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## Homework Statement

I need to find the discrete time equivalent of the following system:

[itex]

\begin{bmatrix}

\ddot{x} \\

\dot{x} \\

\ddot{\theta} \\

\dot{\theta}

\end{bmatrix} = \begin{bmatrix}

0 & 0 & 0 & 4.2042857 \\

1 & 0 & 0 & 0 \\

0 & 0 & 0 & 105.1071428 \\

0 & 0 & 1 & 0

\end{bmatrix} \begin{bmatrix}

\dot{x} \\

x \\

\dot{\theta} \\

\theta

\end{bmatrix}

[/itex]

this can be written as

[itex]

\vec{\dot{x}} = A \vec{x}

[/itex]

This requires that I find the matrix exponential of "A".

## Homework Equations

The discrete time equivalent matrix, [itex]A_k[/itex], is computed as

[itex]A_k = e^{A \tau}[/itex]

where [itex]\tau[/itex] is the sampling period of the discrete time system.

## The Attempt at a Solution

I tried to diagonalize the A matrix by pre- and post-multiplying by its matrix of eigenvectors, but said matrix seems like it might be singular which means I can't diagonalize the A matrix. The A matrix is not full rank, and I don't know if this is why it's causing me problems. I read somehwere that a matrix can only be diagonalized if it has non-repeating eigenvalues, and mine has two zero eigenvalues, so is this why it won't work? If not, how can I transform my system into a system that can be discretized?