Transfer Function

Even with the same coefficients, responses depend on the interconnection and cancellation conditions. Practice predicting behavior from coefficients.

• num: coefficient array of the numerator polynomial.
• den: coefficient array of the denominator polynomial.
• Array order: highest power first.

Starting conditions

• A: $\mathrm{num}=[2]$, $\mathrm{den}=[5,1]$.
• B: $\mathrm{num}=[1,2]$, $\mathrm{den}=[1,3,2]$.
• C: $\mathrm{num}=[1,1]$, $\mathrm{den}=[1,2,5]$.

Procedure

1. Run A as the baseline example. Write the numerator and denominator as polynomials, and calculate using np.roots and ctl.poles/ctl.zerosroots and pole/zero.
2. Apply the same procedure to B and C. Distinguish numerator and denominator degrees from the number of finite zeros, and record both real and imaginary parts of complex roots.
3. Plot the poles and zeros of all three models on one complex plane. Use × for poles and ○ for zeros, with a different color for each model.

Results to retain

• 1 table with one row per model and 1 pole–zero map.
• Example level for A: H(s)=2/(5s+1), pole -0.2, no finite zeros. Complete B and C yourself.
• Briefly explain the relationship between coefficients and roots.

Starting conditions

$H_1(s)=\frac2s$, $H_2(s)=\frac3{4s+1}$. Use negative unity feedback for $H_1H_2$.

Procedure

1. Derive expressions for $H_s=H_1H_2$, $H_p=H_1+H_2$, and $T=\frac{H_s}{1+H_s}$.
2. Use ctl.series, ctl.parallel, ctl.feedbackseries, parallel, feedback to build the same models, and compare with the directly derived expressions at ω=[0.1,1,10] rad/s.
3. Plot the unit-step response of T over $0\le t\le 50$ s. Distinguish the poles at the origin of Hs and Hp from the meaning of DC gain.

Results to retain

• 1 coefficient table for the 3 interconnections and 1 step plot of T.
• Record the error as the absolute difference between complex responses at the same frequency.

Starting conditions

Create model D by changing B's numerator from [1,2] to [1,2.1]. Both models have denominator [1,3,2].

Procedure

1. Record the original pole–zero maps of B and D alongside the models simplified with minreal.
2. Plot the difference between the two responses to the same unit-step input over $0\le t\le 10$ s.
3. Distinguish exact from near cancellation, and explain why the model before cancellation should also be retained.

Results to retain

• 1 model-order table and a response-difference plot.
• A brief explanation of the tolerance used to identify cancellation and its interpretation.