Episodic Gaussian Process-Based Learning Control with Vanishing Tracking Errors

Due to the increasing complexity of technical systems, accurate first principle models can often not be obtained. Supervised machine learning can mitigate this issue by inferring models from measurement data. Gaussian process regression is particularly well suited for this purpose due to its high data-efficiency and its explicit uncertainty representation, which allows the derivation of prediction error bounds. These error bounds have been exploited to show tracking accuracy guarantees for a variety of control approaches, but their direct dependency on the training data is generally unclear. We address this issue by deriving a Bayesian prediction error bound for GP regression, which we show to decay with the growth of a novel, kernel-based measure of data density. Based on the prediction error bound, we prove time-varying tracking accuracy guarantees for learned GP models used as feedback compensation of unknown nonlinearities, and show to achieve vanishing tracking error with increasing data density. This enables us to develop an episodic approach for learning Gaussian process models, such that an arbitrary tracking accuracy can be guaranteed. The effectiveness of the derived theory is demonstrated in several simulations

Evaluation of an offshore wind farm computational fluid dynamics model against operational site data

Modelling wind turbine wake effects at a range of wind speeds and directions with actuator disk (AD) models can provide insight but also be challenging. With any model it is important to quantify the level of error, but this can also present a challenge when comparing a steady-state model to measurement data with scatter. This paper models wind flow in a wind farm at a range of wind speeds and directions using an AD implementation. The results from these models are compared to data collected from the actual farm being modelled. An extensive comparison is conducted, constituted from 35 cases where two turbulence models, the standard k-ε and k-ω SST are evaluated. The steps taken in building the models as well as processes for comparing the AD computational fluid dynamics (CFD) results to real-world data using the regression models of ensemble bagging and Gaussian process are outlined. Turbine performance data and boundary conditions are determined using the site data. Modifications to an existing opensource AD code are shown so that the predetermined turbine performance can be implemented into the CFD model. Steady state solutions are obtained with the OpenFOAM CFD solver. Results are compared in terms of velocity deficit at the measurement locations. Using the standard k-ε model, a mean absolute error for all cases together of roughly 8% can be achieved, but this error changes for different directions and methods of evaluating it

Uncertainty quantification in DIC with Kriging regression

© 2015 Elsevier Ltd. All rights reserved. A Kriging regression model is developed as a post-processing technique for the treatment of measurement uncertainty in classical subset-based Digital Image Correlation (DIC). Regression is achieved by regularising the sample-point correlation matrix using a local, subset-based, assessment of the measurement error with assumed statistical normality and based on the Sum of Squared Differences (SSD) criterion. This leads to a Kriging-regression model in the form of a Gaussian process representing uncertainty on the Kriging estimate of the measured displacement field. The method is demonstrated using numerical and experimental examples. Kriging estimates of displacement fields are shown to be in excellent agreement with 'true' values for the numerical cases and in the experimental example uncertainty quantification is carried out using the Gaussian random process that forms part of the Kriging model. The root mean square error (RMSE) on the estimated displacements is produced and standard deviations on local strain estimates are determined

Simultaneous inference for Berkson errors-in-variables regression under fixed design

In various applications of regression analysis, in addition to errors in the dependent observations also errors in the predictor variables play a substantial role and need to be incorporated in the statistical modeling process. In this paper we consider a nonparametric measurement error model of Berkson type with fixed design regressors and centered random errors, which is in contrast to much existing work in which the predictors are taken as random observations with random noise. Based on an estimator that takes the error in the predictor into account and on a suitable Gaussian approximation, we derive %uniform confidence statements for the function of interest. In particular, we provide finite sample bounds on the coverage error of uniform confidence bands, where we circumvent the use of extreme-value theory and rather rely on recent results on anti-concentration of Gaussian processes. In a simulation study we investigate the performance of the uniform confidence sets for finite samples

Gaussian Processes with Errors in Variables: Theory and Computation

Covariate measurement error in nonparametric regression is a common problem in nutritional epidemiology and geostatistics, and other fields. Over the last two decades, this problem has received substantial attention in the frequentist literature. Bayesian approaches for handling measurement error have only been explored recently and are surprisingly successful, although the lack of a proper theoretical justification regarding the asymptotic performance of the estimators. By specifying a Gaussian process prior on the regression function and a Dirichlet process Gaussian mixture prior on the unknown distribution of the unobserved covariates, we show that the posterior distribution of the regression function and the unknown covariates density attain optimal rates of contraction adaptively over a range of H\"{o}lder classes, up to logarithmic terms. This improves upon the existing classical frequentist results which require knowledge of the smoothness of the underlying function to deliver optimal risk bounds. We also develop a novel surrogate prior for approximating the Gaussian process prior that leads to efficient computation and preserves the covariance structure, thereby facilitating easy prior elicitation. We demonstrate the empirical performance of our approach and compare it with competitors in a wide range of simulation experiments and a real data example

Development of A Tool Condition Monitoring System for Flank Wear in Turning Process Using Machine Learning

Computer-numerical control (CNC) machining has become the norm in most manufacturing businesses. In order to maximize machining efficiency, prevent unintended damage, and maintain product quality all at once, tool wear measurement is essential. Wear on the tools is typically an inevitable side effect of machining. Because it endangers the machining process, flank wear, the most frequent type of tool wear, should be avoided. This study's objective is to fill this growing need by developing a system that can use machine learning to monitor tool condition during the turning process using MATLAB. The regression method with boosted decision trees and SVR with Gaussian kernels are applied to predict the flank wear based on the vibration signal and cutting parameters. This study found that the regression with boosted decision tree method has a lower mean average percentage error, 6.43%, while SVR is 10.11%. Plus, R2 for regression is slightly better than SVR. It shows that the system successfully produced an accurate prediction of flank wear