# 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 and Related Fields, doi: 10.3934/mcrf.2021041
##### References:
 [1] M. Armand and J.-M. Tarascon, Building better batteries, Nature, 451 (2008), 652-657.  doi: 10.1038/451652a. [2] M. J. Balas, Finite-dimensional controllers for linear distributed parameter systems: Exponential stability using residual mode filters, J. Math. Anal. Appl., 133 (1988), 283-296.  doi: 10.1016/0022-247X(88)90401-5. [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer-Verlag New York, 2011. [4] N. A. Chaturvedi, R. Klein, J. Christensen, J. Ahmed and A. Kojic, Algorithms for advanced battery-management systems, IEEE Control Systems Magazine, 30 (2010), 49-68.  doi: 10.1109/MCS.2010.936293. [5] J.-P. Corriou, Nonlinear control of reactors with state estimation, Process Control, Springer International Publishing, Cham, (2018), 769–791. doi: 10.1007/978-3-319-61143-3_19. [6] R. F. Curtain, Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input, IEEE Trans. Automat. Control, 27 (1982), 98-104.  doi: 10.1109/TAC.1982.1102875. [7] J. Deutscher, A backstepping approach to the output regulation of boundary controlled parabolic PDEs, Automatica J. IFAC, 57 (2015), 56-64.  doi: 10.1016/j.automatica.2015.04.008. [8] J. Deutscher, Backstepping design of robust output feedback regulators for boundary controlled parabolic PDEs, IEEE Trans. Automat. Control, 61 (2016), 2288-2294.  doi: 10.1109/TAC.2015.2491718. [9] C. Harkort and J. Deutscher, Finite-dimensional observer-based control of linear distributed parameter systems using cascaded output observers, Internat. J. Control, 84 (2011), 107-122.  doi: 10.1080/00207179.2010.541942. [10] H. K. Khalil, Nonlinear Systems, Macmillan Publishing Company, New York, 1992. [11] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Advances in Design and Control, 16. Society for Industrial and Applied Mathematic, 2008. doi: 10.1137/1.9780898718607. [12] S. Limmer, Dynamic pricing for electric vehicle charging-a literature review, Energies, 12, 3574. doi: 10.3390/en12183574. [13] T. Meurer, Control of Higher-Dimensional PDEs : Flatness and Backstepping Designs, Communications and Control Engineering Series, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-30015-8. [14] T. Meurer and A. Kugi, Tracking control for boundary controlled parabolic PDEs with varying parameters: Combining backstepping and differential flatness, Automatica J. IFAC, 45 (2009), 1182-1194.  doi: 10.1016/j.automatica.2009.01.006. [15] T. Meurer and M. Zeitz, Feedforward and feedback tracking control of nonlinear diffusion-convection-reaction systems using summability methods, Industrial and Engineering Chemistry Research, 44 (2005), 2532-2548.  doi: 10.1021/ie0495729. [16] S. Moura, Scott-Moura/SPMeT: The full SPMeT. doi: 10.5281/zenodo.221376. [17] S. J. Moura, Estimation and control of battery electrochemistry models: A tutorial, in 2015 54th IEEE Conference on Decision and Control (CDC), (2015), 3906–3912. [18] S. J. Moura, F. B. Argomedo, R. Klein, A. Mirtabatabaei and M. Krstic, Battery state estimation for a single particle model with electrolyte dynamics, IEEE Transactions on Control Systems Technology, 25 (2017), 453-468.  doi: 10.1109/TCST.2016.2571663. [19] S. J. Moura, N. A. Chaturvedi and M. Krstic, PDE estimation techniques for advanced battery management systems — Part I: SOC estimation, in American Control Conference (ACC), Montréal, Canada, (2012), 559–565. doi: 10.1109/ACC.2012.6315019. [20] S. J. Moura, N. A. Chaturvedi and M. Krstić, Adaptive partial differential equation observer for battery state-of-charge/state-of-health estimation via an electrochemical model, Journal of Dynamic Systems, Measurement, and Control, 136 (2014), 011015, 11 pp. doi: 10.1115/1.4024801. [21] N. Petit, P. Rouchon, J.-M. Boueilh, F. Guérin and P. Pinvidic, Control of an industrial polymerization reactor using flatness, Journal of Process Control, 12 (2002), 659–665, URL http://www.sciencedirect.com/science/article/pii/S095915240100049X. [22] T. Reis and T. Selig, Funnel control for the boundary controlled heat equation, SIAM J. Control Optim., 53 (2015), 547-574.  doi: 10.1137/140971567. [23] S. Santhanagopalan, Q. Guo, P. Ramadass and R. E. White, Review of models for predicting the cycling performance of lithium ion batteries, Journal of Power Sources, 156 (2006), 620-628.  doi: 10.1016/j.jpowsour.2005.05.070. [24] A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic pdes-part II: Estimation-based designs, Automatica J. IFAC, 43 (2007), 1543-1556.  doi: 10.1016/j.automatica.2007.02.014. [25] A. Terrand-Jeanne, V. Andrieu, V. D. S. Martins and C.-Z. Xu, Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems, IEEE Trans. Automat. Control, 65 (2020), 4481-4492.  doi: 10.1109/TAC.2019.2957349. [26] K. E. Thomas, J. Newman and R. M. Darling, Mathematical Modeling of Lithium Batteries, Springer US, Boston, MA, (2002), 345–392. doi: 10.1007/0-306-47508-1_13. [27] R. Vazquez and M. Krstic, Boundary control and estimation of reaction–diffusion equations on the sphere under revolution symmetry conditions, Internat. J. Control, 92 (2019), 2-11.  doi: 10.1080/00207179.2017.1286691. [28] C.-Z. Xu and H. Jerbi, A robust PI-controller for infinite-dimensional systems, Internat. J. Control, 61 (1995), 33-45.  doi: 10.1080/00207179508921891.

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##### References:
 [1] M. Armand and J.-M. Tarascon, Building better batteries, Nature, 451 (2008), 652-657.  doi: 10.1038/451652a. [2] M. J. Balas, Finite-dimensional controllers for linear distributed parameter systems: Exponential stability using residual mode filters, J. Math. Anal. Appl., 133 (1988), 283-296.  doi: 10.1016/0022-247X(88)90401-5. [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer-Verlag New York, 2011. [4] N. A. Chaturvedi, R. Klein, J. Christensen, J. Ahmed and A. Kojic, Algorithms for advanced battery-management systems, IEEE Control Systems Magazine, 30 (2010), 49-68.  doi: 10.1109/MCS.2010.936293. [5] J.-P. Corriou, Nonlinear control of reactors with state estimation, Process Control, Springer International Publishing, Cham, (2018), 769–791. doi: 10.1007/978-3-319-61143-3_19. [6] R. F. Curtain, Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input, IEEE Trans. Automat. Control, 27 (1982), 98-104.  doi: 10.1109/TAC.1982.1102875. [7] J. Deutscher, A backstepping approach to the output regulation of boundary controlled parabolic PDEs, Automatica J. IFAC, 57 (2015), 56-64.  doi: 10.1016/j.automatica.2015.04.008. [8] J. Deutscher, Backstepping design of robust output feedback regulators for boundary controlled parabolic PDEs, IEEE Trans. Automat. Control, 61 (2016), 2288-2294.  doi: 10.1109/TAC.2015.2491718. [9] C. Harkort and J. Deutscher, Finite-dimensional observer-based control of linear distributed parameter systems using cascaded output observers, Internat. J. Control, 84 (2011), 107-122.  doi: 10.1080/00207179.2010.541942. [10] H. K. Khalil, Nonlinear Systems, Macmillan Publishing Company, New York, 1992. [11] M. Krstic and A. Smyshlyaev, Boundary Control of PDEs: A Course on Backstepping Designs, Advances in Design and Control, 16. Society for Industrial and Applied Mathematic, 2008. doi: 10.1137/1.9780898718607. [12] S. Limmer, Dynamic pricing for electric vehicle charging-a literature review, Energies, 12, 3574. doi: 10.3390/en12183574. [13] T. Meurer, Control of Higher-Dimensional PDEs : Flatness and Backstepping Designs, Communications and Control Engineering Series, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-30015-8. [14] T. Meurer and A. Kugi, Tracking control for boundary controlled parabolic PDEs with varying parameters: Combining backstepping and differential flatness, Automatica J. IFAC, 45 (2009), 1182-1194.  doi: 10.1016/j.automatica.2009.01.006. [15] T. Meurer and M. Zeitz, Feedforward and feedback tracking control of nonlinear diffusion-convection-reaction systems using summability methods, Industrial and Engineering Chemistry Research, 44 (2005), 2532-2548.  doi: 10.1021/ie0495729. [16] S. Moura, Scott-Moura/SPMeT: The full SPMeT. doi: 10.5281/zenodo.221376. [17] S. J. Moura, Estimation and control of battery electrochemistry models: A tutorial, in 2015 54th IEEE Conference on Decision and Control (CDC), (2015), 3906–3912. [18] S. J. Moura, F. B. Argomedo, R. Klein, A. Mirtabatabaei and M. Krstic, Battery state estimation for a single particle model with electrolyte dynamics, IEEE Transactions on Control Systems Technology, 25 (2017), 453-468.  doi: 10.1109/TCST.2016.2571663. [19] S. J. Moura, N. A. Chaturvedi and M. Krstic, PDE estimation techniques for advanced battery management systems — Part I: SOC estimation, in American Control Conference (ACC), Montréal, Canada, (2012), 559–565. doi: 10.1109/ACC.2012.6315019. [20] S. J. Moura, N. A. Chaturvedi and M. Krstić, Adaptive partial differential equation observer for battery state-of-charge/state-of-health estimation via an electrochemical model, Journal of Dynamic Systems, Measurement, and Control, 136 (2014), 011015, 11 pp. doi: 10.1115/1.4024801. [21] N. Petit, P. Rouchon, J.-M. Boueilh, F. Guérin and P. Pinvidic, Control of an industrial polymerization reactor using flatness, Journal of Process Control, 12 (2002), 659–665, URL http://www.sciencedirect.com/science/article/pii/S095915240100049X. [22] T. Reis and T. Selig, Funnel control for the boundary controlled heat equation, SIAM J. Control Optim., 53 (2015), 547-574.  doi: 10.1137/140971567. [23] S. Santhanagopalan, Q. Guo, P. Ramadass and R. E. White, Review of models for predicting the cycling performance of lithium ion batteries, Journal of Power Sources, 156 (2006), 620-628.  doi: 10.1016/j.jpowsour.2005.05.070. [24] A. Smyshlyaev and M. Krstic, Adaptive boundary control for unstable parabolic pdes-part II: Estimation-based designs, Automatica J. IFAC, 43 (2007), 1543-1556.  doi: 10.1016/j.automatica.2007.02.014. [25] A. Terrand-Jeanne, V. Andrieu, V. D. S. Martins and C.-Z. Xu, Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems, IEEE Trans. Automat. Control, 65 (2020), 4481-4492.  doi: 10.1109/TAC.2019.2957349. [26] K. E. Thomas, J. Newman and R. M. Darling, Mathematical Modeling of Lithium Batteries, Springer US, Boston, MA, (2002), 345–392. doi: 10.1007/0-306-47508-1_13. [27] R. Vazquez and M. Krstic, Boundary control and estimation of reaction–diffusion equations on the sphere under revolution symmetry conditions, Internat. J. Control, 92 (2019), 2-11.  doi: 10.1080/00207179.2017.1286691. [28] C.-Z. Xu and H. Jerbi, A robust PI-controller for infinite-dimensional systems, Internat. J. Control, 61 (1995), 33-45.  doi: 10.1080/00207179508921891.
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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