## 1 Introduction  
Lithium-ion batteries (LIBs) have been extensively applied in various fields, highlighting their critical role in future energy sustainability [1]. Among these applications, road transportation is where LIBs have the greatest impact. While LIBs play a critical role in electric vehicles, the aging of these batteries presents significant challenges for end-of-life management. Retired batteries that no longer meet the requirements must be urgently addressed to prevent safety issues [2]. For the reliable use of LIBs in electric vehicles, developing an advanced battery management system (BMS) for accurately estimating battery states is essential [3]. Research in this area has become increasingly prominent in recent years, with a significant rise in publications focusing on state-of-charge (SOC) estimation [4; 5; 6; 7]. Although directly measuring SOC based on Coulomb counting is possible, achieving high accuracy requires dedicated current sensors, which can be as costly as USD 1000. Such sensors are unaffordable for commercial BMSs that are under significant cost pressure. Therefore, inexpensive but reliable estimation methods are required. Despite being estimates, the SOC must be as accurate as measurements obtained from current sensors enhanced with internal compensation. For use in electric vehicles, an SOC error within 5% over the full range is targeted. Among estimation methods, Kalman filters (KFs) have proven particularly effective, with more than half of model-based SOC estimators relying on KFs and their variants [4]. Notwithstanding the widespread use of Kalman filtering, estimating the SOC for aged batteries remains challenging, as it requires the concurrent estimation of SOC and capacity [8; 9; 10]. Despite the strong correlation between SOC and capacity, existing methods typically estimate them separately, resulting in relatively limited comprehensive investigations into their simultaneous estimation [11]. Even when simultaneous estimation is explored, new or minimally used batteries are typically applied [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. This gap in research limits the development of BMSs for electric vehicles, where state estimation accuracy must be maintained throughout the battery's lifespan. Besides KFs, alternative approaches such as using ultrasonic reflection waves for state-of-charge and temperature joint estimation have also been reported. In this method, a piezoelectric transducer attached to the battery surface transmits ultrasonic signals, allowing for the real-time monitoring of internal battery conditions in a non-destructive manner [26]. Depending on how states and parameters in a battery model are estimated and updated, KFs are divided into two main groups. The first group uses a single filter to estimate states, with parameters updated separately through an optimization technique [27; 28; 29; 30]. However, this method is burdened by higher computational demands due to the execution of two separate implementations. Determining the frequency of parameter updates can be another challenge in this method. The second group uses a dual filter for simultaneous state estimation and parameter updating [31; 32; 33; 34; 35]. Robust Kalman filtering techniques, such as those presented in [36], emphasize the strength of the dual extended Kalman filter (DEKF) in addressing model uncertainties, enhancing robustness and accuracy in state and parameter estimation. While this method allows for more comprehensive investigations of battery aging, it poses significant challenges due to the complexity of the variables involved. This complexity is further compounded by the limited number of direct measurements, namely, terminal voltage compared to the number of estimates. Consequently, the limited measurements can lead to less accurate estimations of battery states and parameters, affecting the reliability of the overall BMS and its decisions.