A linear time-invarient system is described by the state variable model
$[\begin{matrix} _{1}_{2} \end{matrix}] = [\begin{matrix} - 1 & 0 0 & - 2 \end{matrix}] [\begin{matrix} x_{1} x_{2} \end{matrix}] + [\begin{matrix} 01 \end{matrix}] u . Y = [\begin{matrix} 1 & 2 \end{matrix}] [\begin{matrix} x_{1} x_{2} \end{matrix}]$

The system is completely controllable

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The system is not completely controllable

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The system is completely observable

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The system is not completely observable

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Solution

The correct option is C The system is completely observable
$A = [\begin{matrix} - 1 & 0 0 & - 2 \end{matrix}] B = [\begin{matrix} 01 \end{matrix}] A B = [\begin{matrix} - 1 & 0 0 & - 2 \end{matrix}] [\begin{matrix} 01 \end{matrix}] = [\begin{matrix} 0 - 2 \end{matrix}] Q_{c} = [\begin{matrix} B & A B \end{matrix}] Q_{c} = [\begin{matrix} 0 & 0 1 & - 2 \end{matrix}] | Q_{c} | = 0 - 0 = 0 S o, t h e g i v e n s y s t e m i s n o t c o m p l e t e l y c o n t r o l l a b l e . A^{T} = [\begin{matrix} - 1 & 0 0 & - 2 \end{matrix}] C = [\begin{matrix} 1 & 2 \end{matrix}] C^{T} = [\begin{matrix} 12 \end{matrix}] A^{T} C^{T} = [\begin{matrix} - 1 & 0 0 & - 2 \end{matrix}] [\begin{matrix} 12 \end{matrix}] = [\begin{matrix} - 1 - 4 \end{matrix}] Q_{0} = [\begin{matrix} C^{T} & A^{T} C^{T} \end{matrix}] Q_{0} = [\begin{matrix} 1 & - 1 2 & - 4 \end{matrix}]$
$| Q_{0} | = - 4 - (- 2) = - 2 \neq 0$
So, thegiven system is completely observable.
(b) The system is not completely controllable.
(c) The system is completely observable.

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