Consider a linear system whose state space representation is -t - Axt-If the initial state vector of the system is x0 - - 1- -2 -then the system response is xt - - e-2t- -2e-2t - If the initial state vector of the system changes to x0 - - 1- -1 - then the system response becomes xt - - e-t- -e-t -The eigen-value and eigen-vector pairs -i- vi for the system are

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State Vector and Matrix

Consider a li...

Question

Consider a linear system whose state space representation is $˙ x (t) = A x (t) .$ If the initial state vector of the system is $x (0) = [\begin{matrix} 1 - 2 \end{matrix}]$ ,then the system response is $x (t) = [\begin{matrix} e^{- 2 t} - 2 e^{- 2 t} \end{matrix}] .$ If the initial state vector of the system changes to $x (0) = [\begin{matrix} 1 - 1 \end{matrix}],$ then the system response becomes $x (t) = [\begin{matrix} e^{- t} - e^{- t} \end{matrix}] .$

The eigen-value and eigen-vector pairs $(λ_{i}, v_{i})$ for the system are

(- 1, [\begin{matrix} 11 \end{matrix}]) and (- 2, [\begin{matrix} 12 \end{matrix}])

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(- 2, [\begin{matrix} 1 - 1 \end{matrix}]) and (- 1, [\begin{matrix} 1 - 2 \end{matrix}])

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(- 1, [\begin{matrix} 1 - 1 \end{matrix}]) and (- 2, [\begin{matrix} 1 - 2 \end{matrix}])

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(- 2, [\begin{matrix} 1 - 1 \end{matrix}]) and (1, [\begin{matrix} 1 - 2 \end{matrix}])

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˙ x (t) = A x (t), x (0) = x_{0}

F o r x_{0} = [\begin{matrix} 1 - 1 \end{matrix}], x (t) = [\begin{matrix} e^{- t} - e^{- t} \end{matrix}] and for x_{0} = [\begin{matrix} 01 \end{matrix}]

x (t) = [\begin{matrix} e^{- t} & e^{- 2 t} - e^{- t} & + 2 e^{- 2 t} \end{matrix}]

W h e n x_{0} = [\begin{matrix} 35 \end{matrix}], x (t) i s

h (t)

δ (t)

h (t) = e^{- t} + e^{- 2 t}

x (t) = [\begin{matrix} 0 & 1 0 & - 3 \end{matrix}] X (t) + [\begin{matrix} 10 \end{matrix}] U (t),

X (0) = [- 13]^{T}

u (t)

x (t) = [\begin{matrix} 0 & 1 0 & - 3 \end{matrix}] X (t) + [\begin{matrix} 10 \end{matrix}] U (t),

X (0) = [- 13]^{T}

u (t)

x (t) = 2 e^{- 3 t} u (t - 1)

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