Intersections of Dragon Curves of the 18th Iteration
It is shown that intersections occur in the dragon curve up to an unfolding angle of 95.828{\deg}. To this end, two specific edges in the 18th iteration are examined.
math.MG↗
arXiv subjects
Zongyi Guo
Publications and source records attributed to Zongyi Guo.
Avoiding Intersections of Dragon Curves
This article proves that there are no self-intersections in the dragon curve when the unfolding angle is greater than 98.195°. This is shown by constructing a hull for the dragon curve that is mapped onto itself by the generating mappings for the dragon curve. The treatment is purely geometric. The proof is supplemented by a conjecture that reduces the boundary for the unfolding angle to 96.241°.
math.MG↗
Enhancing Robustness of Control Barrier Function: A Reciprocal Resistance-based Approach
In this note, a new reciprocal resistance-based control barrier function (RRCBF) is developed to enhance the robustness of control barrier functions for disturbed affine nonlinear systems, without requiring explicit knowledge of disturbance bounds. By integrating a reciprocal resistance-like term into the conventional zeroing barrier function framework, we formally establish the concept of the reciprocal resistance-based barrier function (RRBF), rigorously proving the forward invariance of its associated safe set and its robustness against bounded disturbances. The RRBF inherently generates a buffer zone near the boundary of the safe set, effectively dominating the influence of uncertainties and external disturbances. This foundational concept is extended to formulate RRCBFs, including their high-order variants. To alleviate conservatism in the presence of complex, time-varying disturbances, we further introduce a disturbance observer-based RRCBF (DO-RRCBF), which exploits disturbance estimates to enhance safety guarantees and recover nominal control performance. The effectiveness of the proposed framework is validated through two simulation studies: a second-order linear system illustrating forward invariance in the phase plane, and an adaptive cruise control scenario demonstrating robustness in systems with high relative degree.
eess.SY↗
Extremum Seeking for PDE Systems using Physics-Informed Neural Networks
Extremum Seeking (ES) is an effective real-time optimization method for PDE systems in cascade with nonlinear quadratic maps. To address PDEs in the feedback loop, a boundary control law and a re-design of the additive probing signal are mandatory. The latter, commonly called "trajectory generation" or "motion planning," involves designing perturbation signals that anticipate their propagation through PDEs. Specifically, this requires solving motion planning problems for systems governed by parabolic and hyperbolic PDEs. Physics-Informed Neural Networks (PINN) is a powerful tool for solving PDEs by embedding physical laws as constraints in the neural network's loss function, enabling efficient solutions for high-dimensional, nonlinear, and complex problems. This paper proposes a novel construction integrating PINN and ES, automating the motion planning process for specific PDE systems and eliminating the need for case-by-case analytical derivations. The proposed strategy efficiently extracts perturbation signals, optimizing the PDE system.
math.OC↗
PUMGPT: A Large Vision-Language Model for Product Understanding
E-commerce platforms benefit from accurate product understanding to enhance user experience and operational efficiency. Traditional methods often focus on isolated tasks such as attribute extraction or categorization, posing adaptability issues to evolving tasks and leading to usability challenges with noisy data from the internet. Current Large Vision Language Models (LVLMs) lack domain-specific fine-tuning, thus falling short in precision and instruction following. To address these issues, we introduce PumGPT, the first e-commerce specialized LVLM designed for multi-modal product understanding tasks. We collected and curated a dataset of over one million products from AliExpress, filtering out non-inferable attributes using a universal hallucination detection framework, resulting in 663k high-quality data samples. PumGPT focuses on five essential tasks aimed at enhancing workflows for e-commerce platforms and retailers. We also introduce PumBench, a benchmark to evaluate product understanding across LVLMs. Our experiments show that PumGPT outperforms five other open-source LVLMs and GPT-4V in product understanding tasks. We also conduct extensive analytical experiments to delve deeply into the superiority of PumGPT, demonstrating the necessity for a specialized model in the e-commerce domain.
cs.CV↗
Maneuvering tracking algorithm for reentry vehicles with guaranteed prescribed performance
This paper presents a prescribed performance-based tracking control strategy for the atmospheric reentry flight of space vehicles subject to rapid maneuvers during flight mission. A time-triggered non-monotonic performance funnel is proposed with the aim of constraints violation avoidance in the case of sudden changes of the reference trajectory. Compared with traditional prescribed performance control methods, the novel funnel boundary is adaptive with respect to the reference path and is capable of achieving stability under disturbances. A recursive control structure is introduced which does not require any knowledge of specific system parameters. By a stability analysis we show that the tracking error evolves within the prescribed error margin under a condition which represents a trade-off between the reference signal and the performance funnel. The effectiveness of the proposed control scheme is verified by simulations.
math.OC↗