Distributed Control of Networked Dynamical Systems: Static Feedback, Integral Action and Consensus (1310.8620v2)

Abstract: This paper analyzes distributed control protocols for first- and second-order networked dynamical systems. We propose a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain, and a sum of nonlinear interaction functions. By using integral Lyapunov functions, we prove the stability of the proposed control protocols, and explicitly characterize the equilibrium set. We also propose a distributed proportional-integral (PI) controller for networked dynamical systems. The PI controllers successfully attenuate constant disturbances in the network. We prove that agents with single-integrator dynamics are stable for any integral gain, and give an explicit tight upper bound on the integral gain for when the system is stable for agents with double-integrator dynamics. Throughout the paper we highlight some possible applications of the proposed controllers by realistic simulations of autonomous satellites, power systems and building temperature control.

• The paper introduces nonlinear consensus protocols that decouple gain and interaction functions, enabling rigorous stability proofs via integral Lyapunov functions.
• The paper presents a distributed PI controller that mitigates constant disturbances with a tight upper bound on the required integral gain for stability.
• Realistic simulations on systems like autonomous satellites and building temperature control validate the practical efficacy of the proposed distributed control methods.

Overview of Distributed Control of Networked Dynamical Systems

This paper presents a comprehensive paper on distributed control protocols tailored for networked dynamical systems with both first- and second-order integrator dynamics. The focus lies in the formulation and analysis of nonlinear consensus controllers, which leverage a control input structure defined as a product of a nonlinear gain and a sum of nonlinear interaction functions. By utilizing integral Lyapunov functions, the stability of these protocols is established, and the equilibrium set is explicitly defined. Additionally, the paper introduces a distributed proportional-integral (PI) controller, adept at mitigating constant disturbances within the networked systems.

Key Contributions

The paper introduces several pivotal contributions to the domain of distributed control:

1. Nonlinear Consensus Protocols: It proposes a class of nonlinear consensus protocols applicable to both first- and second-order networked dynamics. The control inputs are decoupled into gain and interaction components, supporting the formulation of a wide array of systems with inherent nonlinear dynamics.
2. Stability Proofs and Equilibrium Characterization: Leveraging integral Lyapunov functions, the stability of these nonlinear consensus protocols is rigorously proven. The equilibrium set is characterized by integral equations, offering predictability regarding the point of convergence.
3. PI Controllers for Disturbance Rejection: The paper introduces a distributed PI controller that effectively handles constant disturbances, ensuring system stability. Notably, it provides a tight upper bound on the integral gain necessary for stability in systems with double-integrator dynamics.
4. Realistic Simulations and Applications: The applicability of the theoretical results is demonstrated through simulations focused on various real-world systems, including autonomous satellites, power systems, and building temperature control, thereby bridging the gap between theory and practice.

Analytical Insights

The paper's strength lies in its detailed analytical treatment of nonlinear feedback mechanisms within distributed protocols. By decoupling nonlinear gain and interaction dynamics, it addresses scenarios unmet by linear consensus approaches. This decoupling facilitates systems with unique properties, such as velocity-dependent mass dynamics or position-dependent damping forces, to converge predictably to consensus despite their nonlinearity.

Numerical Results and Implications

The simulations provide strong numerical support for the practicality of the proposed protocols. Key systems, like thermal regulation in buildings, exhibit energy preservation and optimized temperature balance, while satellite formations achieve coherent consensus without centralized control. These results underscore the efficacy of the discussed methods in practical, large-scale networked environments.

Theoretical and Practical Implications

From a theoretical perspective, this investigation paves the way for extending Lyapunov-based methods to nonlinear, integrator-based dynamics in networked settings. Practically, it opens avenues for implementing decentralized control strategies in power systems and autonomous vehicle networks, which are highly sensitive to disturbances and require rapid, reliable consensus.

Future Developments

Looking to the future, this work provides a foundation for further exploration into nonlinear and adaptive control frameworks. It suggests potential research directions including the integration of learning-based frameworks to augment traditional control protocols, thereby enhancing their adaptability in dynamically changing environments. Additionally, extending towards robustness guarantees in the presence of more complex disturbances and uncertainties remains an attractive avenue for follow-up studies.

In summary, this paper embodies a significant step towards understanding and implementing distributed nonlinear control in dynamical systems underpinned by networked interactions, with a particular focus on robustness and stability in the presence of disturbances.