# طراحی کنترل‌کننده مدلغزشی مبتنی بر پایداری رازومیخین و نامساوی ماتریسی خطی برای سیستم‌های غیر‌خطی مرتبه کسری تاخیری

نوع مقاله : مقاله پژوهشی

نویسندگان

دانشکده علوم ریاضی، دانشگاه پیام نور، تهران، ایران

چکیده

ﺩﺭ ﺍﻳﻦ ﻣﻘﺎﻟﻪ، ﺍﺯ ﻳﮏ ﮐﻨﺘﺮﻝ ﮐﻨﻨﺪﻩ ﻣﺪﻟﻐﺰﺷﻲ ﺑﺮﺍﻱ ﺳﻴﺴﺘﻢ ﻫﺎﻱ ﻏﻴﺮ ﺧﻄﻲ ﻭ ﻣﺮﺗﺒﻪ ﮐﺴﺮﻱ ﺗﺎﺧﻴﺮﻱ ، ﺍﺳﺘﻔﺎﺩﻩ ﺷﺪﻩ

ﺍﺳﺖ. ﺳﻴﺴﺘﻢ ﺗﺤﺖ ﻣﻄﺎﻟﻌﻪ ﺩﺭ ﺣﻀﻮﺭ ﺍﻏﺘﺸﺎﺵ ﻭ ﻧﺎﻣﻌﻴﻨﻲ ﺍﺳﺖ. ﻫﺪﻑ ﻣﻘﺎﻟﻪ ﻃﺮﺍﺣﻲ ﮐﻨﺘﺮﻝ ﮐﻨﻨﺪﻩ ﻣﺪ ﻟﻐﺰﺷﻲ ﺑﻪﮔﻮﻧﻪﺍﻱ

ﺍﺳﺖ ﮐﻪ ﺳﻴﺴﺘﻢ ﻏﻴﺮ ﺧﻄﻲ ﭘﺎﻳﺪﺍﺭ ﻣﺠﺎﻧﺒﻲ ﺷﺪﻩ ﻭ ﻣﺘﻐﻴﺮﻫﺎﻱ ﺳﻴﺴﺘﻢ ﺩﺭ ﻳﮏ ﺯﻣﺎﻥ ﻣﺘﻨﺎﻫﻲ ﺑﻪ ﺳﻄﺢ ﻟﻐﺰﺵ ﺑﺮﺳﻨﺪ. ﺑﺎ ﺍﺳﺘﻔﺎﺩﻩ

ﺍﺯ ﻗﻀﻴﻪ ﺭﺍﺯﻭﻣﻴﺨﻴﻦ ﺑﺮﺍﻱ ﭘﺎﻳﺪﺍﺭﻱ ﺳﻴﺴﺘﻢ ﻫﺎﻱ ﮐﺴﺮﻱ ﺗﺎﺧﻴﺮﻱ ﻭ ﻧﺎﻣﺴﺎﻭﻱ ﻣﺎﺗﺮﻳﺴﻲ ﺧﻄﻲ، ﺷﺮﺍﻳﻂ ﻻﺯﻡ ﺑﺮﺍﻱ ﭘﺎﻳﺪﺍﺭﻱ ﺭﺍ

ﺑﻪ ﺩﺳﺖ ﻣﻲ ﺁﻭﺭﻳﻢ. ﺩﺭ ﻧﻬﺎﻳﺖ ، ﺑﺎ ﺩﻭ ﻣﺜﺎﻝ ﻋﺪﺩﻱ ﺻﺤﺖ ﻧﺘﺎﻳﺞ ﻭ ﮐﺎﺭﺍﻳﻲ ﺭﻭﺵ ﭘﻴﺶ ﻧﻬﺎﺩﻱ ﺭﺍ ﺑﺮﺭﺳﻲ ﮐﺮﺩﻩﺍﻳﻢ.

کلیدواژه‌ها

موضوعات

عنوان مقاله [English]

### Design of sliding mode control based on Razumikhin approach and linear matrix inequality for nonlinear fractional time-varying delay systems

نویسندگان [English]

• Seyed Mehdi Mirhosseini-Alizamini
Department of Mathematics, Payame Noor University (PNU), Tehran, Iran
چکیده [English]

In this paper, a sliding mode control for systems that are nonlinear, having fractional order, and

involve delay is used. The aim of this paper is to design a sliding mode controller, such that the closed­

looped nonlinear system becomes asymptotically stable and its trajectory can be driven onto the sliding

surface in finite time. By using the fractional Razumikhin theorem for the stability of fractional­ order

systems including delay and a linear matrix inequality, necessary conditions on asymptotic stabilization

are obtained. Some numerical examples are given to illustrate the effectiveness of the proposed results.

کلیدواژه‌ها [English]

• Delay Systems
• Fractional order systems
• Sliding mode control
• Linear Matrix Inequality

#### مراجع

[1] Balachandran K., Park J.Y. and Trujillo J.J., Controllability of nonlinear fractional dynamical systems,
Nonlinear Analysis: Theory, Methods and Applications, 75 (2012) 1919–1926.
[2] Boyd S., Ghaoui L.E.l., Feron E. and Balakrishnan V., Linear Matrix Inequalities in System and
[3] Chen L., Wu R., Cheng Y. and Chen Y., Delay dependent and order dependent stability and stabilization
of fractional order linear systems with time varying delay, IEEE Transactions on Circuits
and Systems II: Express Briefs, (2019).
[4] Doye I.N., Voos H. and Darouach M., Observer-based approach for fractional-order chaotic synchronization
and secure communication, IEEE Journal on Emerging and Selected Topics in Circuits
and Systems, 3 (2013) 442–450.
[5] Duarte-Mermoud M.A., Aguila-Camacho N., Gallegos J.A. and Castro-Linares R., Using general
quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Communications
in Nonlinear Science and Numerical Simulation, 22 (2015) 650–659.
[6] Efe M.O., Fractional order sliding mode control with reaching law approach, Turkish Journal of
Electrical Engineering and Computer Sciences, 18 (2010) 731–747.
[7] Eker I., Second-order sliding mode control with experimental application, ISA Trans, 49 (2010)
394–405 .
[8] Han Y., Kao Y. , Gao C., Zhao J. and Wang C., H∞ sliding mode control of discrete switched systems
with time-varying delays, ISA Trans, 89 (2019) 12–19.
[9] He S. and Song J., Finite-time Sliding Mode Control Design for a Class of Uncertain Conic Nonlinear
Systems,Journal of Automatic Sinica, 4 (2017) 809-816.
[10] Hu J. B., Lu G. P., Zhang S. B. and Zhao L. D., Lyapunov stability theorem about fractional system
without and with delay, Communications in Nonlinear Science and Numerical Simulation, 20 (3)
(2015) 905-913.
[11] Khargonekar P., Petersen I. and Zhou k., Robust stabilization of uncertain linear systems:quadratic
stabilization and H1 control theory,IEEE Transactions on Automatic Control, 35 (1990) 356-361.
[12] Kilbas A.A., Srivastava H.M. and Trujillo J.J., Theory and Application of Fractional Differential
Equations, Elsevier, New York (2006).
[13] Li M. and Wang J., Exploring delayed mittaglefer type matrix functions to study finite time stability of
fractional delay differential equations. Applied Mathematics and Computation, 324 (2018) 254-265.
[14] Li M. and Wang J., Finite time stability of fractional delay differential equations, Applied Mathematics
Letters, 64 (2017) 170-176.
[15] Jiang B., Gao C. and Xie J., Passivity based sliding mode control of uncertain singular Markovian
jump systems with time-varying delay and nonlinear perturbations, Applied Mathematics and Computation,
271 (2015) 187–200.
[16] Ji Y., Du M. and Guo Y., Stabilization of non-linear fractional-order uncertain systems, Asian Journal
of Control, 20 (2018) 669–677.
[17] Joice Nirmala R. , Balachandran K. and Rodriguez- Germa L. and Trujillo J.J. , Controllability of
nonlinear fractional delay dynamical systems, Reports on Mathematical Physics, 77 (2016) 87–104.
[18] Li H., Wang J., Lam H.K., Zhou Q. and Du H., Adaptive sliding mode control for interval type-2
fuzzy systems, IEEE Transactions on Systems, Man, and Cybernetics, 46 (2016) 1654–1663.
[19] Matignon D., Stability results on fractional differential equations to control processing, Proceedings
Computational Engineering in Systems and Application multiconference, (1996) 963-968.
[20] Mirhosseini-Alizamani S. M., Effati S., and Heydari A., An iterative method for suboptimal control
of a class of nonlinear time-delayed systems, International Journal of Control, 92 (2019) ) 2869-2885.
[21] Mirhosseini-Alizamani S. M., Effati S. and Heydari A., Solution of linear time-varying multi-delay
systems via variational iteration method. Journal of Mathematics and Computer Science, 16 (2016)
282-297.
[22] Mirhosseini-Alizamini S. M. , Effati S. and Heydari A., Solution of linear time-varying multi-delay
systems via variational iteration method, J. Math. Comput. Sci., 16 (2016) 282-297.
[23] Mirhosseini-Alizamini S. M., Numerical Solution of the Controlled Harmonic Oscillator by Homotopy
Perturbation Method, Control Optim. Appl. Math., 2(1) (2017) 77-91.
[24] Mirhosseini-Alizamini S. M., Solving linear optimal control problems of the time-delayed systems
by Adomian decomposition method, Iranian J. Num. Anal. Optim., 9(2) (2019) 165-183.
[25] Monje C.A., Chen Y.Q., Vinagre B.M., Xue D. and Feliu V., Fractional-order systems and controls,
Springer, Newyork, 2010.
[26] Naifar O., Makhlouf A. B. and Hammami M. A., Comments on ‘Lyapunov stability theorem about
fractional system without and with delay, Communications in Nonlinear Science and Numerical Simulation,
30 (2016) 360-361.
[27] Saad W., Anissellami A. and Garcia G., Robust sliding mode- H∞control approach for a class of
nonlinear systems affected by unmatched uncertainties using a poly-quadratic Lyapunov function,
International Journal of Control, Automation and Systems, 14 (2016) 1464–1474.
[28] Sakthivel R., Kaviarasan B., Selvaraj P. and Karimi H.R., EID based sliding mode investment policy
design for fuzzy stochastic jump financial systems, Nonlinear Analysis: Hybrid Systems, 31 (2018)
100–108 .
[29] Shen H., Li F., Cao J., Wu Z.G. and Lu G., Fuzzy-model-based output feedback reliable control for
network-based semi-Markov jump nonlinear systems subject to redundant channels, IEEE Transactions
on Cybernetics (2019).
[30] Shen H., Men Y., Wu Z.G., Cao J. and Lu G., Network-based quantized control for fuzzy singularly
perturbed semi-Markov jump systems and its application, IEEE Transactions on Circuits and Systems
I: Regular Papers, 66 (2019) 1130–1140.
[31] Wang Y. and Li T., Stability analysis of fractional-order nonlinear systems with delay, Mathematical
Problems in Engineering, 2014 (2014) 1–8.
[32] Wen Y., Zhou X., Zhang Z. and Liu S., Lyapunov method for nonlinear fractional differential systems
with delay, Nonlinear Dynamics, 82 (2015) 1015–1025.
[33] Xia Y. and Jia Y., Robust sliding mode control for uncertain time-delay systems: an LMI approach,
IEEE T. Automat. Contr, 48 (2003) 1086–1091.
[34] Yin C., Chen Y. and Zhong S., LMI based design of a sliding mode controller for a class of uncertain
fractional order nonlinear systems, American Control Conference (ACC), (2013) 6526-6531.
[35] Yousefi M., Binazadeh T., Delay-independent sliding mode control of time-delay linear fractional
order systems , Transactions of the Institute of Measurement and Control, (2018) 1212-1222.
[36] Zhang H., Ye R., Cao J., Ahmed A., Li X. and Y. Wan., Lyapunov functional approach to stability
analysis of Riemann-Liouville fractional neural networks with time-varying delays, Asian Journal of
Control, 20 (2018) 1938-1951.

### سابقه مقاله

• تاریخ دریافت: 30 شهریور 1400
• تاریخ بازنگری: 18 خرداد 1401
• تاریخ پذیرش: 18 تیر 1401