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Damping ratio \(\xi\) and peak overshoot Mp are measures of
1. relative stability
2. absolute stability
3. speed of response
4. steady state error

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Correct Answer - Option 1 : relative stability

Damping ratio:

The ratio of ti me constant of critical damping to that of actual damping is known as damping ratio(ζ).

Time constant for critical damping = 1 / ωn 

Times constant for actual damping = 1 / ζωn 

\(ζ = \frac{{\frac{1}{{{ω _n}}}}}{{\frac{1}{{ζ {ω _n}}}}}\)

ωn = Natural frequency 

Maximum overshoot (MP):

  • It is straight way difference between the magnitude of the highest peak of time response and magnitude of its steady state.
  • Maximum overshoot is expressed in term of percentage of steady-state value of the response.
  • As the first peak of response is normally maximum in magnitude, maximum overshoot is simply normalized difference between first peak and steady-state value of a response.
  • The amount of maximum overshoot directly indicates the relative stability of the system.

\({M_P} = \frac{{c\left( {{t_P}} \right) \;-\; c\left( ∞ \right)}}{{c\left( ∞ \right)}}\)

c(tP) = response at rise time

c(∞) = response at steady state

Peak overshoot: It is the maximum error at the output.

\({{M}_{P}}={{e}^{-\left( \frac{ζ \pi }{\sqrt{1-{{ζ }^{2}}}} \right)}}\)

It is indicative of damping in the system. The peak overshoot is more for fewer values of the damping factor.

ζ ↓ → MP ↑  ⇒ which is not good for the system

so Peak overshoot and damping ratio are the measures of relative stability.

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