Cramér-Rao Lower Bounds for Battery Estimation

Other conference contribution, 2016

To secure safety, reliability and performance of an electri- fied vehicle, it is important to monitor the State of Charge (SoC) of its battery. Today, there are no sensors that can measure SoC directly. Instead, it is usually estimated with an algorithmic filter. Since batteries are nonlinear, all feasible filters are only able to approximate the posterior densities which, in other words, means that their perfor- mances will be more or less suboptimal (Särkkä, 2013). To be able to evaluate the performance of a filter, it is of great value to know how well a parameter or a state can be estimated. It can then be decided if it is worth spending time on tuning the filter, or implementing a more advanced filter. Furthermore, analyzing the achievable accuracy can be a way to better understand the application. One suitable measure for benchmarking the performance is the Cramér-Rao Lower Bound (CRLB), which is a lower bound on the Mean Square Error (MSE) of any estimator. In this paper we adopt a method to numerically determine the posterior CRLBs with a Monte Carlo-based algorithm. The posterior CRLBs are calculated for combined esti- mation of the states and the parameters of a commonly used equivalent circuit model. It is investigated how the posterior CRLBs depend on the amplitude and the fre- quency of the current. Furthermore, the posterior CRLBs are computed for a commercially available lithium- ion battery using data from laboratory experiments, and the results are compared to the MSEs of an Extended Kalman Filter (EKF). It is shown that the MSEs of the EKF are close to the posterior CRLBs, which means that the EKF seems to be a good observer for this application.