arXiv subjects

Samuel Coogan

Publications and source records attributed to Samuel Coogan.

At least 73 records · Page 4Linked to original sources

arXiv2021

Improving the Fidelity of Mixed-Monotone Reachable Set Approximations via State Transformations

Mixed-monotone systems are separable via a decomposition function into increasing and decreasing components, and this decomposition function allows for embedding the system dynamics in a higher-order monotone embedding system. Embedding the system dynamics in this way facilitates the efficient over-approximation of reachable sets with hyperrectangles, however, unlike the monotonicity property, which can be applied to compute, e.g., the tightest hyperrectangle containing a reachable set, the application of the mixed-monotonicity property generally results in conservative reachable set approximations. In this work, explore conservatism in the method and we consider, in particular, embedding systems that are monotone with respect to an alternative partial order. This alternate embedding system is constructed with a decomposition function for a related system, formed via a linear transformation of the initial state-space. We show how these alternate embedding systems allow for computing reachable sets with improved fidelity, i.e., reduced conservatism.

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arXiv2021

Safety With Limited Range Sensing Constraints For Fixed Wing Aircraft

In this paper we discuss how to use a barrier function that is subject to kinematic constraints and limited sensing in order to guarantee that fixed wing unmanned aerial vehicles (UAVs) will maintain safe distances from each other at all times despite being subject to limited range sensing constraints. Prior work has shown that a barrier function can be used to guarantee safe system operation when the state can be sensed at all times. However, in this paper we show that this construction does not guarantee safety when the UAVs are subject to limited range sensing. To resolve this issue, we introduce a method for constructing a new barrier function that accommodates limited sensing range from a previously existing barrier function that may not necessarily accommodate limited range sensing. We show that, under appropriate conditions, the newly constructed barrier function ensures system safety even in the presence of limited range sensing. We demonstrate the contribution of this paper in a scenario of 20 fixed wing aircraft, where because of the proposed algorithm, the vehicles are able to maintain safe distances from each other even though the vehicles are subject to limited range sensing.

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arXiv2020

A Numerical Method to Compute Stability Margins of Switching Linear Systems

Stability margins for linear time-varying (LTV) and switched-linear systems are traditionally computed via quadratic Lyapunov functions, and these functions certify the stability of the system under study. In this work, we show how the more general class of homogeneous polynomial Lyapunov functions is used to compute stability margins with reduced conservatism, and we show how these Lyapunov functions aid in the search for periodic trajectories for marginally stable LTV systems. Our work is premised on the recent observation that the search for a homogeneous polynomial Lyapunov function for some LTV systems is easily encoded as the search for a quadratic Lyapunov function for a related LTV system, and our main contribution is an intuitive algorithm for generating upper and lower bounds on the system's stability margin. We show also how the worst-case switching scheme - which draws an LTV system closest to a periodic orbit - is generated. Three numerical examples are provided to aid the reader and demonstrate the theoretical contributions of the work.

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arXiv2020

Resource Aware Pricing for Electric Vehicle Charging

Electric vehicle charging facilities offer electric charge and parking to users for a fee. Both parking availability and electric charge capacity are constrained resources, and as the demand for charging facilities grows with increasing electric vehicle adoption, so too does the potential for exceeding these resource limitations. In this paper, we study how prices set by the charging facility impact the likelihood that resource constraints are exceeded. Specifically, we present probabilistic bounds on the number of charging spots and the total power supply needed at a facility based on the characteristics of the arriving vehicles. We assume the charging facility either offers a set of distinct and fixed charging rates to each user or allows the user to decide a charging deadline, from which a charging rate is determined. Users arrive randomly, requiring a random amount of charge. Additionally, each user has a random impatience factor that quantifies their value of time, and a random desired time to stay at a particular location. Assuming rational user behavior, and with knowledge of the probability distribution of the random parameters, we present high-confidence bounds on the total number of vehicles parked at the station and the aggregate power use of all vehicles actively charging. We demonstrate how these bounds can be used by a charging facility to determine appropriate prices and investigate through a Monte-Carlo simulation case study the tightness of the bounds.

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arXiv2020

Abstraction-based Synthesis for Stochastic Systems with Omega-Regular Objectives

This paper studies the synthesis of controllers for discrete-time, continuous state stochastic systems subject to omega-regular specifications using finite-state abstractions. We present a synthesis algorithm for minimizing or maximizing the probability that a discrete-time stochastic system with finite number of modes satisfies an omega-regular property. Our approach uses a finite-state abstraction of the underlying dynamics in the form of a Bounded-parameter Markov Decision Process (BMDP) arising from a finite partition of the system's domain. Such abstractions allow for a range of transition probabilities between states for each action. Our method analyzes the product between the abstraction and a Deterministic Rabin Automaton encoding the specification. Synthesis is decomposed into a qualitative problem, where the greatest permanent winning components of the product are created, and a quantitative problem, which requires maximizing the probability of reaching this component. We propose a metric for the quality of the controller with respect to the abstracted states and devise a domain partition refinement technique to reach a quality target. Next, we present a method for computing controllers for stochastic systems with a continuous input set. The system is assumed to be affine in input and disturbance, and we derive a technique for solving the qualitative and quantitative problems in the abstractions of such systems called Controlled Interval-valued Markov Chains. The greatest permanent component of such abstractions are found by partitioning the input space to generate a BMDP accounting for all possible qualitative transitions between states. Maximizing the probability of reaching this component is cast as an optimization problem. Quality of the synthesized controller and a refinement scheme are described for this framework.

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arXiv2020

Performance Analysis and Non-Quadratic Lyapunov Functions for Linear Time-Varying Systems

Performance analysis for linear time-invariant (LTI) systems has been closely tied to quadratic Lyapunov functions ever since it was shown that LTI system stability is equivalent to the existence of such a Lyapunov function. Some metrics for LTI systems, however, have resisted treatment via means of quadratic Lyapunov functions. Among these, point-wise-in-time metrics, such as peak norms, are not captured accurately using these techniques, and this shortcoming has prevented the development of tools to analyze system behavior by means other than e.g. time-domain simulations. This work demonstrates how the more general class of homogeneous polynomial Lyapunov functions can be used to approximate point-wise-in-time behavior for LTI systems with greater accuracy, and we extend this to the case of linear time-varying (LTV) systems as well. Our findings rely on the recent observation that the search for homogeneous polynomial Lyapunov functions for LTV systems can be recast as a search for quadratic Lyapunov functions for a related hierarchy of time-varying Lyapunov differential equations; thus, performance guarantees for LTV systems are attainable without heavy computation. Numerous examples are provided to demonstrate the findings of this work.

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arXiv2020

Computing Robustly Forward Invariant Sets for Mixed-Monotone Systems

This work presents new tools for studying reachability and set invariance for continuous-time mixed-monotone dynamical systems subject to a disturbance input. The vector field of a mixed-monotone system is decomposable via a decomposition function into increasing and decreasing components, and this decomposition enables embedding the original dynamics in a higher-dimensional embedding system. While the original system is subject to an unknown disturbance input, the embedding system has no disturbances and its trajectories provide bounds for finite-time reachable sets of the original dynamics. Our main contribution is to show how one can efficiently identify robustly forward invariant and attractive sets for mixed-monotone systems by studying certain equilibria of this embedding system. We show also how this approach, when applied to the backward-time dynamics, establishes different robustly forward invariant sets for the original dynamics. Lastly, we present an independent result for computing decomposition functions for systems with polynomial dynamics. These tools and results are demonstrated through several examples and a case study.

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arXiv2020

Enforcing Safety at Runtime for Systems with Disturbances

Safety for control systems is often posed as an invariance constraint; the system is said to be safe if state trajectories avoid some unsafe region of the statespace for all time. An assured controller is one that enforces safety online by filtering a desired control input at runtime, and control barrier functions (CBFs) provide an assured controller that renders a safe subset of the state-space forward invariant. Recent extensions propose CBF-based assured controllers that allow the system to leave a known safe set so long as a given backup control strategy eventually returns to the safe set, however, these methods have yet to be extended to consider systems subjected to unknown disturbance inputs. In this work, we present a problem formulation for CBF-based runtime assurance for systems with disturbances, and controllers which solve this problem must, in some way, incorporate the online computation of reachable sets. In general, computing reachable sets in the presence of disturbances is computationally costly and cannot be directly incorporated in a CBF framework. To that end, we present a particular solution to the problem, whereby reachable sets are approximated via the mixed-monotonicity property. Efficient algorithms exist for overapproximating reachable sets for mixed-monotone systems with hyperrectangles, and we show that such approximations are suitable for incorporating into a CBF-based runtime assurance framework.

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arXiv2020

Continuous Reachability Task Transition Using Control Barrier Functions

In this paper, a method to achieve smooth transitions between sequential reachability tasks for a continuous time mobile robotic system is presented. Control barrier functions provide formal guarantees of forward invariance of safe sets and finite-time reachability and are able to enforce task execution. Barrier functions used in quadratic programs result in implementation of controllers with real-time performance guarantees. Existing approaches for multi-objective task execution using control barrier functions leverage discretely switched, sequential quadratic programs to achieve successive tasks. However, discrete switching can lead to control input discontinuities which can affect a robot's performance. Hence, we propose a method which ensures continuous transitions between sequential quadratic programs. In particular, a time varying component to the barrier function constraint is introduced which allows for a smooth transition between objectives. Robotic implementation results are also provided.

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arXiv2020

Characterizing Safety: Minimal Barrier Functions from Scalar Comparison Systems

Verifying set invariance has classical solutions stemming from the seminal work by Nagumo, and defining sets via a smooth barrier function constraint inequality results in computable flow conditions for guaranteeing set invariance. While a majority of these historic results on set invariance consider flow conditions on the boundary, recent results on control barrier functions extended these conditions to the entire set, although they required regularity conditions on the barrier function. This paper fully characterizes set invariance through \emph{minimal barrier functions} by directly appealing to a comparison result to define a flow condition over the entire domain of the system. A considerable benefit of this approach is the removal of regularity assumptions of the barrier function. This paper also outlines necessary and sufficient conditions for a valid differential inequality condition, giving the minimum conditions for this type of approach. We also show when minimal barrier functions are necessary and sufficient for set invariance.

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arXiv2020

Tight Decomposition Functions for Continuous-Time Mixed-Monotone Systems with Disturbances

The vector field of a mixed-monotone system is decomposable via a decomposition function into increasing (cooperative) and decreasing (competitive) components, and this decomposition allows for, e.g., efficient computation of reachable sets and forward invariant sets. A main challenge in this approach, however, is identifying an appropriate decomposition function. In this work, we show that any continuous-time dynamical system with a Lipschitz continuous vector field is mixed-monotone, and we provide a construction for the decomposition function that yields the tightest approximation of reachable sets when used with the standard tools for mixed-monotone systems. Our construction is similar to that recently proposed by Yang and Ozay for computing decomposition functions of discrete-time systems [1] where we make appropriate modifications for the continuous-time setting and also extend to the case with unknown disturbance inputs. As in [1], our decomposition function construction requires solving an optimization problem for each point in the state-space; however, we demonstrate through example how tight decomposition functions can sometimes be calculated in closed form. As a second contribution, we show how under-approximations of reachable sets can be efficiently computed via the mixed-monotonicity property by considering the backward-time dynamics.

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arXiv2020

Control of Mobile Robots Using Barrier Functions Under Temporal Logic Specifications

In this paper, we propose a framework for the control of mobile robots subject to temporal logic specifications using barrier functions. Complex task specifications can be conveniently encoded using linear temporal logic. In particular, we consider a fragment of linear temporal logic which encompasses a large class of motion planning specifications for a robotic system. Control barrier functions have recently emerged as a convenient tool to guarantee reachability and safety for a system. In addition, they can be encoded as affine constraints in a quadratic program. In this paper, a fully automatic framework which translates a user defined specification in temporal logic to a sequence of barrier function based quadratic programs is presented. In addition, with the aim of alleviating infeasibility scenarios, we propose methods for composition of barrier functions as well as a prioritization based control method to guarantee feasibility of the controller. We prove that the resulting system trajectory synthesized by the proposed controller satisfies the given specification. Robotic simulation and experimental results are provided in addition to the theoretical framework.

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arXiv2020

Synthesis of Control Barrier Functions Using a Supervised Machine Learning Approach

Control barrier functions are mathematical constructs used to guarantee safety for robotic systems. When integrated as constraints in a quadratic programming optimization problem, instantaneous control synthesis with real-time performance demands can be achieved for robotics applications. Prevailing use has assumed full knowledge of the safety barrier functions, however there are cases where the safe regions must be estimated online from sensor measurements. In these cases, the corresponding barrier function must be synthesized online. This paper describes a learning framework for estimating control barrier functions from sensor data. Doing so affords system operation in unknown state space regions without compromising safety. Here, a support vector machine classifier provides the barrier function specification as determined by sets of safe and unsafe states obtained from sensor measurements. Theoretical safety guarantees are provided. Experimental ROS-based simulation results for an omnidirectional robot equipped with LiDAR demonstrate safe operation.

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arXiv2020

A Sequential Composition Framework for Coordinating Multi-Robot Behaviors

A number of coordinated behaviors have been proposed for achieving specific tasks for multi-robot systems. However, since most applications require more than one such behavior, one needs to be able to compose together sequences of behaviors while respecting local information flow constraints. Specifically, when the inter-agent communication depends on inter-robot distances, these constraints translate into particular configurations that must be reached in finite time in order for the system to be able to transition between the behaviors. To this end, we develop a distributed framework based on finite-time convergence control barrier functions that enables a team of robots to adjust its configuration in order to meet the communication requirements for the different tasks. In order to demonstrate the significance of the proposed framework, we implemented a full-scale scenario where a team of eight planar robots explore an urban environment in order to localize and rescue a subject.

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arXiv2020

Lyapunov Differential Equation Hierarchy and Polynomial Lyapunov Functions for Switched Linear Systems

This work studies the problem of searching for homogeneous polynomial Lyapunov functions for stable switched linear systems. Specifically, we show an equivalence between polynomial Lyapunov functions for systems of this class and quadratic Lyapunov functions for a related hierarchy of Lyapunov differential equations. This creates an intuitive procedure for checking the stability properties of switched linear systems and a computationally competitive algorithm is presented for generating high-order homogeneous polynomial Lyapunov functions in this manner. Additionally, we provide a comparison between polynomial Lyapunov functions generated with our proposed approach and polynomial Lyapunov functions generated with a more traditional sum-of-squares based approach.

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arXiv2020

Mixed Autonomy in Ride-Sharing Networks

We consider ride-sharing networks served by human-driven vehicles (HVs) and autonomous vehicles (AVs). We propose a model for ride-sharing in this mixed autonomy setting for a multi-location network in which a ride-sharing platform sets prices for riders, compensations for drivers of HVs, and operates AVs for a fixed price with the goal of maximizing profits. When there are more vehicles than riders at a location, we consider three vehicle-to-rider assignment possibilities: rides are assigned to HVs first; rides are assigned to AVs first; rides are assigned in proportion to the number of available HVs and AVs. Next, for each of these priority possibilities, we establish a nonconvex optimization problem characterizing the optimal profits for a network operating at a steady-state equilibrium. We then provide a convex problem which we show to have the same optimal profits, allowing for efficient computation of equilibria, and we show that all three priority possibilities result in the same maximum profits for the platform. Next, we show that, in some cases, there is a regime for which the platform will choose to mix HVs and AVs in order to maximize its profit, while in other cases, the platform will use only HVs or only AVs, depending on the relative cost of AVs. For a specific class of networks, we fully characterize these thresholds analytically and demonstrate our results on an example.

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arXiv2020

Specification-Guided Verification and Abstraction Refinement of Mixed Monotone Stochastic Systems

This paper addresses the problem of verifying discrete-time stochastic systems against omega-regular specifications using finite-state abstractions. Omega-regular properties allow specifying complex behavior and encompass, for example, linear temporal logic. We focus on a class of systems with mixed monotone dynamics. This class has recently been show to be amenable to efficient reachable set computation and models a wide-range of physically relevant systems. In general, finite-state abstractions of continuous state stochastic systems give rise to augmented Markov Chains wherein the probabilities of transition between states are restricted to an interval. We present a procedure to compute a finite-state Interval-valued Markov Chain abstraction of discrete-time, mixed monotone stochastic systems subject to affine disturbances given a rectangular partition of the state-space. Then, we suggest an algorithm for performing verification against omega-regular properties in IMCs. Specifically, we aim to compute bounds on the probability of satisfying the specification of interest from any initial state in the IMC. This is achieved by solving a reachability problem on sets of so-called winning and losing components in the Cartesian product between the IMC and a Rabin automaton representing the specification. Next, the verification of IMCs may yield a set of states whose acceptance status is undecided with respect to the specification, requiring a refinement of the abstraction. We describe a specification-guided approach that compares the best-case and worst-case behaviors of accepting paths in the IMC and targets the appropriate states accordingly. Finally, we show a case study.

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arXiv2020

Extent-Compatible Control Barrier Functions

Safety requirements in dynamical systems are commonly enforced with set invariance constraints over a safe region of the state space. Control barrier functions, which are Lyapunov-like functions for guaranteeing set invariance, are an effective tool to enforce such constraints and guarantee safety when the system is represented as a point in the state space. In this paper, we introduce extent-compatible control barrier functions as a tool to enforce safety for the system including its volume (extent) in the physical world. In order to implement the extent-compatible control barrier functions framework, a sum-of-squares based optimization program is proposed. Since sum-of-squares programs can be computationally prohibitive, we additionally introduce a sampling based method in order to retain the computational advantage of a traditional barrier function based quadratic program controller. We prove that the proposed sampling based controller retains the guarantee for safety. Simulation and robotic implementation results are also provided.

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