Page "State observer" Paragraph 63

So, for sufficiently large gains, all observer estimated states reach the actual states in finite time.

In fact, increasing allows for convergence in any desired finite time so long as each function can be bounded with certainty.

Hence, the requirement that the map is a diffeomorphism ( i. e., that its Jacobian linearization is invertible ) asserts that convergence of the estimated output implies convergence of the estimated state.

That is, the requirement is an observability condition.