This paper provides an introduction and overview of recent work on control barrier functions and their use to verify and enforce safety properties in the context of (optimization based) safety-critical controllers. We survey the main technical results and discuss applications to several domains including robotic systems.

We present an efficient computational procedure for finite abstraction of discrete-time mixed monotone systems by considering a rectangular partition of the state space. Mixed monotone systems are decomposable into increasing and decreasing components, and significantly generalize the well known class of monotone systems. We tightly overapproximate the one-step reachable set from a box of initial conditions by computing a decomposition function at only two points, regardless of the dimension of the state space. We apply our results to verify the dynamical behavior of a model for insect population dynamics and to synthesize a signaling strategy for a traffic network.

We propose to synthesize a control policy for a Markov decision process (MDP) such that the resulting traces of the MDP satisfy a linear temporal logic (LTL) property. We construct a product MDP that incorporates a deterministic Rabin automaton generated from the desired LTL property. The reward function of the product MDP is defined from the acceptance condition of the Rabin automaton. This construction allows us to apply techniques from learning theory to the problem of synthesis for LTL specifications even when the transition probabilities are not known a priori. We prove that our method is guaranteed to find a controller that satisfies the LTL property with probability one if such a policy exists, and we suggest empirically with a case study in traffic control that our method produces reasonable control strategies even when the LTL property cannot be satisfied with probability one.

We propose a macroscopic traffic network flow model suitable for analysis as a dynamical system, and we qualitatively analyze equilibrium flows as well as convergence. Flows at a junction are determined by downstream supply of capacity as well as upstream demand of traffic wishing to flow through the junction. This approach is rooted in the celebrated Cell Transmission Model for freeway traffic flow. Unlike related results which rely on certain system cooperativity properties, our model generally does not possess these properties. We show that the lack of cooperativity is in fact a useful feature that allows traffic control methods, such as ramp metering, to be effective. Finally, we leverage the results of the paper to develop a linear program for optimal ramp metering.1 the few existing results on equilibria and convergence such as [9] and we present a simple linear program for obtaining a ramp metering control strategy that achieves the maximum possible steady-state network throughput.Our work is related to the dynamical flow networks recently proposed in [10,11] and further studied in [12]. In [10,11], downstream supply is not considered and thus downstream congestion does not affect upstream flow, an unrealistic assumption for traffic modeling. In [12], the authors allow flow to depend on the density of downstream links, but the paper focuses on throughput optimality of a particular class of routing policies that ensure the resulting dynamics are cooperative [13,14]. In contrast, the model proposed here is generally not cooperative. Furthermore, the adaptation to the CTM described briefly in [12, Section II.C] differs from our model in the following important respects: the model as discussed in [12, Section II.C] assumes a path graph network topology, requires identical links (i.e., identical supply and demand functions), and only considers trajectories in the region in which supply does not restrict flow (that is, α v (ρ) = 1 for all v ∈ V in our model), which is shown to be positively invariant given their assumptions. In this work, we generalize each of these restrictions.In a separate direction of research, many network models attempt to apply single road PDE models such as [15,16] directly to networks, see [17] for a thorough treatment. Recent results such as [18] and [19] provide analytical tools for traffic network estimation and modeling using PDE models. The CTM and related models, including our proposed model, can be considered to be a discretization of an appropriate PDE model [6]. Alternatively, these models and the model we propose in this work fit into the broad class of compartmental systems that model the flow of a substance among interconnected "compartments" [20,21,22].We first proposed a compartmental model of traffic flow in [23]. Here, we expand on the conference version by discussing the general lack of cooperativity for our proposed model, identifying how lack of cooperativity can be exploited to increase throughput via ramp metering, and providing an explicit optimization problem for...

We propose a framework for generating a signal control policy for a traffic network of signalized intersections to accomplish control objectives expressible using linear temporal logic. By applying techniques from model checking and formal methods, we obtain a correct-by-construction controller that is guaranteed to satisfy complex specifications. To apply these tools, we identify and exploit structural properties particular to traffic networks that allow for efficient computation of a finite state abstraction. In particular, traffic networks exhibit a componentwise monotonicity property which allows reach set computations that scale linearly with the dimension of the continuous state space.

In this paper, we propose a traffic network flow model particularly suitable for qualitative analysis as a dynamical system. Flows at a junction are determined by downstream supply of capacity (lack of congestion) as well as upstream demand of traffic wishing to flow through the junction. This approach is rooted in the celebrated Cell Transmission Model for freeway traffic flow, and we analyze resulting equilibrium flows and convergence properties. I. INTRODUCTIONCompartmental systems are a broad modeling paradigm to study fluid-like flow of a single substance among interconnected "compartments" [1], [2]. The main contribution of this paper is to propose and analyze a compartmental model of freeway traffic networks that is amenable to analysis as a dynamical system. Existing approaches are often well-suited for simulations or for validation/fitting with empirical data, but the available literature often gives little insight into the network-level, qualitative properties of the dynamics. For example, models such as [3], [4] and the celebrated Cell Transmission Model (CTM) [5], [6] were primarily developed for simulation with few analytical results available. The primary exception is [7] which provides a thorough investigation of the CTM when modeling a stretch of highway with onramp queues but does not consider more general networks.We propose a model that encompasses the CTM and extends the model to general nonlinear supply and demand functions and to more general network topologies. In our proposed model, we consider a traffic network composed of road links interconnected at junctions. In keeping with the philosophy of the CTM, the flow of traffic through a junction is determined by the available supply of downstream road space into which vehicles can flow and upstream demand of vehicles wishing to flow into a given link.Our work is related to the dynamical flow networks recently proposed in [8], [9] and further studied in [10]. In [8], [9], downstream supply is not considered and therefore the flow exiting a link is equal to the link's demand. Thus downstream congestion does not affect upstream flow, an unrealistic assumption for traffic modeling. In [10], the authors allow flow to depend on the density of downstream links, but the paper focuses on throughput optimality of a particular class of routing policies that does not accomodate most models of traffic flow, including the proportional-priority, first-in-first-out rule considered in this paper, and limited theoretical results are given for general routing policies.

In this paper, a method to synthesize controllers using finite time convergence control barrier functions guided by linear temporal logic specifications for continuous time multi-agent dynamical systems is proposed. Finite time convergence to a desired set in the state space is guaranteed under the existence of a suitable finite time convergence control barrier function. In addition, these barrier functions also guarantee forward invariance once the system converges to the desired set. This allows us to formulate a theoretical framework which synthesizes controllers for the multi-agent system. These properties also enable us to solve the reachability problem in continuous time by formulating a theorem on the composition of multiple finite time convergence control barrier functions. This approach is more flexible than existing methods and also allows for a greater set of feasible control laws. Linear temporal logic is used to specify complex task specifications that need to be satisfied by the multi-agent system. With this solution methodology, a control law is synthesized that satisfies the given temporal logic task specification. Robotic experiments are provided which were performed on the Robotarium multi-robot testbed at Georgia Tech.

We consider global stability of a flow network model for vehicular traffic. Standard approaches which rely on monotonicity of flow networks for stability analysis do not immediately apply to traffic networks with diverging junctions. In this paper, we show that the network model nonetheless exhibits a mixed monotonicity property. Mixed monotonicity allows us to prove global asymptotic stability by embedding the system in a larger system that is monotone.