Multiple Active Material Lithium-Ion Batteries: Finite-Dimensional Modeling and Constrained State Estimation

Lithium-ion batteries with composite electrodes are being developed in the industry for their improved power, energy, and durability properties over classical single active material accumulators. To make the best out of multiple active material batteries, their management system needs to be fed with reliable information on the internal states, which are not available by measurements. It appears that tools developed for this purpose for single active material batteries are not suitable for multiple active material ones due to their distinctive electrical coupling rules within the electrodes. In this context, we first present a new finite-dimensional model of lithium-ion batteries for which the cathode is represented by two particles made of two different materials, and the anode is described by two graphite particles. We then use the model to design a nonlinear observer based on polytopic techniques that estimate the internal battery variables and thus the state of charge (SOC). The convergence of the observer is guaranteed under a linear matrix inequality (LMI) condition. We then exploit a technique recently proposed in the literature to modify the observer so that its state estimates remain in given plausible range at all times, while preserving its original convergence properties. This allows avoiding aberrant estimated values in the transient and may also favor the future implementation of the observer on embedded devices. Simulation results using literature data illustrate the efficiency of the observer to estimate internal battery variables.