PID Stabilization of Linear Neutral Time-Delay Systems in a Numerical Approach

Affiliation(s)

Faculty of Electrical & Computer Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran.

Faculty of Electrical & Computer Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran.

ABSTRACT

In this paper, the stabilization of neutral time-delay systems is investigated. An efficient numerical approach is presented in an algorithm to establish results so that stability of such systems is achieved and stabilizing PID parameters are determined directly. It is based on determining the rightmost characteristic roots and Nyquist plot. The Newton-Raphson’s iterative method based on Lambert W function is used for the calculation of these stabilizing roots directly from the closed-loop characteristic equation of the neutral time-delay system and then stability is checked by Nyquist plot and step response of closed-loop system. Two numerical examples are included to illustrate the effectiveness of the proposed approach.

In this paper, the stabilization of neutral time-delay systems is investigated. An efficient numerical approach is presented in an algorithm to establish results so that stability of such systems is achieved and stabilizing PID parameters are determined directly. It is based on determining the rightmost characteristic roots and Nyquist plot. The Newton-Raphson’s iterative method based on Lambert W function is used for the calculation of these stabilizing roots directly from the closed-loop characteristic equation of the neutral time-delay system and then stability is checked by Nyquist plot and step response of closed-loop system. Two numerical examples are included to illustrate the effectiveness of the proposed approach.

Cite this paper

H. Moghadam, N. Vasegh and S. Moussavi, "PID Stabilization of Linear Neutral Time-Delay Systems in a Numerical Approach,"*Intelligent Control and Automation*, Vol. 3 No. 4, 2012, pp. 313-318. doi: 10.4236/ica.2012.34036.

H. Moghadam, N. Vasegh and S. Moussavi, "PID Stabilization of Linear Neutral Time-Delay Systems in a Numerical Approach,"

References

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[7] K. Gu, V. L. Kharitonov and J. Chen, “Stability of Time-Delay Systems,” Birkhauser, Boston, 2003. doi:10.1007/978-1-4612-0039-0

[8] J. R. Partingtona and C. Bonnet, “H∞ and BIBO Stabilization of Delay Systems of Neutral Type,” Systems & Control Letters, Vol. 52, No. 3, 2004, pp. 283-288. doi:10.1016/j.sysconle.2003.09.014

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[13] V. Chellaboina, A. Kamath and W. M. Haddad, “TimeDomain Sufficient Conditions for Stability Analysis of Linear Neutral Time-Delay Systems,” Proceedings of the 2007 American Control Conference, New York, 9-13 July 2007, pp. 4917-4918.

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[17] C. Dey and R. K. Mudi, “An Improved Auto-Tuning Scheme for PID Controllers,” ISA Transactions, Vol. 48, No. 4, 2009, pp. 396-408. doi:10.1016/j.isatra.2009.07.002

[18] B. Fang, “Computation of Stabilizing PID Gain Regions Based on the Inverse Nyquist Plot,” Journal of Process Control, Vol. 20, No. 10, 2010, pp. 1183-1187. doi:10.1016/j.jprocont.2010.07.004

[19] N. Tan, “Computation of Stabilizing PI and PID Controllers for Processes with Time Delay,” ISA Transactions, Vol. 44, No. 2, 2005, pp. 213-223. doi:10.1016/S0019-0578(07)90000-2

[20] K. W. Ho, A. Datta and S. P. Bhattacharya, “Generalizations of the Hermite-Biehler Theorem,” Linear Algebra and Its Applications, Vol. 302-303, 1999, pp. 135-153. doi:10.1016/S0024-3795(99)00069-5

[21] K. W. Ho, A. Datta and S. P. Bhattacharya, “PID Stabilization of LTI Plants with Time-Delay,” Proceedings of 42nd IEEE Conference on Decision and Control, Maui, 9-12 December 2003, pp. 4038-4043.

[22] G. J. Silva, A. Datta and S. P. Bhattacharyya, “PID Controllers for Time-Delay Systems,” Birkh?user, Boston, 2005.

[23] W. Michiels, K. Engelborghs, P. Vansevenant and D. Roose, “Continuous Pole Placement Method for Delay Equations,” Automatica, Vol. 38, No. 5, 2002, pp. 747761. doi:10.1016/S0005-1098(01)00257-6

[24] Z. H. Wang and H. Y. Hu, “Calculation of the Rightmost Characteristic Root of Retarded Time-Delay Systems via Lambert W Function,” Journal of Sound and Vibration, Vol. 318, No. 4-5, 2008, pp. 757-767. doi:10.1016/j.jsv.2008.04.052

[25] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, “On the Lambert W function,” Advances in Computational Mathematics, Vol. 5, No. 4, 1996, pp. 329-359. doi:10.1007/BF02124750

[26] H. Y. Hu and Z. H. Wang, “Dynamics of Controlled Mechanical Systems with Delayed Feedback,” SpringerVerlag, Berlin Heidellberg, 2002.

[1] J. P. Richard, “Time-Delay Systems: An Overview of Some Recent Advances and Open Problems,” Automatica, Vol. 39, No. 10, 2003, pp. 1667-1694. doi:10.1016/S0005-1098(03)00167-5

[2] J. E. Normey-Rico and E. F. Camacho, “Control of Dead-Time Processes,” Springer-Verlag, London, 2007.

[3] V. B. Kolmanovskii and J. P. Richard, “Stability of Some Linear Systems with Delays,” IEEE Transactions on Automatic Control, Vol. 44, No. 5, 1999, pp. 984-989. doi:10.1109/9.763213

[4] J. K. Hale and S. M. V. Lunel, “Introduction to Functional Differential Equations,” Springer-Verlag, New York, 1993.

[5] L. Dugard and E. E. Verriest, “Stability and Control of Time-Delay Systems,” Springer, New York, 1998. doi:10.1007/BFb0027478

[6] S. I. Niculescu and R. Lozano, “On the Passivity of Linear Delay Systems,” IEEE Transactions on Automatic Control, Vol. 46, No. 3, 2001, pp. 460-464. doi:10.1109/9.911424

[7] K. Gu, V. L. Kharitonov and J. Chen, “Stability of Time-Delay Systems,” Birkhauser, Boston, 2003. doi:10.1007/978-1-4612-0039-0

[8] J. R. Partingtona and C. Bonnet, “H∞ and BIBO Stabilization of Delay Systems of Neutral Type,” Systems & Control Letters, Vol. 52, No. 3, 2004, pp. 283-288. doi:10.1016/j.sysconle.2003.09.014

[9] S. I. Niculescu, “Delay Effects on Stability: A Robust Control Approach,” Springer, New York, 2001.

[10] V. B. Kolmanovskii and A. D. Myshkis, “Applied Theory of Functional Differential Equations,” Kluwer, Dordrecht, 1992.

[11] V. B. Kolmanovskii and V. R. Nosov, “Stability of Functional Differential Equations,” Academic Press, New York, 1986.

[12] J. H. Park and O. Kwon, “On New Stability Criterion for Delay-Differential Systems of Neutral Type,” Applied Mathematics and Computation, Vol. 162, No. 2, 2005, pp. 627-637. doi:10.1016/j.amc.2004.01.001

[13] V. Chellaboina, A. Kamath and W. M. Haddad, “TimeDomain Sufficient Conditions for Stability Analysis of Linear Neutral Time-Delay Systems,” Proceedings of the 2007 American Control Conference, New York, 9-13 July 2007, pp. 4917-4918.

[14] Z. H. Wang, “Numerical Stability Test of Neutral Delay Differential Equations,” Hindawi Publishing Corporation, Cairo, 2008, pp. 1-10.

[15] J. G. Ziegler and N. B. Nichols, “Optimum Settings for Automatic Controllers,” Transactions on ASME, Vol. 64, 1942, pp. 759-768.

[16] S. Yamamoto and I. Hashimoto, “Present Status and Future Needs: The View from Japanese Industry,” Chemical Process Control—CPCIV: Proceedings of 4th International Conference on Chemical Process Control, Padre Island, 17-22 February 1991, pp. 1-28.

[17] C. Dey and R. K. Mudi, “An Improved Auto-Tuning Scheme for PID Controllers,” ISA Transactions, Vol. 48, No. 4, 2009, pp. 396-408. doi:10.1016/j.isatra.2009.07.002

[18] B. Fang, “Computation of Stabilizing PID Gain Regions Based on the Inverse Nyquist Plot,” Journal of Process Control, Vol. 20, No. 10, 2010, pp. 1183-1187. doi:10.1016/j.jprocont.2010.07.004

[19] N. Tan, “Computation of Stabilizing PI and PID Controllers for Processes with Time Delay,” ISA Transactions, Vol. 44, No. 2, 2005, pp. 213-223. doi:10.1016/S0019-0578(07)90000-2

[20] K. W. Ho, A. Datta and S. P. Bhattacharya, “Generalizations of the Hermite-Biehler Theorem,” Linear Algebra and Its Applications, Vol. 302-303, 1999, pp. 135-153. doi:10.1016/S0024-3795(99)00069-5

[21] K. W. Ho, A. Datta and S. P. Bhattacharya, “PID Stabilization of LTI Plants with Time-Delay,” Proceedings of 42nd IEEE Conference on Decision and Control, Maui, 9-12 December 2003, pp. 4038-4043.

[22] G. J. Silva, A. Datta and S. P. Bhattacharyya, “PID Controllers for Time-Delay Systems,” Birkh?user, Boston, 2005.

[23] W. Michiels, K. Engelborghs, P. Vansevenant and D. Roose, “Continuous Pole Placement Method for Delay Equations,” Automatica, Vol. 38, No. 5, 2002, pp. 747761. doi:10.1016/S0005-1098(01)00257-6

[24] Z. H. Wang and H. Y. Hu, “Calculation of the Rightmost Characteristic Root of Retarded Time-Delay Systems via Lambert W Function,” Journal of Sound and Vibration, Vol. 318, No. 4-5, 2008, pp. 757-767. doi:10.1016/j.jsv.2008.04.052

[25] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, “On the Lambert W function,” Advances in Computational Mathematics, Vol. 5, No. 4, 1996, pp. 329-359. doi:10.1007/BF02124750

[26] H. Y. Hu and Z. H. Wang, “Dynamics of Controlled Mechanical Systems with Delayed Feedback,” SpringerVerlag, Berlin Heidellberg, 2002.