arXiv subjects

Abraham P. Vinod

Publications and source records attributed to Abraham P. Vinod.

27 records · Page 2Linked to original sources

arXiv2020

Blending Controllers via Multi-Objective Bandits

Safety and performance are often two competing objectives in sequential decision-making problems. Existing performant controllers, such as controllers derived from reinforcement learning algorithms, often fall short of safety guarantees. On the contrary, controllers that guarantee safety, such as those derived from classical control theory, require restrictive assumptions and are often conservative in performance. Our goal is to blend a performant and a safe controller to generate a single controller that is safer than the performant and accumulates higher rewards than the safe controller. To this end, we propose a blending algorithm using the framework of contextual multi-armed multi-objective bandits. At each stage, the algorithm observes the environment's current context alongside an immediate reward and cost, which is the underlying safety measure. The algorithm then decides which controller to employ based on its observations. We demonstrate that the algorithm achieves sublinear Pareto regret, a performance measure that models coherence with an expert that always avoids picking the controller with both inferior safety and performance. We derive an upper bound on the loss in individual objectives, which imposes no additional computational complexity. We empirically demonstrate the algorithm's success in blending a safe and a performant controller in a safety-focused testbed, the Safety Gym environment. A statistical analysis of the blended controller's total reward and cost reflects two key takeaways: The blended controller shows a strict improvement in performance compared to the safe controller, and it is safer than the performant controller.

eess.SY↗

arXiv2020

Trust-based user-interface design for human-automation systems

We present a method for dynamics-driven, user-interface design for a human-automation system via sensor selection. We define the user-interface to be the output of a MIMO LTI system, and formulate the design problem as one of selecting an output matrix from a given set of candidate output matrices. Sufficient conditions for situation awareness are captured as additional constraints on the selection of the output matrix. These constraints depend upon the level of trust the human has in the automation. We show that the resulting user-interface design problem is a combinatorial, set-cardinality minimization problem with set function constraints. We propose tractable algorithms to compute optimal or sub-optimal solutions with suboptimality bounds. Our approaches exploit monotonicity and submodularity present in the design problem, and rely on constraint programming and submodular maximization. We apply this method to the IEEE 118-bus, to construct correct-by-design interfaces under various operating scenarios.

eess.SY↗

arXiv2018

Voronoi Partition-based Scenario Reduction for Fast Sampling-based Stochastic Reachability Computation of LTI Systems

In this paper, we address the stochastic reach-avoid problem for linear systems with additive stochastic uncertainty. We seek to compute the maximum probability that the states remain in a safe set over a finite time horizon and reach a target set at the final time. We employ sampling-based methods and provide a lower bound on the number of scenarios required to guarantee that our estimate provides an underapproximation. Due to the probabilistic nature of the sampling-based methods, our underapproximation guarantee is probabilistic, and the proposed lower bound can be used to satisfy a prescribed probabilistic confidence level. To decrease the computational complexity, we propose a Voronoi partition-based to check the reach-avoid constraints at representative partitions (cells), instead of the original scenarios. The state constraints arising from the safe and target sets are tightened appropriately so that the solution provides an underapproximation for the original sampling-based method. We propose a systematic approach for selecting these representative cells and provide the flexibility to trade-off the number of cells needed for accuracy with the computational cost.

math.OC↗

arXiv2018

Lagrangian Approximations for Stochastic Reachability of a Target Tube

In this paper we examine how Lagrangian techniques can be used to compute underapproximations and overapproximation of the finite-time horizon, stochastic reach-avoid level sets for discrete-time, nonlinear systems. This approach is applicable for a generic nonlinear system without any convexity assumptions on the safe and target sets. We examine and apply our methods on the reachability of a target tube problem, a more generalized version of the finite-time horizon reach-avoid problem. Because these methods utilize a Lagrangian (set theoretic) approach, we eliminate the necessity to grid the state, input, and disturbance spaces allowing for increased scalability and faster computation. The methods scalability are currently limited by the computational requirements for performing the necessary set operations by current computational geometry tools. The primary trade-off for this improved extensibility is conservative approximations of actual stochastic reach set. We demonstrate these methods on several examples including the standard double-integrator, a chain of integrators, and a 4-dimensional space vehicle rendezvous docking problem.

eess.SY↗

arXiv2018

Probabilistic Occupancy Function and Sets Using Forward Stochastic Reachability for Rigid-Body Dynamic Obstacles

We present theory and algorithms for the computation of probability-weighted "keep-out" sets to assure probabilistically safe navigation in the presence of multiple rigid body obstacles with stochastic dynamics. Our forward stochastic reachability-based approach characterizes the stochasticity of the future obstacle states in a grid-free and recursion-free manner, using Fourier transforms and computational geometry. We consider discrete-time Markovian switched systems with affine parameter-varying stochastic subsystems (DMSP) as the obstacle dynamics, which includes Markov jump affine systems and discrete-time affine parameter-varying stochastic systems (DPV). We define a probabilistic occupancy function, to describe the probability that a given state is occupied by a rigid body obstacle with stochastic dynamics at a given time; keep-out sets are the super-level sets of this occupancy function. We provide sufficient conditions that ensure convexity and compactness of these keep-out sets for DPV obstacle dynamics. We also propose two computationally efficient algorithms to overapproximate the keep-out sets --- a tight polytopic approximation using projections, and an overapproximation using Minkowski sum. For DMSP obstacle dynamics, we compute a union of convex and compact sets that covers the potentially non-convex keep-out set. Numerical simulations show the efficacy of the proposed algorithms for a modified version of the classical unicycle dynamics, modeled as a DMSP.

eess.SY↗

arXiv2017

Scalable Underapproximation for the Stochastic Reach-Avoid Problem for High-Dimensional LTI Systems using Fourier Transforms

We present a scalable underapproximation of the terminal hitting time stochastic reach-avoid probability at a given initial condition, for verification of high-dimensional stochastic LTI systems. While several approximation techniques have been proposed to alleviate the curse of dimensionality associated with dynamic programming, these techniques are limited and cannot handle larger, more realistic systems. We present a scalable method that uses Fourier transforms to compute an underapproximation of the reach-avoid probability for systems with disturbances with arbitrary probability densities. We characterize sufficient conditions for Borel-measurability of the value functions. We exploit fixed control sequences parameterized by the initial condition (an open-loop control policy) to generate the underapproximation. For Gaussian disturbances, the underapproximation can be obtained using existing efficient algorithms by solving a convex optimization problem. Our approach produces non-trivial lower bounds and is demonstrated on a chain of integrators with 40 states.

eess.SY↗

arXiv2017

Underapproximation of Reach-Avoid Sets for Discrete-Time Stochastic Systems via Lagrangian Methods

We examine Lagrangian techniques for computing underapproximations of finite-time horizon, stochastic reach-avoid level-sets for discrete-time, nonlinear systems. We use the concept of reachability of a target tube in the control literature to define robust reach-avoid sets which are parameterized by the target set, safe set, and the set in which the disturbance is drawn from. We unify two existing Lagrangian approaches to compute these sets and establish that there exists an optimal control policy of the robust reach-avoid sets which is a Markov policy. Based on these results, we characterize the subset of the disturbance space whose corresponding robust reach-avoid set for the given target and safe set is a guaranteed underapproximation of the stochastic reach-avoid level-set of interest. The proposed approach dramatically improves the computational efficiency for obtaining an underapproximation of stochastic reach-avoid level-sets when compared to the traditional approaches based on gridding. Our method, while conservative, does not rely on a grid, implying scalability as permitted by the known computational geometry constraints. We demonstrate the method on two examples: a simple two-dimensional integrator, and a space vehicle rendezvous-docking problem.

eess.SY↗

arXiv2017

Forward Stochastic Reachability Analysis for Uncontrolled Linear Systems using Fourier Transforms

We propose a scalable method for forward stochastic reachability analysis for uncontrolled linear systems with affine disturbance. Our method uses Fourier transforms to efficiently compute the forward stochastic reach probability measure (density) and the forward stochastic reach set. This method is applicable to systems with bounded or unbounded disturbance sets. We also examine the convexity properties of the forward stochastic reach set and its probability density. Motivated by the problem of a robot attempting to capture a stochastically moving, non-adversarial target, we demonstrate our method on two simple examples. Where traditional approaches provide approximations, our method provides exact analytical expressions for the densities and probability of capture.

eess.SY↗

arXiv2016

Computation of forward stochastic reach sets: Application to stochastic, dynamic obstacle avoidance

We propose a method to efficiently compute the forward stochastic reach (FSR) set and its probability measure for nonlinear systems with an affine disturbance input, that is stochastic and bounded. This method is applicable to systems with an a priori known controller, or to uncontrolled systems, and often arises in problems in obstacle avoidance in mobile robotics. When used as a constraint in finite horizon controller synthesis, the FSR set, and its probability measure facilitates probabilistic collision avoidance, in contrast to methods which presume the obstacles act in a worst-case fashion and generate hard constraints that cannot be violated. We tailor our approach to accommodate rigid body constraints, and show convexity is assured so long as the rigid body shape of each obstacle is also convex. We extend methods for multi-obstacle avoidance through mixed integer linear programming (with linear robot and obstacle dynamics) to accommodate chance constraints that represent the FSR set probability measure. We demonstrate our method on a rigid-body obstacle avoidance scenario, in which a receding horizon controller is designed to avoid several stochastically moving obstacles while reaching the desired goal. Our approach can provide solutions when approaches that presume a worst-case action from the obstacle fail.

eess.SY↗