Course materials · Control Engineering I

Week 12 · Damping Ratio and Natural Frequency of a Second-Order System

English translation

Before practice

01 Response speed and oscillation

We want fast motion without oscillation after passing the target. Which parameters should be adjusted to reduce overshoot and shorten settling time?

Description
  • A standard 2nd-order model uses natural frequency and damping ratio to examine the response time scale and oscillation characteristics.
  • Change each value and compare the graphs to examine the relation between model coefficients and actual motion.

02 Design to satisfy multiple conditions

You have specified the desired speed and allowable oscillation. How can you find and verify a model satisfying both conditions?

Description
  • Multiple parameter combinations can satisfy the conditions
  • Find candidates using an approximation, then verify the conditions using the actual model response

Practice

1. Calculate the performance conditions and design range

  • Change the code’s initial wn value from 4 to the value 5 specified for this activity.

    For the first run of the “4. Verify the design” code in the practice Live Script, use ζ=0.7,ωn=5\zeta=0.7,\quad\omega_n=5.
  • First calculate overshoot and settling time by hand using Mp=100e−πζ/1−ζ2M_p=100e^{-\pi\zeta/\sqrt{1-\zeta^2}} [%] and Ts≈4ζωnT_s\approx\frac{4}{\zeta\omega_n}.
  • Use e−0.7π/0.51≈0.046e^{-0.7\pi/\sqrt{0.51}}\approx0.046 for numerical comparison.
  • Determine whether Mp≤10%M_p\le10\% and Ts≤2T_s\le2 s are each satisfied. Keeping ζ=0.7\zeta=0.7 fixed, derive an inequality for the range of ωn\omega_n satisfying TsT_s.
  • Then change ωn\omega_n to 5 and run stepinfo to compare the approximation with the measured results.
  • Record this condition as one of the existing design combinations.

2. Convert to standard form

  1. Calculate natural frequency ωn=K1\omega_n=\sqrt{K_1} and damping ratio ζ=K2/(2ωn)\zeta=K_2/(2\omega_n) for the following two combinations: (K1,K2)=(9,3)(K_1,K_2)=(9,3), (16,8)(16,8).
  2. Record ωn\omega_n, ζ\zeta, and damping type for each combination in one row. Determine damping type from the calculated ζ\zeta.

3. Compare damping ratios

  1. Keep ζ\zeta fixed, double only ωn\omega_n, and obtain stepinfo for the models before and after the change.
  2. Record ζ\zeta, ωn\omega_n, overshoot, rise time, and settling time in two rows for before and after the change.
  3. Check whether overshoot stays nearly the same and both time metrics become about half their original values. State which metrics natural frequency primarily changes when damping ratio is fixed.

4. Verify the design

  1. In addition to the baseline, choose two combinations of ζ\zeta and ωn\omega_n that satisfy the existing conditions: overshoot ≤10%\le10\% and settling time ≤2\le2 s.
  2. Read overshoot and settling time from each model's stepinfo and state separately whether each condition is satisfied. Compare with the model metrics rather than deciding from the calculated approximation alone.
  3. Also record one combination that fails the conditions. Use columns for ζ\zeta, ωn\omega_n, overshoot, settling time, and whether each condition is satisfied; mark the failed metric.

Submission and saving

Include code, actual outputs and plots, and your own interpretation. Submit through KLAS; check the semester schedule and latest announcements for the deadline.

Filename: Name_StudentID_Week12.mlx.

Restart the runtime, run all code from the top, then save.

Optional practice

Example: omitting natural frequency from a second-order denominator

  1. Keep zeta=.7 and wn=4, and create a separate model by changing the denominator from [1 2*zeta*wn wn^2] to [1 2*zeta wn^2].
  2. Record both models' poles and the settling time and overshoot of stepinfo side by side. Change only the second denominator coefficient to 2*zeta*wn and recalculate.