  In this paper we show that inverses of well-conditioned, finite-time Gramians
and impulse response matrices of large-scale interconnected systems described
by sparse state-space models, can be approximated by sparse matrices. The
approximation methodology established in this paper opens the door to the
development of novel methods for distributed estimation, identification and
control of large-scale interconnected systems. The novel estimators
(controllers) compute local estimates (control actions) simply as linear
combinations of inputs and outputs (states) of local subsystems. The size of
these local data sets essentially depends on the condition number of the
finite-time observability (controllability) Gramian. Furthermore, the developed
theory shows that the sparsity patterns of the system matrices of the
distributed estimators (controllers) are primarily determined by the sparsity
patterns of state-space matrices of large-scale systems. The computational and
memory complexity of the approximation algorithms are $O(N)$, where $N$ is the
number of local subsystems of the interconnected system. Consequently, the
proposed approximation methodology is computationally feasible for
interconnected systems with an extremely large number of local subsystems.
