## Kalman Filtering and Zonotopic State Bounding for Robust Fault Detection

The satisfaction of advanced monitoring and fault tolerant control requirements heavily depends on the reliable estimation of physical values in many engineering systems. However, the information that can be inferred about such values is necessarily partial since measurements are usually corrupted by noise and since the knowledge brought by physical models can never be a complete and perfectly accurate image of reality. The explicit characterization of uncertainties is thus of prior importance in the context of model-based fault detection to ensure the reliability of the computed decisions. Indeed, such characterization strongly influences the computation of consistent confidence domains and/or explicit decision thresholds.

Two usually distinct paradigms exist to model uncertainties. The stochastic one deals with random uncertainties. Strong assumptions about their probability distributions are often made, especially when online computation is required. For instance, assuming independent Gaussian distributions is common practice when using standard versions of the famous Kalman Filter. Though efficient to deal with measurement noises, the assumption of known probability distributions finds its limits to characterize epistemic uncertainties. Indeed, the lack of a precise knowledge about disturbances such as the load torque of a motor may be better characterized by interval bounds with no assumption about the values distribution. This is the usual motivation for using the set-membership paradigm to model uncertainties. Though explicitly computed sets can achieve a so-called guaranteed robustness to the worst-cases resulting from the specified uncertainty bounds, the bounded-error paradigm however usually suffers from a poor management of random measurement noises.

In this talk, a joint Zonotopic and Gaussian Kalman Filter (ZGKF) will be presented. It will be shown to provide a solution for the robust fault detection of uncertain discrete-time systems simultaneously subject to bounded disturbances and Gaussian noises. The covariation of a zonotope will be introduced as a set-membership analog to covariance, making it possible to compute a time-varying optimal observer gain jointly minimizing both kinds of uncertainties: bounded/zonotopic and Gaussian. Then, given a maximal probability of false alarms, an innovation-based detection test will be shown to merge the usually mutually exclusive beneﬁts granted by set-membership techniques (robustness to worst-case within speciﬁed bounds, domain computations) and stochastic approaches (taking noise distribution into account, probabilistic evaluation of tests). Numerical simulations will illustrate the signiﬁcantly improved tradeoﬀ between sensitivity to faults and robustness to disturbances/noises, while the computations (prediction/update, optimal gain, conﬁdence domains, adaptive thresholds, detection test) remain explicit and can be eﬃciently implemented.