## 7 Concluding Remarks  
This paper presents estimation strategies that utilize the interacting multiple model (IMM) algorithm with dual-filtering strategies to accurately estimate the state of charge (SOC) and the battery capacity of a lithium-ion battery (LiB) under cycling conditions. This study is of particular importance for electric vehicles. As the battery is subject to cycling, its capacity degrades over time. The proposed strategies made use of the standard Kalman filter (KF) and the robust sliding innovation filter (SIF) in a dual-filter strategy (dual-KF and dual-SIF) to estimate the SOC and degradation of the battery capacity. Furthermore, the IMM was combined with the dual filter to allow access to different degradation rates of the battery's parameter values (battery capacity and internal resistance). Including the IMM structure led to two new strategies, namely, the dual-KF-IMM and dual-SIF-IMM algorithms. The two proposed strategies were evaluated using two experimental datasets. The first dataset was used as a baseline to test the performance of the algorithm under normal aging conditions and the second dataset was used to test the algorithms under a faster aging condition. Moreover, the battery model parameters and OCV curve were identified using the baseline dataset and remained constant in the accelerated aging experiment. Lastly, the proposed algorithms were only given access to the internal resistance and battery capacity parameters to account for the faster aging conditions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **RMSE** \\ \hline Dual-KF & 0.3229 \\ Dual-KF-IMM & 0.2581 \\ Dual-SIF & 0.0831 \\ Dual-SIF-IMM & 0.0458 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: B034 RMSE: Battery Capacity Using the Minimum at each Estimation cycle. Figure 22: (a) Dual-KF-IMM mode probability. (b) Dual-SIF-IMM mode probability. The proposed algorithms demonstrated good accuracy for the baseline case (B036 dataset). For the accelerated aging case, the dual-IMM algorithms showed greater accuracy than their dual-only counterparts. Moreover, the dual-KF-IMM showed the most accuracy and better adaptation to the faster aging conditions. Finally, the dual-IMM algorithms showed great accuracy based on the RMSE values when the minimum value in the battery capacity estimation process was selected to generate a comparison across all algorithms. Lastly, in terms of mode probability, neither dual-IMM algorithm made a complete switch to the faster aging conditions to which they had access. A reason behind that could be the indirect effect of the parameter in the estimation process. Further study of the dual-IMM algorithms is encouraging. A new potential study could implement the dual-IMM algorithms with a neural network strategy that could update the OCV curve for different aging conditions. Updating the OCV curve to account for the battery's new dynamics would significantly increase the accuracy of the algorithms and relax the use of the battery capacity parameter to account for the degradation rate of the battery. Additionally, other nonlinear estimation strategies such as the extended and unscented Kalman filters, particle filters, or the H-infinity filter could be implemented and studied. Conceptualization, R.B. and S.A.G. ; methodology, R.B. ; software, R.B. ; validation, R.B. and S.A.G. ; formal analysis, R.B. ; investigation, R.B. ; resources, S.A.G. ; data curation, R.B. ; writing--original draft preparation, R.B. ; writing--review and editing, S.A.G., M.A.-S. and S.M. ; visualization, R.B. ; supervision, S.A.G. ; project administration, S.A.G. ; funding acquisition, S.A.G. All authors have read and agreed to the published version of the manuscript. This research was funded by the Government of Ontario (Gadsden, ER17-13-056) and the Natural Sciences and Engineering Research Council of Canada (Gadsden, RGPIN-2017-04087). Not applicable. Non-standard-  
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