A local characterization of observability

We consider time-dependent linear systems of the form , y = Cx with state x Rn, control (input) u Rm, and output y Rp. The main results are local characterizations of observability and strong observability (or observability with unknown inputs) of (A, C) and (A, B, C). These criteria are pointwise rank conditions on a certain matrix, which is explicitly built up from the first n − 2 derivatives of A and B and the first n − 1 derivatives of C. The results generalize well-known theorems for time-invariant systems. The proofs lead also to observers (with and without the input), and the main tool is a generalized product rule for the differentiation of a product of matrices, where only one factor and the product itself are known to be differentiable.