arXiv subjects

Paulo Tabuada

Publications and source records attributed to Paulo Tabuada.

At least 19 recordsLinked to original sources

arXiv2026

A Duality-Based Optimization Formulation of Safe Control Design with State Uncertainties

State estimation uncertainty is prevalent in real-world applications, hindering the application of safety-critical control. Existing methods address this by strengthening a Control Barrier Function (CBF) condition either to handle actuation errors induced by state uncertainty, or to enforce stricter, more conservative sufficient conditions. In this work, we take a more direct approach and formulate a robust safety filter by analyzing the image of the set of all possible states under the CBF dynamics. We first prove that convexifying this image set does not change the set of possible inputs. Then, by leveraging duality, we propose an equivalent and tractable reformulation for cases where this convex hull can be expressed as a polytope or ellipsoid. Simulation results show the approach in this paper to be less conservative than existing alternatives.

eess.SY↗

arXiv2026

Event-Triggered Newton Extremum Seeking for Multivariable Optimization

This paper presents a static event-triggered control strategy for multivariable Newton-based extremum seeking. The proposed method integrates event-triggered actuation into the Newton-based optimization framework to reduce control updates while maintaining rapid convergence to the extremum. Unlike traditional gradient-based extremum seeking, where the convergence rate depends on the unknown Hessian of the cost function, the proposed approach employs a dynamic estimator of the Hessian inverse, formulated as a Riccati equation, enabling user-assignable convergence rates. The event-triggering mechanism is designed to minimize unnecessary actuation updates while preserving stability and performance. Using averaging theory, we establish local stability results and exponential convergence to a neighborhood of the unknown extremum point. Additionally, numerical simulations illustrate the benefits of the proposed approach over gradient-based and continuously actuated Newton-based extremum seeking, showing improved convergence rates and reduced control update frequency, leading to more efficient implementation in real-time optimization scenarios.

math.OC↗

arXiv2025

Safe Navigation under State Uncertainty: Online Adaptation for Robust Control Barrier Functions

Measurements and state estimates are often imperfect in control practice, posing challenges for safety-critical applications, where safety guarantees rely on accurate state information. In the presence of estimation errors, several prior robust control barrier function (R-CBF) formulations have imposed strict conditions on the input. These methods can be overly conservative and can introduce issues such as infeasibility, high control effort, etc. This work proposes a systematic method to improve R-CBFs, and demonstrates its advantages on a tracked vehicle that navigates among multiple obstacles. A primary contribution is a new optimization-based online parameter adaptation scheme that reduces the conservativeness of existing R-CBFs. In order to reduce the complexity of the parameter optimization, we merge several safety constraints into one unified numerical CBF via Poisson's equation. We further address the dual relative degree issue that typically causes difficulty in vehicle tracking. Experimental trials demonstrate the overall performance improvement of our approach over existing formulations.

eess.SY↗

arXiv2025

Safety Under State Uncertainty: Robustifying Control Barrier Functions

Safety-critical control is a crucial aspect of modern systems, and Control Barrier Functions (CBFs) have gained popularity as the framework of choice for ensuring safety. However, implementing a CBF requires exact knowledge of the true state, a requirement that is often violated in real-world applications where only noisy or estimated state information is available. This paper introduces the notion of Robust Control Barrier Functions (R-CBF) for ensuring safety under such state uncertainty without requiring prior knowledge of the magnitude of uncertainty. We formally characterize the class of robustifying terms that ensure robust closed-loop safety and show how a robustly safe controller can be constructed. We demonstrate the effectiveness of this approach through simulations and compare it to existing methods, highlighting the additional robustness and convergence guarantees it provides.

eess.SY↗

arXiv2025

Secure Safety Filter Design for Sampled-data Nonlinear Systems under Sensor Spoofing Attacks

This paper presents a secure safety filter design for nonlinear systems under sensor spoofing attacks. Existing approaches primarily focus on linear systems which limits their applications in real-world scenarios. In this work, we extend these results to nonlinear systems in a principled way. We introduce exact observability maps that abstract specific state estimation algorithms and extend them to a secure version capable of handling sensor attacks. Our generalization also applies to the relaxed observability case, with slightly relaxed guarantees. More importantly, we propose a secure safety filter design in both exact and relaxed cases, which incorporates secure state estimation and a control barrier function-enabled safety filter. The proposed approach provides theoretical safety guarantees for nonlinear systems in the presence of sensor attacks. We numerically validate our analysis on a unicycle vehicle equipped with redundant yet partly compromised sensors.

eess.SY↗

arXiv2025

Secure Safety Filter: Towards Safe Flight Control under Sensor Attacks

Modern autopilot systems are prone to sensor attacks that can jeopardize flight safety. To mitigate this risk, we proposed a modular solution: the secure safety filter, which extends the well-established control barrier function (CBF)-based safety filter to account for, and mitigate, sensor attacks. This module consists of a secure state reconstructor (which generates plausible states) and a safety filter (which computes the safe control input that is closest to the nominal one). Differing from existing work focusing on linear, noise-free systems, the proposed secure safety filter handles bounded measurement noise and, by leveraging reduced-order model techniques, is applicable to the nonlinear dynamics of drones. Software-in-the-loop simulations and drone hardware experiments demonstrate the effectiveness of the secure safety filter in rendering the system safe in the presence of sensor attacks.

cs.RO↗

arXiv2025

Computationally Efficient Safe Control of Linear Systems under Severe Sensor Attacks

Cyber-physical systems are prone to sensor attacks that can compromise safety. A common approach to synthesizing controllers robust to sensor attacks is secure state reconstruction (SSR) -- but this is computationally expensive, hindering real-time control. In this paper, we take a safety-critical perspective on mitigating severe sensor attacks, leading to a computationally efficient solution. Namely, we design feedback controllers that ensure system safety by directly computing control actions from past input-output data. Instead of fully solving the SSR problem, we use conservative bounds on a control barrier function (CBF) condition, which we obtain by extending the recent eigendecomposition-based SSR approach to severe sensor attack settings. Additionally, we present an extended approach that solves a smaller-scale subproblem of the SSR problem, taking on some computational burden to mitigate the conservatism in the main approach. Numerical comparisons confirm that the traditional SSR approaches suffer from combinatorial issues, while our approach achieves safety guarantees with greater computational efficiency.

eess.SY↗

arXiv2025

Categorical Lyapunov Theory I: Stability of Flows

Lyapunov's theorem provides a fundamental characterization of the stability of dynamical systems. This paper presents a categorical framework for Lyapunov theory, generalizing stability analysis with Lyapunov functions categorically. Core to our approach is the set of axioms underlying a setting for stability, which give the necessary ingredients for ``doing Lyapunov theory'' in a category of interest. With these minimal assumptions, we define the stability of equilibria, formulate Lyapunov morphisms, and demonstrate that the existence of Lyapunov morphisms is necessary and sufficient for establishing the stability of flows. To illustrate these constructions, we show how classical notions of stability, e.g., for continuous and discrete time dynamical systems, are captured by this categorical framework for Lyapunov theory. Finally, to demonstrate the extensibility of our framework, we illustrate how enriched categories, e.g., Lawvere metric spaces, yield settings for stability enabling one to ``do Lyapunov theory'' in enriched categories.

math.DS↗

arXiv2025

Event-Triggered Newton-Based Extremum Seeking Control

This paper proposes the incorporation of static event-triggered control in the actuation path of Newton-based extremum seeking and its comparison with the earlier gradient version. As in the continuous methods, the convergence rate of the gradient approach depends on the unknown Hessian of the nonlinear map to be optimized, whereas the proposed event-triggered Newton-based extremum seeking eliminates this dependence, becoming user-assignable. This is achieved by means of a dynamic estimator for the Hessian's inverse, implemented as a Riccati equation filter. Lyapunov stability and averaging theory for discontinuous systems are applied to analyze the closed-loop system. Local exponential practical stability is guaranteed to a small neighborhood of the extremum point of scalar and static maps. Numerical simulations illustrate the advantages of the proposed approach over the previous gradient method, including improved convergence speed, followed by a reduction in the amplitude and updating frequency of the control signals.

math.OC↗

arXiv2024

The Asymptotic Behavior of Attention in Transformers

The transformer architecture has become the foundation of modern Large Language Models (LLMs), yet its theoretical properties are still not well understood. As with classic neural networks, a common approach to improve these models is to increase their size and depth. However, such strategies may be suboptimal, as several works have shown that adding more layers yields increasingly diminishing returns. More importantly, prior studies have shown that increasing depth may lead to model collapse, i.e., all the tokens converge to a single cluster, undermining the ability of LLMs to generate diverse outputs. Building on differential equation models for the transformer dynamics, we prove that all the tokens in a transformer asymptotically converge to a cluster as depth increases. At the technical level we leverage tools from control theory, including consensus dynamics on manifolds and input-to-state stability (ISS). We then extend our analysis to autoregressive models, exploiting their structure to further generalize the theoretical guarantees.

cs.AI↗

arXiv2024

Safety of Linear Systems under Severe Sensor Attacks

Cyber-physical systems can be subject to sensor attacks, e.g., sensor spoofing, leading to unsafe behaviors. This paper addresses this problem in the context of linear systems when an omniscient attacker can spoof several system sensors at will. In this adversarial environment, existing results have derived necessary and sufficient conditions under which the state estimation problem has a unique solution. In this work, we consider a severe attacking scenario when such conditions do not hold. To deal with potential state estimation uncertainty, we derive an exact characterization of the set of all possible state estimates. Using the framework of control barrier functions, we propose design principles for system safety in offline and online phases. For the offline phase, we derive conditions on safe sets for all possible sensor attacks that may be encountered during system deployment. For the online phase, with past system measurements collected, a quadratic program-based safety filter is proposed to enforce system safety. A 2D-vehicle example is used to illustrate the theoretical results.

eess.SY↗

arXiv2024

Meanings and Feelings of Large Language Models: Observability of Latent States in Generative AI

We tackle the question of whether Large Language Models (LLMs), viewed as dynamical systems with state evolving in the embedding space of symbolic tokens, are observable. That is, whether there exist multiple 'mental' state trajectories that yield the same sequence of generated tokens, or sequences that belong to the same Nerode equivalence class ('meaning'). If not observable, mental state trajectories ('experiences') evoked by an input ('perception') or by feedback from the model's own state ('thoughts') could remain self-contained and evolve unbeknown to the user while being potentially accessible to the model provider. Such "self-contained experiences evoked by perception or thought" are akin to what the American Psychological Association (APA) defines as 'feelings'. Beyond the lexical curiosity, we show that current LLMs implemented by autoregressive Transformers cannot have 'feelings' according to this definition: The set of state trajectories indistinguishable from the tokenized output is a singleton. But if there are 'system prompts' not visible to the user, then the set of indistinguishable trajectories becomes non-trivial, and there can be multiple state trajectories that yield the same verbalized output. We prove these claims analytically, and show examples of modifications to standard LLMs that engender such 'feelings.' Our analysis sheds light on possible designs that would enable a model to perform non-trivial computation that is not visible to the user, as well as on controls that the provider of services using the model could take to prevent unintended behavior.

cs.AI↗

arXiv2024

Heat Death of Generative Models in Closed-Loop Learning

Improvement and adoption of generative machine learning models is rapidly accelerating, as exemplified by the popularity of LLMs (Large Language Models) for text, and diffusion models for image generation. As generative models become widespread, data they generate is incorporated into shared content through the public web. This opens the question of what happens when data generated by a model is fed back to the model in subsequent training campaigns. This is a question about the stability of the training process, whether the distribution of publicly accessible content, which we refer to as "knowledge", remains stable or collapses. Small scale empirical experiments reported in the literature show that this closed-loop training process is prone to degenerating. Models may start producing gibberish data, or sample from only a small subset of the desired data distribution (a phenomenon referred to as mode collapse). So far there has been only limited theoretical understanding of this process, in part due to the complexity of the deep networks underlying these generative models. The aim of this paper is to provide insights into this process (that we refer to as "generative closed-loop learning") by studying the learning dynamics of generative models that are fed back their own produced content in addition to their original training dataset. The sampling of many of these models can be controlled via a "temperature" parameter. Using dynamical systems tools, we show that, unless a sufficient amount of external data is introduced at each iteration, any non-trivial temperature leads the model to asymptotically degenerate. In fact, either the generative distribution collapses to a small set of outputs or becomes uniform over a large set of outputs.

cs.LG↗

arXiv2024

A Framework for Time-Varying Optimization via Derivative Estimation

Optimization algorithms have a rich and fundamental relationship with ordinary differential equations given by its continuous-time limit. When the cost function varies with time -- typically in response to a dynamically changing environment -- online optimization becomes a continuous-time trajectory tracking problem. To accommodate these time variations, one typically requires some inherent knowledge about their nature such as a time derivative. In this paper, we propose a novel construction and analysis of a continuous-time derivative estimation scheme based on "dirty-derivatives", and show how it naturally interfaces with continuous-time optimization algorithms using the language of ISS (Input-to-State Stability). More generally, we show how a simple Lyapunov redesign technique leads to provable suboptimality guarantees when composing this estimator with any well-behaved optimization algorithm for time-varying costs.

math.OC↗

arXiv2023

Taming AI Bots: Controllability of Neural States in Large Language Models

We tackle the question of whether an agent can, by suitable choice of prompts, control an AI bot to any state. To that end, we first introduce a formal definition of ``meaning'' that is amenable to analysis. Then, we characterize ``meaningful data'' on which large language models (LLMs) are ostensibly trained, and ``well-trained LLMs'' through conditions that are largely met by today's LLMs. While a well-trained LLM constructs an embedding space of meanings that is Euclidean, meanings themselves do not form a vector (linear) subspace, but rather a quotient space within. We then characterize the subset of meanings that can be reached by the state of the LLMs for some input prompt, and show that a well-trained bot can reach any meaning albeit with small probability. We then introduce a stronger notion of controllability as {\em almost certain reachability}, and show that, when restricted to the space of meanings, an AI bot is controllable. We do so after introducing a functional characterization of attentive AI bots, and finally derive necessary and sufficient conditions for controllability. The fact that AI bots are controllable means that an adversary could steer them towards any state. However, the sampling process can be designed to counteract adverse actions and avoid reaching undesirable regions of state space before their boundary is crossed.

cs.AI↗

arXiv2022

Joint Continuous and Discrete Model Selection via Submodularity

In model selection problems for machine learning, the desire for a well-performing model with meaningful structure is typically expressed through a regularized optimization problem. In many scenarios, however, the meaningful structure is specified in some discrete space, leading to difficult nonconvex optimization problems. In this paper, we connect the model selection problem with structure-promoting regularizers to submodular function minimization with continuous and discrete arguments. In particular, we leverage the theory of submodular functions to identify a class of these problems that can be solved exactly and efficiently with an agnostic combination of discrete and continuous optimization routines. We show how simple continuous or discrete constraints can also be handled for certain problem classes and extend these ideas to a robust optimization framework. We also show how some problems outside of this class can be embedded within the class, further extending the class of problems our framework can accommodate. Finally, we numerically validate our theoretical results with several proof-of-concept examples with synthetic and real-world data, comparing against state-of-the-art algorithms.

math.OC↗

arXiv2022

Let's Talk Through Physics! Covert Cyber-Physical Data Exfiltration on Air-Gapped Edge Devices

Although organizations are continuously making concerted efforts to harden their systems against network attacks by air-gapping critical systems, attackers continuously adapt and uncover covert channels to exfiltrate data from air-gapped systems. For instance, attackers have demonstrated the feasibility of exfiltrating data from a computer sitting in a Faraday cage by exfiltrating data using magnetic fields. Although a large body of work has recently emerged highlighting various physical covert channels, these attacks have mostly targeted open-loop cyber-physical systems where the covert channels exist on physical channels that are not being monitored by the victim. Network architectures such as fog computing push sensitive data to cyber-physical edge devices--whose physical side channels are typically monitored via state estimation. In this paper, we formalize covert data exfiltration that uses existing cyber-physical models and infrastructure of individual devices to exfiltrate data in a stealthy manner, i.e., we propose a method to circumvent cyber-physical state estimation intrusion detection techniques while exfiltrating sensitive data from the network. We propose a generalized model for encoding and decoding sensitive data within cyber-physical control loops. We evaluate our approach on a distributed IoT network that includes computation nodes residing on physical drones as well as on an industrial control system for the control of a robotic arm. Unlike prior works, we formalize the constraints of covert cyber-physical channel exfiltration in the presence of a defender performing state estimation.

cs.CR↗

arXiv2022

From LTL to rLTL Monitoring: Improved Monitorability through Robust Semantics

Runtime monitoring is commonly used to detect the violation of desired properties in safety critical cyber-physical systems by observing its executions. Bauer et al. introduced an influential framework for monitoring Linear Temporal Logic (LTL) properties based on a three-valued semantics: the formula is already satisfied by the given prefix, it is already violated, or it is still undetermined, i.e., it can still be satisfied and violated by appropriate extensions. However, a wide range of formulas are not monitorable under this approach, meaning that they have a prefix for which satisfaction and violation will always remain undetermined no matter how it is extended. In particular, Bauer et al. report that 44% of the formulas they consider in their experiments fall into this category. Recently, a robust semantics for LTL was introduced to capture different degrees by which a property can be violated. In this paper we introduce a robust semantics for finite strings and show its potential in monitoring: every formula considered by Bauer et al. is monitorable under our approach. Furthermore, we discuss which properties that come naturally in LTL monitoring - such as the realizability of all truth values - can be transferred to the robust setting. Lastly, we show that LTL formulas with robust semantics can be monitored by deterministic automata and report on a prototype implementation.

cs.FL↗