Robust and Adaptive Fractional-Order Observer Design for One-sided Lipschitz Systems

Document Type : Original Article


Electrical and Computer Engineering, Faculty of Engineering Department of Kharazmi university, Tehran, Iran


This paper presents adaptive observer design for one-sided Lipschitz systems. One-sided Lipschitz systems are a wide branch of nonlinear systems that include Lipschitz systems. Designed observer simultaneously estimate states and unknown parameters of the system and it is robust against input perturbation and limited observer gain disturbance. As using  observer will results in having desirable performance besides observer stability, a non-fragile  observer is presented and its stability is investigated based on Lyapunov theorem. Using linear matrix inequality causes minimizing the effect of disturbance on the estimation error addition to calculating the observer’s gain systematically. Two examples are presented to show the efficiently and performance of the proposed observer and comparison with a new research in this field. Finally the conclusion of the paper and the useful suggestions for future researches in this field is presented.


[1]      Y. Li, Y. Q. Chen and H. S. Ahn, “Fractional-order iterative learning control for fractional-order linear systems,” Asian J. Control, vol. 13, no. 1, pp. 54–63, 2011.
[2]      A. S. Ammour, S. Djennoune, W. Aggoune and M. Bettayeb, “Stabilization of fractional-order linear systems with state and input delay,” Asian J. Control, vol. 17, no. 5, pp. 1946–1954, 2015.
[3]      Y. Luo, Y. Q. Chen, H. S. Ahn and Y. G. Pi, “Fractional order robust control for cogging effect compensation in PMSM position servo systems: Stability analysis and experiments,” Control Eng. Practice, vol. 18, no. 9, pp. 1022–1036, 2010.
[4]      T. Sangpet and S. Kuntanapreeda, “Force control of an electrohydraulic actuator using a fractional order controller,” Asian J. Control, vol. 15, no. 3, pp. 764–772, 2012.
[5]      I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[6]      R. Hilfer, Application of Fractional Calculus in Physics, World Science Publishing, Singapore, 2000.
[7]      A. Zemouche and M. Boutayeb, “Nonlinear-observer-based synchronization and unknown input recovery,” IEEE Transaction on Circuits Systems, vol. 56, no. 8, pp. 1720-1731, 2008.
[8]      I. N’doye, M. Zasadzinski, M. Darouach and N. E. Radhy, “Observer-based control for fractional-order continuous - time systems,” Proc of the 48h IEEE Conference on Decision and Control (CDC),pp. 1932-1937, Shanghai, China, December 2009.
[9]      D. Matignon and B. D. Novel, “Observer-based controllers for fractional differential systems,” Proceedings of the 36th Conference on Decision & Control, vol. 5, pp. 4967-4972, December 1997.
[10]      بهروز صفری‌نژادیان و مجتبی اسد، «ارائه دو فیلتر کالمن مرتبه کسری جدید برای سیستم‌های مرتبه کسری خطی در حضور نویز اندازه‌گیری خطی»، مجله مهندسی برق دانشگاه تبریز، جلد 47، شماره 2، 607-595، 1396.
[11]      E. A. Boroujeni and H. R. Momeni, “Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems,” Signal Process, vol. 92, no. 10, pp. 2365-2370, 2012.
[12]      E. A. Boroujeni and H. R. Momeni, “Lyapunov based design of a nonlinear resilient fractional order observer,” In Proc: 21st Iranian Conference on Electrical Engineering, pp. 1-5, Mashhad, Iran, May 2013.
[13]      E A. Boroujeni and H. R. Momeni, “An iterative method to design optimal non-fragile  observer for Lipschitz nonlinear fractional order systems,” Nonlinear Dynamics, vol. 80, no. 4, pp. 1801-1810, 2015.
[14]      E A. Boroujeni, H. R. Momeni and M. Sojoodi, “An LMI approach to resilient  fractional order observer design for Lipschitz fractional order nonlinear systems using continuous frequency distribution,” Modares Journal of Electrical Engineering, vol. 12, no. 1, pp. 1-10, 2012.
[15]      I. N’doye and M. Darouach, “  adaptive observer for fractional-order systems,” Int. J. Adapt. Control Signal Process, vol. 31, no. 3, pp. 314-331,  2017.
[16]      C. S. Jeong, E. E. Yaz, A. Bahakeem and Y. I. Yaz, “Resilient design of observers with general criteria using LMIs,” The 25th American Control Conference, pp. 111-116, Minnesota. USA, June 2006.
[17]      M. Pourgholi and V. Johari Majd, “A novel robust proportional-integral (PI) adaptive observer design for chaos synchronization,” Chin. Phys. B, vol. 20, no. 12, 120503, pp.1-7, 2011.
[18]      M. Abbaszadeh and H. J. Marquez, “Design of nonlinear state observers for one-Sided Lipschitz systems,” February 2013,
[19]      M. Karkhaneh and M. Pourgholi, “Adaptive observer design for one-Sided Lipschitz class of nonlinear systems,” Modares Journal of Electrical Engineering, vol. 11, no. 4, pp. 45-51, 2012.
[20]      Y. H. Lan, L. L. Wang, L. Ding and Y. Zhou, “Full-order and reduced-order observer design for a class of fractional-order nonlinear systems,” Asian Journal of Control, vol. 18, no. 5,pp. 1-11, 2016.
[21]      Y. H. Lan, W. J. Li, Y. Zhou and Y. P. Luo, “Non-fragile observer design for fractional-order one-sided Lipschitz nonlinear systems,” International Journal of Automation and Computing, vol. 10, no. 4, pp. 296-302, 2013.
[22]      A. Jmal, O. Naifar, A. B. Makhlouf, N. Derbel and M. A. Hammami, “On observer design for nonlinear Caputo fractional-order systems,” vol. 20, no. 5, pp. 1-8, 2018.
[23]      M. Benallouch, M. Boutayeb and M. Zasadzinski, “Observer design for one-sided Lipschitz discrete-time systems,” Systems and Control Letters, vol. 61, no. 9, pp. 879–886, 2012.
[24]      M. Pourgholi and V. J. Majd, “Robust  adaptive observer design for Lipschitz  class of  nonlinear systems,” International Journal of Computer, Electrical, Automation, Control and Information Engineering, vol. 6, no. 3, pp. 275-279, 2012.
[25]      محمدمهدی عارفی، «طراحی یک رویتگر مقاوم برای دسته وسیعی از سیستم‌های غیرخطی در حضور دینامیک‌های مدل‌نشده»، مجله مهندسی برق دانشگاه تبریز، جلد 47، شماره 2، 627-621، 1396.
[26]      M. H. Asemani, V. J. Majd and S. Mobayen, “Robust  observer - based control of uncertain T-S fuzzy systems with control constraints,” 20th Iranian Conference on Electrical Engineering, pp. 910-915, Tehran, Iran, May 2012.
[27]      M. Y. Xu, G. D. Hu and Y. B. Zhao, “Reduced-order observer design for one-sided Lipschitz non-linear systems,” IMA Journal of Mathematical Control & Information, vol. 26, no. 3, pp. 299–317, 2009.
[28]      J. C. Trigeassou, N. Maamri, J. Sabatier and A. Oustaloup, “A Lyapunov approach to the stability of fractional differential equations,” Signal Processing 91, vol. 91, no.3, pp. 437–445, 2011.
[29]      F. Chen and W. Zhang, “LMI criteria for robust chaos synchronization of a class of chaotic systems,” Nonlinear Analysis TMA 67, vol. 67, no. 12, pp. 3384–3393, 2007.
[30]      W. Rudin, Principles of Mathematical Analysis, New York: McGraw-Hill, 1976.
[31]      E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1993.
[32]      S. Boyd, L.E. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, Society for Industrial and Applied Mathematics, Philadelphia, 1994.
[33]      J. J. E. Slotine and W. Li, Applied Nonlinear Control, Englewood Cliffs: Prentice-Hall, 1991.
[34]      F. Cacace, A. Germani and C. Manes, “An observer for a class of nonlinear systems with time varying observation delay,” Systems & Control Letters, vol. 51, no. 5, pp. 305-312, 2010.
[35]      W. Zhang, H. Su, H. Wang and Z. Han, “Full-order and reduced-order observers for one-side nonlinear systems using Riccati equations,” Commun Nonlinear Sci Numer Simulat, vol. 17, no. 12, pp. 4968–4977, 2012.