Browsing by Subject "Dynamical systems"

It is currently an exciting time to be doing research at the intersection of sports and engineering. Advances in wearable sensor technology now enable large quantities of physiological and biomechanical data to be collected from athletes with minimal obstruction and cost. These technological advances, combined with an increased public awareness of the relationship between exercise, fitness, and health, has created an environment where engineering principles can be integrated with biomechanics, exercise physiology, and sports science to dramatically improve methods for physiological assessment, injury prevention, and athletic performance.

The first part of this dissertation develops a new method for analyzing heart rate (HR) and oxygen uptake (VO2) dynamics. A dynamical system model was derived based on the equilibria and stability of the HR and VO2 responses. The model accounts for nonlinear phenomena and person-specific physiological characteristics. A heuristic parameter estimation algorithm was developed to determine model parameters from experimental data. An artificial neural network (ANN) was developed to predict VO2 from HR and exercise intensity data. A series of experiments was performed to validate: 1) the ability of the dynamical system model to make accurate time series predictions for HR and VO2; 2) the ability of the dynamical system model to make accurate submaximal predictions for maximum heart rate (HRmax) and maximal oxygen uptake (VO2max); 3) the ability of the ANN to predict VO2 from HR and exercise intensity data; and 4) the ability of a system comprising an ANN, dynamical system model, and heuristic parameter estimation algorithm to make submaximal predictions for VO2max without requiring VO2 data collection. The dynamical system model was successfully validated through comparisons with experimental data. The model produced accurate time series predictions for HR and VO2 and, more importantly, the model was able to accurately predict HRmax and VO2max using data collected during submaximal exercise. The ANN was successfully able to predict VO2 responses using HR and exercise intensity as system inputs. The system comprising an ANN, dynamical system model, and heuristic parameter estimation algorithm was able to make accurate submaximal predictions for VO2max without requiring VO2 data collection.

The second part of this dissertation applies a support vector machine (SVM) to classify lower extremity movement patterns that are associated with increased lower extremity injury risk. Participants for this study each performed a jump-landing task, and experimental data was collected using two video cameras, two force plates and a chest-mounted single-axis accelerometer. The video data was evaluated to classify the lower extremity movement patterns of the participants as either excellent or poor using the Landing Error Scoring System (LESS) assessment method. Two separate linear SVM classifiers were trained using the accelerometer data and the force plate data, respectively, with the LESS assessment providing the classification labels during training and evaluation. The same participants from this study also performed several bouts of treadmill running, and an additional set of linear SVM classifiers were trained using accelerometer data and gyroscope data to classify movement patterns, with the LESS assessment again providing the classification labels during training and evaluation. Both sets of SVM's performed with a high level of accuracy, and the objective and autonomous nature of the SVM screening methodology eliminates the subjective limitations associated with many current clinical assessment tools.

This work introduces geometric and topological data analysis (TDA) tools that can be used in conjunction with sliding window transformations, also known as delay-embeddings, for discovering structure in time series and dynamical systems in an unsupervised or supervised learning framework. For signals of unknown period, we introduce an intuitive topological method to discover the period, and we demonstrate its use in synthetic examples and real temperature data. Alternatively, for almost-periodic signals of known period, we introduce a metric called Geometric Complexity of an Almost Periodic signal (GCAP), based on a topological construction, which allows us to continuously measure the evolving variation of its periods. We apply this method to temperature data collected from over 200 weather stations in the United States and describe the novel patterns that we observe. Next, we show how geometric and TDA tools can be used in a supervised learning framework. Seizure-detection using electroencephalogram (EEG) data is formulated as a binary classification problem. We define new collections of geometric and topological features of multi-channel data, which utilizes temporal and spatial context of EEG, and show how it results in better overall performance of seizure detection than using the usual time-domain and frequency domain features. Finally, we introduce a novel method to sonify persistence diagrams, and more generally any planar point cloud, using a modified version of the harmonic table. This auditory display can be useful for finding patterns that visual analysis alone may miss.

When simulating multiscale stochastic differential equations (SDEs) in high-dimensions, separation of timescales and high-dimensionality can make simulations expensive. The computational cost is dictated by microscale properties and interactions of many variables, while interesting behavior often occurs on the macroscale with few important degrees of freedom. For many problems bridging the gap between the microscale and macroscale by direct simulation is computationally infeasible, and one would like to learn a fast macroscale simulator. In this paper we present an unsupervised learning algorithm that uses short parallelizable microscale simulations to learn provably accurate macroscale SDE models. The learning algorithm takes as input: the microscale simulator, a local distance function, and a homogenization scale. The learned macroscale model can then be used for fast computation and storage of long simulations. I will discuss various examples, both low- and high-dimensional, as well as results about the accuracy of the fast simulators we construct, and its dependency on the number of short paths requested from the microscale simulator.

Subtractive machining operations such as milling, turning, and drilling are an essential part of many manufacturing processes. Unfortunately, under certain combinations of machine settings, the motion of the cutting tool can become unstable, due to feedback between consecutive passes of the tool. This phenomenon is known as chatter. Mathematical models, specifically delay differential equations (DDEs), can describe the motion of the cutting tool and predict this instability. While these models are useful, estimates of the models' parameters are necessary in order to apply them to real systems. Unfortunately, estimating the parameters directly can be time-consuming, expensive, and difficult. The objective of this research is to develop automated methods to estimate these parameters indirectly, from time series measurements of the tool's motion which can be collected in a few minutes with sensors attached to the machine. The estimated parameters can then be used to predict when chatter will occur so that the machine operator can select appropriate settings.

One way to estimate the parameters of a dynamics model is to match the characteristic multipliers (CMs) predicted by the model to CMs estimated from time series data. CMs describe the behavior, such as stability, of a dynamical system near a limit cycle. While existing CM estimation methods are available, practical challenges such as measurement noise, limited time series length, and repeated CMs can substantially reduce their accuracy. The first part of this dissertation presents improved methods for estimating CMs from time series. Numerical validation studies demonstrate that the improved methods consistently provide more accurate CM estimates than existing methods in a variety of scenarios.

The second part of this dissertation introduces improvements to CM matching and trajectory matching methods for estimating the parameters of DDEs from noisy time series data. For CM matching, it incorporates the empirical CM estimation improvements from the previous part, and it introduces a way to match multiple CM estimates for each time series. For trajectory matching, it describes how to handle multivariate observations and prior knowledge in a principled way; it uses the spectral element method to provide a convenient representation of the initial interval and reduce the computational cost of computing the objective function; and it fits multiple time series simultaneously. Simulation results demonstrate that these improved methods work well in practice, although CM matching has some limitations which are not a problem for the trajectory matching method.

The final part of this dissertation introduces a new approach to estimate the parameters of a DDE model for milling from noisy time series data, based on the trajectory matching approach described in the previous part. It extends models from the literature to more closely fit the time series data, and it describes a procedure to estimate the unknown parameters in stages, without having to solve a global optimization algorithm for all the parameters simultaneously. Additionally, it adapts the spectral element method to make predictions for this model. Experimental results using time series data collected on an instrumented milling machine demonstrate that the model and fitting procedure successfully estimate parameters for which the predicted stability boundaries approximate the true stability boundaries.

High dimensional random dynamical systems are ubiquitous, including-but not limited to- cyber-physical systems, daily return on different stocks of S\&P 1500 and velocity profile of interacting particle systems around McKeanVlasov limit. Mathematically speaking, observed time series data can be captured via a stable $n-$ dimensional linear transformation `$A$' and additive randomness. System identification aims at extracting useful information about underlying dynamical system, given a length $N$ trajectory from it (corresponds to an $n \times N$ dimensional data matrix). We use spectral theorem for non-Hermitian operators to show that spatio-temperal correlations are dictated by the \emph{discrepancy between algebraic andgeometric multiplicity of distinct eigenvalues} corresponding to state transition matrix. Small discrepancies imply that original trajectory essentially comprises of multiple \emph{lower dimensional random dynamical systems living on $A$ invariant subspaces and are statistically independent of each other}. In the process, we provide first quantitative handle on decay rate of finite powers of state transition matrix $\|A^{k}\|$ . It is shown that when a stable dynamical system has only one distinct eigenvalue and discrepancy of $n-1$: $\|A\|$ has a dependence on $n$, resulting dynamics are \emph{spatially inseparable} and consequently there exist at least one row with covariates of typical size $\Theta\big(\sqrt{N-n+1}$ $e^{n}\big)$ i.e., even under stability assumption, covariates can \emph{suffer from curse of dimensionality }.

In the light of these findings we set the stage for non-asymptotic error analysis in estimation of state transition matrix $A$ via least squares regression on observed trajectory by showing that element-wise error is essentially a variant of well-know Littlewood-Offord problem and(can be extremely sensitive to dimension of the state space and number of iterations). We also show that largest singular value of the data matrix can be cursed by dimensionality even when state-transition matrix is stable. Overarching theme of this thesis is new theoretical results on spectral theorem for non-Hermitian operators, non-asymptotic behavior of high dimensional dynamical systems , which we incorporate with the work of Talagrand on concentration of measure phenomenon to better understand behavior of the structured random matrices(data matrix) and subsequently the performance of different learning algorithms with dependent data. Besides, we also show that there exists stable linear Gaussians with process level Talagrands' inequality linear in dimension of the state space(previously an open problem), along with deterioration of mixing times with increase in discrepancy between algebraic and geometric multiplicity of $A$.

Within the past decade, advances in small-scale electronics have reduced power consumption requirements such that mechanisms for harnessing ambient kinetic energy for self-sustenance are a viable technology. Such devices, known as energy harvesters, may enable self-sustaining wireless sensor networks for applications ranging from Tsunami warning detection to environmental monitoring to cost-effective structural health diagnostics in bridges and buildings. In particular, flexible electroelastic materials such as lead-zirconate-titanate (PZT) are sought after in designing such devices due to their superior efficiency in transforming mechanical energy into the electrical domain in comparison to induction methods. To date, however, material and dynamic nonlinearities within the most popular type of energy harvester, an electroelastically laminated cantilever beam, has received minimal attention in the literature despite being readily observed in laboratory experiments.

In the first part of this dissertation, an experimentally validated first-principles based modeling framework for quantitatively characterizing the intrinsic nonlinearities and moderately large amplitude response of a cantilevered electroelastic generator is developed. Nonlinear parameter identification is facilitated by an analytic solution for the generator's dynamic response alongside experimental data. The model is shown to accurately describe amplitude dependent frequency responses in both the mechanical and electrical domains and implications concerning the conventional approach to resonant generator design are discussed. Higher order elasticity and nonlinear damping are found to be critical for correctly modeling the harvester response while inclusion of a proof mass is shown to invigorate nonlinearities a much lower driving amplitudes in comparison to electroelastic harvesters without a tuning mass.

The second part of the dissertation concerns dynamical systems design to purposefully engage nonlinear phenomena in the mechanical domain. In particular, two devices, one exploiting hysteretic nonlinearities and the second featuring homoclinic bifurcation are investigated. Both devices exploit nonlinear magnet interactions with piezoelectric cantilever beams and a first principles modeling approach is applied throughout. The first device is designed such that both softening and hardening nonlinear resonance curves produces a broader response in comparison to the linear equivalent oscillator. The second device makes use of a supercritical pitchfork bifurcation wrought by nonlinear magnetic repelling forces to achieve a bistable electroelastic dynamical system. This system is also analytically modeled, numerically simulated, and experimentally realized to demonstrate enhanced capabilities and new challenges. In addition, a bifurcation parameter within the design is examined as a either a fixed or adaptable tuning mechanism for enhanced sensitivity to ambient excitation. Analytical methodologies to include the method of Harmonic Balance and Melnikov Theory are shown to provide superior insight into the complex dynamics of the bistable system in response to deterministic and stochastic excitation.

We consider stochastic hybrid systems that stem from evolution equations with right-hand sides that stochastically switch between a given set of right-hand sides. To begin our study, we consider a linear ordinary differential equation whose right-hand side stochastically switches between a collection of different matrices. Despite its apparent simplicity, we prove that this system can exhibit surprising behavior.

Next, we construct mathematical machinery for analyzing general stochastic hybrid systems. This machinery combines techniques from various fields of mathematics to prove convergence to a steady state distribution and to analyze its structure.

Finally, we apply the tools from our general framework to partial differential equations with randomly switching boundary conditions. There, we see that these tools yield explicit formulae for statistics of the process and make seemingly intractable problems amenable to analysis.