A Model-Based Framework for Lithium-Ion Battery SoC Estimation Using a Tuning-Light Discrete-Time Sliding-Mode Observer

Abstract

Reliable state-of-charge (SoC) estimation is crucial for safe and efficient battery management. However, it is challenging in practice. Terminal-voltage sensitivity becomes weak in open-circuit-voltage (OCV) plateau regions. Model uncertainty also persists at practical sampling periods. To tackle this issue, this paper proposes a discrete-time, model-based SoC estimation framework. This framework combines a dual-polarization equivalent-circuit model with a tuning-light sliding-mode observer. It is specifically designed for digitally sampled battery management systems. The modeling stage includes: (i) a discrete-time DP representation suitable for embedded use, (ii) a shape-preserving PCHIP reconstruction of the OCV–SoC curve and its derivative, and (iii) an effective-slope regularization mechanism that maintains non-vanishing output sensitivity even in flat OCV regions. On top of this structure, a boundary-layer SMO is developed with output-error shaping, model-driven gain scaling, and simple bias-compensation terms based on integral correction and leaky Coulomb counting. A discrete-time Lyapunov analysis is conducted directly on the surface dynamics. This analysis shows finite-time reaching to the boundary layer and a practical limit on the steady-state error that depends on the sampling period, disturbance level, and boundary-layer width. Numerical tests on a DP model identified from experimental data indicate that the proposed method achieves SoC accuracy similar to a switching-gain adaptive SMO. The results confirm the benefits of a model-centric design. The discrete-time formulation and convergence proof, which do not depend on high sampling rates, provide robustness advantages over traditional sliding-mode methods. The proposed method also performs better than a tuned EKF in plateau regions, requiring much less tuning effort.

1. Introduction

Modeling lithium-ion batteries is essential for designing reliable battery management systems (BMSs), effective charging strategies, and control that considers battery health. Among the internal battery states, the state of charge (SoC) is crucial for safety, performance, and lifespan management, yet it cannot be measured directly and must be estimated using a model-based approach. The reliability of this process is closely connected to the accuracy and structure of the battery model [1,2,3]. In practice, lithium-ion batteries exhibit nonlinear behavior due to polarization dynamics, diffusion effects, and temperature variations, all of which influence the terminal voltage response. Moreover, the open-circuit voltage (OCV)–SoC relationship plays a central role in estimation accuracy but is often affected by operating conditions and parameter uncertainty, motivating modeling approaches that can preserve or adapt OCV characteristics. Consequently, an effective battery model must capture nonlinear OCV behavior, polarization and diffusion dynamics, and temperature dependence while remaining suitable for real-time BMS implementation [4,5,6,7,8].

Modeling uncertainties, parameter drift caused by aging and operating conditions, and the loss of observability linked to the near-zero OCV and SoC slope in voltage plateau regions are ongoing challenges for reliable SoC estimation. This is especially true under dynamic load and temperature changes. Small voltage variations can result in significant estimation errors or reduced sensitivity [9,10,11,12,13,14]. These challenges have led to significant research on improving model structures and observer robustness. Within the group of equivalent-circuit models (ECMs), the dual-polarization (DP) topology provides a good balance between accuracy and simplicity. It captures quick charge-transfer effects and slower diffusion-driven relaxation while being manageable for real-time implementation and observer design [3,15,16].

However, even with this structure, estimating SoC becomes challenging in OCV-flat regions. In these areas, the sensitivity of terminal voltage to SoC is small, and observability is weak.

A broad range of SoC estimators comes from ECMs. This includes extended Kalman filters (EKFs) [17,18], dual or augmented Kalman filters [11,19], H-infinity observers [10,12], and sliding-mode observers (SMOs) [15,20,21,22,23]. EKF-based methods are sensitive to model mismatch and linearization errors, especially in areas where the open-circuit voltage (OCV) and state-of-charge (SOC) relationship is flat. While H-infinity observers and sliding mode observers (SMOs) provide better robustness under uncertainty, they usually depend on continuous-time formulations, adjusting gain, or assumptions like small sampling intervals. These needs may not fit the limits of the computational and hardware capabilities of embedded battery management systems (BMSs). Recent studies also indicate that the performance of observers is greatly affected by the model’s discretization, structure, and how uncertainty is represented. This illustrates the importance of discrete-time and sampled-data implementations for practical use [24,25].

Recent work highlights the importance of sensitivity and structural observability in battery-oriented models [4,6,26]. It also discusses the development of robust and H∞-type observer structures for electrochemical energy storage systems [10,12,13]. These findings suggest that improving the model through methods like slope regularization, monotonic interpolation, or discrete-time restructuring is as crucial as changing the observer design to achieve reliable state estimation.

Motivated by these insights, this paper takes a modeling-first approach to SoC estimation. A discrete-time DP model is formulated along with a shape-preserving OCV representation and an effective-slope regularization that reduces OCV–SoC overlap in flat areas. Based on this structure, a tuning-light boundary-layer sliding-mode observer is developed. Its stability is analyzed directly in discrete time, without depending on small-sampling assumptions.

Overall, the proposed framework shows how structural improvements in battery modeling, especially monotone OCV interpolation and slope regularization, can significantly boost the performance of nonlinear observers for SoC estimation without relying on a high sampling rate.

Despite significant progress in observer design, there is still a need for SoC estimators that combine (i) explicit discrete-time model formulation, (ii) strong resistance to OCV–SoC degeneracy, and (iii) simple tuning requirements suitable for embedded BMS platforms. As discussed later in Section 2, the DP model faces slope degeneracy in mid-range OCV plateaus. This issue reduces observability and makes the model more sensitive to errors [7,13].

This work develops a unified model-estimation framework. It restructures the DP model and integrates effective slope regularization directly into the observer design. The new method improves observability in plateau regions and produces a discrete-time SMO with straightforward tuning rules and clear stability conditions.

Compared to EKF- and SMO-based SoC estimators, the proposed method is designed in discrete time. It improves robustness in low-slope OCV regions and reduces tuning complexity. This drives the contributions summarized below.

The main contributions include:

• Model-centric robustness: a discrete-time DP model with shape-preserving PCHIP OCV reconstruction and an effective-slope regularization that prevents OCV, SoC slope degeneracy, and strengthens observability [27,28].
• Tuning-light observer structure: A boundary-layer SMO with model-driven gain scaling, output-error shaping, and simple bias-compensation mechanisms. This approach avoids adaptive-gain dynamics or terminal surfaces [22,23].
• Discrete-time stability guarantees: a Lyapunov-based analysis developed entirely in discrete time offers finite-time reaching and practical convergence limits under bounded disturbances and model uncertainty.

Simulations under standard pulse discharge at a sampling period of $T_s=1s$ show that the proposed method achieves SoC accuracy comparable to a switching-gain adaptive SMO [21] and an EKF [19], while requiring less tuning.

2. Battery System Modeling

State estimation requires a modeling framework that captures both the electrochemical behavior of the lithium-ion cell and the key dynamic phenomena relevant for digital BMS implementation. Physics-based electrochemical models offer high fidelity but are often too complex and uncertain for real-time use. Equivalent-circuit models (ECMs) provide a more manageable approach and have been widely adopted for embedded estimation and control [1,3,9,17]. Among these, the dual-polarization (DP) architecture offers a satisfactory balance between model complexity and computational cost. It captures rapid polarization and slow diffusion effects across a wide range of operating conditions [4,5,6].

2.1. Continuous-Time DP Model

The DP model, shown schematically in Figure 1, consists of a nonlinear OCV source $V_{\mathrm{o}\mathrm{c}}\left(z\right)$, an ohmic resistance R, and two parallel resistor–capacitor (RC) branches representing charge-transfer and diffusion dynamics.

The continuous-time terminal voltage is modeled as

$$V\left(t\right)=V_{\mathrm{o}\mathrm{c}}\left(z\left(t\right)\right)-v_p\left(t\right)-v_s\left(t\right)-R\hspace{0.17em}I\left(t\right),$$

(1)

where $I\left(t\right)$ represents the applied current (positive for discharge), $v_p\left(t\right)$ and $v_s\left(t\right)$ are the polarization and diffusion voltages, and $z\left(t\right)$ is the SoC.

The internal RC dynamics are first-order linear subsystems driven by the same current:

$$\begin{array}{rr}{\dot{v}}_p\left(t\right)& =-{\displaystyle \frac{1}{R_p{C}_p}}\hspace{0.17em}{v}_p\left(t\right)+{\displaystyle \frac{1}{C_p}}\hspace{0.17em}I\left(t\right),\\ {\dot{v}}_s\left(t\right)& =-{\displaystyle \frac{1}{R_s{C}_s}}\hspace{0.17em}{v}_s\left(t\right)+{\displaystyle \frac{1}{C_s}}\hspace{0.17em}I\left(t\right),\end{array}$$

(2)

where $\left(R_p,{ C}_p\right)$ represent fast transients and $\left(R_s,C_s\right)$ control slower relaxation. The SoC changes according to

$$\dot{z}\left(t\right)={\displaystyle \frac{I\left(t\right)}{C_n}},$$

(3)

with nominal capacity $C_n$.

This structure can effectively reproduce voltage relaxation, diffusion transients, and current-induced dynamics under realistic profiles. It is also suitable for observer synthesis and discrete-time implementation [15,29].

2.2. Nonlinear OCV–SoC Modeling

The OCV $V_{\mathrm{o}\mathrm{c}}\left(z\right)$ is a nonlinear, monotonic function of SoC. Its slope varies greatly throughout the charge range. Large plateau regions exhibit a very small derivative.

$$d\left(z\right)={\displaystyle \frac{d{V}_{\mathrm{o}\mathrm{c}}}{dz}},$$

(4)

leading to weak observability and reduced estimation sensitivity. Because the OCV–SoC curve directly governs the output sensitivity of the DP model, its representation is critical in ECM-based SoC estimation [4,6].

In this work, this relationship is modeled using a shape-preserving cubic Hermite interpolant (PCHIP) fitted to experimentally measured OCV and SoC data points. Unlike standard cubic spline interpolation, PCHIP keeps local monotonicity. It avoids overshoot and artificial oscillations, which can be issues in plateau regions and near steep end-of-charge transitions. Oscillatory or non-monotone interpolants can produce non-physical or sign-changing estimates of $d{V}_{\mathrm{o}\mathrm{c}}/dz$, which breaks the monotonicity assumptions needed for observer convergence and reduces observability in flat regions [27,28]. In this work, PCHIP offers a smooth, physically consistent, and numerically strong representation of both $V_{\mathrm{o}\mathrm{c}}\left(z\right)$ and its derivative.

The variability of $d\left(z\right)$ across SoC is crucial in the observer design discussed in Section 3. The near-zero slopes found in plateau regions lead to the introduction of an effective-slope regularization. This ensures that output sensitivity does not vanish and improves discrete-time observability.

2.3. Model Assumptions and Practical Considerations

For the discrete-time observer design, the following modeling assumptions are adopted in this paper:

• The OCV function $V_{\mathrm{o}\mathrm{c}}\left(z\right)$ is continuously differentiable and monotone, which is consistent with experimentally measured OCV, SoC curves for lithium-ion chemistries.
• The DP parameters $\left(R,R_p,R_s,C_p,{and C}_s\right)$ stay the same during each estimation window. However, they can vary from nominal values because of aging or temperature changes. This reflects parametric uncertainty.
• The measured terminal voltage is affected by limited disturbances and sensor noise. These effects are modeled as additive bounded disturbance $w_k$ in the discrete-time surface dynamics, satisfying $\left|w_k\right|\le \rho$.
• The sampling period $T_s$ is fixed and consistent with digital BMS hardware, such as 0.1 to 1 s. The stability analysis does not depend on small $T_s$ limits.

These assumptions reflect real-world BMS applications. They also align with nonlinear state estimation frameworks created for ECM-based lithium-ion models [4,6].

2.4. Motivation for a Discrete-Time Formulation

Since real BMSs operate in discrete time, it is better to use a direct discrete-time representation of the DP model instead of starting with continuous-time designs and then converting them to discrete time. Traditional SMO constructions are often created in continuous time and then converted, which may not maintain stability margins or robustness during real sampling periods. In contrast, this work develops the observer directly in discrete time. This approach ensures that the estimation dynamics match the model representation and that the Lyapunov analysis remains valid during practical sampling [25].

This modeling choice establishes the basis for the observer framework presented next.

3. Model-Based Observer Formulation

A discrete-time sliding-mode observer (SMO) is formulated directly on the DP model from Section 2. The goal is to build an estimator whose design and convergence benefits come from the model itself, not from continuous-time approximations. The observer design starts by defining a voltage-error-based sliding surface. It shapes the innovation signal to improve robustness in low-slope OCV regions [15]. It also establishes practical convergence using a discrete-time Lyapunov analysis.

The DP model is considered with state vector

$$x={\left[\begin{array}{ccc}z& v_p& v_s\end{array}\right]}^{\mathrm{\top }},$$

(5)

Current input $I$ and terminal-voltage output

$$y=h\left(x,I\right)=V_{\mathrm{o}\mathrm{c}}\left(z\right)-v_p-v_s-RI.$$

(6)

The function $V_{\mathrm{o}\mathrm{c}}\left(z\right)$ and its slope $d\left(z\right)$ are defined using the PCHIP interpolation from Section 2. This ensures a smooth, differentiable OCV, SoC mapping.

3.1. Sliding Surface and Output Error Shaping

The discrete-time sliding surface is defined as the terminal-voltage error:

$$s_k=y_k-{\widehat{y}}_k=h\left(x_k,I_k\right)-h\left({\widehat{x}}_k,I_k\right).$$

(7)

This sliding-surface formulation follows standard output error-based sliding mode observer designs [30] that are often used in battery state-of-charge (SoC) estimation and in creating reliable observers [22,23,24,25]. To make the system more resistant to noise and measurement outliers, smoothing the sliding variable, and to reduce chattering, the proposed method uses three operations:

1.

Clipping to bound large spikes;

2.

Exponentially weighted moving average (EWMA) filtering;

3.

Boundary-layer smoothing via a saturation function.

These steps are written as

$${\tilde{s}}_k=\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{p}\left(s_k;\pm s_{\mathrm{m}\mathrm{a}\mathrm{x}}\right),$$

(8)

$$s_k^{\mathrm{f}}={\alpha }_s{\tilde{s}}_k+\left(1-{\alpha }_s\right)s_{k-1}^{\mathrm{f}},$$

(9)

$$u_k^{\mathrm{s}\mathrm{g}\mathrm{n}}=\mathrm{s}\mathrm{a}\mathrm{t}\left({\displaystyle \frac{s_k^{\mathrm{f}}}{\varphi }}\right)$$

(10)

where $\varphi >0$ defines the boundary-layer width.

The boundary-layer saturation is introduced to reduce chattering. It also ensures practical convergence in discrete-time implementations of sliding-mode observers [24].

3.2. Effective-Slope Scaling and Gain Structure

The output sensitivity of the DP model is determined by $d\left(z\right)=d{V}_{\mathrm{o}\mathrm{c}}/dz$ which directly links variations in the terminal voltage to changes in the SoC state. In OCV plateau regions, this sensitivity becomes very small, leading to weak voltage–SoC coupling and practical loss of observability for voltage-based observers. To address this issue and maintain reliable state estimation across the full SoC range, an effective slope is introduced

$$d_{\mathrm{e}\mathrm{f}\mathrm{f}}\left({\widehat{z}}_k\right)=d\left({\widehat{z}}_k\right)+d_0,\hspace{1em}{d}_0>0,$$

(11)

which guarantees non-vanishing sensitivity for all ${\widehat{z}}_k$. Here, $d_0$ is a small positive design constant introduced to regularize the OCV slope in plateau regions, ensuring a nonzero effective sensitivity and preserving output observability.

The observer gains use this effective slope as

$$\begin{array}{rrrr}{L}_k& =\left[\begin{array}{c}{k}_{z,\mathcal{l}}\hspace{0.17em}{d}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left({\widehat{z}}_k\right)\\ -k_{p,\mathcal{l}}\\ -k_{s,\mathcal{l}}\end{array}\right],& G_k& =\left[\begin{array}{c}{k}_{z,s}\hspace{0.17em}{d}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left({\widehat{z}}_k\right)\\ -k_{p,s}\\ -k_{s,s}\end{array}\right],\end{array}$$

(12)

where $L_k$ shapes the continuous correction and $G_k$ shapes the discontinuous term. The signs ensure that positive voltage error increases the estimated SoC and decreases the estimated polarization voltages.

3.3. Observer Update and Bias Compensation

The core observer update law is

$$\begin{array}{r}{\widehat{x}}_{k+1}={\widehat{x}}_k+T_s\left(f\left({\widehat{x}}_k,I_k\right)+L_k{s}_k^{\mathrm{f}}+G_k{u}_k^{\mathrm{s}\mathrm{g}\mathrm{n}}\right),\end{array}$$

(13)

where $f\left({\widehat{x}}_k,I_k\right)$ indicates the DP model drift calculated at the estimated state. Since boundary-layer smoothing introduces a slight steady-state bias, this method includes two compensators.

First, an integral correction for SoC:

$${\widehat{z}}_{k+1}\leftarrow {\widehat{z}}_{k+1}+T_s{k}_{z,\mathrm{i}\mathrm{n}\mathrm{t}}\hspace{0.17em}{\gamma }_k\hspace{0.17em}{s}_k^{\mathrm{f}},$$

(14)

which reduces residual SoC error inside the boundary layer.

Second, a leaky Coulomb-counting correction:

$$z_{\mathrm{c}\mathrm{c},k+1}={\Pi }_{\left[0,1\right]}\left(z_{\mathrm{c}\mathrm{c},k}-{\displaystyle \frac{T_s{I}_k}{C_n}}\right),$$

(15)

$${\widehat{z}}_{k+1}\leftarrow {\widehat{z}}_{k+1}+T_s{k}_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}}\left(z_{\mathrm{c}\mathrm{c},k+1}-{\widehat{z}}_{k+1}\right)$$

(16)

which compensates long-term drift and constrains SoC to $\left[0,1\right]$ via projection ${\Pi }_{\left[0,1\right]}\left(\cdot \right)$. Together, these mechanisms yield a robust, tuning-light observer compatible with embedded implementations.

3.4. Surface Increment and Discrete-Time Stability

Define the Jacobian of the output map as

$$h_x\left({\widehat{x}}_k\right)=\left[\begin{array}{ccc}d\left({\widehat{z}}_k\right)& -1& -1\end{array}\right].$$

(17)

The exact discrete-time increment of the sliding variable is

$$\mathit{\Delta }{s}_k=s_{k+1}-s_k=-T_s{\alpha }_k{s}_k^{\mathrm{f}}-T_s{\beta }_k{u}_k^{\mathrm{s}\mathrm{g}\mathrm{n}}+T_s{w}_k,$$

(18)

where $\left|w_k\right|\le \rho$ captures modeling and measurement mismatch, and

$${\alpha }_k=k_{z,\mathcal{l}}\hspace{0.17em}{d}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left({\widehat{z}}_k\right)\hspace{0.17em}d\left({\widehat{z}}_k\right)+k_{p,\mathcal{l}}+k_{s,\mathcal{l}},$$

(19)

$${\beta }_k=k_{z,s}\hspace{0.17em}{d}_{\mathrm{e}\mathrm{f}\mathrm{f}}\left({\widehat{z}}_k\right)\hspace{0.17em}d\left({\widehat{z}}_k\right)+k_{p,s}+k_{s,s}$$

(20)

Using the Lyapunov function $V_s\left(k\right)=1/2s_k^2$, the discrete-time limits for the reaching phase and boundary-layer behavior are derived. Outside the boundary layer, finite-step reaching occurs whenever ${\beta }_k>\rho$. Inside, the shaped innovations ensure that the error trajectory is exponentially stable. It has a practical bound that relates to the layer width $\varphi$, disturbance level $\rho$, and sampling period $T_s$. The details are in Appendix A and depend on discrete-time Lyapunov reasoning [24,25,30].

4. Model Validation and Numerical Experiments

This section confirms the proposed observer by using a numerical experiment framework that matches the modeling assumptions from Section 2. The goals are to (i) check that the discrete-time observer works consistently with the DP model under realistic operating conditions and (ii) show how the modeling choices, specifically the PCHIP OCV representation and effective-slope regularization, affect estimation performance. All simulations and plotting were performed in MATLAB (Version 224a, MathWorks, Natick, MA, USA) in discrete time, reflecting embedded BMS operation.

4.1. DP Parameters and Operating Conditions

The DP model parameters come from experimental characterization of a lithium-ion cell. For simulation studies, representative parameter values were selected to reflect the typical characteristics of commercial 21700-format lithium-ion cells and to be consistent with the Samsung INR21700–50E cell used in the experimental validation. The ohmic resistance, short-term polarization branch $\left(R_p,C_p\right)$, long-term polarization branch $\left(R_s,C_s\right)$, and the nominal capacity $C_n$ are given by

$$\begin{gathered} R=102.5\hspace{0.17em}\mathrm{m}\mathsf{\Omega }, R_p=4.96\hspace{0.17em}\mathrm{m}\mathsf{\Omega }, R_s=2.86\hspace{0.17em}\mathrm{m}\mathsf{\Omega }, \\ C_p=4.93\hspace{0.17em}\mathrm{k}\mathrm{F}, C_s=14.33\hspace{0.17em}\mathrm{k}\mathrm{F}, C_n=5\hspace{0.17em}\mathrm{A}\mathrm{h}. \end{gathered}$$

(21)

These values represent the main time constants that affect battery dynamics. They include the immediate ohmic drop, quick polarization, and slow diffusion-driven relaxation.

The OCV function and its derivative are built using the PCHIP method from Section 2. This approach guarantees a smooth and monotonic relationship between voltage and SoC, making it suitable for observer design.

4.2. OCV Curve and Excitation Profile

Figure 2 shows the interpolated OCV–SoC curve used in all experiments. It also includes the pulsed-discharge current profile. The 5 A excitation pulses have a period of 500 s and a duty cycle of 30%. This is a standard stress pattern for checking observability and state-estimation performance in DP-based models.

This profile excites both fast and slow RC dynamics and creates voltage relaxation segments that are challenging for SoC estimation, especially in plateau regions.

4.3. Observer Benchmarking and Comparative Evaluation

The proposed discrete-time SMO is evaluated against two benchmark SoC estimators widely used in the battery-modeling literature:

• Extended Kalman filter (EKF): a linearization-based method sensitive to small OCV slopes and model mismatch.
• Switching-gain adaptive SMO (SGASMO) [21]: a nonlinear observer with adaptive-gain tuning that offers strong robustness at the cost of higher complexity.

All estimators operate on the same DP plant driven by the current profile of Figure 2. Figure 3 compares the terminal voltage. The proposed SMO converges faster in SOC estimation, and it closely tracks the true voltage with smooth transitions and limited chattering, whereas the EKF shows higher error in plateau regions where $d{V}_{\mathrm{o}\mathrm{c}}/dz$ is small. SGASMO also performs well, but it has more switching activity due to its adaptive gain mechanism.

Figure 4 shows the estimated SoC trajectories. The proposed SMO keeps high accuracy throughout the entire SoC range, including the plateau areas. EKF convergence slows down when $d\left(z\right)$ is small. SGASMO achieves similar accuracy, but it requires more complex tuning and is sensitive to discretization.

Finally, Figure 5 compares the reconstructed internal polarization voltages. Accurate recovery of the fast ($v_p$) and slow ($v_s$) dynamics is important for model-consistent SoC estimation. The proposed SMO yields stable and accurate internal-state estimates, outperforming the EKF and closely matching SGASMO.

4.4. Discussion of Modeling-Driven Behavior

The numerical results support two main modeling insights:

• Effective-slope regularization is crucial. It prevents loss of observability in OCV plateau regions and ensures that the sliding-mode correction remains active even when $d\left(z\right)$ is small.
• Discrete-time formulation improves consistency. SGASMO, originally developed in continuous time, shows mild discretization artifacts, whereas the proposed SMO remains stable and consistent at every sampling instant because its analysis is carried out in discrete time.

Overall, the experiments confirm that the proposed observer provides model-consistent, tuning-light, and computationally efficient SoC estimation suitable for real-time embedded BMS implementations.

5. Experimental Validation and Parameter Identification

This section presents the experimental workflow used to obtain real cell parameters for the dual-polarization (DP) model and to validate the accuracy of the proposed framework with real cell parameters. A commercially available Samsung INR21700–50E lithium-ion cell (nominal capacity 4900–5000 mAh; NCA chemistry; Samsung SDI Co., Ltd., Yongin-si, South Korea) was tested using a Squidstat Cycler (Admiral Instruments, Tempe, AZ, USA), shown in Figure 6. All experiments were conducted at room temperature (25 ± 1 °C).

5.1. Experimental Setup

The Squidstat Cycler offers high-resolution control of current and voltage with sub-millivolt and milliampere precision. This allows for reproducible hybrid pulse power characterization (HPPC) measurements and OCV–SoC mapping. The cell was placed in a precise cell holder, which ensures consistent contact resistance and proper thermal conduction.

5.2. OCV–SoC Characterization

The cell was charged with a constant-current (CC) step of 70 mA to 4.20 V to generate the static open-circuit-voltage (OCV) curve, illustrated in Figure 7. A cut-off of 20 mA was used. After charging, there was a 28 min rest period to let the terminal voltage relax. Then, the cell was discharged in 70 mA CC steps down to lower SoC levels, reaching a voltage limit of 2.5 V and a cut-off of −20 mA. Each discharge step included a 28 min rest period. During these rests, the data-sampling interval was shortened to 1 min to monitor the approach to equilibrium.

Repeating this charge-rest-discharge-rest cycle across the full SoC range generated a set of equilibrium voltage measurements. These measurements were used to create the OCV lookup table. As mentioned in Section 2.2, these values were fitted using a shape-preserving PCHIP interpolant. This method guaranteed monotonicity and smooth, physically consistent OCV–SoC derivatives for the discrete-time observer.

5.3. HPPC Test Protocol

A hybrid pulse power characterization (HPPC) profile was carried out to identify the dual-polarization (DP) model parameters. The experiment took place on the Samsung 21700-50E cell using the Squidstat Cycler. It followed a repeated sequence of 10 s galvanostatic discharge pulses at about 4.9 A (approximately 1C equivalent), each followed by a 50 s rest period. This pulse-rest structure was repeated throughout the usable state-of-charge (SoC) range as the terminal voltage dropped under load. This produced a representative set of voltage transients and relaxation curves, which were suitable for estimating the ohmic and polarization parameters.

The resulting current and terminal-voltage waveforms are shown in Figure 7. For completeness, the right side of the figure shows the OCV–SoC curve obtained earlier from the dedicated OCV characterization procedure described in Section 5.2.

5.4. Parameter Identification Using PSO + LSQ

To extract the DP model parameters

$$\theta =\left[R_0,\hspace{0.17em}{R}_1,\hspace{0.17em}{\tau }_1,\hspace{0.17em}{R}_2,\hspace{0.17em}{\tau }_2\right],$$

This study employed a two-stage identification strategy combining global and local optimization:

• Global search: Particle Swarm Optimization (PSO);
• Local refinement: Least-squares (LSQs)/Levenberg–Marquardt.

The approach follows best practices reported in the PSO and ECM literature [31]. PSO explores the nonconvex search space and provides coarse estimates; LSQ refines these parameters to minimize weighted voltage residuals.

5.5. Identified Parameters

Table 1. Identified DP parameters for the Samsung 21700–50E cell.

Parameter	${\mathit{R}}_0$ ($\mathbf{m}\mathsf{\Omega }$)	${\mathit{R}}_1$ ($\mathbf{m}\mathsf{\Omega }$)	${\mathit{C}}_1$ (F)	${\mathit{\tau }}_1$ (s)	${\mathit{R}}_2$ ($\mathbf{m}\mathsf{\Omega }$)	${\mathit{C}}_2$ (F)	${\mathit{\tau }}_2$ (s)
PSO Result	73.98	2.8334	586.57	1.662	14.129	2311.5	32.659
LSQ Result	73.98	2.8339	586.69	1.6626	14.129	2311.7	32.663

reports the final DP model parameters averaged over valid pulses, pulses meeting the voltage window. Both PSO and LSQ results are included. Fitting each pulse separately allowed for the evaluation of parameter sensitivity to pulse selection. The observed consistency across pulses indicates that the final averaged parameters are not significantly impacted by the inclusion of individual pulses.

5.6. Pulse Reconstruction Accuracy

To evaluate the quality of the identified DP parameters, the measured HPPC pulse response was compared with the reconstructed voltage obtained from the PSO and LSQ parameter sets. Figure 8 shows the measured voltage for a representative pulse, together with the simulated responses of the DP model using PSO-only parameters and the LSQ-refined set.

As expected, both methods capture the fast voltage drop and subsequent relaxation dynamics, while the LSQ refinement produces a noticeably closer match in the first few seconds after the current transition, where $R_0$, $R_1$, and $C_1$ dominate the transient. The slow relaxation tail, which is mostly controlled by $\left(R_2,{\tau }_2\right)$, is also a good fit for both parameter sets, with LSQ showing somewhat better tracking. Overall, the improvement introduced by LSQ is modest but consistent, indicating that PSO already converges near the optimum and that LSQ primarily performs a local fine-tuning of the identified parameters.

These results confirm that the identified DP parameters accurately reproduce the pulse dynamics and are suitable for use in the numerical observer validation presented in Section 4. Local least-squares refinement also reduced the overall voltage fitting error by a small but consistent margin relative to PSO-only estimates, with improvements most apparent immediately after the current step and during the subsequent relaxation. This behavior is consistent with the LSQ stage fine-tuning parameters governing fast polarization dynamics and long-term relaxation without altering the global optimum identified by PSO.

5.7. Long-Duration Pulse-Discharge Validation

To validate the observer under a simple and sustained excitation profile, the cell was subjected to a 3000 s pulse-discharge test consisting of repeated 120 s discharge pulses at 1.5 A followed by 120 s rest intervals.

Figure 9 shows the resulting SoC estimation performance. The proposed SMO maintains accurate, stable tracking throughout all pulse-rest cycles, while the EKF exhibits noticeable drift and SGASMO shows higher fluctuation due to switching-gain behavior.

Figure 10 illustrates the reconstructed internal RC polarization states. The proposed SMO accurately recovers both fast and slow polarization components across the entire test, with reduced noise and drift compared with EKF and SGASMO.

These results demonstrate that the identified DP model and the proposed discrete-time SMO remain stable, drift-resistant, and accurate under extended pulse-rest cycling. The observer preserves consistent SoC accuracy and internal state reconstruction over long horizons, confirming suitability for real-time embedded BMS operation.

6. Conclusions

This paper presents a unified model-based framework for estimating the state of charge (SoC) in lithium-ion batteries. It combines discrete-time dual polarization (DP) modeling, structural regularization, and nonlinear observer design. The approach starts with modeling. The DP equivalent circuit is formulated in discrete time, and its open-circuit voltage (OCV) to SoC relationship is reconstructed using a shape-preserving PCHIP interpolant. This ensures that the relationship remains monotonic, with physically consistent derivatives. This representation introduces effective slope regularization to address observability loss in the OCV plateau regions. This phase in the modeling process is very important for getting reliable behavior from a discrete-time observer.

Using this structured model, this study proposes a tuning-light boundary-layer sliding-mode observer with output-error shaping, model-driven gain scaling, and simple bias-compensation terms. Unlike continuous-time SMO designs, this method is analyzed completely in the discrete domain. This way, the Lyapunov-based convergence guarantees connect directly with digitally sampled BMS hardware. The resulting estimator avoids adaptive-gain dynamics, reduces chattering, and makes tuning much simpler compared to traditional sliding-mode and Kalman-based methods.

Numerical experiments indicated that the observer accurately reconstructs terminal voltage, internal RC-state dynamics, and SoC across the full charge range, including areas of weak observability. Its performance is similar to that of an adaptive SMO with less chattering and faster than an EKF with less steady-state error. These results highlight how clear modeling choices, particularly slope regularization and monotone OCV interpolation, can significantly improve estimation performance with moderate algorithm complexity.

The current study is based on several practical assumptions. Specifically, the dual-polarization model parameters are identified offline under fixed ambient conditions. Factors like temperature dependence, hysteresis, aging-related parameter drift, and online adaptation are not explicitly considered. These elements may affect long-term performance and will be addressed in future research.

Combining these effects with the proposed discrete-time observer may allow for unified modeling and control strategies for improved charging protocols, health-aware BMS operation, and embedded real-time implementations.