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Research Papers

Finite Time Fractional-Order Sliding Mode-Based Tracking for a Class of Fractional-Order Nonholonomic Chained System

[+] Author and Article Information
Deepika

Department of Electrical Engineering,
Punjab Engineering College (PEC)
(Deemed to be University),
Chandigarh 160012, India
e-mail: sharmadeepika504@gmail.com

Shiv Narayan

Professor
Department of Electrical Engineering,
Punjab Engineering College (PEC)
(Deemed to be University),
Chandigarh 160012, India

Sandeep Kaur

Department of Electrical Engineering,
Punjab Engineering College (PEC)
(Deemed to be University),
Chandigarh 160012, India

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 7, 2017; final manuscript received March 7, 2018; published online April 2, 2018. Assoc. Editor: Bernard Brogliato.

J. Comput. Nonlinear Dynam 13(5), 051006 (Apr 02, 2018) (9 pages) Paper No: CND-17-1408; doi: 10.1115/1.4039581 History: Received September 07, 2017; Revised March 07, 2018

This paper formulates a fractional-order finite time tracking scheme for a perturbed n-dimensional fractional-order extended chained form of nonholonomic system. First, the hierarchical sliding surfaces are properly selected, and then, a novel fractional-order sliding mode methodology is derived mathematically to accomplish the prescribed goal. With the introduction of new virtual controls, the proposed strategy renders finite time tracking of a fractional-order reference model in-spite of various system uncertainties. Further, the closed loop system stability is procured with the proposed method through fractional-Lyapunov and Mittag–Leffler-based stability theorems. It has also been verified analytically that the error dynamics converge to the chosen sliding surfaces in finite time. Finally, two examples including an engineering application of mobile robot are illustrated through matlab simulations to demonstrate the usefulness of the introduced control technique.

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References

Chen, H. , Wang, C. , Zhang, B. , and Zhang, D. , 2013, “ Saturated Tracking Control for Nonholonomic Mobile Robots With Dynamic Feedback,” Trans. Inst. Meas. Control, 35(2), pp. 105–116. [CrossRef]
Chen, H. , 2014, “ Robust Stabilization for a Class of Dynamic Feedback Uncertain Nonholonomic Mobile Robots With Input Saturation,” Int. J. Control, Autom. Syst., 12(6), pp. 1216–1224. [CrossRef]
Murray, R. M. , and Sastry, S. S. , 1993, “ Nonholonomic Motion Planning: Steering Using Sinusoids,” IEEE Trans. Autom. Control, 38(5), pp. 700–716. [CrossRef]
Brockett, R. W. , 1983, “ Asymptotic Stability and Feedback Stabilization,” Differential Geometric Control Theory, R. W. Brockett , R. S. Millman , and H. J. Sussmann , eds., Birkhauser, Boston, MA, pp. 181–208.
Ryan, E. , 1994, “ On Brockett's Condition for Smooth Stabilizability and Its Necessity in a Context of Non-Smooth Feedback,” SIAM J. Control Optim., 32(6), pp. 1597–1604. [CrossRef]
Lefeber, E. , Robertsson, A. , and Nijmeijer, H. , 2000, “ Linear Controllers for Exponential Tracking of Systems in Chained-Form,” Int. J. Robust Nonlinear Control, 10(4), pp. 243–263. [CrossRef]
Lefeber, E. , 2000, Tracking Control of Nonlinear Mechanical Systems, University of Twente, Enschede, The Netherlands.
Sordalen, O. J. , and Egeland, O. , 1995, “ Exponential Stabilization of Nonholonomic Chained Systems,” IEEE Trans. Autom. Control, 40(1), pp. 35–49. [CrossRef]
Ge, S. S. , Wang, Z. , and Lee, T. H. , 2003, “ Adaptive Stabilization of Uncertain Non-Holonomic Systems by State and Output Feedback,” Automatica, 39(8), pp. 1451–1460. [CrossRef]
Huang, J. , Wen, C. , Wang, W. , and Jiang, Z. P. , 2013, “ Adaptive Stabilization and Tracking Control of a Nonholonomic Mobile Robot With Input Saturation and Disturbance,” Syst. Control Lett., 62(3), pp. 234–241. [CrossRef]
Jiang, Z. P. , and Nijmeijer, H. , 1999, “ A Recursive Technique for Tracking Control of Nonholonomic Systems in the Chained Form,” IEEE Trans. Autom. Control, 44(2), pp. 265–279. [CrossRef]
Tian, Y. P. , and Cao, K. C. , 2007, “ Time‐Varying Linear Controllers for Exponential Tracking of Non‐Holonomic Systems in Chained Form,” Int. J. Rob. Nonlinear Control, 17(7), pp. 631–647. [CrossRef]
Oldham, K. , and Spanier, J. , 1974, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York.
Li, C. , and Deng, W. , 2007, “ Remarks on Fractional Derivatives,” Appl. Math. Comput., 187(2), pp. 777–784.
Hamamci, S. E. , 2007, “ An Algorithm for Stabilization of Fractional-Order Time Delay Systems Using Fractional-Order PID Controllers,” IEEE Trans. Autom. Control, 52(10), pp. 1964–1969. [CrossRef]
Delavari, H. , Ghaderi, R. , Ranjbar, A. , and Momani, S. , 2010, “ Fuzzy Fractional Order Sliding Mode Controller for Nonlinear Systems,” Commun. Nonlinear Sci. Numer. Simul., 15(4), pp. 963–978. [CrossRef]
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(2), p. 021010. [CrossRef]
Young, K. D. , Utkin, V. I. , and Ozguner, U. , 1996, “ A Control Engineer's Guide to Sliding Mode Control,” IEEE International Workshop on Variable Structure Systems (VSS'96), Tokyo, Japan, Dec. 5–6, pp. 1–14.
Bhat, S. P. , and Bernstein, D. S. , 2000, “ Finite-Time Stability of Continuous Autonomous Systems,” SIAM J. Control Optim., 38(3), pp. 751–766. [CrossRef]
Wu, Y. , Yu, X. , and Man, Z. , 1998, “ Terminal Sliding Mode Control Design for Uncertain Dynamic Systems,” Syst. Control Lett., 34(5), pp. 281–287. [CrossRef]
Li, T. H. S. , and Huang, Y. C. , 2010, “ MIMO Adaptive Fuzzy Terminal Sliding-Mode Controller for Robotic Manipulators,” Inf. Sci., 180(23), pp. 4641–4660. [CrossRef]
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. [CrossRef]
Gao, F. , and Yuan, F. , 2015, “ Adaptive Finite-Time Stabilization for a Class of Uncertain High Order Nonholonomic Systems,” ISA Trans., 54, pp. 75–82. [CrossRef] [PubMed]
Mobayen, S. , 2015, “ Finite-Time Tracking Control of Chained-Form Nonholonomic Systems With External Disturbances Based on Recursive Terminal Sliding Mode Method,” Nonlinear Dyn., 80(1–2), pp. 669–683. [CrossRef]
Yuqiang, W. , Wang, B. , and Zong, G. D. , 2005, “ Finite-Time Tracking Controller Design for Nonholonomic Systems With Extended Chained Form,” IEEE Trans. Circuits Syst. II: Express Briefs, 52(11), pp. 798–802. [CrossRef]
Bayat, F. , Mobayen, S. , and Javadi, S. , 2016, “ Finite-Time Tracking Control of Nth-Order Chained-Form Non-Holonomic Systems in the Presence of Disturbances,” ISA Trans., 63, pp. 78–83. [CrossRef] [PubMed]
Mobayen, S. , and Javadi, S. , 2017, “ Disturbance Observer and Finite-Time Tracker Design of Disturbed Third-Order Nonholonomic Systems Using Terminal Sliding Mode,” J. Vib. Control, 23(2), pp. 181–189. [CrossRef]
Chen, M. , Wu, Q. X. , and Cui, R. X. , 2013, “ Terminal Sliding Mode Tracking Control for a Class of SISO Uncertain Nonlinear Systems,” ISA Trans., 52(2), pp. 198–206. [CrossRef] [PubMed]
Mobayen, S. , 2015, “ Fast Terminal Sliding Mode Tracking of Non-Holonomic Systems With Exponential Decay Rate,” IET Control Theory Appl., 9(8), pp. 1294–1301. [CrossRef]
Chen, H. , Yan, D. , Chen, X. , Lei, Y. , and Wang, Y. , 2016, “ Robust Finite-Time Stabilization of Fractional-Order Extended Nonholonomic Chained Form Systems,” Chinese Intelligent Systems Conference (CISC), Yangzhou, China, pp. 1–15.
Li, Y. , Chen, Y. , and Podlubny, I. , 2009, “ Mittag–Leffler Stability of Fractional Order Nonlinear Dynamic Systems,” Automatica, 45(8), pp. 1965–1969. [CrossRef]
Oustaloup, A. , Levron, F. , Mathieu, B. , and Nanot, F. M. , 2000, “ Frequency-Band Complex Noninteger Differentiator: Characterization and Synthesis,” IEEE Trans. Circuits Syst. I: Fundam. Theory Appl., 47(1), pp. 25–39. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Simulations with the proposed method versus time in seconds for example 1: (a) tracking errors: x1e,x2e,x3e,x4e, (b) input errors with u1d=exp(−3t) and u2d=−3, and (c) convergence of sliding surfaces for both subsystems

Grahic Jump Location
Fig. 3

Simulations for the proposed method versus time in seconds, for example, 2: (a) x1e, (b) x2e, (c) x3e, (d) input tracking error 2 with u2d=1, (e) input tracking error 1 with u1d=1−e−t, and (f) convergence of all the sliding surfaces

Grahic Jump Location
Fig. 2

Tracking desired controls versus time in seconds, for example, 1: (a) u1 tracks u1d=exp(−3t) and (b) u2 tracks u2d=−3

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