Design of an Estimator Based on Duality Principle for Unknown and Time-varying Delay Estimation in LTI Systems

Document Type : Power Article

Authors

1 Department of Electrical and Computer Engineering, University of Birjand, Birjand, Iran

2 Department of electrical and computer engineering, university of birjand, birjand, iran

3 department of electrical and computer engineering, university of birjand, birjand, iran

Abstract

Abstract: In this article, a novel observer has been designed for the online estimation of time delay in SISO-LTI continuous time systems with uncertain and time variant delay in the control input. It is obvious that Laplace transfer function of a delayed system includes a time delay factor. In this paper, it is assumed that the only uncertain and time variant parameter in the system is the very system’s time delay parameter. The main idea used in designing the proposed observer is based on the establishment of duality principle between controller and observer, such that a direct adaptive controller structure (MRAS) is indirectly used for designing an estimator. For this, the main sections in an MRAS control system are organized the way that designing the controller will lead to designing the delay estimator in dual problem. In fact, adaptation rule in designing the controller will express the same estimator mechanism in the dual problem. Also, in computing the estimation mechanism, the two methods, one based on Lyapunov Theory and the other based on MIT rule, are used. Eventually, simulation results are indicative of the rather desirable performance of the proposed estimator in dealing with uncertain and time varying delays.

Keywords


[1] J. E. Normey-Rico and E. F. Camacho, "Control of Dead-time Processes", Springer-Verlag, 1th ed, London, UK, 2007.
[2] Q. C. Zhong, "Robust Control of Time-delay Systems", Springer-Verlag, 1th ed, London, UK, 2006.
[3] M. Wu, Y. He, and J. H. She, "Stability Analysis and Robust Control of Time-Delay Systems", Science Press Beijing and Springer-Verlag Berlin Heidelberg, 2010.
[4] S. Bjorklund and L. Ljung, "A review of time-delay estimation techniques", 42nd IEEE International Conference on Decision and Control, (IEEE Cat. No.03CH37475), Maui, HI, Vol. 3, 2003, pp. 2502-2507
[5] A. O. Dwyer and R. Gao, "Comparison of two B-polynomial Methods Application to the Identification of Time delayed Processes", Proceedings of the Irish Signals and Systems Conference, NUI Maynooth, Ireland, June 2001, pp. 105-111.
[6] J. Roe, R. Gao and A. Dwyer, "Identification of a Time-delayed Process Model using an Over parameterization Method", Proceedings of the China-Ireland International Conference on Information and Communications Technologies (CIICT), DCU, August 2007.
[7] K. Taarita, L. Belkoura, M. Ksouri and J.P. Richard, "A Fast Identification Algorithm For Systems With Delayed Inputs", International Journal of Systems Science, Taylor & Francis, Vol. 42, No. 3, pp. 449-456. 2011.
[8] L. Belkoura, J. P. Richard and M. Fliess, "On-line identification of systems with delayed inputs", 17th Symposium on Mathematical Theory of Networks and Systems (MTNS), Kyoto, Japon, July 2006.
[9] A. O'Dwyer and J. V. Ringwood, "Model Parameter And Time Delay Estimation Using Gradient Methods", Proceedings of the Irish Colloquium on DSP and Control, Dublin City University, July 1994, pp. 211-218.
[10] D. Etter and S. Stearns, "Adaptive estimation of time delays in sampled data systems", in IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 29, No. 3, June 1981, pp. 582-587.
[11] S. Ahmed, B. Huang and S. L. Shah, "Parameter and delay estimation of continuous-time models using a linear filter", Journal of Process Control, Vol. 16, No. 4 , April 2006, pp. 323–331.
[12] S.V. Drakunov, W. Perruquetti, J.-P. Richard and L. Belkoura, "Delay identification in time-delay systems using variable structure observers", Annual Reviews in Control, Vol. 30, Issue 2, 2006, pp. 143-158
[13] J. Kozłowski and Z. Kowalczuk. "On-line Parameter and Delay Estimation of Continuous-Time Dynamic Systems", Int. J. Appl. Math. Comput. Sci, Vol. 25, Issue 2, June 2015, pp. 223-232.
[14] V. Léchappé, E. Moulay and F. Plestan, "Dynamic observation-prediction for LTI systems with a time-varying delay in the input", 2016 IEEE 55th Conference on Decision and Control (CDC), Las Vegas, NV, 2016, pp. 2302-2307.
[15] C. Lai and P. Hsu, "Design the Remote Control System With the Time-Delay Estimator and the Adaptive Smith Predictor", in IEEE Transactions on Industrial Informatics, Vol. 6, No. 1, Feb. 2010, pp. 73-80.
[16] R.M.C.De Keyser, "ADAPTIVE DEAD-TIME ESTIMATION", 2nd IFAC Workshop on Adaptive Systems in Control and Signal Processing, Lund, Sweden, Vol. 20, Issue. 2, July 1986, pp. 385-389.
[17] J. Tuch, A. Feuer and Z. J. Palmor, "Time delay estimation in continuous linear time-invariant systems," in IEEE Transactions on Automatic Control, Vol. 39, No. 4, April 1994, pp. 823-827.
[18] V. Léchappé, J. De León, E. Moulay, F. Plestan and A. Glumineau, "Delay and state observer for SISO LTI systems", 2015 American Control Conference (ACC), Chicago, IL, 2015, pp. 4585-4590.
[19] X. Hong and Q. Zhu, "An on-line algorithm of uncertain time delay estimation in a continuous system," 2009 International Conference on Networking, Sensing and Control, Okayama, 2009, pp. 498-501.
[20] M. Krstic, "Lyapunov Stability of Linear Predictor Feedback for Time-Varying Input Delay", in IEEE Transactions on Automatic Control, Vol. 55, No. 2, Feb 2010, pp. 554-559.
[21] N. Nguyen and E. Summers, "On Time Delay Margin Estimation for Adaptive Control and Robust Modification Adaptive Laws", AIAA Guidance, Navigation, and Control Conference, Guidance, Navigation, and Control and Co-located Conferences, Portland, Oregon, 2011.
[22] Y. Liu, L. –S. Hu and P. Shi, "A novel approach on stabilization for linear systems with time-varying input delay", Applied Mathematics and Computation, Vol. 218, No. 10, 2012 pp. 5937-5947.
[23] F. Cacace, F. Conte and A. Germani, "State Feedback Stabilization of Linear Systems with Unknown Input Time Delay", IFAC-Papers On Line, Vol. 50, Issue. 1, 2017, pp. 1245-1250.
[24] Y. Wei and Z. Lin, "A delay-independent output feedback for linear systems with time-varying input delay", International Journal of Robust and Nonlinear Control, 2018, pp. 1-11.
[25] D. Yue and Q. L. Han, "Delayed feedback control of uncertain systems with time-varying input delay", Automatica, Vol. 41, Issue. 2, 2005, pp. 233-240.
[26] C. Y. Kao and B. Lincoln, "Simple Stability Criteria For Systems With time-Varying Delays", Automatica, Vol. 40, No. 8, August 2004, pp. 1429–1434.
[27] W. -A. Zhang and L. Yu, "A robust control approach to stabilization of networked control systems with time-varying delays", Automatica, Vol. 45, No. 10, October 2009, pp. 2440-2445.
[28] A. Polyakov, A. Poznyak and J. Richard, "Robust output stabilization of time-varying input delay systems using attractive ellipsoid method", 52nd IEEE Conference on Decision and Control, Florence, 2013, pp. 934-939.
[29] C. Yuan and F. Wu, "ℋ∞ state-feedback control of linear systems with time-varying input delays," 2016 IEEE 55th Conference on Decision and Control (CDC), Las Vegas, NV, 2016, pp. 586-591.
[30] D. B. Pietri, F. Mazenc and N. Petit, "Robust compensation of a chattering time-varying input delay with jumps", Automatica, Vol. 92, 2018, pp. 225-234.
[31] S. Roy and I. N. Kar, "Robust Control of Uncertain Euler-Lagrange Systems with Time-Varying Input Delay," In Proceedings of the Advances in Robotics (AIR '17), ACM, New York, NY, USA, Article 16, 2017, 6 pages.
[32] R. Matusu and R. Prokop, "Control of systems with time-varying delay: A comparison study", 12th WSEAS International Conference on Automatic Control, Modelling and Simulation, ACMOS 10, 2010, pp. 125-130.
[33] J. G. Dawson, "Fuzzy logic control of linear systems with variable time delay", Proceedings of 1994 9th IEEE International Symposium on Intelligent Control, Columbus, OH, USA, 1994, pp. 5-10.
[34] D. Srinivasagupta, H. Schättlerb and B. Joseph, "Time-stamped model predictive control: an algorithm for control of processes with random delays", Computers and Chemical Engineering, Vol. 28, No. 8, July 2004, pp. 1337–1346.
[35] S. Y. Yoon and Z. Lin, "Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay", Systems & Control Letters, Vol. 62, Issue. 10, 2013, pp. 837-844.
[36] F. Cacace , A. Germani and C. Manes, "Predictor-based control of linear systems with large and variable measurement delays", International Journal of Control, Taylor & Francis, Vol. 87, No. 4, 2014, pp. 704-714.
[37] V. Léchappé and E. Moulay and F. Plestan "Prediction-based control for LTI systems with uncertain time-varying delays and partial state knowledge", International Journal of Control, Taylor & Francis, Vol. 91, No. 6, 2018, pp. 1403-1414.
[38] X. Han, E. Fridman and S.K. Spurgeon, "Sliding mode control in the presence of input delay: A singular perturbation approach", Automatica, Vol. 48, Issue. 8, 2012, pp. 1904-1912.
[39] Y. Farid and N. Bigdeli, "Robust adaptive intelligent sliding model control for a class of uncertain chaotic systems with unknown time-delay", Nonlinear Dyn, Vol. 67, No. 3, February 2012, pp. 2225–2240.
[40] F. Carravetta, P. Palumbo and P. Pepe, "Quadratic Optimal control of linear systems with time-varying input delay," 49th IEEE Conference on Decision and Control (CDC), Atlanta, GA, 2010, pp. 4996-5000.
[41] F. Cacace, F. Conte and A. Germani, "Memoryless Approach to the LQ and LQG Problems With Variable Input Delay", IEEE Transactions on Automatic Control, Vol. 61, No. 1, 2016, pp. 216-221.
[42] F. Cacace, F. Conte, A. Germani and G. Palombo, "Optimal control of linear systems with large and variable input delays", Systems & Control Letters, Vol. 89, pp. 1-7, 2016.
[43] J. K. Pieper, B. W. Surgenor and J. Z. Liu, "On Self-Tuning Control of Processes with Time Varying Dead Time" , American Control Conference 1991, Boston, MA, USA, 26-28 June 1991, pp. 2166-2171,
[44] H. kurzt and W. goedecke, "Digital Parameter-Adaptive Control of Processes with Unknown Dead Time", Automatica, Vol. 17, No. I, January 1981, pp. 245-252.
[45] G. A. Dumont, A. Elnaggar and A. Elshafelt, "Adaptive Predictive Control Of Systems With Time-Varying Time Delay", International Journal Of Adaptive Control And Signal Processing, Vol. 7, No. 2, March 1993, pp. 91-101.
[46] C. Chandra Prasad, V. Hahn, H. Unbehauen and U. Keuchel, "Adaptive Control of a Variable Dead Time Process with an Integrator", IFAC Proceedings Volumes, Vol. 18, Issue. 15, 1985, pp. 71-75.
[47] J. P. Nelson and M. J. Balas, "Direct model reference adaptive control of linear systems with input/output delays", Numerical Algebra, Control & Optimization, Vol. 3, No. 3, 2013, pp. 445-462.
 
[48] M. T. Nihtilä, "Adaptive control of a continuous-time system with time-varying input delay", Systems & Control Letters, Vol. 12, Issue. 4, 1989,  pp. 357-364.
[49] P. S. Agachi, Z. K..Nagy, M. V. Cristea and A. L. Imre-Lucaci, "Model Based Control", WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2006.
[50] N. B. Liberis and M. Krstic, "Nonlinear Control Under Nonconstant Delays (Advances in Design and Control)", SIAM-Society for Industrial and Applied Mathematics, 24 September 2013.
[51] K. J. Astrom and B. Wittenmark, "Adaptive Control", Dover Publications; Second ed, Mineola, New York, December 2008.
 
]52[ حسین شریف­زاده و مصطفی جزائری، " طراحی تخمین­گر حالت و آشکار­ساز داده غلط سیستم‌­های قدرت با استفاده از شبکه عصبی پرسپترون"، نشریه مدل­‌سازی در مهندسی، دوره 9، شماره 26، پائیز 1390، صفحه 13- 22.
]53[ مرضیه حسین­آبادی، عبدالحسین حداد و حسین نادر­پور، " استفاده از شبکه­‌های عصبی مصنوعی در تخمین ظرفیت بار­بری شالوده ­های سطحی واقع بر بستر­های چند لایه چسبنده "، نشریه مدل­‌سازی در مهندسی، دوره 9، شماره 24، بهار 1390، صفحه 65- 82.
]54[ حسین بخشی، غلامرضا قدرتی امیری، محمد خراسانی، محمد­رضا فدوی و محمد­علی برخورداری، " تخمین پارامتر­های شتاب، سرعت و جابجایی ماکزیمم زمین با استفاده از شبکه عصبی مصنوعی "، نشریه مدل­‌سازی در مهندسی، دوره 5، شماره 19، زمستان 1388، صفحه 11- 21.
]55[ نیما امجدی، آشنایی با سیستم­های هوشمند. چاپ اول، انتشارات دانشگاه سمنان، ایران، 1381.