If the characteristic equation of a closed loop system is S2 + 2S + 2 = 0, then the system is
underdamped
Compare the equation with the standard second-order form and read off the damping ratio.
\(s^{2}+2\zeta\omega_{n}s+\omega_{n}^{2}=0\)
Matching against \(s^{2}+2s+2=0\):
\(\omega_{n}^{2}=2\ \Rightarrow\ \omega_{n}=\sqrt{2}=1.414\ \text{rad/s}\)
\(2\zeta\omega_{n}=2\ \Rightarrow\ \zeta=\dfrac{1}{\sqrt{2}}=0.707\)
Since \(0\lt\zeta\lt1\), the system is underdamped — option 3.
| ζ | Roots | Response |
|---|---|---|
| ζ = 0 | Purely imaginary | Undamped — sustained oscillation |
| 0 < ζ < 1 | Complex conjugate | Underdamped — decaying oscillation |
| ζ = 1 | Real, repeated | Critically damped |
| ζ > 1 | Real, distinct | Overdamped |
The roots confirm it directly. Solving the quadratic,
\(s=\dfrac{-2\pm\sqrt{4-8}}{2}=-1\pm j1\)
A complex conjugate pair means the response contains \(e^{-t}\cos t\) and \(e^{-t}\sin t\) terms — it oscillates, while the negative real part makes the envelope decay. Both facts together are precisely what "underdamped" means, and the negative real part also confirms the system is stable.
Why this particular value is famous. \(\zeta=0.707\) is the standard design target, and for good reason. Overshoot is
\(M_{p}=e^{-\pi\zeta/\sqrt{1-\zeta^{2}}}=4.3\%\)
which is small, while settling time \(4/\zeta\omega_{n}=4\) s remains short. A critically damped system reaches its final value without any overshoot but takes noticeably longer; an overdamped one is slower still. In the frequency domain, \(\zeta=0.707\) is also the largest damping ratio that produces no resonant peak — the maximally flat or Butterworth response.
Hence, the system is underdamped.
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