How do I find time continuous A, B matrices from a discrete state space model?

Question

The easiest way to compute a discrete state space model from time continuous state space model is by this method.

$$ \begin{bmatrix} A_d & B_d \\ 0 & I \end{bmatrix} = \exp\left( \begin{bmatrix} A & B \\ 0 & 0 \end{bmatrix} h\right), $$

But how would it be if I have the discrete state space model and I want to find the time continuous state space model?

Edit: I'm just guessing now. Is this the right method?

$$\frac {1}{h}\ln \left(\begin{bmatrix} A_d & B_d \\ 0 & I \end{bmatrix}\right) = \begin{bmatrix} A & B \\ 0 & 0 \end{bmatrix}$$

Answer 1

If you want to find $T=\begin{bmatrix} A & B \\ 0 & 0 \end{bmatrix}$ such that $\exp(hT)=\begin{bmatrix} A_d & B_d \\ 0 & I \end{bmatrix}=:M$, then you can compute $\log(M)/h$ where $\log$ is the matrix logarithm. The matrix logarithm, like the complex logarithm, is multi-valued, but the principal one can be obtained by the Matlab function logm.