Finite state abstraction and formal methods for traffic flow networks

Formal methods from computer science have emerged as a powerful suite of tools that, under appropriate modifications, are applicable to a wide range of physical control systems. These methods promise automated algorithms for verification and synthesis of controllers to accomplish specifications and objectives that are not accommodated by traditional approaches. However, formal methods require a finite abstraction of the underlying physical process, and challenges in obtaining scalable abstraction techniques have impeded the applicability of these automated tools. This tutorial paper exploits structural properties in a class of networked systems motivated by traffic flow networks to overcome some of these challenges and points towards new directions of research. The first aim of this tutorial paper is to review a broad technique for synthesizing correct-by-design controllers for dynamical systems by first obtaining a finite state abstraction and then applying a game-based algorithm for synthesizing a control strategy to satisfy a linear temporal logic specification. The second objective is to review vehicular traffic flow models and to characterize a general model amenable to formal control synthesis. This general model is shown to be mixed monotone, a generalization of monotone dynamical systems. Using properties of the mixed monotone dynamics, a finite state abstraction is efficiently computed by overapproximating the set of states that are one-step reachable under the traffic flow dynamics. These results are demonstrated on a case study which leads to a discussion of open problems and avenues for future research.