## 1 Introduction
The model introduced in [28] was used to estimate capacity loss during cyclic aging, based on the values of different current rates using the incremental capacity analysis (ICA). The results showed a capacity reduction with aging, with a prediction error below 3.2%. Moreover, a commercial LiFePO4/graphite type aging cycle was tested in [29] via nineteen points, based on combinations of C-rate, temperature, depth of discharge, and SOC. SOH prediction approach was introduced in [30] based on grey Markov chain (GMC), considering the battery internal resistance using a designed monitor device. The results showed that the model is efficient in SOH estimation. Moreover, in [31], a SOH relative evaluation method was proposed based on the ohmic resistance. First, a dual extended Kalman filter (DEKF) was used for the ohmic resistance estimation along with the SOC. Then, the relative SOH was estimated based on the proposed method. The method was verified using a LiFePO4 battery. An optimization technique using non-dominated sorting genetic algorithm II (NSGA-II) was introduced in [32] to estimate the SOH. The optimizer considered both accuracy and measurement cost. The validation was done using the measurement from two LiFePO\({}_{4}\)/C batteries, providing the grid with the primary frequency regulation (PFR) service. A combination of SOC and SOH estimation was introduced in [33]; a sequential algorithm to improve estimation performance was proposed. It used frequency-scale separation and estimated parameters/states sequentially by different frequencies of currents injection. The proposed algorithm was verified by experiments at different temperatures for varying battery capacities. As a combination of an electro-thermal model and a semi-empirical cycle-life model, a coupled electro-thermal-aging model was introduced in [34], which demonstrated the system dynamics for LiFePO\({}_{4}\) batteries. It provided an assumption for the behaviors of the battery, which would be discussed in the section of the mathematical model, as well as an open-loop observer for both SOC and SOH. As stated above, considerable studies have focused on the estimation problem in LiFePO\({}_{4}\) batteries, where various approaches were investigated. Several existing approaches implement assumptions due to the complexity of the problem, such as the simplified linear models, which affect the accuracy rates and produce unreliable estimations for SOC and SOH of LiFePO\({}_{4}\) batteries. To cover the gap in the literature, this paper proposes a reliable and robust observer, which estimates the SOC and SOH of LiFePO\({}_{4}\) batteries in a simultaneous manner with high accuracy. To do so, comprehensive modeling of LiFePO\({}_{4}\) batteries was achieved via an equivalent-circuit model whose input is the ambient temperature and the current, and its output was assumed to be the voltage. Furthermore, a comprehensive model was proposed, which combines three parts: a thermal model, an electrical model, and an aging model. To assure high performance of the proposed observer, the use of the dual extended Kalman filter (DEKF) is adopted for the SOC and SOH estimation of LiFePO\({}_{4}\) batteries with current and terminal voltage sensors. A robust voltage-temperature (VT) observer and an open-loop observer were also built to provide an efficient method to monitor SOC and SOH for the battery. The system was constructed and simulated using MATLAB. Furthermore, the robustness of the proposed observer was verified by assessing its performance against ill-conditions that include wrong initial estimates and wrong model parameters. The intensive results reveal the reliability and robustness of the proposed observer for estimating both SOC and SOH of LiFePO4 batteries simultaneously, thereby ensuring safe operation for them. The mathematical model is explained in Section 2. The algorithms and implementation are presented in Section 3. Section 4 introduces the results as well as their analysis. Finally, conclusions are provided in Section 5.