The 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 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 filter is introduced, and it is shown that the gain from the noise to the estimation error is bounded in the sense of the norm. Based on the extended bounded real lemma, the 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 filter and the fractional-order sliding-mode observer is illustrated with consideration of the initialization functions.
Issue Section:
Research Papers
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
.10.1109/94.3266544.
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)90207-25.
Wang
, J. C.
, 1987
, “Realizations of Generalized Warburg Impedance With RC Ladder Networks and Transmission Lines
,” J. Electrochem. Soc.
, 134
(8
), pp. 1915
–1920
.10.1149/1.21007896.
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.2010.09.0047.
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.2010.09.0138.
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.2009.04.00311.
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
.10.1115/1.400405912.
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
.10.1115/1.402306613.
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
.10.1115/1.400607214.
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.2010.031415.
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.2012.030916.
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.2012.031217.
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.2013.09.00618.
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.2012.222543419.
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.2013.227984320.
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.156521.
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
.10.1115/1.283358524.
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.2009.08.00326.
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.18.260-27127.
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.459004040329.
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/cvx32.
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.1987.110453033.
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.2008.200257137.
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.2012.03.09940.
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.2009.05.07041.
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
.10.1115/1.402386542.
Slotine
, J. J. E.
, and Li
, W.
, 1991
, Applied Nonlinear Control
, Prentice-Hall
, Englewood Cliffs, NJ
.Copyright © 2014 by ASME
You do not currently have access to this content.
