Optimal Control of Vehicular Formations with Nearest Neighbor Interactions

TL;DR

This paper develops methods to design optimal localized feedback controllers for vehicle formations, improving coherence and scalability by considering symmetric and non-symmetric gains, with solutions derived via convex optimization and homotopy methods.

Contribution

It introduces a novel approach to optimize localized feedback gains for vehicular formations, including non-symmetric controllers, using convex analysis and perturbation techniques.

Findings

Localized non-symmetric controllers outperform symmetric ones.

Optimal controllers improve formation coherence as the number of vehicles increases.

Explicit scaling laws for formation coherence with large vehicle numbers.

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.