Skip to main content
SpringerLink
Log in

Designing a nonlinear controller based on the sum of square relaxation to obtain an optimal controller with a stability guarantee applied to the proton exchange membrane fuel cell nonlinear model

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

Proton exchange membrane fuel cell (PEMFC) is an environmentally friendly technology that eliminates the pollution caused by burning fossil fuels. Minimizing power consumption through the air-feed systems of PEMFCs and maximizing the net power output have led to the most optimal control results. Hence, the present paper is conducted to control these parameters optimally. Most previous works used linear models of PEMFC to deal with control problems. Linear models cannot precisely model the nonlinear behavior of the PEMFC stacks. Accordingly, the open-loop nonlinear system is presented as state-dependent polynomial equations, and the state feedback method is employed to design the controller. Furthermore, semi-definite programming based on the sum of square relaxation is used to solve the state-dependent polynomial nonlinear model numerically. On the other hand, due to the high response time of sum of square relaxation (SOS)-based control design methods, the feasibility problem of the SOS-based technique is conceived as an optimization problem to decrease the response time. The most important advantage of the proposed method is that this method guarantees stability. A comparison is also made between the results obtained in this study and the results of one of the most valuable studies. Accordingly, the proposed method could outperform its competitor in the accuracy and efficiency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data availability

The datasets generated during the current study are not publicly available but are available from the corresponding author.

References

  1. Han, Y., Lai, C., Li, J., Zhang, Z., Zhang, H., Hou, S., et al.: Elastocaloric cooler for waste heat recovery from proton exchange membrane fuel cells. Energy 238, 121789 (2022)

    Article  Google Scholar 

  2. Anyanwu, I.S., Xie, X., Liu, Z., Jiao, K.: Experimental investigation and optimization of proton exchange membrane fuel cell using different flow fields. Energy 217, 119313 (2021)

    Article  Google Scholar 

  3. Ahmadi, S., Ghaebi, H., Shokri, A.: A comprehensive thermodynamic analysis of a novel CHP system based on SOFC and APC cycles. Energy 186, 115899 (2019). https://doi.org/10.1016/j.energy.2019.115899

    Article  Google Scholar 

  4. Zhang, G., Yuan, H., Wang, Y., Jiao, K.: Three-dimensional simulation of a new cooling strategy for proton exchange membrane fuel cell stack using a non-isothermal multiphase model. Appl. Energy 255, 113865 (2019)

    Article  Google Scholar 

  5. Ashrafi, M., Shams, M.: The effects of flow-field orientation on water management in PEM fuel cells with serpentine channels. Appl. Energy 208, 1083–1096 (2017)

    Article  Google Scholar 

  6. Pukrushpan, J.T., Stefanopoulou, A.G., Peng, H.: Control of Fuel Cell Power Systems: Principles, Modeling, Analysis and Feedback Design. Springer Science & Business Media, Berlin (2004)

    Book  MATH  Google Scholar 

  7. Laghrouche, S., Harmouche, M., Ahmed, F.S., Chitour, Y.: Control of PEMFC air-feed system using Lyapunov-based robust and adaptive higher order sliding mode control. IEEE Trans. Control Syst. Technol. 23, 1594–1601 (2014)

    Article  Google Scholar 

  8. Zeng, G.-Q., Xie, X.-Q., Chen, M.-R., Weng, J.: Adaptive population extremal optimization-based PID neural network for multivariable nonlinear control systems. Swarm Evol. Comput. 44, 320–334 (2019)

    Article  Google Scholar 

  9. Jia, Z., Yu, J., Mei, Y., Chen, Y., Shen, Y., Ai, X.: Integral backstepping sliding mode control for quadrotor helicopter under external uncertain disturbances. Aerosp. Sci. Technol. 68, 299–307 (2017)

    Article  Google Scholar 

  10. Liaqat, K., Rehman, Z., Ahmad, I.: Nonlinear controllers for fuel cell, photovoltaic cell and battery based hybrid energy management system. J. Energy Storage 32, 101796 (2020)

    Article  Google Scholar 

  11. Li, Y., Zhao, D., Chen, Y., Podlubny, I., Zhang, C.: Finite energy Lyapunov function candidate for fractional order general nonlinear systems. Commun. Nonlinear Sci. Numer. Simul. 78, 104886 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shahri, E.S.A., Alfi, A., Machado, J.A.T.: Lyapunov method for the stability analysis of uncertain fractional-order systems under input saturation. Appl. Math. Model. 81, 663–672 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, A., Sun, J.: Stability of nonlinear system under distributed Lyapunov-based economic model predictive control with time-delay. ISA Trans. 99, 148–153 (2020)

    Article  Google Scholar 

  14. Rakhshan, M., Vafamand, N., Shasadeghi, M., Dabbaghjamanesh, M., Moeini, A.: Design of networked polynomial control systems with random delays: sum of squares approach. Int. J. Autom. Control 10, 73–86 (2016)

    Article  Google Scholar 

  15. Yang, D., Pan, R., Wang, Y., Chen, Z.: Modeling and control of PEMFC air supply system based on TS fuzzy theory and predictive control. Energy 188, 116078 (2019)

    Article  Google Scholar 

  16. Deng, H., Li, Q., Cui, Y., Zhu, Y., Chen, W.: Nonlinear controller design based on cascade adaptive sliding mode control for PEM fuel cell air supply systems. Int. J. Hydrog. Energy 44, 19357–19369 (2019)

    Article  Google Scholar 

  17. Aylward, E.M., Parrilo, P.A., Slotine, J.J.E.: Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming. Automatica 44, 2163–2170 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ataei, A., Wang, Q.: Non-linear control of an uncertain hypersonic aircraft model using robust sum-of-squares method. IET Control Theory Appl. 6, 203–215 (2012)

    Article  MathSciNet  Google Scholar 

  19. Khodadadi, L., Samadi, B., Khaloozadeh, H.: Estimation of region of attraction for polynomial nonlinear systems: a numerical method. ISA Trans. 53, 25–32 (2014)

    Article  Google Scholar 

  20. Zečević, A.I., Šiljak, D.D.: Estimating the region of attraction for large-scale systems with uncertainties. Automatica 46, 445–451 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pozo, F., Rodellar, J.: Robust stabilisation of polynomial systems with uncertain parameters. Int. J. Syst. Sci. 41, 575–584 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Martins, L., Varela, M.L.R., Fernandes, N.O., Carmo-Silva, S., Machado, J.: Literature review on autonomous production control methods. Enterp. Inf. Syst. 14, 1219–1231 (2020)

    Article  Google Scholar 

  23. Josz, C., Maeght, J., Panciatici, P., Gilbert, J.C.: Application of the moment-SOS approach to global optimization of the OPF problem. IEEE Trans. Power Syst. 30, 463–470 (2014)

    Article  Google Scholar 

  24. Sanchez, T., Cruz-Zavala, E., Moreno, J.A.: An SOS method for the design of continuous and discontinuous differentiators. Int. J. Control 91, 2597–2614 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, H., Wang, Y., Wang, Y., Zhang, J.: A novel sliding mode control for a class of stochastic polynomial fuzzy systems based on SOS method. IEEE Trans. Cybern. 50, 1037–1046 (2019)

    Article  Google Scholar 

  26. Yu J, Wang J, Li Q, Chen L, Wang H. Nonlinear controller synthesis for ship course with actuator using sum-of-squares optimization. In: Proceedings 33rd Chinese Control Conference, IEEE; 2014, p. 6060–4.

  27. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, New Delhi (1994)

    Book  MATH  Google Scholar 

  28. Jaadari, A.: Continuous quasi-LPV Systems: how to leave the quadratic framework. Université de Valenciennes et du Hainaut-Cambresis; Universidad politécnica de Valencia (Espagne) (2013)

  29. Gao, Y.-F., Sun, X.-M., Wen, C., Wang, W.: Adaptive tracking control for a class of stochastic uncertain nonlinear systems with input saturation. IEEE Trans. Automat. Contr. 62, 2498–2504 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Papachristodoulou A, Prajna S. On the construction of Lyapunov functions using the sum of squares decomposition. In: Proceedings 41st IEEE Conference on Decision and Control, vol. 3, IEEE; 2002, p. 3482–7.

  31. Zusai D. On forward invariance in Lyapunov stability theorem for local stability. 2020; Available from https://arxiv.org/abs/2006.04280.

  32. Mahapatra, S., Subudhi, B.: Nonlinear matrix inequality approach based heading control for an autonomous underwater vehicle with experimental realization. IFAC J. Syst. Control 16, 100138 (2021)

    Article  MathSciNet  Google Scholar 

  33. Doyle J, Glover K, Khargonekar P, Francis B. State-space solutions to standard H 2 and H∞ control problems. In: American Control Conference , IEEE; 1988, p. 1691–6.

  34. Cloutier JR. State-dependent Riccati equation techniques: an overview. In Proceedings of the 1997 American Control Conference (Cat. No. 97CH36041), vol. 2, IEEE; 1997, p. 932–6.

  35. Huang Y, Lu W-M. Nonlinear optimal control: alternatives to Hamilton-Jacobi equation. In: Proceedings of 35th IEEE Conference on Decision and Control, vol. 4, IEEE; 1996, p. 3942–7.

  36. Yang, H., Wang, Z., Zhang, T., Du, F.: A review on vibration analysis and control of machine tool feed drive systems. Int. J. Adv. Manuf. Technol. 107, 503–525 (2020)

    Article  Google Scholar 

  37. Wu, F., Yang, X.H., Packard, A., Becker, G.: Induced L2-norm control for LPV systems with bounded parameter variation rates. Int. J. Robust Nonlinear Control 6, 983–998 (1996)

    Article  MATH  Google Scholar 

  38. Coutinho, D.F., Trofino, A., Fu, M.: Nonlinear H-infinity control: an LMI approach. IFAC Proc. 35, 91–96 (2002)

    Article  Google Scholar 

  39. Jarvis-Wloszek Z, Feeley R, Tan W, Sun K, Packard A. Some controls applications of sum of squares programming. In: 42nd IEEE International Conference on Decision and Control (IEEE Cat. No. 03CH37475), vol. 5, IEEE; 2003, p. 4676–81.

  40. Javanmardi, H., Dehghani, M., Mohammadi, M., Vafamand, N.: Bilinear matrix inequality-based nonquadratic controller design for polytopic-linear parameter varying systems. Int. J. Robust Nonlinear Control 30, 7655–7669 (2020)

    Article  MathSciNet  Google Scholar 

  41. Prajna S, Papachristodoulou A, Wu F. Nonlinear control synthesis by sum of squares optimization: a Lyapunov-based approach. In: 2004 5th Asian control conference (IEEE Cat. No. 04EX904), vol. 1, IEEE; 2004, p. 157–65.

  42. Zhang, W., Xu, L., Li, J., Ouyang, M., Liu, Y., Han, Q., et al.: Comparison of daily operation strategies for a fuel cell/battery tram. Int. J. Hydrog. Energy 42, 18532–18539 (2017)

    Article  Google Scholar 

  43. Shao, Y., Yin, G., Wang, Z., Gao, Y.: Proton exchange membrane fuel cell from low temperature to high temperature: material challenges. J. Power Sources 167, 235–242 (2007)

    Article  Google Scholar 

  44. Meidanshahi, V., Karimi, G.: Dynamic modeling, optimization and control of power density in a PEM fuel cell. Appl. Energy 93, 98–105 (2012)

    Article  Google Scholar 

  45. Niknezhadi, A., Allué-Fantova, M., Kunusch, C., Ocampo-Martínez, C.: Design and implementation of LQR/LQG strategies for oxygen stoichiometry control in PEM fuel cells based systems. J. Power Sources 196, 4277–4282 (2011)

    Article  Google Scholar 

  46. Matraji, I., Laghrouche, S., Wack, M.: Pressure control in a PEM fuel cell via second order sliding mode. Int. J. Hydrog. Energy 37, 16104–16116 (2012)

    Article  Google Scholar 

Download references

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Faculty of Electrical and Computer Engineering, Babol Noshirvani University of Technology, Babol, 47148-71167, Iran

    Behnam Sobhani & Zahra Rahmani

Authors
  1. Behnam Sobhani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zahra Rahmani
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study's conception and design. Material preparation, data collection, and analysis were performed by BS and ZR. BS wrote the first draft of the manuscript, and ZR commented on the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Zahra Rahmani.

Ethics declarations

Competing interests

The authors have no relevant financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sobhani, B., Rahmani, Z. Designing a nonlinear controller based on the sum of square relaxation to obtain an optimal controller with a stability guarantee applied to the proton exchange membrane fuel cell nonlinear model. Nonlinear Dyn 111, 6431–6448 (2023). https://doi.org/10.1007/s11071-022-08172-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-08172-1

Keywords

Search

Navigation