Interconnected Observers for Robust Decentralized Estimation With Performance Guarantees and Optimized Connectivity Graph

Motivated by the need for observers that are both robust to disturbances and guarantee fast convergence to zero of the estimation error, we propose an observer for linear time-invariant systems with a noisy output that consists of a combination of $N$-coupled observers over a connectivity graph. At each node of the graph, the output of these interconnected observers is defined as the average of the estimates obtained using local information. The convergence rate and the robustness to measurement noise of the proposed observer's output are characterized in terms of $\mathcal{KL}$ bounds. Several optimization problems are formulated to design the proposed observer in order to satisfy a given rate of convergence specification while minimizing the $H_{\infty}$ gain from noise to estimates or the size of the connectivity graph. It is shown that the interconnected observers relax the well-known tradeoff between the rate of convergence and noise amplification, which is a property attributed to the proposed innovation term, that over the graph, couples the estimates between the individual observers. Sufficient conditions involving information of the plant only, ensuring that the estimate obtained at each node of the graph outperforms the one obtained with a single, standard Luenberger observer are given. The results are illustrated in several examples throughout this paper.