Trouble writing state equations for RC network

Discussion in 'Homework Help' started by Obanion, Oct 18, 2010.

  1. Obanion

    Thread Starter New Member

    Nov 26, 2009
    I'm having a serious amount of trouble solving this problem. I can't seem to eliminate one of the state variables so I can create the A, B and C matrices. I am using C1, C2, C3 and R in place of the numerical values for now.


    From writing the KCL and KVL equations I have:



    But as you can see, even if I substitute these equations into another I won't be able to get rid of the needed state variable. I will always have two in it.

    Is there something that I'm missing that's really simple? It seems like it should be trivial but I am struggling with this.

    Any hints would be appreciated.
  2. steveb

    Senior Member

    Jul 3, 2008
    There are a couple of different ways to do this.

    1. Use X2 as a state and use the total capacitance from all three capacitors in the equation. That is Ct=C2+(C1C3)/(C1+C3) and the state equation is

    Ct dX2/dt =(U-X2)/R.

    Then generate another state equation for Y (which equals X3, btw).

    2. Use the more general form for state equations using the E matrix, as follows.

    E dX/dt = A' x +B' u

    Then convert to standard form by multiplying through by the inverse of the E matrix to get

    dX/dt = E^(-1) A' x + E^(-1) B' u

    The final form is obtained as

    dX/dt = A x + B u

    where A=E^(-1) A' and B=E^(-1) B'

    The second method is a trick to use when you just can't figure out a direct method. I believe that method 1 will work for you. The state equation for X2 is fine, and the state equation for Y can be put in proper form without too much trouble, I think.
    Obanion likes this.
  3. Obanion

    Thread Starter New Member

    Nov 26, 2009
    Thanks! That helps me out a lot. Appreciate it.
  4. hamidjaan

    New Member

    Dec 4, 2010