Robust Time-Delay Filter Design for Precision Motion Control with Nonzero Initial Conditions

This paper presents a comprehensive analysis and design of time-delay filters for systems with nonzero initial conditions, focusing on robustness in the presence of initial positions and velocities. A novel approach is introduced to account for nonzero initial conditions by deriving a closed-form representation of the time-delay filters using Weierstrass substitution in the frequency domain, enabling precise and efficient implementation. The proposed methodology is validated through numerical examples, including a standard second-order form system and a velocity-driven gantry crane payload model, where the robust algorithm is applied to the latter and demonstrates the effectiveness of the filter. This work provides new insights into the role of initial conditions in time-delay filters design for their practical application in systems requiring high precision and robustness.