Kalman filtering under unknown inputs and norm constraints

Due to its potential applications in robotics and navigation, recent years have witnessed some progress in Kalman filter (KF) with norm constraints on the state. A noticeable discovery of the existing literature is that the KF gain has an analytical expression, and the brute-force normalization (i.e., estimation without considering the norm constraints, followed by a normalization operation) is optimal in the mean-square sense. Although there are some extensions of the former works to situations with uncertainties, existing results are only limited to cases where models/bounds or statistical properties of the disturbances are known. The paper considers the design of KF for systems subject to norm constraints on the state and unknown inputs, whose models or statistical properties are not assumed to be available. Both cases with and without direct feedthrough will be discussed. For systems without direct feedthrough, we show that the KF gain can be derived explicitly and the brute-force normalization is optimal in the mean-square sense, thereby generalizing the above-mentioned works on norm-constrained KF. However, for the case with direct feedthrough, the KF gain does not seem to admit an analytical solution, as to be shown via a counterexample.