Computationally Efficient Encrypted Neuroadaptive Optimal Control for Euler–Lagrange Systems With Unknown Dynamics

Euler–Lagrange (E–L), widely used in robotics, industrial automation, and autonomous vehicles, face increasing demands for both privacy protection and computational efficiency. Although traditional control methods can handle dynamic uncertainties of the system, they cannot fully address these demands. To fill this gap, this study develops a computationally efficient encrypted neuroadaptive optimal control (CE²NOC) method for E–L systems with unknown dynamics, which consists of three novel components: 1) an explicit-critic reinforcement learning (ECRL) algorithm that approximates the optimal control law without solving the HamiltonJacobiBellman equation by exploiting E–L system properties; 2) a finite impulse response-based encrypted neuroadaptive controller that compensates for dynamic uncertainties using fully homomorphic encryption, thus ensuring privacy without introducing the finite-lifespan issue; and 3) a dynamic event-triggered mechanism (DETM) that reduces communication and computation costs while maintaining closed-loop stability. Theoretical analysis and two practical examples validate the effectiveness of CE²NOC. Compared to conventional time-triggered schemes, CE²NOC ensures stability and data privacy while demonstrating a maximum reduction of 96% in communication and computation usage.