E Plant

Romain Postoyan
email: romain.postoyan@univ-lorraine.fr.

S Ra Ël

Youssef Jebroun
email: jebrounyoussef@gmail.com.

S Benjamin

Saft Daniel Monier Reyes

S Ébastien Benjamin

Multiple active material Lithium-ion batteries: finite-dimensional modeling and constrained state estimation

Keywords: Lithium-ion batteries, observer, Lyapunov methods, constrained estimation

Lithium-ion batteries with composite electrodes are being developed in the industry for their improved power, energy, and durability properties over classical single active material accumulators. To make the best out of multiple active material batteries, their management system needs to be fed with reliable information on the internal states, which are not available by measurements. It appears that tools developed for this purpose for single active material batteries are not suitable for multiple active material ones due to their distinctive electrical coupling rules within the electrodes. In this context, we first present a new finite-dimensional model of lithium-ion batteries for which the cathode is represented by two particles made of two different materials and the anode is described by two graphite particles. We then use the model to design a nonlinear observer based on polytopic techniques, that estimates the internal battery variables, and thus the state of charge. The convergence of the observer is guaranteed under a linear matrix inequality condition. We then exploit a technique recently proposed in the literature to modify the observer so that its state estimates remain in given plausible range at all times, while preserving its original convergence properties. This allows to avoid aberrant estimated values in the transient and may also favour the future implementation of the observer on embedded devices. Simulation results using literature data illustrate the efficiency of the observer to estimate internal battery variables.

I. INTRODUCTION

Among the different energy storage technologies available, lithium-ion accumulators offer many advantages in terms of volume capacity, weight, power density, as well as the absence of memory effect, which explains their popularity. On the other hand, they require a so-called battery management system (BMS) for a safe and efficient usage. The BMS thus plays a key role on the battery performances and lifespan. In this context, it is essential to feed the BMS with accurate data on the actual state of the battery. The challenge is that few information on the battery variables is directly accessible by measurements, typically the current, the voltage and possibly the temperature. The concentration of lithium in each electrode for instance, which is directly related to the state of charge (SOC) and is thus of primary importance, cannot be measured on-line and needs to be estimated.

A common way to estimate unmeasured battery data is to design an observer based on a mathematical model of the internal dynamics. The problem is non-trivial due to the nonlinear relationships between the measured variables and the internal ones. Various methods are available in the literature that address this challenge, see, e.g., [START_REF] He | Comparison study on the battery models used for the energy management of batteries in electric vehicles[END_REF], [START_REF] Meng | Overview of lithium-ion battery modeling methods for state-ofcharge estimation in electrical vehicles[END_REF] and the references therein. The simplest approach consists in abstracting the battery dynamics by an equivalent circuit model (ECM) made of few resistors and capacitors and then to design an observer for it, see, e.g., [START_REF] Barillas | A comparative study and validation of state estimation algorithms for Li-ion batteries in battery management systems[END_REF], [START_REF] Beelen | Joint estimation of battery parameters and state of charge using an extended Kalman filter: a single-parameter tuning approach[END_REF], [START_REF] Dai | An improved SOC estimator using timevarying discrete sliding mode observer[END_REF], [START_REF] Lee | State-of-charge and capacity estimation of lithium-ion battery using a new open-circuit voltage versus state-of-charge[END_REF], [START_REF] Xia | A novel method for state of charge estimation of lithium-ion batteries using a nonlinear observer[END_REF]. This apparent simplicity has a price nevertheless: the model requires a nontrivial parameterization to provide accurate data in general. An alternative consists in developing electrochemical models based on coupled partial differential equations (PDE) involving space and time, which describe locally the physics of lithiumion cell operation: lithium diffusion within electrode active materials, electron migration in electrodes, ion migration and diffusion within the electrolyte, and electrochemical kinetics of lithium insertion/de-insertion at electrode/electrolyte interface. Primary works in this field led to a pseudo-two dimensional formulation, presently known as Doyle-Fuller-Newman model [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF], [START_REF] Fuller | Simulation and optimization of the dual lithium ion insertion cell[END_REF]. In the last decade, Doyle-Fuller-Newman local formulation was extended to reduced models dedicated to on-board system state estimation and diagnosis, the most popular approach being the so-called single particle model (SPM); see [START_REF] Di Domenico | Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter[END_REF] for a thorough presentation. Electrochemical models are often more complex than ECM but they provide a more faithful description of the dynamics. We concentrate on electrochemical models for this reason in this work.

In this paper, we first present, like in [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF], [START_REF] Camacho-Solorio | State estimation for an electrochemical model of multiplematerial lithium-ion batteries[END_REF], an electrochemical battery model based on SPM formulation, with composite electrodes. Lithium-ion batteries with composite electrodes are being developed in the industry for their improved power, energy, and durability properties over classical single active material accumulators. It appears that existing models for single active material batteries are not suitable for multiple active material batteries due to their specific electrical coupling rules within the electrodes. There is therefore a need for models and associated estimation algorithms for this battery technology. In the model we propose, lithium solid diffusion is treated by means of a set of ordinary differential equations obtained by spatial discretization of electrode particles [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF]. Contrary to [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF], [START_REF] Camacho-Solorio | State estimation for an electrochemical model of multiplematerial lithium-ion batteries[END_REF], both electrodes are composite as two graphite particles are considered for the negative electrode to further improve the model accuracy. Furthermore and importantly, we derive the analytical solution of the nonlinear algebraic relationships that govern current distribution between the different electrode materials, thereby avoiding possible inaccuracies due to the linearization of these relationships as done in [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF], [START_REF] Camacho-Solorio | State estimation for an electrochemical model of multiplematerial lithium-ion batteries[END_REF], which may be non-negligible for high currents and/or low temperatures as we show in simulation. At last, another crucial difference with [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF] is the model detectability, obtained by means of the lithium conservation in the whole solid phase of the battery, as demonstrated in [START_REF] Klein | Electrochemical model based observer design for a lithium-ion battery[END_REF]. This additional equation between the model states enables to reduce the model order, and then to design an observer for the full battery state vector, so that we do not have to run an openloop model to estimate the variables of the negative electrode as done in [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF].

In particular, we design a Luenberger-like observer in the sense that its dynamics are given by a copy of the battery nonlinear dynamics with the addition of a linear correction term. The constant matrix gain is designed based on the resolution of a linear matrix inequality (LMI). As a result, when the latter holds, the state estimates generated by the observer are guaranteed to locally converge to the battery states asymptotically as time grows. The convergence analysis of the observer relies on polytopic and Lyapunov-based tools, like in [START_REF] Blondel | Observer design for an electrochemical model of Lithium ion batteries based on a polytopic approach[END_REF], [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF], [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H8 performance analysis[END_REF]. Compared to (extended) Kalman filters used in [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF], the observer has a fixed gain and thus does not require solving on-line a Riccati equation. Moreover, its convergence is guaranteed a priori under the LMI condition over a given region. Afterwards, we add an extra term to the observer equations to constrain its state estimates to always lie in a given set for all times. We apply for this purpose the recent general results in [START_REF] Astolfi | Constrained state estimation for nonlinear systems: a redesign approach based on convexity[END_REF]. To constrain the state estimates has several benefits. First, and most importantly, it avoids having estimated values that are not physically plausible during the transient such as negative estimated concentrations. Second, to constrain the state estimates is also relevant for practical implementation purpose. Indeed, we often need to specify a priori the range of each variable on embedded devices. The proposed constrained observer allows doing so, contrary to the approach that consists in saturating a posteriori the estimates generated by the observer, without altering the convergence properties of the initial observer. Simulations results for standard parameters values as customarily found in the literature are provided that illustrate the convergence and the good performances of the proposed observer on the electrochemical model.

To summarize, the contributions of this work are the following. First, a new finite-dimensional electrochemical model of multiple-material lithium-ion batteries is developed, which does not rely on the linearization of the algebraic relationship between the overpotentials and which allows for composite negative electrodes contrary to [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF], [START_REF] Camacho-Solorio | State estimation for an electrochemical model of multiplematerial lithium-ion batteries[END_REF], thereby extending its domain of validity. The relevance of the model goes beyond the state estimation problem investigated in this work, as it can be used for other purposes, such as designing charging control strategies. Second, we exploit the model to construct an observer with guaranteed convergence, whose implementation is lighter than Kalman filters as the correction gain is constant and no Jacobian matrices need to be evaluated on line, and whose convergence is guaranteed under a LMI condition, which may be verified off line. Third, the observer generates estimates for all the states in each particle, as opposed to [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF]. Fourth, we show how to modify the observer to generate estimates in a plausible set for all positive times by applying the recent general approach of [START_REF] Astolfi | Constrained state estimation for nonlinear systems: a redesign approach based on convexity[END_REF].

The rest of the paper is organised as follows. The model of the considered multiple-material batteries is presented in Section II. We then synthesize the observer in Section III. Simulation results are provided in Section IV. Finally, Section V concludes the paper. Technical developments regarding the solution to the aforementioned algebraic constraint between the overpotentials are postponed to the appendix.

Notation. Let R be the set of real numbers, R ě0 :" r0, 8q, Z be the set of integers and Z ą0 :" t1, 2, . . .u. For n, m P Z ą0 , I n stands for the identity matrix of dimension n and 0 nˆm for the zero matrix of R nˆm , whose indices may be dropped when clear from the context. Given square matrices A 1 , . . . , A n , diagpA 1 , . . . , A n q is the block-diagonal matrix, whose block-diagonal components are A 1 , . . . , A n . Given a positive definite matrix P , λ min pP q and λ max pP q respectively denote its minimum and maximum eigenvalue, which are strictly positive. For x P R n and y P R m with n, m P Z ą0 , px, yq stands for px J , y J q J . Given c P R and n P Z ą0 , c n denotes the vector column cp1, . . . , 1q J P R n . The notation B nx stands for the closed unit ball centered at the origin of R nx , and we will often drop the index n x when the dimension is clear from the context.

II. BATTERY MODELLING

In this section, we first state the assumptions we make to construct the model. We then derive the dynamical equations governing the solid phase. Afterwards, we model the electrochemical kinetics, which gives rise to an algebraic nonlinear constraint that we solve. The output voltage equation is then derived and the overall model is finally given in a state-space form. All the parameters are summarized in Table I, where k P t1, 2u and elec P tpos, negu.

A. Standing assumptions

The model we present relies on the next standing assumptions.

Standing Assumption 1 (SA1): Both electrodes are composite, each being composed of two types of active materials. A binary blend positive electrode is considered, such as the LiCo-NMC electrode in Figure 1, and a single insertion material is used for the negative electrode with two particles of different size. l SA1 allows considering various batteries technology currently available. It is important to note that different particle sizes are allowed in the model development. Hence, we also consider the negative electrode as a binary blend electrode made with graphite particles of two different sizes, as illustrated in Figure 1.

We make the next assumption on lithium (de-)insertion, which is customary in single particle models, see, e.g., [START_REF] Di Domenico | Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter[END_REF].

Standing Assumption 2 (SA2): The lithium insertion or deinsertion reaction is homogeneous along both electrodes. l SA2 implies a constant volume rate of current generation, proportional to the cell current for each electrode. Consequently, active particles of a given electrode material can be reduced to an equivalent single particle, in which the solid diffusion PDE, which governs lithium transport, is then decoupled from transport processes within the electrolyte, and can therefore be directly solved. As customarily done in electrochemical modeling of lithium-ion batteries, material equivalent particles are supposed spherical.

We make the next assumption on the electrolyte.

Standing Assumption 3 (SA3):

The dynamics of transport processes within the electrolyte is neglected. l As in [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF], the numerical simulations we have run for a standard set of parameters show that the impact of the electrolyte on the lithium concentrations and the output voltage variations is negligible at standard currents, and can thus be ignored in this case. The electrolyte will therefore be represented by a pure resistance. For high current rates, the multi-material lithium-ion battery model we propose can be enhanced by taking into account lithium ion concentration in the electrolyte phase and associated overpotentials, as done in, e.g., [START_REF] Rahimian | Extension of physicsbased single particle model for higher charge-discharge rates[END_REF], [START_REF] Tanim | State of charge estimation of a lithium ion cell based on a temperature dependent and electrolyte enhanced single particle model[END_REF] for standard SPM under minor changes. Such changes do not lead to any obstacle in terms of observer design, see [START_REF] Benzine | Systematic observer redesign for lithium-ion battery models to account for the electrolyte dynamics[END_REF].

Finally, we work at constant temperature as formalized next to focus on the main challenges induced by the modeling and the estimation of the considered multi-material batteries.

Standing Assumption 4 (SA4): The temperature T of the battery cell is supposed homogeneous and constant. l When some of the parameters vary with the temperature, the proposed model and observer can be adapted mutatis mutandis.

B. Solid phase

Battery solid phase modeling describes the lithium transport within the active material particles. In view of Section II-A, each electrode is composed of two spheres whose radius are denoted R mat ą 0, where the index "mat" stands for the considered material, i.e., mat P pos 1 , pos 2 , neg 1 , neg 2 ( in view of Figure 1. The equation relating the diffusive flux density of each material ϕ mat to its lithium concentration C mat depends on the radial coordinate r P r0, R mat s and is derived from Fick's first law

ϕ mat prq " ´Dmat BC mat prq Br , (1)
where D mat ą 0 is the solid diffusion coefficient of the considered material. The solid phase lithium transport in one electrode for a given material is then described by the mass conservation equation, for any t ě 0 and r P r0, R mat s,

BC mat prq Bt " 1 r 2 B Br ´Dmat r 2 BC mat prq Br ¯, (2)
with the boundary conditions ϕ mat p0q " 0, ϕ mat pR mat q " j Li mat {pa mat F q, where j Li mat P R is the electrochemical reaction rate, a mat :" 3 mat {R mat is the active surface per volume unit, mat ą 0 is the volume fraction of the material "mat" within the considered electrode, and F ą 0 is Faraday's constant. Each material is sampled in N mat P Z ą0 sphere crowns of internal radius r mat,n´1 ě 0, of external radius r mat,n ą 0, of volume V mat,n " 4 3 πpr 3 mat,n ´r3 mat,n´1 q, and of external surface S mat,n " 4πr 2 mat,n for n P t1, . . . , N mat u. Integrating the mass conservation (2) over the sphere crown n P t1, . . . , N mat u of the material "mat" leads to

d dt ş rmat,n
rmat,n´1 4πr 2 C mat prqdr " S mat,n´1 ϕ mat pr mat,n´1 q ´Smat,n ϕ mat pr mat,n q.

(3)

Then, by making a zero-order approximation function for C mat prq, i.e., by assuming that C mat prq " C mat,n for r P pr mat,n´1 , r mat,n q and in view of the expression of V mat,n above, (3) leads to

dC mat,n dt " S mat,n´1
V mat,n ϕ mat pr mat,n´1 q ´Smat,n V mat,n ϕ mat pr mat,n q.

(4) Hence, we obtain the next equations using the finite difference method to express the diffusive flux density, except at r " 0 and r " R mat for which boundary conditions on flux density are exploited in the following,

dC mat,1 dt " S mat,1 V mat,1 D mat C mat,2 ´Cmat,1 r mat,2 ´rmat,1 , dC mat,n dt " S mat,n V mat,n D mat C mat,n`1 ´Cmat,n r mat,n`1 ´rmat,n ´Smat,n´1 V mat,n D mat C mat,n ´Cmat,n´1 r mat,n ´rmat,n´1 dC mat,Nmat dt " ´Smat,Nmat´1 V mat,Nmat D mat ˆCmat,Nmat ´Cmat,Nmat´1 r mat,Nmat ´rmat,Nmat´1 ´Smat,Nmat V mat,Nmat j Li mat a mat F . (5)
Regarding the sphere discretization, it is important to have a fine discretization of the particle surfaces as the electrode open circuit voltages (OCV) depend on the corresponding local lithium concentrations, i.e., at r " R mat . This can be done by finely discretizing the whole particle for each material, however this would unnecessarily increase the dimension N mat of the state vector x mat . A fair compromise between accuracy and model order is obtained by considering a uniform volume discretization for n P t1, . . . , N mat u

V mat,n " V mat :" 1 N mat 4 3 πR 3 mat , (6)
which leads to the radius samples, for any n P t1, . . . , N mat u,

r mat,n " ˆn N mat ˙1{3 R mat , (7)
and r mat,0 " 0. This discretization gives a crown thickness finer at the particle surface than at its center, resulting in a higher accuracy compared to the standard uniform radius discretization for a given model order, or equivalently in a lower model order (approximately twice smaller) for a given accuracy.

As a result, we can write (5) as, with x mat :" pC mat,1 , . . . , C mat,Nmat q P R Nmat ,

9 x mat " A mat x mat `Bmat j Li mat , (8)
where A mat is given in (9) and B mat :" ´0, . . . , 0, ´Smat,Nmat {pV mat a mat F q ¯J. The resolution of (8) requires the knowledge of the boundary condition on flux density at r " R mat . The latter relies on the determination of the volume rate of current generation j Li mat associated with the material "mat": this is the purpose of the next section.

C. Electrical coupling constraint within electrodes

The volume rate of current generation j Li mat associated with the material "mat" is given by, according to Butler-Volmer law, and considering that oxidation and reduction charge transfer coefficients are both equal to 1{2, j Li mat " 2a mat j 0,mat sinh

´ηmat 2u T ¯, (10)
where j 0,mat P R is the exchange current density, η mat P R is the kinetic overvoltage, and u T " RT {F P R is the thermal voltage. The two materials of each electrode are electrically in parallel. They are therefore at the same potential, and the sum of their volume rates of current generation is equal to the total generation term, which is proportional to the cell current (as a consequence of the SPM assumption). Hence, the electrical coupling between the two electrode materials is governed by, for elec P tpos, negu,

$ ' & ' % j Li elec1 `jLi elec2 " ς elec I cell A cell δ elec OCV elec1 pH elec1 x elec1 q `ηelec1 " OCV elec2 pH elec2 x elec2 q `ηelec2 (11)
with I cell P R the cell current, A cell ą 0 the cell area, δ elec ą 0 the electrode thickness. OCV elec k pH elec k x elec k q with k P t1, 2u is the open circuit voltages of the electrode materials elec k , which continuously depend on the surface lithium concentrations within the corresponding particles, thereby justifying its dependency on H elec k x elec k as H elec k P R 1ˆNelec k is a line vector full of zeros except the element of the last column, which is 1. In generator convention (i.e., I cell ą 0 in discharge regime), ς neg " 1 for the negative electrode, and ς pos " ´1 for the positive electrode. As the OCV terms depend on particle surface concentrations (as well as, in theory, on the exchange current densities that are involved in Butler-Volmer law), ( 11) is a nonlinear algebraic constraint between the two particle surface concentrations, i.e., between the last term of the first material state vector, C elec1,Nelec 1 , and the last term of the second material state vector, C elec2,Nelec 2 . This is a distinctive feature of composite electrode lithium-ion battery models [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF], [START_REF] Camacho-Solorio | State estimation for an electrochemical model of multiplematerial lithium-ion batteries[END_REF], compared to standard SPM models. A common way to solve this nonlinear algebraic constraint is to linearize Butler-Volmer law, see [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF], [START_REF] Camacho-Solorio | State estimation for an electrochemical model of multiplematerial lithium-ion batteries[END_REF]. This approximation introduces an error, which may become significant at high current and/or low temperature as illustrated in Section IV-A. To avoid this possible shortcoming, we explain how to derive an exact solution to [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF] in the next section.

D. Solution to (10)-(11)

Let elec P tpos, negu, k P t1, 2u denotes one of the two materials in the given electrode and z elec k :" sinh ´ηelec k 2u T ¯. As explained in the appendix, the expressions of the volume rates of current generation [START_REF] Di Domenico | Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter[END_REF], [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF] lead to

α elec k z elec k `βelec k b 1 `z2 elec k " γ elec , (12)
A mat :"

¨´ν
‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' ν mat,k :" S mat,k V mat D mat 1 r mat,k`1 ´rmat,k
for any k P t1, . . . , N mat ´1u

µ mat,k 1 :" S mat,k 1 ´1 V mat D mat 1 r mat,k 1 ´rmat,k 1 ´1 for any k 1 P t2, . . . , N mat u (9) with $ ' ' ' ' ' ' & ' ' ' ' ' ' % α elec k :" a elec k j 0,elec k `aelec 3´k j 0,elec 3´k ˆcosh ´OCVelec k pHelec k xelec k q´OCVelec 3´k pHelec 3´k xelec 3´k q 2u T βelec k :" a elec 3´k j 0,elec 3´k ˆsinh ´OCVelec k pHelec k xelec k q´OCVelec 3´k pHelec 3´k xelec 3´k q 2u T γelec :" ςelecIcell 2Acellδelec . (13)
The unknown z elec k is therefore solution to the next second degree polynomial equation

pα 2 elec k ´β2 elec k qz 2 elec k ´2α elec k γ elec z elec k `γ2 elec ´β2 elec k " 0. ( 14
) The quantity α 2 elec k ´β2
elec k is strictly positive by the properties of sinh and cosh, hence [START_REF] He | Comparison study on the battery models used for the energy management of batteries in electric vehicles[END_REF] has two solutions, namely z elec k ,1 :"

αelec k γelec`bβ 2 elec k pα 2 elec k ´β2 elec k `γ2 elec q α 2 elec k ´β2 elec k and z elec k ,2 :" αelec k γelec´bβ 2 elec k pα 2 elec k ´β2 elec k `γ2 elec q α 2 elec k ´β2 elec k
. These two solutions are both equal to γ elec {α elec k when β elec k " 0 i.e., when OCV elec k pH elec k x elec k q " OCV elec 3´k pH elec 3´k x elec 3´k q.

To select a physically plausible solution, we first note that pz elec k ,1 ´γelec {α elec k qpz elec k ,2 ´γelec {α elec k q ď 0 as a consequence of a property of second degree polynomial equations with real coefficients 1 . Using α 2 elec k ´β2

elec k ą 0 and z elec k ,1 ě z elec k ,2 , we derive that z elec k ,1 ě γ elec {α elec k and z elec k ,2 ď γ elec {α elec k . Moreover, (12) can be written as

β elec k b 1 `z2 elec k " α elec k pγ elec {α elec k ´zelec k q.
As α elec k ą 0, we deduce that the physically plausible solution to [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF] satisfies β elec k pγ elec {α elec k ´zelec k q ě 0, from 1 More precisely, the two roots to [START_REF] He | Comparison study on the battery models used for the energy management of batteries in electric vehicles[END_REF], z elec k ,1 and z elec k ,2 , respectively upper and lower bound the double root γ elec {α elec k , which arises when

β elec k " 0.
which we derive that η elec k " ψ elec k px 1 , x 2 , I cell q with

ψ elec k px elec1 , x elec2 , I cell q :" $ ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' % 2u T asinhpz elec k ,2 q
when OCV elec k pH elec k x elec k q ą OCV elec 3´k pH elec 3´k x elec 3´k q 2u T asinhpγ elec {α elec k q when OCV elec k pH elec k x elec k q " OCV elec 3´k pH elec 3´k x elec 3´k q 2u T asinhpz elec k ,1 q when OCV elec k pH elec k x elec k q ă OCV elec 3´k pH elec 3´k x elec 3´k q.

(15)

Equation ( 15) thus gives η elec1 and η elec2 , from which we derive the volume rates of current generation j Li elec1 and j Li elec2 by using Butler-Volmer law [START_REF] Di Domenico | Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter[END_REF].

Remark 1: As mentioned earlier, the exact solution of the algebraic constraint [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF] is especially relevant for high currents and/or low temperatures. Otherwise, accurate approximate solutions can be obtained by replacing sinh ´ηmat 2u T ¯by ηmat 2u T in [START_REF] Di Domenico | Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter[END_REF] like in [START_REF] Bartlett | Electrochemical model-based state of charge and capacity estimation for a composite electrode lithium-ion battery[END_REF], [START_REF] Camacho-Solorio | State estimation for an electrochemical model of multiplematerial lithium-ion batteries[END_REF]. In this case, we obtain

η elec k " u T ´aelec k j 0,elec k `aelec 3´k j 0,elec 3´k ¯´1
ˆ" `OCV elec 3´k pH elec 3´k x elec 3´k q ´OCV elec k pH elec k x elec k q ȃelec 3´k j 0,elec 3´k {u T `ςelec I cell {pA cell δ elec q ı for elec P tpos, negu and k P t1, 2u. l

E. Model reduction

The multi-material model of the battery is reduced by taking advantage of the fact that the quantity of lithium inserted in the cell solid phases is constant, under the assumption that ageing can be ignored, like in e.g., [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF], [START_REF] Klein | Electrochemical model based observer design for a lithium-ion battery[END_REF]. Let Q Li ą 0 be this quantity. The lithium quantity balance in all the materials can be formulated as

Q Li :" Q pos1 `Qpos2 `Qneg1 `Qneg2 , (16)
where Q mat ą 0 (in Ah) is the quantity of lithium in a material, which is expressed as

Q mat :" xCy mat mat A cell δ elec F 3600 , ( 17
)
where xCy mat is the average concentration of lithium in a particle. Using a constant volume sampling for particle discretization, as defined by [START_REF] Blondel | Observer design for an electrochemical model of Lithium ion batteries based on a polytopic approach[END_REF], xCy mat in a particle is given by

xCy mat " 1 N mat Nmat ÿ k"1 C mat,k . (18)
Let

ω mat :" mat A cell δ elec F 3600N mat , (16) becomes
Q Li " ω pos 1 ř Npos 1 k"1 C pos 1 ,k `ωpos 2 ř Npos 2 k"1 C pos 2 ,k `ωneg 1 ř Nneg 1 k"1 C neg 1 ,k `ωneg 2 ř Nneg 2 k"1 C neg 2 ,k . (19)
We can then extract any sample lithium concentration from [START_REF] Tanim | State of charge estimation of a lithium ion cell based on a temperature dependent and electrolyte enhanced single particle model[END_REF] as a function of the total quantity of lithium Q Li and the remaining concentrations C mat,k 's. We select the concentration of the first sample of the first positive electrode material, i.e., C pos 1 ,1 , which is thus given by

C pos 1 ,1 :" QLi ωpos 1 ´řNpos 1 k"2 C pos 1 ,k ´ωpos 2 ωpos 1 ř Npos 2 k"1 C pos 2 ,k ´ωneg 1 ωpos 1 ř Nneg 1 k"1 C neg 1 ,k ´ωneg 2 ωpos 1 ř Nneg 2 k"1 C neg 2 ,k . (20)
In view of (20), C pos 1 ,1 is not needed in the state-space description of the overall system as it can be recovered from the other concentrations. This reduction is essential in the following for the convergence of the designed observer in Section III. In the following, we denote xpos 1 :" pC pos 1 ,2 , . . . , C pos 1 ,Npos 1 q P R N 1 pos 1 with N 1 pos 1 :" N pos ´1. In words, xpos 1 corresponds to x pos 1 truncated from its first component.

F. Output voltage

The last step before presenting the overall state-space model is the derivation of the output voltage equation. We have that

y :" OCV pos 1 p s H pos 1 xpos 1 q `ηpos 1 ´OCV neg 1 pH neg 1 x neg 1 q ´ηneg 1 `ρI cell , (21)
where s H pos 1 P R 1ˆN 1 pos 1 is the line vector made of zeros except the element of its last column, which is 1, and ρ ě 0 is a resistive term. We derive from [START_REF] Klein | Electrochemical model based observer design for a lithium-ion battery[END_REF] y " OCV pos 1 p s H pos 1 xpos 1 q `ψpos 1 px pos 1 , x pos 2 , I cell q ´OCV neg 1 pH neg 1 x neg 1 q ´ψneg 1 px neg 1 , x neg 2 , I cell q `ρI cell .

(22)

G. State-space representation

We are ready to present the overall state-space model. We introduce for this purpose the state vector x :" px pos 1 , x pos 2 , x neg 1 , x neg 2 q P R nx with n x :" N 1 pos 1 `Npos 2 Ǹneg 1 `Nneg 2 . In view of ( 8), ( 11), ( 15), ( 20) and (22), 9

x " Ax `φpx, I cell q y " OCV pos 1 p s H pos 1 xpos 1 q `ψpos 1 px pos 1 , x pos 2 , I cell q ´OCV neg 1 pH neg 1 x neg 1 q ´ψneg 1 px neg 1 , x neg 2 , I cell q.

(23) where A :" diagpA pos 1 , A pos 2 , A neg 1 , A neg 2 q `D with A pos 2 , A neg 1 , A neg 2 in (9), and A pos 1 , D and φ in 2 Table II.

III. OBSERVER DESIGN

In this section, we design an observer to estimate the state x of system (23).

A. Polytopic approach

To proceed with the design of the observer, we write system (23) as 9

x " Ax `B pxq `K1 pI cell q `∆1 pHxq y " C pxq `K2 pI cell q `∆2 pHxq,

where the corresponding matrices and functions are defined in Table II. In words, Ax`B pxq`K 1 pI cell q in (24) corresponds to the vector field we would obtain for the x-system by approximating sinh ´ηmat 2u T ¯with ηmat 2u T in [START_REF] Di Domenico | Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter[END_REF], see Remark 1. Similarly, C pxq `K2 pI cell q is the output equation we obtain under this approximation. Hence, ∆ 1 and ∆ 2 respectively correspond to the differences between the vector field of the x-system and the output equation with and without the above linear approximation in [START_REF] Di Domenico | Lithium-ion battery state of charge and critical surface charge estimation using an electrochemical model-based extended Kalman filter[END_REF]. Since ∆ 1 and ∆ 2 are included in (24), it is important to notice that (24) exactly corresponds to (23).

The observer takes the form 9

x " Ax `φpx, I cell q `Lpy ´ŷq " Ax `B pxq `K1 pI cell q `∆1 pHxq `Lpy ´ŷq ŷ " C pxq `K2 pI cell q `∆2 pHxq, (25) where x P R nx is the state estimate, L P R nxˆ1 is a matrix gain to be constructed and ŷ P R is the estimated output. The observer dynamics is a copy of the battery model (24) with the addition of the linear correction term Lpy ´ŷq.

We define the estimation error e :" x ´x. In view of (24) and (25), the estimation error dynamics is 9 e " Ae `pB ´LCq p pxq ´ pxqq `r ∆px, xq,

with r ∆px, xq :" ∆ 1 pHxq ´∆1 pHxq Ĺ p∆ 2 pHxq ´∆2 pHxqq. Our objective is to design L such that the origin of (26) satisfies a uniform, exponential stability property. We make the next assumption on the OCV's for this purpose.

Assumption 1: For any mat P tpos 1 , pos 2 , neg 1 , neg 2 u, there exist O 1 mat , O 2 mat P R such that for any z, z 1 P R, OCV mat pzq ´OCV mat pz 1 q " O mat pz, z 1 qpz ´z1 q,

where O mat pz, z 1 q :" λ 1 mat pz, z 1 qO 1 mat `λ2 mat pz, z 1 qO 2 mat with λ i mat pz, z 1 q P r0, 1s for i P t1, 2u and λ 1 mat pz, z 1 q`λ 2 mat pz, z 1 q " 1.

l Assumption 1 means that each OCV mat lies in a polytope generated by scalars O 1 mat and O 2 mat with mat P tpos 1 , pos 2 , neg 1 , neg 2 u, like in [START_REF] Blondel | Observer design for an electrochemical model of Lithium ion batteries based on a polytopic approach[END_REF]. This condition is often verified in practice. Indeed, OCV's are generally defined on the interval r0, 1s (after a normalization step, if needed) where they can be well-approximated by a piecewise continuously differentiable function. Then, it suffices to extrapolate the OCV on r1, 8q (respectively, p´8, 0s) by using zero-order or first-order approximations based on the value of the OCV at 1 (respectively, at 0) for Assumption 1 to hold. The constants φpx, I cell q := ´µpos 1 ,2 Q Li {ωpos 1 , 0 N pos 1 ´3 , ψpos,1 px, I cell q, 0 N pos 2 ´1 , ψpos,2 px, I cell q, 0 N neg 1 ´1 , ψneg,1 px, I cell q, 0 N neg 2 ´1 , ψneg,2 px, I cell q ¯J ψmatpx, I cell q :" ´Smat,Nmat {pVmatF q ˆ2j 0,mat sinh pψmatpx, I cell q{p2u T qq System (24)

B :" u ´1 T ¨0Npos 1 ´2 0 Npos 1 ´2 0 Npos 1 ´2 0 Npos 1 ´2 ´ϑpos 1 apos 2 j 0,pos 2 ϑpos 1 apos 2 j 0,pos 2 0 0 0 Npos 2 ´1 0 Npos 2 ´1 0 Npos 2 ´1 0 Npos 2 ´1
ϑpos 2 apos 1 j 0,pos 1 ´ϑpos 2 apos 1 j 0,pos

1 0 0 0 Nneg 1 ´1 0 Nneg 1 ´1 0 Nneg 1 ´1 0 Nneg 1 ´1 0 0 ´ϑneg 1 aneg 2 j 0,neg 2 ϑneg 1 aneg 2 j 0,neg 2 0 Nneg 2 ´1 0 Nneg 2 ´1 0 Nneg 2 ´1 0 Nneg 2 ´1 0 0 ϑneg 2 aneg 1 j 0,neg 1 ´ϑneg 2 aneg 1 j 0,neg 1 ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ‹ ' C :" ´1
´`apos 1 j 0,pos 1 `apos 2 j 0,pos 2 ˘´1 apos 2 j 0,pos 2 , `apos 1 j 0,pos 1 `apos 2 j 0,pos 2 ˘´1 apos 2 j 0,pos 2 , ´1 ``aneg 1 j 0,neg 1 `aneg 2 j 0,neg 2 ˘´1 aneg 2 j 0,neg 2 , ´`aneg 1 j 0,neg 1 `aneg 2 j 0,neg 2 ˘´1 aneg 2 j 0,neg 2 ¯ pxq :" pOCVpos 1 p s Hpos 1 xpos 1 q, OCVpos 2 pHpos 2 xpos 2 q, OCVneg 1 pHneg

1 xneg 1 q, OCVneg 2 pHneg 2 xneg 2 qq J K 1 pI cell q :" ´µpos 1 ,2 Q Li {ωpos 1 , 0 N pos 1 ´3 , ϑpos 1 {pA cell δposqI cell , 0 N pos 2 ´1 , ϑpos 2 {pA cell δposqI cell , 0 N neg 1 ´1 , ϑneg 1 {pA cell δnegqI cell , 0 N neg 2 ´1 , ϑneg 2 {pA cell δnegqI cell ¯J K 2 pI cell q :" u T {A cell
ˆ´`a pos 1 j 0,pos 1 `apos 2 j 0,pos 2 ˘´1 {δpos ´`aneg 1 j 0,neg 1 `aneg 2 j 0,neg 2 ˘´1 {δneg ¯Icell `ρI cell ∆ 1 pHxq :" φpx, I cell q ´B pxq ´K1 pI cell q ∆ 2 pHxq :" OCVpos 1 p s Hpos 1 xpos 1 q `ψpos 1 pxpos 1 , xpos 2 , I cell q ´OCVneg 1 pHneg 1 xneg 1 q ´ψneg 1 pxneg 1 , xneg 2 , I cell q ´C pxq ´K2 pI cell q ϑ elec k :" ´Selec k ,N elec k {pV elec k F q ˆj0,elec k ´aelec k j 0,elec k `aelec 3´k j 0,elec 3´k

¯´1

for elec P tpos, negu, k P t1, 2u

TABLE II MATRICES AND FUNCTIONS IN SYSTEMS (23) AND (24).

O 1 mat and O 2 mat in this case correspond to the maximum and the minimum slopes of OCV mat , respectively 3 .

A consequence of Assumption 1 is that, for any x, x P R nx , the term pxq ´ pxq arising in (26) (recall that p¨q is defined in Table II) can be written as

pxq ´ pxq " Opx, xqHpx ´xq, ( 28
)
where Opx, xq :" diag ´Opos 1 px pos 1 , xpos 1 q, O pos 2 px pos 2 ,

xpos 2 q, O neg 1 px neg 1 , xneg 1 q, O neg 2 px neg 2 , xneg 2 q ¯,
and H is defined in Table II. In view of Assumption 1, the term Opx, xq belongs to the convex hull of the set of matrices M defined 3 In practice, OCVs are often given by look-up tables. It then suffices to evaluate the maximum and minimum slopes of two consecutive points to derive O 1 mat and O 2 mat . If the matrix inequality in (31) appearing in Theorem 1 stated later is not satisfied for these matrices, it may help to evaluate O 1 mat and O 2 mat by considering not every two successive points, but each point with its n th successor for n P Z ą0 . In that way, the observer design is based on an approximation of the OCVs, which is tight for small n. as M :"

! diag ´Oipos 1 pos 1 , O ipos 2 pos 2 , O ineg 1 neg 1 , O ineg 2 neg 2
¯:

i mat P t1, 2u with mat P tpos 1 , pos 2 , neg 1 , neg 2 u ) . ( 29
)
The idea is to design L in (25) such that A `pB ´LCqM H has eigenvalues with strictly negative real parts together with a common Lyapunov for any M P M, and to treat the term r ∆px, xq in (26) as a vanishing perturbation. We note for this purpose that ∆ 1 and ∆ 2 , which appear in the definition of r ∆, are locally Lipschitz in view of their expressions and Assumption 1, and are thus Lipschitz on any compact set. Consequently, for any compact set S Ă R nx , there exist κ i pSq ě 0 for i P t1, 2u such that for any x, x P S,

|∆ i pHxq ´∆i pHxq| ď κ i pSq|Hpx ´xq|. ( 30
)
We use this property in the following to cope with the term r ∆ in the stability analysis of (26).

The next theorem provides a condition for the design of L in (25), which ensures a local, uniform, exponential stability property for the origin of system (26).

Theorem 1: Given a compact set S Ă R nx , θ ą 0 and κ i ě 0 with i P t1, 2u as in (30) for compact set 4 S `θB, we suppose the following holds.

(i) Assumption 1 is satisfied.

(ii) There exist P P R nxˆnx symmetric, positive definite, L P R nxˆ1 , ε 1 , ε 2 ą 0 such that for any M P M with M given in (29),

¨N pM q `pε 1 κ 1 `ε2 κ 2 qH J H P ´P L P ´ε1 I nx 0 nxˆ1 ´LJ P 0 1ˆnx ´ε2 'ă 0

(31) with N pM q :" pA `pB ´LCqM Hq J P P pA `pB ´LCqM Hq. There exist m 1 ě 1 and θ 0 , m 2 ą 0 such that for any solution x to (24) with input I cell : R ě0 Ñ R satisfying xptq P S for any t ě 0, and any solution x to (25) such that pxp0q xp0qq J P pxp0q ´xp0qq ď θ 0 , it holds that

|xptq ´xptq| ď m 1 e ´m2t |xp0q ´xp0q| (32a) xptq P S `θB (32b)
for any t ě 0. l Proof: We first write system (26) as, in view of (28), 9 e " Ae `pB ´LCq Opx, xqHe `r ∆px, xq.

(

Let V peq :" e J P e for any e P R nx with P from item (ii) of Theorem 1. Let x P S, x P S `θB, e " x ´x and 9 e " Ae `pB ´LCq Opx, xqHe `r ∆px, xq with some abuse of notation. We have x∇V peq, 9 ey " 2e J P pA `pB ´LCqOpx, xqHq e `2e J P ∆px, xq.

Let r ∆ i :" ∆ i pHxq ´∆i pHxq for i P t1, 2u, and χ :" pe, r ∆ 1 , r ∆ 2 q. As mentioned before (29), Opx, xq belongs to the convex hull of M in (29). Numbering the elements of M as M 1 , . . . , M 2 4 , we thus have that there exist r λ j px, xq P r0, 1s for j P t1, . . . , 2 4 u such that ř jPt1,...,2 4 u r λ j px, xq " 1 and Opx, xq " ř jPt1,...,2 4 u r λ j px, xqM j . As a consequence of these two properties x∇V peq, 9

ey " ÿ jPt1,...,2 4 u r λ j px, xq ´2e J P pA `pB ´LCqM j Hq e 2e

J P ∆px, xq ¯, which we write as x∇V peq, 9 ey " ÿ jPt1,...,2 4 u r λ j px, xqχ J ¨N pM j q P ´P L P 0 0 ´LJ P 0 0 'χ, (35) where N pM i q is defined in item (ii) of Theorem 1. We deduce from the strict matrix inequality in (31) that there exists ε ą 0 (independent of x, x, S and θ) such that

x∇V peq, 9 ey ď χ J diag `´pε 1 κ 1 `ε2 κ 2 qH J H, ε 1 I nx , ε 2 ˘χ ´ε|χ| 2 .
(36) 4 We omit the dependency of κ i , i P t1, 2u, on S `θB in this theorem.

On the other hand, because x P S and x P S `θB, (30) holds and is equivalent to

" χ J diag `´κ 1 H J H, I nx , 0 ˘χ ď 0 χ J diag `´κ 2 H J H, 0, 1 ˘χ ď 0. (37)
We thus obtain from (36) and (37) that x∇V peq, 9 ey ď ´ε|χ| 2 ď ´ε|e| 2 ď ´ε λ max pP q V peq, (38) as V peq ď λ max pP q|e| 2 . Consider now any solution x to (24) with input I cell such that xptq P S for all t ě 0, and let x be a solution to (25) with xp0q such that V pxp0q ´xp0qq ď θ 0 :" λ min pP qθ 2 . Using the fact that λ min pP q|xp0q ´xp0q| 2 ď V pxp0q ´xp0qq, we obtain that |xp0q ´xp0q| ď θ and thus xp0q P S `θI nx , as xp0q P S. We note that x∇V pep0qq, 9 ep0qy ď ´ε|ep0q| 2 from (38), with ep0q " xp0q ´xp0q. We thus deduce from (38) that 9 V pxptq ´xptqq ď 0, which implies that V pxptq ´xptqq ď V pxp0q ´xp0qq ď θ 2 0 and thus that xptq P S `θB for any t in the domain of x, which is actually r0, 8q. We have proved (32b) holds.

Property (32a) is obtained by integrating (38) over time, which we can as xp¨q and xp¨q lie in S and S`θB, respectively. We then use that λ min pP q|e| 2 ď V peq ď λ max pP q|e| 2 for any e P R nx to obtain the desired result.

Theorem 1 states that the estimation error uniformly, exponentially converges to the origin whenever: (a) the solution to (24) lies in a given compact set S for all positive times, (b) the matrix inequality (31) holds, and (c) the initial value of x is sufficiently close to the initial value of x. We discuss each of these requirements below.

Regarding (a), this property is always verified in practice. Indeed, we recall that the state vector x corresponds to lithium concentrations in different regions of each particle. These concentrations are always between 0 and C max,mat , which is the maximum concentration in particle "mat" for mat P tpos 1 , pos 2 , neg 1 , neg 2 u. A possible definition of S is thus r0, C max,pos 1 s N 1 pos 1 ˆr0, C max,pos 2 s Npos 2 ˆr0, C max,neg 1 s Nneg 1 r0, C max,neg 2 s Nneg 2 . We will refine the definition of the set where the solutions to (24) lie in practice in Section III-B.

Concerning (b), the matrix inequality in (31) is linear when considering P , ε 1 , ε 2 and W :" P L as unknowns. We then retrieve L by defining as P ´1W . Efficient numerical solvers are available to check whether this condition holds. We note that (31) involves κ 1 and κ 2 . These parameters can be ignored when current I cell is not too high and the temperature not too low. Indeed, in this case, the ∆ i 's are small, which lead to small κ i 's in (30). Then, the satisfaction of (31) for κ i " 0 induces its satisfaction for some small enough κ i ą 0, i P t1, 2u. Note that when κ i " 0 for i P t1, 2u, (31) reduces to N pM q ă 0 for any M P M.

Finally, the local condition (c) is due to the term r ∆ in (24). Indeed, this term may become very large, in view of its expression, when x and x are initialized far from each other at t " 0, which may prevent the state estimate to converge to the battery state. Nevertheless, the numerical simulations reported in Section IV show that convergence occurs even when xp0q and xp0q are significantly different and the current is high, thereby illustrating the efficiency of the proposed observer. Note that, when the current is not too high and the temperature not too low, the ∆ i 's can be ignored and the observer converges for any initialization xp0q in view of the proof of Theorem 1.

Contrary to an extended Kalman filter, which may also ensure a local convergence property, the proposed observer is simpler to implement as the gain L is constant, and not derived from the solution of a Riccati equation, and no Jacobian matrices need to be evaluated on-line, which may computationally demanding for system (23). In addition, we know a priori whether convergence occurs thanks to the conditions of Theorem 1.

B. Constraining the state estimates

Theorem 1 provides conditions under which the state estimate generated by the observer in (25) converges to the state and remains in a set S `θB for all positive times. Still, it may happen that the estimated state has aberrant values in the transient like negative concentrations or concentrations far bigger than the maximum admissible one in the corresponding electrode, even it remains in S `θB as θ may actually be large. To address this potential issue, we apply the technique recently proposed in [1, Section III] to constrain the state estimates to remain in a given prescribed plausible set p X for all positive times.

We first note, as already mentioned at the end of Section III-A, that the solutions to system (24) always lie in the set r0, C max,pos 1 s N 1 pos 1 ˆr0, C max,pos 2 s Npos 2 ˆr0, C max,neg 1 s Nneg 1 r0, C max,neg 2 s Nneg 2 in practice. This set gives rise to the next constraints, for any i P t1, . . . , n x u, with x P R nx ,

" a ì x ď b ì a í x ď b í , (39)
where a ì P R 1ˆnx is full of zeros except its i th element, which is 1, a í :" ´aì , b ì :" C max,pos 1 when i P t1, . . . , N 1 pos 1 u, b ì :" C max,pos 2 when i P tN 1 pos 1 `1, N 1 pos 1 `Npos 2 u, b ì :" C max,neg 1 when i P tN 1 pos 1 `Npos 2 `1, N 1 pos 1 `Npos 2 `Nneg 1 u, b ì :" C max,neg 2 when i P tn x ´Nneg 2 , n x u, and b í :" 0.

We also exploit [START_REF] Xia | A novel method for state of charge estimation of lithium-ion batteries using a nonlinear observer[END_REF] and the fact that C pos 1 ,1 P r0, C max,pos 1 s to derive the next extra constraints

" a red x ď b red x a ŕed x ď b ŕed x, (40)
where the index "red" stands for "reduction" as these constraints are related to the model reduction step in Section II-E, a red :" ´pω pos 1 , . . . , ω pos 1 , ω pos 2 , . . . , ω pos 2 , ω neg 1 , . . . , ω neg 1 , ω neg 2 , . . . , ω neg 2 q P R 1ˆnx , a ŕed :" ´ar ed , b red :" ω pos 1 C max,pos 1 ´QLi and b ŕed :" Q Li .

In view of (39) and (40), we define the set X , where the solutions to (24) lie as X :"

!

x P R nx : a ì x ď b ì , a í x ď b í for i P t1, . . . , n x u,

a red x ď b red , a ŕed x ď b ŕed ) . (41)
We note that set X is compact, as required by [START_REF] Astolfi | Constrained state estimation for nonlinear systems: a redesign approach based on convexity[END_REF]SA1], and convex. To apply the results of [1, Section III], we also constrain the input I cell to take values in U :" rI min,cell , I max,cell s for some I min,cell ă I max,cell , which is a reasonable requirement in practice.

To define the set p X where we want to constrain the state estimate, we introduce, as in 5 [1, Lemma 7], the function c : R nx Ñ R 2nx`2 ě0 defined as, for any x P R nx , cpxq :" pc 1 pxq, . . . , c 2nx`2 pxqq J c i pxq :" $ ' ' & ' ' % maxta ì x ´bì , 0u 2 , i P t1, . . . , n x u maxta í x ´bí , 0u 2 , i P tn x `1, . . . , 2n x u maxta red x ´br ed , 0u 2 , i " 2n x `1 maxta ŕed x ´bŕ ed , 0u 2 , i " 2n x `2.

(42) Given r ą 0, we define p X :" tx P R nx : |cpxq| ď ru. Noting that X :" tx P R nx : cpxq " 0u in view of (41), we have X Ĺ p X . We have therefore inflated the original set X , but this inflation can be made as close as desired to the set X by taking r ą 0 small.

To ensure that the state estimates remain in p X for all positive times, when initialized in this set, we need to modify observer (25). We assume the next assumption for this purpose.

Assumption 2: Item (ii) of Theorem 1 holds with κ 1 and κ 2 in (30) computed for set p X . l Assumption 2 means that we can design a gain L and an associated symmetric, positive definite matrix P for the original observer in (25). We modify observer (25) as 9

x " Ax `B pxq `K1 pI cell q `∆1 pHxq `Lpy ´ŷq ´gP ´1 dcpxq dx

J cpxq ŷ " C pxq `K2 pI cell q `∆2 pHxq, (43)
where P is obtained for item (ii) of Theorem 1, which holds in view of Assumption 2, and g ą 0 is a design parameter. Constrained observer (43) thus corresponds to (25) with the addition of the extra term ´gP ´1 dcpxq dx J cpxq for the xsystem.

The next proposition provides conditions under which the state estimates generated by (43) remain in p X for all positive times, and the convergence property established in Theorem 1 is preserved.

Proposition 1: Suppose Assumptions 1-2 hold. There exist g ‹ ą 0 and θ 0 ą 0 such that for any g ą g ‹ , for any solution x to (24) such that xptq P X and I cell ptq P U for all t ě 0, any solution x to (43) initialized in X and such that pxp0q xp0qq J P pxp0q ´xp0qq ď θ 0 satisfy (32a) and xptq P p X for all t ě 0. l Sketch of proof: The proof follows similar lines as the proof of [1, Theorem 1] but some minor modifications are required due to the fact that the original observer in (25) converges locally, and not (semi)globally.

We introduce the set Y :" C pX q `K2 pUq `∆2 pHX q. We also define F max :" sup ! |Ax `φpx, I cell q `Lpy ýq| : I cell P U, y P Y, x P R nx such that cpxq " r ) , g ‹ :" pF max λ max pP qq{d min with d min :" inf

x such that cpxq " r ) . Let g ą g ‹ and x be a solution to (24) such that xptq P X and its input I cell verifies I cell ptq P U for all t ě 0. Let x be a solution to (43) initialized in X and such that pxp0q xp0qq J P pxp0q ´xp0qq ď θ 0 where θ 0 ą 0 is fixed below. By following the same lines as in the proof of [1, Theorem 1], we derive that xptq P p X for any t ě 0 in the domain of existence of x.

Then, still be following [1, Theorem 1] and in view of (38), we derive that by taking θ 0 ą 0 sufficiently small like in the proof of Theorem 1, x is defined for all positive times and converges exponentially to x, so that (32a) holds and xptq P p X for all t ě 0.

Proposition 1 ensures that the exponential stability property established in Theorem 1 is preserved with the additional benefit of maintaining the estimated state in the set p X at all times. We need for this purpose to select g in (43) sufficiently large. An explicit condition is provided in the proof of Proposition 1.

An handy way to proceed in practice with the design of observer ( 43) is as follows. First, solve the matrix inequality in (31) for some (small enough) κ i ě 0 with i P t1, 2u to construct L and P . Second, progressively increase the value of g until satisfactory results are obtained in simulations. The idea for this second step is that the larger g, the smaller r in the definition of p X , and vice versa.

IV. SIMULATION RESULTS

We simulate system (24) with the numerical values provided in Table I. The considered OCV's are depicted in Figure 2, where we can see that the same function is taken for both negative particles. We consider the sequence of 10C-discharge / rest / 10C-charge shown in Figure 3 for I cell .

We initialize system (23) at equilibrium, meaning that all the initial local concentrations within the same particle are equal, with a state-of-charge (SOC) of 100%. This corresponds to the initial state xp0q " 10 4 p1.18591 N 1 pos 1 , 0.83091 Npos 2 , 0.82251 Nneg 1 `Nneg 2 q. Because of the composite nature of the electrodes, the SOC is defined as SOC :"

SOC pos 1 Qu pos 1 `SOC pos 2 Qu pos 2 Q cell ( 44
)
where Qu pos k ą 0 is the useful capacity of particle pos k , Q cell ą 0 is the cell capacity and with, for k P t1, 2u, SOC pos k :" 100

1 Npos k ř Npos k j"1 C pos k ,j {C max,pos k ´ιpos k ,soc0 ι pos k ,soc100 ´ιpos k ,soc0 , (45)
where ι pos1,soc0 , ι pos1,soc100 ą 0 are the insertion rates in particle pos k at SOC " 0% and SOC " 100%, respectively. We consider the positive electrode in (44) but the definition is equivalent and thus gives the same SOC when considering the negative electrode. The time evolution of the corresponding local concentrations is shown in Figure 4.

In the following, we illustrate the benefit of exactly solving (10)- [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF] to construct model ( 23) compared to a model based the condition on the trace of P at the price of a slower convergence speed. Regarding (43), we select g " 10.

We initialize observers ( 25) and ( 43) with the same value, in particular we assume the battery is at rest, so that the local concentrations within the same particle are identical as mentioned above, with a corresponding SOC estimate of 0%. This corresponds to xp0q " 10 4 p2.36561 N 1 pos 1 , 2.36941 Npos 2 , 0 Nneg 1 `Nneg 2 q We therefore initialize the observer with 100% of error on the SOC. This allows to illustrate good convergence properties of the observers even in this worst case scenario. The norms of the corresponding estimation errors are depicted in Figure 7 and the actual SOC and the estimated ones are reported in Figure 8. In all cases, we see that the estimated variables converge quickly to states of system (23). Furthermore, we notice that the estimates generated by (25) and (43) cannot be distinguished. This comes from the fact that the solution to (25) always belongs to X . As a result, the additional term ´gP ´1 dcpxq dx J cpxq in (43) is always equal to zero and the solutions to (25) and (43) coincide. This is no longer the case when considering the initial value xp0q " 10 4 p0.906, 1.763, 0.547, 1.640, 0.139, 0.352, 0.574, 1.436, 2.348, 1.6101 Nneg 1 , 0 Nneg 1 q, which does belong to X . In this case, the solution to the unconstrained observer (25) leaves X in the transient, which activates the extra term in (43) to bring back the estimates in the plausible set. This is illustrated in Figures 9 and10. These figures show that, while both observers still generate converging estimates, those given by the constrained observer are more accurate in the transient, thereby demonstrating its additional benefit over observer (25).

V. CONCLUSION

We have presented a finite-dimensional electrochemical model of lithium-ion batteries, whose positive electrode is modeled as two particles of different active materials and the negative electrode is described by two graphite particles, possibly of different size. Compared to the related works of the literature, an interesting feature of the model is that it relies on the exact solution to the algebraic relationship between the overpotentials, and not on the solution of its linearization, and is thus valid on a larger range of currents and temperatures. We have then exploited this model to design a (constrained) observer with guaranteed convergence whenever a LMI holds, to reconstruct the lithium concentrations in each particles. Simulations results using the literature data illustrate the efficiency of the approach.

A possible future direction would be to develop a sampleddata version of the proposed estimation scheme to ease its digital implementation on a BMS.

VI. APPENDIX

We explain in this section how to obtain [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF]. Let k P t1, 2u. We write the first equation in [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF] as, in view of (10), a elec k j 0,k sinh ´ηelec k 2u T ¯`a 3´k j 0,3´k sinh ´η3´k ānd that η elec 3´k ´ηelec k " OCV elec k pH elec k x elec k q ÓCV elec 3´k pH elec 3´k x elec 3´k q.

Fig. 1 .

 1 Fig. 1. Battery cell schematic.

Fig. 2 .

 2 Fig. 2. OCV for each particle as a function of the insertion rate.

Fig. 3 .

 3 Fig. 3. Current profile for I cell .

Fig. 4 .

 4 Fig. 4. Concentrations generated by model (23) in pos 1 (top left), pos 2 (bottom left), neg 1 (top right), neg 2 (bottom right).

Fig. 5 .

 5 Fig. 5. Differences between the surface concentrations generated by model (23) and its approximate version following Remark 1 in pos 1 (top left), pos 2 (bottom left), neg 1 (top right), neg 2 (bottom right).

Fig. 6 .

 6 Fig. 6. Output voltages of model (23) and the approximate model given by Remark 1.

Fig. 7 .

 7 Fig. 7. Norm of the estimation errors.

Fig. 8 .

 8 Fig. 8. State of charge and its estimates.

Fig. 9 .

 9 Fig. 9. Norm of the estimation errors for observer (25) (red) and for constrained observer (43) (yellow).

Fig. 10 .

 10 Fig. 10. State of charge (blue), and its estimates generated by observer (25) (red), and by constrained observer (43) (yellow).

  2u T ¯" σ Icell 2Acellδelec . (46) We obviously have sinh ´η3´k 2u T ¯" sinh ´η3´k ´ηelec k `ηelec k 2u T ¯. Since sinhpa `bq " sinhpaqcoshpbq sinhpbqcoshpaq for any a, b P R, we derive that sinh ´η3´k 2u T ¯" sinh ´ηelec k 2u T ¯cosh ´η3´k ´ηelec k 2u T ¯sinh ´η3´k ´ηelec k 2u T ¯cosh ´ηelec k 2u T ¯. Hence, (46) gives a elec k j 0,k sinh ´ηelec k 2u T ¯à 3´k j 0,3´k sinh ´ηelec k 2u T ¯cosh ´η3´k ´ηelec k 2u T ¯à 3´k j 0,3´k sinh ´η3´k ´ηelec k 2u T ¯cosh ´ηelec k 2u T ¯" σ Icell 2Acellδelec . We obtain (12) by noting that cosh ´ηelec k 2u T ¯" c 1 `sinh 2 ´ηelec k 2u T

TABLE I PARAMETER

 I DEFINITION AND NUMERICAL VALUES USED IN SECTION IV Insertion rate of pos 1 at SOC " 100% [-] 0.3477 ι pos2,soc0 Insertion rate of pos 2 at SOC " 0% [-] 0.9898 ι pos2,soc100 Insertion rate of pos 2 at SOC " 100% [-] 0.4962 ρ Resistive term in (22) rΩs 0

	A cell	Cell area [m 2 ]	0.8
	F	Faraday's constant [C{mol]	96485
	R	Gas constant [J{K{mol]	8.3145
	T	Temperature [K]	298.15
	N elec k " N Number of samples for the positive	5
		(resp. negative) electrode materials [-]
	δpos	Thickness of the positive electrode [µm]	36.4
	δneg	Thickness of the negative electrode [µm]	50
	Dpos k Dneg k Cmax,pos k Cmax,neg k Rpos 1 Rpos 2 Rneg 1 Rneg 2 j 0,pos k j 0,neg k	Solid phase diffusive coefficient [m 2 {s] Solid phase diffusive coefficient [m 2 {s] Maximum concentration [mol{m 3 ] Maximum concentration [mol{m 3 ] Particle radius of pos 1 [µm] Particle radius of pos 2 [µm] Particle radius of neg 1 [µm] Particle radius of neg 2 [µm] Exchange current density [A{m 2 ] Exchange current density [A{m 2 ]	3.7 ˆ10 ´15 2 ˆ10 ´15 23900 16100 4 2 2 4 0.26 0.36
	pos k	Volume fraction of the material within the positive electrode [-]	0.283
	neg k	Volume fraction of the material within the negative electrode [-]	0.340
	ςpos		´1
	ςneg		1
	Q Li	Lithium quantity in cell solid phases rAhs 10.452
	Qupos 1 Qupos 2 Q cell	Useful capacity of pos 1 [Ah] Useful capacity of pos 2 [Ah] Cell capacity [Ah]	3.3959 2.6041 6
	ι pos1,soc0	Insertion rate of pos 1 at SOC " 0% [-]	0.9914
	ι pos1,soc100

  ´µpos 1 ,2 , 0 nx´1 ¯J ˆ´´1 Npos 1 ´1, ´ωpos 2 {ωpos 1 1 Npos 2 , ´ωneg 1 {ωpos 1 1 Nneg 1 , ´ωneg 2 {ωpos 1 1 Nneg 2 Hpos 1 , Hpos 2 , Hneg 1 , Hneg 2 ˘J

	System (23)
		¨´ν mat,2 ´µmat,2 µ mat,3	ν mat,2 ´νmat,3 ´µmat,3 ν mat,3 0	. . . 0	. . . . . .	0 . . .	‹ ‹ ‹
	Apos 1 :"	0 . . .	. . .	. . . . . .	. . .		. . . 0	‹ ‹ ‹ ‹ ‹ ‹ ‹
		. . .			µ mat,Nmat´1 ´νmat,Nmat´1 ´µmat,Nmat´1 ν mat,Nmat´1	‹ ‹ '
		0	. . .	. . .	0	µ mat,Nmat	´µmat,Nmat
	D :"

H

:" `s

We write ψmat as a function of x (and I cell ) in TableIIfor the sake of convenience.

There is a typo in[1, (62)]: it should be cn c pxq and not cn x pxq.

on the approximation discussed in Remark 1; see Section IV-A. Afterwards, we design observers by following Section III and illustrate their good convergence properties in Section IV-B.

A. Effect of the exact solution to (10)- [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF]

on model (23)

We simulate the model derived from Remark 1 with the same input and the same initial condition as specified above. We observe appreciable differences on the surface concentrations of each particle and the output voltage, as shown in Figures 5 and6, respectively. These figures illustrate the better accuracy due to the exact solution ( 10)- [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF] in model (23).

B. State estimation

We design observer (25) and its constrained version in (43) by solving (31) with κ 1 " 10 ´8 and κ 2 " 10 ´9.

In particular, we impose that the left hand-side of (31) is less than ´0.1I nx and we constrain the trace of P to be less than 2 ˆ10 3 . The obtained gain matrix is L " 10 5 p´0.069, ´0.762, ´0.872, ´1.276, ´1.198, ´1.237, ´1.309, ´1.458, ´2.101, 0.507, 0.782, 1.041, 1.276, 1.380, 0.069, 0.268, 0.389, 0.433, 0.939q J . The condition on the trace of P leads to large coefficients for L on purpose: in that way, the convergence of the estimation error to zero is fast. Smaller coefficients for L can be obtained by relaxing