# A Regression-Based Technique for Capacity Estimation of Lithium-Ion Batteries
## 1 Introduction
High computational load and cost are the main obstacles in implementing the NN algorithm. Another limitation to obtaining accurate answers is data training that must work with a considerable amount of different data [17]. Sample entropy (SE) is a method that records oscillation, instability, and complication of the battery's voltage response in capacity fade duration. Therefore, it is a kind of diagnostic tool that can record battery capacity. This method commonly uses other ways to have more accurate results [18]. The third group is empirical-based methods that describe battery behavior by employing phenomenological elements such as equivalent circuit models (ECM). The accuracy of these strategies is not as high as physics-based methods, but the computational load is much lower. In addition, empirically based approaches do not face the aforementioned challenges of the data-driven process. An example of these kinds of models is the impedance spectroscopy method that estimates SOH using a broad frequency spectrum. Computing equivalent circuit model parameters make estimation easier [19]. Algorithms and filters are used to achieve faster and more accurate results. Kalman Filter (KF) is a powerful algorithm that can precisely estimate SOH parameters. For nonlinear systems, the KF is more reasonable. In addition, nonlinear systems use some modified types of this algorithm, such as dual extended Kalman filter (DEKF) and unscented Kalman filter (UKF) [20]. Particle filter (PF) uses sampling and resampling (Sequential Monte Carlo) algorithm, and a collection of weighted particles to solve filtering problems is used to estimate the probability density function (PDF) [21]. The main factor determining whether electric vehicles (EVs) earn their place in daily applications is batteries lifetime. Choosing a mathematical model that models the physical dynamics and constraints of a battery with high precision is crucial to increase estimation accuracy. Therefore, the chosen model should be able to cover errors from uncontrollable working conditions, sensor inaccuracies, and other noise sources to enable an accurate estimation. This paper used an ECM of the battery, which gave the relationship between SOC and the total capacity \(Q\). In addition, the SOC data were obtained by providing voltage and current data as input to this model. Then, this relationship was rearranged to a form of \(y=Qx\) that made it possible to estimate \(Q\) by the regression (least squares) method. Different forms of the least-squares method are proposed in this article, and the results are compared. The main contributions of the proposed methods are as follow:  
* Current measurement error and SOC estimation error are considered in the proposed method. * The proposed methods are closed-form and recursive. This means they do not need high computational effort and high memory space. * In addition, a proposed method is fading memory. This gives weight to recent data and fades the effect of early data. This makes proposed methods applicable in online applications and increases capacity estimation during several cycles for SOH estimation. The following three sections of the article provide some reasons for the ageing of lithium-ion batteries, describe the capacity estimation method, and show the validation of the model using experimental data. In the end, some conclusions are given.