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Erschienen in: Soft Computing 12/2020

17.10.2019 | Methodologies and Application

Terminal observer and disturbance observer for the class of fractional-order chaotic systems

verfasst von: Mohammad Reza Soltanpour, Mehrdad Shirkavand

Erschienen in: Soft Computing | Ausgabe 12/2020

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Abstract

In this paper, a terminal fractional-order observer and a terminal disturbance observer is proposed to estimate internal states and external disturbances of the class of fractional-order chaotic systems. The estimation of states within fixed time is achieved by employing a nonlinear feedback in terms of the observer error. The fixed convergence time is not relevant to the initial conditions and can be adjusted to any desired values by tuning the designable parameters. Finally, the numerical simulations are performed on fractional-order chaotic Liu, Chen, and Financial systems to validate the theoretical results. Moreover, some numerical simulations are provided to compare the obtained theoretical results with the other methods in the literature.

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Literatur
Zurück zum Zitat Aghababa MP, Hashtarkhani B (2015) A new adaptive observer design for a class of nonautonomous complex chaotic systems. Complexity 21(2):145–153MathSciNetCrossRef Aghababa MP, Hashtarkhani B (2015) A new adaptive observer design for a class of nonautonomous complex chaotic systems. Complexity 21(2):145–153MathSciNetCrossRef
Zurück zum Zitat Boroujeni EA, Momeni HR (2012) Observer based control of a class of nonlinear fractional-order systems using LMI. Int J Sci Eng Investig 1(1):48–52 Boroujeni EA, Momeni HR (2012) Observer based control of a class of nonlinear fractional-order systems using LMI. Int J Sci Eng Investig 1(1):48–52
Zurück zum Zitat Boulkroune A, Bouzeriba A, Bouden T, Azar AT (2016a) Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. In: Advances in chaos theory and intelligent control. Springer, pp 681–697 Boulkroune A, Bouzeriba A, Bouden T, Azar AT (2016a) Fuzzy adaptive synchronization of uncertain fractional-order chaotic systems. In: Advances in chaos theory and intelligent control. Springer, pp 681–697
Zurück zum Zitat Boulkroune A, Hamel S, Azar AT, Vaidyanathan S (2016b) Fuzzy control-based function synchronization of unknown chaotic systems with dead-zone input. In: Advances in chaos theory and intelligent control. Springer, pp 699-718 Boulkroune A, Hamel S, Azar AT, Vaidyanathan S (2016b) Fuzzy control-based function synchronization of unknown chaotic systems with dead-zone input. In: Advances in chaos theory and intelligent control. Springer, pp 699-718
Zurück zum Zitat Butzer PL, Westphal U (2000) An introduction to fractional calculus. In: Applications of Fractional Calculus in Physics. pp 1-85 Butzer PL, Westphal U (2000) An introduction to fractional calculus. In: Applications of Fractional Calculus in Physics. pp 1-85
Zurück zum Zitat Cafagna D, Grassi G (2012) Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems. Nonlinear Dyn 68(1):117–128MATHCrossRef Cafagna D, Grassi G (2012) Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems. Nonlinear Dyn 68(1):117–128MATHCrossRef
Zurück zum Zitat Chen M, Zhou D, Shang Y (2005) A new observer-based synchronization scheme for private communication. Chaos, Solitons Fractals 24(4):1025–1030MATHCrossRef Chen M, Zhou D, Shang Y (2005) A new observer-based synchronization scheme for private communication. Chaos, Solitons Fractals 24(4):1025–1030MATHCrossRef
Zurück zum Zitat Cheng Z-F, Shi D-P (2010) Chaos in a fractional-order nonlinear financial system. In: Intelligent Systems and Applications (ISA), 2010 2nd International Workshop on, 2010. IEEE, pp 1-3 Cheng Z-F, Shi D-P (2010) Chaos in a fractional-order nonlinear financial system. In: Intelligent Systems and Applications (ISA), 2010 2nd International Workshop on, 2010. IEEE, pp 1-3
Zurück zum Zitat Cruz-Victoria JC, Martínez-Guerra R, Pérez-Pinacho CA, Gómez-Cortés GC (2015) Synchronization of nonlinear fractional order systems by means of PI rα reduced order observer. Appl Math Comput 262:224–231MathSciNetMATH Cruz-Victoria JC, Martínez-Guerra R, Pérez-Pinacho CA, Gómez-Cortés GC (2015) Synchronization of nonlinear fractional order systems by means of PI rα reduced order observer. Appl Math Comput 262:224–231MathSciNetMATH
Zurück zum Zitat Defoort M, Polyakov A, Demesure G, Djemai M, Veluvolu K (2015) Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics. IET Control Theory Appl 9(14):2165–2170MathSciNetCrossRef Defoort M, Polyakov A, Demesure G, Djemai M, Veluvolu K (2015) Leader-follower fixed-time consensus for multi-agent systems with unknown non-linear inherent dynamics. IET Control Theory Appl 9(14):2165–2170MathSciNetCrossRef
Zurück zum Zitat Delavari H, Senejohnny D, Baleanu D (2012) Sliding observer for synchronization of fractional order chaotic systems with mismatched parameter. Open Physics 10(5):1095–1101CrossRef Delavari H, Senejohnny D, Baleanu D (2012) Sliding observer for synchronization of fractional order chaotic systems with mismatched parameter. Open Physics 10(5):1095–1101CrossRef
Zurück zum Zitat Diethelm K (2010) The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer, Diethelm K (2010) The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer,
Zurück zum Zitat Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1–4):3–22MathSciNetMATHCrossRef Diethelm K, Ford NJ, Freed AD (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1–4):3–22MathSciNetMATHCrossRef
Zurück zum Zitat Diethelm K, Ford NJ, Freed AD, Luchko Y (2005) Algorithms for the fractional calculus: a selection of numerical methods. Comput Methods Appl Mech Eng 194(6–8):743–773MathSciNetMATHCrossRef Diethelm K, Ford NJ, Freed AD, Luchko Y (2005) Algorithms for the fractional calculus: a selection of numerical methods. Comput Methods Appl Mech Eng 194(6–8):743–773MathSciNetMATHCrossRef
Zurück zum Zitat Djeghali N, Djennoune S, Bettayeb M, Ghanes M, Barbot J-P (2016) Observation and sliding mode observer for nonlinear fractional-order system with unknown input. ISA Trans 63:1–10CrossRef Djeghali N, Djennoune S, Bettayeb M, Ghanes M, Barbot J-P (2016) Observation and sliding mode observer for nonlinear fractional-order system with unknown input. ISA Trans 63:1–10CrossRef
Zurück zum Zitat Filali RL, Benrejeb M, Borne P (2014) On observer-based secure communication design using discrete-time hyperchaotic systems. Commun Nonlinear Sci Numer Simul 19(5):1424–1432MathSciNetMATHCrossRef Filali RL, Benrejeb M, Borne P (2014) On observer-based secure communication design using discrete-time hyperchaotic systems. Commun Nonlinear Sci Numer Simul 19(5):1424–1432MathSciNetMATHCrossRef
Zurück zum Zitat Guce IK (2013) On fractional derivatives: the non-integer order of the derivative. International Journal of Scientific & Engineering, Research 4(3):1 Guce IK (2013) On fractional derivatives: the non-integer order of the derivative. International Journal of Scientific & Engineering, Research 4(3):1
Zurück zum Zitat Hardy G, Littlewood J, Polya G (1952) Inequalities. Press, Cambridge UMATH Hardy G, Littlewood J, Polya G (1952) Inequalities. Press, Cambridge UMATH
Zurück zum Zitat Hilfer R (2000) Applications of fractional calculus in physics. World Scientific, Hilfer R (2000) Applications of fractional calculus in physics. World Scientific,
Zurück zum Zitat Jun-Jie L, Chong-Xin L (2007) Realization of fractional-order Liu chaotic system by circuit. Chin Phys 16(6):1586CrossRef Jun-Jie L, Chong-Xin L (2007) Realization of fractional-order Liu chaotic system by circuit. Chin Phys 16(6):1586CrossRef
Zurück zum Zitat Khan H, Abou SC, Sepehri N (2005) Nonlinear observer-based fault detection technique for electro-hydraulic servo-positioning systems. Mechatronics 15(9):1037–1059CrossRef Khan H, Abou SC, Sepehri N (2005) Nonlinear observer-based fault detection technique for electro-hydraulic servo-positioning systems. Mechatronics 15(9):1037–1059CrossRef
Zurück zum Zitat Kilbas A, Srivastava H, Trujillo J (2006) Theory and applications of fractional differential equations Elsevier. Amsterdam, Kilbas A, Srivastava H, Trujillo J (2006) Theory and applications of fractional differential equations Elsevier. Amsterdam,
Zurück zum Zitat Laghrouche S, Liu J, Ahmed FS, Harmouche M, Wack M (2015) Adaptive second-order sliding mode observer-based fault reconstruction for PEM fuel cell air-feed system. IEEE Trans Control Syst Technol 23(3):1098–1109CrossRef Laghrouche S, Liu J, Ahmed FS, Harmouche M, Wack M (2015) Adaptive second-order sliding mode observer-based fault reconstruction for PEM fuel cell air-feed system. IEEE Trans Control Syst Technol 23(3):1098–1109CrossRef
Zurück zum Zitat Lan Y-H, Zhou Y (2013) Non-fragile observer-based robust control for a class of fractional-order nonlinear systems. Syst Control Lett 62(12):1143–1150MathSciNetMATHCrossRef Lan Y-H, Zhou Y (2013) Non-fragile observer-based robust control for a class of fractional-order nonlinear systems. Syst Control Lett 62(12):1143–1150MathSciNetMATHCrossRef
Zurück zum Zitat Lan Y-H, Huang H-X, Zhou Y (2012) Observer-based robust control of a (1 ≤ a < 2) fractional-order uncertain systems: a linear matrix inequality approach. IET Control Theory Appl 6(2):229–234MathSciNetCrossRef Lan Y-H, Huang H-X, Zhou Y (2012) Observer-based robust control of a (1 ≤ a < 2) fractional-order uncertain systems: a linear matrix inequality approach. IET Control Theory Appl 6(2):229–234MathSciNetCrossRef
Zurück zum Zitat Lan Y-H, Gu H-B, Chen C-X, Zhou Y, Luo Y-P (2014) An indirect Lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks. Neurocomputing 136:235–242CrossRef Lan Y-H, Gu H-B, Chen C-X, Zhou Y, Luo Y-P (2014) An indirect Lyapunov approach to the observer-based robust control for fractional-order complex dynamic networks. Neurocomputing 136:235–242CrossRef
Zurück zum Zitat Li C, Deng WJAM, Computation (2007) Remarks on fractional derivatives. 187 (2):777-784 Li C, Deng WJAM, Computation (2007) Remarks on fractional derivatives. 187 (2):777-784
Zurück zum Zitat Li H, Gao Y, Shi P, Lam H-K (2016a) Observer-based fault detection for nonlinear systems with sensor fault and limited communication capacity. IEEE Trans Autom Control 61(9):2745–2751MathSciNetMATHCrossRef Li H, Gao Y, Shi P, Lam H-K (2016a) Observer-based fault detection for nonlinear systems with sensor fault and limited communication capacity. IEEE Trans Autom Control 61(9):2745–2751MathSciNetMATHCrossRef
Zurück zum Zitat Li L, Ding SX, Qiu J, Yang Y, Zhang Y (2016b) Weighted fuzzy observer-based fault detection approach for discrete-time nonlinear systems via piecewise-fuzzy Lyapunov functions. IEEE Trans Fuzzy Syst 24(6):1320–1333CrossRef Li L, Ding SX, Qiu J, Yang Y, Zhang Y (2016b) Weighted fuzzy observer-based fault detection approach for discrete-time nonlinear systems via piecewise-fuzzy Lyapunov functions. IEEE Trans Fuzzy Syst 24(6):1320–1333CrossRef
Zurück zum Zitat Luenberger DG (1964) Observing the state of a linear system. IEEE transactions on military electronics 8(2):74–80CrossRef Luenberger DG (1964) Observing the state of a linear system. IEEE transactions on military electronics 8(2):74–80CrossRef
Zurück zum Zitat Matignon D, d’Andréa-Novel B Some results on controllability and observability of finite-dimensional fractional differential systems. In: Computational engineering in systems applications, 1996. Citeseer, pp 952-956 Matignon D, d’Andréa-Novel B Some results on controllability and observability of finite-dimensional fractional differential systems. In: Computational engineering in systems applications, 1996. Citeseer, pp 952-956
Zurück zum Zitat N’Doye I, Darouach M, Voos H Observer-based approach for fractional-order chaotic synchronization and communication. In: 2013 European Control Conference (ECC), 2013a. IEEE, pp 4281-4286 N’Doye I, Darouach M, Voos H Observer-based approach for fractional-order chaotic synchronization and communication. In: 2013 European Control Conference (ECC), 2013a. IEEE, pp 4281-4286
Zurück zum Zitat N’Doye I, Voos H, Darouach M (2013) Observer-based approach for fractional-order chaotic synchronization and secure communication. IEEE Journal on Emerging and Selected Topics in Circuits and Systems 3(3):442–450CrossRef N’Doye I, Voos H, Darouach M (2013) Observer-based approach for fractional-order chaotic synchronization and secure communication. IEEE Journal on Emerging and Selected Topics in Circuits and Systems 3(3):442–450CrossRef
Zurück zum Zitat Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic press, Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, vol 198. Academic press,
Zurück zum Zitat Polyakov A (2012) Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans Autom Control 57(8):2106–2110MathSciNetMATHCrossRef Polyakov A (2012) Nonlinear feedback design for fixed-time stabilization of linear control systems. IEEE Trans Autom Control 57(8):2106–2110MathSciNetMATHCrossRef
Zurück zum Zitat Polyakov A Fixed-time stabilization of linear systems via sliding mode control. In: Variable Structure Systems (VSS), 2012 12th International Workshop on, 2012a. IEEE, pp 1-6 Polyakov A Fixed-time stabilization of linear systems via sliding mode control. In: Variable Structure Systems (VSS), 2012 12th International Workshop on, 2012a. IEEE, pp 1-6
Zurück zum Zitat Rahme S, Meskin N (2015) Adaptive sliding mode observer for sensor fault diagnosis of an industrial gas turbine. Control Engineering Practice 38:57–74CrossRef Rahme S, Meskin N (2015) Adaptive sliding mode observer for sensor fault diagnosis of an industrial gas turbine. Control Engineering Practice 38:57–74CrossRef
Zurück zum Zitat Senejohnny DM, Delavari H (2012) Active sliding observer scheme based fractional chaos synchronization. Commun Nonlinear Sci Numer Simul 17(11):4373–4383MathSciNetMATHCrossRef Senejohnny DM, Delavari H (2012) Active sliding observer scheme based fractional chaos synchronization. Commun Nonlinear Sci Numer Simul 17(11):4373–4383MathSciNetMATHCrossRef
Zurück zum Zitat Shao S, Wheeler PW, Clare JC, Watson AJ (2013) Fault detection for modular multilevel converters based on sliding mode observer. IEEE Trans Power Electron 28(11):4867–4872CrossRef Shao S, Wheeler PW, Clare JC, Watson AJ (2013) Fault detection for modular multilevel converters based on sliding mode observer. IEEE Trans Power Electron 28(11):4867–4872CrossRef
Zurück zum Zitat Shao S, Chen M, Yan X (2016) Adaptive sliding mode synchronization for a class of fractional-order chaotic systems with disturbance. Nonlinear Dyn 83(4):1855–1866MathSciNetMATHCrossRef Shao S, Chen M, Yan X (2016) Adaptive sliding mode synchronization for a class of fractional-order chaotic systems with disturbance. Nonlinear Dyn 83(4):1855–1866MathSciNetMATHCrossRef
Zurück zum Zitat Slotine J, Li W (1998) Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, 1991. Google Scholar Slotine J, Li W (1998) Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, 1991. Google Scholar
Zurück zum Zitat Smith AH, Monti A, Ponci F (2007) Indirect measurements via a polynomial chaos observer. IEEE Trans Instrum Meas 56(3):743–752CrossRef Smith AH, Monti A, Ponci F (2007) Indirect measurements via a polynomial chaos observer. IEEE Trans Instrum Meas 56(3):743–752CrossRef
Zurück zum Zitat Tornambè A (1992) Asymptotic observers for non-linear systems. Int J Syst Sci 23(3):435–442MATHCrossRef Tornambè A (1992) Asymptotic observers for non-linear systems. Int J Syst Sci 23(3):435–442MATHCrossRef
Zurück zum Zitat Wang H, Zhu X-J, Gao S-W, Chen Z-Y (2011) Singular observer approach for chaotic synchronization and private communication. Commun Nonlinear Sci Numer Simul 16(3):1517–1523CrossRef Wang H, Zhu X-J, Gao S-W, Chen Z-Y (2011) Singular observer approach for chaotic synchronization and private communication. Commun Nonlinear Sci Numer Simul 16(3):1517–1523CrossRef
Zurück zum Zitat Wiener N (1949) Extrapolation, interpolation, and smoothing of stationary time series, vol 7. MIT press Cambridge, MAMATHCrossRef Wiener N (1949) Extrapolation, interpolation, and smoothing of stationary time series, vol 7. MIT press Cambridge, MAMATHCrossRef
Zurück zum Zitat Yao J, Jiao Z, Ma D (2014) Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE Trans Industr Electron 61(11):6285–6293CrossRef Yao J, Jiao Z, Ma D (2014) Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping. IEEE Trans Industr Electron 61(11):6285–6293CrossRef
Zurück zum Zitat Yu J, Ma Y, Yu H, Lin C (2016) Reduced-order observer-based adaptive fuzzy tracking control for chaotic permanent magnet synchronous motors. Neurocomputing 214:201–209CrossRef Yu J, Ma Y, Yu H, Lin C (2016) Reduced-order observer-based adaptive fuzzy tracking control for chaotic permanent magnet synchronous motors. Neurocomputing 214:201–209CrossRef
Zurück zum Zitat Zhang R, Gong J (2014) Synchronization of the fractional-order chaotic system via adaptive observer. Systems Science & Control Engineering: An Open Access Journal 2(1):751–754CrossRef Zhang R, Gong J (2014) Synchronization of the fractional-order chaotic system via adaptive observer. Systems Science & Control Engineering: An Open Access Journal 2(1):751–754CrossRef
Zurück zum Zitat Zuo Z, Tie L (2016) Distributed robust finite-time nonlinear consensus protocols for multi-agent systems. Int J Syst Sci 47(6):1366–1375MathSciNetMATHCrossRef Zuo Z, Tie L (2016) Distributed robust finite-time nonlinear consensus protocols for multi-agent systems. Int J Syst Sci 47(6):1366–1375MathSciNetMATHCrossRef
Metadaten
Titel
Terminal observer and disturbance observer for the class of fractional-order chaotic systems
verfasst von
Mohammad Reza Soltanpour
Mehrdad Shirkavand
Publikationsdatum
17.10.2019
Verlag
Springer Berlin Heidelberg
Erschienen in
Soft Computing / Ausgabe 12/2020
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-019-04418-0

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