arXiv subjects

Akash Harapanahalli

Publications and source records attributed to Akash Harapanahalli.

17 recordsLinked to original sources

arXiv2026

Certified Robust Invariant Polytope Training in Neural Controlled ODEs

We propose a framework for training neural network controllers with certified robust forward invariant polytopes. First, we parameterize a family of lifted control systems in a higher dimensional space, where the original neural controlled system evolves on an invariant subspace of each lifted system. We use interval analysis and neural network verifiers to further construct a family of lifted embedding systems, carefully capturing the knowledge of this invariant subspace. If the vector field of any lifted embedding system satisfies a sign constraint at a single point, then a certain convex polytope of the original system is robustly forward invariant. Treating the neural network controller and the lifted system parameters as variables, we propose an algorithm to train controllers with certified forward invariant polytopes in the closed-loop control system. Through two examples, we demonstrate how the simplicity of the sign constraint allows our approach to scale with system dimension to over $50$ states, and outperform state-of-the-art Lyapunov-based sampling approaches in runtime.

cs.LG↗

arXiv2026

Learning Certified Neural Network Controllers Using Contraction and Interval Analysis

We present a novel framework that jointly trains a neural network controller and a neural Riemannian metric with rigorous closed-loop contraction guarantees using formal bound propagation. Directly bounding the symmetric Riemannian contraction linear matrix inequality causes unnecessary overconservativeness due to poor dependency management. Instead, we analyze an asymmetric matrix function $G$, where $2^n$ GPU-parallelized corner checks of its interval hull verify that an entire interval subset $X$ is a contraction region in a single shot. This eliminates the sample complexity problems encountered with previous Lipschitz-based guarantees. Additionally, for control-affine systems under a Killing field assumption, our method produces an explicit tracking controller capable of exponentially stabilizing any dynamically feasible trajectory using just two forward inferences of the learned policy. Using JAX and \immrax{} for linear bound propagation, we apply this approach to a full 10-state quadrotor model. In $<$10 minutes of post-JIT training, we simultaneously learn a control policy $π$, a neural contraction metric $Θ$, and a verified 10-dimensional contraction region $X$.

eess.SY↗

arXiv2026

linrax: A JAX Compatible, Simplex Method Linear Program Solver

We present linrax, the first simplex based linear program (LP) solver compatible with the JAX ecosystem. In many control algorithms, LPs are often automatically generated and frequently solved either offline or online in the control loop. This motivates the design of linrax, which is especially suited for compilation into a complex JAX-based pipeline as a subroutine. We discuss the challenges associated with implementing a general purpose LP solver under strict design requirements from JAX. Notably, we can solve general problems which may include dependent constraints-something not possible with existing JAX-compatible LP solvers that use first-order techniques and may fail to converge. We demonstrate the utility of linrax through several examples, including a robust control synthesis pipeline for a nonlinear vehicle model using automatic differentiation through a LP-based reachable set framework.

eess.SY↗

arXiv2026

Differentiable Invariant Sets for Hybrid Limit Cycles with Application to Legged Robots

For hybrid systems exhibiting periodic behavior, analyzing the invariant set containing the limit cycle is a natural way to study the robustness of the closed-loop system. However, computing these sets can be computationally expensive, especially when applied to contact-rich cyber-physical systems such as legged robots. In this work, we extend existing methods for overapproximating reachable sets of continuous systems using parametric embeddings to compute a forward-invariant set around the nominal trajectory of a simplified model of a bipedal robot. Our three-step approach (i) computes an overapproximating reachable set around the nominal continuous flow, (ii) catalogs intersections with the guard surface, and (iii) passes these intersections through the reset map. If the overapproximated reachable set after one step is a strict subset of the initial set, we formally verify a forward invariant set for this hybrid periodic orbit. We verify this condition on the bipedal walker model numerically using immrax, a JAX-based library for parametric reachable set computation, and use it within a bi-level optimization framework to design a tracking controller that maximizes the size of the invariant set.

eess.SY↗

arXiv2025

Efficient Norm-Based Reachable Sets via Iterative Dynamic Programming

In this work, we present a numerical optimal control framework for reachable set computation using \emph{normotopes}, a new set representation as a norm ball with a shaping matrix. In reachable set computations, we expect to continuously vary the shape matrix as a function of time. Incorporating the shape dynamics as an input, we build a \emph{controlled embedding system} using a linear differential inclusion overapproximating the dynamics of the system, where a single forward simulation of this embedding system always provides an overapproximating reachable set of the system, no matter the choice of \emph{hypercontrol}. By iteratively solving a linear quadratic approximation of the nonlinear optimal hypercontrol problem, we synthesize less conservative final reachable sets, providing a natural tradeoff between runtime and accuracy. Terminating our algorithm at any point always returns a valid reachable set overapproximation.

math.OC↗

arXiv2025

Automatic and Scalable Safety Verification using Interval Reachability with Subspace Sampling

Interval refinement is a technique for reducing the conservatism of traditional interval based reachability methods by lifting the system to a higher dimension using new auxiliary variables and exploiting the introduced structure through a refinement procedure. We present a novel, efficiently scaling, automatic refinement strategy based on a subspace sampling argument and motivated by reducing the number of interval operations through sparsity. Unlike previous methods, we guarantee that refined bounds shrink as additional auxiliary variables are added. This additionally encourages automation of the lifting phase by allowing larger groups of auxiliary variables to be considered. We implement our strategy in JAX, a high-performance computational toolkit for Python and demonstrate its efficacy on several examples, including regulating a multi-agent platoon to the origin while avoiding an obstacle.

eess.SY↗

arXiv2025

Parametric Reachable Sets Via Controlled Dynamical Embeddings

In this work, we propose a new framework for reachable set computation through continuous evolution of a set of parameters and offsets which define a parametope, through the intersection of constraints. This results in a dynamical approach towards nonlinear reachability analysis: a single trajectory of an embedding system provides a parametope reachable set for the original system, and uncertainties are accounted for through continuous parameter evolution. This is dual to most existing computational strategies, which define sets through some combination of generator vectors, and usually discretize the system dynamics. We show how, under some regularity assumptions of the dynamics and the set considered, any desired parameter evolution can be accommodated as long as the offset dynamics are set accordingly, providing a virtual "control input" for reachable set computation. In a special case of the theory, we demonstrate how closing the loop for the parameter dynamics using the adjoint of the linearization results in a desirable first-order cancellation of the original system dynamics. Using interval arithmetic in JAX, we demonstrate the efficiency and utility of reachable parametope computation through two numerical examples.

math.OC↗

arXiv2025

A Linear Differential Inclusion for Contraction Analysis to Known Trajectories

Infinitesimal contraction analysis provides exponential convergence rates between arbitrary pairs of trajectories of a system by studying the system's linearization. An essentially equivalent viewpoint arises through stability analysis of a linear differential inclusion (LDI) encompassing the incremental behavior of the system. In this note, we use contraction tools to study the exponential stability of a system to a particular known trajectory, deriving a new LDI characterizing the error between arbitrary trajectories and this known trajectory. As with classical contraction analysis, this new inclusion is constructed via first partial derivatives of the system's vector field, and convergence rates are obtained with familiar tools: uniform bounding of the logarithmic norm and LMI-based Lyapunov conditions. Our LDI is guaranteed to outperform a usual contraction analysis in two special circumstances: i) when the bound on the logarithmic norm arises from an interval overapproximation of the Jacobian matrix, and ii) when the norm considered is the $\ell_1$ norm. Finally, we demonstrate how the proposed approach strictly improves an existing framework for ellipsoidal reachable set computation.

eess.SY↗

arXiv2025

A Global Coordinate-Free Approach to Invariant Contraction on Homogeneous Manifolds

In this work, we provide a global condition for contraction with respect to an invariant Riemannian metric on reductive homogeneous spaces. Using left-invariant frames, vector fields on the manifold are horizontally lifted to the ambient Lie group, where the Levi-Civita connection is globally characterized as a real matrix multiplication. By linearizing in these left-invariant frames, we characterize contraction using matrix measures on real square matrices, avoiding the use of local charts. Applying this global condition, we provide a necessary condition for a prescribed subset of the manifold to possibly admit a contracting system, which accounts for the underlying geometry of the invariant metric. Applied to the sphere, this condition implies that no great circle can be contained in a contraction region. Finally, we apply our results to compute reachable sets for an attitude control problem.

eess.SY↗

arXiv2024

Efficient Reachable Sets on Lie Groups Using Lie Algebra Monotonicity and Tangent Intervals

In this paper, we efficiently compute overapproximating reachable sets for control systems evolving on Lie groups, building off results from monotone systems theory and geometric integration theory. We consider intervals in the tangent space, which describe real sets on the Lie group through the exponential map. A local equivalence between the original system and a system evolving on the Lie algebra allows existing interval reachability techniques to apply in the tangent space. Using interval bounds of the Baker-Campbell-Hausdorff formula, these reachable set estimates are extended to arbitrary time horizons in an efficient Runge-Kutta-Munthe-Kaas integration algorithm. The algorithm is demonstrated through consensus on a torus and attitude control on $SO(3)$.

eess.SY↗

arXiv2024

Efficient Interaction-Aware Interval Analysis of Neural Network Feedback Loops

In this paper, we propose a computationally efficient framework for interval reachability of systems with neural network controllers. Our approach leverages inclusion functions for the open-loop system and the neural network controller to embed the closed-loop system into a larger-dimensional embedding system, where a single trajectory over-approximates the original system's behavior under uncertainty. We propose two methods for constructing closed-loop embedding systems, which account for the interactions between the system and the controller in different ways. The interconnection-based approach considers the worst-case evolution of each coordinate separately by substituting the neural network inclusion function into the open-loop inclusion function. The interaction-based approach uses novel Jacobian-based inclusion functions to capture the first-order interactions between the open-loop system and the controller by leveraging state-of-the-art neural network verifiers. Finally, we implement our approach in a Python framework called ReachMM to demonstrate its efficiency and scalability on benchmarks and examples ranging to $200$ state dimensions.

eess.SY↗

arXiv2024

$\texttt{immrax}$: A Parallelizable and Differentiable Toolbox for Interval Analysis and Mixed Monotone Reachability in JAX

We present an implementation of interval analysis and mixed monotone interval reachability analysis as function transforms in Python, fully composable with the computational framework JAX. The resulting toolbox inherits several key features from JAX, including computational efficiency through Just-In-Time Compilation, GPU acceleration for quick parallelized computations, and Automatic Differentiability. We demonstrate the toolbox's performance on several case studies, including a reachability problem on a vehicle model controlled by a neural network, and a robust closed-loop optimal control problem for a swinging pendulum.

eess.SY↗

arXiv2023

Interval Signal Temporal Logic from Natural Inclusion Functions

We propose an interval extension of Signal Temporal Logic (STL) called Interval Signal Temporal Logic (\ISTL). Given an STL formula, we consider an interval inclusion function for each of its predicates. Then, we use minimal inclusion functions for the $\min$ and $\max$ functions to recursively build an interval robustness that is a natural inclusion function for the robustness of the original STL formula. The resulting interval semantics accommodate, for example, uncertain signals modeled as a signal of intervals and uncertain predicates modeled with appropriate inclusion functions. In many cases, verification or synthesis algorithms developed for STL apply to \ISTL with minimal theoretic and algorithmic changes, and existing code can be readily extended using interval arithmetic packages at negligible computational expense. To demonstrate \ISTL, we present an example of offline monitoring from an uncertain signal trace obtained from a hardware experiment and an example of robust online control synthesis enforcing an STL formula with uncertain predicates.

eess.SY↗

arXiv2023

Contraction-Guided Adaptive Partitioning for Reachability Analysis of Neural Network Controlled Systems

In this paper, we present a contraction-guided adaptive partitioning algorithm for improving interval-valued robust reachable set estimates in a nonlinear feedback loop with a neural network controller and disturbances. Based on an estimate of the contraction rate of over-approximated intervals, the algorithm chooses when and where to partition. Then, by leveraging a decoupling of the neural network verification step and reachability partitioning layers, the algorithm can provide accuracy improvements for little computational cost. This approach is applicable with any sufficiently accurate open-loop interval-valued reachability estimation technique and any method for bounding the input-output behavior of a neural network. Using contraction-based robustness analysis, we provide guarantees of the algorithm's performance with mixed monotone reachability. Finally, we demonstrate the algorithm's performance through several numerical simulations and compare it with existing methods in the literature. In particular, we report a sizable improvement in the accuracy of reachable set estimation in a fraction of the runtime as compared to state-of-the-art methods.

eess.SY↗

arXiv2023

Forward Invariance in Neural Network Controlled Systems

We present a framework based on interval analysis and monotone systems theory to certify and search for forward invariant sets in nonlinear systems with neural network controllers. The framework (i) constructs localized first-order inclusion functions for the closed-loop system using Jacobian bounds and existing neural network verification tools; (ii) builds a dynamical embedding system where its evaluation along a single trajectory directly corresponds with a nested family of hyper-rectangles provably converging to an attractive set of the original system; (iii) utilizes linear transformations to build families of nested paralleletopes with the same properties. The framework is automated in Python using our interval analysis toolbox $\texttt{npinterval}$, in conjunction with the symbolic arithmetic toolbox $\texttt{sympy}$, demonstrated on an $8$-dimensional leader-follower system.

eess.SY↗

arXiv2023

Interval Reachability of Nonlinear Dynamical Systems with Neural Network Controllers

This paper proposes a computationally efficient framework, based on interval analysis, for rigorous verification of nonlinear continuous-time dynamical systems with neural network controllers. Given a neural network, we use an existing verification algorithm to construct inclusion functions for its input-output behavior. Inspired by mixed monotone theory, we embed the closed-loop dynamics into a larger system using an inclusion function of the neural network and a decomposition function of the open-loop system. This embedding provides a scalable approach for safety analysis of the neural control loop while preserving the nonlinear structure of the system. We show that one can efficiently compute hyper-rectangular over-approximations of the reachable sets using a single trajectory of the embedding system. We design an algorithm to leverage this computational advantage through partitioning strategies, improving our reachable set estimates while balancing its runtime with tunable parameters. We demonstrate the performance of this algorithm through two case studies. First, we demonstrate this method's strength in complex nonlinear environments. Then, we show that our approach matches the performance of the state-of-the art verification algorithm for linear discretized systems.

eess.SY↗

arXiv2023

A Toolbox for Fast Interval Arithmetic in numpy with an Application to Formal Verification of Neural Network Controlled Systems

In this paper, we present a toolbox for interval analysis in numpy, with an application to formal verification of neural network controlled systems. Using the notion of natural inclusion functions, we systematically construct interval bounds for a general class of mappings. The toolbox offers efficient computation of natural inclusion functions using compiled C code, as well as a familiar interface in numpy with its canonical features, such as n-dimensional arrays, matrix/vector operations, and vectorization. We then use this toolbox in formal verification of dynamical systems with neural network controllers, through the composition of their inclusion functions.

eess.SY↗