Raktim Bhattacharya

Kooktae Lee, Raktim Bhattacharya

In this paper, we study a convergence condition for asynchronous consensus problems in multi-agent systems. The convergence in this context implies the asynchronous consensus value converges to the synchronous one and is unique. Although it is reported in the literature that the consensus value under asynchronous communications may not coincide with the synchronous consensus value, it has not received much attention. In some applications, the discrepancy between them may result in serious consequences. For such applications it is critical to determine under what conditions the asynchronous consensus value is the same as the synchronous consensus value. We illustrate these issues with a few examples and then provide a condition, which guarantees that the asynchronous consensus value converges to the synchronous one. The validity of the proposed result is verified with simulations.

Raktim Bhattacharya

Designing the sensing architecture for large-scale spatio-temporal systems is hard when accuracy requirements are specified but sensor models are uncertain or unavailable. Classical design treats sensor placement and estimation sequentially, requiring valid forward models for each sensing modality. This paper inverts the design flow: given an error budget, synthesize the measurement likelihood that enforces it while injecting minimal information beyond the dynamical prior. The likelihood is constructed by constrained optimization: among all posteriors satisfying a prescribed accuracy bound relative to a target, select the one minimizing Kullback-Leibler divergence from the prior. The solution is a maximum-entropy posterior in relative-entropy form, and the induced likelihood is the Radon-Nikodym derivative. The framework accommodates arbitrary discrepancies and is instantiated for Wasserstein distance, maximum mean discrepancy, $f$-divergences, moment constraints, and hybrid metrics. For each, we derive the discrete particle-level problem, analyze its convex or convex-relaxed structure, and present solvers with complexity scaling. A closed-form solution exists for the symmetric exponential-tilt case, and a distillation procedure converts nonparametric likelihood samples into parametric forms. A two-layer sensor design architecture embeds the synthesized likelihood in the recursive predict-update loop, connecting accuracy budgets to physical sensor placement, precision, and configuration. Numerical experiments comparing four metrics on unimodal and multimodal scenarios confirm the accuracy constraints are reliably enforced and reveal how metric choice determines the amount and spatial distribution of injected information.

Raktim Bhattacharya, Felix Biertümpfel

This paper develops a Finsler-based LMI for robust $\Hinf$ observer design with integral quadratic constraints (IQCs) and block-structured uncertainty. By introducing a slack variable that relaxes the coupling between the Lyapunov matrix, the observer gain, and the IQC multiplier, the formulation addresses two limitations of the standard block-diagonal approach: the LMI requirement $\He{PA} \prec 0$ (which fails for marginally stable dynamics), and a multiplier--Lyapunov trade-off that causes infeasibility for wide uncertainty ranges. For marginally stable dynamics, artificial damping in the design model balances certified versus actual performance. The framework is demonstrated on quaternion attitude estimation with angular velocity uncertainty and mass-spring-damper state estimation with uncertain physical parameters.

Tanay Kumar, Raktim Bhattacharya

Attitude stabilization of unmanned aerial vehicles (UAVs) in uncertain environments presents significant challenges due to nonlinear dynamics, parameter variations, and sensor limitations. This paper presents a comparative study of $\mathcal{H}_\infty$ and classical PID controllers for multi-rotor attitude regulation in the presence of wind disturbances and gyroscope noise. The flight dynamics are modeled using a linear parameter-varying (LPV) framework, where nonlinearities and parameter variations are systematically represented as structured uncertainties within a linear fractional transformation formulation. A robust controller based on $\mathcal{H}_\infty$ formulation is designed using only gyroscope measurements to ensure guaranteed performance bounds. Nonlinear simulation results demonstrate the effectiveness of the robust controllers compared to classical PID control, showing significant improvement in attitude regulation under severe wind disturbances.

Rui Tuo, Haoyuan Chen, Raktim Bhattacharya

We propose a novel theoretical and methodological framework for Gaussian process regression subject to privacy constraints. The proposed method can be used when a data owner is unwilling to share a high-fidelity supervised learning model built from their data with the public due to privacy concerns. The key idea of the proposed method is to add synthetic noise to the data until the predictive variance of the Gaussian process model reaches a prespecified privacy level. The optimal covariance matrix of the synthetic noise is formulated in terms of semi-definite programming. We also introduce the formulation of privacy-aware solutions under continuous privacy constraints using kernel-based approaches, and study their theoretical properties. The proposed method is illustrated by considering a model that tracks the trajectories of satellites and a real application on a census dataset.

Vaishnav Tadiparthi, Raktim Bhattacharya

In this paper, we propose a refinement strategy to the well-known Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs) based on the concept of Optimal Transport (OT). Conventional black-box PINNs solvers have been found to suffer from a host of issues: spectral bias in fully-connected architectures, unstable gradient pathologies, as well as difficulties with convergence and accuracy. Current network training strategies are agnostic to dimension sizes and rely on the availability of powerful computing resources to optimize through a large number of collocation points. This is particularly challenging when studying stochastic dynamical systems with the Fokker-Planck-Kolmogorov Equation (FPKE), a second-order PDE which is typically solved in high-dimensional state space. While we focus exclusively on the stationary form of the FPKE, positivity and normalization constraints on its solution make it all the more unfavorable to solve directly using standard PINNs approaches. To mitigate the above challenges, we present a novel training strategy for solving the FPKE using OT-based sampling to supplement the existing PINNs framework. It is an iterative approach that induces a network trained on a small dataset to add samples to its training dataset from regions where it nominally makes the most error. The new samples are found by solving a linear programming problem at every iteration. The paper is complemented by an experimental evaluation of the proposed method showing its applicability on a variety of stochastic systems with nonlinear dynamics.