The problem of stabilization of nonlinear fractional systems in spite of system uncertainties is investigated in this paper. First, a proper fractional derivative type sliding manifold with desired stability and convergence properties is designed. Then, the fractional stability theory is adopted to derive a robust sliding control law to force the system trajectories to attain the proposed sliding manifold and remain on it evermore. The existence of the sliding motion is mathematically proven. Furthermore, the sign function in the control input, which is responsible to the being of harmful chattering, is transferred into the fractional derivative of the control input. Therefore, the resulted control input becomes smooth and free of the chattering. Some numerical simulations are presented to illustrate the efficient performance of the proposed chattering-free fractional variable structure controller.

References

References
1.
Müller
,
S.
,
Kästner
,
M.
,
Brummund
,
J.
, and
Ulbricht
,
V.
,
2011
, “
A Nonlinear Fractional Viscoelastic Material Model for Polymers
,”
Comput. Mater. Sci.
,
50
, pp.
2938
2949
.10.1016/j.commatsci.2011.05.011
2.
Jesus
,
I. S.
, and
Machado
,
J. A. T.
,
2009
, “
Development of Fractional Order Capacitors Based on Electrolyte Processes
,”
Nonlinear Dyn.
,
56
, pp.
45
55
.10.1007/s11071-008-9377-8
3.
Aghababa
,
M. P.
, and
Aghababa
,
H. P.
,
2013
, “
The Rich Dynamics of Fractional-Order Gyros Applying a Fractional Controller
,”
Proc. Inst. Mech. Eng., Part I: J. Syst. Control Eng.
,
227
, pp.
588
601
.10.1177/0959651813492326
4.
Rivero
,
M.
,
Trujillo
,
J. J.
,
Vàzquez
,
L.
, and
Velasco
,
M. P.
,
2011
, “
Fractional Dynamics of Populations
,”
Appl. Math. Comput.
,
218
, pp.
1089
1095
.10.1016/j.amc.2011.03.017
5.
Aghababa
,
M. P.
,
2012
, “
Chaos in a Fractional–Order Micro–Electro–Mechanical Resonator and its Suppression
,”
Chin. Phys. B
,
21
,
100505
.10.1088/1674-1056/21/10/100505
6.
Dadras
,
S.
, and
Momeni
,
H. R.
,
2010
, “
Control of a Fractional-Order Economical System via Sliding Mode
,”
Phys. A
,
389
, pp.
2434
2442
.10.1016/j.physa.2010.02.025
7.
Hamamci
,
S. E.
,
2008
, “
Stabilization Using Fractional-Order PI and PID Controllers
,”
Nonlinear Dyn.
,
51
, pp.
329
343
.10.1007/s11071-007-9214-5
8.
Tatar
,
N.
,
2010
, “
On a Boundary Controller of Fractional Type
,”
Nonlinear Anal. Theory, Methods Appl.
,
72
, pp.
3209
3215
.10.1016/j.na.2009.12.017
9.
Orsoni
,
B.
,
Melchior
,
P.
, and
Oustaloup
,
A.
,
2002
, “
Fractional Motion Control: Application to an XY Cutting Table
,”
Nonlinear Dyn.
,
29
, pp.
297
314
.10.1023/A:1016561916189
10.
Manabe
,
S.
,
2002
, “
A Suggestion of Fractional-Order Controller for Flexible Spacecraft Attitude Control
,”
Nonlinear Dyn.
,
29
, pp.
251
268
.10.1023/A:1016566017098
11.
Ding
,
Y.
,
Wang
,
Z.
, and
Ye
,
H.
,
2012
, “
Optimal Control of a Fractional-Order HIV-Immune System With Memory
,”
IEEE Trans. Control Syst. Tech.
,
20
, pp.
763
769
.10.1109/TCST.2011.2153203
12.
Ozbay
,
H.
,
Bonnet
,
C.
, and
Fioravanti
,
A. R.
,
2012
, “
PID Controller Design for Fractional-Order Systems With Time Delays
,”
Syst. Control Lett.
,
61
, pp.
18
23
.10.1016/j.sysconle.2011.09.011
13.
Aghababa
,
M. P.
, and
Akbari
,
M. E.
,
2012
, “
A Chattering-Free Robust Adaptive Sliding Mode Controller for Synchronization of Two Different Chaotic Systems With Unknown Uncertainties and External Disturbances
,”
Appl. Math. Comput.
,
218
, pp.
5757
5768
.10.1016/j.amc.2011.11.080
14.
Aghababa
,
M. P.
, and
Aghababa
,
H. P.
,
2012
, “
A Novel Finite-Time Sliding Mode Controller Applied to Synchronize Chaotic Systems With Input Nonlinearity
,”
Arabian J. Sci. Eng.
,
38
, pp.
3221
3232
.10.1007/s13369-012-0459-z
15.
Aghababa
,
M. P.
,
2013
, “
A Novel Terminal Sliding Mode Controller for a Class of Non-Autonomous Fractional-Order Systems
,”
Nonlinear Dyn.
,
73
, pp.
679
688
.10.1007/s11071-013-0822-y
16.
Tavazoei
,
M. S.
, and
Haeri
,
M.
,
2008
, “
Synchronization of Chaotic Fractional-Order Systems via Active Sliding Mode Controller
,”
Phys. A
,
387
, pp.
57
70
.10.1016/j.physa.2007.08.039
17.
Wang
,
X.
,
Zhang
,
X.
, and
Ma
,
C.
,
2012
, “
Modified Projective Synchronization of Fractional-Order Chaotic Systems via Active Sliding Mode Control
,”
Nonlinear Dyn.
,
69
, pp.
511
517
.10.1007/s11071-011-0282-1
18.
Aghababa
,
M. P.
,
2012
, “
Robust Finite-Time Stabilization of Fractional-Order Chaotic Systems Based on Fractional Lyapunov Stability Theory
,”
ASME J. Comput. Nonlinear Dyn.
,
7
,
021010
.10.1115/1.4005323
19.
Aghababa
,
M. P.
,
2012
, “
Robust Stabilization and Synchronization of a Class of Fractional-Order Chaotic Systems via a Novel Fractional Sliding Mode Controller
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
, pp.
2670
2681
.10.1016/j.cnsns.2011.10.028
20.
Aghababa
,
M. P.
,
2012
, “
Finite-Time Chaos Control and Synchronization of Fractional-Order Chaotic (Hyperchaotic) Systems via Fractional Nonsingular Terminal Sliding Mode Technique
,”
Nonlinear Dyn.
,
69
, pp.
247
261
.10.1007/s11071-011-0261-6
21.
Aghababa
,
M. P.
,
2013
, “
No-Chatter Variable Structure Control for Fractional Nonlinear Complex Systems
,”
Nonlinear Dyn.
,
73
, pp.
2329
2342
.10.1007/s11071-013-0944-2
22.
Yin
,
C.
,
Zhong
,
S.
, and
Chen
,
W.
,
2012
, “
Design of Sliding Mode Controller for a Class of Fractional-Order Chaotic Systems
,”
Commun. Nonlinear Sci. Numer. Simul.
,
17
, pp.
356
366
.10.1016/j.cnsns.2011.04.024
23.
Chen
,
D.
,
Liu
,
Y.
,
Ma
,
X.
, and
Zhang
,
R.
,
2012
, “
Control of a Class of Fractional-Order Chaotic Systems via Sliding Mode
,”
Nonlinear Dyn.
,
67
, pp.
893
901
.10.1007/s11071-011-0002-x
24.
Chen
,
D.
,
Li
,
Y.
,
Ma
,
X.
, and
Zhang
,
R.
,
2011
, “
No-Chattering Sliding Mode Control in a Class of Fractional-Order Chaotic Systems
,”
Chin. Phys. B
,
20
,
120506
.10.1088/1674-1056/20/12/120506
25.
Zhang
,
R.
, and
Yang
,
S.
,
2011
, “
Adaptive Synchronization of Fractional-Order Chaotic Systems via a Single Driving Variable
,”
Nonlinear Dyn.
,
66
, pp.
831
837
.10.1007/s11071-011-9944-2
26.
Qi
,
D.
,
Wang
,
Q.
, and
Yang
,
J.
,
2011
, “
Comparison Between Two Different Sliding Mode Controllers for a Fractional-Order Unified Chaotic System
,”
Chin. Phys. B
,
20
,
100505
.10.1088/1674-1056/20/10/100505
27.
Balochian
,
S.
,
2013
, “
Sliding Mode Control of Fractional Order Nonlinear Differential Inclusion Systems
,”
Evolving Syst.
,
4
, pp.
145
152
.10.1007/s12530-012-9048-3
28.
Zhang
,
B.
,
Pi
,
Y.
, and
Luo
,
Y.
,
2012
, “
Fractional Order Sliding-Mode Control Based on Parameters Auto-Tuning for Velocity Control of Permanent Magnet Synchronous Motor
,”
ISA Trans.
,
51
, pp.
649
656
.10.1016/j.isatra.2012.04.006
29.
Lee
,
H.
, and
Utkin
,
V. I.
,
2007
, “
Chattering Suppression Methods in Sliding Mode Control Systems
,”
Annu. Rev. Control
,
31
, pp.
179
188
.10.1016/j.arcontrol.2007.08.001
30.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic
,
New York
.
31.
Shuqin
,
Z.
,
2009
, “
Monotone Iterative Method for Initial Value Problem Involving Riemann–Liouville Fractional Derivatives
,”
Nonlinear Anal. Theory, Methods Appl.
,
71
, pp.
2087
2093
.10.1016/j.na.2009.01.043
32.
Matignon
,
D.
,
1996
, “
Stability Results of Fractional Differential Equations With Applications to Control Processing
,”
Proceedings of the Computational Engineering in Systems and Application Multiconference
,
IMACS
,
Lille, France
, pp.
963
968
.
33.
Hu
,
J. B.
,
Han
,
Y.
, and
Zhao
,
L. D.
,
2009
, “
A Novel Stability Theorem for Fractional Systems and its Applying in Synchronizing Fractional Chaotic System Based on Back-Stepping Approach
,”
Acta. Phys. Sin.
,
58
, pp.
2235
2239
.
34.
Efe
,
M. O.
, and
Kasnakoglu
,
C.
,
2008
, “
A Fractional Adaptation Law for Sliding Mode Control
,”
Int. J. Adapt. Control Signal Process.
,
22
, pp.
968
986
.10.1002/acs.1062
35.
Utkin
,
V. I.
,
1992
,
Sliding Modes in Control Optimization
,
Springer Verlag
,
Berlin
.
36.
Lubich
,
C.
,
1986
, “
Discretized Fractional Calculus
,”
SIAM J. Math. Anal.
,
17
, pp.
704
719
.10.1137/0517050
37.
Lu
,
J. G.
,
2005
, “
Chaotic Dynamics and Synchronization of Fractional-Order Arneodo's Systems
,”
Chaos, Soliton Fract
,
26
, pp.
1125
1133
.10.1016/j.chaos.2005.02.023
38.
Guo
,
L. J.
,
2005
, “
Chaotic Dynamics and Synchronization of Fractional-Order Genesio-Tesi Systems
,”
Chin. Phys. B
,
14
, pp.
1517
1521
.10.1088/1009-1963/14/8/007
You do not currently have access to this content.