A continuous-time delayed observer: The case study of chatter regenerative vibrations

This paper presents the derivation of a novel continuous time-delayed observer. Compared to observers present in the literature, the new formulation includes time-delayed observations. In order to include delayed observations, two gains are introduced, one managing the current error and one the delayed error. The gains are computed by minimising the upper bound of the error and state covariance matrices. The resulting current gain resembles a weighted Kalman gain, while the conceived observer is unbiased. Both time-variant and steady-state observer formulations are derived. The applicability of the two formulations is tested and validated for chatter regenerative vibrations in milling. The observers, tuned on stable machining conditions, are applied even to unstable machining conditions to estimate unknown input cutting forces and tool-tip vibrations. Although the time-varying observer performs better, both observers obtained excellent results, with root mean square estimation errors below 68.7 N and 97.9 N under stable and unstable conditions, respectively (normalised correlations were above 99.7% and 97.9%).