  This paper studies observability for non-uniform hypergraphs with inputs and
outputs. To capture higher-order interactions, we define a canonical
non-homogeneous dynamical system with nonlinear outputs on hypergraphs. We then
construct algebraic necessary and sufficient conditions based on polynomial
ideals and varieties for global observability at an initial state of
hypergraphs. An example is given to illustrate the proposed criteria for
observability. Further, necessary and sufficient conditions for local
observability are derived based on rank conditions of observability matrices,
which provide a framework to study local observability for non-uniform
hypergraphs. Finally, the similarity of observability for hypergraphs is
proposed using similarity of tensors, which reveals the relation of
observability between two hypergraphs and helps to check the observability
intuitively.
