Simultaneous Input and State Estimation Under Output Quantization: A Gaussian Mixture Approach

Simultaneous Input and State Estimation (SISE) enables the reconstruction of unknown inputs and internal states in dynamical systems, with applications in fault detection, robotics, and control. While various methods exist for linear systems, extensions to systems with output quantization are scarce, and no formal connections to limit Kalman filters are known in this context. This letter addresses these gaps by proposing a novel SISE algorithm for linear systems with quantized output measurements. The proposed algorithm introduces a Gaussian mixture model formulation of the observation model, which leads to closed-form recursive equations in the form of a Gaussian sum filter. In the absence of input prior knowledge, the recursions are shown to converge to a limit-case SISE algorithm, implementable as a bank of linear SISE filters running in parallel. A simulation example is presented to illustrate the effectiveness of the proposed approach.