Consider a state-variable model of a system

$[\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] = [\begin{matrix} 0 & 1 \\ - α & - 2 β \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] + [\begin{matrix} 0 \\ α \end{matrix}] r$

$y = [\begin{matrix} 1 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]$

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

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Consider a state-variable model of a system

$[\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] = [\begin{matrix} 0 & 1 \\ - α & - 2 β \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] + [\begin{matrix} 0 \\ α \end{matrix}] r$

$y = [\begin{matrix} 1 & 0 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \end{matrix}]$

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

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$ξ = \frac{β}{\sqrt{α}}; ω_{n} = \sqrt{α}$
$ξ = \sqrt{α}; ω_{n} = \frac{β}{\sqrt{α}}$
$ξ = \frac{\sqrt{α}}{β}; ω_{n} = \sqrt{β}$
$ξ = \sqrt{β}; ω_{n} = \sqrt{α}$

Answer 1

From the given state model

$\begin{matrix} A = [\begin{matrix} 0 & 1 \\ - α & - 2 β \end{matrix}] & B = [\begin{matrix} 0 \\ α \end{matrix}] & C = [\begin{matrix} 1 & 0 \end{matrix}] \end{matrix}$

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

$[S I - A] = S [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] - [\begin{matrix} 0 & 1 \\ - α & - 2 β \end{matrix}]$

$= [\begin{matrix} s & - 1 \\ α & s + 2 β \end{matrix}]$

$= {[s I - A]}^{- 1} = \frac{1}{s (s + 2 β) α} [\begin{matrix} s + 2 β & 1 \\ - α & s \end{matrix}]$

$\frac{T}{F} = \frac{[\begin{matrix} 1 & 0 \end{matrix}] [\begin{matrix} s + 2 β & 1 \\ - α & s \end{matrix}] [\begin{matrix} 0 \\ α \end{matrix}]}{(s (s + 2 β) + α)}$

$= \frac{1}{(s^{2} + 2 β s + α)} [\begin{matrix} 1 & 0 \end{matrix}] [\begin{matrix} α \\ s α \end{matrix}]$

$= \frac{α}{(s^{2} + 2 β s + α)}$

By comparing the above transfer function with the standard second order transfer function $\frac{ω_{n}^{2}}{s^{2} + 2 ξ ω_{n} s}$

$ω_{n} = \sqrt{α}$

And 2ξω_n = 2

$\Rightarrow ξ = \frac{β}{\sqrt{α}}$

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Consider a state-variable model of a system [x˙1x˙2]=[01−α−2β][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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