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A nonlinear asymptotic observer for a discrete-time nonlinear system is considered. The observer is based on a Kalman filter that uses the state dependent Riccati equation (SDRE) to obtain the filter gain. Unlike the extended Kalman filter, the SDRE-based Kalman filter does not involve the evaluation of a Jacobian at every time step. The convergence properties of the SDRE-based Kalman filter when used as an observer in a deterministic setting are analyzed. A few simulation examples are provided to demonstrate the performance and implementation of the SDRE-based observer in both deterministic and stochastic settings.