V-Formation via Model Predictive Control

Abstract

We present recent results that demonstrate the power of viewing the problem of V-formation in a flock of birds as one of Model Predictive Control (MPC). The Vformation-MPC marriage can be understood in terms of the problem of synthesizing an optimal plan for a continuous-space and continuous-time Markov decision process (MDP), where the goal is to reach a target state that minimizes a given cost function.The first result we consider is ARES, an efficient approximation algorithm for generating optimal plans (action sequences) that take an initial state of an MDP to a state whose cost is below a specified (convergence) threshold. ARES uses Particle Swarm Optimization, with adaptive sizing for both the receding horizon and the particle swarm. Inspired by Importance Splitting, the length of the horizon and the number of particles are chosen such that at least one particle reaches a next-level state, i.e., a state where the cost decreases by a required delta from the previous-level state. The level relation on states and the plans constructed by ARES implicitly define a Lyapunov function and an optimal policy, respectively, both of which could be explicitly generated by applying ARES to all states of the MDP, up to some topological equivalence relation. We assess the effectiveness of ARES by statistically evaluating its rate of success in generating optimal plans for V-formation.ARES can alternatively be viewed as a model-predictive control (MPC) algorithm that utilizes an adaptive receding horizon, a technique we refer to as Adaptive MPC (AMPC). We next present Distributed AMPC (DAMPC), a distributed version of AMPC that works with local neighborhoods. We introduce adaptive neighborhood resizing, whereby the neighborhood size is determined by the cost-based Lyapunov function evaluated over a global system state. Our approach applies to reachability problems for any collection of entities that seek convergence from an arbitrary initial state to a desired goal state, where a notion of distance to the goal state(s) can be suitably defined. Our experimental evaluation shows that DAMPC can perform almost as well as centralized AMPC, while using only local information and a form of distributed consensus in each time step.Finally, inspired by security attacks on cyber-physical systems (CPS), we introduce controller-attacker games, where two players, a controller and an attacker, have antagonistic objectives. To highlight the power of adaptation, we formulate a special case of controllerattacker games called V-formation games, where the attacker's goal is to prevent the controller from attaining V-formation. We demonstrate how adaptation in the design of the controller helps in overcoming certain attacks.

An STL-based Formulation of Resilience in Cyber-Physical Systems

“…We consider a 500-second simulation (trajectory) of a 30-boid flock with a time-step of 0.1 sec in a 2-D plane; so, T = 500 secs. We subject 20 of the boids to an intermittent uniformly random displacement [11] from [0, M ] × [0, 2π], where M = 20 meters and 2π are the maximum magnitude and direction of the displacement, respectively. The subset of 20 boids is chosen uniformly at random during the intervals [100, 150], [250, 300], and [400, 450] in seconds.…”