State variable formulation of circuit with Vx and Vy

Circuit (as read from the figure). V_x in series with R feeds node N. The inductor L is connected from N to ground. The capacitor C is connected from N to a node P, and V_y is connected from P to ground (+ at P). Choose the inductor current i_L and capacitor voltage v_C (positive at N) as state variables; V_x and V_y are inputs.

Equations

KVL across the capacitor path: the voltage at node N equals the inductor voltage, and v_C = v_L − V_y, so

v_L = v_C + V_y

Inductor: L di_L/dt = v_L = v_C + V_y

KCL at N: current in R equals i_L plus the capacitor current.

i_R = (V_x − v_L)/R = (V_x − v_C − V_y)/R
C dv_C/dt = i_R − i_L = (V_x − v_C − V_y)/R − i_L

State equations

di_L/dt = (1/L) v_C + (1/L) V_y
dv_C/dt = −(1/C) i_L − (1/RC) v_C + (1/RC) V_x − (1/RC) V_y

Matrix form

| i_L' |   |  0      1/L    | | i_L |   |  0       1/L    | | Vx |
|      | = |                | |     | + |                 | |    |
| v_C' |   | −1/C  −1/(RC)  | | v_C |   | 1/(RC) −1/(RC)  | | Vy |

ẋ = A x + B u with x = [i_L, v_C]ᵀ, u = [V_x, V_y]ᵀ,

A = [[0, 1/L], [−1/C, −1/RC]], B = [[0, 1/L], [1/RC, −1/RC]].

Outputs such as the resistor current are obtained from i_R = (V_x − v_C − V_y)/R. The system has two energy-storing elements, so the order is 2.

Check: the characteristic equation of A is s² + s/(RC) + 1/(LC) = 0, which is the standard parallel RLC (R, L, C) denominator, as expected.