Real-time anomaly detection of the stochastically excited systems on spherical (S2) manifold球面流形随机激励系统的实时异常检测
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Panda S, Fitzgerald B, Hazra B, 2024. Real-time anomaly detection of the stochastically excited systems on spherical (S2) manifold. Probabilistic Engineering Mechanics, 78: 103689.DOI: 10.1016/j.probengmech.2024.103689
先进的分析工具对研究当今复杂系统变化至关重要。实时主测地线分析 (real-time principal geodesic analysis, RPGA) 是一种新技术，为分析可微流形上非线性数据提供了独特视角。传统线性方法在探索此类数据的复杂性时往往不够充分。主成分分析 (principal component analysis, PCA) 和主测地线分析等正交变换技术对于机械、航空航天和土木工程等领域的随机激励系统状态监测非常理想。然而，需要有效的分析方法对不确定性和动态波动进行早期变化检测，以确保安全、性能和成本效益。线性正交变换技术 (如主成分分析及其相应递归技术) 的局限性使其只能应用于平面 Euclid 空间中数据不变的非线性情形。近几十年来，该领域取得了重大进展，数据驱动的实时算法 (如实时主成分分析、递归典型相关分析和 递归奇异谱分析) 为复杂多维问题提供了可靠的求解方案。然而，对于曲面空间，非线性实时主测地线分析技术是核心方法，这主要归因于它可有效地从复杂数据流中提取有意义信息。本文探讨了从线性到非线性数据分析转变的基础概念和方法。通过研究球面随机几何振子和粗糙面导航倒置球形摆车系统等算例，我们说明了实时主测地线分析这一可靠的实时损伤检测技术的重要性。关键词: 状态监测, 流形, 随机动力学, Lie 代数, 主测地线分析, 微分几何Advanced analytical tools have become crucial in today’s constantly changing and complex systems. Real-time Principal Geodesic Analysis (RPGA) is a novel technique that provides a unique perspective for analyzing nonlinear data on differentiable manifolds. Traditional linear methods are often inadequate when exploring the complexities of such data. Orthogonal transformation techniques such as Principal Component Analysis (PCA) and Principal Geodesic Analysis (PGA) are highly desirable for condition monitoring stochastically excited systems in domains like mechanical, aerospace, and civil engineering. However, uncertainties and dynamic fluctuations necessitate robust analytical methods for early change detection to ensure safety, performance, and cost-effectiveness. Limitations posed by linear orthogonal transformation techniques such as PCA and its recursive counterparts make it imperative to adapt these techniques to nonlinear situations where data does not evolve in a flat Euclidean space. Significant advancements have been made in this field over recent decades, with data-driven real-time algorithms such as RPCA, RCCA, and RSSA providing reliable solutions for complex multidimensional problems. However, for curved space, the nonlinear RPGA technique takes center stage. It is known for its effectiveness in extracting meaningful information from the complex data stream. This paper explores the foundational concepts and methodologies underlying the transition from linear to nonlinear data analysis. By examining examples such as stochastic geometric oscillator on S2, and the inverted spherical pendulum cart system navigating a rough surface, we illustrate the significance of reliable, real-time damage detection techniques provided by tools such as RPGA.Keywords: Condition monitoring; Manifolds; Stochastic dynamical; Lie algebra; PGA; Differential geometry.Real-time Principal Geodesic Analysis (RPGA) for manifold-based data is proposed.
RPGA surpasses limitations of linear techniques, enhancing analysis of nonlinear systems.
- Continuous monitoring and adaptation facilitated by RPGA for accurate anomaly detection.
- Numerical validations on diverse systems showcase RPGA’s efficacy in anomaly detection.
Sensitivity analysis is provided that explores parameter variations’ impact on system behavior.
Fig. 1. Computing of the extrinsic mean
Fig. 2. An illustration of the intrinsic mean computation
Fig. 3. A schematic of the proposed framework showcasing the functionality of the condition indicator
图 4: 随机线性系统: 基于实时主测地线分析框架的条件指标表示Fig. 4. Stochastic linear system: Representation of the condition indicator using RPCA framework
图 5: 几何随机线性系统: 基于流形实时主测地线分析框架的条件指标表示Fig. 5. Geometric stochastic linear system: Representation of the condition indicator using manifold based RPGA framework
图 6: 随机摆车系统: 初始条件下三维实数空间中车与二维球面上摆的全局轨迹Fig. 6. Stochastic pendulum cart system: Global trajectory of the cart in R^3 (red line) and pendulum on S^2 (black line) with the initial conditions x (0) = [0, 0, 0], x (0) = [0, 0, 0], q = [0, 0, -1], ω = [0.2, 0.3, 0.5]
图 7: 随机摆车系统: 初始条件下二维球面上摆的局部轨迹Fig. 7. Stochastic pendulum cart system: Local trajectory of the pendulum on S^2 with the initial conditions
图 8: 随机摆车系统: 基于流形实时主测地线分析框架的条件指标表示Fig. 8. Stochastic pendulum cart system: Representation of the condition indicator using manifold based RPGA framework
图 9: 随机摆车系统: 基于流形主测地线分析框架的条件指标表示Fig. 9. Stochastic pendulum cart system: Representation of the condition indicator using manifold based PGA framework
图 10: 随机摆车系统: 20%, 15%, 14%, 10% 损伤敏感性测试Fig. 10. Stochastic pendulum cart system: Damage sensitivity test for 20%, 15%, 14% and 10% damage
作者信息 | Authors
Satyam Panda, 通讯作者 (Corresp.)爱尔兰都柏林三一学院 (Trinity College Dublin) 土木结构与环境工程系爱尔兰都柏林三一学院 (Trinity College Dublin) 土木结构与环境工程系
印度理工学院古瓦哈提分校 (Indian Institute of Technology Guwahati) 土木工程系
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
