Optimal memory-based feedback control for time-delayed systems with application to suppressing regenerative chatter

Chatter, as a self-excited vibration in the milling process, leads to detrimental effects such as increased workpiece surface roughness and tool wear. Therefore, suppressing this phenomenon, characterized as a time-delay system, is critical. This paper proposes an optimal delay-dependent memory-based controller designed using a general Lyapunov functional. The controller incorporates both the current (instantaneous) system state and its time-delayed counterparts as memory-based state feedback, and it is formulated as a Linear Quadratic Regulator (LQR) problem. Optimal strategy is derived from complex partial differential-algebraic Riccati equations and the novelty of this research lies in presenting an efficient approach to obtaining optimal control gains by approximating matrix functions with finite delay-dependent matrices using Linear Matrix Inequalities (LMIs). The proposed controller outperforms memoryless feedback controllers by ensuring asymptotic stability of the closed-loop system and substantially expanding the stability region, enabling greater axial cutting depths at each spindle speed. Moreover, its superiority lies in effectively suppressing chatter under nonlinear dynamic modeling.