The state-variable formulation of a system is ẋ = Ax + Bu; y = [1 0] x where $A = [\begin{matrix} - 3 & 1 \\ 0 & - 2 \end{matrix}], B = [\begin{matrix} 2 \\ 1 \end{matrix}]$ The system transfer function would be

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The state-variable formulation of a system is ẋ = Ax + Bu; y = [1 0] x where $A = [\begin{matrix} - 3 & 1 \\ 0 & - 2 \end{matrix}], B = [\begin{matrix} 2 \\ 1 \end{matrix}]$ The system transfer function would be

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$\frac{s + 2}{s^{2} + 5 s - 6}$
$\frac{2 s + 5}{s^{2} + 5 s + 6}$
$\frac{2 s - 5}{s^{2} + 5 s - 6}$
$\frac{s + 1}{s^{2} + 5 s + 6}$

Answer 1

Concept:

X’ = Ax + Bu → (1)

Y = Cx + Du → (2)

Apply Laplace transform, to equation (1)

S x(s) = A x(s) B u(s)

⇒ B u(s) = (SI - A) X(s)

⇒ X(s) = (SI - A)-1 B u(s)

Now apply Laplace transform to equation (2)

Y(s) = C x(s) + D u(s)

⇒ Y(s) = C[(SI - A)-1] B u(s) + D u(s)

\Rightarrow \frac{Y (s)}{U (s)} = C {(S I - A)}^{- 1} B + D

Calculation:

From the given state-space representation,

$A = [\begin{matrix} - 3 & 1 \\ 0 & - 2 \end{matrix}], B = [\begin{matrix} 2 \\ 1 \end{matrix}], C = [\begin{matrix} 1 & 0 \end{matrix}], D = 0$

$(s I - A) = [\begin{matrix} s + 3 & - 1 \\ 0 & s + 2 \end{matrix}]$

${(s I - A)}^{- 1} = \frac{1}{(s + 3) (s + 2)} [\begin{matrix} s + 2 & 1 \\ 0 & s + 3 \end{matrix}]$

$= \frac{1}{s^{2} + 5 s + 6} [\begin{matrix} s + 2 & 1 \\ 0 & s + 3 \end{matrix}]$

$T F = [\begin{matrix} 1 & 0 \end{matrix}] \frac{1}{s^{2} + 5 s + 6} [\begin{matrix} s + 2 & 1 \\ 0 & s + 3 \end{matrix}] [\begin{matrix} 2 \\ 1 \end{matrix}] + 0$

$= \frac{2 s + 5}{s^{2} + 5 s + 6}$

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The state-variable formulation of a system is ẋ = Ax + Bu; y = [1 0] x where $A = [\begin{matrix} - 3 & 1 \\ 0 & - 2 \end{matrix}], B = [\begin{matrix} 2 \\ 1 \end{matrix}]$ The system transfer function would be

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The state-variable formulation of a system is ẋ = Ax + Bu; y = [1 0] x where A=[−310−2],B=[21] The system transfer function would be

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The state-variable formulation of a system is ẋ = Ax + Bu; y = [1 0] x where $A = [\begin{matrix} - 3 & 1 \\ 0 & - 2 \end{matrix}], B = [\begin{matrix} 2 \\ 1 \end{matrix}]$ The system transfer function would be