In this paper, a novel fractional-order (FO) backstepping sliding-mode control is proposed for a class of FO nonlinear systems with mismatched disturbances. Here the matched/mismatched disturbances are estimated by an FO nonlinear disturbance observer (NDO). This FO NDO is proposed based on FO backstepping algorithm to estimate the mismatched disturbances. The stability of the closed-loop system is proved by the new extension of Lyapunov direct method for FO systems. Exponential reaching law considerably decreases the chattering and provides a high dynamic tracking performance. Finally, three simulation examples are presented to show the features and the effectiveness of the proposed method. Results show that this observer approximates the unknown mismatched disturbances successfully.

References

References
1.
Machado
,
J. T.
,
Kiryakova
,
V.
, and
Mainardi
,
F.
,
2011
, “
Recent History of Fractional Calculus
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
3
), pp.
1140
1153
.
2.
Moreles
,
M. A.
, and
Lainez
,
R.
,
2017
, “
Mathematical Modelling of Fractional Order Circuit Elements and Bioimpedance Applications
,”
Commun. Nonlinear Sci. Numer. Simul.
,
46
, pp.
81
88
.
3.
Lazopoulos
,
K. A.
,
Karaoulanis
,
D.
, and
Lazopoulos
,
Α. Κ.
,
2016
, “
On Fractional Modelling of Viscoelastic Mechanical Systems
,”
Mech. Res. Commun.
,
78
(
Pt. A
), pp.
1
5
.
4.
Antonopoulos
,
C. S.
,
Kantartzis
,
N.
, and
Rekanos
,
I. T.
,
2017
, “
FDTD Method for Wave Propagation in Havriliak-Negami Media Based on Fractional Derivative Approximation
,”
IEEE Trans. Magn.
,
53
(
6
), pp.
1
4
.
5.
Ameen
,
I.
, and
Novatib
,
P.
,
2017
, “
The Solution of Fractional Order Epidemic Model by Implicit Adams Methods
,”
Appl. Math. Modell.
,
43
, pp.
78
84
.
6.
Babiarz
,
A.
,
Czornik
,
A.
,
Klamka
,
J.
, and
Niezabitowski
,
M.
,
2017
, “
Theory and Applications of Non-Integer Order Systems
,”
Lecture Notes Electrical Engineering
, Vol.
407
,
Springer
,
New York
.
7.
Yin
,
C.
,
Chen
,
Y. Q.
, and
Zhong
,
S. M.
,
2014
, “
Fractional-Order Sliding Mode Based Extremum Seeking Control of a Class of Nonlinear Systems
,”
Automatica
,
50
(
12
), pp.
3173
3181
.
8.
Delavari
,
H.
,
Baleanu
,
D.
, and
Sadati
,
S. J.
,
2012
, “
Stability Analysis of Caputo Fractional-Order Nonlinear Systems Revisited
,”
Nonlinear Dyn.
,
67
(
4
), pp.
2433
2439
.
9.
Duarte-Mermoud
,
M. A.
,
Aguila-Camacho
,
N.
,
Gallegos
,
J. A.
, and
Ma
,
X. Y.
,
2015
, “
Using General Quadratic Lyapunov Functions to Prove Lyapunov Uniform Stability for Fractional Order Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
22
(
1–3
), pp.
650
659
.
10.
Li
,
Y.
,
Chen
,
Y. Q.
, and
Podlubny
,
I.
,
2010
, “
Stability of Fractional-Order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag Leffler Stability
,”
Comput. Math. Appl.
,
59
(
5
), pp.
1810
1821
.
11.
Tavakoli
,
M.
, and
Tabatabaei
,
M.
,
2017
, “
Controllability and Observability Analysis of Continuous-Time Multi-Order Fractional Systems
,”
Multidimens. Syst. Signal Process.
,
28
(
2
), pp.
427
450
.
12.
Tseng
,
C. C.
, and
Lee
,
S. L.
,
2017
, “
Closed-Form Designs of Digital Fractional Order Butterworth Filters Using Discrete Transforms
,”
Signal Process.
,
137
, pp.
80
97
.
13.
Wei
,
Y. H.
,
Peter
,
W. T.
,
Du
,
B.
, and
Wang
,
Y.
,
2016
, “
An Innovative Fixed-Pole Numerical Approximation for Fractional Order Systems
,”
ISA Trans.
,
62
, pp.
94
102
.
14.
Tang
,
Y.
,
Li
,
N.
,
Liu
,
M.
,
Lu
,
Y.
, and
Wang
,
W.
,
2017
, “
Identification of Fractional-Order Systems With Time Delays Using Block Pulse Functions
,”
Mech. Syst. Signal Process.
,
91
, pp.
382
394
.
15.
Tepljakov
,
A.
,
2017
,
Fractional-Order Modeling and Control of Dynamic Systems
,
1st ed.
,
Springer
,
New York
.
16.
Delavari
,
H.
,
Heydarinejad
,
H.
, and
Baleanu
,
D.
, 2018, “
Adaptive Fractional Order Blood Glucose Regulator Based on High Order Sliding Mode Observer
,”
IET Syst. Biol.
, (epub).
17.
Aghababa
,
M. P.
,
2013
, “
No-Chatter Variable Structure Control for Fractional Nonlinear Complex Systems
,”
Nonlinear Dyn.
,
73
(
4
), pp.
2329
2342
.
18.
Delavari
,
H.
,
2017
, “
A Novel Fractional Adaptive Active Sliding Mode Controller for Synchronization of Non-Identical Chaotic Systems With Disturbance and Uncertainty
,”
Int. J. Dyn. Control
,
5
(
1
), pp.
102
114
.
19.
Delavari
,
H.
,
Senejohnny
,
D.
, and
Baleanu
,
D.
,
2012
, “
Sliding Observer for Synchronization of Fractional Order Chaotic Systems With Mismatched Parameter
,”
Open Phys.
,
10
(
5
), pp.
1095
1101
.https://www.degruyter.com/view/j/phys.2012.10.issue-5/s11534-012-0073-4/s11534-012-0073-4.xml
20.
Mobayen
,
S.
,
2014
, “
Design of CNF-Based Nonlinear Integral Sliding Surface for Matched Uncertain Linear Systems With Multiple State-Delays
,”
Nonlinear Dyn.
,
77
(
3
), pp.
1047
1054
.
21.
Jakovljević
,
B.
,
Pisano
,
A.
,
Rapaić
,
M. R.
, and
Usai
,
E.
,
2016
, “
On the Sliding-Mode Control of Fractional-Order Nonlinear Uncertain Dynamics
,”
Int. J. Robust Nonlinear Control
,
26
(
4
), pp.
782
798
.
22.
Aghababa
,
M. P.
,
2013
, “
A Novel Terminal Sliding Mode Controller for a Class of Non-Autonomous Fractional-Order Systems
,”
Nonlinear Dyn.
,
73
(
1–2
), pp.
679
688
.
23.
Yina
,
C.
,
Dadras
,
S.
,
Zhong
,
S. M.
, and
Chen
,
Y. Q.
,
2013
, “
Control of a Novel Class of Fractional-Order Chaotic Systems Via Adaptive Sliding Mode Control Approach
,”
Appl. Math. Modell.
,
37
(
4
), pp.
2469
2483
.
24.
Pashaei
,
S.
, and
Badamchizadeh
,
M.
,
2016
, “
A New Fractional-Order Sliding Mode Controller Via a Nonlinear Disturbance Observer for a Class of Dynamical Systems With Mismatched Disturbances
,”
ISA Trans.
,
63
, pp.
39
48
.
25.
Gao
,
Z.
, and
Liao
,
X. Z.
,
2013
, “
Integral Sliding Mode Control for Fractional Order Systems With Mismatched Uncertainties
,”
Nonlinear Dyn.
,
72
(
1–2
), pp.
27
35
.
26.
Yang
,
J.
, and
Zheng
,
W. X.
,
2014
, “
Offset-Free Nonlinear MPC for Mismatched Disturbance Attenuation With Application to a Static Var Compensator
,”
IEEE Trans. Circuits Syst. II: Express Briefs
,
61
(
1
), pp.
49
53
.
27.
Wang
,
J.
,
Li
,
S.
,
Yang
,
J.
,
Wu
,
B.
, and
Li
,
Q.
,
2015
, “
Extended State Observer-Based Sliding Mode Control for PWM-Based DC–DC Buck Power Converter Systems With Mismatched Disturbances
,”
IET Control Theory Appl.
,
9
(
4
), pp.
579
586
.
28.
Wei
,
Y.
,
Chen
,
Y.
,
Liang
,
S.
, and
Wang
,
Y.
,
2015
, “
A Novel Algorithm on Adaptive Backstepping Control of Fractional Order Systems
,”
Neurocomputing
,
165
, pp.
395
402
.
29.
Wei
,
Y.
,
Peter
,
W. T.
,
Yao
,
Z.
, and
Wang
,
Y.
,
2016
, “
Adaptive Backstepping Output Feedback Control for a Class of Nonlinear Fractional Order Systems
,”
Nonlinear Dyn.
,
86
(
2
), pp.
1047
1056
.
30.
Sheng
,
D.
,
Wei
,
Y.
,
Cheng
,
S.
, and
Shuai
,
J.
,
2017
, “
Adaptive Backstepping Control for Fractional Order Systems With Input Saturation
,”
J. Franklin Inst.
,
354
(
5
), pp.
2245
2268
.
31.
Chen
,
W. H.
,
2004
, “
Disturbance Observer Based Control for Nonlinear Systems
,”
IEEE/ASME Trans. Mechatronics
,
9
(
4
), pp.
706
710
.
32.
Chen
,
W. H.
,
Ballanc
,
D. J.
,
Gawthrop
,
P. J.
, and
O’Reilly
,
J.
,
2000
, “
A Nonlinear Disturbance Observer for Robotic Manipulators
,”
IEEE Trans. Ind. Electron.
,
47
(
4
), pp.
932
938
.
33.
Yang
,
J.
,
Chen
,
W. H.
, and
Li
,
S.
,
2011
, “
Non-Linear Disturbance Observer-Based Robust Control for Systems With Mismatched Disturbances/Uncertainties
,”
IET Control Theory Appl.
,
5
(
18
), pp.
2053
2062
.
34.
Ginoya
,
D.
,
Shendge
,
P. D.
, and
Phadke
,
S. B.
,
2015
, “
Disturbance Observer Based Sliding Mode Control of Nonlinear Mismatched Uncertain Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
26
(
1–3
), pp.
89
107
.https://www.sciencedirect.com/science/article/pii/S1007570415000519
35.
Taghia
,
J.
,
Wang
,
X.
,
Lam
,
S.
, and
Katupitiya
,
J.
,
2017
, “
A Sliding Mode Controller With a Nonlinear Disturbance Observer for a Farm Vehicle Operating in the Presence of Wheel Slip
,”
Auton. Rob.
,
41
(
1
), pp.
71
88
.
36.
Monje
,
C. A.
,
Chen
,
Y.
,
Vinagre
,
B. M.
,
Xue
,
D.
, and
Feliu
,
V.
,
2010
,
Fractional-Order Systems and Controls
,
1st ed.
,
Springer
,
London
.
37.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA
.
38.
Li
,
Y.
,
Chen
,
Y. Q.
, and
Podlubny
,
I.
,
2009
, “
Mittag–Leffler Stability of Fractional Order Nonlinear Dynamic Systems
,”
Automatica
,
45
(
8
), pp.
1965
1969
.
39.
Li
,
C.
, and
Deng
,
W.
,
2007
, “
Remarks on Fractional Derivatives
,”
Appl. Math. Comput.
,
187
(
2
), pp.
777
784
.https://www.sciencedirect.com/science/article/abs/pii/S0096300306012094
40.
Aguila-Camacho
,
N.
,
Duarte-Mermoud
,
M. A.
, and
Gallegos
,
J. A.
,
2014
, “
Lyapunov Functions for Fractional Order Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
19
(
9
), pp.
2951
2957
.
41.
Shao
,
S.
,
Chen
,
M.
,
Chen
,
S.
, and
Wu
,
Q.
,
2016
, “
Adaptive Neural Control for an Uncertain Fractional-Order Rotational Mechanical System Using Disturbance Observer
,”
IET Control Theory Appl.
,
10
(
16
), pp.
1972
1980
.
42.
Singh
,
V. K.
, and
Kumar
,
V.
,
2014
, “
Nonlinear Design for Inverted Pendulum Using Backstepping Control Technique
,”
Int. J. Sci. Res. Eng. Technol.
,
2
(
11
), pp.
807
810
.http://www.ijsret.org/pdf/120469.pdf
43.
Afshari
,
M.
,
Mobayen
,
S.
,
Hajmohammadi
,
R.
, and
Baleanu
,
D.
,
2018
, “
Global Sliding Mode Control Via Linear Matrix Inequality Approach for Uncertain Chaotic Systems With Input Nonlinearities and Multiple Delays
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
3
), pp.
1
14
.
44.
Faieghi
,
M. R.
, and
Delavari
,
H.
,
2012
, “
Chaos in Fractional-Order Genesio–Tesi System and Its Synchronization
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
(
2
), pp.
731
741
.
45.
Fallaha
,
C. J.
,
Saad
,
M.
,
Kanaan
,
H. Y.
, and
Al-Haddad
,
K.
,
2011
, “
Sliding-Mode Robot Control With Exponential Reaching Law
,”
IEEE Trans. Ind. Electron.
,
58
(
2
), pp.
600
610
.
46.
Delavari
,
H.
,
Asadbeigi
,
A.
, and
Heydarnia
,
O.
,
2015
, “
Synchronization of Micro-Electro-Mechanical-Systems in Finite Time
,”
Discontinuity Nonlinearity Complexity
,
4
(
2
), pp.
173
185
.
47.
Valério
,
D.
, and
Sáda Costa
,
J.
,
2004
, “
Ninteger: A Non-Integer Control Toolbox for Matlab
,” Proceedings of the Fractional Differentiation and its Applications, Bordeaux, France.
48.
Luo
,
R.
, and
Zeng
,
Y.
,
2017
, “
The Control and Synchronization of Fractional-Order Genesio–Tesi System
,”
Nonlinear Dyn.
,
88
(
3
), pp.
2111
2121
.
49.
Aghababa
,
M. P.
,
2012
, “
Chaos in a Fractional-Order Micro-Electro-Mechanical Resonator and Its Suppression
,”
Chin. Phys. B
,
21
(
10
), pp.
1
9
.http://iopscience.iop.org/article/10.1088/1674-1056/21/10/100505/meta
You do not currently have access to this content.