# Differential Equation to state spaces representation

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

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

• Can you provide an attempt at the solution? Commented Nov 3, 2021 at 19:31

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.

• Thank you, that makes sense. Commented Nov 3, 2021 at 20:06
• 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. Commented Nov 26, 2021 at 2:07