The H and sliding mode observers are important in integer-order dynamic systems. However, these observers are not well explored in the field of fractional-order dynamic systems. In this paper, the H filter and the fractional-order sliding mode unknown input observer are developed to estimate state of the linear time-invariant fractional-order dynamic systems with consideration of proper initial memory effect. As the first result, the fractional-order H filter is introduced, and it is shown that the gain from the noise to the estimation error is bounded in the sense of the H norm. Based on the extended bounded real lemma, the H filter design is formulated in a linear matrix inequality form, and it will be seen that numerical methods to solve convex optimization problems are feasible in fractional-order systems (FOSs). As the second result of this paper, not only state but also unknown input disturbance are estimated by fractional-order sliding-mode unknown input observer, simultaneously. In this paper, it is shown that the design and stability analysis of the two estimation techniques are not related with the initial history. Through two numerical examples, the performance of the fractional-order H filter and the fractional-order sliding-mode observer is illustrated with consideration of the initialization functions.

References

References
1.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic
,
San Diego
.
2.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order
,
Academic
,
New York
.
3.
Westerlund
,
S.
, and
Ekstam
,
L.
,
1994
, “
Capacitor Theory
,”
IEEE Trans. Dielectr. Electr. Insul.
,
1
(
5
), pp.
826
839
.10.1109/94.326654
4.
Mehaute
,
A. Le
.
, and
Crepy
,
G.
,
1983
, “
Introduction to Transfer and Motion in Fractal Media: The Geometry of Kinetics
,”
Solid State Ionics
,
9
, pp.
17
30
.10.1016/0167-2738(83)90207-2
5.
Wang
,
J. C.
,
1987
, “
Realizations of Generalized Warburg Impedance With RC Ladder Networks and Transmission Lines
,”
J. Electrochem. Soc.
,
134
(
8
), pp.
1915
1920
.10.1149/1.2100789
6.
Jumarie
,
G.
,
2010
, “
Fractional Multiple Birthdeath Processes With Birth Probabilities λi(δt)α+O((δt)α)
,”
J. Franklin Inst.
,
347
(
10
), pp.
1797
1813
.10.1016/j.jfranklin.2010.09.004
7.
Gabano
,
J. D.
, and
Poinot
,
T.
,
2011
, “
Estimation of Thermal Parameters Using Fractional Modelling
,”
Signal Process.
,
91
(
4
), pp.
938
948
.10.1016/j.sigpro.2010.09.013
8.
Matignon
,
D.
,
1996
, “
Stability Results for Fractional Differential Equations With Applications to Control Processing
,”
CESA’96 IMACS Multiconference: Computational Engineering in Systems Applications
, Vol.
2
, pp.
963
968
.
9.
Matignon
,
D.
, and
D'andrea-Nobel
,
B.
,
1996
, “
Some Results on Controllability and Observability of Finite-Dimensional Fractional Differential Systems
,”
CESA’96 IMACS Multiconference: Computational Engineering in Systems Applications
, Vol.
2
, pp.
952
956
.
10.
Li
,
Y.
,
Chen
,
Y. Q.
, and
Podlubny
,
I.
,
2009
, “
Mittag-Leffler Stability of Fractional Order Nonlinear Dynamic Systems
,”
Automatica
,
45
(
8
), pp.
1965
1969
.10.1016/j.automatica.2009.04.003
11.
Tavakoli-Kakhki
,
M.
, and
Haeri
,
M.
,
2011
, “
Temperature Control of a Cutting Process Using Fractional Order Proportional-Integral-Derivative Controller
,”
ASME J. Dyn. Syst., Meas., Control
,
133
(
5
),
p. 051014
.10.1115/1.4004059
12.
Tavazoei
,
M. S.
,
2013
, “
Optimal Tuning for Fractional-Order Controllers: An Integer-Order Approximating Filter Approach
,”
ASME J. Dyn. Syst., Meas., Control
,
135
, p.
021017
.10.1115/1.4023066
13.
Meng
,
L.
, and
Xue
,
D.
,
2012
, “
A New Approximation Algorithm of Fractional Order System Models Based Optimization
,”
ASME J. Dyn. Syst., Meas., Control
,
134
, p.
044504
.10.1115/1.4006072
14.
Luo
,
Y.
,
Chao
,
H.
,
Di
,
L.
, and
Chen
,
Y. Q.
,
2011
, “
Lateral Directional Fractional Order (PI)α Control of a Small Fixed-Wing Unmanned Aerial Vehicles: Controller Designs and Flight Tests
,”
IET Control Theory Appl.
,
5
(
18
), pp.
2156
2167
.10.1049/iet-cta.2010.0314
15.
Charef
,
A.
,
Assabaa
,
M.
,
Ladaci
,
M.
, and
Loiseau
,
J. J.
,
2013
, “
Fractional Order Adaptive Controller for Stabilised Systems Via High-Gain Feedback
,”
IET Control Theory Appl.
,
7
(
6
), pp.
822
828
.10.1049/iet-cta.2012.0309
16.
Li
,
C.
,
Wang
,
J.
, and
Lu
,
J.
,
2012
, “
Observer-Based Robust Stabilisation of a Class of Non-Linear Fractional-Order Uncertain Systems: An Linear Matrix Inequalitie Approach
,”
IET Control Theory Appl.
,
6
(
18
), pp.
2757
2764
.10.1049/iet-cta.2012.0312
17.
Zhang
,
H.
,
Wang
,
J.
, and
Shi
,
Y.
,
2013
, “
Robust h∞ Sliding-Mode Control for Markovian Jump Systems Subject to Intermittent Observations and Partially Known Transition Probabilities
,”
Syst. Control Lett.
,
62
(
12
), pp.
1114
1124
.10.1016/j.sysconle.2013.09.006
18.
Zhang
,
H.
,
Shi
,
Y.
, and
Liu
,
M.
,
2013
, “
h∞ Step Tracking Control for Networked Discrete-Time Nonlinear Systems With Integral and Predictive Actions
,”
IEEE Trans. Ind. Inf.
,
9
(
1
), pp.
1551
3203
.10.1109/TII.2012.2225434
19.
Shuai
,
Z.
,
Zhang
,
H.
,
Wang
,
J.
,
Li
,
J.
, and
Ouyang
,
M.
,
2013
, “
Combined AFS and DYC Control of Four-Wheel-Independent-Drive Electric Vehicles Over CAN Network With Time-Varying Delays
,”
IEEE Trans. Veh. Technol.
,
63
(
2
), pp.
591
602
.10.1109/TVT.2013.2279843
20.
Pisano
,
A.
,
Rapaić
,
M. R.
,
Jeličić
,
Z. D.
, and
Usai
,
E.
,
2010
, “
Sliding Mode Control Approaches to the Robust Regulation of Linear Multivariable Fractional-Order Dynamics
,”
Int. J. Rob. Nonlinear Control
,
20
(
18
), pp.
2045
2056
.10.1002/rnc.1565
21.
Dadras
,
S.
, and
Momeni
,
H. R.
,
2011
, “
Fractional Sliding Mode Observer Design for a Class of Uncertain Fractional Order Nonlinear Systems
,”
50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC)
, IEEE, pp.
6925
6930
.
22.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
,
2011
, “
Time-Varying Initialization and Laplace Transform of the Caputo Derivative: With Order Between Zero and One
,”
Proceedings of IDET/CIE FDTA’ 2011 Conference
, Washington DC, Vol. 3, pp.
163
168
23.
Lorenzo
,
C. F.
, and
Hartley
,
T. T.
,
2008
, “
Initialization of Fractional-Order Operators and Fractional Differential Equations
,”
ASME J. Comput. Nonlinear Dyn.
,
3
(
2
), p.
021101
.10.1115/1.2833585
24.
Gautschi
,
W.
,
1998
, “
The Incomplete Gamma Functions Since Tricomi
,”
Atti dei Con. Linci
,
147
, pp.
203
237
.
25.
Sabatier
,
J.
,
Moze
,
M.
, and
Farges
,
C.
,
2010
, “
LMI Stability Conditions for Fractional-Order Systems
,”
Comput. Math. Appl.
,
59
(
5
), pp.
1594
1609
.10.1016/j.camwa.2009.08.003
26.
Sabatier
,
J.
,
Farges
,
C.
,
Merveillaut
,
M.
, and
Feneteau
,
L.
,
2012
, “
On Observability and Pseudo State Estimation of Fractional Order Systems
,”
European J. Control
,
18
(
3
), pp.
260
271
.10.3166/ejc.18.260-271
27.
Moze
,
M.
,
Sabatier
,
J.
, and
Oustaloup
,
A.
,
2008
, “
On Bounded Real Lemma for Fractional Systems
,”
Proceedings of the 2008 IFAC World Congress
, Vol.
17
, pp.
15267
15272
.
28.
Gahinet
,
P.
, and
Apkarian
,
P.
,
1994
, “
A Linear Matrix Inequality Approach to H∞ Control
,”
Int. J. Rob. Nonlinear Control
,
4
(
4
), pp.
421
448
.10.1002/rnc.4590040403
29.
Dullerud
,
G. E.
, and
Paganini
,
F. G.
,
2005
,
A Course in Robust Control Theory: A Convex Approach
, Vol.
36
,
Springer
,
New York
.
30.
Lofberg
,
J.
,
2004
, “
YALMIP: A Toolbox for Modeling and Optimization in MATLAB
,”
2004 IEEE International Symposium on Computer Aided Control Systems Design
, IEEE, pp.
284
289
.
31.
Grant
,
M.
,
Boyd
,
S.
, and
Ye
,
Y.
,
2008
, “
CVX: Matlab Software for Disciplined Convex Programming
,” (Online), Available: http://www.stanford.edu/∼boyd/cvx
32.
Walcott
,
B. L.
, and
Zak
,
S. H.
,
1987
, “
State Observation of Nonlinear Uncertain Dynamical Systems
,”
IEEE Trans. Autom. Control
,
32
(
2
), pp.
166
170
.10.1109/TAC.1987.1104530
33.
Hui
,
S.
, and
Zak
,
S. H.
,
2005
, “
Observer Design for Systems With Unknown Inputs
,”
Int. J. Appl. Math. Comput. Sci.
,
15
(
4
), pp.
431
446
.
34.
Monje
,
C. A.
,
Chen
,
Y. Q.
,
Vinagre
,
B. M.
,
Xue
,
D.
, and
Feliu
,
V.
,
2010
,
Fractional-Order Systems and Controls: Fundamentals and Applications
,
Springer
,
New York
.
35.
Boyd
,
S.
,
Ghaoui
,
L. El.
,
Feron
,
E.
, and
Balakrishnan
,
V.
,
1994
,
Linear Matrix Inequalities in System and Control Theory
,
SIAM
,
Philadelphia
, PA.
36.
Wen
,
X. J.
,
Wu
,
Z. M.
, and
Lu
,
J. G.
,
2008
, “
Stability Analysis of a Class of Nonlinear Fractional-Order Systems
,”
IEEE Trans. Circuits Syst. II: Express Briefs
,
55
(
11
), pp.
1178
1182
.10.1109/TCSII.2008.2002571
37.
N'doye
,
I.
,
Zasadzinski
,
M.
,
Darouach
,
M.
, and
Radhy
,
N. E.
,
2009
, “
Observer-Based Control for Fractional-Order Continuous-Time Systems
,”
Proceedings of the 48th IEEE Conference on Decision and Control, Held Jointly With the 28th Chinese Control Conference
, CDC/CCC, IEEE, pp.
1932
1937
.
38.
Utkin
,
V. I.
,
1992
,
Sliding Modes in Control and Optimization
,
Springer
,
Berlin, Germany
.
39.
Trigeassou
,
J. C.
,
Maamri
,
N.
,
Sabatier
,
J.
, and
Oustaloup
,
A.
,
2012
, “
State Variables and Transients of Fractional Order Differential Systems
,”
Comput. Math. Appl.
,
64
(
10
), pp.
3117
3140
.10.1016/j.camwa.2012.03.099
40.
Sabatier
,
J.
,
Merveillaut
,
M.
,
Malti
,
R.
, and
Oustaloup
,
A.
,
2010
, “
How to Impose Physically Coherent Initial Conditions to a Fractional System?
,”
Commun. Nonlinear Sci. Num. Simul.
,
15
(
5
), pp.
1318
1326
.10.1016/j.cnsns.2009.05.070
41.
Hartley
,
T. T.
,
Lorenzo
,
C. F.
, and
Trigeassou
,
J. -C.
, and
Maamri
,
N.
,
2013
, “
Equivalence of History-Function Based and Infinite-Dimensional-State Initializations for Fractional-Order Operators
,”
ASME J. Comput. Nonlinear Dyn.
,
8
(
4
), p.
041014
.10.1115/1.4023865
42.
Slotine
,
J. J. E.
, and
Li
,
W.
,
1991
,
Applied Nonlinear Control
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
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