Weixuan Wang
email: weixuanw@student.unimelb.edu.au

Alejandro I Maass
email: alejandro.maass@unimelb.edu.au

Dragan Nešić

Ying Tan
email: yingt@unimelb.edu.au

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

W P M H Heemels

Stability of nonlinear systems with two time scales over a single communication channel*

This paper studies the stabilisation problem for a class of nonlinear systems with two time scales, where only a single communication channel is available to allocate both low and high-frequency transmissions from slow and fast subsystems, respectively. A clock mechanism is proposed to govern the transmissions, and the closed-loop system is modelled by a hybrid singularly perturbed system. Singular perturbationbased analysis is used to obtain individual maximum allowable transmission intervals for both slow and fast transmissions, and also to guarantee semi-global practical asymptotic stability with respect to the minimum allowable transmission interval of slow transmissions. We illustrate the results via a numerical example.

I. INTRODUCTION

Networked control systems (NCSs) are feedback control systems whose loops are closed through real-time communication networks. The rapid development of network technologies provides NCSs with widespread Internet of Things (IoT) application scenarios where two or more time scales are often involved, such as manufacturing automation, smart transportation, telemedicine and space and terrestrial exploration [START_REF] Xu | A survey on industrial internet of things: A cyber-physical systems perspective[END_REF]. Most of the state-of-the-art NCS design methodologies, e.g., [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF], [START_REF] Heijmans | Computing minimal and maximal allowable transmission intervals for networked control systems using the hybrid systems approach[END_REF], do not directly exploit the multiple-scale structure due to the oversimplified system models, as a consequence, they potentially require excessive transmission rates for stability. As IoT devices are often wireless and battery-supported, with limited resources such as bandwidth, a high transmission rate for the overall system may be infeasible. By exploiting the multiple time-scale property of the system with the singularly perturbed method [START_REF] Khalil | Nonlinear Systems[END_REF], it is possible to obtain stability and robustness guarantees while mitigating redundant transmissions for the slow dynamics. Singularly Perturbed NCSs (SPNCSs) have garnered significant attention in recent years due to their practical importance in various engineering applications. Researchers have proposed a few control and analysis approaches for both linear and nonlinear SPNCSs. For instance, [START_REF] Wang | Observer-based sliding mode control for networked fuzzy singularly perturbed systems under weighted try-once-discard protocol[END_REF] and [START_REF] Song | Dynamic event-triggered sliding mode control: Dealing with slow sampling singularly perturbed systems[END_REF] proposed sliding mode control strategies for linear discrete-time SPNCSs, while [START_REF] Cheng | Ultimate boundedness control for networked singularly perturbed systems with deception attacks: A markovian communication protocol approach[END_REF] explored ultimate boundedness control for linear discrete SPNCSs with communication constraints and deception attacks. In the case of nonlinear SPNCSs, [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] developed a stabilizing event-triggered feedback law for SPNCSs with fast plant dynamics assumed to be stable. Meanwhile, [START_REF] Heijmans | Singularly perturbed networked control systems[END_REF] introduced sufficient conditions that guarantee stability for time-triggered nonlinear SPNCSs, where only plant output is transmitted via network. More precisely, [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] and [START_REF] Heijmans | Singularly perturbed networked control systems[END_REF] adopted an emulation-based approach proposed in [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF] to design the NCS. That is, a controller is initially designed to guarantee stability in the absence of communication constrains. Then, an event/time-triggered condition is determined to preserve the stability of the closed-loop system when implemented over the network. Additionally, they cast the overall problem as a hybrid SPS with the formalism of [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]. While [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] only transmit slow states by assuming stable fast plant dynamics, [START_REF] Heijmans | Singularly perturbed networked control systems[END_REF] made no assumption on the stability of the plant. Moreover, it required two separate channels to transmit fast and slow states respectively, which may be restrictive since two separate channels may not be available in practice. These studies highlight the importance of considering both communication constraints and the two time scale nature of SPNCSs in developing effective control strategies for practical applications.

In this paper, we propose an emulation-based approach for the design of two-time-scale nonlinear SPNCS, where only a single channel is available to transmit both slow and fast signals, which is easier to implement in practice. Our approach leads to three main contributions. Firstly, we introduce a mechanism, which reduces the demand on resources. It involves two clocks to govern the transmissions of slow and fast signals over a single channel. Secondly, we present a stability analysis of hybrid SPS when its flow and jump sets depend on the time scale separation parameter, which is commonly denoted by ϵ in the literature. Thirdly, we consider more general nonlinear NCS scenarios than those considered in [START_REF] Heijmans | Singularly perturbed networked control systems[END_REF]. While only plant states are transmitted in [START_REF] Heijmans | Singularly perturbed networked control systems[END_REF], we consider the transmission of both plant output and controller input in our NCS. In addition to the plant and state-feedback controller form introduced in [START_REF] Heijmans | Singularly perturbed networked control systems[END_REF], we consider a dynamical output feedback controller, which draws inspiration from the linear works [START_REF] Yoo | New designs of linear observers and observerbased controllers for singularly perturbed linear systems[END_REF], [START_REF] Lin | Composite observer-based feedback design for singularly perturbed systems via LMI approach[END_REF] and the nonlinear work [START_REF] Christofides | Output feedback control of nonlinear twotime-scale processes[END_REF] that study dynamic controllers for SPSs in the absence of network. We also consider NCS with scheduling protocols, whereas [START_REF] Song | Dynamic event-triggered sliding mode control: Dealing with slow sampling singularly perturbed systems[END_REF] and [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] only assumed the sampled-data structure. Moreover, while [START_REF] Wang | Observer-based sliding mode control for networked fuzzy singularly perturbed systems under weighted try-once-discard protocol[END_REF]- [START_REF] Abdelrahim | Event-triggered control of nonlinear singularly perturbed systems based only on the slow dynamics[END_REF] assumed periodic transmissions, we allow transmissions to be aperiodic. As a summary, our proposed approach provides a more general and flexible framework for designing SPNCSs with single communication channel.

Notation: The sets of real numbers and integers larger than or equal to a real number n are denoted by R ≥n and Z ≥n , respectively. For vectors

v i ∈ R n , i ∈ {1, 2, • • • , N }, we denote the vector [v T 1 v T 2 • • • v T N ] T by (v 1 , v 2 , • • • , v N )
, and the inner product by ⟨•, •⟩. Given a vector x ∈ R nx and a non-empty closed set A ⊆ R nx , the distance from x to A is denoted by |x| A , and it is defined by |x| A := min y∈A |x-y|. We use U • to denote the Clarke generalized derivative [START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF]Eqn. (20)] of a function U . We denote the n by m zero matrix by 0 n×m .

II. PROBLEM SETTING

In this paper, we consider a two-time-scale nonlinear NCS that is designed by an emulation-based approach. Specifically, a dynamical output-feedback controller is initially designed to guarantee the stability of a reduced system (slow) and a boundary-layer system (fast) in the absence of a network. Subsequently, this controller is implemented over the network, and the design consists in selecting two different maximum allowable transmission intervals (MATIs) to preserve the stability of the networked reduced and boundary-layer systems, respectively. Figure 1 illustrates the configuration of the considered time-triggered SPNCS. Subsequent sections will elaborate on each individual element.

     ẋp = f p (x p , z p , û) ϵ żp = g p (x p , z p , û) y p = (y s , y f ) = k ps (x p ), k p f (x p , z p ) , (1)
where x p ∈ R nx p and z p ∈ R nz p denote the slow and fast plant states, respectively, while y s ∈ R ny s and y f ∈ R ny f represent the slow and fast outputs. Additionally, û = (û s , ûf ) refers to the latest received control input from the network. It is assumed that k ps and k p f are continuously differentiable, and f p and g p are locally Lipschitz in their arguments.

Controller (C): We consider a class of dynamic controllers of the form

C :      ẋc = f c (x c , z c , ŷp ) ϵ żc = g c (x c , z c , ŷp ) u = (u s , u f ) = k cs (x c ), k c f (x c , z c ) , (2)
where ϵ comes from (1), and x c ∈ R nx c and z c ∈ R nz c denote the slow and fast controller states, respectively. Additionally, u s ∈ R nu s and u f ∈ R nu f are control inputs that depend on the slow and fast controller states, respectively, while ŷp = (ŷ s , ŷf ) refers to the most recently received output of the plant transmitted after the network. Lastly, k cs and k c f are continuously differentiable, and f c and g c are locally Lipschitz in their arguments. We also assume that n ys + n us ∈ Z ≥1 and n y f + n u f ∈ Z ≥1 , which ensures that both fast and slow signals are present in the system. Description of communication (N ): To provide a mathematical representation of the single channel network, we first assume that the slow inputs and outputs will never be transmitted at the same instant as the fast inputs and outputs. Define T s := {t s 0 , t s 1 , t s 2 , • • • } as the unbounded set of transmission times at which slow inputs and outputs are transmitted, and

T f := {t f 0 , t f 1 , t f 2 , • • • }
as the unbounded set of transmission times at which fast inputs and outputs are transmitted, such that

T s ∩ T f = ∅. Then, let T := T s ∪ T f = {t 0 , t 1 , t 2 , • • • } denote the set
of all transmission instances, with its elements arranged in ascending time order. Moreover, we assume that the transmission intervals satisfy

τ s miati ≤ t s k+1 -t s k ≤ τ s mati , ∀t s k , t s k+1 ∈ T s , k ∈ Z ≥0 , τ miati ≤ t ℓ+1 -t ℓ ≤ τ mati , ∀t ℓ , t ℓ+1 ∈ T , ℓ ∈ Z ≥0 , (3)
where 0 < τ miati ≤ τ mati denote, respectively, the minimum allowable transmission interval (MIATI) and MATI between any two consecutive transmissions. Similarly, τ s miati and τ s mati are the MIATI and MATI between two consecutive slow updates. Note that, τ miati ≤ 1 2 τ s mati , τ mati < τ s miati and (3) prevent successive transmissions of slow inputs/outputs in T , which implies fast transmissions exist between every two consecutive slow transmissions. We further assume τ s miati ≤ τ s mati -τ miati to simplify the analysis. A change of variable useful for analysis is the so-called network-induced error, which we define as e ys := ŷsy s , e y f := ŷfy f , e us := ûsu s and e u f := ûfu f . For simplicity, (ŷ s , ŷf , ûs , ûf ) are assumed to be constant between any two successive transmission times (i.e. zeroorder hold behaviour). Other type of network-processing may be implemented if desired, see, e.g., [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF]. Define x := (x p , x c ) ∈ R nx , z := (z p , z c ) ∈ R nz , e s := (e ys , e us ) ∈ R ne s and e f := (e y f , e u f ) ∈ R ne f , with n x := n xp + n xc , n z := n zp + n zc , n es := n ys + n us and n e f := n y f + n u f .

A channel may consist of multiple network nodes, each representing a group of sensor and/or actuator states. However, only one node can transmit its data at each transmission time, and the access to the channel is regulated by the scheduling protocol of the channel. At each transmission time t s k for slow updates, a node, which is a group of elements in y s and u s are sampled and transmitted, and the values (ŷ s , ŷf , ûs , ûf ) are updated according to

ŷs ((t s k ) + ), ûs ((t s k ) + ) = y s (t s k ), u s (t s k ) + h s (k, e s (t s k )) ŷf ((t s k ) + ), ûf ((t s k ) + ) = (ŷ f (t s k ), ûf (t s k )
) , where h s : Z ≥0 × R ne s → R ne s models the scheduling protocol for the slow updates. Similarly, for each t f ℓ ∈ T f , we have that ŷs ((

t f ℓ ) + ), ûs ((t f ℓ ) + ) = ŷs (t f ℓ ), ûs (t f ℓ ) ŷf ((t f ℓ ) + ), ûf ((t f ℓ ) + ) = y f (t f ℓ ), u f (t f ℓ ) + h f ℓ, e f (t f ℓ ) ,
where the function h f : Z ≥0 × R ne f → R ne f is the scheduling protocol for the update of fast components.

III. A HYBRID MODEL FOR THE SPNCS

A. Closed-loop System

We now present a hybrid system model for the SPNCS described in Section II, in the formalism of [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]. For this purpose, we introduce two clocks and two counters, namely τ s , τ ∈ R ≥0 and κ s , κ f ∈ Z ≥0 . In particular, τ s records the time elapsed since the last slow transmission, and τ describes the inter-transmission time between any two successive transmissions, therefore, τ s resets to zero at each slow transmission, and τ resets to zero at any transmission. Meanwhile, κ s and κ f count the number of slow and fast transmissions, respectively. Moreover, f x , g z , f es and g e f are defined in [START_REF] Khalil | Nonlinear Systems[END_REF] in the next page, where we use f x,ι and g z,ι , ι ∈ {1, 2}, to denote the ι-th component of f x and g z , respectively. Let ξ := (x, e s , τ s , κ s , z, e f , τ, κ f ) ∈ X, with

X := R nx × R ne s × R ≥0 × Z ≥0 × R nz × R ne f × R ≥0 × Z ≥0 ,
denote the full state of the hybrid system. Consequently, the SPNCS can now be expressed as the following hybrid model

H 1 : ξ = F (ξ), ξ ∈ C ϵ 1 , ξ + ∈ G(ξ), ξ ∈ D ϵ s ∪ D ϵ f , (4)
where

F (ξ) := f x (x, z, e s , e f ), f es (x, z, e s , e f ), 1, 0, 1 ϵ g z (x, z, e s , e f ), 1 ϵ g e f (x, z, e s , e f , ϵ), 1 ϵ , 0 , and
G(ξ) :=      G s (ξ), ξ ∈ D ϵ s \ D ϵ f , G f (ξ), ξ ∈ D ϵ f \ D ϵ s , {G s (ξ), G f (ξ)}, ξ ∈ D ϵ s ∩ D ϵ f .
The jump maps are defined such that G s (ξ) := (x, h s (κ s , e s ), 0, κ s + 1, z, e f , 0, κ f ) and G f (ξ) := (x, e s , τ s , κ s , z, h f (κ f , e f ), 0, κ f + 1). The jump and flow sets are defined as

D ϵ s := {ξ ∈ X | τ s ∈ [τ s miati , τ s mati ] ∧ ϵτ ∈ [τ miati , τ mati ]} , D ϵ f := ∈ X | τ s ∈ [τ miati , τ s mati -τ miati ] ∧ ϵτ ∈ [τ miati , τ mati ]} , C ϵ 1 :=D ϵ s ∪ D ϵ f ∪ C ϵ 1,a ∪ C ϵ 1,b
where ∧ denotes the logical conjunction, In contrast to the two-channel case where the fast and slow jumps are essentially determined solely by their fast and slow timer [START_REF] Heijmans | Singularly perturbed networked control systems[END_REF], in the single-channel case these are determined by both the fast and slow timers (i.e., τ and τ s ). Similarly, for the single-channel case, the conditions on τ and τ s in the flow set depend on both timers, e.g., τ s ∈ [max{0, ϵτ -(τ mati -τ miati )}, τ miati ], whereas the fast and slow timers are decoupled in the two-channel case. This can be seen from the fact that slow and fast signals are transmitted over the single-channel, and there exist a minimum of τ miati time units between any two consecutive transmissions.

C ϵ 1,a := {ξ ∈ X | τ s ∈ [0, τ miati ] ∧ ϵτ ∈ [0, τ s + (τ mati -τ miati )]} and C ϵ 1,b := {ξ ∈ X | τ s ∈ [τ miati , ϵτ + (τ s mati -τ miati )] ∧ ϵτ ∈ [0, τ miati ]}.

Fig. 2: Flow set and jump set

To simplify the analysis, we introduce H 2 as the hybrid system with dynamics as per (4), i.e., same dynamics as H 1 , but with the "patched" flow set defined as

C ϵ 2 := {ξ ∈ X | τ s ∈ [0, τ s mati ] ∧ ϵτ ∈ [0, τ mati ]}
, which aligns with the entirety of the colored area depicted in Fig. 2. We note that H 2 contains H 1 in the sense that all solutions of H 1 are also solutions to H 2 , since C ϵ 1 ⊆ C ϵ 2 and they have identical flow map, jump map and jump set [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]Section 3.4].

B. Boundary Layer System and Reduced System of H 2

The goal of this stage is to study the stability of H 2 . Since this is a hybrid SPS, we adopt a similar approach to [5, Section 11.5] to show stability, but generalised to hybrid systems. Particularly, we will first derive a system H y 2 by changing the z-coordinate of H 2 , and determining its stability through a boundary layer and reduced system.

To that end, we now derive the quasi-steady-state of H 2 , under the following assumption.

Standing Assumption 1 For any x ∈ R nx , e s ∈ R ne s and z ∈ R nz , equation 0 = g z (x, z, ēs , 0) has a unique real solution z = H(x, ēs ), where H is continuously differentiable and 0 = H(0, 0).

The quasi-steady-state equilibrium z and ēf for the fast states z and e f obtained as follows: ēf is equal to zero, as for sufficiently high frequency of fast-output transmissions, e f converges to zero; and z corresponds to the unique solution z = H(x, ēs ) as per S.A. 1. Next, to derive H y 2 , we define y := z -H(x, e s ) and the full state of H y 2 , namely ξ y := (ξ s , ξ f ) := (x, e s , τ s , κ s ), (y, e f , τ, κ f ) , where

ξ y ∈ X, ξ s ∈ R n ξs := R nx ×R ne s ×R ≥0 ×Z ≥0 and ξ f ∈ R n ξ f := R nz ×R ne f ×R ≥0 ×Z ≥0 . Then, H y 2 is
given by

H y 2 : ξy = F y (ξ y ), ξ y ∈ C y,ϵ 2 , ξ y + ∈ G y (ξ y ), ξ y ∈ D y,ϵ s ∪ D y,ϵ f .
The flow map is F y (ξ y ) = ( ξs , ξf ) = F y s (x, y, e s , e f ), Additionally, we have

1 ϵ F y f (x,
G y (ξ y ) :=      G y s (ξ y ), ξ y ∈ D y,ϵ s \ D y,ϵ f , G y f (ξ y ), ξ y ∈ D y,ϵ f \ D y,ϵ s , {G y s (ξ y ), G y f (ξ y )}, ξ y ∈ D y,ϵ s ∩ D y,ϵ f
, where G y s (ξ y ) := x, h s (κ s , e s ), 0, κ s + 1, y, e f , 0, κ f and G y f (ξ y ) := x, e s , τ s , κ s , y, h f (κ f , e f ), 0, κ f + 1 . For analysis purposes, we write τ mati = ϵT * for some

T * ∈ R >0 , τ miati = aτ mati for some a ∈ (0, 1]. Since ϵ > 0, ϵτ ∈ [τ miati , τ mati ] is equivalent to τ ∈ [aT * , T * ].
Then the jump and flow sets are defined by

D y,ϵ s :={ξ y ∈ X | τ s ∈ [τ s miati , τ s mati ] ∧ τ ∈ [aT * , T * ]}, D y,ϵ f :={ξ y ∈ X | τ s ∈ [ϵaT * , τ s mati -ϵaT * ] ∧ τ ∈ [aT * , T * ]}, C y,ϵ 2 :={ξ y ∈ X | τ s ∈ [0, τ s mati ] ∧ τ ∈ [0, T * ]}.
We have changed the coordinate from z to y, and we are now ready to derive the reduced system H r and boundary layer system H bl associated with H y 2 . We first define the fast time scale σ = t ϵ , where ∂ ∂σ = ϵ ∂ ∂t . Then we set ϵ = 0. In the perspective of H bl (i.e., fast dynamics), the slow dynamics are now frozen. Meanwhile, the jump and flow sets of H bl contain the condition τ s ∈ [0, τ s mati ], which can always be satisfied. Therefore, the jumps and flows of H bl are essentially only determined by τ . We thus write

H bl : ( ∂ξs ∂σ , ∂ξ f ∂σ ) = (0 n ξs ×1 , F y f (x, y, e s , e f , 0)), ξ y ∈ C y,0 2,bl , ξ y + = G y f (ξ y ), ξ y ∈ D y,0 f , where C y,0 2,bl := {ξ y ∈ X | τ ∈ [0, T * ]} and D y,0 f := {ξ y ∈ X | τ ∈ [aT * , T * ]}.
From the perspective of H r (i.e., slow dynamics), the fast dynamics evolve infinitely fast. Therefore, for any τ s ∈ [0, τ s mati ], the waiting time for the condition τ ∈ [aT * , T * ] to be satisfied approaches to zero, and the flows and jumps of H r are essentially determined only by τ s . We assume H bl is asymptotically stable at its quasi-steady state, which we formalise later. Then in H r , y = 0 and e f = 0, that is

H r : ξs = F y s (x, 0, e s , 0), ξ y ∈ C y,0 2,r , ξ + s = (x, h s (κ s , e s ), 0, κ s + 1), ξ y ∈ D y,0 s , where C y,0 2,r := {ξ y ∈ X | τ s ∈ [0, τ s mati ]} and D y,0 s := {ξ y ∈ X | τ s ∈ [τ s miati , τ s mati ]}.

IV. STABILITY ANALYSIS

In this section, we start with the assumptions and preliminaries that are necessary to ensure stability property for H 1 . Particularly, Assumptions 1 and 2 introduced below provide sufficient conditions to guarantee asymptotic stability properties for H r and H bl , respectively. These assumptions are commonly adopted in the NCS literature, see [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF], [START_REF] Heijmans | Singularly perturbed networked control systems[END_REF].

Assumption 1 There exist a function W s : Z ≥0 × R ne s → R ≥0 that is locally Lipschitz in its second argument uniformly in its first argument, a continuous function H s : R nx × R ne s → R ≥0 , K ∞ -functions α Ws , α Ws , constants λ s ∈ [0, 1) and L s ≥ 0 such that, for all κ s ∈ Z ≥0 and e s ∈ R ne s , the following properties hold:

α Ws (|e s |) ≤ W s (k s , e s ) ≤ α Ws (|e s |) , (6)
W s (κ s + 1, h s (κ s , e s )) ≤ λ s W s (κ s , e s ). (7)
For all x ∈ R nx , κ s ∈ Z ≥0 and almost all e s ∈ R ne s , ∂Ws(κs,es) ∂es , f es (x, H(x, e s ), e s , 0)

≤ L s W s (κ s , e s ) + H s (x, e s ). (8)
Moreover, there exist a locally Lipschitz, positive definite and radially unbounded function V s : R nx → R ≥0 , positive definite function ρ s , and real number γ s > 0, such that for all e s ∈ R ne s , all κ s ∈ Z ≥0 , and almost all x ∈ R nx , the following inequality holds

∂Vs(x) ∂x , f x (x, H(x, e s ), e s , 0) ≤ -ρ s (|x|) -ρ s (W s (κ s , e s )) -H 2 s (x, e s ) + γ 2 s W 2 s (κ s , e s ). (9)
Assumption 2 There exist a function

W f : Z ≥0 × R ne f → R ≥0 that is locally Lipschitz in its second argument uni- formly in its first argument, a continuous function H f : R nx × R ne f → R ≥0 , K ∞ -functions α W f , α W f , constants λ f ∈ [0, 1
) and L f ≥ 0 such that, for all κ f ∈ Z ≥0 and e f ∈ R ne f , the following properties hold:

α W f (|e f |) ≤ W f (k f , e f ) ≤ α W f (|e f |) , (10)
W f (κ f + 1, h f (κ f , e f )) ≤ λ f W f (κ f , e f ). (11)
For all x ∈ R nx , κ f ∈ Z ≥0 and almost all e f ∈ R ne f ,

∂W f (κ f ,e f ) ∂e f
, g e f (x, y + H(x, e s ), e s , e f , 0)

≤ L f W f (κ f , e f ) + H f (y, e f ). (12)
Moreover, there exists a locally Lipschitz, positive definite and radially unbounded function V f : R nx × R nz → R ≥0 , positive definite function ρ f , and real number γ f > 0, such that for all e f ∈ R ne f , all κ f ∈ Z ≥0 , and almost all y ∈ R nz , the following holds

∂V f (x,y) ∂y , g z (x, y + H(x, e s ), e s , e f ) ≤ -ρ f (|y|) -ρ f (W f (κ f , e f )) -H 2 f (y, e f ) + γ 2 f W 2 f (κ f , e f ). (13)
In Assumption 1 (similarly with Assumption 2), ( 6)-( 7) relate to exponentially stable protocols [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], and (9) relates to the L 2 stability of H r from W s to H s , which is typically ensured at the first stage of emulation. We refer the reader to [2, Proposition 3] for more details on how to find Lyapunov functions to satisfy Assumptions 1 and 2.

We next provide a lemma as a preliminary to our main result. We define Lyapunov functions U s : R n ξs → R ≥0 and

U f : R n ξs × R n ξ f → R ≥0 , just as in [3, Eqn. (25)] U s (ξ s ) = V s (x) + γ s ϕ s (τ s )W 2 s (κ s , e s ) (14a) U f (ξ s , ξ f ) = V f (x, y) + γ f ϕ f (τ )W 2 f (κ f , e f ) (14b)
where

φ⋆ = -2L ⋆ ϕ ⋆ -γ ⋆ (ϕ 2 ⋆ + 1), ϕ ⋆ (0) = 1/λ * ⋆ , ⋆ ∈ {s, f }. By abuse of notation, we write U f (ξ y ) = U f (ξ s , ξ f ).
Lemma 1 Suppose Assumptions 1 and 2 hold. For any L ≥ 0, λ ∈ (0, 1) and γ > 0, we define the following mapping:

T (L, γ, λ) :=              1 Lr tan -1 r(1-λ) 2 λ 1+λ γ L -1 +1+λ , γ > L 1 L 1-λ 1+λ , γ = L 1 Lr tanh -1 r(1-λ) 2 λ 1+λ γ L -1 +1+λ , γ < L, where r := | γ L 2 -1|. Let (L s , γ s , λ s ) and (L f , γ f , λ f )
come from Assumption 1 and 2, respectively, and U s and U f come from [START_REF] Christofides | Output feedback control of nonlinear twotime-scale processes[END_REF] with some λ * s ∈ (λ s , 1) and λ * f ∈ (λ f , 1). For all τ s mati ≤ T (L s , γ s , λ * s ) and T * ≤ T (L f , γ f , λ * f ), there exist K ∞ -functions α Us , α Us , α U f , α U f , continuous positive definite functions ψ 1 , ψ 2 and positive constants a 1 , a 2 such that (15a) holds for all ξ s ∈ C y,0 2,r ∪ D y,0 s , (15b) holds for all ξ s ∈ C y,0 2,r , (15c) holds for all ξ s ∈ D y,0 s , (16a) holds for all ξ f ∈ C y,0 2,bl ∪ D y,0 f , (16b) holds for all ξ f ∈ C y,0 2,bl and (16c) holds for all

ξ f ∈ D y,0 f , α Us (|(x, e s )|) ≤ U s (ξ s ) ≤ α Us (|(x, e s )|) , (15a) U • s (ξ s ; F y s (x, 0, e s , 0)) ≤ -a 1 ψ 2 1 (|(x, e s )|) , (15b) U s ((x, h s (κ s , e s ), 0,κ s + 1)) ≤ U s (ξ s ), (15c)
α U f (|(y, e f )|) ≤ U f (ξ s , ξ f ) ≤ α U f (|(y, e f )|) , (16a)
U • f (ξ s , ξ f ); (0 n ξs×1 , F y f (x,y, e s , e f , 0)) ≤ -a 2 ψ 2 2 (|(y, e f )|) , (16b)
U f (G y f (ξ y )) ≤ U f (ξ s , ξ f ). (16c)
Proof: The proof follows similarly to [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]Theorem 1]. Lemma 1 asserts that, under Assumptions 1 and 2, we can establish upper bounds on τ s mati and T * in a manner such that, when both bounds are met, we can construct Lyapunov functions U s and U f that guarantee stability for H r and H bl , respectively. These Lyapunov functions will play a crucial role in the proof of our main result (namely Theorem 1 below), since we will conclude stability of H y 2 by considering H r , H bl , and their interconnection induced by nonzero ϵ. Assumption 3 specifies the interconnection condition between the slow and fast dynamics during flow, which is analogous to the continuous-time case as described in [5, pp. 451].

< ϵ < ϵ * , τ s mati ≤ T (L s , γ s , λ * s ) and τ mati ≤ ϵT (L f , γ f , λ * f )
, the following statement holds: There exists a KL-function β, such that for all ∆, ν > 0, there exists a τ s, * miati > 0, such that if τ s, * miati ≤ τ s matiτ miati , any solution ξ with |ξ(0, 0)| E < ∆ satisfies |ξ(t, j)| E ≤ β(|ξ(0, 0)| E , t + j) + ν for any (t, j) ∈ dom ξ.

Remark 1 From Theorem 1, we note that, to stabilise a SPNCS (in a semi-global practical sense) over one single communication channel, not only fast and slow transmissions need to be sent sufficiently fast, but also slow transmissions should not occur too often. Otherwise, there may not be enough bandwidth to stabilise the fast dynamics. This is in contrast to the two-channel case presented in [START_REF] Heijmans | Singularly perturbed networked control systems[END_REF], where transmitting fast and slow dynamics sufficiently fast over two separate channels was enough for stability.

V. ILLUSTRATIVE EXAMPLE

This section provides an example to show how the quasisteady state, and the reduced and boundary layer systems can be obtained, and how to determine the stability of the system using Theorem 1. Consider an SPNCS with the plant and the controller from [START_REF] Kokotović | Singular perturbation methods in control: analysis and design[END_REF]pp. 150,Example 7.4], given by P :

         ẋp = x p + z 1 + 2û ϵ ż1 = -z 1 - û ϵ ż2 = x p -z 1 -z 2 - û y p = x p + z 2 C : ẋc = -ax c + aŷ p u = -kx c
where a = 2.088 and k = 0.7563. We define the network induced error as e s := ûu and e f := ŷpy p . Then the flow map of the hybrid model H 1 is given by

                             ẋp = x p + z 1 + 2(-kx c + e s ) ẋc = -ax c + a(x p + z 2 + e f ) ϵ ż1 = -z 1 -(-kx c + e s ) ϵ ż2 = x p -z 1 -z 2 -(-kx c + e s ) ės = -akx c + ak(x p + z 2 + e f ) ϵ ėf = -ϵ x p + z 1 + 2(-kx c + e s ) -x p -z 1 -z 2 -(-kx c + e s ) τs = 1, ϵ τ = 1, κs = 0, κf = 0
and the flow set is C ϵ 1 . The enlarged system H 2 will have the same flow map as H 1 and a flow set C ϵ 2 . Moreover, since both u and y p are one dimensional variables, we have h s (κ s , e s ) = 0 and h f (κ f , e f ) = 0.

By setting ϵ = 0, we have the quasi-steady states e f = 0 and H = (z 1 , z 2 ) = (kx c -e s , x p ). We define (y 1 , y 2 ) := (z 1 -z 1 , z 2 -z 2 ), then we can derive the boundary layer system H bl of H 2 by setting ϵ = 0 and substituting (z 1 , z 2 ) = (y 1 + z , y 2 + z 2 ) into H 2 , which is given by x + = x, e + s = e s , τ + s = τ s , κ + s = κ s y + = y, e + f = 0, τ + = 0, κ + f = κ f + 1 when ξ y ∈ D y,0 f . Moreover, the reduced system H r can be obtained by setting ϵ = 0 and substituting e f and H into H 2 , and it is given by Now that we have verified all the assumptions, we can compute ϵ * = 0.000151, T (L s , γ s , λ * s ) = 0.0707 and T (L f , γ f , λ * f ) = 0.6929, and the required MATIs to stabilise the system are given by τ s mati ≤ 0.0707 and τ mati ≤ 0.6929ϵ.

H bl :                 

VI. CONCLUSION

We studied stability of two-time-scale Singularly Perturbed Networked Control Systems (SPNCSs) under one single communication channel. Compared to previous works that assumed availability of two channels to transmit fast and slow dynamics separately, we consider a more general setting and provide a more resource-aware strategy to stabilise the SPNCS when only one channel is available. Future work will focus on event-triggered strategies to manage fast and slow dynamics over a single-channel.

Fig. 1 :

 1 Fig. 1: NCS Block Diagram

Fig. 2

 2 depicts all the components of C ϵ 1 , and the jump sets D ϵ s and D ϵ f are indicated by the orange and green regions, respectively. Additionally, C ϵ 1,a and C ϵ 1,b are the regions where a jump is not allowed due to a recent transmission of slow and fast signals, respectively.

Assumption 3 Theorem 1

 31 There exist b 1 , b 2 , b 3 ≥ 0 such that ∂Us ∂ξs , F y s (x, y, e s , e f ) -F y s (x, 0, e s , 0) ≤ b 1 ψ 1 (|(x, e s )|) ψ 2 (|(y, e f )|) , y s (x, y, e s , e f ) ≤ b 2 ψ 1 (|(x, e s )|) ψ 2 (|(y, e f )|) + b 3 ψ 2 2 (|(y, e f )|) (17b)hold for almost all ξ y ∈ C y,ϵ 2 , where k(x, z) := (k pf (x p , z p ), k cf (x c , z c )).By introducing the setE := {ξ ∈ X | x = 0 ∧ e s = 0 ∧ z = 0 ∧ e f = 0}, we are now in a position to state our main result. Considering system H 1 and suppose Assumptions 1-3 hold. Let b 1 , b 2 , and b 3 come from Assumption 3 and a 1 and a 2 come from Lemma 1. Then, there exists ϵ * = a1a2 a1b3+b1b2 such that for all 0

  ∂x ∂σ =0, ∂es ∂σ = 0, ∂τs ∂σ = 0, ∂κs ∂σ = 0∂y1 ∂σ = -y 1 , ∂y2 ∂σ = -y 1 -y 2 ∂e f ∂σ =y 1 + y 2 , ∂τ ∂σ = 1,

  p + (-kx c + e s ) ẋc = 2ax p -ax c , ės = 2akx p -akx c x + = x, e + s = 0 τ + s = τ s , κ + s = κ s + 1 when ξ y ∈ D y,0 s . We now verify each adopted assumption. Assumption 1: Let W s (κ s , e s ) := |e s |, then (6)-(8) hold for α Ws (s) = α Ws (s) = s, λ s = 0, L s = 0 and H s (x, e s ) = |A 21 x|, where A 21 = [ 2ak -ak ] and x = [ xp xc ]. Let P s = 16.476 -5.615 -5.615 3.037 , ρ s (s) = s and γ s = 11.45, then (9) is satisfied withV s (x) = x T P s x. Assumption 2: Let W f (κ f , e f ) := |e f |, then (10)-(12) hold for α W f (s) = α W f (s) = s, λ f = 0, L f = 0 and H f (y, e f ) = |y 1 + y 2 |, where y = [ y1 y2 ]. Let P f = [ 1 0 0 1 ], ρ f (s) = 0.5s and γ f = √ 0.5, then (13) is satisfied with V f (y) = y T P f y. Assumption 3: Let ϕ ⋆ = -2L ⋆ ϕ ⋆ -γ ⋆ (ϕ 2 ⋆ + 1), ϕ ⋆ (0) = λ * ⋆ and U ⋆ = V ⋆ + γ ⋆ ϕ ⋆ (τ ⋆ )W 2⋆ (e ⋆ ) for ⋆ ∈ {s, f }, and we let λ * s = 0.4 and λ * f = 0.6. By Assumptions 1, 2 and the definition of ϕ ⋆ , inequalities[START_REF] Teel | On assigning the derivative of a disturbance attenuation control Lyapunov function[END_REF] and (16) hold with a 1 = 1, a 2 = 0.5, ψ 1 (s) = s, ψ 2 (s) = s. Then inequality[START_REF] Kokotović | Singular perturbation methods in control: analysis and design[END_REF] holds with b 1 = 293.61, b 2 = 11.25, b 3 = 4.36.

  x (x, z, e s , e f ) := f p (x p , z p , (k cs (x c ) + e us , k c f (x c , z c ) + e u f )), f c (x c , z c , (k ps (x p ) + e ys , k p f (x p , z p ) + e y f )) g z (x, z, e s , e f ) := g p (x p , z p , (k cs (x c ) + e us , k c f (x c , z c ) + e u f )), g c (x c , z c , (k ps (x p ) + e ys , k p f (x p , z p ) + e y f )) f es (x, z, e s , e f ) := -∂kp s (xp) ∂xp f x,1 (x, z, e s , e f ), -

	∂kc s (xc) ∂xc	f x,2 (x, z, e s , e f )	(5)
	g e ∂kp f (xp,zp) ∂zp	g z,1 (x, z, e s , e f ),
	-ϵ	∂kc f (xc,zc) ∂xc	f x,2 (x, z, e s , e f ) -	∂kc f (xc,zc) ∂zc	g z,2 (x, z, e

y, e s , e f , ϵ) , where F y s (x, y, e s , e f ) := f x (x, y + H(x, e s ), e s , e f ), f es (x, y + H(x, e s ), e s , e f ), 1, 0 , F y f (x, y, e s , e f , ϵ) := ϵ ∂y ∂t , g e f (x, y + H(x, e s ), e s , e f , ϵ), 1, 0 and ϵ ∂y ∂t = g z (x, y + H(x, e s ), e s , e f ) -ϵ ∂H ∂ξs F y s (x, y, e s , e f ). f f (x, z, e s , e f , ϵ) := -ϵ ∂kp f (xp,zp) ∂xp f x,1 (x, z, e s , e f )s , e f ) .

*This work was supported by the Australian Research Council under the Discovery Project DP200101303.


Mira Khalil

Romain Postoyan

Stéphane Raël

State estimation for enhanced low dimensional electrochemical models of lithium-ion batteries

Safe and efficient operation of lithium-ion batteries requires an accurate estimation of the internal states. One approach is to design an observer based on an electrochemical model of the battery internal dynamics. However, electrochemical models, and their associated observers, typically require a high dimension to generate accurate variables. In this paper, we explain how to alleviate this limitation by correcting the lithium concentrations generated by a finite-dimensional electrochemical model derived from the spatial discretizations of partial differential equations (PDE). We show that the corrected concentrations asymptotically match those generated by the original PDEs for constant input currents, irrespectively of the order of the considered finite-dimensional model. We then exploit this fact to derive a new state space model for which we design an observer, whose global, robust convergence is supported by a Lyapunov analysis provided a linear matrix inequality holds. The estimated concentrations are then corrected to asymptotically match those of the original PDEs in absence of disturbances for constant currents. Simulation results show improvements both in terms of modeling and estimation accuracy as a result of the proposed corrections.

I. INTRODUCTION

Lithium-ion batteries present many advantages in terms of volume capacity, weight, power density, as well as the absence of memory effect. Nevertheless, this type of batteries require a battery management system (BMS) to make them safe, reliable and efficient. The BMS needs to be fed with accurate information on the state of the battery for this purpose. Unfortunately, some key battery variables cannot be measured directly with sensors, which means that they have to be estimated.

A popular method to estimate the internal state of a lithium-ion battery consists in designing an observer based on a mathematical model of the battery internal dynamics. Several types of battery models are available in the literature for this purpose, 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-of-charge estimation in electrical vehicles[END_REF]. The simplest ones consist in representing the battery dynamics by an equivalent circuit model made of few resistors and capacitors, 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]. This apparent simplicity has a price nevertheless: the model requires a non-trivial parameterization to provide accurate data in general, see e.g., [START_REF] Lin | A lumped-parameter electro-thermal model for cylindrical batteries[END_REF]. An alternative is based on electrochemical models, which locally describe the physics of lithium-ion cell operation using partial differential equations (PDE), see e.g., [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]. Electrochemical models often rely on the assumption that the particles within an *This work was funded by Lorraine Université d'Excellence LUE. 1 Université de Lorraine, CNRS, CRAN, F-54000 Nancy,

France.

(mira.khalil@univ-lorraine.fr, romain.postoyan@univ-lorraine.fr).

2 Université de Lorraine, GREEN, F-54000 Nancy, France. (stephane.rael@univ-lorraine.fr).

electrode behave like an average particle, we talk of single particle models (SPM) as in e.g., [START_REF] Blondel | Observer design for an electrochemical model of lithium-ion batteries based on a polytopic approach[END_REF]- [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] Moura | PDE estimation techniques for advanced battery management systems -Part II: SOH identification[END_REF], [START_REF] Planté | Multiple active material lithium-ion batteries: Finitedimensional modeling and constrained state estimation[END_REF]. The corresponding infinite-dimensional model can then be spatially discretized to obtain a finite-dimensional model, which is convenient to design and implement a state observer. Finitedimensional models typically require a high dimension to generate accurate battery variables, as the latter are affected by the errors resulting from the spatial discretization of the original PDEs. This may be problematic as the derived observer may then be of high dimension, which may make its design numerically challenging and may be an obstacle for its implementation.

In this work, we present a method to alleviate the need for finite-dimensional models of high dimension to generate accurate variables. We consider for this purpose the finitedimensional model of [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF], whose dimension is directly related to the number of samples of the PDEs and can therefore be freely selected. We then present a technique to systematically correct the concentrations generated by this model so that these asymptotically match the concentrations given by the original PDEs for constant currents. Hence, for any given model order, the corrected concentrations from the finite-dimensional model asymptotically tend to the actual concentrations of the original infinite-dimensional model for constant inputs thereby asymptotically eliminating the errors induced by spatial discretization. Although the purpose of these corrections is to eliminate asymptotic errors for constant inputs, the provided simulation results show that significant improvements are also obtained for short time horizons with a rapidly changing current profile. We then exploit these corrected concentrations to derive a new output voltage equation, which leads to a new state space model.

Afterwards, we design an observer for this new model. We present a method for this purpose based on polytopic and Lyapunov-based tools similarly to e.g., [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 H ∞ performance analysis[END_REF]. This method guarantees the robust convergence of the state estimates generated by the observer to the actual battery states provided a linear matrix inequality holds. We then explain how to correct the estimated concentrations to asymptotically track those of the original PDEs in absence of disturbances and for constant inputs. Simulation results are presented to illustrate the improvements brought by the corrected model and the associated estimation scheme. The definitions and numerical values of all the parameters, as well as all the proofs are given in the extended version of this work in [START_REF] Khalil | Enhancing accuracy of finitedimensional models for lithium-ion batteries, observer design and experimental validation[END_REF]. Notation. Let R be the set of real numbers, R >0 := (0, ∞), R ≥0 := [0, ∞), R <0 := (-∞, 0), Z be the set of integers, Z >0 := {1, 2, 3, ...}. We use I n to denote the identity matrix of dimension n and 1 n×m the matrix of R n×m whose elements are all equal to 1, with n, m ∈ Z >0 . Given square matrices A 1 , ..., A n , diag(A 1 , ..., A n ) is the block diagonal matrix, whose block diagonal components are A 1 , ..., A n and diag(A 1 , ..., A n ) (diag(A 1 , ..., A n )) is the lower (upper) block diagonal matrix, whose lower (upper) block diagonal components are A 1 , ..., A n . Given a vector x ∈ R n , x ⊤ denotes the transpose of x. Given x ∈ R n and y ∈ R m with n, m ∈ Z >0 , we use the notation (x, y) to denote (x ⊤ , y ⊤ ) ⊤ . Given f : R → R n with n ∈ Z >0 , ( f ) ∞ stands for lim t→∞ f (t) when it exists. For a vector x ∈ R N , |x| denotes its Euclidean norm. For a matrix A ∈ R n×m , ∥A∥ stands for its 2-induced norm and for any i ∈ R n , j ∈ R m , (A) i j represents the j-th element of the i-th row of matrix A.

Let f : R ≥0 → R N , ∥ f ∥ L 2 ,[0,t) denotes the L 2 norm of f on the interval [0,t), where t ∈ [0, ∞]. We write f ∈ L 2 , when ∥ f ∥ L 2 ,[0,∞) < ∞.

II. BATTERY MODEL

We recall the SPM model in [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF], and we explain how to correct its lithium concentrations. Those concentrations are then exploited to derive a new output equation and the new model is finally given in a state space form.

A. Preliminaries on the model in [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF] We first briefly recall the main elements of a lithiumion cell, namely: the positive electrode, the separator and the negative electrode, which are all immersed in the electrolyte, and two current collectors. The electrolyte is an ionic solution that ensures ion transport within the battery. The porous separator is an electrical insulator that does not allow electrons to flow between the two electrodes. However, being porous, it allows the passage of ions via the electrolyte. The positive and negative electrodes consist of almost spherical particles of porous materials. The electrodes structure creates a surface of contact between the electrodes and the electrolyte producing electrochemical couples between them and thus introducing a potential difference between the positive and negative electrode.

The considered electrochemical model relies on the next assumption.

Standing Assumption 1 (SA1): The following holds: (i) lithium insertion or de-insertion reactions are homogeneous along the thickness of each electrode; (ii) the electrolyte dynamics is neglected; (iii) the temperature of the cell is constant and homogeneous. □ Item (i) implies that each electrode can be reduced to a single particle, whose size is equal to the average size of all the particles that compose the actual electrode, we talk of SPM assumption, see e.g., [START_REF] Blondel | Observer design for an electrochemical model of lithium-ion batteries based on a polytopic approach[END_REF]- [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] Moura | PDE estimation techniques for advanced battery management systems -Part II: SOH identification[END_REF], [START_REF] Planté | Multiple active material lithium-ion batteries: Finitedimensional modeling and constrained state estimation[END_REF]. As customarily done in electrochemical modeling of lithium-ion batteries, electrodes material particles are supposed spherical. In item (ii), we ignore the electrolyte dynamics, which is reasonable for moderate currents and moderate temperatures. For high current rates and/or low temperatures, item (ii) can be relaxed and electrolyte dynamics can be added to the presented model and observer by applying the results of [START_REF] Benzine | Systematic observer redesign for lithium-ion battery models to account for the electrolyte dynamics[END_REF]. In item (iii), we suppose that the temperature is constant however, when the temperature varies, and is measured we can adapt the model and the developed observer of Section III.A to take into account the temperature variation like in [START_REF] Blondel | Observer design for an electrochemical model of lithium-ion batteries based on a polytopic approach[END_REF], [START_REF] Planté | Multiple active material lithium-ion batteries: Finitedimensional modeling and constrained state estimation[END_REF]. As for the temperature homogeneity assumption, it is reasonable at moderate and high temperatures. For low temperatures, it may become invalid and be interpreted as a parametric uncertainty, which can be handled by the observer of Section III.A if the uncertainty is small enough, see [START_REF] Planté | Multiple active material lithium-ion batteries: Finitedimensional modeling and constrained state estimation[END_REF].

Given SA1, the main dynamical phenomenon is the lithium diffusion in the electrodes active particles. This phenomenon is described using the next PDEs (see [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF]), for any t ≥ 0 and r ∈ [0, R s ], where R s > 0 is the radius of the particle in electrode s ∈ {neg, pos}, with neg and pos denoting the negative and positive electrode, respectively,

ϕ s (r,t) = -D s ∂ c s (r,t) ∂ r ∂ c s (r,t) ∂t = 1 r 2 ∂ ∂ r D s r 2 ∂ c s (r,t) ∂ r , (1)
where c s is the local concentration of lithium, ϕ s is the lithium flux density and D s > 0 is the diffusion coefficient of lithium, along with two boundary conditions ϕ s (0,t) = 0 and

ϕ s (R s ,t) = j Li s a s F
, where j Li s ∈ R is the electrochemical reaction rate, a s := 3ε s R s is the active surface per volume unit, ε s > 0 is the volume fraction of the active material particle and F > 0 is Faraday's constant.

To derive a set of ordinary differential equations (ODE) from ( 1), a spatial discretization with uniform volume is performed. In particular, each particle is discretized into N s ∈ Z >0 samples each having the same volume, where s ∈ {neg, pos}. A zero-order approximation is made, i.e., we assume that the lithium concentration in each sample, c s,n for n ∈ {1, ..., N s } and s ∈ {neg, pos}, is homogeneous. From the obtained set of ODEs, we derive the next state space equation, with the index s ∈ {neg, pos}

ẋs = A s x s + B s m s , (2)
where

x s := (c s,1 , . . . , c s,N s ) ∈ R N s is the concatenation of the concentrations in electrode s and m s := -j Li s ε s F ∈ R is the input. The matrices A s ∈ R N s ×N s and B s ∈ R N s ×1 are defined as A s := diag(-µ s 1 , -υ s 2 , . . . , -υ s N s -1 , -μs N s ) + diag( μs 2 , . . . , μs N s ) + diag(µ s 1 , . . . , µ s N s -1 ), B s := 0 . . . 0 N s ⊤ , where µ s i := S s i r s i+1 -r s i D s V s for any i ∈ {1, . . . , N s -1}, μs i := S s i-1 r s i -r s i-1 D s V s for any i ∈ {2, . . . , N s }, υ s i := μs i + µ s i for any i ∈ {2, . . . , N s -1}, V s := 1 N s 4 3 πR 3 s , r s i := i N s 1/3 R s and S s i := 4π(r s i ) 2 .

B. Concentrations correction

The spatial discretization of (1) to obtain (2) generates errors on the concentrations given by (2). These errors can be reduced by increasing the number of samples N s , s ∈ {neg, pos}, but this leads to a high-dimensional system in [START_REF] Beelen | Joint estimation of battery parameters and state of charge using an extended Kalman filter: a single-parameter tuning approach[END_REF], which may lead to computational issues, especially when using the model in (2) for observer design. We present in this section an alternative method to reduce these errors by correcting the lithium concentrations generated by [START_REF] Beelen | Joint estimation of battery parameters and state of charge using an extended Kalman filter: a single-parameter tuning approach[END_REF] so that they asymptotically match those given by (1) for constant inputs, as formalized in the sequel. We can already emphasize that, although these corrections are established by considering the asymptotic behavior of (1) and (2) for constant inputs, these allow improving the accuracy of the concentrations given by (2) even for rapidly changing inputs as illustrated in Section IV. In this section, we denote by c s,(1) the lithium concentrations generated from the PDEs in (1) and c s,(2) := x s the lithium concentrations generated by model (2) for electrode s, with s ∈ {neg, pos}.

We denote the corrected lithium concentrations of model (2) as c s,cor , which are defined by, for j ∈ {1, . . . , N s } and s ∈ {neg, pos},

c s,cor, j := c s,mean -K s j (c s,mean -c s,(2), j ), (3)
where c s,mean := 1

N s ∑ N s n=1 c s,( 2
)
,n is the lithium-ion mean concentration in electrode s given by model (2) and K s j ∈ R is a static correction coefficient given in [12, (4)].

We present the next result on the error between the corrected concentrations of model ( 2) and the original concentrations of the PDEs in (1) when time tends to infinity for constant inputs.

Theorem 1: For a constant input m s and an initial uniform profile for c s, [START_REF] Barillas | A comparative study and validation of state estimation algorithms for Li-ion batteries in battery management systems[END_REF] , in the sense that there exists c 0 ∈ R ≥0 such that for any r ∈ [0, R s ] c s,(1) (r, 0) = c 0 , any corresponding solution c s,(1) to (1) and c s,(2) to (2) with c s,(2) (0) = 1 N s ×1 c 0 satisfy for any j ∈ {1, . . . , N s } and s ∈ {neg, pos}

(c s,cor, j -c s,(1) (r s j , •)) ∞ = 0, (4)
where c s,cor is defined in (3). □ Theorem 1 implies that, as times tends to infinity, the corrected concentrations defined in (3) match those generated by the PDEs in (1) when the input is constant and c s,(1) (•, 0) is uniform. It is important to note that Theorem 1 imposes no conditions on the number of samples with which the PDEs in (1) are discretized, and thus no conditions on the dimension of ( 2) for (4) to hold.

Remark 1: In Theorem 1, the lithium concentrations c s,(1) have an initial uniform profile. We claim that this is not restrictive as there are always periods of time in the life of a lithium-ion battery where c s, [START_REF] Barillas | A comparative study and validation of state estimation algorithms for Li-ion batteries in battery management systems[END_REF] is homogeneous in electrode s, with s ∈ {neg, pos}. □

C. Towards a state space model

Before we present the new output equation derived from (3), we recall the relation between m s in (2) and the cell current I cell and we perform a model reduction, which is essential to ensure the system detectability. Given SA1, the electrochemical reaction rate is homogeneous within each electrode. Therefore, a proportional relationship can be established between I cell and j Li s in particular j Li neg := I cell A cell d neg and j Li pos := -I cell A cell d pos , where I cell is in generator convention (i.e., I cell > 0 in discharge), A cell is the electrode surface and d s is the thickness of electrode s. On the other hand, we have m s := -j Li s ε s F . Hence, we obtain m neg := -

I cell ε neg A cell d neg F
and m pos :=

I cell ε pos A cell d pos F .
On the other hand, model ( 2) is reduced just 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], [START_REF] Planté | Multiple active material lithium-ion batteries: Finitedimensional modeling and constrained state estimation[END_REF] by adopting the next assumption, which is essential later for the observer convergence.

Standing Assumption 2 (SA2): The quantity of lithium inserted in battery electrodes is constant and known. □ SA2 is reasonable over short periods of time. Factors such as cell degradation or side reactions can cause capacity loss over time, resulting in a reduction in the total quantity of lithium and the violation of SA2. In this case, if there is a small uncertainty regarding the quantity of lithium, the battery state and its estimation would exhibit asymptotic small errors. Conversely, if the uncertainty is big and thus the quantity of lithium needs to be estimated, state of health estimation algorithms may be employed. SA2 allows to write a lithium mass conservation. Hence, the quantity of lithium is defined as

Q := α neg N neg ∑ i=1 c neg,i V neg + α pos N pos ∑ i=1 c pos,i V pos , (5)
and is constant over time, where α s := F 3600 ε s A cell d s V total s and V total s := N s V s is the volume of the particle of electrode s. From (5), we express the lithium concentration at the center of the negative electrode c neg,1 as a linear combination of all the other sampled concentration in solid phase

c neg,1 = K - N neg ∑ i=2 c neg,i - α pos V pos α neg V neg N pos ∑ i=1 c pos,i , (6)
where K := Q α neg V neg . In view of (6), c neg,1 is no longer needed in the state space representation as it can be recovered from the other concentrations.

D. Corrected output equation

We are ready to present the new output voltage equation. The output equation of model ( 2) is obtained by decomposition of the cell voltage V cell . The main components of V cell are the potential differences between the electrodes and the electrolyte called open circuit voltages (OCV) denoted OCV s for s ∈ {neg, pos}, which depend on the surface insertion rates ζ s defined by ζ s := c s,Ns c s max for s ∈ {neg, pos}, where c s max is the maximum lithium concentration of electrode s and c s,N s is the surface concentration generated by model [START_REF] Beelen | Joint estimation of battery parameters and state of charge using an extended Kalman filter: a single-parameter tuning approach[END_REF]. Given the lithium concentrations correction made in Section II.B, instead of using c s,N s to define the surface insertion rates, we use the corrected surface concentration c s,cor,N s defined in (3) to derive the corrected surface insertion rates ζ s,cor . As a result, we obtain the output equation for y

:= V cell y = OCV pos (ζ pos,cor ) -OCV neg (ζ neg,cor ) + g(u), (7)
where ζ s,cor := I].

E. State space form

We present the overall state space representation. We introduce for this purpose the state vector x := (c neg,2 , ..., c neg,N neg , c pos,1 , ..., c pos,N pos ) ∈ R N with N := N neg -1 + N pos , the input u = I cell ∈ R, the output y = V cell ∈ R, and w ∈ R n w and v ∈ R n v represent additive exogenous perturbations and measurement noise respectively. We derive the next state space equation

ẋ = Ax + Bu + K + Ew y = h cor (x) + g(u) + v, (8)
where the expressions of the matrices A ∈ R N×N , B ∈ R N×1 and K ∈ R N×1 are found after [12, (24)]. The function h cor : R N → R is defined as, for any x ∈ R N , h cor (x) := OCV pos (H pos,cor x) -OCV neg (H neg,cor x + K 1 ), such as ζ pos,cor := H pos,cor x and ζ neg,cor := H neg,cor x + K 1 , with H s,cor ∈ R 1×N and K 1 are given in [12, (25)].

We are now ready to proceed with the observer design for system [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF].

III. STATE ESTIMATION

In this section, we first synthesize an observer for system [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF]. Then, we correct the estimated concentrations generated by the observer so that they match the concentrations given by the PDEs in (1) asymptotically for constant inputs.

A. Polytopic approach

The proposed observer takes the form

ẋ = A x + Bu + K + L(y -ŷ) ŷ = h cor ( x) + g(u), (9)
where x ∈ R N is the state vector estimate, L ∈ R N is the observation matrix gain to be designed and ŷ is the estimated (corrected) output. We define the estimation error e := xx. The dynamics of the estimation error follows from ( 8) and ( 9),

ė = Ae + Ew -L(h cor (x) -h cor ( x)) -Lv. ( 10
)
We make the next assumption on the OCVs as in e.g., [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] Planté | Multiple active material lithium-ion batteries: Finitedimensional modeling and constrained state estimation[END_REF]. Assumption 1: For any s ∈ {neg, pos}, there exist constant matrices C s,1 ,C s,2 ∈ R such as for any z, z ′ ∈ R,

OCV s (z) -OCV s (z ′ ) = C s (z, z ′ )(z -z ′ ), (11)
where

C s (z, z ′ ) := λ s 1 (z, z ′ )C s,1 + λ s 2 (z, z ′ )C s,2 with λ s i (z, z ′ ) ∈ [0, 1] for i ∈ {1, 2} and λ s 1 (z, z ′ ) + λ s 2 (z, z ′ ) = 1.
□ Assumption 1 means that each OCV s lies in a polytope defined by C s,1 ,C s,2 with s ∈ {neg, pos}. This condition is often verified in practice. Indeed, the OCVs are generally defined on the interval [0, 1] and are typically well-approximated by a piecewise continuously differentiable and thus globally Lipschitz function. Then, it suffices to extrapolate the OCVs on [1, ∞) (resp. on (-∞, 0]) by using zero order or first order approximations based on the value of the OCVs at 1 (resp. at 0) for Assumption 1 to hold. Then, C s,1 and C s,2 represent the minimum and maximum slopes of OCV s , respectively. Assumption 1 implies that h cor , defined in Section II.E, satisfies the next property.

Lemma 1: Suppose Assumption 1 holds and consider h cor as defined in Section II.E, we have for any

x, x ′ ∈ R N h cor (x) -h cor (x ′ ) = C(x, x ′ )(x -x ′ ), (12)
where

C(x, x ′ ) := ∑ 4 i=1 Λ i (x, x ′ )C i , with C i ∈ R N defined in [12, (41)], Λ i (x, x ′ ) ∈ [0, 1] for i ∈ {1, 2, 3, 4} and ∑ 4 i=1 Λ i (x, x ′ ) = 1.
□ Lemma 1 implies that h cor lies in a polytope defined by the vertices C i with i ∈ {1, 2, 3, 4}.

In view of ( 12), the estimation error dynamics can be written as ė = (A -LC(x, x))e + Ew -Lv.

The next theorem provides a sufficient condition to design gain L ∈ R N under which e = 0 is globally exponentially stable in absence of noise v and disturbance w, and satisfies L 2 -stability properties when the latter are non-zero. Theorem 2: Suppose Assumption 1 holds and there exist ε, µ w , µ v ∈ R >0 , L ∈ R N and P ∈ R N×N symmetric and positive definite such that for any i ∈ {1, ..., 4}

  H i + εI N PE -PL E ⊤ P -µ w I n w 0 -L ⊤ P 0 -µ v I n v   ≤ 0, (14)
with H i := (A -LC i ) T P + P(A -LC i ) then the following hold.

• System (8), ( 9) is L 2 -stable from (w, v) to e with gain less or equal to µ w ε and µ v ε , respectively, in particular, there exist c ≥ 0 such that for w, v ∈ L 2 , any solution (x, e) to ( 8), [START_REF] Klein | Electrochemical model based observer design for a lithium-ion battery[END_REF] with u a Lebesgue measurable, locally essentially bounded input satisfies ∥e∥ L 2 ,[0,t) ≤ c|e(0)|+ µ w ε ∥w∥ L 2 ,[0,t) + µ v ε ∥v∥ L 2 ,[0,t) for any t ≥ 0. • {(x, e) : e = 0} is uniformly globally exponentially stable when w=0 and v=0, i.e., there exist γ 1 ≥ 1, γ 2 > 0 such that for any solution (x, e) to ( 8), [START_REF] Klein | Electrochemical model based observer design for a lithium-ion battery[END_REF] with u a Lebesgue measurable, locally essentially bounded input satisfies |e(t)| ≤ γ 1 |e(0)|e -γ 2 t for any t ≥ 0. □ The matrix inequality in ( 14) is not linear, however it becomes linear after a standard change of variables, namely W = PL. Condition ( 14) can be easily tested given the model parameters. Also, the order reduction performed in Section II.C appears to be essential for its feasibility. From x, which represents the concatenation of the estimated concentrations generated by observer [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF] x := ( ĉneg,2 , . . . , ĉneg,N neg , ĉpos,1 , . . . , ĉpos,N pos ), we can retrieve ĉneg,1 by replacing the concentrations in (6) by their estimates.

B. Corrected estimated concentrations

Observer (9) generates estimated lithium concentrations, which can also be corrected along with ĉneg,1 so that they asymptotically match the concentrations of the PDEs in (1) for constant input currents as seen in Section II.B. We denote in the following c s,(1) the concentrations generated by the PDEs in [START_REF] Barillas | A comparative study and validation of state estimation algorithms for Li-ion batteries in battery management systems[END_REF] as in Section II.B and ĉs the estimated concentrations of electrode s, with s ∈ {pos, neg}.

We denote the corrected estimated concentrations as ĉs,cor , which are given by, for j ∈ {1, . . . , N s } and s ∈ {neg, pos}, ĉs,cor,j := ĉs,mean -K s j ( ĉs,meanĉs, j ), [START_REF] Meng | Overview of lithium-ion battery modeling methods for state-of-charge estimation in electrical vehicles[END_REF] where ĉs,mean := 1 N s ∑ N s i=1 ĉs,i and K s j as defined in (3). The next theorem states an asymptotic property of the error between the corrected estimated concentrations and the concentrations generated by the original PDEs in (1) when time tends to infinity for constant inputs.

Theorem 3: Consider system (8) and its observer [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF] and suppose e = xx = 0 is globally exponentially stable when w = 0 and v = 0. Then, for any constant input current I cell and any c 0 ∈ R ≥0 such that c s,(1) (r, 0) = c 0 for all r ∈ [0, R s ], any corresponding solution x to (9) and c s,(1) to ( 1) satisfy

( ĉcor -c (1) ) ∞ = 0, ( 16
)
where □ Theorem 3 implies that, under the conditions of Theorem 2, the estimated corrected lithium concentrations asymptotically match the lithium concentrations of (1) in absence of noise and disturbance provided that a constant input current is applied and c s,(1) (r, 0) is uniform for all r ∈ [0, R s ].

IV. NUMERICAL CASE STUDY

We simulate model [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF] and the model in [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF] with the parameters values given in [START_REF] Khalil | Enhancing accuracy of finitedimensional models for lithium-ion batteries, observer design and experimental validation[END_REF]Table I]. We assume that there is no measurement noise and no perturbation. The considered OCVs curves are given in Figure 1, which satisfy Assumption 1. The used input current u is a Plug-in Hybrid Electrical Vehicle (PHEV) discharge current on the time interval [0, 1800], a PHEV charge current on [1800, 3600] and 0 on [3600, 4500], as illustrated in Figure 2. The current profile is thus rapidly varying on [0,3600], during which we will see that improvements are obtained with the corrections presented in Section II.B. We initialize both models at equilibrium, meaning that all the initial concentrations within the same electrode are equal, with a state of charge (SOC) equal to 100%. The SOC is defined by for s ∈ {pos, neg}

SOC s := 100 c s,mean -c s 0 c s 100 -c s 0 , (17)
where c s 0 , c s 100 are the lithium concentration of electrode s at SOC equal 0% and 100%, respectively. Given that SOC pos and SOC neg are equal, we use the notation SOC instead. As for the reference model, it is obtained by solving the PDEs of (1) using a finite elements method and simulated with the parameters given in [START_REF] Khalil | Enhancing accuracy of finitedimensional models for lithium-ion batteries, observer design and experimental validation[END_REF]Table I], see [START_REF] Raël | Using electrical analogy to describe mass and charge transport in lithium-ion batteries[END_REF] for details.

We have compared the surface concentrations c s,surf and the output voltages V cell generated by the model in [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF] and model [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF], with those generated by the reference infinitedimensional model over the whole interval of time [0,4500] as well as [0,3600] where the current is rapidly varying in Table I. We have computed the mean absolute error (MAE) and the root mean square error (RMSE) of the voltage error e V cell between V cell generated by ( 1) and V cell generated by the model in [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF] and by [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF], respectively and the normalized surface concentrations error e c s,surf between the surface concentrations generated by ( 1) and those generated by the model in [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF] and model [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF], respectively. We see significant improvement of the output voltage and the surface concentrations given by model ( 8) compared to the one in [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF] on the interval [0,4500] and particularly on [0,3600], when the current is rapidly varying. [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF] AND MODEL [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF].

We have then designed an observer for system [START_REF] Doyle | Modeling of galvanostatic charge and discharge of the lithium/polymer/insertion cell[END_REF] and one for the system in [START_REF] Blondel | Nonlinear circle-criterion observer design for an electrochemical battery model[END_REF] by applying the results of Section III.A. We have been able to obtain the same observer gain L in both cases so that the only difference between the two observers is the output equation used to synthesize them. In particular, L = 10 4 (3.2354,3.5397,3.3862,-5.0338,-5.7366, -5.3258,-5.4284). We have initialized both observers such that all estimated concentrations within the same particle are equal and correspond to a SOC estimate, denoted SOC, of 0%. We note that SOC is obtained by replacing c s,mean in (17) by its estimate ĉs,mean . Both observers are fed by the voltage output generated by the reference model in [START_REF] Raël | Using electrical analogy to describe mass and charge transport in lithium-ion batteries[END_REF].

Figure 3 reports the actual SOC given by the infinitedimensional model and the estimated ones, as well as the corresponding norm of the estimation errors on the SOC e SOC = SOC -SOC obtained with the observer in [START_REF] Blondel | Observer design for an electrochemical model of lithium-ion batteries based on a polytopic approach[END_REF], observer (9) and observer [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF] with the correction of its estimated concentrations ĉ := ( ĉ1 neg , x) as done in Theorem 3. We see that observer [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF] based on the corrected model ( 8) provides a more accurate SOC. This improvement is quantified by computing the MAE and the RMSE of the SOC estimation errors e SOC for the observer in [START_REF] Blondel | Observer design for an electrochemical model of lithium-ion batteries based on a polytopic approach[END_REF], observer (9) and observer [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF] with ĉcor , respectively, averaged over 20 simulations for initial SOC estimates ranging from {0%, 5%, ..., 100%}. The results are given in Table II. We see that observer [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF] generates more accurate results in terms of SOC estimation and this improvement is of the order of percent, which is significant for lithium-ion batteries.

We have also computed in Table II the average MAE and the average RMSE over the same 20 scenarios, of the norm of the normalized estimated concentrations error e c s for s ∈ {pos, neg} for the observer in [START_REF] Blondel | Observer design for an electrochemical model of lithium-ion batteries based on a polytopic approach[END_REF], for observer [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF] and for observer [START_REF] Dreef | LMI-based robust observer design for battery state-of-charge estimation[END_REF]

V. CONCLUSION

We have designed an observer based on a finitedimensional electrochemical model, for which we corrected its lithium concentrations to asymptotically eliminate the errors induced by the PDE discretization for constant currents. Simulations results show the improvement of the model and the state estimates as a result of the proposed corrections.

  u) := -η r,pos (u)η pos (u)η neg (u)η r,neg (u)η r,sep (u) for any u := I cell ∈ R, where η s (u) := 2 RT F asinh R s 6ε s j s 0 A cell d s u , η r,s (u) := 1 2A cell d s σ s,eff + d s κ s u, η r,sep (u) := 1 A cell d sep κ sep u, κ s := κ e ε 1.5 e,s , σ s,eff := σ s ε s , κ sep := κ e ε 1.5 e,sep , with R, T , j s 0 , κ e , ε e,s , ε e,sep , ε s and σ s defined in[START_REF] Khalil | Enhancing accuracy of finitedimensional models for lithium-ion batteries, observer design and experimental validation[END_REF] Table

  ĉcor := ( ĉneg,cor,1 , ĉneg,cor,2 , . . . , ĉneg,cor,N neg , ĉpos,cor,1 , . . . , ĉpos,cor,N pos ) := (c neg,(1) (r neg 1 , •), c neg,(1) (r neg 2 , •), . . . , c neg,(1) (r neg N neg , •), c pos,(1) (r pos 1 , •), . . . , c pos,(1) (r pos N pos , •)).

Fig. 1 .

 1 Fig. 1. OCVs curves.

Fig. 2 .

 2 Fig. 2. Input current profile.

TABLE II AVERAGE

 II with ĉcor . More accurate estimated concentrations are obtained as a result of the correction of the estimated concentrations. MAE AND RMSE OVER 20 SIMULATIONS OF THE SOC ESTIMATION AND OF THE ESTIMATED CONCENTRATIONS ERRORS.

		MAE RMSE
	e SOC : observer in [4] [%]	1.92	2.76
	e SOC : observer (9) [%]	0.81	1.34
	e SOC : observer (9) + ĉcor [%] 1.59	2.28
	e cpos : observer in [4] [%]	1.16	1.87
	e cpos : observer (9) [%]	1.78	2.35
	e cpos : observer (9) + ĉcor [%]	1.02	1.64
	e cneg : observer in [4] [%]	5.82	6.51
	e cneg : observer (9) [%]	7.28	8.19
	e cneg : observer (9) + ĉcor [%]	4.99	5.58