Jump Control of Probability Densities With Applications to Autonomous Vehicle Motion

Abstract

We investigate the problem of controlling the probability density of the state of a process that is observed by the controller via a fixed but unknown function of the state. The goal is to control the process so that its probability density at a point in the state space becomes proportional to the value of the function observed at that point. Our solution, inspired by bacterial chemotaxis, involves a randomized controller that switches among different deterministic modes. We show that under appropriate controllability conditions, this controller guarantees convergence of the probability density to the desired function. The results can be applied to the problem of in loco optimization of a measurable signal using a team of autonomous vehicles that use point measurements of the signal but do not have access to position measurements. Alternative applications in the area of mobile robotics include deployment and environmental monitoring.

Supervisory control theory applied to swarm robotics

“…Hybrid system theory (HST) offers an alternative, where the system can be represented by both discrete and continuous states. HST has been used in the context of swarm robotics, multirobot systems, and other multi-agent systems (Tomlin et al 1998;Fierro et al 2001;McNew and Klavins 2006;McNew et al 2007;Zavlanos et al 2009;Mesquita 2010;Mesquita and Hespanha 2012). Belta et al (2007), for example, use HST in a motion planning problem.…”

Performance Bounds on Spatial Coverage Tasks by Stochastic Robotic Swarms

“…ADR PDE models, in particular, have been used by the authors to design robot control policies that achieve target spatial distributions of robot activity over a bounded domain [13] and that drive the swarm to a distribution that is proportional to a locally measured scalar field [12]. ADR PDEs have also been used to control the probability density functions (pdfs) of multidimensional stochastic processes [2], develop multiagent coverage and search strategies that are inspired by bacterial chemotaxis [22], and maximize the probability of swarm robotic presence in a desired region [23]. Other work on PDE-based analysis and design of agent control laws includes a study of multiagent consensus protocols in an Eulerian framework [8]; strategies for confining a population of agents, represented as a continuum, with a few discrete leader agents [10]; and an approach to flocking control for a group of agents governed by the kinetic Cucker-Smale model [26].…”

Construction of Lyapunov functions for piecewise-deterministic Markov processes

“…This problem was extended in [10], where it was posed as the problem of controlling the probability density of a PDP by selecting the jump intensity λ and the jump kernel Q as a function of an output. Applications were provided in the area of mobile robotics, where the method can be used to solve problems such as search, deployment and monitoring.…”

“…All other parameters are kept the same. Under the framework of [10], one can verify that q remains the unique stationary density of the process as long as ρ is some locally Lipschitz smooth function of x and independent of v.…”

“…Along with convexity, another important property of the functional µ, −e −U Le U is that it depends affinely of the differential operator f · ∇. As a consequence, this operator does not appear in the first variation optimality condition (10) or in the second variation of µ, −e −U Le U . This allows one to exploit the compactness properties typically present in jump kernels.…”