Additive Fault Tolerant Control Applied to Delayed Singularly Perturbed System

ABSTRACT

The additive fault tolerant control (FTC) for delayed system is studied in this work. To design the additive control, two steps are necessary; the first one is the estimation of the sensor fault amplitude using a Luenberger observer with delay, and the second one consists to generate the additive fault tolerant control law and to add it to the nominal control of delayed system. The additive control law must be in function of fault term, then, in the absence of fault the expression of additive control equal to zero. The generation of nominal control law consist to determinate the state feedback gain by using the Lambert W method. Around all these control tools, we propose an extension of the additive FTC to delayed singularly perturbed systems (SPS). So, this extension consists to decompose the delayed SPS in two parts: delayed slow subsystem (delayed SS) and fast subsystem (FS) without time delay. Next, we consider that the delayed SPS is affected at its steady-state, and we apply the principal of FTC to the delayed SS and finally we combine them with the feedback gain control of FS by using the principal of composite control.

The additive fault tolerant control (FTC) for delayed system is studied in this work. To design the additive control, two steps are necessary; the first one is the estimation of the sensor fault amplitude using a Luenberger observer with delay, and the second one consists to generate the additive fault tolerant control law and to add it to the nominal control of delayed system. The additive control law must be in function of fault term, then, in the absence of fault the expression of additive control equal to zero. The generation of nominal control law consist to determinate the state feedback gain by using the Lambert W method. Around all these control tools, we propose an extension of the additive FTC to delayed singularly perturbed systems (SPS). So, this extension consists to decompose the delayed SPS in two parts: delayed slow subsystem (delayed SS) and fast subsystem (FS) without time delay. Next, we consider that the delayed SPS is affected at its steady-state, and we apply the principal of FTC to the delayed SS and finally we combine them with the feedback gain control of FS by using the principal of composite control.

Cite this paper

N. Abdelkrim, A. Tellili and M. Naceur Abdelkrim, "Additive Fault Tolerant Control Applied to Delayed Singularly Perturbed System,"*Journal of Software Engineering and Applications*, Vol. 5 No. 4, 2012, pp. 217-224. doi: 10.4236/jsea.2012.54027.

N. Abdelkrim, A. Tellili and M. Naceur Abdelkrim, "Additive Fault Tolerant Control Applied to Delayed Singularly Perturbed System,"

References

[1] P. V. Kokotovic, H. K. Khalil and J. O’Reilly, “Singular Perturbations Method in Control: Analysis and Design,” Academic Press, London, 1986.

[2] M. N. Abdelkrim, “Sur la Modélisation et la Synthèse des Systèmes Singulièrement Perturbés, ” Thèse de Doctorat, de l’Ecole Nationale d’Ingénieurs de Tunis, Tunisie, 1985.

[3] M. Benrejeb, M. Gasmi and M. N. Abdelkrim, “Nouvelle Méthode de Modélisation des Systèmes Linéaire Singulièrement Perturbés-Méthode du Cercle,” IMACS-IFACSymposium, New Haven, 6-8 August 1986, pp. 569-571.

[4] M. N. Abdelkrim and M. Benrejeb, “Sur la Sensibilité d’Un Modèle Réduit d’Un Système Singulièrement Perturbé,” JTEA’9, Monastir, 9-11 December 1988, pp. 1-4.

[5] F. H. Hasio, S. T. Pan and C. C. Teng, “D-Stability Bound Analysis for Discrete Multiparameter Singularly Perturbed Systems,” IEEE Transactions on Circuit and Systems (I): Fundamental Theory and Applications, Vol. 44, No. 4, 1997, pp. 347-350. doi:10.1109/81.563624

[6] S.-T. Pan and C.-F. Chen, “Robust Stability Analysis of Discrete Uncertain Singularly Perturbed Time-Delay Systems,” Applied Mathematics Letters, Vol. 19, No. 2, 2006, pp. 197-205. doi:10.1106/j.aml.2005.05.005

[7] M. S. Mahmoud and M. G. Singh, “Large Scale Systems Modeling,” Pergamon Press, Oxford, 1981.

[8] E. Fridman, “Stability of Singularly Perturbed Differential-Difference Systems: A LMI Approach,” Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, Vol. 9, 2002, pp. 201-212.

[9] B. L. Zhang and M. Q. Fan, “Near Optimal Control for Singularly Perturbed Systems with Small Time-Delay,” Proceedings of the 7th World Congress on Intelligent Control and Automation, Chongqing, 25-27 June 2008, pp. 7212-7216. doi:10.1109/WCICA.2008.4594039

[10] L. L. Liu, J. P. Peng and B. W. Wu, “Robust Stability of Singularly Perturbed Systems with State Delays,” Proceedings of the 2009 International Workshop on Information Security and Application (IWISA 2009), Qingdao, 21-22 November 2009, pp. 19-21. doi:10.1049/ip-cta:20030018

[11] S. T. Pan, C. F. Chen and J.-G. Hsieh, “Stability Analysis for a Class of Singularly Perturbed Systems with Multiple Time Delays,” Transactions of the ASME, Vol. 126, 2004, pp: 462-466,.

[12] N. Abdelkrim, A. Tellili and M. N. Abdelkrim, “Composite Control of a Delayed Singularly Perturbed System by Using Lambert-W Function,” Proceedings of 2010 7th International Multi-Conference on Systems Signals and Devices (SSD), Amman, 27-30 June 2010, pp. 1-4. doi:10.1109/SSD.2010.5585577

[13] N. Abdelkrim, A. Tellili and M. N. Abdelkrim, “Additive Fault Tolerant Composite State Feedback Control of Singularly Perturbed System,” 6th International Conference on Electrical Systems and Automatic Control, Tunisia, 26-28 March 2010, pp. 26-28.

[14] H. Noura, “Méthode d’Accommodation aux Défauts: Théories et Applications,” Habilitation à Diriger des Recherches, l’Université Henri Poincaré, Nancy, 2002.

[15] H. Noura, D. Theilliol, J. C. Ponsart and A. Chamseddine, “Fault-Tolerant Control Systems: Design and Practical Application,” 1st Edition, Springer, Heidelberg 2009.

[16] G. Y. Tang, B.-L. Zhang, Y.-D. Zhao and S.-M. Zhang, “Optimal Sinusoidal Disturbances Damping for Singularly Perturbed Systems with Time-Delay,” Journal of Sound and Vibration, Vol. 300, No. 1-2, 2007, pp. 368-378. doi:10.1016/j.jsv.2006.08.034

[17] A. Tellili, M. N. Abdelkrim and M. Benrejeb, “Model Based Fault Diagnosis of Two-Time Scales Singularly Perturbed Systems,” Proceedings of 2004 First International Symposium on Control, Communications and Signal, Hammamet, 21-24 March 2004, pp. 819-822. doi:10.1109/ISCCSP.2004.1296571

[18] Y. Sun, P. W. Nelson and A. G. Ulsoy, “Feedback Control via Eigenvalues Assignment for Time Delayed Systems Using the Lambert W Function,” Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE 2007), Las Vegas, 4-7 September 2007, pp. 783-792. doi:10.1115/DETC2007-35711

[19] Y. Sun and A. G. Ulsoy, “Solution of a System of Linear Delay Differential, Equations Using the Matrix Lambert Function,” Proceedings of the 2006 Conference on American Control, Minneapolis, 14-16 June 2006, p. 6. doi:10.1109/ACC.2006.1656585

[20] K. M. Pietarila and F. Roger, “Developing and Automating Time Delay System Stability Analysis of Dynamic Systems Using the Matrix Lambert W (MLW) Function Method,” Ph.D. Thesis, University of Missouri, Columbia, 2009.

[1] P. V. Kokotovic, H. K. Khalil and J. O’Reilly, “Singular Perturbations Method in Control: Analysis and Design,” Academic Press, London, 1986.

[2] M. N. Abdelkrim, “Sur la Modélisation et la Synthèse des Systèmes Singulièrement Perturbés, ” Thèse de Doctorat, de l’Ecole Nationale d’Ingénieurs de Tunis, Tunisie, 1985.

[3] M. Benrejeb, M. Gasmi and M. N. Abdelkrim, “Nouvelle Méthode de Modélisation des Systèmes Linéaire Singulièrement Perturbés-Méthode du Cercle,” IMACS-IFACSymposium, New Haven, 6-8 August 1986, pp. 569-571.

[4] M. N. Abdelkrim and M. Benrejeb, “Sur la Sensibilité d’Un Modèle Réduit d’Un Système Singulièrement Perturbé,” JTEA’9, Monastir, 9-11 December 1988, pp. 1-4.

[5] F. H. Hasio, S. T. Pan and C. C. Teng, “D-Stability Bound Analysis for Discrete Multiparameter Singularly Perturbed Systems,” IEEE Transactions on Circuit and Systems (I): Fundamental Theory and Applications, Vol. 44, No. 4, 1997, pp. 347-350. doi:10.1109/81.563624

[6] S.-T. Pan and C.-F. Chen, “Robust Stability Analysis of Discrete Uncertain Singularly Perturbed Time-Delay Systems,” Applied Mathematics Letters, Vol. 19, No. 2, 2006, pp. 197-205. doi:10.1106/j.aml.2005.05.005

[7] M. S. Mahmoud and M. G. Singh, “Large Scale Systems Modeling,” Pergamon Press, Oxford, 1981.

[8] E. Fridman, “Stability of Singularly Perturbed Differential-Difference Systems: A LMI Approach,” Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, Vol. 9, 2002, pp. 201-212.

[9] B. L. Zhang and M. Q. Fan, “Near Optimal Control for Singularly Perturbed Systems with Small Time-Delay,” Proceedings of the 7th World Congress on Intelligent Control and Automation, Chongqing, 25-27 June 2008, pp. 7212-7216. doi:10.1109/WCICA.2008.4594039

[10] L. L. Liu, J. P. Peng and B. W. Wu, “Robust Stability of Singularly Perturbed Systems with State Delays,” Proceedings of the 2009 International Workshop on Information Security and Application (IWISA 2009), Qingdao, 21-22 November 2009, pp. 19-21. doi:10.1049/ip-cta:20030018

[11] S. T. Pan, C. F. Chen and J.-G. Hsieh, “Stability Analysis for a Class of Singularly Perturbed Systems with Multiple Time Delays,” Transactions of the ASME, Vol. 126, 2004, pp: 462-466,.

[12] N. Abdelkrim, A. Tellili and M. N. Abdelkrim, “Composite Control of a Delayed Singularly Perturbed System by Using Lambert-W Function,” Proceedings of 2010 7th International Multi-Conference on Systems Signals and Devices (SSD), Amman, 27-30 June 2010, pp. 1-4. doi:10.1109/SSD.2010.5585577

[13] N. Abdelkrim, A. Tellili and M. N. Abdelkrim, “Additive Fault Tolerant Composite State Feedback Control of Singularly Perturbed System,” 6th International Conference on Electrical Systems and Automatic Control, Tunisia, 26-28 March 2010, pp. 26-28.

[14] H. Noura, “Méthode d’Accommodation aux Défauts: Théories et Applications,” Habilitation à Diriger des Recherches, l’Université Henri Poincaré, Nancy, 2002.

[15] H. Noura, D. Theilliol, J. C. Ponsart and A. Chamseddine, “Fault-Tolerant Control Systems: Design and Practical Application,” 1st Edition, Springer, Heidelberg 2009.

[16] G. Y. Tang, B.-L. Zhang, Y.-D. Zhao and S.-M. Zhang, “Optimal Sinusoidal Disturbances Damping for Singularly Perturbed Systems with Time-Delay,” Journal of Sound and Vibration, Vol. 300, No. 1-2, 2007, pp. 368-378. doi:10.1016/j.jsv.2006.08.034

[17] A. Tellili, M. N. Abdelkrim and M. Benrejeb, “Model Based Fault Diagnosis of Two-Time Scales Singularly Perturbed Systems,” Proceedings of 2004 First International Symposium on Control, Communications and Signal, Hammamet, 21-24 March 2004, pp. 819-822. doi:10.1109/ISCCSP.2004.1296571

[18] Y. Sun, P. W. Nelson and A. G. Ulsoy, “Feedback Control via Eigenvalues Assignment for Time Delayed Systems Using the Lambert W Function,” Proceedings of the ASME 2007 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE 2007), Las Vegas, 4-7 September 2007, pp. 783-792. doi:10.1115/DETC2007-35711

[19] Y. Sun and A. G. Ulsoy, “Solution of a System of Linear Delay Differential, Equations Using the Matrix Lambert Function,” Proceedings of the 2006 Conference on American Control, Minneapolis, 14-16 June 2006, p. 6. doi:10.1109/ACC.2006.1656585

[20] K. M. Pietarila and F. Roger, “Developing and Automating Time Delay System Stability Analysis of Dynamic Systems Using the Matrix Lambert W (MLW) Function Method,” Ph.D. Thesis, University of Missouri, Columbia, 2009.