Frequency domain representation
Your notation is confusing.

Let \$ \small y(t)\rightarrow Y(s)\$ be the system output signal from the integrator, and let the system input signal be \$\small u(t) \rightarrow U(s)\$. The CLTF is then:
$$\small\frac{Y(s)}{U(s)}=\frac{2(s+1)}{3s^2+7s+2}$$

To proceed via controller canonical form (other forms are available - see literature), introduce a dummy variable, \$\small x(t) \rightarrow X(s)\$:
$$\small\frac{Y(s)}{U(s)}=\frac{Y(s)}{X(s)}.\frac{X(s)}{U(s)}=\frac{2(s+1)}{3s^2+7s+2}=\frac{\frac{2}{3}s+\frac{2}{3}}{s^2+\frac{7}{3}s+\frac{2}{3}}$$

Now, partion the TF as below:

$$\small\frac{X(s)}{U(s)}=\frac{1}{s^2+\frac{7}{3}s+\frac{2}{3}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \small \frac{Y(s)}{X(s)}=\frac{2}{3}s+\frac{2}{3}$$

Derive the differential equation from each expression (\$\small sX(s)\rightarrow\dot{x}\:\:etc.)\$:

$$\ddot{x}= u-\small\frac{7}{3}\dot{x}-\small\frac{2}{3}x \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:y=\frac{2}{3}\dot{x}+\small\frac{2}{3}x$$
Finally, express in state space form:
$$
\begin{bmatrix}
    \ddot{x} \\
    \dot{x}
  \end{bmatrix}
=\begin{bmatrix}
    -\frac{7}{3} &amp; -\frac{2}{3}  \\
    1 &amp; 0  \\
  \end{bmatrix}
\begin{bmatrix}
    \dot{x} \\
    x
  \end{bmatrix}
+
\begin{bmatrix}
    1 \\
    0
  \end{bmatrix}u
$$

$$y
=\begin{bmatrix}
    \frac{2}{3} &amp; \frac{2}{3}   
  \end{bmatrix}
\begin{bmatrix}
    \dot{x} \\
    x
  \end{bmatrix} $$