Adding 0's to state variable matrices?

Discussion in 'Homework Help' started by LordOfTheStrings, Apr 17, 2012.

  1. LordOfTheStrings

    Thread Starter New Member

    Mar 6, 2012
    1
    0
    Greetings everyone, this is my first post. Been lurking awhile, some great stuff on this forum.

    I have, what i hope is a simple question. I'm given this question:


    The equations shown are derived from a system with 2 inputs and 2 outputs. Derive the state
    equations and H(s) of the system.


     x`_1= -x_1-2x_2+z_2
     x`_2= -3x_1-4x_2+z_1
     x`_3= -5x_3

    and

     y_1=x_1+x_3+z_1
     y_2=x_3+z_2

    To solve this, we generally use the equations:

     x`=Ax+Bx
     y =Cx+Dz

    and solve with:

     H(s) = Y(s)/Z(s)=C(sI-A)^{-1}B+D

    I believe i can setup my matrices properly, this is where i begin to question myself. I get:


    A =  \begin{Bmatrix}<br />
  -1 & -2 & 0 \\<br />
  -3 & -5 & 0 \\<br />
   0 & 0 & -5<br />
\end{Bmatrix}

    B =  \begin{Bmatrix}<br />
     0 & 1 \\<br />
     1 & 0 \\<br />
     0 & 0<br />
\end{Bmatrix}

    C =  \begin{Bmatrix}<br />
1 & 0 & 1\\<br />
0 & 0 & 1<br />
\end{Bmatrix}

    D =  \begin{Bmatrix}<br />
0 & 1 \\<br />
1 & 0<br />
\end{Bmatrix}

    I've seen a few fellow students suggest adding rows to the C and D matrices, of 0's to match the system matrix, but this doesn't seem right to me. I know 0's need to be added to the bottom of the B matrix, as Z1 and Z2 are simply 0's for X3. but since A is 3x3, B needs to have 3 rows. Am i doing this properly? Or do i have some serious review to do? Normally we're given the whole circuit, never the equations like this. I'm just curious to know I'm on the right track setting up t his problem.

    thanks in advance everyone.
     
  2. t_n_k

    AAC Fanatic!

    Mar 6, 2009
    5,448
    782
    Presumably you meant

    \bold{\dot {x}}=\bold{Ax+Bz}

    Everything looks fine to me except I think

    \bold{D}=\left[ \begin{array}{cc}1 \ 0 \\ 0 \ 1 \end{array} \right]
     
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