# Control Systems: How would you find the State Equations for this simple circuit?

• Engineering
amr55533

## Homework Statement

Consider the electrical circuit shown:

http://imageshack.us/a/img525/8163/p1circuit.png [Broken]

Let the state variables be x1(t)=Vc(t), x2(t)=iL(t), and x3(t)=Vc(t); output is Vo(t). Write the state-space equations in matrix form and find the transfer function, T(s)=Vo(s)/Vi(s).

KCL and KVL

## The Attempt at a Solution

State Variables:

x1(t)=Vc(t)
x2(t)=iL(t)
x3(t)=Vo(t)

Outputs:

Vo(t)

Inputs

Vi(t)

Differential Equations for State Variables:

X1'=dV1/dt=i2
X2'=di4/dt=V2
X3'=dVo/dt=i5

Now this is the part that I am stuck at. I know that I have to solve for X1', X2', and X3' in terms of the state variables and inputs only. However, I can't seem to reduce the equations enough to get it into this format.

Basically, I am trying to solve for i2, i5, and V2 in terms of i4, V1, Vo, and Vi only (the state variables and inputs). Once I have these equations, I can easily put them into matrix form and solve using MATLAB. I solved a problem earlier that was exactly the same, only the first capacitor was replaced with an inductor. So, I think it is the capacitor that is giving me problems.

A few equations that I found:

Vi=i1+i3+i5+Vo

i3=i1-i2

i5=i3-i4

V1=Vi-i1

V2=V1-i3

Vo=V2-i5

Thanks for the help!

Edit:

I looked over the problem again, and it seems that I can't solve for i3 without it containing a V2 or an i2. Is there any way to solve for i3 with a combination of only i4, V1, Vo, and Vi? Once I find this, I will be able to solve the problem.

Last edited by a moderator: