# Finding the state space representation of a circuit

I want to find the state space equations for the following circuit.

simulate this circuit – Schematic created using CircuitLab

The equations I find are:

1. Ir = I1 + I2 = (U-Vn)/2
2. I2 = I3
3. I1 = 2*X1'
4. I2 = 1*X2'
5. I3 = 3*X3'
6. -X1 = X2+X3
7. Y = X3

Plugging in (3) and (4) to (1) I get: (U+X1)/2 = 2*X1'+ X2'

I need to decouple the derivatives, but I don't see a relation that can do that.

• What are you considering your state? What is it dimension? – jDAQ Nov 14 '19 at 0:32
• I consider the voltages across my capacitors my states. Respectively they're X1,X2,X3. – Kevin Silken Nov 14 '19 at 0:42

The current going through $$\C_2\$$ and $$\C_3\$$ is the same, so, by, $$Q = Cv \leftrightarrow i=C\dot{v}$$ For that branch,
$$\frac{\dot{v_3} - \dot{v_2} }{C_2} = \frac{-\dot{v_3}}{C_3}.$$
Also, the current that goes through the resistor either goes through $$\C_1\$$ or the branch with $$\C_2, C_3\$$. By associating them in series and then parallel, we find that,
$$\frac{v_1 - v_2 }{R} = \frac{ \dot{v_2}}{C_1+(C_2^{-1}+C_3^{-1})^{-1}}.$$
I did not go through the whole process of figuring out the state equations, but these should show that $$\\dot{v_3} = k \dot{v_2}\$$, those derivatives are related by a constant.