# Time-domain system identification using fractional models from non-zero initial conditions applied to Li-ion Batteries

Abderrahmane Adel, Rachid Malti, Olivier Briat

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Abderrahmane Adel, Rachid Malti, Olivier Briat. Time-domain system identification using fractional models from non-zero initial conditions applied to Li-ion Batteries. Annual Reviews in Control, 2025, 60, pp.101017. 10.1016/j.arcontrol.2025.101017. hal-04843985v2

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# Time-domain system identification using fractional models from non-zero initial conditions applied to Li-ion Batteries☆

Abderrahmane Adel*, Rachid Malti, Olivier Briat

Univ. Bordeaux, CNRS, Bordeaux-INP, IMS – UMR 5218, 351, cours de la Libération, Talence, 33400, France

## ARTICLE INFO

### Keywords:

System identification

Equivalent circuit model

Initial conditions

Parameters estimation

Time-domain identification

Constant phase element

Fractional model

## ABSTRACT

The main contribution of this paper is to present two distinct algorithms for fractional system identification using non-zero initial conditions, by assuming the input signal prior to $t = 0$ and the input/output signals after $t = 0$ known. Addressing this problem is particularly important, in the context of short-time data acquisition, mainly because the effect of the free response is important compared to the forced one and because the time response of fractional systems converge polynomially, as compared to the exponential convergence of rational systems. The first developed algorithm uses a two-stage iterative procedure that computes system forced response at the upper stage, and system parameters at the lower stage using the forced response. The second one uses the simultaneous contribution of system free and forced responses. The efficacy of both algorithms is first assessed using Monte Carlo simulations with significant signal to noise ratios. The proposed algorithms allow solving a technical issue on commercial battery cells: their identification using input-output data whatever their history, i.e. the battery cells need not be in a completely relaxed state (with zero initial conditions) prior to collecting system identification data, contrary to the actual practice.

## 1. Introduction

There are various system identification methods that use fractional models and zero initial conditions (Deniz et al., 2020; Gabano et al., 2015; Lai et al., 2017; Victor et al., 2013; Wang et al., 2015). The initial conditions are often assumed to be zero for the sake of simplicity, which might have a limited effect when the amount of data is big. However, in case of short-time data acquisition, ignoring initial conditions leads to significantly biased parameter estimation when the initial conditions are varied (Du et al., 2016; Lorenzo & Hartley, 2008; Trigeassou & Maamri, 2011).

Handling initial conditions in fractional calculus has been solved in the general case by Lorenzo and Hartley (2008) and Trigeassou and Maamri (2011), see also the book by the same authors (Trigeassou & Maamri, 2019).

Yet, most existing system identification methods with non-zero initial conditions rely on the Caputo derivative (Chikrii & Matychyn, 2011; Piotrowska & Rogowski, 2019; Vatsala & Sambandham, 2024), which inherently limits the initial conditions to the very specific case of constant past output and may lead to biased parameter estimation when system past is non-constant. The algorithms developed in this paper overcome this limitation by considering any past signal and hence a more generalized initialization framework.

The developed methodology has straightforward applications in the identification of Li-ion batteries for the sake of predicting their State Of Charge (SOC) or State Of Health (SOH) (Wang et al., 2022). When battery cells operate in dynamic conditions or are brought to specific SOC levels, a long relaxation period is typically required before traditional methods can be applied (Adel, Malti, Briat, 2024; Barai et al., 2015; Koga & Krstic, 2022; Messing et al., 2019; Wang et al., 2016). The developed methods allow avoiding such a long relaxation periods and provide a more accurate parameter estimation, making it particularly suitable for real-time battery diagnosis and monitoring.

In parallel with physics-based identification, recent contributions address battery state estimation and parameter tracking using complementary methodologies. Wang et al. (2024) combine improved parameter identification with a radial-basis correction differential support vector machine to increase the accuracy of state-of-charge estimation under dynamic profiles. Zeng et al. (2024) develop an asynchronous fractional-order thermo-electric coupling model with frequency-characteristic separation to enhance robustness at low temperatures. Wang et al. (2025) integrate an improved particle swarm optimizer with an adaptive dual extended Kalman filter and feedback correction for joint estimation of state of charge and state of energy. These works emphasize estimator design and thermal coupling, whereas the present

<footnote>☆ This article is part of a Special issue entitled: 'FDA-Modeling and System identification' published in Annual Reviews in Control.</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: abderrahmane.adel@ims-bordeaux.fr (A. Adel), rachid.malti@ims-bordeaux.fr (R. Malti), olivier.briat@ims-bordeaux.fr (O. Briat).</footnote>
study focuses on time-domain identification of fractional models with non-zero initial conditions.

This paper presents two system identification algorithms able to compute unbiased parameters in presence of non-zero initial conditions, whatever past input and output signals (not necessarily constant). It is organized as follows. First of all, some basics on handling initial conditions for rational and fractional systems are recalled and the difficulty of handling the fractional case is explained. Then, the problem is formulated under some hypotheses and the main contributions of the paper are stated. The case-study application is also introduced in this section prior to describing the main contribution of the paper with two developed algorithms in Section 2. Their performance is tested in Monte Carlo simulation in Section 3 and applied to a real case study in Section 4 before concluding.

## 1.1. Initial conditions in fractional calculus

Consider a linear system (rational or fractional) characterized by a transfer function $H(s)$. Basically, its time-domain response to an input signal $u(t)$ is governed by the following convolution integral, which clearly points out the contribution to the future output $y(t)$ for $t \ge 0$, of both the free response due to the past input $u(t)$ for $t < 0$, and the forced response due to the future input $u(t)$ for $t \ge 0$ (Trigeassou & Maamri, 2019):

$$y(t) = \underbrace{\int_{a}^{0} h(t-x)u(x)dx}_{\varphi(t) \equiv \text{free response}} + \underbrace{\int_{0}^{t} h(t-x)u(x)dx}_{\text{forced response}} \quad (1)$$

The concept of free response, known also as the initial condition response, refers to how a system reacts to displacements of its variables from their equilibrium position. In rational (integer-order) systems, it is well established that the initial values are all defined at $t = 0$, whatever the past trajectories ($t < 0$) of system variables which allowed reaching these initial values (Ljung, 2010). Take for example the RC-circuit of Fig. 1 subject to the input signal, plotted in red, in Fig. 2. Then, its transfer function, computed from the circuit impedance, writes:

$$H(s) = \frac{Y(s)}{U(s)} = \frac{1}{1 + \tau s} \quad (2)$$

where $Y(s)$ and $U(s)$ are the Laplace transforms of the voltages $y(t)$ and $u(t)$, indicated in Fig. 1, and $\tau = RC$. By computing the inverse Laplace transform of (2), one may obtain the impulse response $h(t) = \frac{1}{\tau} \exp(-t/\tau)$, and consequently reduce (1) to:

$$y(t) = y_0 \exp\left(\frac{-t}{\tau}\right) + 1 - \exp\left(\frac{-t}{\tau}\right) \underbrace{\text{free response}}_{\text{forced response}} \quad (3)$$

where $y_0 = (1 - \exp(a/\tau))$ represents the initial condition in this example. It is fully characterized by the initial value $y(0) = y_0$, whatever the possible past trajectories of $y(t)$, $t < 0$. When dealing with fractional differential equations, extra care is necessary due to their infinite memory behavior and the polynomial convergence of their free response as compared to the exponential convergence of rational systems (Martynyuk & Ortigueira, 2015; Poinot & Trigeassou, 2003). The free response not only depends on the initial values of system variables but also on the past trajectory of these variables. Instead of a finite number of initial conditions, fractional systems require using an infinite number of initial values, usually gathered in an initialization function. Two kinds of initialization functions are proposed in the literature. The first one is based on a time-domain “history-function” (Lorenzo, 2000; Lorenzo & Hartley, 2008) and the second one on the frequency-domain “infinite state approach” (Trigeassou & Maamri, 2011). Both approaches show similar results on numerous examples (Hartley et al., 2013). They are presented in the sequel on a simple example borrowed from Hartley et al. (2013). Consider the circuit of Fig. 3 with a constant phase element (CPE) having the impedance $\frac{1}{Qs^\alpha}$, with $0 < \alpha < 1$. The

<figure><img src="image_2.png" /><figcaption>Fig. 1. RC-circuit.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Input signal $u(t)$ in red and output signal $y(t)$ in blue, the blue dot indicates the initial condition $y_0$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3. R-CPE-circuit.</figcaption></figure>

case $\alpha = 1$ describes a capacitor, $\alpha = 0$ a resistor, and $\alpha = 0.5$ Warburg impedance encountered in battery modeling (Cruz-Manzo & Greenwood, 2020; dos Santos et al., 2022). This circuit is governed by the fractional differential equation:

$$\tau_c \frac{d^\alpha}{dt^\alpha} y(t) + y(t) = u(t) \quad (4)$$

where $\tau_c = RQ$. Its transfer function may be computed from (4):

$$H(s) = \frac{Y(s)}{U(s)} = \frac{1}{1 + \tau_c s^\alpha} \quad (5)$$

Let $h(t) = \mathcal{L}^{-1}\{H(s)\}$ be the impulse response of the new circuit. The convolution product (1) yet holds for fractional systems. However, it becomes more difficult to compute the free response integral in the fractional case. The main approach developed in the literature is based on the infinite state approach.

### 1.1.1. Infinite-state approach (Trigeassou and Maamri)

In this approach (Trigeassou & Maamri, 2011), the free response of the fractional differential equation (4), is solution of the homogeneous
equation (without the input $u(t)$, for $t > 0$):

$$ \tau_c \frac{d^a}{dt^a} y(t) + y(t) = u'(t) \text{ with } u'(t) = \begin{cases} u(t) & \text{for } t < 0 \\ 0 & \text{for } t \ge 0 \end{cases} \qquad (6) $$

It may be expressed as a distributed differential system:

$$ \left\{ \begin{aligned} \frac{\partial z(\omega, t)}{\partial t} &= -\omega z(\omega, t) - \frac{1}{\tau_c} y(t) \\ y(t) &= \int_0^\infty \mu_a(\omega) z(\omega, t) d\omega \end{aligned} \right. \qquad (7) $$

where $\mu_a(\omega)$ is a weighting function defined in the case of an integrator by

$$ \mu_a(\omega) = \frac{\sin(\alpha\pi)}{\pi} \omega^{-\alpha} \qquad (8) $$

The distributed initial conditions are gathered in the function $z(\omega, 0)$, which depends on all the frequencies $\omega$ and verifies the initial value:

$$ y(0) = \int_{0}^{\infty} \mu_{a}(\omega) z(\omega, 0) d\omega \qquad (9) $$

### 1.1.2. History-function (Hartley and Lorenzo)

The history function approach, due to Lorenzo (2000) and Lorenzo and Hartley (2008), corresponds to the application of the input/output methodology. The authors compute the convolution integral

$$ \varphi(t) = \int_{a}^{0} h(t-x)u(x)dx \qquad (10) $$

in the case of elementary systems of the first kind (Malti, Moreau et al., 2011) by setting constant and/or ramp history functions in order to be able to formulate analytical expressions, based on the Mittag-Leffler and the incomplete Gamma functions, of $\varphi(t)$ and its Laplace transform.

### 1.2. Problem formulation

The problem addressed in this paper is related to system identification using linear fractional models from non-zero initial conditions under the following hypothesis:

(H1) The past ($t < 0$) and the future ($t \ge 0$) of the input signal $u(t)$ are known.

(H2) Only the future ($t \ge 0$) of the output signal $y(t)$ is known and the past of $y(t)$ ($t < 0$) is supposed to be unknown.

(H3) Model structure is known.

(H4) Input signal is noise-free; output signal may be affected by noise.

Hence, model parameters and differentiation orders are unknown and estimated. When model structure is unknown (i.e. when (H3) is invalidated), searching for the model structure may be achieved by using some AIC-like criteria. If input data are noisy (hypothesis (H4) invalidated), then the proposed algorithms may be extended by using Error-In-Variables (EIV) methodology.

### 1.3. Main contributions

The main novelty of this paper lies in the development of two distinct algorithms for time-domain fractional system identification of Li-ion batteries, that explicitly take into account non-zero initial conditions. Unlike conventional approaches that require long relaxation periods for the transient effects to vanish, the proposed methods estimate system parameters by handling initial condition effects.

Hence, the key contributions of this paper are:

* Two methods are developed to solve the problem formulated in 1.2.

- The first one is based on a two stage algorithm allowing separation of the free and the forced responses on each stage, progressively refining parameter estimates,

- the second one jointly estimates parameters by considering both free and forced responses simultaneously, leading to an unbiased estimation.

* The accuracy and the computational efficiency of both methods is assessed.

* A technical issue related to Li-ion batteries is solved. It avoids waiting for long periods of time for the batteries to be in a completely relaxed state, with a vanished free response, prior to collecting identification/validation data.

* Experimental validation is conducted on three commercial batteries.

Through the development and validation of these innovative techniques, this study aims to advance the field of battery characterization with broader applications and improved performance.

### 1.4. Motivations in Li-ion battery modeling

One of the most used system identification techniques is based on Electrochemical Impedance Spectroscopy (EIS) which consists in performing a harmonic analysis with a sine excitation current and measuring the output voltage across the electrodes. Then, the current and voltage signals are used for computing the impedance spectrum, fitted to a fractional order equivalent circuit model (FO-ECM) (Aghdam et al., 2022; Eddahech et al., 2012; Reed et al., 2022). A typical impedance spectrum of a commercial battery cell is plotted in Fig. 4. Low frequencies $0.1 < f < 1$ Hz, characterizing the diffusive Warburg phenomenon, may be modeled by a Constant Phase Element, $CPE_2$, in Fig. 5 which impedance is:

$$ Z_d(s) = CPE_2 = \frac{1}{Q_d s^\beta} \qquad (11) $$

where $Q_d$ is a coefficient, and $\beta$ is an integration order equal to 0.5 when Warburg impedance is considered and might be further adjusted from input-output data. Mid-range frequencies $1 < f < 10^3$ Hz, related to charge transfer phenomenon, are usually modeled by another constant phase element, $CPE_1$, see e.g. Arahbi et al. (2023), characterized by a constant $Q_{d1}$ and a differentiation order $\alpha$, in parallel with the resistor $R_{ct}$:

$$ \begin{aligned} Z_{ct}(s) &= R_{ct} \parallel CPE_1, \quad \text{with} \quad CPE_1 = \frac{1}{Q_{d1} s^{\alpha}} \\ Z_{ct}(s) &= \frac{R_{ct}}{1 + R_{ct} Q_{d1} s^{\alpha}} \end{aligned} \qquad (12) $$

The high frequency $f \approx 10^3$ Hz, corresponding to the intersection of the impedance with the real axis in Fig. 5, corresponds to the ohmic resistor of the battery, $R_0$ in Fig. 5. Beyond this frequency, $f > 10^3$ Hz, an inductive phenomenon appears characterized in Fig. 4 by negative values of the y-coordinate (or positive values of $\mathcal{I}m(z)$). The inductive phenomenon is not considered in this paper and the batteries are modeled in a frequency range up to $10^3$ Hz. Hence, the overall impedance of the Li-ion battery, as illustrated in Fig. 5, is the combination of the three considered effects:

$$ \begin{aligned} Z(s) &= R_0 + Z_{ct}(s) + Z_d(s) \\ &= R_0 + \frac{R_{ct}}{1 + R_{ct} Q_{d1} s^{\alpha} + \frac{1}{Q_d s^{\beta}}} \\ &= \frac{R_0 R_{ct} Q_{d1} s^{\beta+\alpha} + (R_0 + R_{ct}) s^{\beta} + \frac{R_{ct} Q_{d1} s^{\alpha}}{Q_d} + \frac{1}{Q_d}}{R_{ct} Q_{d1} s^{\beta+\alpha} + s^{\beta}}. \end{aligned} \qquad (13) $$

Let battery parameters be gathered in a vector

$$ \theta = [R_0, R_{ct}, Q_{d1}, \alpha, Q_d, \beta]^T. \qquad (14) $$
<figure><img src="image_5.png" /><figcaption>Fig. 4. Nyquist plot of a Samsung “INR18650-25R” battery impedance measured in GEIS mode at SOC 80%, 25 °C.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Global impedance model of the Li-ion battery. The rectangle highlights $Z_{ct}$, consisting of $R_{ct}$ and $CPE_1$.</figcaption></figure>

The fractional-order equivalent circuit model (FO-ECM) with two constant phase elements, as in (13) and in Fig. 5, has been used in Aghdam et al. (2022) to study the structural identifiability of such models by solving algebraic equations. One of the major issues with EIS data is the long acquisition time, as the method sweeps through 5 decades of frequencies ranging from 0.01 Hz to 1 kHz (see e.g. Fig. 4). An interesting alternative to EIS data acquisition consists in using chronopotentiometry time-domain measurements, which consists in applying an excitation current (such as a pseudo-random-binary-signal (PRBS)) and measuring the output voltage across the electrodes. Multiple Refs. Adel et al. (2023), Aghdam et al. (2022), Alavi et al. (2015), Nasser-Eddine et al. (2020) and Nasser-Eddine et al. (2019) suggest using the FO-ECM developed in Section 1.4 as impedance models and time domain data. Fractional models may be simulated in the time domain using Oustaloup (1995) approximation of fractional operators, implemented in the CRONE Toolbox (Malti & Victor, 2015). Nasser-Eddine et al. (2020) use an output-error model combined to a non-linear programming techniques to compute the model parameters. Such techniques effectively reduce acquisition time (Alavi et al., 2015; Nasser-Eddine et al., 2020); however, the battery cells yet need be at a relaxed state, as explained below.

### 1.4.1. Impact of the relaxation period

To obtain accurate impedance models, frequency-domain and/or time-domain measurements require thermodynamic steady-state conditions, which are usually obtained after a significant relaxation time when the battery is active or brought to a given SOC. Indeed, voltage relaxation (VR) stems from an electrochemical process occurring after interrupting the current flow into or from the battery which results

in a gradient of lithium ions through electrodes (Theuerkauf & Swan, 2022). The bigger the relaxation time, the lesser the gradient of ions across the electrodes. Various Refs. Baghdadi et al. (2016), Barai et al. (2015), Messing et al. (2019) and Wang et al. (2016) study the impact of the relaxation time in modeling Li-ion batteries. Wang et al. (2016) show the influence of relaxation-time on the time-domain responses to a prescribed input. Messing et al. (2019) explore the variation of cell impedance for different relaxation periods (0, 30, 60, 180, 300, 420)min, based Potentiostatic-EIS (on voltage variations at the input and current measurement at the output). It is highlighted in the aforementioned paper the necessity to wait for a sufficiently long period (up to 420 min) for a battery cell (Messing et al., 2019) to be completely relaxed. Barai et al. (2015) study the influence of relaxation time on 5 different commercial batteries, stressing the fact that the relaxation time might be as big as 900 min for certain cells. Further, characterizing batteries at different SOC-levels requires the battery to be in a relaxed state every time it is brought to the desired SOC-level, whatever the acquisition: in frequency-domain or time-domain.

This paper aims at developing system identification algorithms able to estimate parameters, using non-relaxed Input-Output data, which constitutes an important breakthrough in Li-ion batteries identification, eliminating relaxation time usually observed for the free response to vanish prior to data acquisition.

### 1.4.2. Fractional models for Li-ion battery modeling

Fractional-order models have been increasingly adopted for Li-ion battery modeling due to their ability to capture complex electrochemical dynamics that traditional integer-order models fail to represent accurately. The primary motivations for employing fractional models in battery identification are:

*   **Long-memory effects:** fractional models inherently incorporate memory effects, which are crucial in modeling the slow relaxation dynamics observed in batteries (Barai et al., 2015).

*   **Diffusive behavior representation:** the Warburg impedance, commonly used to model diffusion processes in battery electrodes, is naturally described by fractional-order elements (Aghdam et al., 2022).

*   **Improved model accuracy:** experimental studies have shown that fractional models provide a better fit to impedance and time-domain response data compared to standard equivalent circuit models (ECMs) (Cruz-Manzo & Greenwood, 2020).

*   **Reduced parametrization complexity:** fractional models require fewer parameters than high-order integer models to achieve similar accuracy, making them more practical for real-time applications (Poinot & Trigeassou, 2003).

In this study, we leverage these advantages to develop an efficient system identification approach that allows accurate battery modeling even when the cells are not in a fully relaxed state.

## 2. Contributions to system identification with non-zero initial conditions

Consider a linear fractional system characterized by a transfer function:

$$H(s, \theta) = \frac{\sum_{i=0}^{M} b_i s^{\beta_i}}{1 + \sum_{j=1}^{N} a_j s^{\alpha_j}} \quad (15)$$

or its impulse response $h(t, \theta) = \mathcal{L}^{-1}\{H(s, \theta)\}$, where the parameter vector

$$\theta = \begin{bmatrix} \rho \\ \mu \end{bmatrix} \quad (16)$$

is composed of a vector of $N + M + 1$ transfer function coefficients,

$$\rho = [b_0, b_1, \dots, b_M, a_1, \dots, a_N]^T, \quad (17)$$
<figure><img src="image_7.png" /><figcaption>Fig. 6. Output error model.</figcaption></figure>

and additionally a vector of $N + M + 1$ ordered differentiation orders,

$$ \mu = [\beta_0, \dots, \beta_M, \alpha_1, \dots, \alpha_N]^T. \qquad (18) $$

System identification using fractional models with zero initial conditions have been treated in the literature, mainly in an output error context (Malti et al., 2008; Poinot & Trigeassou, 2004), or using least squares estimates (Arahi et al., 2023) and the optimal instrumental variable (Victor et al., 2013). In the output error context, model parameters are computed by minimizing the squared $L_2$-norm

$$ J(\theta) = \|\epsilon(t, \theta)\|^2 \qquad (19) $$

of the output error:

$$ \epsilon(t, \theta) = y(t) - \hat{y}(t, \theta). \qquad (20) $$

as illustrated in Fig. 6. In case of the impedance model (13), the parameter vector $\theta$ is defined in (14). Eq. (19) can be written in a discretized form based on the forward rule as:

$$ J(\theta) \approx T_s \sum_{k=0}^{K-1} \epsilon^2(kT_s, \theta) = T_s E^T(\theta)E(\theta) \qquad (21) $$

where $T_s$ is the sampling period, $K$ is the total number of samples, and the error vector is defined as: $E(\theta) = [\epsilon(0, \theta) \ \epsilon(T_s, \theta) \ \dots \ \epsilon((K-1)T_s, \theta)]^T$.

The optimal parameter vector, minimizing (21),

$$ \hat{\theta} = \arg \min_{\theta} J(\theta), \qquad (22) $$

is usually obtained using a nonlinear optimization algorithm. The iterative Levenberg-Marquardt algorithm (Marquardt, 1963) is often used. In case of rational systems, when system output is a combination of a free and a forced response, then the initial conditions, constituted of consecutive derivatives of $y$ up to system order minus one ($n-1$) all evaluated at $t=0$, gathered in $\mathcal{Y}_0 = [y(0), \frac{dy}{dt}(0), \dots, \frac{d^{n-1}y}{dt^{n-1}}(0)]$, may be estimated along with system parameters

$$ (\hat{\theta}, \mathcal{Y}_0) = \arg \min_{\theta} J(\theta, \mathcal{Y}_0), \qquad (23) $$

by minimizing the quadratic criterion

$$ J(\theta, \mathcal{Y}_0) = \|\epsilon(t, \theta, \mathcal{Y}_0)\|^2 = \|y(t) - \hat{y}(t, \theta, \mathcal{Y}_0)\|^2. \qquad (24) $$

Take for instance, the RC-circuit of Fig. 1 and its global response in (3), then the quadratic error may be expressed in terms of the initial condition $y_0$ along with the transfer function parameter $\tau$:

$$ J(\tau, y_0) = \|\epsilon(\tau, \tau, y_0)\|^2 = \|y(t) - \hat{y}(\tau, \tau, y_0)\|^2. \qquad (25) $$

Such a formulation becomes impossible for fractional systems, because the free response depends on an infinite number of initial conditions, as explained in Section 1.1. Hence, this section introduces two novel approaches for parameter estimation of fractional models with non zero initial conditions: the separate free and forced response method and the simultaneous free and forced response method.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Flowchart of the 1st method (separate free and forced responses) estimating the fractional model of the battery considering historical data.</figcaption></figure>

## 2.1. Parameter estimation using separate free and forced responses

In this section, an iterative two-stage algorithm is proposed to minimize the output error when initial conditions are unknown. Two-stage algorithms, have been used in various domains. For instance, in the context of structure identification of Polynomial NARX models (Spinelli et al., 2006) utilize a two-stage algorithm with at the first stage coarsely identifies the model structure, and the second stage iteratively refines it. Victor et al. (2013) employ a two-stage algorithm, estimating transfer function parameters using least squares at one stage and differentiation orders using a gradient-based algorithm at the other stage. The proposed algorithm, sketched in Fig. 7, is detailed below.

### 2.1.1. Initialization

Start by initializing $iter$ at 0, and consider that the free response equals zero (the undetermined $\hat{\theta}_0$ is not required).

### 2.1.2. First stage

The main idea is to subtract from the output signal, at the first stage and from the second iteration on, the free response and to keep only the forced one:

$$ y(t, \hat{\theta}_{iter-1}) \leftarrow y(t) - \varphi(t, \hat{\theta}_{iter-1}) \qquad (26) $$

At the very first iteration, as $\varphi(t, \hat{\theta}_0) = 0$, system output is considered to be only due to the forced response which is non-consistent, yielding
biased results, in presence of free response. The estimation is less biased as the number of iterations gets bigger.<fnref n="1" />

### 2.1.3. Second stage

The basic idea of the second stage is to compute the parameter vector and once the model is known, compute its free response. Hence, estimate the parameter vector $\hat{\theta}_{iter}$ using the Levenberg-Marquardt algorithm from (19)–(22) by considering $y(t, \hat{\theta}_{iter-1})$ instead of $y(t)$ in (20):

$$ \hat{\theta}_{iter} = \arg \min_{\theta} J(\theta). \qquad (27) $$

Once $\hat{\theta}_{iter}$ is known, the transfer function and its impulse response are completely defined $\hat{H}(s, \hat{\theta}_{iter}) = \mathcal{L}\{\hat{h}(t, \hat{\theta}_{iter})\}$. Hence the model free response, is computed:

$$ \varphi(t, \hat{\theta}_{iter}) = \underbrace{\int_{a}^{0} \hat{h}(t - x, \hat{\theta}_{iter})u(x)dx}_{\text{free response}} \qquad (28) $$

and the algorithm is iterated back to the first stage by separating the free response from the whole system response as in (26).

### 2.1.4. Convergence test

The procedure is iterated as long as the norm of the difference between two successive normalized estimates is greater than a threshold $\epsilon$ ($\ell_1$-norm is used here, with $\epsilon = 10^{-4}$):

$$ \sum_{\ell=1}^{L} \left| \frac{\hat{\theta}_{iter}^{\ell} - \hat{\theta}_{iter-1}^{\ell}}{\hat{\theta}_{iter-1}^{\ell}} \right| > \epsilon, \qquad (29) $$

or a maximum number of iterations is not reached yet

$$ iter \le iter_{max}. \quad with \quad iter_{max} = 100. \qquad (30) $$

here $\ell = 1, 2, \dots, L$ represents the number of estimated parameters, $\hat{\theta}_{iter}^{\ell}$ is the $\ell$th element of the estimated parameter vector at the iteration $iter$. If the algorithm does not converge after reaching $iter_{max}$, the algorithm can either be restarted with a new initialization point to mitigate potential convergence issues associated to local minima or alternatively restarted from the estimated parameters obtained from the previous run, if considering that $iter_{max}$ is too low and the algorithm did not converge.

### 2.1.5. Convergence notes

To the best of authors' knowledge, convergence of such two-stage algorithms cannot be proven, although it is noted in multiple references (see e.g. Bianchi et al. (2020), Jiao et al. (2024) and Victor et al. (2013)) that they do often converge to a minimum.

### 2.2. Parameter estimation using simultaneous free and forced responses

In this section, a second algorithm is suggested, estimating model parameters on the basis of a simultaneous estimation of the free and the forced responses by considering straightforwardly system whole response (1). The algorithm, illustrated in Fig. 8, begins by initializing the parameter vector $\theta \leftarrow \theta_0$. Then, the global system response, constituted of the free and forced responses, is computed:

$$ y(t, \theta) \leftarrow \underbrace{\int_{a}^{0} h(t - x, \theta)u(x)dx}_{\varphi(t, \theta) \equiv \text{free response}} + \underbrace{\int_{0}^{t} h(t - x, \theta)u(x)dx}_{\text{forced response}} \qquad (31) $$

where $h(t - x, \theta)$ represents the system impulse response parameterized by the model parameter vector $\theta$. Once the total system output is

<figure><img src="image_9.png" /><figcaption>Fig. 8. Flowchart of the 2nd method (simultaneous free and forced responses) estimating the fractional model of the battery considering historical data.</figcaption></figure>

computed, the algorithm proceeds by estimating the parameter vector $\hat{\theta}$ using the Levenberg-Marquardt algorithm from (19)–(22) by considering $y(t, \theta)$ of (31) instead of $y(t)$:

$$ \hat{\theta} = \arg \min_{\theta} J(\theta). \qquad (32) $$

This method allows simultaneous consideration of both free and forced responses in estimating system fractional model, making it particularly suitable for applications like battery modeling where the past history (free response) significantly influences future behavior.

### 2.3. Significance of the proposed algorithms

It may appear at a first glance that the proposed methods are trivial extensions of existing fractional system identification techniques. However, this is not the case. Traditional methods either neglect initial conditions or treat them as additional unknown parameters to be estimated (in the case of rational systems). In fractional systems, where the free response depends on the entire past and not just a finite set of initial values, this is not possible.

The first proposed algorithm iteratively refines the free response estimation and progressively improves parameter accuracy. Parameter are estimated at each iteration using the output error method. The full model response, including both free and forced responses, is simulated using the lsim function from the CRONE toolbox, with the input composed of historical data ($t < 0$) and future data ($t \ge 0$). The free response is evaluated by setting $u(t \ge 0) = 0$, while maintaining $u(t < 0)$, and then subtracted from the total response to isolate the forced response. This estimation-refinement loop continues until parameters convergence.

The second algorithm improves the previous one by incorporating both the free and forced responses directly into the cost function, enabling simultaneous estimation of parameters and the effect of non-zero initial conditions. At each iteration of the optimization routine (lsqnonlin), both the free and forced responses are computed internally using the same simulation principles. This integration enhances convergence and estimation accuracy.

The fractional models in both approaches are simulated using Oustaloup's recursive approximation over a predefined frequency range, as provided by the CRONE toolbox (Malti, Melchior et al., 2011; Malti & Victor, 2015; Oustaloup, 1995).

<footnote n="1">¹ See Section 2.1.5 for convergence notes.</footnote>
<figure><img src="image_10.png" /><figcaption>Fig. 9. (a) Historical input signal. (b) Input signal used for system characterization. (c) Zoom on the PRBS signal.</figcaption></figure>

Furthermore, the experimental results demonstrate that using conventional output-error methods without explicitly considering free response effects leads to highly biased parameter estimates. This highlights the practical importance of explicitly accounting for initial condition effects and justifies the need for the proposed algorithms.

### 3. Simulation example on a battery impedance model

The objective of this section is to validate both algorithms described in Sections 2.1 and 2.2 on a simulation example with non-zero initial conditions. The advantages of a simulation example are (i) evaluating the effectiveness of the method, by testing whether the true parameters are recovered (ii) using Monte Carlo simulation, with 100 runs, to determine the accuracy of parameter estimation algorithms and their robustness in presence of noisy data with a signal to noise ratio of 20 dB and then 10 dB.

#### 3.1. Simulation example of a battery system

The battery model structure (13) is chosen by setting its parameters to values close to experimental ones computed in (Adel et al., 2023):

$$R_0 = 13.8 \text{ m}\Omega, \quad Q_{dl} = 6.47 \text{ F s}^{1-\alpha}, \quad \alpha = 0.7,$$

$$R_{ct} = 5 \text{ m}\Omega, \quad Q_d = 333 \text{ F s}^{1-\beta}, \quad \beta = 0.6.$$

The input current signal, plotted in Fig. 9, is constituted of a historical input signal (for $t < 0$) a charging sequence with a current amplitude of 1 A and a duration of ten minutes, followed by an identification signal constituted of fast and slow dynamics. An 8-s Pseudo-Random Binary Sequence (PRBS) is implemented for current excitation, with an amplitude of 200 mA, followed by an 8-s charge/discharge sequence with a current alternating between -200 and 200 mA. Finally, a 4 s rest period is observed. The entire sequence lasts 20 s. The sampling time is set to $T_s = 2$ ms.

In this simulation example, the output voltage is assumed to be nil before $t = -10$ min, as shown in Fig. 10(a). Theoretically, a fractional derivative depends on the entire history of the signal. However, as demonstrated in Podlubny (1998), the further the past signal, the less it contributed to the free response. Specifically, Eq. (7.5) in Podlubny (1998) provides an upper bound on the error introduced when shifting the initialization point in fractional differential equations. This concept has been successfully employed in Rauh and Kersten (2021) and Rauh and Malti (2022) to reset interval-based solution procedures and mitigate overestimation due to long simulation horizons. Therefore, assuming a steady-state prior to $t = -10$ min is practically sufficient to approximate the free response with high accuracy.

<figure><img src="image_11.png" /><figcaption>Fig. 10. Simulated battery model without relaxation period: (a) Free response. (b) Battery model and estimated model responses. (c) Zoom on the battery model and estimated model responses. (d) The estimated forced response. (e) Zoom on the forced response.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 11. Comparison on validation data between battery model output y and estimated model outputs with different initializations $\hat{y}_i$. (a) Results from the first approach. (b) Results from the second approach.</figcaption></figure>

#### 3.2. Results of both algorithms

The results are plotted in Fig. 10 and further illustrated in Table 1 which show that the parameters converge for both algorithms to the true values with very low standard deviation.

**For a SNR of 20 dB.** Both methods, described in Sections 2.1 and 2.2, converge for all simulations with a 100% convergence rate. Moreover, both algorithms provide accurate parameter estimates, with low standard deviations for all parameters, indicating good precision under moderate noise conditions.

**For a SNR of 10 dB.** The method described in Section 2.2 demonstrates superior robustness, achieving a 100% convergence rate, while the two-stage algorithm only reaches 81%. The algorithm fails to converge, when during the iterative optimization process, two differentiation orders have exactly the same value. Then, a loss of observability occurs and the algorithm fails. When the algorithm of Section 2.1 converges, it shows highly biased results on some parameters, whereas the algorithm of Section 2.2 still exhibits unbiased results and low standard deviation.
<table><caption>**Table 1**
A comparison of parameter estimation accuracy obtained from using both algorithms in case of SNR equal to 20 dB and 10 dB.</caption><thead><tr><th rowspan="3">Parameter</th><th rowspan="3">True</th><th colspan="4">Algo. Section 2.1</th><th colspan="4">Algo. Section 2.2</th></tr><tr><th colspan="2">20 dB</th><th colspan="2">10 dB</th><th colspan="2">20 dB</th><th colspan="2">10 dB</th></tr><tr><th>θ̂</th><th>σ̂<sub>θ</sub></th><th>θ̂</th><th>σ̂<sub>θ</sub></th><th>θ̂</th><th>σ̂<sub>θ</sub></th><th>θ̂</th><th>σ̂<sub>θ</sub></th></tr></thead><tbody><tr><td>R<sub>0</sub> (mΩ)</td><td>13.8</td><td>13.791</td><td>0.015</td><td>13.762</td><td>0.021</td><td>13.802</td><td>0.009</td><td>13.814</td><td>0.017</td></tr><tr><td>R<sub>ex</sub> (mΩ)</td><td>5</td><td>5.11</td><td>0.08</td><td>26.34</td><td>4.12</td><td>5.01</td><td>0.01</td><td>5.07</td><td>0.02</td></tr><tr><td>Q<sub>dl</sub> (F s<sup>1-α</sup>)</td><td>6.47</td><td>6.49</td><td>0.44</td><td>1.42</td><td>0.85</td><td>6.46</td><td>0.18</td><td>6.49</td><td>0.47</td></tr><tr><td>α</td><td>0.7</td><td>0.7041</td><td>0.0134</td><td>0.7108</td><td>0.0201</td><td>0.7012</td><td>0.0135</td><td>0.7083</td><td>0.0141</td></tr><tr><td>Q<sub>d</sub> (F s<sup>1-β</sup>)</td><td>333</td><td>320</td><td>10</td><td>287</td><td>26</td><td>334</td><td>3</td><td>330</td><td>8</td></tr><tr><td>β</td><td>0.6</td><td>0.6031</td><td>0.0044</td><td>0.5921</td><td>0.0065</td><td>0.6011</td><td>0.0041</td><td>0.6034</td><td>0.0043</td></tr><tr><td>Convergence</td><td></td><td>100/100</td><td></td><td>81/100</td><td></td><td>100/100</td><td></td><td>100/100</td><td></td></tr><tr><td>Non convergence</td><td></td><td>-</td><td></td><td>19/100</td><td></td><td>-</td><td></td><td>-</td><td></td></tr></tbody></table>

<figure><img src="image_13.png" /><figcaption>**Fig. 12.** Current and voltage profiles during preconditioning and relaxation at 85% SOC for SGg Cell. The cell is charged using a Constant Current (CC) from –55 min to –45 min and then a Constant Voltage (CV) up to –15 min and then discharged using a constant current of –1.25 A from –15 min to 0 min.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 13.** (a) Response of SGg cell to a PRBS input with no relaxation time (in red) and a PRBS with a relaxation time of 60 min (in yellow). (b) Zoom of (a) in the indicated interval. (c) Zoom of (b) in the indicated interval. (d) Zoom of (a) in the indicated interval. (e) Zoom of (d) in the indicated interval. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Input signal used for system validation.</figcaption></figure>

### 3.3. Convergence analysis for different initializations

To analyze the convergence properties of both algorithms with respect to initialization, a Monte Carlo simulation consisting of 50 runs is conducted on the battery model. For each run, both algorithms are applied on simulated data generated using the same noise sequence within the run, with a SNR of 20 dB. A different noise sequence is used at each Monte Carlo run. Each run also uses a different initial parameter vector, randomly generated from a Gaussian distribution centered around the nominal parameters used in the simulation example, with a standard deviation of 20% of each parameter value, i.e., $\sigma = 0.2 \times \theta$.

The comparison on validation data, in Fig. 11, demonstrates that both approaches exhibit good performance across different initializations.

## 4. Experimental results on real batteries

Both algorithms are applied in this section for estimating parameters of three commercial battery cells which characteristics are provided in Table 2.

### 4.1. Experimental setup

An electrochemical workstation constituted of a Biologic-BCS-815 galvanostat/potentiostat, is used to perform charge and discharge cycles. The input signal used for identification is the Pseudo-Random Binary Sequence (PRBS) of Fig. 9(b–c). Temperature variations are negligible for such short durations of experiment. As illustrated in Fig. 12, each cell is charged using a CC-CV (Constant Current Constant Voltage) protocol and then discharged at a C-rate of C/2, corresponding to a current of 1.25 A, until it is brought to a SOC level of 85% prior to data acquisition. The sampling period is set to $T_s = 2$ ms. Two Pseudo-Binary-Random-Sequences (PRBS) are generated (i) once at $t = 0$, i.e. with no relaxation time after the discharge sequence of Fig. 12, and (ii) once at $t = 60$ min, when the cell is considered to be completely relaxed (free response completely vanished). Hence, the latter experiment is considered to provide reference values for the estimated parameters. Both experimental conditions are illustrated in Fig. 12 with multiple zooms in Fig. 13. The objectives of the former experiment are to apply:
<figure><img src="image_16.png" /><figcaption>Fig. 15. Comparison on validation data between system output (y), model output considering that the whole output is due only to the forced response ($\hat{y}_0^0$) as in Adel, Malti, Vinassa et al. (2024), model output obtained by the algorithm of Section 2.1 ($\hat{y}_0^1$), model output obtained by the algorithm of Section 2.2 ($\hat{y}_0^2$), and model output obtained using a relaxation period of 60 min ($\hat{y}_{60}$).</figcaption></figure>

<table><caption>Table 2
Characteristics of three commercial battery cells used in this experiment.</caption><thead><tr><th>Cell ref</th><th>SGg</th><th>LGb</th><th>SGp</th></tr></thead><tbody><tr><td>Reference</td><td>INR1865025R</td><td>INR18650HG2</td><td>ICR1865026F</td></tr><tr><td>Manufacturer</td><td>Samsung</td><td>LG Chem</td><td>Samsung</td></tr><tr><td>Chemistry</td><td>NCA</td><td>NMC</td><td>LCO, NMC</td></tr><tr><td>V<sub>nom</sub> (V)</td><td>3.6</td><td>3.6</td><td>3.7</td></tr><tr><td>C<sub>nom</sub> (Ah)</td><td>2.5</td><td>3</td><td>2.6</td></tr><tr><td>Standard charge CC-CV</td><td>1.25 A 4.2 V 125 mA</td><td>1.5A 4.2V 50 mA</td><td>1.3 A 4.2 V 520 mA</td></tr></tbody></table>

* the output error system identification method that assumes the presence of only the forced response, to show how biased it is,

* the system identification algorithms with non-zero initial conditions developed in this paper to show that the bias is eliminated.

The output identification data obtained for one of the three cells are plotted in Fig. 13 (data of other cells exhibit quite similar dynamics). This figure presents data from both experiments: without and with a 60 min relaxation period. Validation input data are depicted in Fig. 14.

## 4.2. Experimental results and discussion

The %fit is used as a metric. It is defined as

$$ \%fit = \left( 1 - \sqrt{\frac{E^T(\theta)E(\theta)}{(y - \text{mean}(y))^T(y - \text{mean}(y))}} \right) 100 \quad (33) $$

where the error vector $E(\theta) = [\epsilon(0, \theta), \dots, \epsilon(K-1, \theta)]^T$ and $\epsilon$ is the output error from (20).

Fig. 15(a,b and c) and Table 3 present the validation results on parameter estimation with both algorithms applied on the three battery cells. The obtained results using a relaxation period of 60 min ($\hat{\theta}_{60}$

in Table 3) have a %fit between 98.4 and 99.9%. In the case of a relaxation period of 60 min, both proposed approaches yield similar estimated parameters, aligned with those obtained by the method presented in Adel, Malti, Vinassa et al. (2024), labeled here as $\theta_{60}$, as shown in Table 3.

Additionally, the results obtained using the two methods presented in this paper are compared to the one that uses the output error method by considering that the whole output is due only to the forced response as in Adel, Malti, Vinassa et al. (2024). In the latter case, the parameter estimation is very biased (see $\hat{\theta}_0^0$ in Table 3) and its corresponding %fit is the worse which explains why a quite long relaxation period is usually observed in the literature prior to data acquisition. The %fit obtained on identification data is lower than the %fit obtained on validation data because identification data cover a wider frequency range, as compared to validation data. Both algorithms developed in this paper perform quite well with a higher %fit rate for the second algorithm, also confirmed in Fig. 15. Additionally, the second algorithm has closer parameters to the ones computed by observing 60 min relaxation period, which provides another way for validating the performance of the proposed algorithms. Furthermore, the zoomed-in view around 8s highlights how well both algorithms converge, demonstrating their ability to accurately capture the system dynamics.
<caption>**Table 3**</caption>

Comparison of parameter estimation accuracy with and without a relaxation period of 60 min for various battery cells using different methods. $\hat{\theta}_{60}$ refers to the results obtained with a 60 min relaxation period, $\hat{\theta}_{0}^{1}$ refers to results obtained without relaxation model output considering that the whole output is due only to the forced response as in Adel, Malti, Vinassa et al. (2024), $\hat{\theta}_{0}^{1}$ refers to results obtained using the method described in Section 2.1, and $\hat{\theta}_{0}^{2}$ refers to results obtained using the method described in Section 2.2.

<table><thead><tr><th rowspan="2">Cell ref<br>Parameters</th><th colspan="4">SGg</th><th colspan="4">LGb</th><th colspan="4">SGp</th></tr><tr><th>θ̂<sub>60</sub></th><th>θ̂<sub>0</sub><sup>0</sup></th><th>θ̂<sub>0</sub><sup>1</sup></th><th>θ̂<sub>0</sub><sup>2</sup></th><th>θ̂<sub>60</sub></th><th>θ̂<sub>0</sub><sup>0</sup></th><th>θ̂<sub>0</sub><sup>1</sup></th><th>θ̂<sub>0</sub><sup>2</sup></th><th>θ̂<sub>60</sub></th><th>θ̂<sub>0</sub><sup>0</sup></th><th>θ̂<sub>0</sub><sup>1</sup></th><th>θ̂<sub>0</sub><sup>2</sup></th></tr></thead><tbody><tr><td>R<sub>0</sub> (mΩ)</td><td>14.2</td><td>14.25</td><td>14.2</td><td>14.2</td><td>15.1</td><td>15.3</td><td>15.1</td><td>15.1</td><td>49.5</td><td>50</td><td>49.5</td><td>49.5</td></tr><tr><td>R<sub>el</sub> (mΩ)</td><td>7.9</td><td>25.8</td><td>18</td><td>7.8</td><td>3.5</td><td>0.36</td><td>3.2</td><td>6.5</td><td>12.2</td><td>18.4</td><td>19.8</td><td>18.6</td></tr><tr><td>Q<sub>d</sub> (F s<sup>1-a</sup>)</td><td>6.81</td><td>3.1</td><td>10.75</td><td>5.98</td><td>1.42</td><td>3.37</td><td>4.8</td><td>1.4</td><td>0.45</td><td>16.33</td><td>1.13</td><td>0.83</td></tr><tr><td>α</td><td>0.67</td><td>0.66</td><td>0.67</td><td>0.66</td><td>0.88</td><td>0.73</td><td>0.88</td><td>0.88</td><td>0.86</td><td>0.89</td><td>0.86</td><td>0.86</td></tr><tr><td>Q<sub>d</sub> (F s<sup>1-β</sup>)</td><td>301</td><td>4000</td><td>437</td><td>340</td><td>348</td><td>3998</td><td>518</td><td>348</td><td>230</td><td>148</td><td>478</td><td>391</td></tr><tr><td>β</td><td>0.65</td><td>0.61</td><td>0.65</td><td>0.65</td><td>0.53</td><td>0.72</td><td>0.53</td><td>0.53</td><td>0.48</td><td>0.008</td><td>0.48</td><td>0.48</td></tr><tr><td>% fit<sub>identif</sub></td><td>98.4</td><td>37.3</td><td>92.5</td><td>94.7</td><td>99.57</td><td>60.89</td><td>96.1</td><td>98.2</td><td>98.5</td><td>17.40</td><td>94.9</td><td>96.3</td></tr><tr><td>% fit<sub>val</sub></td><td>99.7</td><td>63.9</td><td>94.6</td><td>99.7</td><td>99.9</td><td>80.94</td><td>96.7</td><td>98.7</td><td>99.9</td><td>89.9</td><td>99.3</td><td>99.5</td></tr></tbody></table>

It is evident that taking the free response (algorithm of Section 2.2) into account significantly improves the identification results. Hence, the proposed algorithms eliminate the need to wait for the system free response to decay, which is even more important in the case of fractional systems which decay polynomially, as compared to the exponential decay of rational systems. Hence, the relaxation period traditionally observed in battery systems prior to collecting identification/validation data can now be squeezed. Both algorithms have successfully been applied to all three commercial battery cells and the identified parameters were quite close to the ones estimated after a relaxation time of 60 min, which allows asserting that the algorithms are performant.

## 5. Conclusions

This paper introduces two novel algorithms for system identification using fractional models with non-zero initial conditions. To the best of authors' knowledge, no such algorithm has ever been proposed in the literature. It has been shown using Monte Carlo simulation and on real data that both algorithms perform well, with a slight superiority of the one-stage algorithm, described in Section 2.2, as compared to the two-stage algorithm, of Section 2.1. The main reason is that the former algorithm uses both the free and the forced responses for parameter estimation, whereas the latter isolates the forced response which is then used for parameter estimation. On the other hand, the first algorithm is more computationally efficient, thanks to its sequential structure. It is particularly suitable for real-time applications and large datasets where rapid estimation is required. This trade-off between accuracy and computational cost makes the two-stage algorithm a practical alternative in time-constrained environments.

It has been shown that the algorithms are particularly useful in Li-ion battery cells identification, as they allow to completely eliminate the relaxation time, usually observed when a battery is driven to a given SOC-level. Although the study focuses on individual battery cells, the proposed identification approach may be applied by extension to battery packs. When integrated into a Battery Management System (BMS), it may enhance cell balancing by enabling accurate estimation of the SOC of each cell and contribute to better charge redistribution, improved safety, and extended battery lifespan.

As a perspective, the case where both the past input and the system model are unknown could be investigated. In such situations, estimating the free response becomes significantly more challenging and would require the development of new identification strategies capable of handling unknown initial conditions and input history.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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