  The coupled Riccati equations are cosisted of multiple Riccati-like equations
with solutions coupled with each other, which can be applied to depict the
properties of more complex systems such as markovian systems or multi-agent
systems. This paper manages to formulate and investigate a new kind of coupled
Riccati equations, called harmonic-coupled Riccati equations (HCRE), from the
matrix iterative law of the consensus on information-based distributed
filtering (CIDF) algortihm proposed in [1], where the solutions of the
equations are coupled with harmonic means. Firstly, mild conditions of the
existence and uniqueness of the solution to HCRE are induced with collective
observability and primitiviness of weighting matrix. Then, it is proved that
the matrix iterative law of CIDF will converge to the unique solution of the
corresponding HCRE, hence can be used to obtain the solution to HCRE. Moreover,
through applying the novel theory of HCRE, it is pointed out that the real
estimation error covariance of CIDF will also become steady-state and the
convergent value is simplified as the solution to a discrete time Lyapunov
equation (DLE). Altogether, these new results develop the theory of the coupled
Riccati equations, and provide a novel perspective on the performance analysis
of CIDF algorithm, which sufficiently reduces the conservativeness of the
evaluation techniques in the literature. Finally, the theoretical results are
verified with numerical experiments.
