arXiv subjects

Arun D. Mahindrakar

Publications and source records attributed to Arun D. Mahindrakar.

6 recordsLinked to original sources

arXiv2026

Stochastic Recursive Inclusions under Biased Perturbations: An Input-to-State Stability Perspective

This paper investigates the asymptotic behavior of stochastic recursive inclusions in the presence of non-zero, non-diminishing bias, a setting that frequently arises in zeroth-order optimization, stochastic approximation with iterate-dependent noise, and distributed learning with adversarial agents. The analysis is conducted through the lens of input-to-state stability of an associated differential inclusion, which serves as the continuous-time limit of the discrete recursion. We first establish that if the limiting differential inclusion is input-to-state stable and the iterates remain almost surely bounded, then the iterates converge almost surely to the neighborhood of desired equilibrium. We then provide a verifiable sufficient condition for almost sure boundedness by assuming that the underlying operator is single-valued and globally Lipschitz. Finally, we show that several zeroth-order variants of stochastic gradient naturally fit within this framework, and we demonstrate their input-to-state stability under standard conditions. Overall, the results provide a unified theoretical foundation for studying almost sure convergence of biased stochastic approximation schemes through the Input to State stability theory of differential inclusions.

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arXiv2023

Dynamical System Approach for Time-Varying Constrained Convex Optimization Problems

Optimization problems emerging in most of the real-world applications are dynamic, where either the objective function or the constraints change continuously over time. This paper proposes projected primal-dual dynamical system approaches to track the primal and dual optimizer trajectories of an inequality constrained time-varying (TV) convex optimization problem with a strongly convex objective function. First, we present a dynamical system that asymptotically tracks the optimizer trajectory of an inequality constrained TV optimization problem. Later we modify the proposed dynamics to achieve the convergence to the optimizer trajectory within a fixed time. The asymptotic and fixed-time convergence of the proposed dynamical systems to the optimizer trajectory is shown via Lyapunov based analysis. Finally, we consider the TV extended Fermat -Torricelli problem (eFTP) of minimizing the sum-of-squared distances to a finite number of nonempty, closed and convex TV sets, to illustrate the applicability of the projected dynamical systems proposed in this paper.

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arXiv2021

Geometric PID Controller for Stabilization of Nonholonomic Mechanical Systems on Lie Groups

The PID controller is an elegant and versatile controller for set point tracking in double integrator systems of which mechanical systems evolving on Euclidean space constitute a large class. But since mechanical systems are typically constrained interconnections of rigid bodies whose configuration space is $SE(3)$, which is not even topologically Euclidean, a geometric PID controller has been developed for mechanical systems evolving on Lie groups. In this work, we extend the framework to such systems which have nonholonomic constraints. It encompasses many practically applicable mechanical systems encountered in robotics as robots are constrained interconnections of rigid bodies where the constraints could either be holonomic or nonholonomic.

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arXiv2021

Event-triggered Control for Nonlinear Systems with Center Manifolds

In this work, we consider the problem of event-triggered implementation of control laws designed for the local stabilization of nonlinear systems with center manifolds. We propose event-triggering conditions which are derived from a local input-to-state stability characterization of such systems. The triggering conditions ensure local ultimate boundedness of the trajectories and the existence of a uniform positive lower bound for the inter-event times. The ultimate bound can be made arbitrarily small, but by allowing for smaller inter-event times. Under certain assumptions on the controller structure, local asymptotic stability of the origin is also guaranteed. Two sets of triggering conditions are proposed, that cater to the cases where the exact center manifold and only an approximation of the center manifold is computable. The closed-loop system exhibits some desirable properties when the exact knowledge of the center manifold is employed in checking the triggering conditions. Three illustrative examples that explore different scenarios are presented and the applicability of the proposed methods is demonstrated. The third example concerns the event-triggered implementation of a position stabilizing controller for the open-loop unstable Mobile Inverted Pendulum (MIP) robot.

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arXiv2017

Position and line-of-sight stabilization of spherical robot using feedforward proportional-derivative geometric controller

In this paper we present a geometric control law for position and line-of-sight stabilization of the nonholonomic spherical robot actuated by three independent actuators. A simple configuration error function with an appropriately defined transport map is proposed to extract feedforward and proportional-derivative control law. Simulations are provided to validate the controller performance.

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arXiv2014

Synchronization and semistability analysis of the Kuramoto model of coupled nonlinear oscillators

An interesting problem in synchronization is the study of coupled oscillators, wherein oscillators with different natural frequencies synchronize to a common frequency and equilibrium phase difference. In this paper, we investigate the stability and convergence in a network of coupled oscillators described by the Kuramoto model. We consider networks with finite number of oscillators, arbitrary interconnection topology, non-uniform coupling gains and non-identical natural frequencies. We show that such a network synchronizes provided the underlying graph is connected and certain conditions on the coupling gains are satisfied. In the analysis, we consider as states the phase and angular frequency differences between the oscillators, and the resulting dynamics possesses a continuum of equilibria. The synchronization problem involves establishing the Lyapunov stability of the fixed points and showing convergence of trajectories to these points. The synchronization result is established in the framework of semistability theory.

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