Abstract
Significant theoretical and practical challenges arise in the cooperative control of distributed nonlinear multi-agent systems (MAS), particularly when they involve nonstrict-feedback interconnections and unknown state-dependent control gains. Conventional neural adaptive controllers, while versatile, often operate as “closed box” models, leading to solutions that may lack physical plausibility and exhibit compromised robustness. This paper addresses this critical gap by introducing a novel Physics-Regularized Adaptive Control (PRAC) framework, implemented via neural backstepping. Central to PRAC is the design of novel Physics-Regularized Neural Networks (PRNNs), which are realized using Radial Basis Function Neural Networks (RBFNNs) as their architectural foundation in this paper. Instead of treating the neural network as a simple approximator, the PRAC methodology embeds physical priors, such as equilibrium conditions, system smoothness, and energy dissipation principles, into the PRNNs’ online adaptive laws as differentiable regularization terms. The gradients of these terms actively constrain the PRNN weight adaptation, transforming the learning process into a Lyapunov-guided constrained optimization. This enhances the physical consistency and interpretability of the learned dynamics while simultaneously improving control performance. By synergistically combining this physics-regularized architecture with Dynamic Surface Control (DSC) to manage computational complexity, the proposed scheme guarantees cooperative uniformly ultimately bounded (CUUB) tracking of a leader’s trajectory. Rigorous Lyapunov analysis substantiates the theoretical guarantees, which are further validated by comprehensive numerical simulations and a practical networked inverted-pendulum example demonstrating superior tracking accuracy and robustness over conventional neural adaptive controllers. Note to Practitioners—This paper introduces a novel control framework for complex multi-agent systems, such as robotic swarms or drone formations, where the internal dynamics of each agent are nonlinear and largely unknown. A primary challenge with using conventional neural network controllers in these scenarios is that they often act as “black boxes.” While powerful, these controllers learn solely based on tracking errors and lack any inherent understanding of the underlying physics. This can lead to physically implausible behaviors, reduced robustness against unexpected disturbances, and a general lack of interpretability, which is a significant concern in safety-critical applications. This paper addresses this gap by proposing a “Physics-Regularized Adaptive Control” (PRAC) strategy. The core idea is to move beyond the “black-box” paradigm by embedding fundamental physical principles directly into the controller’s online learning process. Specifically, principles such as equilibrium conditions, system smoothness, and energy dissipation are mathematically formulated as soft constraints. These constraints actively guide the neural network’s adaptation, effectively penalizing any learning steps that would violate physical laws. This transforms the learning process into a constrained optimization that is guided by both tracking performance and physical consistency. The result is a controller that not only learns to be accurate but also behaves in a way that is physically sensible and more robust. The paper provides the rigorous mathematical proofs to guarantee system stability and demonstrates through simulations and a practical networked inverted-pendulum example that the proposed PRAC framework significantly outperforms conventional neural controllers, yielding faster convergence, superior tracking precision, and more interpretable behavior, especially in the presence of uncertainties and external disturbances.