As the core of modern energy storage technology, lithium battery health state estimation is widely used in electric vehicles, mobile devices and other fields. With the growth of the use time its performance degradation problem gradually appears, which not only affects the storage capacity of the battery, but also may lead to safety hazards, so it is particularly important to accurately estimate the health state of lithium batteries. Empirical modeling is a common solution in the theory of lithium battery degradation. The double-exponential model in the empirical model can be evolved to derive the power-exponential driven GM(1,1,$ \mathrm{e}^{\mathit{\lambda}t} $) model. A state-space model is built on the basis of the GM(1,1,$ \mathrm{e}^{\mathit{\lambda}t} $) model, and the extended Kalman filter model is integrated to form the grey extended Kalman filter model. The proposed grey extended Kalman filter model is applied to the problem of nonlinear degradation estimation of the health state of lithium batteries. Publicly available data from the University of Oxford is adopted for single cell batteries. On the basis of a single cell, we use the proposed model in the estimation of lithium battery health state of 20 electric vehicles to further validate the applicability of the proposed model in real-world scenarios.