Hello all,

My task here is to take the state space model equations (from this thread http://forum.allaboutcircuits.com/showthread.php?t=62884) into an equivalent discrete time state space model, assuming a zero order hold is used.

the formulas that will be used are:
y[k] = Cx[k] + Du[k].

From the answer from the previous forum, this give: (where t = k)

C is:
\(\left{[ \begin{array}{lml}
-1 & \, & 0\\
\end{array} \right\]
\)

and D is:
\(\left[ \begin{array}{lml}
0
\end{array} \right\]
\)

My problem lies on the next part:
The general formula is x[k+1] =\(\Phi\)x[k]+\(\Gamma u[k]\)
\(\Phi = exponential(AT)\)
so:\( \Phi = exponential{ \left[ \begin{array}{lml}
\frac{-1}{RC} & \, &\frac{-1}{2RC} \\
\frac{1}{2RC} & \, & 0 \\

\end{array} \right\]}T
\)

i dont think I can simplify this anymore so this will be my value on \( \Phi\)

the formula for \(\Gamma=\int ^T_0 exponential((AT)dt)B\)

This gives the result:

\(\Gamma = \frac{1}{RC}\int ^T_0 {exponential{ \left[ \begin{array}{lml}
\frac{-1}{RC} & \, &\frac{-1}{2RC} \\
\frac{1}{2RC} & \, & 0 \\

\end{array}\right\]
\)B

where B is \(\left[ \begin{array}{lml}
-1
0
\end{array} \right\]
\)

Im not sure how to deal with the exponential part to deal with my anlysis. Im thinking laplace but just cant spot how to simplify it. Any hints will be fab!

 

I wanted to revive this thread. Anyone have some suggestions?