Linear time-invariant (LTI) controllers suffer from inherent limitations as the waterbed effect and Bode's gain-phase relationship. Reset control, a nonlinear strategy involving the reset of linear controllers, offers a potential solution to overcome these traditional LTI limitations. Based on the describing function analysis, the reset controller provides the advantage of less phase lag compared to its linear counterpart. In this study, we explore the application of the Proportional Clegg Integrator (PCI), a reset element, to replace the Proportional Integrator (PI) within the PID framework for a wire bonder, aiming to break LTI limitations and therefore improve the system's performance. However, due to the reset action of the PCI, it cannot provide a constant buffer force as the PI does to compensate for the machine's nonlinearity and external disturbances. This lack of compensation can lead to undesired limit-cycling behavior, preventing the system from achieving zero steady-state error. To address this issue, a PI is added after the PCI to form a PCI-PID framework for completely eliminating the limit cycles. To optimize the performance of the PCI within the PCI-PID framework, a tuning algorithm is proposed, leveraging the machine's frequency response function (FRF) data. Frequency-domain analytical tools, including open-loop Higher-Order Sinusoidal Input Describing Functions (HOSIDFs) and the pseudo-sensitivity function from approximate closed-loop HOSIDFs, are applied during the optimization process. However, the introduction of the PI for the PCI leads to a trade-off, causing a reduction in phase margin or a decrease in the nonlinear benefits gained from the PCI. To overcome this limitation, a novel integrator called the Generalized First Order Reset Element-based Integrator (GFbI) is introduced. The GFbI, as a single integrator, has the ability to internally incorporate a PI within its structure, allowing it to achieve the desired zero steady-state error without externally introducing an additional linear integrator. This unique characteristic allows the GFbI to attain zero steady-state error, without being confined by the limitations associated with introducing an additional PI. Furthermore, the GFbI can be automatically tuned based on a proposed constraint regarding the number of reset instants, permitting the use of machines' FRF data. Finally, with the novel GFbI structure and its tuning algorithm, the experimental data obtained from an industrial motion platform demonstrates a decrease (30.8%) in the root-mean-square of the settling error.