arXiv subjects

Vignesh Sivaramakrishnan

Publications and source records attributed to Vignesh Sivaramakrishnan.

9 recordsLinked to original sources

arXiv2026

Autonomous Satellite Rendezvous via Hybrid Feedback Optimization

As satellites have proliferated, interest has increased in autonomous rendezvous, proximity operations, and docking (ARPOD). A fundamental challenge in these tasks is the uncertainties when operating in space, e.g., in measurements of satellites' states, which can make future states difficult to predict. Another challenge is that satellites' onboard processors are typically much slower than their terrestrial counterparts. Therefore, to address these challenges we propose to solve an ARPOD problem with feedback optimization, which computes inputs to a system by measuring its outputs, feeding them into an optimization algorithm in the loop, and computing some number of iterations towards an optimal input. We focus on satellite rendezvous, and satellites' dynamics are modeled using the continuous-time Clohessy-Wiltshire equations, which are marginally stable. We develop an asymptotically stabilizing controller for them, and we use discrete-time gradient descent in the loop to compute inputs to them. Then, we analyze the hybrid feedback optimization system formed by the stabilized Clohessy-Wiltshire equations with gradient descent in the loop. We show that this model is well-posed and that maximal solutions are both complete and non-Zeno. Then, we show that solutions converge exponentially fast to a ball around a rendezvous point, and we bound the radius of that ball in terms of system parameters. Simulations show that this approach provides up to a 98.4\% reduction in the magnitude of disturbances across a range of simulations, which illustrates the viability of hybrid feedback optimization for autonomous satellite rendezvous.

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arXiv2024

Stochastic Reachability of Uncontrolled Systems via Probability Measures: Approximation via Deep Neural Networks

This paper poses a theoretical characterization of the stochastic reachability problem in terms of probability measures, capturing the probability measure of the state of the system that satisfies the reachability specification for all probabilities over a finite horizon. We achieve this by constructing the level sets of the probability measure for all probability values and, since our approach is only for autonomous systems, we can determine the level sets via forward simulations of the system from a point in the state space at some time step in the finite horizon to estimate the reach probability. We devise a training procedure which exploits this forward simulation and employ it to design a deep neural network (DNN) to predict the reach probability provided the current state and time step. We validate the effectiveness of our approach through three examples.

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arXiv2024

Forward Reachability for Discrete-Time Nonlinear Stochastic Systems via Mixed-Monotonicity and Stochastic Order

We present a method to overapproximate forward stochastic reach sets of discrete-time, stochastic nonlinear systems with interval geometry. This is made possible by extending the theory of mixed-monotone systems to incorporate stochastic orders, and a concentration inequality result that lower-bounds the probability the state resides within an interval through a monotone mapping. Then, we present an algorithm to compute the overapproximations of forward reachable set and the probability the state resides within it. We present our approach on two aerospace examples to show its efficacy.

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arXiv2024

SAVER: A Toolbox for Sampling-Based, Probabilistic Verification of Neural Networks

We present a neural network verification toolbox to 1) assess the probability of satisfaction of a constraint, and 2) synthesize a set expansion factor to achieve the probability of satisfaction. Specifically, the tool box establishes with a user-specified level of confidence whether the output of the neural network for a given input distribution is likely to be contained within a given set. Should the tool determine that the given set cannot satisfy the likelihood constraint, the tool also implements an approach outlined in this paper to alter the constraint set to ensure that the user-defined satisfaction probability is achieved. The toolbox is comprised of sampling-based approaches which exploit the properties of signed distance function to define set containment.

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arXiv2022

Probabilistic Verification of ReLU Neural Networks via Characteristic Functions

Verifying the input-output relationships of a neural network so as to achieve some desired performance specification is a difficult, yet important, problem due to the growing ubiquity of neural nets in many engineering applications. We use ideas from probability theory in the frequency domain to provide probabilistic verification guarantees for ReLU neural networks. Specifically, we interpret a (deep) feedforward neural network as a discrete dynamical system over a finite horizon that shapes distributions of initial states, and use characteristic functions to propagate the distribution of the input data through the network. Using the inverse Fourier transform, we obtain the corresponding cumulative distribution function of the output set, which can be used to check if the network is performing as expected given any random point from the input set. The proposed approach does not require distributions to have well-defined moments or moment generating functions. We demonstrate our proposed approach on two examples, and compare its performance to related approaches.

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arXiv2021

Distribution Steering for Discrete-Time Linear Systems with General Disturbances using Characteristic Functions

We propose to solve a constrained distribution steering problem, i.e., steering a stochastic linear system from an initial distribution to some final, desired distribution subject to chance constraints. We do so by characterizing the cumulative distribution function in the chance constraints and by using the absolute distance between two probability density functions using the corresponding characteristic functions. We consider discrete-time, time-varying linear systems with affine feedback. We demonstrate the proposed approach on a 2D double-integrator perturbed by various disturbances and initial conditions.

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arXiv2020

Fast, Convexified Stochastic Optimal Open-Loop Control For Linear Systems Using Empirical Characteristic Functions

We consider the problem of stochastic optimal control in the presence of an unknown disturbance. We characterize the disturbance via empirical characteristic functions, and employ a chance constrained approach. By exploiting properties of characteristic functions and underapproximating cumulative distribution functions, we can reformulate a nonconvex problem by a conic, convex under-approximation. This results in extremely fast solutions that are assured to maintain probabilistic constraints. We construct algorithms for optimal open-loop control using piecewise linear approximations of the empirical characteristic function, and demonstrate our approach on two examples.

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arXiv2020

Approximate Stochastic Reachability for High Dimensional Systems

We present a method to compute the stochastic reachability safety probabilities for high-dimensional stochastic dynamical systems. Our approach takes advantage of a nonparametric learning technique known as conditional distribution embeddings to model the stochastic kernel using a data-driven approach. By embedding the dynamics and uncertainty within a reproducing kernel Hilbert space, it becomes possible to compute the safety probabilities for stochastic reachability problems as simple matrix operations and inner products. We employ a convergent approximation technique, random Fourier features, in order to alleviate the increased computational requirements for high-dimensional systems. This technique avoids the curse of dimensionality, and enables the computation of safety probabilities for high-dimensional systems without prior knowledge of the structure of the dynamics or uncertainty. We validate this approach on a double integrator system, and demonstrate its capabilities on a million-dimensional, nonlinear, non-Gaussian, repeated planar quadrotor system.

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arXiv2020

Convexified Open-Loop Stochastic Optimal Control for Linear Non-Gaussian Systems

We consider stochastic optimal control of linear dynamical systems with additive non-Gaussian disturbance. We propose a novel, sampling-free approach, based on Fourier transformations and convex optimization, to cast the stochastic optimal control problem as a difference-of-convex program. In contrast to existing moment based approaches, our approach invokes higher moments, resulting in less conservatism. We employ piecewise affine approximations and the well-known convex-concave procedure, to efficiently solve the resulting optimization problem via standard conic solvers. We demonstrate that the proposed approach is computationally faster than existing particle based and moment based approaches, without compromising probabilistic safety constraints.

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