The system represented by the state-variable model

$\dot{X} = [\begin{matrix} 0 & - 1 \\ 1 & - 2 \end{matrix}] X + [\begin{matrix} 1 \\ 2 \end{matrix}] U$

C = [1 1] is

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This question was previously asked in

ESE Electronics 2012 Paper 1: Official Paper

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Answer 1

Transfer function = C (sI - A)-1 B + D

In the given question,

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

$C = [\begin{matrix} 1 & 1 \end{matrix}], D = 0$

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

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

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

The characteristics equation is:

s² + 2s + 1 = 0

(s + 1)² = 0

The roots are -1, -1