arXiv subjects

Umesh Vaidya

Publications and source records attributed to Umesh Vaidya.

At least 19 recordsLinked to original sources

arXiv2026

A Data-Driven Koopman-Behavioral Distance for Nonlinear Dynamical Systems

Comparing nonlinear dynamical systems directly from trajectory data remains challenging because finite-horizon trajectory representations are generally coordinate dependent. Motivated by recent behavioral subspace approaches for linear systems, this paper introduces a Koopman-behavioral distance for nonlinear stochastic systems using multi-rollout trajectory data. State measurements are lifted through a common observable dictionary, and dominant lifted behavioral subspaces are compared using Grassmannian distances without explicit Koopman operator identification. We establish coordinate invariance, connection with recently proposed linear behavioral distances under exact lifted closure, and robustness to approximate closure. Numerical examples demonstrate coordinate invariance and parameter discrimination.

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arXiv2026

Spectral Koopman-Hopf Formula for Reachability with Adversary

This paper develops a spectral Koopman-Hopf framework for adversarial reachability analysis of nonlinear systems. By lifting the nonlinear drift dynamics into Koopman eigenfunction coordinates, the proposed approach transforms the original state-dependent Hamilton-Jacobi-Isaacs (HJI) equation into an approximate state-independent optimization problem in spectral coordinates. The transformed control and disturbance directions are approximated using least-squares spectral projections, and upper and lower bounds on the transformed Hamiltonian are used to derive corresponding bounds on the value function and backward reachable sets. Koopman eigenfunctions are computed using a path-integral-based Galerkin framework that avoids spatial discretization of the associated eigenfunction PDEs. Numerical examples demonstrate tractable approximation of backward reachable sets for nonlinear adversarial systems.

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arXiv2026

Discovering the Kalman-Bucy-Koopman Filter

This paper introduces the Kalman-Bucy-Koopman (KBK) filter, a novel framework for nonlinear state estimation grounded in Koopman operator spectral theory. The nonlinear estimation problem is formulated as a maximum-likelihood (Mortensen) estimator whose solution is characterized by a Hamilton-Jacobi (HJ) partial differential equation. The proposed KBK filter provides a spectral, operator-theoretic realization of this nonlinear filtering problem by parameterizing the HJ value function in terms of principal Koopman eigenfunctions. This transformation converts the nonlinear estimation problem into a Riccati-type evolution in Koopman coordinates, yielding a linear-operator analogue of the classical Kalman-Bucy filter while preserving nonlinear structure in the original state variables. We develop a path-integral formulation for computing principal Koopman eigenfunctions and introduce a dynamics-informed, characteristics-inspired basis construction for their approximation. Theoretical error bounds are derived for value-function and state-estimation approximations. Simulation results demonstrate improved performance over the extended Kalman filter and illustrate the ability of the KBK framework to operate in data-driven settings without explicit model linearization.

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arXiv2026

Spectral Koopman Approach for Reconstructing State-space Geometry of Cislunar Restricted 3-Body Problem

In this work, we propose a novel approach, based on the path integral formulation of Koopman spectrum, to discover the phase-space geometry of the planar Cislunar Restricted 3 Body Problem (CR3BP). In contrast to existing techniques, which use trajectory-based (usually) local analysis, we leverage the Koopman operator framework, which generates a global linear \emph{representation} of the system, to reconstruct the global phase space geometry of the CR3BP. In particular, we compute the principal eigenfunctions of the Koopman operator via the path integral approach and show how the zero level curves of these eigenfunctions encode the phase space characteristics of the planar CR3BP.

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arXiv2026

RK-MPC: Residual Koopman Model Predictive Control for Quadruped Locomotion in Offroad Environments

This paper presents Residual Koopman MPC (RK-MPC), a Koopman-based, data-driven model predictive control framework for quadruped locomotion that improves prediction fidelity while preserving real-time tractability. RK-MPC augments a nominal template model with a compact linear residual predictor learned from data in lifted coordinates, enabling systematic correction of model mismatch induced by contact variability and terrain disturbances with provable bounds on multi-step prediction error. The learned residual model is embedded within a convex quadratic-program MPC formulation, yielding a receding-horizon controller that runs onboard at 500 Hz and retains the structure and constraint-handling advantages of optimization-based control. We evaluate RK-MPC in both Gazebo simulation and Unitree Go1 hardware experiments, demonstrating reliable blind locomotion across contact disturbances, multiple gait schedules, and challenging off-road terrains including grass, gravel, snow, and ice. We further compare against Koopman/EDMD baselines using alternative observable dictionaries, including monomial and $SE(3)$-structured bases, and show that the residual correction improves multi-step prediction and closed-loop performance while reducing sensitivity to the choice of observables. Overall, RK-MPC provides a practical, hardware-validated pathway for data-driven predictive control of quadrupeds in unstructured environments. See https://sriram-2502.github.io/rk-mpc for implementation videos.

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arXiv2026

Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristics

We present a unified theoretical and computational framework for constructing reproducing kernels tailored to transport equations and adapted to Koopman eigenfunctions of nonlinear dynamical systems. These eigenfunctions satisfy a transport-type partial differential equation (PDE) that we invert using three analytically grounded methods: (i) A Lions-type variational principle in a reproducing kernel Hilbert space (RKHS), (ii) convolution with a Green's function, and (iii) a resolvent operator constructed via Laplace transforms along characteristic flows. We prove that these three constructions yield identical kernels under mild smoothness and causality assumptions. We further show that the associated kernel eigenfunctions (Mercer modes) converge in L^2 to true Koopman eigenfunctions when the latter lie in the RKHS. Our approach is numerically realized through a mesh-free, convex optimization framework, enhanced with boundary regularization to handle eigenfunction blow-up. A multiple-kernel learning (MKL) scheme selects kernels automatically via residual minimization. Finally, we demonstrate that the same framework applies verbatim to a broader class of linear transport PDEs, including the advection, continuity, and Liouville equations. The unification of variational principles, Green's functions, and the method of characteristics enables the development of novel schemes for approximating eigenfunctions of transport equations, including those of the Koopman operator, and introduces a data-driven approach for learning kernels tailored to these approximations. Numerical experiments confirm the practical utility and robustness of the method.

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arXiv2026

Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation

We extend the unified kernel framework for transport equations and Koopman eigenfunctions, developed in previous work by the authors for deterministic systems, to stochastic differential equations (SDEs). In the deterministic setting, three analytically grounded constructions-Lions-type variational principles, Green's function convolution, and resolvent operators along characteristic flows--were shown to yield identical reproducing kernels. For stochastic systems, the Koopman generator includes a second-order diffusion term, transforming the first-order hyperbolic transport equation into a second-order elliptic-parabolic PDE. This fundamental change necessitates replacing the method of characteristics with probabilistic representations based on the Feynman--Kac formula. Our main contributions include: (i) extension of all three kernel constructions to stochastic systems via Feynman--Kac path-integral representations; (ii) proof of kernel equivalence under uniform ellipticity assumptions; (iii) a collocation-based computational framework incorporating second-order differential operators; (iv) error bounds separating RKHS approximation error from Monte Carlo sampling error; (v) analysis of how diffusion affects numerical conditioning; and (vi) connections to generator EDMD, diffusion maps, and kernel analog forecasting. Numerical experiments on Ornstein--Uhlenbeck processes, nonlinear SDEs with varying diffusion strength, and multi-dimensional systems validate the theoretical developments and demonstrate that moderate diffusion can improve numerical stability through elliptic regularization.

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arXiv2026

Kernel-Based LMI Approaches to Solving the Hamilton-Jacobi-Bellman Equation and Nonlinear Optimal Control

We present a kernel-based linear matrix inequality (LMI) approach for the approximate solution of Hamilton--Jacobi--Bellman (HJB) equations arising in nonlinear optimal control. The method represents the gradient of the value function in a reproducing kernel Hilbert space (RKHS) and uses a Schur-complement reformulation to convert the quadratic HJB inequality into an LMI that is linear in the kernel coefficients, yielding a convex semidefinite program. The novel ingredient is an explicit Riccati--Hessian \emph{equality} constraint at the equilibrium, which removes the trivial solution and forces the Hessian of the approximation to match the algebraic Riccati equation solution of the linearised system. We give a suboptimality bound $J(x_0;\hat u) - V^*(x_0)\le \varepsilon\,T(x_0)$ in which $T(x_0)$ depends only on the problem data and the working domain (not on the approximation), and an RKHS approximation rate. Numerical experiments on a corrected 1D polynomial benchmark and on the Van der Pol oscillator measure $\varepsilon$, the RKHS approximation error, and the closed-loop cost $J(x_0;\hat u)$ versus the optimal value $V^*(x_0)$. On the 1D problem with $V^*$ in the polynomial-kernel RKHS the method recovers $V^*$ to within $3\times10^{-7}$ and achieves $0.000\%$ suboptimality. On Van der Pol it achieves the smallest HJB residual ($\varepsilon\approx 2.62$) of any method tested, beats LQR on every initial condition, and is within $0.42\%$ of the best per-IC cost (Albrekht order 6). When $V^*$ is not in the chosen RKHS, the method degrades gracefully: residuals stop improving with more centres but suboptimality remains bounded ($\le 13\%$ on the 1D test).

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arXiv2025

Safety Critical Model Predictive Control Using Discrete-Time Control Density Functions

This paper presents MPC-CDF, a new approach integrating control density functions (CDFs) within a model predictive control (MPC) framework to ensure safety-critical control in nonlinear dynamical systems. By using the dual formulation of the navigation problem, we incorporate CDFs into the MPC framework, ensuring both convergence and safety in a discrete-time setting. These density functions are endowed with a physical interpretation, where the associated measure signifies the occupancy of system trajectories. Leveraging this occupancy-based perspective, we synthesize safety-critical controllers using the proposed MPC-CDF framework. We illustrate the safety properties of this framework using a unicycle model and compare it with a control barrier function-based method. The efficacy of this approach is demonstrated in the autonomous safe navigation of an underwater vehicle, which avoids complex and arbitrary obstacles while achieving the desired level of safety.

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arXiv2025

When Koopman Meets Hamilton and Jacobi

In this paper, we establish a connection between the spectral theory of the Koopman operator and the solution of the Hamilton Jacobi (HJ) equation. The HJ equation occupies a central place in systems theory, and its solution is of interest in various control problems, including optimal control, robust control, and input-output analysis. A Hamiltonian dynamical system can be associated with the HJ equation and the solution of the HJ equation can be extracted from the Hamiltonian system in the form of Lagrangian submanifold. One of the main contributions of this paper is to show that the Lagrangian submanifolds can be obtained using the spectral analysis of the Koopman operator. We present two different procedures for the approximation of the HJ solution. We utilize the spectral properties of the Koopman operator associated with the uncontrolled dynamical system and Hamiltonian systems to approximate the HJ solution. We present a convex optimization-based computational framework with convergence analysis for approximating the Koopman eigenfunctions and the Lagrangian submanifolds. Our solution approach to the HJ equation using Koopman theory provides for a natural extension of results from linear systems to nonlinear systems. We demonstrate the application of this work for solving the optimal control problem. Finally, we present simulation results to validate the paper's main findings and compare them against linear quadratic regulator and Taylor series based approximation controllers.

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arXiv2024

Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator

The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct computations. In this paper, we introduce a kernel-based method to construct the principal eigenfunctions of the Koopman operator without explicitly computing the operator itself. These principal eigenfunctions are associated with the equilibrium dynamics, and their eigenvalues match those of the linearization of the nonlinear system at the equilibrium point. We exploit the structure of the principal eigenfunctions by decomposing them into linear and nonlinear components. The linear part corresponds to the left eigenvector of the system's linearization at the equilibrium, while the nonlinear part is obtained by solving a partial differential equation (PDE) using kernel methods. Our approach avoids common issues such as spectral pollution and spurious eigenvalues, which can arise in previous methods. We demonstrate the effectiveness of our algorithm through numerical examples.

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arXiv2024

Extensions of the Path-integral formula for computation of Koopman eigenfunctions

Representing nonlinear dynamical systems using the Koopman Operator and its spectrum has distinct advantages in terms of linear interpretability of the model as well as in analysis and control synthesis through the use of well-studied techniques from linear systems theory. As such, efficient computation of Koopman eigenfunctions is of paramount importance towards enabling such Koopman-based constructions. To this end, several approaches have been proposed in literature, including data-driven, convex optimization, and Deep Learning-based methods. In our recent work, we proposed a novel approach based on path-integrals that allowed eigenfunction computations using a closed-form formula. In this paper, we present several important developments such as finite-time computations, relaxation of assumptions on the distribution of the principal Koopman eigenvalues, as well as extension towards saddle point systems, which greatly enhance the practical applicability of our method.

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arXiv2024

Safe Navigation in Dynamic Environments using Density Functions

This work presents a density-based framework for safe navigation in dynamic environments characterized by time-varying obstacle sets and time-varying target regions. We propose an analytical construction of time-varying density functions that enables the synthesis of a feedback controller defined as the positive gradient of the resulting density field. The primary contribution of this paper is a rigorous convergence proof demonstrating almost-everywhere safe navigation under the proposed framework, specifically for systems governed by single-integrator dynamics. To the best of our knowledge, these are the first analytical guarantees of their kind for navigation in dynamic environments using density functions. We illustrate the applicability of the framework to systems with more complex dynamics, including multi-agent systems and robotic manipulators, using standard control design techniques such as backstepping and inverse dynamics. These results provide a foundation for extending density-based navigation methods to a broad class of robotic systems operating in time-varying environments.

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arXiv2024

Digital Twins Meet the Koopman Operator: Data-Driven Learning for Robust Autonomy

Contrary to on-road autonomous navigation, off-road autonomy is complicated by various factors ranging from sensing challenges to terrain variability. In such a milieu, data-driven approaches have been commonly employed to capture intricate vehicle-environment interactions effectively. However, the success of data-driven methods depends crucially on the quality and quantity of data, which can be compromised by large variability in off-road environments. To address these concerns, we present a novel methodology to recreate the exact vehicle and its target operating conditions digitally for domain-specific data generation. This enables us to effectively model off-road vehicle dynamics from simulation data using the Koopman operator theory, and employ the obtained models for local motion planning and optimal vehicle control. The capabilities of the proposed methodology are demonstrated through an autonomous navigation problem of a 1:5 scale vehicle, where a terrain-informed planner is employed for global mission planning. Results indicate a substantial improvement in off-road navigation performance with the proposed algorithm (5.84x) and underscore the efficacy of digital twinning in terms of improving the sample efficiency (3.2x) and reducing the sim2real gap (5.2%).

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arXiv2024

Control Density Function for Robust Safety and Convergence

We introduce a novel approach for safe control design based on the density function. A control density function (CDF) is introduced to synthesize a safe controller for a nonlinear dynamic system. The CDF can be viewed as a dual to the control barrier function (CBF), a popular approach used for safe control design. While the safety certificate using the barrier function is based on the notion of invariance, the dual certificate involving the density function has a physical interpretation of occupancy. This occupancy-based physical interpretation is instrumental in providing an analytical construction of density function used for safe control synthesis. The safe control design problem is formulated using the density function as a quadratic programming (QP) problem. In contrast to the QP proposed for control synthesis using CBF, the proposed CDF-based QP can combine both the safety and convergence conditions to target state into single constraints. Further, we consider robustness against uncertainty in system dynamics and the initial condition and provide theoretical results for robust navigation using the CDF. Finally, we present simulation results for safe navigation with single integrator and double-gyre fluid flow-field examples, followed by robust navigation using the bicycle model and autonomous lane-keeping examples.

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arXiv2024

Synthesizing Controller for Safe Navigation using Control Density Function

We consider the problem of navigating a nonlinear dynamical system from some initial set to some target set while avoiding collision with an unsafe set. We extend the concept of density function to control density function (CDF) for solving navigation problems with safety constraints. The occupancy-based interpretation of the measure associated with the density function is instrumental in imposing the safety constraints. The navigation problem with safety constraints is formulated as a quadratic program (QP) using CDF. The existing approach using the control barrier function (CBF) also formulates the navigation problem with safety constraints as QP. One of the main advantages of the proposed QP using CDF compared to QP formulated using CBF is that both the convergence/stability and safety can be combined and imposed using the CDF. Simulation results involving the Duffing oscillator and safe navigation of Dubin car models are provided to verify the main findings of the paper.

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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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arXiv2023

Safe Motion Planning for Quadruped Robots Using Density Functions

This paper presents a motion planning algorithm for quadruped locomotion based on density functions. We decompose the locomotion problem into a high-level density planner and a model predictive controller (MPC). Due to density functions having a physical interpretation through the notion of occupancy, it is intuitive to represent the environment with safety constraints. Hence, there is an ease of use to constructing the planning problem with density. The proposed method uses a simplified model of the robot into an integrator system, where the high-level plan is in a feedback form formulated through an analytically constructed density function. We then use the MPC to optimize the reference trajectory, in which a low-level PID controller is used to obtain the torque level control. The overall framework is implemented in simulation, demonstrating our feedback density planner for legged locomotion. The implementation of work is available at \url{https://github.com/AndrewZheng-1011/legged_planner}

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