Course materials · Control Engineering I

Damping Ratio and Natural Frequency of a Second-Order System

English translation

Control Engineering I · Week 12

This week's goal · Calculate natural frequency and damping ratio, and verify them using response metrics.

Run: After signing in to MathWorks, open .mlx in your workspace. Click inside the code → Live Editor → Run Section to run one section at a time. Run runs everything. After changing a value, rerun the relevant section. Edit the text in ⟦enter here⟧ response fields; do not execute them. You can use Ctrl+Enter on Windows/Linux and ⌘+Enter on Mac.

Compare response speed and oscillation

First practice interpreting transient responses by changing the natural frequency and damping ratio of a standard 2nd-order model. Next, compare P and PI controllers to examine system type and steady-state error. Here, focus on reading response speed and oscillation separately.

1. Match coefficients to the standard form

In T(s)=ωn2s2+2ζωns+ωn2T(s)=\frac{\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}, ωn is the natural frequency and ζ is the damping ratio. Applying unity negative feedback to G=K1s(s+K2)G=\frac{K_1}{s(s+K_2)} gives T=G1+G=K1s2+K2s+K1T=\frac G{1+G}=\frac{K_1}{s^2+K_2s+K_1}. Comparing coefficients of matching powers with the standard form gives ωn2=K1,2ζωn=K2\omega_n^2=K_1,\quad2\zeta\omega_n=K_2. Thus, for K1>0, we obtain ωn=K1\omega_n=\sqrt{K_1} and ζ=K22K1\zeta=\frac{K_2}{2\sqrt{K_1}}.

Match coefficients to the standard form

If the s² coefficient in the denominator is not 1, divide the entire numerator and denominator by that coefficient before comparing with the standard form. Simply taking the square root of the constant term gives an incorrect ωn.

2. Compare one parameter at a time

Start with the baseline K1=25,K2=4. Then fix ωn=5 and compare ζ=0.1,0.3,0.7,1.2. Label ζ in the legend on the same plot to identify each curve.

Compare one parameter at a time

0<ζ<1 is underdamped, ζ=1 is critically damped, and ζ>1 is overdamped. These categories are defined for the standard order-2 model; use care when summarizing an entire higher-order model with one ζ.

3. Distinguish approximations from actual measurements

Ts≈4/(ζωn) is a design approximation for the 2% settling criterion. For 0<ζ<1, Mp=100exp(−ζπ/√(1−ζ²)). Verify your selected values again with MATLAB stepinfo.

Separate approximations from actual measurements

The MATLAB return fields are Overshoot and SettlingTime. Do not use the field name PercentOvershoot. Explicitly set the settling threshold to SettlingTimeThreshold=0.02.

4. Design means satisfying conditions simultaneously

The design exercise targets are Ts≤2T_s\le2 seconds and Mp≤10%M_p\le10\%. Reduce oscillation using the damping ratio and adjust speed using the natural frequency, then check both conditions. Record whether they are met and explain your choices.

Design means satisfying conditions simultaneously

The actual settling time may differ even if the approximation meets the requirement. Choose values with margin from the boundary and check whether the graph and numerical results agree.

Check your understanding

  1. What values should go into the code, and what does each row and column of the resulting array represent? If calculating physical quantities, have you also checked the units?
  2. Can you identify which parts to rerun when you change one condition?
  3. Do the results match your expectations? If they differ, can you explain why using the output values or plots?

Official MATLAB documentation · Control System Toolbox

Practice materials