Fractional order sliding mode control with reaching law approach

Fractional order sliding mode control is studied in this paper. The control objectives are achieved by adopting the reaching law approach of sliding mode control. The main contribution of this work is to show that the philosophy of integer order sliding mode control is valid also for the systems represented by fractional order operators. A sufficient condition and its implications for stability are given. Matched and unmatched uncertainties are studied. The attractor nature of the switching manifold is analyzed together with a stable sliding subspace design condition. The claims are justified through a set of simulations and the results obtained are found promising.

Anahtar Kelimeler:

Sliding mode control, fractional order control, reaching law approach

Fractional order sliding mode control with reaching law approach

Keywords:

Sliding mode control, fractional order control, reaching law approach,

PDF

___

K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, 1974.
I. Podlubny, Fractional Diﬀerential Equations, Elsevier Science & Technology Books, 1st Ed., 1998.
S. Das, Functional Fractional Calculus for System IdentiŞcation and Controls, Springer, 1st Ed., 2008.
M.D. Ortigueira, “Introduction to Fractional Linear Systems. Part 1: Continuous Time Case,” IEE Proceedings on Visual Image Signal Processing, v.147, n.1, pp.62-70, 2000.
M.D. Ortigueira, “Introduction to Fractional Linear Systems. Part 2: Discrete Time Case,” IEE Proc. Vis. Image Signal Processing, v.147, n.1, pp.71-78, 2000.
S.E. Hamamci, “Stabilization Using Fractional Order PI and PID Controllers,” Nonlinear Dynamics, v.51, no.1-2, pp.329-343, 2007.
D. Val´erio, J. Sa da Costa, “Tuning of Fractional PID Controllers Ziegler-Nichols Type Rules,” Signal Processing, v.86, pp.2771-2784, 2006.
D. Val´erio, J. Sa da Costa, “Time Domain Implementation of Fractional Order Controllers,” IEE Proc. Control Theory and Appls., v.152, n.5, pp.539-552, 2005.
C.-C. Tseng, “Design of Fractional Order Digital FIR Diﬀerentiators,” IEEE Signal Processing Letters, v.8, n.3, pp.77-79, 2001.
D. Sierociuk, A. Dzielinski, “Fractional Kalman Filter Algorithm for the States, Parameters and Order of Fractional System Estimation,” Int. J. Appl. Math. Comput. Sci., v.16, n.1, pp.129-140, 2006.
B.M. Vinagre, C.A. Monje, A.J. Calderon, “Fractional Order Systems and Fractional Order Control Actions,” 41st IEEE Int. Conf. on Decision and Control, Las Vegas, NV, 10-13 December, pp.15-38, 2002.
D. Matignon, B. d’Andera-Novel, “Observer Based Controllers for Fractional Diﬀerential Systems,” Int. Conf. on Decision and Control, San Diego, California, pp.4967-4972, 1997.
B.M. Vinagre, A.J. Calderon, “On Fractional Sliding Mode Control,” 7th Portuguese Conference on Automatic Control (CONTROLO’2006), Lisbon, Portugal, Sep. 11-13, 2006.
A.J. Calderon, B.M. Vinagre, V. Feliu, “Fractional Order Control Strategies for Power Electronic Buck Converters,” Signal Processing, v.86, pp.2803–2819, 2006.
R. El-Khazali, “Output Feedback Sliding Mode Control of Fractional Systems,” 12th IEEE International Conference on Electronics, Circuits and Systems, Dec. 11-14, pp.1-4, 2005.
A. Si-Ammour, S. Djennoune, M. Bettayeb, “A Sliding Mode Control for Linear Fractional systems With Input and State Delays,” Communications in Nonlinear Science and Numerical Simulation, v.14, pp.2310–2318, 2009.
H. Delavari, R. Ghaderi, A. Ranjbar, S. Momani, “Fuzzy Fractional Order Sliding Mode Controller for Nonlinear Systems,” Communications in Nonlinear Science and Numerical Simulation, v.15, pp.963–978, 2010. M. ¨
O. Efe, “A Fractional Order Adaptation Law for Integer Order Sliding Mode Control of a 2DOF Robot,” New Trends in Nanotechnology and Fractional Calculus Applications, Ed. Z.B. G¨uven¸c, D. Baleanu, J.A. Tenreiro Machado, 2010. M. ¨
O. Efe, “Fractional Fuzzy Adaptive Sliding Mode Control of a 2 DOF Direct Drive Robot Arm,” IEEE Transac- tions on Systems, Man and Cybernetics Part B: Cybernetics, v.28, no.6, pp.1561-1570, December 2008. M. ¨
O. Efe, “ADALINE Based Robust Control in Robotics: A Riemann-Liouville Fractional Diﬀerintegration Based Learning Scheme,” Soft Computing, v.13, no.1, pp.23-29, 2009. M. ¨
O. Efe and C. Kasnako˘glu, “A Fractional Adaptation Law for Sliding Mode Control,” International Journal of Adaptive Control and Signal Processing, v.22, no.10, pp.968-986, December 2008. M. ¨
O. Efe, “Fractional Order Sliding Mode Controller Design for Fractional Order Dynamic Systems,” New Trends in Nanotechnology and Fractional Calculus Applications, Ed. Z.B. G¨uven¸c, D. Baleanu, J.A. Tenreiro Machado, D. Val´erio, Fractional control toolbox for MatLab, 2005.