I need to check that I'm solving this question correctly and where to go afterwards. I'm reducing the given equation to a controllable and observable equation.

From the picture, it is clear the A matrix is in Jordan form where λ1 is in a 3×3 Jordan block and λ2 is in a 2×2 Jordan block. The B matrix that corresponds to λ1 is the top 3 elements. Since the third element is 0, λ1 is not controllable.

The B matrix that corresponds to λ2 is the bottom 2 elements. Since the 5th element is 1, it is controllable.

For Observability, I need to check the C matrix. Since the first element is 0, λ1 is not observable, and since the fourth element is 0, λ2 is not observable.

So I know λ1 is not controllable or observable and λ2 is controllable but not observable. How do I write the corresponding state equation?

## Best Answer

I think the problem wants you to do the Kalman decomposition and since they are in a Jordan normal form it should be quick to do. From the wording of the problem itself it is not really clear, but you should probably only represent the part of the system that is both controllable and observable (and discard the rest of it?).