Observability of a Linear System Under Sparsity Constraints

Consider an n-dimensional linear system where it is known that there are at most nonzero components in the initial state. The observability problem, that is the recovery of the initial state, for such a system is considered. We obtain sufficient conditions on the number of available observations to be able to recover the initial state exactly for such a system. Both deterministic and stochastic setups are considered for system dynamics. In the former setting, the system matrices are known deterministically, whereas in the latter setting, all of the matrices are picked from a randomized class of matrices. The main message is that one does not need to obtain full n observations to be able to uniquely identify the initial state of the linear system, even when the observations are picked randomly, when the initial condition is known to be sparse.