According to some aspects, methods and systems are presented to reduce noise, vibration, and harshness during start or restart of an engine. In some embodiments, a torque source such as an electric machine provides a torque to an internal combustion engine during restart to counteract vibrations of the system caused by the output torque of the internal combustion engine. The torque provided by the torque source can be expressed as a sum of a non-linear component and an input shaped component. A perturbation technique can be utilized for separating the scales and isolating the non-linear response of the system. Command shaping can be applied to the remaining, linear response of the system. Parameters used in the modeling of the internal combustion engine and the system may be pre-determined based on vehicle design and operating conditions, or may be iteratively estimated based on previous restarts during vehicle operation.