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Delays endanger safety of autonomous systems operating in a rapidly changing environment, such as nondeterministic surrounding traffic participants in autonomous driving and high-speed racing. Unfortunately, delays are typically not considered during the conventional controller design or learning-enabled controller training phases prior to deployment in the physical world. In this paper, the computation delay from nonlinear optimization for motion planning and control, as well as other unavoidable delays caused by actuators, are addressed systematically and unifiedly. To deal with all these delays, in our framework: 1) we propose a new filtering approach with no prior knowledge of dynamics and disturbance distribution to adaptively and safely estimate the time-variant computation delay; 2) we model actuation dynamics for steering delay; 3) all the constrained optimization is realized in a robust tube model predictive controller. For the application merits, we demonstrate that our approach is suitable for both autonomous driving and autonomous racing. Our approach is a novel design for a standalone delay compensation controller. In addition, in the case that a learning-enabled controller assuming no delay works as a primary controller, our approach serves as the primary controller's safety guard.