State equations of RLC network, 2 F capacitor output

Choose the energy-storing elements' variables as states: capacitor voltages and inductor current.

• $x_1 = v_1$: voltage across the 1 F capacitor (node A)
• $x_2 = i_L$: current in the 1 H inductor (from A to B)
• $x_3 = v_2$: voltage across the 2 F capacitor (node B) $= V_o$

  Vs  1 ohm   A   1 H      B
 (+)-/\/\/--+--UUUU--+------+---o Vo
  |         |   iL   |      |
  |        === 1F   1 ohm  === 2F
  |         |        |      |
 (-)--------+--------+------+---o

KCL at node A

Current from the source through 1 Ω $= (V_s - v_1)/1$:

$$1\cdot\frac{dv_1}{dt} = (V_s - v_1) - i_L \;\Rightarrow\; \dot{x}_1 = -x_1 - x_2 + V_s$$

KVL around the inductor

$$1\cdot\frac{di_L}{dt} = v_1 - v_2 \;\Rightarrow\; \dot{x}_2 = x_1 - x_3$$

KCL at node B

$$2\frac{dv_2}{dt} = i_L - \frac{v_2}{1} \;\Rightarrow\; \dot{x}_3 = 0.5x_2 - 0.5x_3$$

State and output equations

$$\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} = \begin{bmatrix} -1 & -1 & 0 \\ 1 & 0 & -1 \\ 0 & 0.5 & -0.5 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} + \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}V_s$$ $$V_o = y = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$$

Check: from this model $\dfrac{V_o(s)}{V_s(s)} = \dfrac{1}{2s^3 + 3s^2 + 4s + 2}$, the same as found directly by nodal analysis with impedances ($1/s$ for 1 F, $s$ for 1 H, $1/(2s)$ for 2 F).