Data-Driven Parameter Estimation of Time Delay Dynamical Systems for Stability Prediction
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.
Delay differential equations
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