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I have to find the transfer function and state-space representation of the following first-order differential equation that represents a dynamic system: $$5\, \dot{x}(t) +x(t) = u(t) \\$$ The first part I managed to do it, I used the Laplace transformation to find the transfer function, but I couldn't get to the state space equation. I tried to reorganize the equation but the answer I got, which was: $$\dot{x}= -0.2x(t) +0.2 u(t)$$ didn't match the right one. The correct answer is correct answer

I would appreciate some guidance on how to solve this problem.

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    \$\begingroup\$ Can you provide an attempt at the solution? \$\endgroup\$
    – Voltage Spike
    Commented Nov 3, 2021 at 19:31

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There is no one correct answer for the state-space representation.

If you choose the state as \$x_1=x\$, then the equations are

$$ x_1'=-0.2x_1+0.2u\\y=x_1$$

If the state is chosen as \$x_1=2.5 x\$, then

\$x_1'=2.5(-x/5+u/5)=-0.2(2.5 x)+0.5u=-0.2x_1+0.5u\$

and

\$y=x=x_1/2.5=0.4 x_1\$

Putting them together we get the answer you want.

$$ x_1'=-0.2x_1+0.5u\\y=0.4x_1$$

The bottom line is there is no one 'correct' answer.

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  • \$\begingroup\$ Thank you, that makes sense. \$\endgroup\$
    – Luz0000
    Commented Nov 3, 2021 at 20:06
  • \$\begingroup\$ While it is true there is no one answer, it is good to mention choosing the state variables carefully may lead to better representation from matrix algebra perspective. \$\endgroup\$
    – CroCo
    Commented Nov 26, 2021 at 2:07

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