This article proposes an adaptive neural output tracking control scheme for a class of nonlinear fractional order (FO) systems in the presence of unknown actuator faults. By means of backstepping terminal sliding mode (SM) control technique, an adaptive fractional state-feedback control law is extracted to achieve finite time stability along with output tracking for an uncertain faulty FO system. The unknown nonlinear terms are approximated by radial-basis function neural network (RBFNN) with unknown approximation error upper bound. Using convergence in finite time and fractional Lyapunov stability theorems, the finite time stability and tracking achievement are proved. Finally, the proposed fault tolerant control (FTC) approach is validated with numerical simulations on two fractional models including fractional Genesio–Tesi and fractional Duffing's oscillator systems.

References

References
1.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
New York
.
2.
Dar
,
M. R.
,
Kant
,
N. A.
, and
Khanday
,
F. A.
,
2017
, “
Electronic Implementation of Fractional-Order Newton–Leipnik Chaotic System With Application to Communication
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
5
), p.
054502
.
3.
Rezaei
,
D.
, and
Tavazoei
,
M. S.
,
2017
, “
Analysis of Oscillations in Relay Feedback Systems With Fractional-Order Integrating Plants
,”
ASME J. Comput. Nonliner Dyn.
,
12
(
5
), p.
051023
.
4.
Si
,
G.
,
Zhu
,
J.
,
Diao
,
L.
, and
Ding
,
Z.
,
2017
, “
Modeling, Nonlinear Dynamic Analysis and Control of Fractional PMSG of Wind Turbine
,”
Nonlinear Dyn.
,
88
(
2
), pp.
985
1000
.
5.
Zhang
,
C.
, and
Xiao
,
J.
,
2017
, “
Chaotic Behavior and Feedback Control of Magnetorheological Suspension System With Fractional-Order Derivative
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
2
), p.
021007
.
6.
Pinto
,
C.
, and
Carvalho
,
A.
,
2018
, “
Fractional Dynamics of an Infection Model With Time-Varying Drug Exposure
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(9), p. 090904.
7.
Yang
,
J. H.
,
Sanjuán
,
M. A. F.
, and
Liu
,
H. G.
,
2017
, “
Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
5
), p.
051011
.
8.
Aghababa
,
M. P.
,
2017
, “
Stabilization of a Class of Fractional-Order Chaotic Systems Using a Non-Smooth Control Methodology
,”
Nonlinear Dyn.
,
89
(2), pp. 1357–1370.
9.
Effati
,
S.
,
Rakhshan
,
S. A.
, and
Saqi
,
S.
,
2018
, “
Formulation of Euler-Lagrange Equations for Multidelay Fractional Optimal Control Problems
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(6), p. 061007.
10.
Wu
,
G. C.
,
Baleanu
,
D.
,
Xie
,
H. P.
, and
Chen
,
F. L.
,
2016
, “
Chaos Synchronization of Fractional Chaotic Maps Based on Stability Results
,”
Phys. A
,
460
(
2016
), pp.
374
383
.
11.
Wu
,
G. C.
,
Baleanu
,
D.
, and
Luo
,
W. H.
,
2017
, “
Lyapunov Functions for Riemann–Liouville-Like Fractional Difference Equations
,”
Appl. Math. Comput.
,
314
(
2017
), pp.
228
236
.
12.
Aghababa
,
M. P.
,
2014
, “
Control of Fractional-Order Systems Using Chatter-Free Sliding Mode Approach
,”
ASME J. Comput. Nonlinear Dyn.
,
9
(
3
), p.
031003
.
13.
Alaviyan-Shahri
,
E. S.
,
Alfi
,
A.
, and
Tenreiro-Machado
,
J. A.
,
2016
, “
Stabilization of Fractional-Order Systems Subject to Saturation Element Using Fractional Dynamic Output Feedback Sliding Mode Control
,”
ASME J. Comput. Nonlinear Dyn.
,
12
(
3
), p.
031014
.
14.
Hajipour
,
A.
,
Hajipour
,
M.
, and
Baleanu
,
D.
,
2018
, “
On the Adaptive Sliding Mode Controller for a Hyperchaotic Fractional-Order Financial System
,”
Phys. A
,
497
, pp. 139–153.
15.
Aghababa
,
M. P.
,
2016
, “
Control of Non-Integer-Order Dynamical Systems Using Sliding Mode Scheme
,”
Complexity
,
21
(
6
), pp.
224
233
.
16.
Duc
,
T. M.
,
Van Hoa
,
N.
, and
Dao
,
T. P.
,
2018
, “
Adaptive Fuzzy Fractional-Order Nonsingular Terminal Sliding Mode Control for a Class of Second-Order Nonlinear Systems
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
3
), p.
031004
.
17.
Narayan
,
S.
, and
Kaur
,
S.
,
2018
, “
Finite Time Fractional-Order Sliding Mode-Based Tracking for a Class of Fractional-Order Nonholonomic Chained System
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
5
), p.
051006
.
18.
Aghababa
,
M. P.
,
2015
, “
Design of Hierarchical Terminal Sliding Mode Control Scheme for Fractional-Order Systems
,”
IET Sci., Meas. Technol.
,
9
(
1
), pp.
122
133
.
19.
Ni
,
J.
,
Liu
,
L.
,
Liu
,
C.
, and
Hu
,
X.
,
2017
, “
Fractional Order Fixed-Time Nonsingular Terminal Sliding Mode Synchronization and Control of Fractional Order Chaotic Systems
,”
Nonlinear Dyn.
,
89
(
3
), pp.
2065
2083
.
20.
Chen
,
M.
,
Shao
,
S.
, and
Shi
,
P.
,
2017
, “
Adaptive Neural Tracking Control for Uncertain Fractional-Order Chaotic Systems Subject to Input Saturation and Disturbance
,”
Robust Adaptive Control for Fractional-Order Systems With Disturbance and Saturation
,
Wiley
,
Chichester, UK
, pp.
107
122
.
21.
Song
,
S.
,
Song
,
X.
, and
Tejado
,
I.
,
2017
, “
Projective Synchronization for Two Non-Identical Time-Delayed Fractional-Order T–S Fuzzy Neural Networks Based on Mixed H∞/Passive Adaptive Sliding Mode Control
,”
Int. J. Mach. Learn. Cybern.
(epub).
22.
Wu
,
H.
,
Zhang
,
X.
,
Xue
,
S.
, and
Niu
,
F.
,
2017
, “
Quasi-Uniform Stability of Caputo-Type Fractional-Order Neural Networks With Mixed Delay
,”
Int. J. Mach. Learn. Cybern.
,
8
(
5
), pp.
1501
1511
.
23.
Liu
,
H.
,
Pan
,
Y.
,
Li
,
S.
, and
Chen
,
Y.
,
2017
, “
Synchronization for Fractional-Order Neural Networks With Full/Under-Actuation Using Fractional-Order Sliding Mode Control
,”
Int. J. Mach. Learn. Cybern.
,
9
(7), p. 1219–1232.
24.
Shen
,
H.
,
Song
,
X.
, and
Wang
,
Z.
,
2013
, “
Robust Fault-Tolerant Control of Uncertain Fractional-Order Systems Against Actuator Faults
,”
IET Control Theory Appl.
,
7
(
9
), pp.
1233
1241
.
25.
Song
,
X.
, and
Shen
,
H.
,
2013
, “
Fault Tolerant Control for Interval Fractional-Order Systems With Sensor Failures
,”
Adv. Math. Phys.
,
2013
, p. 836743.
26.
Pettinari
,
S.
, and
Corradini
,
M. L.
,
2014
, “
Fault Tolerant Control Allocation for Fractional-Order Systems
,”
IEEE European Control Conference
(
ECC
), Strasbourg, France, June 24–27, pp.
1969
1974
.
27.
Talange
,
D.
, and
Joshi
,
S.
,
2016
, “
Fractional Order Fault Tolerant Controller for AUV
,”
Seventh International Conference on Circuits, Systems, Control, Signals (CSCS '16)
, Venice, Italy, Jan. 29–31, pp.
287
292
.
28.
Melendez-Vazquez
,
F.
,
Martinez-Fuentes
,
O.
, and
Martinez-Guerra
,
R.
,
2017
, “
Fractional Fault-Tolerant Dynamical Controller for a Class of Commensurate-Order Fractional Systems
,”
Int. J. Syst. Sci.
,
49
(
1
), pp.
196
210
.
29.
Farid
,
Y.
,
Johari-Majd
,
V.
, and
Ehsani-Seresht
,
A.
,
2018
, “
Fractional-Order Active Fault-Tolerant Force-Position Controller Design for the Legged Robots Using Saturated Actuator With Unknown Bias and Gain Degradation
,”
Mech. Syst. Signal Process.
,
104
, pp.
465
486
.
30.
Nasimullah
,
M. A.
,
2018
, “
Performance Comparison of Wind Turbine Based Doubly Fed Induction Generator System Using Fault Tolerant Fractional and Integer Order Controllers
,”
Renewable Energy
,
116
(
Pt. B
), pp.
244
264
.
31.
Li
,
Y.
,
Chen
,
Y. Q.
, and
Podlubny
,
I.
,
2009
, “
Mittag–Leffler Stability of Fractional Order Nonlinear Dynamic Systems
,”
Automatica
,
45
(
8
), pp.
1965
1969
.
32.
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
.
33.
Peng
,
X.
,
Wu
,
H.
,
Song
,
K.
, and
Shi
,
J.
,
2017
, “
Global Synchronization in Finite Time for Fractional-Order Neural Networks With Discontinuous Activations and Time Delays
,”
Neural Networks
,
94
, pp.
46
54
.
34.
Ge
,
S. S.
, and
Wang
,
C.
,
2004
, “
Adaptive Neural Control of Uncertain MIMO Nonlinear Systems
,”
IEEE Trans. Neural Networks
,
15
(
3
), pp.
674
692
.
35.
Smart
,
D. R.
,
1974
,
Fixed Point Theorems
,
Cambridge University Press
,
London
.
36.
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
.
37.
Hosseinnia
,
S. H.
,
Ghaderi
,
R.
,
Ranjbar
,
A.
,
Mahmoudian
,
M.
, and
Momani
,
S.
,
2010
, “
Sliding Mode Synchronization of an Uncertain Fractional Order Chaotic System
,”
Comput. Math. Appl.
,
59
(
5
), pp.
1637
1643
.
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