# Low-Order Electrochemical State Estimation for Li-Ion Batteries
###### Abstract
Moreover, these approaches are popular because they allow to improve the prediction by diverse paths of model training. Other approaches, such as support vector machines and fuzzy logic present attractive low-error representations of essential information due to their classification and regression methods. The different advantages of first-principles and data-driven modeling approaches can be found in [24]. What should be highlighted here is that any method that only considers input-output data, even treating hidden dynamics by means of lateral variables can not be verified with respect to the internal electrochemical state predictions, unless it is compared-during training-to detailed first-principle models, which at this moment are the main source of information about what happens inside the battery. This motivates to focus on different systematic model simplification and model order reduction techniques. This has been addressed, e.g., using singular perturbation approaches [4], numerical approximation using, e.g., finite-differences and averaging [25; 26], frequency domain approaches coupled with low-order state space models [27] and Galerkin projections [28; 29], polynomial projections [30] and others (see, e.g., [31]). A further powerful tool for low-order model identification has been gaining importance in different application fields due to its capacity to systematically identify low-dimensional relations of complex systems in discrete-time setups using snapshots of the states, namely _Dynamic Mode Decomposition_ (DMD) [32; 33] and its extensions to include external control inputs, known as _Dynamic Mode Decomposition with control_ (DMDc) [34]. Having in its core singular value decomposition and Koopman operator theory [35], DMDc has shown a big potential use in fields ranging from the analysis of turbulent flows [32; 35; 36] to shallow water propagation in channels [37]. Accordingly, it is to be expected that DMDc also provides an excient basis for low-order prediction model approximations for batteries, using which one can determine the essential (i.e., dominant) modes that govern the electrochemical dynamics and create a reduced order model (ROM) on this basis, which inherits not only the accuracy of the detailed model but is also simple enough to be used in real-time applications. DMDc was recently used for battery model identification [38] with a restriction on the consideration of input-output and SOC data but without any relation to the electrochemical dynamics within the cell. In contrast to this, a ROM based on a detailed electrochemical model and obtained via DMDc was proposed in [39]. In a similar fashion as for the other model-order reduction techniques commented above, the main advantage relies in the fact that essential effects of the electrochemistry can still be represented and explicitely considered, paving the way, e.g., for monitoring and control applications based on the electrochemical state information. An advantage of DMDc from the application point of view is that the choice of functions on which the dynamics are projected takes place automatically on the basis of the data, i.e., the snapshots of the states, obtained either from measurements or from simulations, while in the other mentioned model-order reduction techniques the associated analytical effort is considerably higher. Independent of the method, any prediction model is just an approximation and thus monitoring and control approaches require some estimator, providing a correction of the predictions on the basis of measured data. For such purposes, basically two main groups of approaches can be found: observers based on deterministic system equations [11; 14; 40; 41; 42] and estimators for stochastic system representations [43]. Within classical methods for the estimator design, the Kalman filter (KF) is a well known tool for stochastic linear systems, and the extended Kalman filter (EKF) for non-linear systems.