Early Prediction of Remaining Useful Life for Lithium-Ion Batteries with the State Space Model

Abstract

In the realm of lithium-ion batteries (LIBs), issues like material aging and capacity decline contribute to performance degradation or potential safety hazards. Predicting remaining useful life (RUL) serves as a crucial method of assessing the health of batteries, thereby enhancing reliability and safety. To reduce the complexity and improve the accuracy and applicability of early RUL predictions for LIBs, we proposed a Mamba-based state space model for early RUL prediction. Due to the impacts of abnormal data, we first use the interquartile range (IQR) method with a sliding window for data cleansing. Subsequently, the top three highest correlated features are selected, and only the first 300 cycling data are used for training. The model has the ability to make forecasts using these few historical data. Extensive experiments are conducted using CALCE CS2 datasets. The MAE, RMSE, and RE are less than 0.015, 0.019, and 0.0261; meanwhile, $R^2$ is higher than 0.99. Compared to the baseline approaches (CNN, BiLSTM, and CNN-BiLSTM), the average MAE, RMSE, and RE of the proposed approach are reduced by at least 29%, 21%, and 36%, respectively. According to the experimental results, the proposed approach performs better in terms of accuracy, robustness, and efficiency.

1. Introduction

Nowadays, an important universal consensus in the realm of energy and environmental policy is the need to develop green and low-carbon energy sources, reduce greenhouse gas emissions, and remediate the impacts of global warming. This consensus reflects a global awareness of the urgent need to address climate change and its potentially catastrophic consequences [1]. The development of non-fossil energy technologies has been rapidly advancing in recent years. For instance, lithium-ion batteries (LIBs) (Figure 1) have become the dominant technology in numerous markets, including consumer electronics, electric vehicles, and renewable energy storage systems, due to their high energy density, long cycle life, and low self-discharge [2]. However, over time, LIBs experience capacity degradation, which can lead to reduced performance and even safety hazards if not properly managed [3]. To reduce the risk of unexpected failures and ensure continuous operation of the system, accurate remaining useful life (RUL) estimation allows for proactive maintenance and replacement of batteries [4].

The RUL of a battery refers to the amount of time a battery can continue to operate reliably under normal conditions before it reaches the end of its useful life (EOL). Normally, RUL is defined as the battery capacity being reduced to 80% of its initial value [5]. Although the methods for LIB prognostics are plentiful, they can be broadly categorized into model-based approaches and data-driven approaches [6].

A model-based approach in the field of battery RUL prediction is important and commonly used. This approach mainly establishes the physical model or empirical model of the battery to describe the aging behavior of the battery [7]. Filtering algorithms, such as Kalman filtering [8], particle filtering [9,10], or support vector regression–particle filtering [11], have been proven to be effective and flexible methods for accommodating the nonlinear properties of LIBs. Almutairi et al. [12] proposed an empirical method for estimating the degradation of LIB cells based on empirical lifetime equations. To verify the accuracy, an accelerated state health experiment was conducted on two different types of LIB iron phosphate (LFP) batteries under different operating conditions. The model parameters were estimated using the improved particle swarm optimization (MPSO) algorithm to enhance the global search and reduce the convergence time compared to the original PSO algorithm. However, these methods still face challenges, such as large computational capacity, parameter sensitivity, and data dependence.

On the other hand, with the development of big data and machine learning technologies, data-driven approaches have recently played an important role in battery RUL predictions since these approaches do not require a deep understanding of the electrochemical characteristics inside the battery and are able to conduct predictions based on the performance data of the battery. Severson et al. [13] proposed a data-driven battery method for predicting the cycle life of batteries. That study collected data on 124 battery packs with different service lives and analyzed the data from these batteries in early cycles. A prediction model was built using machine learning methods, and its effectiveness and accuracy were experimentally verified. Inspired by Severson’s work, a novel snapshot ensemble learning strategy [14] combining handcrafted features with domain knowledge and latent features learned by deep networks was proposed to boost the performance of early RUL prediction. Moreover, Peter et al. [15] applied a statistical learning method to rapidly develop a data-driven battery lifetime prediction model using the datasets of Severson et al. [13].

Further, by leveraging artificial intelligence algorithms, quick battery RUL predictions using fewer sampling data have been achieved. Tian et al. [16] proposed a deep neural network (DNN) model that enables the accurate prediction of a complete constant current charging curve during battery aging while using only a small number of charging data fragments, specifically, using 30 data points collected in less than 10 mins. Wang et al. [17] proposed a prediction method based on the convolutional neural network (CNN). That method required no expert knowledge and could learn multiple aging features from limited raw data. The validation results based on 113 hyper-capacitors showed that the method could accurately predict RUL using data over five consecutive cycles with an RMS of 501 cycles. Huzaifi et al. [18] established a battery RUL prediction method based on a hybrid deep model of convolutional neural networks (CNNs) and long short-term memory (LSTM). CNN was used to extract key features in the battery charging process, such as the charging curve and voltage change; meanwhile, LSTM was used to capture the changing trend of those features over time. Inputting the features extracted by the CNN into LSTM, this method highlighted the ability to learn the complex mode of battery performance degradation. Sun et al. [19] proposed a method combining incremental capacity analysis (ICA) and bidirectional long short-term memory (Bi-LSTM) neural networks based on health characteristic parameters. That method had no requirement for constructing complex physical models but relied only on observable data, such as battery charging curves. Lu et al. [20] developed a deep learning framework integrating a swarm of deep neural networks equipped with domain adaptation to produce accurate estimation. That work also examined the possibility and power of deep learning in precluding degradation experiments. Additionally, the raw data collected by sensors were uncompleted and noisy in practice. To remedy those data issues, Wang et al. [21] proposed a method to classify the missing data into random missing, cycle missing, time-step missing, and interpolate data accordingly.

In fact, battery RUL predictions necessitate either an accurate and robust battery degradation mechanism or complete battery performance data. The battery degradation mechanism is a complex and multi-dimensional process involving the battery’s internal chemical reaction, material characteristics, usage environment, and other aspects. The battery degradation process is caused by multiple factors as well. Hence, demonstrating the degradation process using equations or formulas is difficult. Although the research findings in battery RUL prediction have been fruitful in recent years, most of the current methods require a large amount of data for training. Hence, using fewer data to make forecasts is a direction of battery RUL prediction studies. Although Severson et al. [13] made efforts in the study of early RUL prediction, the features selected were difficult to collect and calculate. Applying that method is difficult. On the other hand, prediction accuracy is normally affected by abnormal input data. Wang et al. [21] proposed a method to overcome this issue, but that method required precise classification of abnormal data and made rectification accordingly. Imprecise classification would affect the accuracy of the result. Therefore, those prediction methods are complex and unstable. Integrating those methods into real-world applications is unfeasible.

In this study, we propose a novel approach based on a state space model for early RUL prediction. We first adopt the interquartile range (IQR) method to identify the abnormal data without classification, followed by data rectification. Further, we calculate the Spearman’s correlation coefficient to measure monotonic relationships between features and capacity. The highest three correlated features are selected for RUL prediction. Only the first 300 cycling data are selected for training, and the model has the ability to make predictions based on a certain amount of data. Finally, the accuracy of this proposed approach is verified by comparing it with other baseline approaches.

The contributions of this study are as follows. Firstly, our approach can predict battery RUL using a small number of input data. The first 300 cycling data and the top three highest correlation coefficient features are sufficient to make early RUL prediction. Secondly, the model integrated into our proposed approach has significant improvement compared to other traditional sequence models. Finally, our proposed approach holds the promise of enhancing the battery health monitoring mechanism and provides a feasible battery RUL prediction solution for real-world applications.

2. Methodology

2.1. Overview

Figure 2 presents the workflow of the proposed method. Data, such as voltage, current, internal resistance, etc., can be extracted or calculated during battery operation. After finishing preprocessing, the data can be considered as well-prepared and used for model training and forecasting. In the proposed approach, the correlations between each feature and capacity are calculated using Spearman’s correlation coefficient and sorted in descending order. The features with the top three highest correlation coefficients are finally selected.

For modeling, historical data are used as input, and the size of historical data is critical, especially for the early prediction model. Based on previous studies, the size of the training dataset is generally 50–70% of the whole dataset [22]. In this study, we use the first 300 cycling data for training, whereas the residual data are used to make forecasts. Due to the battery regeneration phenomenon, the capacity curve fluctuates. Spline fitting is used to smooth the capacity curve before training.

2.2. Proposed Method Based on Mamba

The structured state space sequence model (S4) [23] is a deep learning model used for sequence modeling and a mathematical framework used to describe the dynamic behavior of a system over time. S4 assumes that the process of generating the observed data is driven by an implied, non-observed sequence of states. Each state corresponds to observation data, and the transitions between the states are modeled by a transition matrix. The S4 computing procedure is defined as follows.

$$\widehat{X}(t)=A(t)X(t)+B(t)U(t)$$

(1)

$$y(t)=C(t)X(t)$$

(2)

where $X\left(t\right)\in R^n,U\left(t\right)\in R^p, \mathrm{a}\mathrm{n}\mathrm{d} y(t)\in R^q$ denote the state vector, input vector, and output vector. $A\left(t\right)\in R^{m\times n},B\left(t\right)\in R^{n\times p}, \mathrm{a}\mathrm{n}\mathrm{d} C(t)\in R^{q\times n}$ represent the state matrix, input matrix, and output matrix. $\widehat{X}\left(t\right)={\displaystyle \frac{d}{dt}}X(t)$.

Moreover, discretization, which is a key step, is the process of transforming a system of continuous-time states into discrete-time states. The zero-order hold (ZOH) is adopted for discretization.

$$X\left(t\right)=\overline{A}\left(t-1\right)+\overline{B}U(t)$$

(3)

$$y(t)=CX(t)$$

(4)

where $\overline{A}=\exp \left(\Delta A\right), \overline{B}={\left(\Delta A\right)}^{-1}(\exp \left(\Delta A\right)-I)·\Delta B$, and $\Delta$ denotes the step size.

Mamba [24], a state space model (SSM) architecture, improves on the S4 architecture and enables optimization of an SSM by introducing selectivity mechanisms and hardware sensing algorithms. Like the stacked layers of the Transformer model, Mamba derived from the Hungry Hungry Hippo (H3) architecture is built by stacking multiple layers of the Mamba block, as shown in Figure 3. A basic Mamba block consists of H3 blocks and gated MLP. The H3 block is the state space model execution mode unique to the Mamba model, which simplifies the structure of the commonly used SSM block. Gated MLP is used to transform the input and add residual connections to prevent the gradient from disappearing. Figure 4 indicates the details of the structure of the Mamba block. The projection represents a linear projection layer to extend the input embedding and is convolved before using the selective SSM to prevent independent token calculations.

3. Experiments

3.1. Experimental Process and Method

The experiments mainly include four steps, namely, data extraction, data preprocessing using the IQR based on sliding window, feature selection based on Spearman’s correlation, and modeling based on Mamba.

The workflow of experiments is indicated in Figure 5. Firstly, the performance features and capacity sequences are extracted from the discharge cycling data. Since the battery data are collected by sensors, noise data and missing data are inevitable. Hence, removing the abnormal data can improve the prediction accuracy. In the preprocessing step, the IQR based on sliding window is used to segregate the outliers. The data within the window need to be sorted to calculate the quartiles, and then Q1, Q2, and Q3 are calculated from the sorted data. Outliers can be detected and handled using quartiles. The IQR, namely, Q3—Q1, is used to define the range of outliers. The data with values less than Q1 − 1.5 IQR or greater than Q3 + 1.5 IQR are defined as outliers and deleted from the dataset. The gaps in the dataset are filled by using linear interpolation in the end. The next step is to calculate Spearman’s correlation coefficients between features and capacity sequences. The features ranking for the top three are selected for training and testing. The last step is to use the Mamba-based prediction model to make forecasts. In the Mamba model, the input data are firstly processed by the selection mechanism, which filters out irrelevant information while retaining necessary and relevant data. Afterward, the data are sent into the simplified state space model architecture for further processing. In the end, the final prediction results are generated. The specific implementation steps are shown in Figure 6.

3.2. Definition of RUL and SOH

Remaining useful life (RUL) and State of Health (SOH) are two of the indicators used to evaluate battery performance and service life. The RUL is crucial for battery fault diagnosis. Accurately predicting the battery RUL plays a key role in ensuring the safety and stability of batteries [25]. Thus, the RUL represents the remaining number of battery cycles, including charging and discharging, before the actual capacity drops to the baseline and reaches the end of life (EOL) [26]. The formula for RUL is defined as follows:

$$\mathrm{RUL}=N_{eol}-N_t$$

(5)

where $N_{eol}$ denotes the number of battery cycles while reaching the end of life and $N_t$ represents the total number of charge–discharge cycles that occurred [25].

SOH is essential to predict the safety and reliability of batteries [27]. SOH is the key health indicator for battery aging and indicates the percentage of current charge–discharge capacity over the nominal capacity [28]. Therefore, the SOH formula is defined as follows:

$$\mathrm{SOH}={\displaystyle \frac{C_m}{C_n}}\times 100\%$$

(6)

where $C_m$ denotes the capacity at the mth cycle and $C_n$ denotes the nominal capacity when the battery is fresh [29].

While reviewing these two indicators, the key point that can be noticed is that the factor of capacity plays a crucial role in both calculations. Furthermore, the RUL can also be considered as ended once the actual capacity drops to 80% of the initial capacity or below [30]. To summarize, when a battery ages and deteriorates, the battery capacity gradually decreases. Once the battery capacity drops to 80% of the initial capacity, the battery deteriorates to the end of its lifespan.

3.3. Datasets

In this study, we used the public battery degradation dataset CALCE CS2 provided by the Advanced Life Cycle Engineering Center (CALCE) at the University of Maryland to conduct experiments. The CALCE CS2 [31,32,33] dataset contains a variety of key parameters of a battery in the process of the charge and discharge cycle, such as voltage, current, charging capacity, discharge capacity, internal resistance, etc. These data provide researchers with comprehensive information on battery performance, facilitating an in-depth analysis of the aging mechanism. Therefore, CALCE CS2 is widely recognized with a high degree of accuracy and reliability. Additionally, we used CS2_35, CS2_36, CS2_37, and CS2_38 for experimental testing. All of them were cycled at a constant current of 1C. Figure 7a shows the degeneration curves for CS2_35, Figure 7b for CS2_36, Figure 7c for CS2_37, and Figure 7d for CS2_38. The initial capacity is 1.1 Ah; hence, the EOL is 0.88, which is 80% of the initial capacity. Because of capacity regeneration, the degeneration curve diminishes with fluctuation. But once the degeneration curve breaks through the EOL, the curve suddenly accelerates the decline. From this point of view, predicting when the degeneration curve reaches the EOL has an important role.

The number of cycles and capacity are the primary features in this study, but they are implicit in the CALCE CS2 dataset. There are nine operation steps in the CALCE CS2 dataset, and these nine steps form a loop, where the second step represents the status of constant current charge, and the seventh step represents the status of constant current discharge. A new charge–discharge cycle starts over once the operation step is adjusted from nine to one. Thus, the number of cycles adds one accordingly. For the capacity calculation, we use the ampere-hour integral method as follows:

$$C_n=\underset{t_1}{\overset{t_2}{\int }}Idt$$

(7)

where $C_n$ stands for the discharge capacity in the nth cycle, $I$ denotes discharge current, $t_1$ represents the start time of discharge in the nth cycle, and $t_2$ represents the end time of discharge in the nth cycle.

3.4. Features Analysis and Selection

Figure 8a shows the time–voltage curve for battery CS2_35. With battery aging and degradation, the voltage–time curve gradually shifts to the left. The battery voltage drops faster within the same discharge time after experiencing more discharge cycles. This left offset phenomenon reflects the increase in internal impedance and decrease in battery capacity, and it is a direct manifestation of battery degradation. Therefore, the change in voltage is a key factor reflecting the battery degradation process. Internal resistance is another important parameter in the battery degradation process, which reflects the change in the internal structural and chemical properties. With the increase in charge–discharge cycles, a series of physical and chemical changes occurs inside the battery, such as the structural changes in electrode materials, the loss and decomposition of electrolytes, etc. Meanwhile, the internal resistance is affected as well. Figure 8b shows the change in internal resistance during the degradation process. The internal resistance increases after experiencing more charge–discharge cycles. On the other hand, the voltage drops due to the increment in resistance since the battery needs to overcome greater resistance in the charge–discharge process. As a result, the battery performance declines, including capacity attenuation, charge–discharge efficiency reduction, etc. A constant voltage charge time (CVCT) [34,35] is obtained by the preliminary extraction of the CACLE CS2 dataset. The voltage added to the two ends of the battery is fixed throughout the charging process. When the charging voltage rises close to the saturation point, the charging mode is switched to the constant voltage mode. During this period, the charging current begins to drop until the charging ends. Figure 8c shows that the CVCT curve starts out relatively flat and rises suddenly and dramatically with the increase in the aging period. Constant current charge time (CCCT) [34,35,36] reflects the charging speed and time of the battery during the constant current charging phase. CCCT is closely related to SOH and can perfectly replace the incremental capacity peak area. CCCT can be used as an indirect observation indicator to assess the battery RUL. The CCCT change trend is indicated in Figure 8d. That curve fades and drops more and more obviously while aging. Additionally, temperature, excluded from the CALCE CS2 dataset, is strongly related to battery aging [37]. High temperatures accelerate the chemistry inside the battery, leading to lower battery performance. Meanwhile, high temperatures also accelerate the volatilization of electrolytes and the fall off of electrode materials, further shortening the battery life.

Spearman’s correlation coefficient [38] is based on the Pearson correlation coefficient of two ranked variables, and it is used to assess the monotonic relationship between the two variables. Spearman’s correlation coefficient is calculated as follows.

$$\rho =1-{\displaystyle \frac{6\sum {d_i}^2}{n(n^2-1)}}$$

(8)

where $d_i$ is the difference between the rank of each pair of observations and $n$ is the number of observations.

The correlations between capacity and battery features for dataset CS2_35 were calculated and are shown in Figure 9. The highest correlation with battery capacity is CCCT, followed by both voltage and resistance with the same correlation values. CVCT occupies the fourth place. Therefore, voltage, internal resistance, and CCCT were selected for prediction.

3.5. Evaluation Metrics

To examine and verify the experimental results, four evaluation metrics were employed. They are the mean absolute error (MAE), root mean square error (RMSE), relative error (RE), and coefficient of determination ($R^2$).

MAE is widely used to evaluate the accuracy of prediction models. The best possible score is 0.0, and a smaller value is better. MAE is calculated as the average of the absolute differences between the predicted values and the actual values [39].

$$\mathrm{MAE}={\displaystyle \frac{\sum _{i=0}^{N-1}\left|y_i-{\widehat{y}}_i\right|}{N}}$$

(9)

where $y_i$ denotes the observed value for the $i^{th}$ observation, ${\widehat{y}}_i$ denotes the predicted value for the $i^{th}$ observation, and $N$ denotes the total sample size.

RMSE is widely used to evaluate the performance of prediction models in various fields. The best possible score is 0.0, and a smaller value is better. A lower RMSE indicates better prediction accuracy. RMSE is calculated as the square root of the average of the squared differences between the predicted values and the actual values [40].

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{(y_i-{\widehat{y}}_i)}^2}$$

(10)

RE is used for the evaluation of regression model accuracy. The best possible score is 0.0, and a smaller value is better. RE is the ratio of the absolute error to the actual value.

$$\mathrm{RE}={\displaystyle \frac{\left|y_i-{\widehat{y}}_i\right|}{\left|y_i\right|}}$$

(11)

$R^2$ is a measure of how well unseen samples are likely to be predicted by the model through the proportion of explained variance. The best possible score is 1.0, and a bigger value is better. $R^2$ represents the proportion of variance that is explained by the independent variables in the model [41].

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^N{(y_i-{\widehat{y}}_i)}^2}{\sum _{i=1}^N{(y_i-\overline{y})}^2}}$$

(12)

where $\overline{y}$ denotes the mean value for the sample.

4. Results and Discussion

4.1. RUL Prediction Results

In this paper, the proposed approach was validated by conducting experiments using CALCE battery data CS2_35, CS2_36, CS3_37, and CS2_38. All of them were tested from the same start point at the 301st cycle, whereas the first 300 cycling data were used for training. Applying the proposed approach, the predicted RUL results are plotted in Figure 9. In Figure 10a, the actual curve for CS2_35 breaks through the threshold at the 556th cycle, while the predicted is at the 557th cycle. In Figure 10b, the actual curve for CS2_36 breaks through the threshold at the 536th cycle, while the predicted is at the 550th cycle. In Figure 10c, the actual curve for CS2_37 breaks through the threshold at the 606th cycle, while the predicted is at the 618th cycle. In Figure 10d, both curves for CS2_38 break through the threshold at the 624th cycle. This reflects that the predicted results are close to the actual results.

4.2. Performance Comparison

The performance results for the CS2 dataset are listed in

Table 1. Performance result for dataset CALCE CS2.

Attribute	CS2_35	CS2_36	CS2_37	CS2_38
MAE	0.0099	0.0097	0.0145	0.0127
RMSE	0.0123	0.0112	0.0189	0.0180
RE	0.0017	0.0261	0.0194	0
$R^2$	0.9964	0.9980	0.9916	0.9920

. The MAE values for CS2_35, CS2_36, CS2_37, and CS2_38 are 0.0099, 0.0097, 0.0145, and 0.0127, respectively, which are all lower than 0.015. The RMSE values (0.0123, 0.0112, 0.0189, 0.0180) are no more than 0.019 as well. Although RE for CS2_36 is 0.0261 and higher than 0.02, RE for CS2_35 is 0.0017, for CS2_37 is 0.0194, and for CS2_38 is 0. This means that the predicted RUL is close to the real one, and both values for CS2_38 are matched. For $R^2$, the values (0.9964, 0.9980, 0.9916, 0.9920) are higher than 0.99. This reflects the high consistency in the prediction curves and the capacity curves. Therefore, based on these performance results, a conclusion can be made that the predicted results for our proposed approach are accurate and efficient.

Figure 11 indicates the deviation for the evaluation metrics. In Figure 11a, the median for MAE is about 0.012, and the range is about 0.0048. The median for RMSE is about 0.015, and the range is about 0.0077. The median for RE is about 0.011, and the range is about 0.0261. In Figure 11b, the median for $R^2$ is about 0.09943, and the range is about 0.064. The ranges for MAE and RMSE are less than 0.008, and for RE and $R^2$, they are no more than 0.07, which means the variability is low. Although the distributions for all the evaluation metrics are right-skewed, their medians are close to the middle of the boxes. This indicates that the performance results are centralized and do not have a high degree of dispersion. This shows that our proposed approach is robust and stable for different datasets. This is mainly because the IQR method is used for data preprocessing. The noise data and missing data are rectified in the first place.

In

Table 2. Performance result comparison.

Data	Method	Evaluation Metrics
		MAE	RMSE	RE	${\mathit{R}}^2$
CS2_35	CNN	0.0138	0.0193	0.0237	0.9813
	BiLSTM	0.0176	0.0221	0.0255	0.9887
	CNN-BiLSTM	0.0259	0.0319	0.0237	0.9764
	Proposed Approach	0.0099	0.0123	0.0017	0.9964
CS2_36	CNN	0.0109	0.0206	0.0101	0.9837
	BiLSTM	0.0131	0.0232	0.0201	0.9820
	CNN-BiLSTM	0.0202	0.0296	0.0101	0.9870
	Proposed Approach	0.0097	0.0112	0.0061	0.9980
CS2_37	CNN	0.0185	0.0191	0.0186	0.9847
	BiLSTM	0.0179	0.0198	0.0186	0.9849
	CNN-BiLSTM	0.0170	0.0228	0.0186	0.9880
	Proposed Approach	0.0145	0.0189	0.0094	0.9916
CS2_38	CNN	0.0231	0.0182	0.0224	0.9820
	BiLSTM	0.0192	0.0243	0.0224	0.9850
	CNN-BiLSTM	0.0347	0.0397	0.1105	0.9620
	Proposed Approach	0.0127	0.0180	0	0.9920

, the performance results for baseline approaches (CNN, BiLSTM, and CNN-BiLSTM) are listed and adopted for performance comparison. Take the performance results for data CS2_35 as an example, MAE and RMSE are 0.0099 and 0.0123, respectively. Meanwhile, those for CNN are the lowest among all baseline approaches, namely, 0.0138 and 0.0193. MAE and RMSE for the proposed approach are 71% and 64% of CNN’s values. For $R^2$, the proposed approach is 0.9964, and CNN, BiLSTM, and CNN-BiLSTM are all lower than 0.99. For data CS2_36, CS2_37, and CS2_38, the comparison results are similar to that of CS2_35, which shows that the proposed approach is better. These results show that our proposed approach has better accuracy and efficiency than the other baseline approaches. Compared to the baseline approaches, our proposed approach is more effective and has practical value because our proposed approach only requires a limited amount of cycling data for training. Additionally, our proposed approach achieves the best prediction results when compared to the other baseline approaches. This gives credit to the selective state space layer in the Mamba model, which allows the model to selectively propagate or suppress information based on the input at each step. This mechanism allows Mamba to more accurately capture the key patterns in the time series, thus improving the prediction accuracy.

Figure 12 presents the evaluation metric (MAE, RMSE, and RE) comparison results from another angle. Take CS2_35 in Figure 12a as an example; the distance to the central point for the MAE of the proposed approach is 0.0099, that for CNN is 0.0138, that for BiLSTM is 0.0176, and that for CNN-BiLSTM is 0.0259. The distance of RMSE for our proposed approach is 0.0123, that for CNN is 0.0193, that for BiLSTM is 0.0221, and that for CNN-BiLSTM is 0.0319. The distance of RE for our proposed approach is 0.0017, that for CNN is 0.0237, that for BiLSTM is 0.0255, and that for CNN-BiLSTM is 0.0237. The distances of those metrics for our proposed approach are the shortest. Applying this analysis to CS2_36 in Figure 12b, CS2_37 in Figure 12c, and CS2_38 in Figure 12d, the comparison results are similar as well. This reflects that our proposed approach has better accuracy and efficiency.

It is important to highlight that our proposed approach offers better performance with fewer hardware resources due to the hardware-oriented parallel scanning algorithm in the Mamba model. This algorithm enables the computational process to perform efficiently and achieves linear time complexity.

5. Conclusions

In this paper, we propose a novel prediction approach based on the state space model. This approach only uses limited historical data, i.e., the first 300 cycling data, for RUL prediction. With the assistance of the preprocessing method, the input is cleansed and further improves the prediction accuracy. To verify this approach, experiments are conducted using CALCE CS2 datasets. The averages of MAE, RMSE, RE, and $R^2$ are 0.0117, 0.0151, 0.0118, and 0.9945, respectively. Compared to the other baseline approaches (CNN, BiLSTM, and CNN-BiLSTM), the average of MAE, RMSE, and RE for our proposed approach are reduced by at least 29%, 21%, and 36%, respectively. The improvement in $R^2$ for our proposed approach is at least 1%. Therefore, our proposed approach is better in terms of accuracy, efficiency, robustness, and adaptability.

In the future, we will use the first 100 cycling data for prediction and archive the satisfied results as well. Meanwhile, Mamba, which is an innovative state space model, has attracted wide attention since it was proposed. Many Mamba-based derivative models have been developed in recent times; we will adopt those state-of-the-art (SOTA) models to upgrade our proposed approach in terms of accuracy, efficiency, robustness, and adaptability.