In this paper, a unified batch-online learning approach is introduced to learn a linear representation of nonlinear system dynamics using the Koopman operator. The presented system modeling approach leverages a novel incremental Koopman-based update law that regains a mini-collection of samples stored in a memory to minimize not only the instantaneous Koopman operator’s identification errors but also the identification errors for the collection of retrieved samples. Discontinuous modifications of gradient flows are presented for the online update law to assure finite-time convergence under easy-to-verify conditions defined on the batch of data. Therefore, this unified online-batch framework allows joint sample- and time-domain analysis to converge the Koopman operator’s parameters. More specifically, it is shown that if the collected mini-batch of samples guarantees a rank condition, then finite-time guarantee in the time domain can be certified, and the settling time depends on the quality of collected samples being reused in the update law. Moreover, the efficiency of the proposed Koopman-based update law is further analyzed by showing that the identification regret in continuous time grows sub-linearly with time. Furthermore, to avoid learning corrupted dynamics due to the selection of an inappropriate set of Koopman observables, a higher-layer meta-learner employs a discrete Bayesian optimization algorithm to obtain the best library of observable functions for the operator. Since finite-time convergence of the Koopman model for each set of observables is guaranteed under a rank condition on stored data, the fitness of each set of observables can be obtained based on the identification error on the stored samples in the proposed framework and even without implementing any controller based on the learned system. Finally, to confirm the effectiveness of the proposed scheme, two simulation examples are presented.

This article presents a fixed-time (FxT) system identifier for continuous-time nonlinear systems. A novel adaptive update law with discontinuous gradient flows of the identification errors is presented, which leverages concurrent learning (CL) to guarantee the learning of uncertain nonlinear dynamics in a fixed time, as opposed to asymptotic or exponential time. More specifically, the CL approach retrieves a batch of samples stored in a memory, and the update law simultaneously minimizes the identification error for the current stream of samples and past memory samples. Rigorous analyses are provided based on FxT Lyapunov stability to certify FxT convergence to the stable equilibria of the gradient descent flow of the system identification error under easy-to-verify rank conditions. The performance of the proposed method in comparison with the existing methods is illustrated in the simulation results.

In this article, we analyze the adverse effects of cyber–physical attacks as well as mitigate their impacts on the event-triggered distributed Kalman filter (DKF). We first show that although event-triggered mechanisms are highly desirable, the attacker can leverage the event-triggered mechanism to cause nontriggering misbehavior, which significantly harms the network connectivity and its collective observability. We also show that an attacker can mislead the event-triggered mechanism to achieve continuous-triggering misbehavior, which not only drains the communication resources but also harms the network’s performance. An information-theoretic approach is presented next to detect attacks on both sensors and communication channels. In contrast to the existing results, the restrictive Gaussian assumption on the attack signal’s probability distribution is not required. To mitigate attacks, a meta-Bayesian approach is presented that incorporates the outcome of the attack detection mechanism to perform second-order inference. The proposed second-order inference forms confidence and trust values about the truthfulness or legitimacy of sensors’ own estimates and those of their neighbors, respectively. Each sensor communicates its confidence to its neighbors. Sensors then incorporate the confidence they receive from their neighbors and the trust they formed about their neighbors into their posterior update laws to successfully discard corrupted information. Finally, the simulation result validates the effectiveness of the presented resilient event-triggered DKF.