Distributed and finite-time estimation in networked systems

This thesis is a broad treatment of the distributed state estimation problem for linear systems. In this setting, a network of observer nodes collectively estimates the state of a dynamical system, since individually they are not able to do so. The proposed solution consists of a distributed observer which uses diffusive coupling and leads to three complementary contributions. The first one considers exponential convergence with arbitrary rates. Various design approaches are put into a unified framework to facilitate their comparison. Differences in the continuous-time and discrete-time case are related with synchronization. The designs are feasible under the necessary condition of distributed observability with respect to the graph of the network, which is akin to observability in centralized state estimation. Furthermore, diagonal stability of the compressed Laplacian matrix and block-diagonal stability of the error dynamics in original coordinates form crucial building blocks for the other contributions of the thesis. It is concluded that a more general design procedure is desirable to reduce the size of exchanged information and to account for delays. The second contribution is the design of distributed observers where the estimates reach the state of the system exactly in a finite time, in contrast to the asymptotic convergence of the preceding linear designs. Taking advantage of local observability decompositions, centralized finite-time observers are used for the observable substate while a nonlinear consensus-term is employed for the unobservable substate. Sufficient bounds on the gain parameters are obtained using the concept of homogeneity. As a third contribution, an advantage of distributed observers is demonstrated by taking into account the specific effects of communications, which are modeled using time-varying delays. With the Lyapunov-Krasovskii functional approach, the selection of gains which guarantee a certain exponential convergence rate in the face of communication are cast into solving a linear matrix inequality. In a numerical example, diffusively coupled observer nodes achieve a better performance compared to the direct transmission of partial outputs. The exchange of artificial outputs with decreasing size shows a gradual decline in performance to the level of output exchange, which is balanced by weaker connectivity requirements. Finally, the vector Lyapunov approach is extended to cope with time-varying delays.