# control systems - state space representation using transfer functions

I am trying to manually get the state space representation of an RLC network. I often end up with a bunch of differential equations that I can't relate to each other. I think it would be easier to just get the transfer function between the output and the input for the system and then convert the transfer function to a state space. Can I always do this ? Using transfer functions always gives a different state space representation and I just want to make sure this is fine. Here's my attempt at using a transfer function to get a state space representation: • Any particular reason why you chose the input voltage and its derivative as the state variables? From what I can see, the system output Y is the input voltage (Vin) itself. That's right? – Dirceu Rodrigues Jr Jan 17 '17 at 14:25
• Furthermore it seems a variation of an example presented in the book of Norman S. Nise - Control Systems Engineering. – Dirceu Rodrigues Jr Jan 17 '17 at 14:27
• I made a mistake here, I actually let Ir = x1 and the derivative of Ir is x2. – Nemo Jan 17 '17 at 14:38
• I was trying to find a solved example from the book so I can compare my answer and check it. I assumed that by getting a transfer function between the output current through the resistor and the input voltage Vin I would get a similar state space representation to that in the book. Apparently my assumption is wrong since I am not using the same state variables. @DirceuRodriguesJr – Nemo Jan 17 '17 at 14:42
• Your first differential equation is incorrect - you have confused the current and voltage variables. Cross-multiply the transfer function, then use: s=d/dt. – Chu Jan 17 '17 at 15:43

There are several ways to convert a transfer function into a state space representation. They lead to apparently different results, but retain the same essential information.

Possible representations:

_ First companion form (controllable canonical form).

_ Jordan canonical form.

_ Alternate first companion form (Toeplitz first companion form).

_ Second companion form (observable canonical form).

There is no single set of state variables which describe a given system. Different sets of variables can be chosen. It is possible to transform one set into another (ie linear combination).

• I added an example so you can see how I used transfer function and tell me if there's anything wrong. – Nemo Jan 17 '17 at 13:50

There's nothing mathematically wrong in your system. But it gives state variables that are not easily detectable in practical circuits. If you select natural gradually changing quantities to be your state variales, your calculations would be interesting also to practical electricians. In this case the natural state variables are the current of the inductor and the voltage of the capacitor. Their derivatives are easily got to the left side and all else to the right side. Easily = by solving the linear equations.