arXiv subjects

Samuel Coogan

Publications and source records attributed to Samuel Coogan.

At least 91 records · Page 5Linked to original sources

arXiv2019

Optimal Tolling for Heterogeneous Traffic Networks with Mixed Autonomy

When people pick routes to minimize their travel time, the total experienced delay, or social cost, may be significantly greater than if people followed routes assigned to them by a social planner. This effect is accentuated when human drivers share roads with autonomous vehicles. When routed optimally, autonomous vehicles can make traffic networks more efficient, but when acting selfishly, the introduction of autonomous vehicles can actually worsen congestion. We seek to mitigate this effect by influencing routing choices via tolling. We consider a network of parallel roads with affine latency functions that are heterogeneous, meaning that the increase in capacity due to to the presence of autonomous vehicles may vary from road to road. We show that if human drivers and autonomous users have the same tolls, the social cost may be arbitrarily worse than when optimally routed. We then prove qualities of the optimal routing and use them to design tolls that are guaranteed to minimize social cost at equilibrium. To the best of our knowledge, this is the first tolling scheme that yields a unique socially optimal equilibrium for parallel heterogeneous network with affine latency functions.

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arXiv2019

A Barrier Function Approach to Finite-Time Stochastic System Verification and Control

This paper studies the problem of enforcing safety of a stochastic dynamical system over a finite-time horizon. We use stochastic control barrier functions as a means to quantify the probability that a system exits a given safe region of the state space in finite time. A barrier certificate condition that bounds the expected value of the barrier function over the time horizon is recast as a sum-of-squares optimization problem for efficient numerical computation. Unlike prior works, the proposed certificate condition includes a state-dependent upper bound on the evolution of the expectation. We present formulations for both continuous-time and discrete-time systems. Moreover, for systems for which the drift dynamics are affine-in-control, we propose a method for synthesizing polynomial state feedback controllers that achieve a specified probability of safety. Several case studies are presented which benchmark and illustrate the performance of our verification and control method in the continuous-time and discrete-time domains.

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arXiv2019

Monitor-Based Runtime Assurance for Temporal Logic Specifications

This paper introduces the safety controller architecture as a runtime assurance mechanism for system specifications expressed as safety properties in Linear Temporal Logic (LTL). The safety controller has three fundamental components: a performance controller, a backup controller, and an assurance mechanism. The assurance mechanism uses a monitor, constructed as a finite state machine (FSM), to analyze a suggested performance control input and search for system trajectories that are bad prefixes of the system specification. A fault flag from the assurance mechanism denotes a potentially dangerous future system state and triggers a sequence of inputs that is guaranteed to keep the system safe for all time. A case study is presented which details the construction and implementation of a safety controller on a non-deterministic cyber-physical system.

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arXiv2019

Verification and Control for Finite-Time Safety of Stochastic Systems via Barrier Functions

This paper studies the problem of enforcing safety of a stochastic dynamical system over a finite time horizon. We use stochastic barrier functions as a means to quantify the probability that a system exits a given safe region of the state space in finite time. A barrier certificate condition that bounds the infinitesimal generator of the system, and hence bounds the expected value of the barrier function over the time horizon, is recast as a sum-of-squares optimization problem for efficient numerical computation. Unlike prior works, the proposed certificate condition includes a state-dependent bound on the infinitesimal generator, allowing for tighter probability bounds. Moreover, for stochastic systems for which the drift dynamics are affine-in-control, we propose a method for synthesizing polynomial state feedback controllers that achieve a specified probability of safety. Two case studies are presented that benchmark and illustrate the performance of our method.

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arXiv2019

Control Barrier Functions: Theory and Applications

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.

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arXiv2019

Ride-Sharing Networks with Mixed Autonomy

We consider ride-sharing networks served byhuman-driven vehicles and autonomous vehicles. First, wepropose a novel model for ride-sharing in this mixed autonomysetting for a multi-location network in which the platformsets prices for riders, compensation for drivers, and operatesautonomous vehicles for a fixed price. Then we study thepossible benefits, in the form of increased profits, to the ride-sharing platform that are possible by introducing autonomousvehicles. We first establish a nonconvex optimization problemcharacterizing the optimal profits for a network operatingat a steady-state equilibrium and then propose a convexproblem with the same optimal profits that allows for efficientcomputation. Next, we study the relative mix of autonomous andhuman-driven vehicles that results at equilibrium for variouscosts of operation for autonomous vehicles. In particular, weshow that there is a regime for which the platform will chooseto mix autonomous and human-driven vehicles in order tooptimize profits. Our results provide insights into how suchride-sharing platforms might choose to integrate autonomousvehicles into their fleet.

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arXiv2019

Koopman Operator Applications in Signalized Traffic Systems

This paper proposes Koopman operator theory and the related algorithm dynamical mode decomposition (DMD) for analysis and control of signalized traffic flow networks. DMD provides a model-free approach for representing complex oscillatory dynamics from measured data, and we study its application to several problems in signalized traffic. We first study a single signalized intersection, and we propose applying this method to infer traffic signal control parameters such as phase timing directly from traffic flow data. Next, we propose using the oscillatory modes of the Koopman operator, approximated with DMD, for early identification of unstable queue growth that has the potential to cause cascading congestion. Then we demonstrate how DMD can be coupled with knowledge of the traffic signal control status to determine traffic signal control parameters that are able to reduce queue lengths. Lastly, we demonstrate that DMD allows for determining the structure and the strength of interactions in a network of signalized intersections. All examples are demonstrated using a case study network instrumented with high resolution traffic flow sensors.

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arXiv2018

Satisfiability Bounds for ω-regular Properties in Interval-valued Markov Chains

We derive an algorithm to compute satisfiability bounds for arbitrary ω-regular properties in an Interval-valued Markov Chain (IMC) interpreted in the adversarial sense. IMCs generalize regular Markov Chains by assigning a range of possible values to the transition probabilities between states. In particular, we expand the automata-based theory of ω-regular property verification in Markov Chains to apply it to IMCs. Any ω-regular property can be represented by a Deterministic Rabin Automata (DRA) with acceptance conditions expressed by Rabin pairs. Previous works on Markov Chains have shown that computing the probability of satisfying a given ω-regular property reduces to a reachability problem in the product between the Markov Chain and the corresponding DRA. We similarly define the notion of a product between an IMC and a DRA. Then, we show that in a product IMC, there exists a particular assignment of the transition values that generates a largest set of non-accepting states. Subsequently, we prove that a lower bound is found by solving a reachability problem in that refined version of the original product IMC. We derive a similar approach for computing a satisfiability upper bound in a product IMC with one Rabin pair. For product IMCs with more than one Rabin pair, we establish that computing a satisfiability upper bound is equivalent to lower-bounding the satisfiability of the complement of the original property. A search algorithm for finding the largest accepting and non-accepting sets of states in a product IMC is proposed. Finally, we demonstrate our findings in a case study.

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arXiv2018

Control of Multi-Agent Systems with Finite Time Control Barrier Certificates and Temporal Logic

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.

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arXiv2018

Sampled-data reachability analysis using sensitivity and mixed-monotonicity

This paper over-approximates the reachable sets of a continuous-time uncertain system using the sensitivity of its trajectories with respect to initial conditions and uncertain parameters. We first prove the equivalence between an existing over-approximation result based on the sign-stability of the sensitivity matrices and a discrete-time approach relying on a mixed-monotonicity property. We then present a new over-approximation result which scales at worst linearly with the state dimension and is applicable to any continuous-time system with bounded sensitivity. Finally, we provide a simulation-based approach to estimate these bounds through sampling and falsification. The results are illustrated with numerical examples on traffic networks and satellite orbits.

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arXiv2018

A Benchmark Problem in Transportation Networks

In this note, we propose a case study of freeway traffic flow modeled as a hybrid system. We describe two general classes of networks that model flow along a freeway with merging onramps. The admission rate of traffic flow from each onramp is metered via a control input. Both classes of networks are easily scaled to accommodate arbitrary state dimension. The model is discrete-time and possesses piecewise-affine dynamics. Moreover, we present several control objectives that are especially relevant for traffic flow management. The proposed model is flexible and extensible and offers a benchmark for evaluating tools and techniques developed for hybrid systems.

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arXiv2017

Contraction-based Observers using non-Euclidean Norms with an Application to Traffic Networks

In this note, we study Luenberger-type full-state observers for nonlinear systems using contraction theory. We show that if the matrix measure of a suitably defined Jacobian matrix constructed from the dynamics of the system-observer interconnection is uniformly negative, then the state estimate converges exponentially to the actual state. This sufficient condition for convergence establishes that the distance between the estimate and state is infinitesimally contracting with respect to some norm on the state-space. In contrast to existing results for contraction-based observer design, we allow for contraction with respect to non-Euclidean norms. Such norms have proven useful in applications. To demonstrate our results, we study the problem of observing vehicular traffic density along a freeway modeled as interconnected, spatially homogenous compartments, and our approach relies on establishing contraction of the system-observer interconnection with respect to the one-norm.

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arXiv2017

A Contractive Approach to Separable Lyapunov Functions for Monotone Systems

Monotone systems preserve a partial ordering of states along system trajectories and are often amenable to separable Lyapunov functions that are either the sum or the maximum of a collection of functions of a scalar argument. In this paper, we consider constructing separable Lyapunov functions for monotone systems that are also contractive, that is, the distance between any pair of trajectories exponentially decreases. The distance is defined in terms of a possibly state-dependent norm. When this norm is a weighted one-norm, we obtain conditions which lead to sum-separable Lyapunov functions, and when this norm is a weighted infinity-norm, symmetric conditions lead to max-separable Lyapunov functions. In addition, we consider two classes of Lyapunov functions: the first class is separable along the system's state, and the second class is separable along components of the system's vector field. The latter case is advantageous for many practically motivated systems for which it is difficult to measure the system's state but easier to measure the system's velocity or rate of change. In addition, we present an algorithm based on sum-of-squares programming to compute such separable Lyapunov functions. We provide several examples to demonstrate our results.

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arXiv2017

The Price of Anarchy for Transportation Networks with Mixed Autonomy

We study routing behavior in transportation networks with mixed autonomy, that is, networks in which a fraction of the vehicles on each road are equipped with autonomous capabilities such as adaptive cruise control that enable reduced headways and increased road capacity. Motivated by capacity models developed for such roads with mixed autonomy, we consider transportation networks in which the delay on each road or link is an affine function of two quantities: the number of vehicles with autonomous capabilities on the link and the number of regular vehicles on the link. We particularly study the price of anarchy for such networks, that is, the ratio of the total delay experienced by selfish routing to the socially optimal routing policy. Unlike the case when all vehicles are of the same type, for which the price of anarchy is known to be bounded, we first show that the price of anarchy can be arbitrarily large for such mixed autonomous networks. Next, we define a notion of asymmetry equal to the maximum possible travel time improvement due to the presence of autonomous vehicles. We show that when the degree of asymmetry of all links in the network is bounded by a factor less than 4, the price of anarchy is bounded. We also bound the bicriteria, which is a bound on the cost of selfishly routing traffic compared to the cost of optimally routing additional traffic. These bounds depend on the degree of asymmetry and recover classical bounds on the price of anarchy and bicriteria in the case when no asymmetry exists. Further, we show with examples that these bound are tight in particular cases.

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arXiv2017

Approximating the Frequency Response of Contractive Systems

We consider contractive systems whose trajectories evolve on a compact and convex state-space. It is well-known that if the time-varying vector field of the system is periodic then the system admits a unique globally asymptotically stable periodic solution. Obtaining explicit information on this periodic solution and its dependence on various parameters is important both theoretically and in numerous applications. We develop an approach for approximating such a periodic trajectory using the periodic trajectory of a simpler system (e.g. an LTI system). Our approximation includes an error bound that is based on the input-to-state stability property of contractive systems. We show that in some cases this error bound can be computed explicitly. We also use the bound to derive a new theoretical result, namely, that a contractive system with an additive periodic input behaves like a low pass filter. We demonstrate our results using several examples from systems biology.

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arXiv2016

Traffic Predictive Control from Low-Rank Structure

The operation of most signalized intersections is governed by predefined timing plans that are applied during specified times of the day. These plans are designed to accommodate average conditions and are unable to respond to large deviations in traffic flow. We propose a control approach that adjusts time-of-day signaling plans based on a prediction of future traffic flow. The prediction algorithm identifies correlated, low rank structure in historical measurement data and predicts future traffic flow from real-time measurements by determining which structural trends are prominent in the measurements. From this prediction, the controller then determines the optimal time of day to apply new timing plans. We demonstrate the potential benefits of this approach using eight months of high resolution data collected at an intersection in Beaufort, South Carolina.

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arXiv2016

Mixed Monotonicity of Partial First-In-First-Out Traffic Flow Models

In vehicle traffic networks, congestion on one outgoing link of a diverging junction often impedes flow to other outgoing links, a phenomenon known as the first-in-first-out (FIFO) property. Simplified traffic models that do not account for the FIFO property result in monotone dynamics for which powerful analysis techniques exist. FIFO models are in general not monotone, but have been shown to be mixed monotone - a generalization of monotonicity that enables similarly powerful analysis techniques. In this paper, we study traffic flow models for which the FIFO property is only partial, that is, flows at diverging junctions exhibit a combination of FIFO and non-FIFO phenomena. We show that mixed monotonicity extends to this wider class of models and establish conditions that guarantee convergence to an equilibrium.

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arXiv2016

Separability of Lyapunov Functions for Contractive Monotone Systems

We consider constructing Lyapunov functions for systems that are both monotone and contractive with respect to a weighted one norm or infinity norm. This class of systems admits separable Lyapunov functions that are either the sum or the maximum of a collection of functions of a single argument. In either case, two classes of separable Lyapunov functions exist: the first class is separable along the system's state, and the second class is separable along components of the system's vector field. The latter case is advantageous for many practically motivated systems for which it is difficult to measure the system's state but easier to measure the system's velocity or rate of change. We provide several examples to demonstrate our results.

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