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A hybrid filtered basis function (FBF) approach is proposed in this paper for feedforward tracking control of linear systems with unmodeled nonlinear dynamics. Unlike most available tracking control techniques, the FBF approach is very versatile; it is applicable to any type of linear system, regardless of its underlying dynamics. The FBF approach expresses the control input to a system as a linear combination of basis functions with unknown coefficients. The basis functions are forward filtered through a linear model of the system's dynamics and the unknown coefficients are selected such that tracking error is minimized. The linear models used in existing implementations of the FBF approach are typically physics-based representations of the linear dynamics of a system. The proposed hybrid FBF approach expands the application of the FBF approach to systems with unmodeled nonlinearities by learning from data. A hybrid model is formulated by combining a physics-based model of the system's linear dynamics with a data-driven linear model that approximates the unmodeled nonlinear dynamics. The hybrid model is used online in receding horizon to compute optimal control commands that minimize tracking errors. The proposed hybrid FBF approach is shown in simulations on a model of a vibration-prone 3D printer to improve tracking accuracy by up to 65.4%, compared to an existing FBF approach that does not incorporate data. 

Servo error pre-compensation (SEP) is commonly used to improve the accuracy of feed drives. Existing SEP approaches often involve the use of physics-based linear models (e.g., transfer functions) to predict servo errors, but suffer from inaccuracies due to unmodeled nonlinear dynamics in feed drives. This paper proposes a linear hybrid model for SEP that combines physics-based and data-driven linear models. The proposed model is shown to approximate nonlinearities unmodeled in physics-based linear models. In experiments on a precision feed drive, the proposed hybrid model improves the accuracy of servo error prediction by up to 38% compared to a physics-based model. 

Abstract Accurate tracking of nonminimum phase (NMP) systems is well known to require large amounts of control effort. It is, therefore, of practical value to minimize the effort needed to achieve a desired level of tracking accuracy for NMP systems. There is growing interest in the use of the filtered basis functions (FBF) approach for tracking the control of linear NMP systems because of distinct performance advantages it has over other methods. The FBF approach expresses the control input as a linear combination of user-defined basis functions. The basis functions are forward filtered through the dynamics of the plant, and the coefficients are selected such that the tracking error is minimized. There is a wide variety of basis functions that can be used with the FBF approach, but there has been no work to date on how to select the best set of basis functions. Toward selecting the best basis functions, the Frobenius norm of the lifted system representation (LSR) of dynamics is proposed as an excellent metric for evaluating the performance of linear time varying (LTV) discrete-time tracking controllers, like FBF, independent of the desired trajectory to be tracked. Using the metric, an optimal set of basis functions that minimize the control effort without sacrificing tracking accuracy is proposed. The optimal set of basis functions is shown in simulations and experiments to significantly reduce control effort while maintaining or improving tracking accuracy compared to popular basis functions, like B-splines.