Iterative Machine Learning for Output Tracking

Abstract

This article develops iterative machine learning (IML) for output tracking. The inputoutput data generated during iterations to develop the model used in the iterative update. The main contribution of this article to propose the use of kernel-based machine learning to iteratively update both the model and the model-inversion-based input simultaneously. Additionally, augmented inputs with persistency of excitation are proposed to promote learning of the model during the iteration process. The proposed approach is illustrated with a simulation example. §1. IntroductionIterative learning methods, initially developed in e.g.,, 1)-3) improve the outputtracking performance by correcting the input based on the measured tracking error. For example, iterative control has led to some of the highest precision for output tracking, e.g., in scanning probe microscopy as demonstrated in, e.g.,. 4)-10) Note that sets of learned trajectories can be used to enable tracking of other trajectories, e.g., by designing the feedforward control input using pre-specified basis functions e.g., using polynomial functions 11) or rational functions of the reference trajectory. 12) Similarly, new desired output can be generated by considering different combinations of previously-learned output segments 13) and the feedforward can be represented using radial basis functions that can be optimized for a range of task parameters. 14) This article proposes a kernel-based machine learning approach to use augmented inputs to iteratively learn not just the inverse input u d needed to track a specified output y d but to also use the data acquired during the iteration process to to estimate both (i) the modelĝ (and its inverse) for the control update, as well as (ii) the model uncertainty needed to establish bounds on the iteration gain for ensuring trackingerror reduction..Conditions for convergence of iterative methods have been well studied in literature, e.g.,. 15)-24) For example, the need to invert the system g to find the perfect input u d for tracking a desired output y d has motivated the use of the inversê g −1 of the known modelĝ of the system g in early iterative control development. 3) Since convergence depends on the size of modeling error, improvements of the model through parameter adaptation with data acquired during the iteration was studied in, e.g., 25) for robotics application using a discrete-time implementation. Here, for each iteration step, the sampled input vector is mapped to the sampled output vector through a lower triangular matrix map. A stochastic version using such a lower-triangular map has been studied in. 26) Even in the ideal case with no mod- * ) S. Devasia is with the Mechanical Engineering Department, U.

Iterative machine learning for precision trajectory tracking with series elastic actuators

“…This has motivated approaches that use the data obtained during the iteration process without an explicit model to improve convergence [22]. Additionally, kernel-based Gaussian process regression has been used to update the models using iteration data [23] and such iterative machine learning (IML) approaches have been experimentally evaluated in [24]. The extension of such IML methods for finding the feedforward input to improve the precision when using SEAs is investigated in this article.…”

Application of Norm Optimal Iterative Learning Control to Quadrotor Unmanned Aerial Vehicle for Monitoring Overhead Power System

“…The properties of linear optimal algorithms have been studied extensively [33][34][35][36][37]. Leading ILC examples are now introduced with their own specific features.…”

MIMO ILC for Precision SEA robots using Input-weighted Complex-Kernel Regression

“…The initial input I 0 = O d was applied to the SEA robot and the output O 0 was measured. 20), and the staircase functions consist of three consecutive 5-second steps starting from t = t H with magnitude equaling to h a , h b and h c , and are zero otherwise. The parameters for each staircase function H j,p are tabulated in Table 2.…”

“…An approach is to not iterate at those specific frequencies where the desired output is small, e.g., [17,19]. Another approach is to inject additional input at such frequencies (where the desired output is small) to ensure persistence of excitation when estimating the model from data, which enables the learned model to be portable and applicable to track new trajectories, e.g., [20]. In either case, provided the uncertainty is sufficiently small, the ILC converges to the input needed for exact output tracking for SISO systems, even in the presence of modeling uncertainties.…”

“…One approach is to use a relatively large number of repetitive experiments for identifying both the model and the uncertainty prior to applying the ILC [15,24]. An alternative approach is to use kernel-based regression approaches to estimate the model and its uncertainty for the SISO case [20] from data obtained from a relativelysparse number (one or two) of the ILC iterations. This use of ILC data allows the data-based model to capture the local, linear model near the work area that includes potential contact-dependent effects, which can be challenging to model from first principles.…”