# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021041
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## A tracking problem for the state of charge in a electrochemical Li-ion battery model

 1 Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Valparaíso, Chile 2 Gipsa-Lab, Université Grenoble Alpes, 11 rue des Mathématiques, Grenoble Campus BP46, Saint Martin D'Heres, France 3 Instituto de Ingeniería Matemática y Computacional, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avda. Vicuña Mackenna 4860, Santiago, Chile

* Corresponding author: Esteban Hernández

Received  June 2019 Revised  November 2020 Early access September 2021

Fund Project: This work has been partially financed by Fondecyt 1180528 (E. Cerpa), ECOS-CONICYT C16E06, PIIC Universidad Técnica Federico Santa María, Basal Project FB0008 AC3E and ANID BECAS/DOCTORADO NACIONAL/2017-21171188.

In this paper the Single Particle Model is used to describe the behavior of a Li-ion battery. The main goal is to design a feedback input current in order to regulate the State of Charge (SOC) to a prescribed reference trajectory. In order to do that, we use the boundary ion concentration as output. First, we measure it directly and then we assume the existence of an appropriate estimator, which has been established in the literature using voltage measurements. By applying backstepping and Lyapunov tools, we are able to build observers and to design output feedback controllers giving a positive answer to the SOC tracking problem. We provide convergence proofs and perform some numerical simulations to illustrate our theoretical results.

Citation: Esteban Hernández, Christophe Prieur, Eduardo Cerpa. A tracking problem for the state of charge in a electrochemical Li-ion battery model. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021041
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##### References:
The continuous function $\tau$ is positive in $[2+\sqrt{6}, \lambda_{\sup} )$
$(Left)$ The input $I_{ref}(t) = 0.5C$. $(Right)$ We compare $SOC(t)$ for the controlled system with the reference $SOC_{ref}(t)$, generated by $I_{ref}(t)$
$(Left)$ The input $I_{ref}(t) = 4.5\, square(\frac{64}{900\pi}t)C$. $(Right)$ We compare $SOC(t)$ for the controlled system with the reference $SOC_{ref}(t)$, generated by $I_{ref}(t)$
(a) We compare $SOC(t)$ for the controlled system with the reference $SOC_{ref}(t)$ generated by $I_{ref}(t) = 0.5C$ (Left) and $I_{ref}(t) = 4.5square(\frac{64}{900\pi}t)C$ (Right)
Model variables and electrochemical parameters
 Model states, inputs and outputs $c^{\pm}$ Lithium concentration in solid phase $[mol/m^3]$ $c_{s}(t)$ Lithium concentration at solid particle surface$[mol/m^2]$ $c_{e}$ Lithium concentration in electrolyte phase $[mol/m^3]$ $T$ Temperature $[K]$ $I$ Applied current, $[A/m^2]$ $V$ Output Voltage $[V]$ Electrochemical model parameters $D^{\pm}$ Diffusivity $[m^2/s]$ $R_{\pm}$ Particle radius in solid phase $[m]$ $F$ Faraday Constant $[C/mol]$ $R$ Universal gas constant $[J/mol\cdot K]$ $\alpha$ Charge transfer coefficient $[-]$ $c^{\pm}_{\max}$ Maximum concentration of solid material $[mol/m^3]$ $U^{\pm}$ Open circuit potential of solid material $[V]$ $R_{f}$ Solid interphase films resistance $[\Omega\cdot m^2]$ $L^{\pm}$ Length of region $[m]$ $A$ Area $[m^2]$ $\phi_{1}$ Heat transfer coefficient $[1/s]$ $\phi_{2}$ Inverse of heat capacity $[J/K]^{-1}$ $\varepsilon^{\pm}$ Volume fraction of solid phase $[-]$
 Model states, inputs and outputs $c^{\pm}$ Lithium concentration in solid phase $[mol/m^3]$ $c_{s}(t)$ Lithium concentration at solid particle surface$[mol/m^2]$ $c_{e}$ Lithium concentration in electrolyte phase $[mol/m^3]$ $T$ Temperature $[K]$ $I$ Applied current, $[A/m^2]$ $V$ Output Voltage $[V]$ Electrochemical model parameters $D^{\pm}$ Diffusivity $[m^2/s]$ $R_{\pm}$ Particle radius in solid phase $[m]$ $F$ Faraday Constant $[C/mol]$ $R$ Universal gas constant $[J/mol\cdot K]$ $\alpha$ Charge transfer coefficient $[-]$ $c^{\pm}_{\max}$ Maximum concentration of solid material $[mol/m^3]$ $U^{\pm}$ Open circuit potential of solid material $[V]$ $R_{f}$ Solid interphase films resistance $[\Omega\cdot m^2]$ $L^{\pm}$ Length of region $[m]$ $A$ Area $[m^2]$ $\phi_{1}$ Heat transfer coefficient $[1/s]$ $\phi_{2}$ Inverse of heat capacity $[J/K]^{-1}$ $\varepsilon^{\pm}$ Volume fraction of solid phase $[-]$
Parameter simulations
 Parameters Values $c(r, 0)$ $1.5c_{0}$ $\widehat{c}(r, 0)$ $1.5c_{0}$ $c_{\max}$ $2.5\cdot 10^4$ $\lambda$ $5$ $\gamma$ $70$
 Parameters Values $c(r, 0)$ $1.5c_{0}$ $\widehat{c}(r, 0)$ $1.5c_{0}$ $c_{\max}$ $2.5\cdot 10^4$ $\lambda$ $5$ $\gamma$ $70$
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