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1. Optimal Selection of Basis Functions for Robust Tracking Control of Uncertain Linear Systems—With Application to Three-Dimensional Printing

   https://doi.org/10.1115/1.4051097  

   Ramani, Keval S.; Okwudire, Chinedum E. (October 2021, Journal of Dynamic Systems, Measurement, and Control)

   null (Ed.)

   Abstract There is growing interest in the use of the filtered basis functions (FBF) approach to track linear systems, especially nonminimum phase (NMP) plants, because of its distinct advantages compared to other tracking control methods in the literature. The FBF approach expresses the control input to the plant as a linear combination of basis functions with unknown coefficients. The basis functions are forward filtered through the plant dynamics, and the coefficients are selected such that tracking error is minimized. Similar to other feedforward control methods, the tracking accuracy of the FBF approach deteriorates in the presence of uncertainties. However, unlike other methods, the FBF approach presents flexibility in terms of the choice of the basis functions, which can be used to improve its accuracy. This paper analyzes the effect of the choice of the basis functions on the tracking accuracy of FBF, in the presence of uncertainties, using the Frobenius norm of the lifted system representation (LSR) of FBF's error dynamics. Based on the analysis, a methodology for optimal selection of basis functions to maximize robustness is proposed, together with an approach to avoid large control effort when it is applied to NMP systems. The basis functions resulting from this process are called robust basis functions. Applied experimentally to a desktop three-dimensional (3D) printer with uncertain NMP dynamics, up to 48% improvement in tracking accuracy is achieved using the proposed robust basis functions compared to B-splines, while utilizing much less control effort. 

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2. Robust Filtered Basis Functions Approach for Feedforward Tracking Control

   https://doi.org/10.1115/DSCC2018-9196  

   Ramani, Keval S.; Okwudire, Chinedum E. (September 2018, ASME 2018 Dynamic Systems and Control Conference)

   This paper proposes a robust filtered basis functions approach for feedforward tracking of linear time invariant systems with dynamic uncertainties. Identical to the standard filtered basis functions (FBF) approach, the robust FBF approach expresses the control trajectory as a linear combination of user-defined basis functions with unknown coefficients. The basis functions are forward filtered using a model of the plant and their coefficients are selected to minimize tracking errors. The standard FBF and robust FBF approaches differ in the filtering process. The robust FBF approach uses an optimal robust filter which is based on minimization of a frequency domain based error cost function over the dynamic uncertainty, whereas, the standard FBF approach uses the nominal model. Simulation examples and experiments on a desktop 3D printer are used to demonstrate significantly more accurate tracking of uncertain plants using robust FBF compared with the standard FBF. 

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3. Optimal Selection of Basis Functions for Minimum-Effort Tracking Control of Nonminimum Phase Systems Using Filtered Basis Functions

   https://doi.org/10.1115/1.4044355  

   Ramani, Keval S.; Duan, Molong; Okwudire, Chinedum E.; Ulsoy, A. Galip (November 2019, Journal of Dynamic Systems, Measurement, and Control)

   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. 

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4. Optimal Selection of Basis Functions for Robust Tracking Control of Linear Systems using Filtered Basis Functions

   https://doi.org/10.23919/ACC45564.2020.9147557  

   Ramani, Keval S.; Okwudire, Chinedum E. (July 2020, 2020 American Control Conference (ACC))

   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. 

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5. A linear hybrid model for enhanced servo error pre-compensation of feed drives with unmodeled nonlinear dynamics

   https://doi.org/https://doi.org/10.1016/j.cirp.2021.04.070  

   Chou, Cheng-Hao; Duan, Molong; Okwudire, Chinedum E. (January 2021, CIRP annals)

   null (Ed.)

   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. 

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