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 = \( \begin{Bmatrix}
-1 & -2 & 0 \\
-3 & -5 & 0 \\
0 & 0 & -5
\end{Bmatrix} \)

B = \( \begin{Bmatrix}
0 & 1 \\
1 & 0 \\
0 & 0
\end{Bmatrix} \)

C = \( \begin{Bmatrix}
1 & 0 & 1\\
0 & 0 & 1
\end{Bmatrix} \)

D = \( \begin{Bmatrix}
0 & 1 \\
1 & 0
\end{Bmatrix} \)

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.

 

Presumably you meant

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

Everything looks fine to me except I think

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