Two time-varying linear state-space representations of the generally accepted physicochemical model (PCM) of a lithium-ion cell are used to estimate local and global states during different charging scenarios. In terms of computational speed and suitability towards recursive state observer models, the solid-phase diffusion in the PCM of an exemplaric MCMB/LiCoO2 lithium-ion cell is derived with the aid of two different numerical reduction methods in the form of a Polynomial Profile and an Eigenfunction Method. As a benchmark, the PCM using the original Duhamel Superposition Integral approximation serves for the comparison of accuracy and computational speed. A modified spatial discretization via the finite volume method improves handling of boundary conditions and guarantees accurate simulation results of the PCM even at a low level of spatial discretization. The Polynomial Profile allows for a significant speed-up in computational time whilst showing a poor prediction accuracy during dynamic load profiles. The Eigenfunction Method shows a comparable accuracy as the benchmark for all load profiles whilst resulting in an even higher computational effort. The two derived observer models incorporate the state-space representation of the reduced PCM applying both the Polynomial and Eigenfunction approach combined with an Extended Kalman Filter algorithm based on a novel initialization algorithm and conservation of lithium mass. The estimation results of both models show robust and quick reduction of the residual errors for both local and global states when considering the applied current and the resulting cell voltage of the benchmark model, as the underlying measurement signal. The carried out state estimation for a 4C constant charge current showed a regression of the cell voltage error to 1 mV within 30 s with an initial SOC error of 42.4% under a standard deviation of 10 mV and including process noise.