For the system described by the state equation- - 0 1 0- 0 0 1- 0-5 1 2 - x - 0- 0- 1 -uIf the control signal u is given by u - -0-5-3-5- x- v- then the eigen values of the closed-loop system will be

Ẋ= [ 0 amp; 1 amp;0; nbsp; nbsp;0 amp;0 nbsp; amp;1; nbsp; nbsp;0.5 amp; nbsp;1 amp; 2 ] x +[ 0; nbsp; 0; nbsp; 1 ] [ ...

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Question

For the system described by the state equation
$˙ x = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 0 0 & 0 & 1 0.5 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦ x + ⎡ ⎢ ⎣ \begin{matrix} 001 \end{matrix} ⎤ ⎥ ⎦ u$
If the control signal $u$ is given by $u = [- 0.5 - 3 - 5] x + v$ , then the eigen values of the closed-loop system will be

0, - 1, - 2

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0, - 1, - 3

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- 1, - 1, - 2

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0, - 1, - 1

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Solution

The correct option is A $0, - 1, - 2$
$˙ x = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 0 0 & 0 & 1 0.5 & 1 & 2 \end{matrix} ⎤ ⎥ ⎦ x + ⎡ ⎢ ⎣ \begin{matrix} 001 \end{matrix} ⎤ ⎥ ⎦ [- 0.5 - 3 - 5] x + v$
$∴ ˙ x = ⎡ ⎢ ⎣ \begin{matrix} 0 & 1 & 0 0 & 0 & 1 0 & - 2 & - 3 \end{matrix} ⎤ ⎥ ⎦ x + v$
Characteristic equation
$λ^{3} + 3 λ^{2} + 2 λ + 0 = 0$
$\Rightarrow λ = 0, - 1, - 2$

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For the system described by the state equation ˙x=⎡⎢⎣0100010.512⎤⎥⎦x+⎡⎢⎣001⎤⎥⎦u If the control signal u is given by u=[−0.5−3−5]x+v, then the eigen values of the closed-loop system will be

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