We propose a novel nonlinear control approach for fast charging of lithium-ion batteries, where health- and safety-related variables, or their time derivatives, are expressed in an input-polynomial form. By converting a constrained optimal control problem into an output tracking problem with multiple tracking references, the required control input, i.e., the charging current, is obtained by computing a series of candidate currents associated with different tracking references. Consequently, an optimization-free nonlinear model inversion-based control algorithm is derived for charging the batteries. We demonstrate the efficacy of our method using a spatially discretized high-fidelity pseudo-two-dimensional (P2D) model with thermal dynamics. Conventional methods require computationally demanding optimization to solve the corresponding fast charging problem for such a high-order system, leading to practical difficulties in achieving low-cost implementation. Results from comparative studies show that the proposed controller can achieve performance very close to nonlinear model predictive control but with much lower computational costs and minimal parameter tuning efforts.