Frequency-Domain Stability

Before simulating all time responses, try assessing stability from the frequency response and open-loop poles.

Starting conditions

$G(s)=\frac{s^2+10s+24}{s^2-8s+15}$, $K=.5$,1. Use negative unity feedback.

Procedure

1. Count P, the number of open-loop right-half-plane poles.
2. Plot the Nyquist plot for each K, and directly calculate Z, the number of closed-loop poles in the right half-plane.
3. State your sign convention for the direction of encirclements of -1, and check the relationship among P, the encirclement count, and Z.

Results to retain

• 2 Nyquist plots and an assessment table.

Starting conditions

For $G(s)=\frac{50}{s^3+9s^2+30s+40}$, compare $K\in\{1,3,5\}$.

Procedure

1. Find the gain margin, phase margin, and corresponding crossover frequencies using margin(K*G).
2. Report gain margin separately as a linear ratio and in dB using 20log10.
3. Examine each set of closed-loop poles and time response, and explain the relationship between margins and stability.

Results to retain

• A margin table and response comparison plot.

Starting conditions

For the same G, compare examining only ω=[.1,1,10] with examining logspace(-2,3,1000).

Procedure

1. Compare gain and phase on both grids.
2. Describe the information missed when assessing crossover points and stability margins using only three sparse points.
3. Record both calculated values and visual evidence to distinguish assessment results from plot resolution.

Results to retain

• A plot comparing resolutions and a brief interpretation.