Phase space warping: nonlinear time-series analysis for slowly drifting systems

2006 ◽  
Vol 364 (1846) ◽  
pp. 2495-2513 ◽  
Author(s):  
D Chelidze ◽  
J.P Cusumano
Keyword(s):  
Phase Space ◽  
Damage Evolution ◽  
Systems Approach ◽  
Reference Model ◽  
Recursive Estimation ◽  
Time To Failure ◽  
Damage State ◽  
Beam Experiment ◽  
Vector Tracking ◽  
Phase Space Warping

A new general dynamical systems approach to data analysis is presented that allows one to track slowly evolving variables responsible for non-stationarity in a fast subsystem. The method is based on the idea of phase space warping , which refers to the small distortions in the fast subsystem's phase space that results from the slow drift, and uses short-time reference model prediction error as its primary measurement of this phenomenon. The basic theory is presented and the issues associated with its implementation in a practical algorithm are discussed. A vector-tracking version of the procedure, based on smooth orthogonal decomposition analysis, is applied to the study of a nonlinear vibrating beam experiment in which a crack propagates to complete fracture. Our method shows that the damage evolution is governed by a scalar process, and we are able to give real-time estimates of the current damage state and identify the governing damage evolution model. Using a final recursive estimation step based on this model, the time to failure is continuously and accurately predicted well in advance of actual failure.

A Dynamical Systems Approach to Failure Prognosis

2004 ◽  
Vol 126 (1) ◽  
pp. 2-8 ◽  
Author(s):  
David Chelidze ◽  
Joseph P. Cusumano
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Damage Evolution ◽  
Systems Approach ◽  
Remaining Useful Life ◽  
Electromechanical System ◽  
Time To Failure ◽  
Damage Tracking ◽  
Failure Prognosis ◽  
Short Time

In this paper, a previously published damage tracking method is extended to provide failure prognosis, and applied experimentally to an electromechanical system with a failing supply battery. The method is based on a dynamical systems approach to the problem of damage evolution. In this approach, damage processes are viewed as occurring in a hierarchical dynamical system consisting of a “fast,” directly observable subsystem coupled to a “slow,” hidden subsystem describing damage evolution. Damage tracking is achieved using a two-time-scale modeling strategy based on phase space reconstruction. Using the reconstructed phase space of the reference (undamaged) system, short-time predictive models are constructed. Fast-time data from later stages of damage evolution of a given system are collected and used to estimate a tracking function by calculating the short time reference model prediction error. In this paper, the tracking metric based on these estimates is used as an input to a nonlinear recursive filter, the output of which provides continuous refined estimates of the current damage (or, equivalently, health) state. Estimates of remaining useful life (or, equivalently, time to failure) are obtained recursively using the current damage state estimates under the assumption of a particular damage evolution model. The method is experimentally demonstrated using an electromechanical system, in which mechanical vibrations of a cantilever beam are dynamically coupled to electrical oscillations in an electromagnet circuit. Discharge of a battery powering the electromagnet (the “damage” process in this case) is tracked using strain gauge measurements from the beam. The method is shown to accurately estimate both the battery state and the time to failure throughout virtually the entire experiment.

Invariant Manifold Detection From Phase Space Trajectories

2008 ◽  
Author(s):  
Joseph Kuehl ◽  
David Chelidze
Keyword(s):  
Phase Space ◽  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Basins Of Attraction ◽  
Detection Methods ◽  
Trajectory Data ◽  
Space Trajectories ◽  
Data Set ◽  
Phase Space Warping ◽  
Phase Space Trajectories

Invariant manifolds provide important information about the structure of flows. When basins of attraction are present, the stable invariant manifold serves as the boundary between these basins. Thus, in experimental applications such as vibrations problems, knowledge of these manifolds is essential to understanding the evolution of phase space trajectories. Most existing methods for identifying invariant manifolds of a flow rely on knowledge of the flow field. However, in experimental applications only knowledge of phase space trajectories is available. We provide modifications to several existing invariant manifold detection methods which enables them to deal with trajectory only data, as well as introduce a new method based on the concept of phase space warping. The method of Stochastic Interrogation applied to the damped, driven Duffing equation is used to generate our data set. The result is a set of trajectory data which randomly populates a phase space. Manifolds are detected from this data set using several different methods. First is a variation on manifold “growing,” and is based on distance of closest approach to a hyperbolic trajectory with “saddle like behavior.” Second, three stretching based schemes are considered. One considers the divergence of trajectory pairs, another quantifies the deformation of a nearest neighbor cloud, and the last uses flow fields calculated from the trajectory data. Finally, the new phase space warping method is introduced. This method takes advantage of the shifting (warping) experienced by a phase space as the parameters of the system are slightly varied. This results in a shift of the invariant manifolds. The region spanned by this shift, provides a means to identify the invariant manifolds. Results show that this method gives superior detection and is robust with respect to the amount of data.

Damage Detection Using Dissimilarity in Phase Space Topology of Dynamic Response of Structure Subjected to Shock Wave Loading

2018 ◽  
Vol 1 (4) ◽  
Author(s):  
Lavish Pamwani ◽  
Amit Shelke
Keyword(s):  
Phase Space ◽  
Damage Detection ◽  
System Analysis ◽  
Singular System ◽  
Second Phase ◽  
Progressive Damage ◽  
Damage State ◽  
Space Topology ◽  
Damage States ◽  
Phase Space Topology

Shockwave is a high pressure and short duration pulse that induce damage and lead to progressive collapse of the structure. The shock load excites high-frequency vibrational modes and causes failure due to large deformation in the structure. Shockwave experiments were conducted by imparting repetitive localized shock loads to create progressive damage states in the structure. Two-phase novel damage detection algorithm is proposed, that quantify and segregate perturbative damage from microscale damage. The first phase performs dimension reduction and damage state segregation using principal component analysis (PCA). In the second phase, the embedding dimension was reduced through empirical mode decomposition (EMD). The embedding parameters were derived using singular system analysis (SSA) and average mutual information function (AMIF). Based, on Takens theorem and embedding parameters, the response was represented in a multidimensional phase space trajectory (PST). The dissimilarity in the multidimensional PST was used to derive the damage sensitive features (DSFs). The DSFs namely: (i) change in phase space topology (CPST) and (ii) Mahalanobis distance between phase space topology (MDPST) are evaluated to quantify progressive damage states. The DSFs are able to quantify the occurrence, magnitude, and localization of progressive damage state in the structure. The proposed algorithm is robust and efficient to detect and quantify the evolution of damage state for extreme loading scenarios.

scholarly journals Application of Phase Space Warping on Damage Tracking for Bearing Fault

2012 ◽  
Vol 364 ◽  
pp. 012025 ◽  
Author(s):  
Bin Fan ◽  
Niaoqing Hu ◽  
Lei Hu ◽  
Fengshou Gu
Keyword(s):  
Phase Space ◽  
Bearing Fault ◽  
Damage Tracking ◽  
Phase Space Warping

Fatigue Detection Using Phase-Space Warping

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Abdullatif Alwasel ◽  
Marcus Yung ◽  
Eihab M. Abdel-Rahman ◽  
Richard P. Wells ◽  
Carl T. Haas
Keyword(s):  
Time Delay ◽  
Phase Space ◽  
Musculoskeletal System ◽  
Physical State ◽  
Muscle Group ◽  
Physiological Signals ◽  
Laboratory Environment ◽  
Sharing Patterns ◽  
Fatigue Detection ◽  
Phase Space Warping

A novel application of phase-space warping (PSW) method to detect fatigue in the musculoskeletal system is presented. Experimental kinematic, force, and physiological signals are used to produce a fatigue metric. The metric is produced using time-delay embedding and PSW methods. The results showed that by using force and kinematic signals, an overall estimate of the muscle group state can be achieved. Further, when using electromyography (EMG) signals the fatigue metric can be used as a tool to evaluate muscles activation and load sharing patterns for individual muscles. The presented method will allow for fatigue evolution measurement outside a laboratory environment, which open doors to applications such as tracking the physical state of players during competition, workers in a plant, and patients undergoing in-home rehabilitation.

Local Flow Variation Method for Damage Identification

2005 ◽  
Author(s):  
Ming Liu ◽  
David Chelidze
Keyword(s):  
Phase Space ◽  
Damage Identification ◽  
Computing Time ◽  
Variation Method ◽  
Fast Time ◽  
Slow Time ◽  
Local Flow ◽  
Flow Variation ◽  
Damage Process ◽  
Phase Space Warping

In this paper, a damage identification method called local flow variation is introduced. It is a practical implementation of a phase space warping concept. A hierarchical dynamical system is considered where a slow-time damage process causes drifts in the parameters of fast-time system describing measurable response of a structure. The method is based on the hypothesis that the probability distribution function of the fast-time trajectory in its phase space is a function of damage state. In this method, an ensemble of estimated expectations of trajectory in different locations of the reconstructed phase space is used as a damage feature vector. Using these feature vectors, damage identification is realized by a smooth orthogonal decomposition. An experiment is conducted to validate the method. A two dimensional slow-time damage process is identified from experimental fast-time data. Although damage identification through the local flow variation is not as accurate as trough phase space warping, the required computing time is about two-orders-of-magnitude shorter.

Tracking Physiological Fatigue in Prolonged Load Carriage Walking Using Phase Space Warping and Smooth Orthogonal Decomposition

2008 ◽  
Author(s):  
David B. Segala ◽  
David Chelidze ◽  
Albert Adams ◽  
Jeffrey M. Schiffman ◽  
Leif Hasselquist
Keyword(s):  
Time Series ◽  
Oxygen Consumption ◽  
Phase Space ◽  
Surface Electromyography ◽  
Orthogonal Decomposition ◽  
Load Carriage ◽  
Leg Muscles ◽  
Smooth Orthogonal Decomposition ◽  
Phase Space Warping ◽  
Gait Variable

The ability to track and predict the onset of physiologic fatigue using easily measurable variables is of great importance to both civilian and military activities. In this paper, biomechanical gait variables are used to reconstruct fatigue evolution in subjects walking with a 40 kg load on a level treadmill for two hours. Fatigue is reconstructed in two steps: (1) phase space warping based feature vectors are estimated from gait variable time series; and (2) smooth orthogonal decomposition is used to extract fatigue related trends from these features. These results are verified using independently obtained measures of fatigue from breath-by-breath oxygen consumption (V˙O2) and surface electromyography (EMG) from a set of leg muscles. V˙O2 based measures for some subjects show no discernable trends. However, for a subject showing monotonically increasing oxygen consumption, the reconstructed dominant fatigue variable closely track V˙O2 measure reflecting global systemic fatigue. For the muscles showing variation in EMG-based fatigue measures, the reconstructed fatigue variables also closely track these local muscle trends. The results show that kinematic angles, which are easier quantities to measure in the field, can be used to track and predict the onset of fatigue.

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