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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 |
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.
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. |
[2] |
A. Benzaouia and A. El Hajjaji, Advanced Takagi-Sugeno Fuzzy Systems: Delay and Saturation, Vol.8, Studies in Systems, Decision and Control, 2014. |
[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. |
[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. |
[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. |
[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. |
[14] |
T. J. Goreau,
Control of atmospheric carbon dioxide, Glob. Environ. Change, 2 (1992), 5-11.
doi: 10.1016/0959-3780(92)90031-2. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
T. J. Ross, Fuzzy Logic with Engineering Application, $3^{rd}Ed$, John Wiley & Sons, Inc, 2010.
doi: 10.1002/9781119994374. |
[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. |
[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. |
[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. |
[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. |
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. |
[2] |
A. Benzaouia and A. El Hajjaji, Advanced Takagi-Sugeno Fuzzy Systems: Delay and Saturation, Vol.8, Studies in Systems, Decision and Control, 2014. |
[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. |
[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. |
[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. |
[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. |
[14] |
T. J. Goreau,
Control of atmospheric carbon dioxide, Glob. Environ. Change, 2 (1992), 5-11.
doi: 10.1016/0959-3780(92)90031-2. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
T. J. Ross, Fuzzy Logic with Engineering Application, $3^{rd}Ed$, John Wiley & Sons, Inc, 2010.
doi: 10.1002/9781119994374. |
[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. |
[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. |
[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. |
[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. |


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