# Systematic Observer Design With Robust Global Convergence Guarantees for a Large Class of Lithium-Ion Battery Models

Mira Khalil, Romain Postoyan, Stéphane Raël

► To cite this version:

Mira Khalil, Romain Postoyan, Stéphane Raël. Systematic Observer Design With Robust Global Convergence Guarantees for a Large Class of Lithium-Ion Battery Models. IEEE Control Systems Letters, 2025, 9, pp.649 - 654. 10.1109/LCSYS.2025.3575681. hal-05096122

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Submitted on 3 Jun 2025

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# Systematic Observer Design With Robust Global Convergence Guarantees for a Large Class of Lithium-Ion Battery Models

Mira Khalil, Student Member, IEEE, Romain Postoyan, Senior Member, IEEE, and Stéphane Raël

**Abstract**— We present a robust nonlinear Luenberger-like observer for a class of systems that captures many of the finite-dimensional lithium-ion cell models found in the literature including electrical equivalent circuit models and electrochemical models. The observer gain is designed by solving a linear matrix inequality (LMI), which guarantees the robust, global convergence of the estimated state to the true state. We then present the main result, that is, conditions ensuring the LMI is feasible and allowing for the explicit construction of a class of suitable observer gains. These conditions are shown to be satisfied by various existing lithium-ion models thereby making the proposed observer design applicable for these models.

**Index Terms**— Energy systems, Lyapunov methods, Observers for nonlinear systems.

## I. INTRODUCTION

LITHIUM-ION batteries are adopted in many applications due to their high energy density and long lifespan compared to other battery technologies. Despite their benefits, lithium-ion batteries are prone to misuse, potentially compromising their safety, longevity, and performance [1]. That is the reason why a battery management system (BMS) is required. The main function of a BMS is to monitor battery variables, ensuring they remain within safe operation limits. Unfortunately, some key battery variables cannot be measured directly via sensors. The BMS must thus estimate these variables from the available measurements, typically the current, voltage, and temperature. A relevant approach in this context is to design an observer based on a mathematical model of the battery internal dynamics. Several models have been developed for this purpose, see e.g., [2] and the references therein, each with its own pros and cons. The choice of the model depends on the needs of the application at hand, such as the required accuracy and the available computational resources. Two noteworthy groups of models are electrical equivalent circuit models (ECM) and electrochemical models. Based on

the selected battery model, an observer is then designed. Since the battery dynamics are nonlinear, nonlinear Kalman filters are often employed, see e.g., [3]–[5]. However, these filters lack (robust, global) convergence guarantees in general, and only local convergence guarantees are typically ensured in the best case e.g., [6]. Moreover, nonlinear Kalman filter gains are varying and evaluated on-line using a Riccati-like equation, which may be computationally demanding. Inspired by works like [7], [8], an alternative is to design a nonlinear Luenberger-like observer, where the gain is constant and chosen off-line based on the solution of a linear matrix equality (LMI) that provides the desired convergence guarantees for the generated estimates, as done in e.g., [9]–[14] for specific battery models. In these works, the LMI is verified numerically for the model at hand and a given set of parameter values. It remains unclear when the LMI is feasible, and whether the approach is suitable for other lithium-ion battery models.

In this work, we consider a general class of finite-dimensional lithium-ion battery models that captures various ECMs and electrochemical models such as those in e.g., [4], [5], [11]–[14]. Then, we design a nonlinear Luenberger-like observer like in e.g., [9]–[11], [13], [14]. The gain of this observer is designed by solving an LMI. When the LMI holds, the observer state estimates robustly, globally converge in the sense that the estimation error system satisfies a global, exponential input-to-state stability property where the inputs are the additive measurement noise and the additive disturbances on the battery dynamics. Afterwards, we present the main novelty of this work that is easy-to-test conditions on the battery model under which the LMI, that needs to be solved to obtain the observer gain, always admits a solution. These conditions rely on the structural properties of the model rather than on the specific set of parameter values. We provide the expression of a class of observer gains with which the LMI is guaranteed to hold. Finally, we illustrate the applicability of our results to the lithium-ion battery models in e.g., [4], [5], [12], [15]. For a numerical illustration of the proposed observer design on a lithium-ion battery electrochemical model, the interested reader can refer to [16]. We stress that these are only a sample of the models for which the proposed method applies.

**Notation:** Let $\mathbb{R}$, $\mathbb{R}_{>0}$, $\mathbb{R}_{\ge 0}$, $\mathbb{R}_{<0}$ be the set of real, strictly positive, non-negative, strictly negative numbers, respectively, and $\mathbb{Z}$, $\mathbb{Z}_{>0}$ be the set of integers, strictly positive integers, respectively. Given a vector $x \in \mathbb{R}^n$ with $n \in \mathbb{Z}_{>0}$, $x^\top$ denotes

This work was supported by the french Programme d'Investissements d'Avenir (PIA) project "Lorraine Université d'Excellence", under grant ANR-15-IDEX-04-LUE.

Mira Khalil is with Université de Lorraine, CNRS, CRAN, GREEN, 54000 Nancy, France (e-mail: mira.khalil@univ-lorraine.fr).

Romain Postoyan is with Université de Lorraine, CNRS, CRAN, 54000 Nancy, France (e-mail: romain.postoyan@univ-lorraine.fr).

Stéphane Raël is with Université de Lorraine, GREEN, 54000 Nancy, France (e-mail: stephane.rael@univ-lorraine.fr).
the transpose of $x$ and $\operatorname{diag}(x)$ is the diagonal matrix, whose diagonal components are the elements of $x$. Given $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$ with $n, m \in \mathbb{Z}_{>0}$, we use $(x, y)$ for $(x^\top, y^\top)^\top$. We use $I_n$ to denote the identity matrix of dimension $n$, $\mathbf{1}_{n \times m}$ to denote the matrix of $\mathbb{R}^{n \times m}$ whose elements are all equal to 1, and $\mathbf{0}_{n \times m}$ to denote the matrix of $\mathbb{R}^{n \times m}$ whose elements are all equal to 0, with $n, m \in \mathbb{Z}_{>0}$. The symbol $*$ in a matrix stands for the symmetric term, i.e., $(A^* B_C) = (B^* B_C)$. Given a real, symmetric matrix $P$, its largest and smallest eigenvalues are denoted by $\lambda_{\max}(P)$ and $\lambda_{\min}(P)$, respectively. For a vector $x \in \mathbb{R}^n$ with $n \in \mathbb{Z}_{>0}$, $|x|$ stands for its Euclidean norm and $(x)_i$ represents the $i$-th element of $x$. For a matrix $A \in \mathbb{R}^{n \times m}$ with $n, m \in \mathbb{Z}_{>0}$, $(A)_{ij}$ represents the $j$-th element of the $i$-th row of $A$. For a matrix $A \in \mathbb{R}^{n \times n}$ with $n \in \mathbb{Z}_{>0}$, $A \prec 0$ ($A \preceq 0$), $A \succ 0$ ($A \succeq 0$) means that $A$ is symmetric negative (semi-)definite and $A$ is symmetric positive (semi-)definite, respectively. Given real, square matrices $A_1, \dots, A_n$, $\operatorname{diag}(A_1, \dots, A_n)$ is the block diagonal matrix, whose block diagonal components are $A_1, \dots, A_n$. Given a Lebesgue measurable and locally essentially bounded function $f: \mathbb{R}_{\ge 0} \to \mathbb{R}^n$ with $n \in \mathbb{Z}_{>0}$, $\|f\|_t := \operatorname{ess.\sup}_{0 \le t' \le t} |f(t')|$. The notation $\mathcal{L}_\infty$ is the set of all functions from $\mathbb{R}_{\ge 0}$ to $\mathbb{R}^n$ for some $n \in \mathbb{Z}_{>0}$ that are Lebesgue measurable and locally essentially bounded.

## II. CLASS OF MODELS

We consider lithium-ion cell models of the form

$$ 
\begin{aligned}
\dot{x} &= Ax + \psi(u, y) + Ew \\
y &= Cx + \phi_1(x) + \phi_2(u) + Fv,
\end{aligned}
\quad (1) 
$$

where $x \in \mathbb{R}^{n_x}$ is the state vector of dimension $n_x \in \mathbb{Z}_{>0}$, $u \in \mathbb{R}$ is the input current and $y \in \mathbb{R}$ is the output voltage, which we both measure. On the other hand, $w \in \mathbb{R}^{n_w}$ and $v \in \mathbb{R}^{n_v}$ represent unknown exogenous disturbance on the dynamics and unknown measurement noise on the output, respectively. System (1) encompasses a wide range of lithium-ion models found in the literature, e.g., [4], [5], [11]–[14], as shown explicitly for some of these references in Section V. The expressions of the matrices $A \in \mathbb{R}^{n_x \times n_x}$, $C \in \mathbb{R}^{1 \times n_x}$, $E \in \mathbb{R}^{n_x \times n_w}$, $F \in \mathbb{R}^{1 \times n_v}$ and the continuous functions $\psi: \mathbb{R} \times \mathbb{R} \to \mathbb{R}^{n_x}$, $\phi_1: \mathbb{R}^{n_x} \to \mathbb{R}$, $\phi_2: \mathbb{R} \to \mathbb{R}$ thus depend on the adopted model. The matrices $E$ and $F$ are defined according to the considered disturbance $w$ and measurement noise $v$, respectively. For example, if each state variable is independently affected by a distinct disturbance (i.e., $w = (w_1, \dots, w_{n_x})$, $n_w = n_x$, and $w_n \in \mathbb{R}$ for any $n \in \{1, \dots, n_x\}$), $E = I_{n_x}$, and if the output voltage is affected by an additive noise (i.e., $v = v_1 \in \mathbb{R}$), $F = 1$.

The function $\phi_1: \mathbb{R}^{n_x} \to \mathbb{R}$ in (1) is such that, for any $x \in \mathbb{R}^{n_x}$, $\phi_1(x) := \sum_{i=1}^M \phi_i(H_i x + g_i)$ with $M \in \mathbb{Z}_{>0}$, $H_i \in \mathbb{R}^{1 \times n_x}$, $g_i \in \mathbb{R}$, and $\phi_i: \mathbb{R} \to \mathbb{R}$ for any $i \in \{1, \dots, M\}$ depend on the considered model. For ECMs, we typically have $M=1$ and the function $\phi_1$ corresponds to the open circuit voltage (OCV) of the cell that varies with the state of charge (SOC) of the cell, see Section V-A. For electrochemical models, we usually have $M=2$ and $\phi_1, \phi_2$ represent the open circuit potential (OCP) of each electrode which varies

with the normalized lithium concentration at the surface of the electrode, see Sections V-B and V-C. We make the next assumption on $\phi_i$ for any $i \in \{1, \dots, M\}$.

**Standing Assumption 1 (SA1):** For any $i \in \{1, \dots, M\}$, there exist $\underline{c}_i$, $\bar{c}_i \in \mathbb{R}$ with $\underline{c}_i < \bar{c}_i$ and $c_i \bar{c}_i > 0$ such that

$$ \phi_i(\eta) - \phi_i(\eta') = c_i(\eta, \eta')(\eta - \eta') \quad \forall \eta, \eta' \in \mathbb{R}, \quad (2) $$

where $c_i(\eta, \eta') := \lambda_i(\eta, \eta')\underline{c}_i + (1 - \lambda_i(\eta, \eta'))\bar{c}_i$ with $\lambda_i(\eta, \eta') \in [0, 1]$. $\square$

SA1 implies that $\phi_i$ is monotone and $c_i(\eta, \eta')$ lies in the interval $[\underline{c}_i, \bar{c}_i]$ where $\underline{c}_i$ and $\bar{c}_i$ are related to the minimum and maximum “slopes” of $\phi_i$, respectively. For many cell models, $\phi_i$ is given by a look-up table over $[0, 1]$ (or $[0, 100]$) depending on the considered scale of its argument) that may be interpolated using a monotone, globally Lipschitz function satisfying (2) on $[0, 1]$ (or on $[0, 100]$). We can then extrapolate the definition of $\phi_i$ over $\mathbb{R}$ while ensuring (2). Given SA1, we derive the next property for $\phi_1$.

**Lemma 1:** For any $x, x' \in \mathbb{R}^{n_x}$, $\phi_1(x) - \phi_1(x') = \sum_{j=1}^{2^M} \Lambda_j(x, x')C_j(x - x')$, where $C_j := \sum_{i=1}^M \kappa_{i,j}H_i \in \mathbb{R}^{1 \times n_x}$ with $\kappa_{i,j} \in \{\underline{c}_i, \bar{c}_i\}$ and for any $j, j' \in \{1, \dots, 2^M\}$ such that $j \neq j'$, there exists $i \in \{1, \dots, M\}$ such that $\kappa_{i,j} \neq \kappa_{i,j'}$, $\Lambda_j(x, x') \in [0, 1]$, and $\sum_{j=1}^{2^M} \Lambda_j(x, x') = 1$. $\square$

**Proof:** Let $x, x' \in \mathbb{R}^{n_x}$. In view of the expression of $\phi_1$, $\phi_1(x) - \phi_1(x') = \sum_{i=1}^M (\phi_i(H_i x + g_i) - \phi_i(H_i x' + g_i))$. By SA1, $\phi_1(x) - \phi_1(x') = \sum_{i=1}^M c_i(H_i x + g_i, H_i x' + g_i)H_i(x - x') = \sum_{i=1}^M [\lambda_i(H_i x + g_i, H_i x' + g_i)\underline{c}_i + (1 - \lambda_i(H_i x + g_i, H_i x' + g_i))\bar{c}_i]H_i(x - x')$. We derive the desired result by the definition of $C_j$ and since $\Lambda_j(x, x') \in [0, 1]$ with $\sum_{j=1}^{2^M} \Lambda_j(x, x') = 1$. $\blacksquare$

**Remark 1:** In some lithium-ion models, $\phi_2$ in (1) depends on the state $x$ (and $u$). We can apply the forthcoming results in this case by considering $\phi_2(x^*, u)$ instead of $\phi_2(u)$ and taking $F=1$ and $v = \phi_2(x, u) - \phi_2(x^*, u)$ for instance, for some given $x^* \in \mathbb{R}^{n_x}$. $\square$

## III. LMI-BASED OBSERVER DESIGN

We propose the Luenberger-like observer

$$ 
\begin{aligned}
\dot{\hat{x}} &= A\hat{x} + \psi(u, y) + L(y - \hat{y}) \\
\hat{y} &= C\hat{x} + \phi_1(\hat{x}) + \phi_2(u),
\end{aligned}
\quad (3) 
$$

where $\hat{x} \in \mathbb{R}^{n_x}$ is the state estimate, $\hat{y} \in \mathbb{R}$ is the output estimate, and $L \in \mathbb{R}^{n_x \times 1}$ is the gain to be designed.

The next proposition provides a sufficient condition to design $L \in \mathbb{R}^{n_x \times 1}$ under which system (1), (3) satisfies a global exponential input-to-state stability property in terms of the estimation error $e := x - \hat{x}$.

**Proposition 1:** Consider system (1), (3). Suppose there exist $L \in \mathbb{R}^{n_x \times 1}$, $P \succ 0$ and $\varepsilon, \varepsilon_w, \varepsilon_v \in \mathbb{R}_{>0}$ such that, for all $j \in \{1, \dots, 2^M\}$,

$$ 
\left( \begin{array}{cccc}
-\mathcal{H}_j + \varepsilon P & P E & -P L F & 0 \\
* & -\varepsilon_w I_{n_w} & 0 & * \\
* & * & -\varepsilon_v I_{n_v} & 0
\end{array} \right) \preceq 0, \quad (4) 
$$
with $\mathcal{H}_j := -(A-L(C+C_j))^\top P - P(A-L(C+C_j))$ and $C_j$ defined in Lemma 1. Then, there exist $c_1, c_2, c_3, c_4 \in \mathbb{R}_{>0}$ such that for any $u, w, v \in \mathcal{L}_\infty$, any corresponding solution $(x, \hat{x})$ to system (1), (3) satisfies, for all $t \ge 0$,

$$|x(t) - \hat{x}(t)| \le c_1|x(0) - \hat{x}(0)|e^{-c_2t} + c_3\|w\|_t + c_4\|v\|_t. \quad (5) \\ \square$$

**Proof:** The dynamics of the estimation error $e = x - \hat{x}$ is in view of (1) and (3),

$$\dot{e} = Ae + Ew - LCe - L(\phi_1(x) - \phi_1(\hat{x}) + Fv). \quad (6)$$

By Lemma 1, we have $\phi_1(x) - \phi_1(\hat{x}) = \sum_{j=1}^{2^M} \Lambda_j(x, \hat{x})C_j(x-\hat{x})$ for any $x, \hat{x} \in \mathbb{R}^{n_x}$. Hence, as $\sum_{j=1}^{2^M} \Lambda_j(x, \hat{x}) = 1$, $\dot{e} = \sum_{j=1}^{2^M} \Lambda_j(x, \hat{x})(A-L(C+C_j))e + Ew-LFv$. We consider the Lyapunov function candidate $V(e) := e^\top Pe$ for any $e \in \mathbb{R}^{n_x}$. Let $x, \hat{x} \in \mathbb{R}^{n_x}$, $e = x - \hat{x}$, $u \in \mathbb{R}$, $w \in \mathbb{R}^{n_w}$, $v \in \mathbb{R}^{n_v}$,

$$\langle \nabla V(e), \sum_{j=1}^{2^M} \Lambda_j(x, \hat{x})(A - L(C + C_j))e + Ew - LFv \rangle \\ = 2 \sum_{j=1}^{2^M} \Lambda_j(x, \hat{x}) (e^\top (A - L(C + C_j))^\top Pe \\ + w^\top E^\top Pe - v^\top F^\top L^\top Pe). \quad (7)$$

By writing the right-hand side of (7) as a quadratic term in $(e, w, v)$, using again $\sum_{j=1}^{2^M} \Lambda_j(x, \hat{x}) = 1$ and invoking (4), we derive $\langle \nabla V(e), \sum_{j=1}^{2^M} \Lambda_j(x, \hat{x})(A - L(C + C_j))e + Ew - LFv \rangle \le -\epsilon V(e) + \epsilon_w |w|^2 + \epsilon_v |v|^2$. Then, using $\lambda_{\min}(P)|e|^2 \le V(e) \le \lambda_{\max}(P)|e|^2$, we derive the desired input-to-state stability property in (5) with $c_1 := \sqrt{\frac{\lambda_{\max}(P)}{\lambda_{\min}(P)}}$, $c_2 := \frac{\epsilon}{2}$, $c_3 := \sqrt{\frac{\epsilon_w}{\epsilon \lambda_{\min}(P)}}$, and $c_4 := \sqrt{\frac{\epsilon_v}{\epsilon \lambda_{\min}(P)}}$ by following similar lines as in [17, Theorem 4.10]. ■

Proposition 1 means that we can design $L$ to ensure the uniform, global exponential convergence of the estimated state to the true state up to an error due to the presence of noise and disturbances whenever (4) is verified. Hence the satisfaction of (4) implies that system (1) is detectable by [18, Lemma 3.1]. The matrix inequality in (4) is not linear due to the terms $PL$ and $L^\top P$, which appear in both $\mathcal{H}_j$ and the off-diagonal blocks, however it becomes linear after a standard change of variables, namely $W = PL$, which is why we write that (4) is an LMI with a slight abuse of terminology. Similar LMI conditions to (4) are used in e.g., [9]–[14] to design observer gains that provide convergence guarantees for the estimation error for specific lithium-ion cell models, and in e.g., [7] for general models. In these works, the LMI condition is tested numerically for a specific model and a given set of parameters. It remains unclear when the LMI is feasible in general. For this reason, in the next section, we give sufficient conditions tailored to a range of lithium-ion cell models, as illustrated in Section V, under which (4) is guaranteed to be feasible.

## IV. SOLUTION TO (4)

We present in this section two assumptions on system (1), thus tailored to lithium-ion battery models, under which we can analytically solve (4).

### A. Assumptions

We first make the next assumption on matrix $A$ in (1).

**Assumption 1:** (i) $A$ is diagonalizable, and (ii) $A$ has $n_x - 1$ strictly negative eigenvalues and one eigenvalue equal to zero. $\square$

Given Assumption 1, there exists a real, invertible matrix $T \in \mathbb{R}^{n_x \times n_x}$ such that $A = TDT^{-1}$ where $D := \text{diag}(\rho, 0) \in \mathbb{R}^{n_x \times n_x}$ with $\rho \in \mathbb{R}^{n_x - 1}$ being the vector of the strictly negative eigenvalues of $A$. We perform the next change of variables

$$z := T^{-1}x, \quad (8)$$

from which we derive, in view of (1),

$$\begin{aligned} \dot{z} &= Dz + T^{-1}\psi(u, y) + T^{-1}Ew \\ y &= CTz + \phi_1(Tz) + \phi_2(u) + Fv. \end{aligned} \quad (9)$$

We then decompose (9) as

$$\begin{aligned} \dot{z}_{\rho} &= D_{\rho}z_{\rho} + \psi_{\rho}(u, y) + E_{\rho}w \\ \dot{z}_{0} &= \psi_{0}(u, y) + E_{0}w \\ y &= C_{\rho}z_{\rho} + c_{0}z_{0} + \tilde{\phi}_{1}(z_{\rho}, z_{0}) + \phi_{2}(u) + Fv, \end{aligned} \quad (10)$$

where $z = (z_\rho, z_0) \in \mathbb{R}^{n_x - 1} \times \mathbb{R}$, $D_\rho := \text{diag}(\rho)$, $T^{-1}\psi = (\psi_\rho, \psi_0)$ with $\psi_\rho : \mathbb{R} \times \mathbb{R} \to \mathbb{R}^{n_x - 1}$, and $\psi_0 : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$. Also, $CT = (C_\rho, c_0)$ with $C_\rho \in \mathbb{R}^{1 \times (n_x - 1)}$, $c_0 \in \mathbb{R}$, and $T^{-1}E = (E_\rho^\top, E_0^\top)^\top$ with $E_\rho \in \mathbb{R}^{(n_x - 1) \times n_w}$, $E_0 \in \mathbb{R}^{1 \times n_w}$. The function $\tilde{\phi}_1$ is, for any $z_\rho \in \mathbb{R}^{n_x - 1}$ and $z_0 \in \mathbb{R}$,

$$\tilde{\phi}_1(z_\rho, z_0) := \sum_{i=1}^{M} \varphi_i(\tilde{H}_{i,\rho}z_\rho + \tilde{h}_{i,0}z_0 + g_i), \quad (11)$$

with $\tilde{H}_{i,\rho} \in \mathbb{R}^{1 \times (n_x - 1)}$, $\tilde{h}_{i,0} \in \mathbb{R}$ are such that $\tilde{H}_i := H_i T = (\tilde{H}_{i,\rho}, \tilde{h}_{i,0})$. We make the next assumption on $\tilde{h}_{i,0}$ in (11) and $c_0$ in (10).

**Assumption 2:** For any $i \in \{1, \dots, M\}$, $\tilde{h}_{i,0}$ has the same sign as $c_i$ (and $\bar{c}_i$) in SA1 and $c_0 \ge 0$. $\square$

The next lemma is a consequence of Assumptions 1 and 2.

**Lemma 2:** Suppose Assumptions 1 and 2 hold. For any $z_\rho \in \mathbb{R}^{n_x - 1}$,

$$c_0 \le \frac{\tilde{\phi}_1(z_\rho, z_0) - \tilde{\phi}_1(z_\rho, z'_0)}{z_0 - z'_0} \le \bar{c}_0 \quad \forall z_0, z'_0 \in \mathbb{R} \quad z_0 \ne z'_0, \quad (12)$$

$$\text{with } \underline{c}_0 := \sum_{i=1}^{M} \min\{\tilde{h}_{i,0}c_i, \tilde{h}_{i,0}\bar{c}_i\} \quad \text{and} \quad \bar{c}_0 := \sum_{i=1}^{M} \max\{\tilde{h}_{i,0}c_i, \tilde{h}_{i,0}\bar{c}_i\}. \quad \square$$

**Proof:** Let $z_\rho \in \mathbb{R}^{n_x - 1}$ and $z_0, z'_0 \in \mathbb{R}$ with $z_0 \ne z'_0$. By (11), $\tilde{\phi}_1(z_\rho, z_0) - \tilde{\phi}_1(z_\rho, z'_0) = \sum_{i=1}^M \varphi_i(\tilde{H}_{i,\rho}z_\rho + \tilde{h}_{i,0}z_0 + g_i) - \varphi_i(\tilde{H}_{i,\rho}z_\rho + \tilde{h}_{i,0}z'_0 + g_i)$. By SA1, $\tilde{\phi}_1(z_\rho, z_0) - \tilde{\phi}_1(z_\rho, z'_0) = \sum_{i=1}^M c_i((\tilde{H}_{i,\rho}z_\rho, \tilde{h}_{i,0}z_0), (\tilde{H}_{i,\rho}z_\rho, \tilde{h}_{i,0}z'_0))\tilde{h}_{i,0}(z_0 - z'_0) = \sum_{i=1}^M (\lambda_i((\tilde{H}_{i,\rho}z_\rho, \tilde{h}_{i,0}z_0), (\tilde{H}_{i,\rho}z_\rho, \tilde{h}_{i,0}z'_0))c_i + (1 - \lambda_i((\tilde{H}_{i,\rho}z_\rho, \tilde{h}_{i,0}z_0), (\tilde{H}_{i,\rho}z_\rho, \tilde{h}_{i,0}z'_0)))\bar{c}_i)\tilde{h}_{i,0}(z_0 - z'_0)$. By Assumption 2, (12) holds as $\tilde{h}_{i,0}c_i > 0$ and $\tilde{h}_{i,0}\bar{c}_i > 0$. ■

Lemma 2 implies that $\tilde{\phi}_1$ is strictly increasing with respect to $z_0$ for a fixed $z_\rho$. Moreover, the difference quotient of $\tilde{\phi}_1(z_\rho, \cdot)$ is lower and upper-bounded by $\underline{c}_0$ and $\bar{c}_0$, respectively. This result is needed in Section IV-B to find a solution to (4) that allows to obtain a robustly globally exponentially converging observer in view of Proposition 1.
## B. Main result

The next theorem ensures the existence of solutions for LMI (4) under Assumptions 1 and 2.

**Theorem 1:** Suppose Assumptions 1 and 2 hold. The inequality in (4) is satisfied for $L = TK$, $K := (0, \dots, 0, k) \in \mathbb{R}^{n_x \times 1}$ with any $k \in \mathbb{R}_{>0}$, $P = (T^\top)^{-1}QT^{-1}$, $Q := \text{diag}(\mu I_{n_x-1}, 1) \in \mathbb{R}^{n_x \times n_x}$, $\mu, \varepsilon_w, \varepsilon_v \in \mathbb{R}_{>0}$ sufficiently large, and $\varepsilon \in \mathbb{R}_{>0}$ sufficiently small. $\square$

**Proof:** For any $j \in \{1, \dots, 2^M\}$, let

$$ M_j := \begin{pmatrix} \mathcal{H}_j - \varepsilon P & -PE & PLF \\ * & \varepsilon_w I_{n_w} & 0 \\ * & * & \varepsilon_v I_{n_v} \end{pmatrix}, \quad (13) $$

with $\mathcal{H}_j = -(A-L(C+C_j))^\top P - P(A-L(C+C_j))$ and $L, P, \varepsilon, \varepsilon_w, \varepsilon_v$ as in Proposition 1. We note that (4) is equivalent to $M_j > 0$ for any $j \in \{1, \dots, 2^M\}$.

Let $j \in \{1, \dots, 2^M\}$, we have $A = TDT^{-1}$, $L = TK$, and $P = (T^\top)^{-1}QT^{-1}$. Noting that $QK = K$ and taking $\tilde{C}_j := C_jT$, $\tilde{C} := CT$,

$$ M_j = \begin{pmatrix} \mathcal{H}_j - \varepsilon P - (T^\top)^{-1}QT^{-1}E (T^\top)^{-1}KF \\ * & \varepsilon_w I_{n_w} & 0 \\ * & * & \varepsilon_v I_{n_v} \end{pmatrix}, \quad (14) $$

and $\mathcal{H}_j = (T^{-1})^\top [-2DQ + K(\tilde{C} + \tilde{C}_j) + (\tilde{C} + \tilde{C}_j)^\top K^\top]T^{-1}$.

We want to show that $M_j > 0$. For this purpose, we consider the Schur complement of $M_j$ $S_{M_j} := \mathcal{H}_j - \varepsilon P - \frac{1}{\varepsilon_w}(T^\top)^{-1}QT^{-1}EE^\top(T^{-1})^\top QT^{-1} - \frac{1}{\varepsilon_v}(T^\top)^{-1}KFF^\top K^\top T^{-1}$. Hence,

$$ S_{M_j} > 0 \Leftrightarrow \mathcal{H}_j > \varepsilon P + \frac{1}{\varepsilon_w}(T^\top)^{-1}QT^{-1}EE^\top(T^{-1})^\top QT^{-1} + \frac{1}{\varepsilon_v}(T^\top)^{-1}KFF^\top K^\top T^{-1}. \quad (15) $$

We have $K(\tilde{C} + \tilde{C}_j) = \begin{pmatrix} \mathbf{0}_{(n_x-1) \times n_x} \\ k\tilde{C} + k\tilde{C}_j \end{pmatrix}$, $\tilde{C} = (C_\rho, c_0)$, and $\tilde{C}_j = C_jT = \sum_{i=1}^M \kappa_{i,j}H_iT = \sum_{i=1}^M \kappa_{i,j}\tilde{H}_i$. We thus get $\mathcal{H}_j = (T^{-1})^\top \mathcal{L}_j T^{-1}$ with $\mathcal{L}_j := \begin{pmatrix} -2\mu D_\rho & B_j \\ B_j^\top & d_{j,0} \end{pmatrix}$, where $B_j := (kC_\rho + k\sum_{i=1}^M \kappa_{i,j}\tilde{H}_{i,\rho})^\top \in \mathbb{R}^{(n_x-1) \times 1}$ and $d_{j,0} := 2k(c_0 + \sum_{i=1}^M \kappa_{i,j}h_{i,0})$. Furthermore, $-2\mu D_\rho > 0$ as $\mu \in \mathbb{R}_{>0}$ and $\rho \in \mathbb{R}^{n_x-1}$, and the Schur complement of $\mathcal{L}_j$ is

$$ S_{\mathcal{L}_j} := d_{j,0} + \frac{1}{2\mu} B_j^\top D_\rho^{-1} B_j. \quad (16) $$

As $d_{j,0} > 0$ by Assumption 2 and since $k \in \mathbb{R}_{>0}$, by taking $\mu \in \mathbb{R}_{>0}$ sufficiently large, $S_{\mathcal{L}_j} > 0$. Hence, for any $j \in \{1, \dots, 2^M\}$, $\mathcal{L}_j > 0$, from which we get $\mathcal{H}_j > 0$. As a consequence, there exist sufficiently small $\varepsilon$ and sufficiently large $\varepsilon_w, \varepsilon_v \in \mathbb{R}_{>0}$ such that $S_{M_j} > 0$ and thus $M_j > 0$, implying that (4) holds. $\blacksquare$

Theorem 1 provides an explicit formula for $L$ in (3) of the form $TK$ with $K = (0, \dots, 0, k)$ for any $k \in \mathbb{R}_{>0}$ with which (4) is guaranteed to always have a solution. In other words, we can always take $L = TK$ for observer (3) and (5) will hold under Assumptions 1 and 2. However, (4) offers greater flexibility in the choice of $L$, which may be exploited to improve the performance of the observer in terms of speed of convergence or robustness for example. Note that the particular form of gain $L = K$ resembles what can be found in e.g., [16], [19] for specific battery models, where $A$ is already a diagonal

matrix that satisfies Assumption 1 and such that Assumption 2 holds. It can be noted that the expressions of $\mu, \varepsilon, \varepsilon_w$, and $\varepsilon_v$ are not provided in Theorem 1. This is only because these expressions are involved and subject to some conservatism. Since we know that such constants exist, numerical LMI tools can be used to estimate for given model parameter values, considering (4) with the data of Theorem 1.

## V. CASE STUDIES

We show in this section that various lithium-ion models from the literature satisfy Assumptions 1 and 2 given SA1. As a result, the LMI in (4) is guaranteed to admit a solution by Theorem 1, meaning that a robustly globally exponentially converging observer of the form of (3) can always be synthesized for the considered model by Proposition 1.

### A. Resistor-capacitor equivalent circuit models

We consider the ECM consisting of $N \in \mathbb{Z}_{>0}$ resistor-capacitor (R-C) parallel branches in series, as in e.g., [4], [15], described by the following equations

$$ \begin{aligned} \frac{dU_{RC,n}}{dt} &= -\frac{1}{\tau_{d,n}}U_{RC,n} + \frac{1}{Q_{d,n}}I, \quad n \in \{1, \dots, N\} \\ \frac{dSOC}{dt} &= -\frac{1}{Q}I \\ V &= -\sum_{n=1}^{N}U_{RC,n} - R_{\text{int}}I + V_{\text{OCV}}(SOC), \end{aligned} \quad (17) $$

where $U_{RC,n} \in \mathbb{R}$ is the voltage across the $n^{th}$ R-C branch with $n \in \{1, \dots, N\}$, $\tau_{d,n} := R_{d,n}Q_{d,n}$ is the $n^{th}$ diffusion time constant, $R_{d,n} \in \mathbb{R}_{>0}$ is the $n^{th}$ diffusion resistance, $Q_{d,n} \in \mathbb{R}_{>0}$ is the $n^{th}$ capacitance associated with the diffusion phenomenon, $V_{OCV}(SOC) \in \mathbb{R}$ is the cell OCV that depends on SOC $\in [0, 100]$, $Q \in \mathbb{R}_{>0}$ is the cell nominal capacity, $R_{\text{int}} \in \mathbb{R}_{>0}$ is the ohmic resistance, $V \in \mathbb{R}$ is the cell voltage, and $I$ is the cell current.

The equations in (17) can be written in the form of (1) by taking $x := (U_{RC,1}, \dots, U_{RC,N}, SOC)$, $n_x := N+1$, $u := I$, $y := V$, $M = 1$, and

$$ \begin{aligned} A &:= \text{diag}\left(-\frac{1}{\tau_{d,1}}, \dots, -\frac{1}{\tau_{d,N}}, 0\right) \\ \psi(u, y) &:= \left(\frac{1}{Q_{d,1}}, \dots, \frac{1}{Q_{d,N}}, -\frac{1}{Q}\right)^\top u \\ C &:= (-1, \dots, -1, 0) \\ \phi_1(x) &:= \phi_1(H_1x), \quad \phi_1(H_1x) := V_{\text{OCV}}(H_1x) \\ \phi_2(u) &:= -R_{\text{int}}u, \end{aligned} \quad (18) $$

with $H_1 := (\mathbf{0}_{1 \times N} \ 1)$. The function $\phi_1$ thus represents the $V_{OCV}$ function, which verifies SA1 with $\underline{c}_1, \bar{c}_1 \in \mathbb{R}_{>0}$ for many battery technologies like lithium nickel manganese cobalt oxide (NMC) and lithium nickel cobalt aluminum oxide (NCA). The next result states that the conditions of Theorem 1 hold for the considered model.

**Proposition 2:** Consider system (1), (18). Suppose $\phi_1$ verifies SA1 with $\underline{c}_1, \bar{c}_1 \in \mathbb{R}_{>0}$. Then, Assumptions 1 and 2 are satisfied. $\square$

**Proof:** By (18), $A$ is diagonal with $N$ strictly negative eigenvalues and one eigenvalue equal 0 as $\tau_{d,n} \in \mathbb{R}_{>0}$ for all $n \in \{1, \dots, N\}$. Thus, Assumption 1 holds for the considered cell model. Take $\rho = (-\frac{1}{\tau_{d,1}}, \dots, -\frac{1}{\tau_{d,N}})$, $T = I_{n_x}$, and
By considering the output voltage equation in [12], we can then write the reduced single particle model in the form of (1) by taking $n_x := n_x - 1$, $x := ((\mathbf{z})_2, \dots, (\mathbf{z})_{n_x})$, $M = 2$, and

$$
\begin{align}
A &:= \operatorname{diag}(\rho, 0) \nonumber \\
\psi(u, y) &:= Gu \nonumber \\
C &:= \mathbf{0}_{1 \times n_x} \nonumber \\
\phi_1(x) &:= \phi_1(H_1 x + g_1) + \phi_2(H_2 x + g_2) \tag{23} \\
\phi_1(H_1 x + g_1) &:= \mathrm{OCP}_{\mathrm{pos}}(S_1 \mathbf{T}(R_1 x + R_2)) \nonumber \\
\phi_2(H_2 x + g_2) &:= -\mathrm{OCP}_{\mathrm{neg}}(S_2 \mathbf{T}(R_1 x + R_2)) \nonumber \\
\phi_2(u) &:= g(u), \nonumber
\end{align}
$$

where G is the vector of the last $n_x$ elements of
$$
\mathbf{T}^{-1}\mathbf{G}, R_1 := \begin{pmatrix} -\frac{\alpha_2}{\alpha_1} & \dots & -\frac{\alpha_{n_x}}{\alpha_1} \\ 1 & 0 & \dots \\ 0 & \ddots & \vdots \\ \vdots & & \vdots \\ 0 & \dots & 0 \end{pmatrix} \in \mathbb{R}^{n_x \times n_x}, R_2 := \frac{Q_{Li}}{\alpha_1}(1, 0, \dots, 0), S_1 := \begin{pmatrix} \mathbf{0}_{1 \times (n_x-1)} & \frac{1}{c_{pos,max}} \\ \mathbf{0}_{1 \times (n_{neg}-1)} & \mathbf{0}_{1 \times n_{pos}} \end{pmatrix}, S_2 := \begin{pmatrix} \mathbf{0}_{1 \times (n_{neg}-1)} & \frac{1}{c_{neg,max}} \end{pmatrix}, c_{s,max} \in \mathbb{R}_{>0} \text{ represents the maximum lithium concentration of electrode s, and the function } g: \mathbb{R} \to \mathbb{R} \text{ for any } u \in \mathbb{R} \text{ follows the same definition as in [12]. The functions } \phi_1 \text{ and } \phi_2 \text{ are the same as in Section V-B. They thus typically satisfy SA1 with } c_1, \bar{c}_1 \in \mathbb{R}_{<0} \text{ and } c_2, \bar{c}_2 \in \mathbb{R}_{>0}, \text{ respectively. The next result is obtained for the reduced single particle model (1), (23).}}
$$

**Proposition 4:** Suppose Assumption 3 holds. Consider system (1), (23). Suppose $\varphi_1$ and $\varphi_2$ verify SA1 with $c_1, \bar{c}_1 \in \mathbb{R}_{<0}$ and $c_2, \bar{c}_2 \in \mathbb{R}_{>0}$, $(\mathbf{T})_{n_x \times n_x} - (\mathbf{T})_{n_x \times 1} \frac{\alpha_{n_x}}{\alpha_1} \in \mathbb{R}_{<0}$, and $(\mathbf{T})_{n_{neg} \times n_x} - (\mathbf{T})_{n_{neg} \times 1} \frac{\alpha_{n_x}}{\alpha_1} \in \mathbb{R}_{>0}$. Then, Assumptions 1 and 2 hold. $\square$

**Proof:** From (23), A is diagonal with $n_x - 1$ strictly negative eigenvalues and one eigenvalue equal to zero. Assumption 1 is thus satisfied. Take $D = A$ and $T = I_{n_x}$. As $H_1 = S_1 \mathbf{T} R_1$, $H_2 = S_2 \mathbf{T} R_1$, and $C = \mathbf{0}_{1 \times n_x}$, we get $\tilde{h}_{1,0} = (H_1 T)_{1 \times n_x} = (H_1)_{1 \times n_x} = \frac{1}{c_{pos,max}}((\mathbf{T})_{n_x \times n_x} - (\mathbf{T})_{n_x \times 1} \frac{\alpha_{n_x}}{\alpha_1})$, $\tilde{h}_{2,0} = \frac{1}{c_{neg,max}}((\mathbf{T})_{n_{neg} \times n_x} - (\mathbf{T})_{n_{neg} \times 1} \frac{\alpha_{n_x}}{\alpha_1})$, and $c_0 = 0$. Thus, since $\tilde{h}_{1,0}$ is negative like $c_1$ ($\bar{c}_1$), $\tilde{h}_{2,0}$ is positive like $c_2$ ($\bar{c}_2$) and $c_0 = 0$, Assumption 2 is satisfied. $\blacksquare$

VI. CONCLUSION

We have designed an LMI-based observer for a large class of lithium-ion cell models, for which the robust, global convergence of the estimation error is guaranteed. Conditions for the feasibility of the LMI have been provided that are satisfied by various existing lithium-ion models. It would be interesting to extend the results to other lithium-ion models involving state nonlinearities, as in [22], to include thermal dynamics, and to account for parametric variations.

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