State space model of a two-stage RC circuit

Take the capacitor voltages as state variables:

$$x_1 = v_2 \ (\text{voltage across } C_1), \qquad x_2 = v_o \ (\text{voltage across } C_2)$$

 Vi   R1     v2    R2     Vo
 o--/\/\/--+--/\/\/--+----o
   i1 ->   |  i2 ->  |
          === C1    === C2
           |         |
 o---------+---------+----o

Circuit equations

Mesh currents: $i_1 = \dfrac{V_i - v_2}{R_1}$ through $R_1$, $i_2 = \dfrac{v_2 - v_o}{R_2}$ through $R_2$.

KCL at node $v_2$ (current in $C_1$ is $i_1 - i_2$):

$$C_1\frac{dv_2}{dt} = \frac{V_i - v_2}{R_1} - \frac{v_2 - v_o}{R_2}$$

KCL at the output node (current in $C_2$ is $i_2$):

$$C_2\frac{dv_o}{dt} = \frac{v_2 - v_o}{R_2}$$

State equations

$$\begin{aligned} \dot{x}_1 &= -\left(\frac{1}{R_1C_1} + \frac{1}{R_2C_1}\right)x_1 + \frac{1}{R_2C_1}x_2 + \frac{1}{R_1C_1}V_i \\ \dot{x}_2 &= \frac{1}{R_2C_2}x_1 - \frac{1}{R_2C_2}x_2 \end{aligned}$$

Matrix form

$$\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \end{bmatrix} = \begin{bmatrix} -\left(\dfrac{1}{R_1C_1} + \dfrac{1}{R_2C_1}\right) & \dfrac{1}{R_2C_1} \\ \dfrac{1}{R_2C_2} & -\dfrac{1}{R_2C_2} \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} \dfrac{1}{R_1C_1} \\ 0 \end{bmatrix}V_i$$ $$V_o = y = \begin{bmatrix} 0 & 1 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$$

This gives $\dfrac{V_o(s)}{V_i(s)} = \dfrac{1}{R_1C_1R_2C_2s^2 + (R_1C_1 + R_2C_2 + R_1C_2)s + 1}$, the known result for a two-stage RC ladder.