Consider a state-variable model of a system

[x˙1x˙2]=[01α2β][x1x2]+[0α]r

y=[10][x1x2]

where y is the output, and r is the input. The damping ratio ξ and the undamped natural frequency ωn (rad/sec) of the system are given by

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  1. ξ=βα;ωn=α
  2. ξ=α;ωn=βα
  3. ξ=αβ;ωn=β
  4. ξ=β;ωn=α

Answer (Detailed Solution Below)

Option 1 : ξ=βα;ωn=α
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Detailed Solution

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From the given state model

A=[01α2β]B=[0α]C=[10]

Transfer function = C [sI - A]-1 B + D

[SIA]=S[1001][01α2β]

=[s1αs+2β]

=[sIA]1=1s(s+2β)α[s+2β1αs]

TF=[10][s+2β1αs][0α](s(s+2β)+α)

=1(s2+2βs+α)[10][αsα]

=α(s2+2βs+α)

By comparing the above transfer function with the standard second order transfer function ωn2s2+2ξωns

ωn=α

And 2ξωn = 2

ξ=βα

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