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## Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model

 1 LASTIMI, High School of Technology in Sale, Mohammed V University in Rabat, Morocco 2 CISIEV, FSJES, Cadi Ayyad University in Marrakech Morocco, Morocco 3 LASTIMI, Mohammedia School of Engineers, Mohammed V University in Rabat Morocco, Morocco

* Corresponding author: elmajidi@gmail.com

Received  August 2020 Revised  February 2021 Published  March 2021

By extending some linear time delay systems stability techniques, this paper, focuses on continuous time delay nonlinear systems (TDNS) dependent delay stability conditions. First, by using the Takagi Sugeno Fuzzy Modeling, a novel relaxed dependent delay stability conditions involving uncommon free matrices, are addressed in Linear Matrix Inequalities (LMI). Then, as application a Nonlinear Carbon Dioxide Model is used and rewritten by a change of coordinate to the interior equilibrium point. Next, by using the non-linearity sector method the model is transformed to a corresponding Fuzzy Takagi Sugeno (TS) multi-model. Also, the maximum delay margin to which the model is stable, is identified. Finally, to prove the analytic results a numerical simulation is also performed and compared to other methods.

Citation: Azeddine Elmajidi, Elhoussine Elmazoudi, Jamila Elalami, Noureddine Elalami. Dependent delay stability characterization for a polynomial T-S Carbon Dioxide model. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021035
##### References:
 [1] J. M. Albertine, W. J. Manning, M. DaCosta, K. A. Stinson, M. L. Muilenberg and C. A. Rogers, Projected carbon dioxide to increase grass pollen and allergen exposure despite higher ozone levels, PLoS ONE, 9 (2014), 1-6.  doi: 10.1371/journal.pone.0111712.  Google Scholar [2] A. Benzaouia and A. El Hajjaji, Advanced Takagi-Sugeno Fuzzy Systems: Delay and Saturation, Vol.8, Studies in Systems, Decision and Control, 2014.  Google Scholar [3] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics SIAM Philadelphia, SIAM Philadelphia, 1994. doi: 10.1137/1.9781611970777.  Google Scholar [4] J.-C. Calvet, A.-L. Gibelin, J.-L. Roujean, E. Martin, P. Le Moigne, H. Douville and J. Noilhan, Past and future scenarios of the effect of carbon dioxide on plant growth and transpiration for three vegetation types of southwestern France, Atmos. Chem. Phys. Discuss., 8 (2008), 397-406.   Google Scholar [5] Y.-Y. Cao and P. M. Frank, Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models, Fuzzy Sets Syst., 124 (2001), 213-229.  doi: 10.1016/S0165-0114(00)00120-2.  Google Scholar [6] M. Chadli, D. Maquin and J. Ragot, Stability and Stabilisability of Continuous Takagi-Sugeno Systems, Journées Doctorales d'Automatique Sep 2001 Toulouse France, 2001. Google Scholar [7] M. Chadli, D. Maquin and J. Ragot, Static Output Feedback for Takaki-Sugeno Systems: An LMI Approach, Proceeding of the 10th Mediterranean conference on control and automation-MED2002, Lisbon, Portugal, 2002. Google Scholar [8] B. Chen and X. Liu, Delay-dependent robust H control for T-S fuzzy systems with time delay, IEEE Transactions on Fuzzy Systems, 13 (2005), 544-556.   Google Scholar [9] S. Devi and R. P. Mishra, Preservation of the forestry biomass and control of increasing atmospheric $\rm CO_2$ using concept of reserved forestry biomass, Int. J. Appl. Comput. Math., 6 (2020), Paper No. 17, 26 pp. doi: 10.1007/s40819-019-0767-z.  Google Scholar [10] A. Elmajidi, H. Elmazoudi, J. Elalami and N. Elalami, Carbon dioxide stability by a fuzzy takagi sugeno model, Proceeding of the 4th Journée Scientifique d'Analyse des Systemes et Traitement de l'Information, Rabat Morocco, (2017), pp.cdrom. Google Scholar [11] A. Elmajidi, E. El Mazoudi, J. Elalami and N. Elalami, New delay dependent stability condition for a carbon dioxide takagi sugeno model, Proceedings of the 6th International Conference on Wireless Technologies, Embedded, and Intelligent Systems (WITS2020), 2020. Google Scholar [12] A. Elmajidi, E. El Mazoudi, J. Elalami and N. Elalami, A fuzzy logic control of a polynomial carbon dioxide model, Ecology, Environment and Conservation, 25 (2019), 876-887.   Google Scholar [13] E. Fridman, Tutorial on Lyapunov-based methods for time-delay systems, Eur. J. Control, 20 (2014), 271-283.  doi: 10.1016/j.ejcon.2014.10.001.  Google Scholar [14] T. J. Goreau, Control of atmospheric carbon dioxide, Glob. Environ. Change, 2 (1992), 5-11.  doi: 10.1016/0959-3780(92)90031-2.  Google Scholar [15] E. Jarlebring, Computing the stability region in delay-space of a TDS using polynomial eigenproblems, IFAC Proceedings Volumes, 39 (2006), 296-301.  doi: 10.3182/20060710-3-IT-4901.00049.  Google Scholar [16] H. K. Khalil, Nonlinear Systems, $3^{rd}$Ed, Prentice Hall, Inc., 2002. Google Scholar [17] K. Kim, J. Joh and W. Kwon, Design of T-S(Takagi-Sugeno) fuzzy control systems under the bound on the output energy, Automation and Systems Engineering, 1 (1999). Google Scholar [18] H. A. Kruthika, A. D. Mahindrakar and R. Pasumarthy, Stability analysis of nonlinear time-delayed systems with application to biological models, Int. J. Appl. Math. Comput. Sci., 27 (2017), 91-103.  doi: 10.1515/amcs-2017-0007.  Google Scholar [19] C. Li, H. Wang and X. Liao, Delay-dependent robust stability of uncertain fuzzy systems with time-varying delays, IEE Proc.-Control Theory Appl., 151 (2004), 417-421.   Google Scholar [20] J. H. Lilly, Fuzzy Control and Identification, $Ed$, John Wiley & Sons, Inc, 2010. doi: 10.1002/9780470874240.  Google Scholar [21] C. Lin, Q.-G. Wang, T. H. Lee and Y. He, LMI Approach to Analysis and Control of Takagi-Sugeno Fuzzy Systems with Time Delay, Vol. 351, Lecture Notes in Control and Information Sciences, 2007.  Google Scholar [22] J. Löfberg, YALMIP: A Toolbox for Modeling and Optimization in Matlab, In Proceedings of the CACSD Conference, 2004. Google Scholar [23] Y. Manai, M. Benrejeb and P. Borne, New Approach of stability for time-delay Takagi-Sugeno fuzzy system based on fuzzy weighting-dependent lyapunov functionals, Applied Mathematics, 2 (2011), 1339-1345.  doi: 10.4236/am.2011.211187.  Google Scholar [24] A. Maria Nagy, Analyse et synthese de multimodeles pour le diagnostic: Application a une station d'epuration, Ph.D Thesis, France. https://tel.archives-ouvertes.fr, 2010. Google Scholar [25] A. K. Misra and M. Verma, A mathematical model to study the dynamics of carbon dioxide gas in the atmosphere, Appl. Math. Comput., 219 (2013), 8595-8609.  doi: 10.1016/j.amc.2013.02.058.  Google Scholar [26] A. K. Misra, M. Verma and E. Venturino, Modeling the control of atmospheric carbon dioxide through reforestation: effect of time delay, Model. Earth Syst. Environ., 1 (2015). doi: 10.1007/s40808-015-0028-z.  Google Scholar [27] Y. S. Moon, P. Park, W. H. Kwon and Y. S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, Internat. J. Control, 74 (2001), 1447-1455.  doi: 10.1080/00207170110067116.  Google Scholar [28] A.-T. Nguyen, K. Tanaka, A. Dequidt and M. Dambrine, Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems, J. Franklin Inst., 354 (2017), 2856-2870.  doi: 10.1016/j.jfranklin.2017.02.017.  Google Scholar [29] P. G. Park, A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Trans. Automat. Control, 44 (1999), 876-877.  doi: 10.1109/9.754838.  Google Scholar [30] T. J. Ross, Fuzzy Logic with Engineering Application, $3^{rd}Ed$, John Wiley & Sons, Inc, 2010. doi: 10.1002/9781119994374.  Google Scholar [31] A. Sala, On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems, Annual Reviews in Control, 33 (2009), 48-58.  doi: 10.1016/j.arcontrol.2009.02.001.  Google Scholar [32] A. Seuret and F. Gouaisbaut, On the use of the Wirtinger inequalities for time-delay systems, Proceedings of the 10-th IFAC Workshop on Time Delay Systems The International Federation of Automatic Control Northeastern University Boston USA, (2012), pp.cdrom. Google Scholar [33] A. Seuret, F. Gouaisbaut and L. Baudouin, Overview of lyapunov methods for time-delay systems, LAAS-CNRS, n 16308 (2016), hal-01369516. Google Scholar [34] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, Readings in Fuzzy Sets for Intelligent Systems, (1993), 387–403. doi: 10.1016/B978-1-4832-1450-4.50045-6.  Google Scholar [35] K. Tanaka and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems, 45 (1992), 135-156.  doi: 10.1016/0165-0114(92)90113-I.  Google Scholar [36] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, $1^st Ed$, John Wiley & Sons, Inc, 2001. Google Scholar [37] K. Tennakone, Stability of the biomass-carbon dioxide equilibrium in the atmosphere: Mathematical model, Applied Mathematics and Computation, 35 (1990), 125-130.   Google Scholar [38] MOSEK modeling cookbook, 3.2.2 (2020). Google Scholar [39] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar

show all references

##### References:
 [1] J. M. Albertine, W. J. Manning, M. DaCosta, K. A. Stinson, M. L. Muilenberg and C. A. Rogers, Projected carbon dioxide to increase grass pollen and allergen exposure despite higher ozone levels, PLoS ONE, 9 (2014), 1-6.  doi: 10.1371/journal.pone.0111712.  Google Scholar [2] A. Benzaouia and A. El Hajjaji, Advanced Takagi-Sugeno Fuzzy Systems: Delay and Saturation, Vol.8, Studies in Systems, Decision and Control, 2014.  Google Scholar [3] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics SIAM Philadelphia, SIAM Philadelphia, 1994. doi: 10.1137/1.9781611970777.  Google Scholar [4] J.-C. Calvet, A.-L. Gibelin, J.-L. Roujean, E. Martin, P. Le Moigne, H. Douville and J. Noilhan, Past and future scenarios of the effect of carbon dioxide on plant growth and transpiration for three vegetation types of southwestern France, Atmos. Chem. Phys. Discuss., 8 (2008), 397-406.   Google Scholar [5] Y.-Y. Cao and P. M. Frank, Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models, Fuzzy Sets Syst., 124 (2001), 213-229.  doi: 10.1016/S0165-0114(00)00120-2.  Google Scholar [6] M. Chadli, D. Maquin and J. Ragot, Stability and Stabilisability of Continuous Takagi-Sugeno Systems, Journées Doctorales d'Automatique Sep 2001 Toulouse France, 2001. Google Scholar [7] M. Chadli, D. Maquin and J. Ragot, Static Output Feedback for Takaki-Sugeno Systems: An LMI Approach, Proceeding of the 10th Mediterranean conference on control and automation-MED2002, Lisbon, Portugal, 2002. Google Scholar [8] B. Chen and X. Liu, Delay-dependent robust H control for T-S fuzzy systems with time delay, IEEE Transactions on Fuzzy Systems, 13 (2005), 544-556.   Google Scholar [9] S. Devi and R. P. Mishra, Preservation of the forestry biomass and control of increasing atmospheric $\rm CO_2$ using concept of reserved forestry biomass, Int. J. Appl. Comput. Math., 6 (2020), Paper No. 17, 26 pp. doi: 10.1007/s40819-019-0767-z.  Google Scholar [10] A. Elmajidi, H. Elmazoudi, J. Elalami and N. Elalami, Carbon dioxide stability by a fuzzy takagi sugeno model, Proceeding of the 4th Journée Scientifique d'Analyse des Systemes et Traitement de l'Information, Rabat Morocco, (2017), pp.cdrom. Google Scholar [11] A. Elmajidi, E. El Mazoudi, J. Elalami and N. Elalami, New delay dependent stability condition for a carbon dioxide takagi sugeno model, Proceedings of the 6th International Conference on Wireless Technologies, Embedded, and Intelligent Systems (WITS2020), 2020. Google Scholar [12] A. Elmajidi, E. El Mazoudi, J. Elalami and N. Elalami, A fuzzy logic control of a polynomial carbon dioxide model, Ecology, Environment and Conservation, 25 (2019), 876-887.   Google Scholar [13] E. Fridman, Tutorial on Lyapunov-based methods for time-delay systems, Eur. J. Control, 20 (2014), 271-283.  doi: 10.1016/j.ejcon.2014.10.001.  Google Scholar [14] T. J. Goreau, Control of atmospheric carbon dioxide, Glob. Environ. Change, 2 (1992), 5-11.  doi: 10.1016/0959-3780(92)90031-2.  Google Scholar [15] E. Jarlebring, Computing the stability region in delay-space of a TDS using polynomial eigenproblems, IFAC Proceedings Volumes, 39 (2006), 296-301.  doi: 10.3182/20060710-3-IT-4901.00049.  Google Scholar [16] H. K. Khalil, Nonlinear Systems, $3^{rd}$Ed, Prentice Hall, Inc., 2002. Google Scholar [17] K. Kim, J. Joh and W. Kwon, Design of T-S(Takagi-Sugeno) fuzzy control systems under the bound on the output energy, Automation and Systems Engineering, 1 (1999). Google Scholar [18] H. A. Kruthika, A. D. Mahindrakar and R. Pasumarthy, Stability analysis of nonlinear time-delayed systems with application to biological models, Int. J. Appl. Math. Comput. Sci., 27 (2017), 91-103.  doi: 10.1515/amcs-2017-0007.  Google Scholar [19] C. Li, H. Wang and X. Liao, Delay-dependent robust stability of uncertain fuzzy systems with time-varying delays, IEE Proc.-Control Theory Appl., 151 (2004), 417-421.   Google Scholar [20] J. H. Lilly, Fuzzy Control and Identification, $Ed$, John Wiley & Sons, Inc, 2010. doi: 10.1002/9780470874240.  Google Scholar [21] C. Lin, Q.-G. Wang, T. H. Lee and Y. He, LMI Approach to Analysis and Control of Takagi-Sugeno Fuzzy Systems with Time Delay, Vol. 351, Lecture Notes in Control and Information Sciences, 2007.  Google Scholar [22] J. Löfberg, YALMIP: A Toolbox for Modeling and Optimization in Matlab, In Proceedings of the CACSD Conference, 2004. Google Scholar [23] Y. Manai, M. Benrejeb and P. Borne, New Approach of stability for time-delay Takagi-Sugeno fuzzy system based on fuzzy weighting-dependent lyapunov functionals, Applied Mathematics, 2 (2011), 1339-1345.  doi: 10.4236/am.2011.211187.  Google Scholar [24] A. Maria Nagy, Analyse et synthese de multimodeles pour le diagnostic: Application a une station d'epuration, Ph.D Thesis, France. https://tel.archives-ouvertes.fr, 2010. Google Scholar [25] A. K. Misra and M. Verma, A mathematical model to study the dynamics of carbon dioxide gas in the atmosphere, Appl. Math. Comput., 219 (2013), 8595-8609.  doi: 10.1016/j.amc.2013.02.058.  Google Scholar [26] A. K. Misra, M. Verma and E. Venturino, Modeling the control of atmospheric carbon dioxide through reforestation: effect of time delay, Model. Earth Syst. Environ., 1 (2015). doi: 10.1007/s40808-015-0028-z.  Google Scholar [27] Y. S. Moon, P. Park, W. H. Kwon and Y. S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, Internat. J. Control, 74 (2001), 1447-1455.  doi: 10.1080/00207170110067116.  Google Scholar [28] A.-T. Nguyen, K. Tanaka, A. Dequidt and M. Dambrine, Static output feedback design for a class of constrained Takagi-Sugeno fuzzy systems, J. Franklin Inst., 354 (2017), 2856-2870.  doi: 10.1016/j.jfranklin.2017.02.017.  Google Scholar [29] P. G. Park, A delay-dependent stability criterion for systems with uncertain time-invariant delays, IEEE Trans. Automat. Control, 44 (1999), 876-877.  doi: 10.1109/9.754838.  Google Scholar [30] T. J. Ross, Fuzzy Logic with Engineering Application, $3^{rd}Ed$, John Wiley & Sons, Inc, 2010. doi: 10.1002/9781119994374.  Google Scholar [31] A. Sala, On the conservativeness of fuzzy and fuzzy-polynomial control of nonlinear systems, Annual Reviews in Control, 33 (2009), 48-58.  doi: 10.1016/j.arcontrol.2009.02.001.  Google Scholar [32] A. Seuret and F. Gouaisbaut, On the use of the Wirtinger inequalities for time-delay systems, Proceedings of the 10-th IFAC Workshop on Time Delay Systems The International Federation of Automatic Control Northeastern University Boston USA, (2012), pp.cdrom. Google Scholar [33] A. Seuret, F. Gouaisbaut and L. Baudouin, Overview of lyapunov methods for time-delay systems, LAAS-CNRS, n 16308 (2016), hal-01369516. Google Scholar [34] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, Readings in Fuzzy Sets for Intelligent Systems, (1993), 387–403. doi: 10.1016/B978-1-4832-1450-4.50045-6.  Google Scholar [35] K. Tanaka and M. Sugeno, Stability analysis and design of fuzzy control systems, Fuzzy Sets and Systems, 45 (1992), 135-156.  doi: 10.1016/0165-0114(92)90113-I.  Google Scholar [36] K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, $1^st Ed$, John Wiley & Sons, Inc, 2001. Google Scholar [37] K. Tennakone, Stability of the biomass-carbon dioxide equilibrium in the atmosphere: Mathematical model, Applied Mathematics and Computation, 35 (1990), 125-130.   Google Scholar [38] MOSEK modeling cookbook, 3.2.2 (2020). Google Scholar [39] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.  doi: 10.1016/S0019-9958(65)90241-X.  Google Scholar
States evolution in accordance to time in years for $\tau$ = 1, 3, 7 and 9 years with intial value $x_0$ = -0.1*$X_e$
States evolution in accordance to time in years for $\tau$ = 7y for different initial conditions
Model Parameter Values
 $Q_0$ $\lambda$ $\alpha$ $\lambda_1$ $s$ $L$ $\theta$ 5 5.76x$10^{-4}$ 1.6x$10^{-2}$ 4.8x$10^{-9}$ 3.2x$10^{-2}$ $10^{5}$ $10^{-6}$ $\pi$ $\phi$ $\mu$ $M$ $\zeta$ $\gamma$ $\sigma_0$ 4x$10^{-5}$ 7.1x$10^{-7}$ 1.3x$10^{-2}$ 7.5x$10^{5}$ 2.6x$10^{-6}$ 8x$10^{-4}$ 2x$10^{-4}$
 $Q_0$ $\lambda$ $\alpha$ $\lambda_1$ $s$ $L$ $\theta$ 5 5.76x$10^{-4}$ 1.6x$10^{-2}$ 4.8x$10^{-9}$ 3.2x$10^{-2}$ $10^{5}$ $10^{-6}$ $\pi$ $\phi$ $\mu$ $M$ $\zeta$ $\gamma$ $\sigma_0$ 4x$10^{-5}$ 7.1x$10^{-7}$ 1.3x$10^{-2}$ 7.5x$10^{5}$ 2.6x$10^{-6}$ 8x$10^{-4}$ 2x$10^{-4}$
Maximal Delay Margin for Different Methods in years
 Method Th. 58 [36] Th. 6.1 [2] Cor. 1 [19] Th. 1 [11] Our Approach $\bar{\tau}_{max}$ Infeasible 8, 561 8, 591 8, 595 8, 555
 Method Th. 58 [36] Th. 6.1 [2] Cor. 1 [19] Th. 1 [11] Our Approach $\bar{\tau}_{max}$ Infeasible 8, 561 8, 591 8, 595 8, 555
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