How do you find the state space representation of $G(s) = \frac {1}{s^2+s+1}$

Question

Let $G(s) = \frac {1}{s^2+s+1}$ be the transfer function of the system

Then $Y(s)(s^2+s+1) = U(s)$

Therefore $y'' + y' + y = u$

After this step, how should I set up my state transition variable $x$ to characterize this system?

Answer 1

You can define $x_1 = y$ and $x_2 = y'$ so that \begin{equation} x_1' = y' = x_2 \end{equation} and \begin{equation} x_2' = y'' = -y' - y + u = -x_2 - x_1 + u. \end{equation}

Now set \begin{equation} x = \left(\begin{array}{c} x_1 \\ x_2 \end{array}\right) \end{equation} so that \begin{equation} x' = \left(\begin{array}{c} x_1' \\ x_2' \end{array}\right) = \left(\begin{array}{rr} 0 & 1 \\ -1 & -1 \end{array}\right)x + \left(\begin{array}{c} 0 \\ 1 \end{array}\right)u, \end{equation} which is in state-space form.

Edit: As dustin noted in the comments below, the output equation here would be \begin{equation} y = \left(\begin{array}{cc} 1 & 0 \end{array}\right)x. \end{equation}