We propose a novel fast-charging control framework for lithium-ion (Li-ion) batteries that can leverage a class of models including the high-dimensional, electrochemical-thermal pseudo-two-dimensional model. The control objective is to find the highest battery current while fulfilling various operating constraints. Conventionally, computationally demanding optimization is needed to solve such a constrained optimal control problem when an electrochemical-thermal model is used, leading to practical difficulties in achieving low-cost implementation. Instead, this paper provides an optimization-free solution to Li-ion battery fast charging by converting the constrained optimal control problem into an output tracking problem with multiple tracking references. The required control input, i.e., the charging current, is derived by inverting the battery model. As a result, a nonlinear inversion-based control algorithm is obtained for Li-ion battery fast charging. Results from comparative studies show that the proposed controller can achieve performance close to nonlinear model predictive control but at significantly reduced computational costs and parameter tuning efforts.