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We provide a novel robust stability analysis for moving horizon estimation (MHE) using a Lyapunov function. Additionally, we introduce linear matrix inequalities (LMIs) to verify the necessary incremental input/output-to-state stability ($δ$-IOSS) detectability condition. We consider an MHE formulation with time-discounted quadratic objective for nonlinear systems admitting an exponential $δ$-IOSS Lyapunov function. We show that with a suitable parameterization of the MHE objective, the $δ$-IOSS Lyapunov function serves as an $M$-step Lyapunov function for MHE. Provided that the estimation horizon is chosen large enough, this directly implies exponential stability of MHE. The stability analysis is also applicable to full information estimation, where the restriction to exponential $δ$-IOSS can be relaxed. Moreover, we provide simple LMI conditions to systematically derive $δ$-IOSS Lyapunov functions, which allows us to easily verify $δ$-IOSS for a large class of nonlinear detectable systems. This is useful in the context of MHE in general, since most of the existing nonlinear (robust) stability results for MHE depend on the system being $δ$-IOSS (detectable). In combination, we thus provide a framework for designing MHE schemes with guaranteed robust exponential stability. The applicability of the proposed methods is demonstrated with a nonlinear chemical reactor process and a 12-state quadrotor model.