Are My State Equations Correct for a Multi-Input System Using KVL and KCL?

[Derill03]

what I've posted is a question from a study guide for our test next week, on the attachment is my work.

The professor only did SISO examples in class and moved really fast, I am not really sure if i found state equations correctly using KVL and KCL. Any help is appreciated

 

[gneill]

Rather than first writing the differential equations for the time domain, you should be able to work directly with the state variable values for impedance (or admittance) for the circuit. Do you know the expressions for the impedances of L and C components? What about unit step functions (to convert the current sources)?

 

[Derill03]

well I do see that you could convert left hand current source to series resistor and volt source, but wouldn't i lose i1? Weve been taught in class don't combine components or transform them if they hold a value your looking for.

One thing I am stuck on and this may help me greatly, does each equation dv/dt and di/dt have to be solely in terms of Vc and iL? or can i have two equations with say one has v and an input variable and other has i and an input variable? Cause right now I am trying to make substitutions to make each equation be solely in terms of state variables

 

[gneill]

well I do see that you could convert left hand current source to series resistor and volt source, but wouldn't i lose i1? Weve been taught in class don't combine components or transform them if they hold a value your looking for.

No need to transform or combine any components. You can write node (or mesh) equations for the circuit as-is, directly with state variable expressions. Can you write the state variable expressions for component impedances and current sources?

One thing I am stuck on and this may help me greatly, does each equation dv/dt and di/dt have to be solely in terms of Vc and iL? or can i have two equations with say one has v and an input variable and other has i and an input variable? Cause right now I am trying to make substitutions to make each equation be solely in terms of state variables

You can have more than one function of the state variable, such as V1(s) and V2(s) and I(s) and so on. Just as in the time domain you'll need enough simultaneous equations to solve for them all.