arXiv subjects

Jorge Cortes

Publications and source records attributed to Jorge Cortes.

At least 19 recordsLinked to original sources

arXiv2026

Timescale Limits of Linear-Threshold Networks

Linear-threshold networks (LTNs) capture the mesoscale behavior of interacting populations of neurons and are of particular interest to control theorists due to their dynamical richness and relative ease of analysis. The aim of this paper is to advance the study of global asymptotic stability in LTNs with asymmetric neural interactions and heterogeneous dissipation under the structural Lyapunov diagonal stability (LDS) condition. To this end, we introduce a one-parameter family of LTNs that preserves the LDS condition and has a parameter-independent equilibrium set. In the fast limit, this family converges to a projected dynamical system (PDS), while in the slow limit, it converges to a discontinuous hard-selector system (HSS). Under LDS, we prove that the fast PDS limit is globally exponentially stable and that the HSS limit is globally asymptotically stable. This alignment suggests that the limiting systems capture essential mechanisms governing stability across the entire LTN family. Together with numerical evidence, these findings indicate that resolving stability at the fast and slow endpoints provides a promising and structurally grounded path toward establishing global stability for LTNs with biologically plausible recurrence and diagonal dissipation.

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arXiv2026

Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors

Data-driven approximations of the infinite-dimensional Koopman operator rely on finite-dimensional projections, where the predictive accuracy of the resulting models hinges heavily on the invariance of the chosen subspace. Subspace pruning systematically discards geometrically misaligned directions to enhance this invariance proximity, which formally corresponds to the largest principal angle between the subspace and its image under the operator. Yet, existing techniques are largely restricted to Euclidean settings. To bridge this gap, this paper presents an approach for computing principal angles and vectors to enable Koopman subspace pruning within a Reproducing Kernel Hilbert Space (RKHS) geometry. We first outline an exact computational routine, which is subsequently scaled for large datasets using randomized Nystrom approximations. Based on these foundations, we introduce the Kernel-SPV and Approximate Kernel-SPV algorithms for targeted subspace refinement via principal vectors. Simulation results validate our approach.

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arXiv2026

A Unified Algebraic Framework for Subspace Pruning in Koopman Operator Approximation via Principal Vectors

Finite-dimensional approximations of the Koopman operator rely critically on identifying nearly invariant subspaces. This invariance proximity can be rigorously quantified via the principal angles between a candidate subspace and its image under the operator. To systematically minimize this error, we propose an algebraic framework for subspace pruning utilizing principal vectors. We establish the equivalence of this approach to existing consistency-based methods while providing a foundation for broader generalizations. To ensure scalability, we introduce an efficient numerical update scheme based on rank-one modifications, reducing the computational complexity of tracking principal angles by an order of magnitude. Finally, we demonstrate the effectiveness of our framework through numerical simulations.

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arXiv2025

Distributed Time-Varying Coverage Control via Singular Perturbations

This paper presents a novel dynamic coverage control algorithm allowing a group of robots to track an optimal-deployment configuration for arbitrary time-varying density functions. Building on singular perturbation theory, the proposed design employs a two-time scale separation approach, with a fast time scale corresponding to communication and a slow time scale corresponding to agent motion. The resulting algorithm is distributed over the 2-hop Delaunay graph and, for small enough values of the perturbation parameter, achieves the same performance as its centralized counterpart. We also introduce three discrete-time versions that rely only on 1-hop communication at the cost of having to use delayed information and formally establish their asymptotic convergence properties. Our technical approach combines computational geometry, singular perturbation theory, generating functions, and linear iterations with delayed updates. Various simulations illustrate the performance of the proposed algorithms.

math.OC↗

arXiv2025

Set-valued regression and cautious suboptimization: From noisy data to optimality

This paper deals with the problem of finding suboptimal values of an unknown function on the basis of measured data corrupted by bounded noise. As a prior, we assume that the unknown function is parameterized in terms of a number of basis functions. Inspired by the informativity approach, we view the problem as the suboptimization of the worst-case estimate of the function. The paper provides closed form solutions and convexity results for this function, which enables us to solve the problem. After this, an online implementation is investigated, where we iteratively measure the function and perform a suboptimization. This nets a procedure that is safe at each step, and which, under mild assumptions, converges to the true optimizer.

math.OC↗

arXiv2025

Learning to Pursue AC Optimal Power Flow Solutions with Feasibility Guarantees

This paper focuses on an AC optimal power flow (OPF) problem for distribution feeders equipped with controllable distributed energy resources (DERs). We consider a solution method that is based on a continuous approximation of the projected gradient flow - referred to as the safe gradient flow - that incorporates voltage and current information obtained either through real-time measurements or power flow computations. These two setups enable both online and offline implementations. The safe gradient flow involves the solution of convex quadratic programs (QPs). To enhance computational efficiency, we propose a novel framework that employs a neural network approximation of the optimal solution map of the QP. The resulting method has two key features: (a) it ensures that the DERs' setpoints are practically feasible, even for an online implementation or when an offline algorithm has an early termination; (b) it ensures convergence to a neighborhood of a strict local optimizer of the AC OPF. The proposed method is tested on a 93-node distribution system with realistic loads and renewable generation. The test shows that our method successfully regulates voltages within limits during periods with high renewable generation.

math.OC↗

arXiv2025

Data-Enabled Predictive Control for Nonlinear Systems Based on a Koopman Bilinear Realization

This paper extends the Willems' Fundamental Lemma to nonlinear control-affine systems using the Koopman bilinear realization. This enables us to bypass the Extended Dynamic Mode Decomposition (EDMD)-based system identification step in conventional Koopman-based methods and design controllers for nonlinear systems directly from data. Leveraging this result, we develop a Data-Enabled Predictive Control (DeePC) framework for nonlinear systems with unknown dynamics. A case study demonstrates that our direct data-driven control method achieves improved optimality compared to conventional Koopman-based methods. Furthermore, in examples where an exact Koopman realization with a finite-dimensional lifting function set of the controlled nonlinear system does not exist, our method exhibits advanced robustness to finite Koopman approximation errors compared to existing methods.

math.OC↗

arXiv2025

Safe Control of Second-Order Systems with Linear Constraints

Control barrier functions (CBFs) offer a powerful tool for enforcing safety specifications in control synthesis. This paper deals with the problem of constructing valid CBFs. Given a second-order system and any desired safety set with linear boundaries in the position space, we construct a provably control-invariant subset of this desired safety set. The constructed subset does not sacrifice any positions allowed by the desired safety set, which can be nonconvex. We show how our construction can also meet safety specification on the velocity. We then demonstrate that if the system satisfies standard Euler-Lagrange systems properties then our construction can also handle constraints on the allowable control inputs. We finally show the efficacy of the proposed method in a numerical example of keeping a 2D robot arm safe from collision.

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arXiv2024

Safety-Critical Control of Discontinuous Systems with Nonsmooth Safe Sets

This paper studies the design of controllers for discontinuous dynamics that ensure the safety of non-smooth sets. The safe set is represented by arbitrarily nested unions and intersections of 0-superlevel sets of differentiable functions. We show that any optimization-based controller that satisfies only the point-wise active safety constraints is generally un-safe, ruling out the standard techniques developed for safety of continuous dynamics. This motivates the introduction of the notion of transition functions, which allow us to incorporate even the inactive safety constraints without falling into unnecessary conservatism. These functions allow system trajectories to leave a component of the nonsmooth safe set to transition to a different one. The resulting controller is then defined as the solution to a convex optimization problem, which we show is feasible and continuous wherever the system dynamics is continuous. We illustrate the effectiveness of the proposed design approach in a multi-agent reconfiguration control problem.

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arXiv2024

Linear-Threshold Network Models for Describing and Analyzing Brain Dynamics

Over the past two decades, an increasing array of control-theoretic methods have been used to study the brain as a complex dynamical system and better understand its structure-function relationship. This article provides an overview on one such family of methods, based on the linear-threshold rate (LTR) dynamics, which arises when modeling the spiking activity of neuronal populations and their impact on each other. LTR dynamics exhibit a wide range of behaviors based on network topologies and inputs, including mono- and multi-stability, limit cycles, and chaos, allowing it to be used to model many complex brain processes involving fast and slow inhibition, multiple time and spatial scales, different types of neural behavior, and higher-order interactions. Here we investigate how the versatility of LTR dynamics paired with concepts and tools from systems and control can provide a computational theory for explaining the dynamical mechanisms enabling different brain processes. Specifically, we illustrate stability and stabilization properties of LTR dynamics and how they are related to goal-driven selective attention, multistability and its relationship with declarative memory, and bifurcations and oscillations and their role in modeling seizure dynamics in epilepsy. We conclude with a discussion on additional properties of LTR dynamics and an outlook on other brain processess that for which they might be play a similar role.

q-bio.NC↗

arXiv2024

Independence of Closed-Loop Equilibria and Stability from the Choice of Control Barrier Function for a Given Safe Set

Control barrier functions (CBFs) play a critical role in the design of safe optimization-based controllers for control-affine systems. Given a CBF associated with a given, predefined ``safe'' set, the typical approach consists in embedding CBF-based constraints into the optimization problem defining the control law to enforce forward invariance of the safe set. While this approach effectively guarantees safety for a given CBF, the CBF-based control law can introduce undesirable equilibrium points (i.e., points that are not equilibria of the original system). Given that there exist many different CBFs associated with a given fixed safe set, open questions remain on how the choice of CBF influences the number and locations of undesirable equilibria and, in general, the dynamics of the closed-loop system. This paper investigates how the choice of CBF impacts the dynamics of the closed-loop system and shows that: (i) The choice of CBF does not affect the number, location, and (local) stability properties of the equilibria in the interior of the safe set; (ii) undesirable equilibria only appear on the boundary of the safe set; and, (iii) the number and location of undesirable equilibria for the closed-loop system do not depend of the choice of the CBF. Additionally, for the well-established \textit{safety filters}, we show that the stability properties of equilibria of the closed-loop system are independent of the choice of the CBF and of the associated extended class-K function, provided that the CBFs are chosen from the same equivalence class.

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arXiv2024

Stabilization of Nonlinear Systems through Control Barrier Functions

This paper proposes a control design approach for stabilizing nonlinear control systems. Our key observation is that the set of points where the decrease condition of a control Lyapunov function (CLF) is feasible can be regarded as a safe set. By leveraging a nonsmooth version of control barrier functions (CBFs) and a weaker notion of CLF, we develop a control design that forces the system to converge to and remain in the region where the CLF decrease condition is feasible. We characterize the conditions under which our controller asymptotically stabilizes the origin or a small neighborhood around it, even in the cases where it is discontinuous. We illustrate our design in various examples.

math.OC↗

arXiv2024

Koopman Operators in Robot Learning

Koopman operator theory offers a rigorous treatment of dynamics and has been emerging as an alternative modeling and learning-based control method across various robotics sub-domains. Due to its ability to represent nonlinear dynamics as a linear (but higher-dimensional) operator, Koopman theory offers a fresh lens through which to understand and tackle the modeling and control of complex robotic systems. Moreover, it enables incremental updates and is computationally inexpensive, thus making it particularly appealing for real-time applications and online active learning. This review delves deeply into the foundations of Koopman operator theory and systematically builds a bridge from theoretical principles to practical robotic applications. We begin by explaining the mathematical underpinnings of the Koopman framework and discussing approximation approaches for incorporating inputs into Koopman-based modeling. Foundational considerations, such as data collection strategies as well as the design of lifting functions for effective system embedding, are also discussed. We then explore how Koopman-based models serve as a unifying tool for a range of robotics tasks, including model-based control, real-time state estimation, and motion planning. The review proceeds to a survey of cutting-edge research that demonstrates the versatility and growing impact of Koopman methods across diverse robotics sub-domains: from aerial and legged platforms to manipulators, soft-bodied systems, and multi-agent networks. A presentation of more advanced theoretical topics, necessary to push forward the overall framework, is included. Finally, we reflect on some key open challenges that remain and articulate future research directions that will shape the next phase of Koopman-inspired robotics. To support practical adoption, we provide a hands-on tutorial with executable code at https://shorturl.at/ouE59.

cs.RO↗

arXiv2024

Characterization of the Dynamical Properties of Safety Filters for Linear Planar Systems

This paper studies the dynamical properties of closed-loop systems obtained from control barrier function-based safety filters. We provide a sufficient and necessary condition for the existence of undesirable equilibria and show that the Jacobian matrix of the closed-loop system evaluated at an undesirable equilibrium always has a nonpositive eigenvalue. In the special case of linear planar systems and ellipsoidal obstacles, we give a complete characterization of the dynamical properties of the corresponding closed-loop system. We show that for underactuated systems, the safety filter always introduces a single undesirable equilibrium, which is a saddle-point. We prove that all trajectories outside the global stable manifold of such equilibrium converge to the origin. In the fully actuated case, we discuss how the choice of nominal controller affects the stability properties of the closed-loop system. Various simulations illustrate our results.

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arXiv2024

Performance-Barrier Event-Triggered Control of a Class of Reaction-Diffusion PDEs

We employ the recent performance-barrier event-triggered control (P-ETC) for achieving global exponential convergence of a class of reaction-diffusion PDEs via PDE backstepping control. Rather than insisting on a strictly monotonic decrease of the Lyapunov function for the closed-loop system, P-ETC allows the Lyapunov function to increase as long as it remains below an acceptable performance-barrier. This approach integrates a performance residual, the difference between the value of the performance-barrier and the Lyapunov function, into the triggering mechanism. The integration adds flexibility and results in fewer control updates than with regular ETC (R-ETC) that demands a monotonic decrease of the Lyapunov function. Our P-ETC PDE backstepping design ensures global exponential convergence of the closed-loop system in the spatial L^2 norm, without encountering Zeno phenomenon. To avoid continuous monitoring of the triggering function that generates events, we develop periodic event-triggered and self-triggered variants (P-PETC and P-STC, respectively) of the P-ETC. The P-PETC only requires periodic evaluation of the triggering function whereas the P-STC preemptively computes the time of the next event at the current event time using the system model and continuously available system states. The P-PETC and P-STC also ensure a Zeno-free behavior and deliver performance equivalent to that of the continuous-time P-ETC which requires continuous evaluation of the triggering function, in addition to the continuous sensing of the state. We provide numerical simulations to illustrate the proposed technique and to compare it with R-ETC associated with strictly decreasing Lyapunov functions.

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arXiv2024

Distributionally Robust Policy and Lyapunov-Certificate Learning

This article presents novel methods for synthesizing distributionally robust stabilizing neural controllers and certificates for control systems under model uncertainty. A key challenge in designing controllers with stability guarantees for uncertain systems is the accurate determination of and adaptation to shifts in model parametric uncertainty during online deployment. We tackle this with a novel distributionally robust formulation of the Lyapunov derivative chance constraint ensuring a monotonic decrease of the Lyapunov certificate. To avoid the computational complexity involved in dealing with the space of probability measures, we identify a sufficient condition in the form of deterministic convex constraints that ensures the Lyapunov derivative constraint is satisfied. We integrate this condition into a loss function for training a neural network-based controller and show that, for the resulting closed-loop system, the global asymptotic stability of its equilibrium can be certified with high confidence, even with Out-of-Distribution (OoD) model uncertainties. To demonstrate the efficacy and efficiency of the proposed methodology, we compare it with an uncertainty-agnostic baseline approach and several reinforcement learning approaches in two control problems in simulation.

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arXiv2023

Optimal Power Flow Pursuit via Feedback-based Safe Gradient Flow

This paper considers the problem of controlling inverter-interfaced distributed energy resources (DERs) in a distribution grid to solve an AC optimal power flow (OPF) problem in real time. The AC OPF includes voltage constraints, and seeks to minimize costs associated with the economic operation, power losses, or the power curtailment from renewables. We develop an online feedback optimization method to drive the DERs' power setpoints to solutions of an AC OPF problem based only on voltage measurements (and without requiring measurements of the power consumption of non-controllable assets). The proposed method - grounded on the theory of control barrier functions - is based on a continuous approximation of the projected gradient flow, appropriately modified to accommodate measurements from the power network. We provide results in terms of local exponential stability, and assess the robustness to errors in the measurements and in the system Jacobian matrix. We show that the proposed method ensures anytime satisfaction of the voltage constraints when no model and measurement errors are present; if these errors are present and are small, the voltage violation is practically negligible. We also discuss extensions of the framework to virtual power plant setups and to cases where constraints on power flows and currents must be enforced. Numerical experiments on a 93-bus distribution system and with realistic load and production profiles show a superior performance in terms of voltage regulation relative to existing methods.

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arXiv2023

Online Regulation of Dynamical Systems to Solutions of Constrained Optimization Problems

This paper considers the problem of regulating a dynamical system to equilibria that are defined as solutions of an input- and state-constrained optimization problem. To solve this regulation task, we design a state feedback controller based on a continuous approximation of the projected gradient flow. We first show that the equilibria of the interconnection between the plant and the proposed controller correspond to critical points of the constrained optimization problem. We then derive sufficient conditions to ensure that, for the closed-loop system, isolated locally optimal solutions of the optimization problem are locally exponentially stable and show that input constraints are satisfied at all times by identifying an appropriate forward-invariant set.

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