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

• Engineering

## 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.

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gneill
Mentor
Hi amr55533, Welcome to Physics Forums.

If I were left to my own devices to solve this problem I might choose to use either mesh analysis to find mesh currents (then use them to find voltages as required), or us nodal analysis to get at the node voltages directly. The state variable versions of impedance are:

R ---> R
C ---> 1/(sC)
L ---> sL