Optimal Control of Vehicular Formations with Nearest Neighbor Interactions

Abstract: We consider the design of optimal localized feedback gains for one-dimensional formations in which vehicles only use information from their immediate neighbors. The control objective is to enhance coherence of the formation by making it behave like a rigid lattice. For the single-integrator model with symmetric gains, we establish convexity, implying that the globally optimal controller can be computed efficiently. We also identify a class of convex problems for double-integrators by restricting the controller to symmetric position and uniform diagonal velocity gains. To obtain the optimal non-symmetric gains for both the single- and the double-integrator models, we solve a parameterized family of optimal control problems ranging from an easily solvable problem to the problem of interest as the underlying parameter increases. When this parameter is kept small, we employ perturbation analysis to decouple the matrix equations that result from the optimality conditions, thereby rendering the unique optimal feedback gain. This solution is used to initialize a homotopy-based Newton's method to find the optimal localized gain. To investigate the performance of localized controllers, we examine how the coherence of large-scale stochastically forced formations scales with the number of vehicles. We establish several explicit scaling relationships and show that the best performance is achieved by a localized controller that is both non-symmetric and spatially-varying.

• The paper develops optimal control strategies for vehicular formations with nearest neighbor interactions, making them behave like rigid lattices using single and double integrator models.
• It demonstrates that non-symmetric feedback gains significantly outperform symmetric ones, achieving better coherence scaling in large formations, computed via homotopy methods.
• These strategies provide practical insights for designing scalable control systems for applications like autonomous vehicle platooning, reducing reliance on global positioning and communication.

Overview of "Optimal Control of Vehicular Formations with Nearest Neighbor Interactions"

The paper focuses on the development of optimal control strategies for vehicular formations, specifically addressing localized feedback with nearest neighbor interactions. It aims to enhance the coherence of vehicular formations, making these formations behave like a rigid lattice. This is fundamentally an optimal control problem under constraints dictated by the need for local interactions, which aligns with real-world technological applications of automated highways and vehicle platooning.

The authors have approached the problem from both single-integrator and double-integrator perspectives. For the single-integrator model, they demonstrate that the problem becomes convex when the feedback gains are symmetric, allowing for efficient computation of globally optimal controllers. They extend the analysis to achieve convexity in double-integrator systems by restricting the gains to symmetric positions and uniform velocities.

Key methodological advances include the use of perturbation analysis to solve parameterized families of control problems, starting from easily solvable cases to those of practical interest. This is performed under a homotopy-based method utilizing Newton’s approach, which allows tracking optimal feedback gains despite the complexity introduced by problem constraints.

Performance evaluation considers how the scaling of coherence in large-scale formations behaves in the presence of stochastic disturbances. The research establishes several explicit scaling relationships and extends its findings to show that non-symmetric and spatially varying gains outperform symmetric strategies in enhancing formation coherence.

Theoretical and Practical Implications

1. Theoretical Implications:
   • The work extends understanding of how structured optimal control problems can be made convex under certain constraints, enhancing the computational feasibility of these problems.
   • By considering both symmetric and non-symmetric designs, it challenges previous norms and shows how strategic asymmetry in gains can optimally enhance performance.
   • A fourth-root asymptotic scaling achieved with optimal non-symmetric gains illustrates significant theoretical advancement beyond symmetric strategies’ square-root scaling.
2. Practical Implications:
   • The insights gained can be directly applied to the design of control systems for traffic management and autonomous vehicle platoons, potentially reducing infrastructure needs for global positioning.
   • By demonstrating that localized control strategies can exhibit strong performance, practical implementations can reduce communication overhead in large networks.
   • The findings suggest practical methods for designing feedback gains that are scalability-aware, enhancing the robustness and efficiency of transportation systems dealing with uncertainty and noise.

Future Directions

Going forward, research could explore exploring other network topologies beyond one-dimensional paths, potentially examining distributed systems over more complex graph structures. Furthermore, applying the frameworks developed to other domains such as robotics and smart grid systems could validate and extend the utility of the presented methodologies. Investigations into adaptive control mechanisms that cater to dynamic changes in formation size and composition would also be valuable, particularly for real-world deployments where predictability of conditions is limited.

Overall, this paper presents a detailed and rigorous approach to designing optimal controllers for vehicular formations, with substantial implications for both theoretical control design and practical vehicular systems.