Adaptive Event-Triggered Optimal Control With Regulative Learning Rate Under Aperiodic DoS Attacks

In this article, the optimal control for a nonlinear affine system under aperiodic Denial-of-service (DoS) attacks is investigated. To solve the Hamilton–Jacobi–Bellman (HJB) equation, an adaptive dynamic programming (ADP) algorithm based on a single critic network is developed. The proposed regulative learning rate strategy outperforms traditional fixed-rate gradient descent approaches found in existing works. With the objective of minimizing the performance index, the optimal value function and the optimal controller are derived from the approximate solution of the HJB equation. To alleviate the resource demand and enhance the flexibility of the threshold function, an adaptive event-triggered (AET) scheme integrating the idea of sampling control and event-triggered strategy is applied to the optimal control initially. Compared with the static event-triggered strategy, the AET method contains increasing engineering value. A piecewise Lyapunov function is constructed based on optimal value function, estimated error introduced by neural network (NN) weight, and the classic Lyapunov-Krasovskii function. Thus, uniform ultimate boundedness for tracking error is proven theoretically. Moreover, the maximum tolerable strength of cyberattacks is provided from the stability analysis. The simulation results exhibit the designed approach’s reliability.