This technical note concerns observer design for Lipschitz nonlinear systems with sampled output. Using reachability analysis, an upper approximation of the attainable set is given. When this approximation is formulated in terms of a convex combination of linear mappings, a sufficient condition is given in terms of linear matrix inequalities (LMIs) which can be solved employing an LMIs solver. This novel approach seems to be an efficient tool to solve the problem of observer synthesis for a class of Lipschitz systems of small dimensions.

This paper is concerned with the design of the distributed prescribed-time unknown input observer (DPTUIO) for a class of linear time-invariant (LTI) systems, which we refer to as the target systems. Results on distributed unknown input observers in the literature did not address the convergence time predefinition problem. To achieve fast estimation, this paper uses weakly unobservable subspace decomposition and introduces a time-scale transformation technology to develop a novel DPTUIO. This design ensures that, for each node, the entire state of the target system can be estimated within a pre-assigned time, despite each node measuring only a small portion of the system output. Detailed discussions on the existence conditions of the DPTUIO are provided. It is established that the observer matching condition, along with the jointly strongly observability condition and a strongly connected communication network, constitutes sufficient conditions for the existence of the DPTUIO. Moreover, the robustness of the developed DPTUIO against measurement and communication noise is analyzed in detail with rigorous deduction. Finally, a practical example is used to verify the effectiveness of the proposed methods.