unstable. System must be stable in open-loop response.

Question 8

Nyquist Diagram

Answer 8

Allows prediction of stability and performance of closed-loop system by observing its open-loop behavior. Can be used for design purposes regardless of open-loop stability. Used in conjunction with bode plots

Question 9

Cauchy Criterion

Answer 9

A closed contour in the complex plane and mapping it through a complex function G(s), the number of times G(s) enxircles the orgin is equal to the number of zeros minus number of poles enclosed by G(s)

Question 10

Closed Loop Performance From Bode Plots Assumptions

Answer 10

1. Stable open-loop system
2. For Canonical 2nd order systems damping ration is equal to PM/100 and PM is between 0 and 60 degrees. Use with caution if PM > 60
3. Special equations for DR, wbw, and settling time for 2nd order canonical systems
4. A very rough estimate of bandwidth is equal to natural frequency

Question 11

Closed Loop Stability from Nyquist

1. Definition
2. Z: Zeros of 1 + G(s)H(s)
3. P: Poles of 1 + G(s)H(s)
4. Positive Encirclement
5. Negative Encirclement

Answer 11

1. For a negative feedback system N = Z - P
2. poles of the closed-loop transfer function
3. poles of the open-loop transfer function
4. Clockwise encirclement of -1
5. Counterclockwise encirclement of -1

Question 12

Gain Margin Calculation

Answer 12

GM = 20 * log(a) dB
G(jw) = -1/a where G(jw) is purely real