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 AM  Vol.2 No.4 , April 2011
Embedding-Based Sliding Mode Control for Linear Time Varying Systems
Abstract: In this paper, a novel strategy using embedding process and sliding surface is proposed. In this method, a state trajectory starting from a given initial point reaches a definite point on a sliding surface in the minimum time, and then tends to the origin along the sliding surface (SS). In the first, a SS is designed, then using an appropriate measure, an embedding is constructed to solve a time optimal control problem such that the system trajectory reaches the SS in minimum time, after that a control is designed such that the system trajectory tends to the origin along the SS. It is well-known that the main disadvantage of the use of sliding mode controls (SMCs) is a phenomenon, the so-called chattering. The proposed SMC here is piecewise continuous and chattering free. Some numerical examples is presented to illustrate the effectiveness and reliability of the proposed method.
Cite this paper: nullM. Zarrabi, M. Farahi, A. Koshkouei, S. Effati and K. Burnham, "Embedding-Based Sliding Mode Control for Linear Time Varying Systems," Applied Mathematics, Vol. 2 No. 4, 2011, pp. 487-495. doi: 10.4236/am.2011.24063.
References

[1]   S. V. Emeyanov, et al., “Variable Structure Control Systems,” Mir Publishers, Moscow, 1967.

[2]   I. Flügge-Lotz, “Discontinuous Automatic Control,” Princeton University Press, New Jersey, 1953.

[3]   A. F. Filippov, “Application of Theory of Differential Equations with Discontinuous Right-Hand Side to Nonlinear Control Problems,” Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities, World Scientific Publishing, Moscow, 1960.

[4]   V. I. Utkin, “Sliding Modes in Control and Optimization,” Springer-Verlag, Berlin, 1992.

[5]   S. V. Emeyanov and V. A. Taran, “On One Class of Variable Structure Control Systems,” Computational Mathematics and Modeling, Vol. 21, No. 3, 2006, pp. 5-26.

[6]   B. Dra?enovi?, “The Invariance Condition in Variable Structure Systems,” Automatica, Vol. 5, No. 3, 1969, pp. 287-295.

[7]   U. Itkis, “Control System of Variable Structure,” Wiley, New York, 1976.

[8]   V. I. Utkin, “Variable Structure Systems with Sliding Modes,” IEEE Transactions on Automatic Control, Vol. 22, No. 2, 1977, pp. 212-222. doi:10.1109/TAC.1977.1101446

[9]   T. Chatchanayuenyong and M. Parnichkun, “Neural Network Based-Time Optimal Sliding Mode Control for an Autonomous Underwater Robot,” Mechatronics, Vol. 16, No. 8, 2006, pp. 471-478. doi:10.1016/j.mechatronics.2006.02.003

[10]   A. J. Koshkouei, “Sliding-Mode Control with Passivity for a Continuously Stirred Tank Reactor,” Proceedings of the Institution of Mechanical Engineers, Vol. 221, No. 5, 2007, pp. 749-755.

[11]   J. Mozaryn and J. E. Kurek, “Design of Decoupled Sliding Mode Control for the PUMA 560 Robot Manipulator,” Proceedings the 3rd International Workshop on Ro- bot Motion and Control, Warsaw, 9-11 November 2002, pp. 45-50.

[12]   N. Yagiz Y. Z. Arslan and Y. Hacioglu, “Sliding Mode Control of a Finger for a Prosthetic Hand,” Vibration and Control, Vol. 13, No. 6, 2007, pp. 733-749.

[13]   A. Nowacka-Leverton and A. Bartoszewicz, “IAE Optimal Sliding Mode Control of Cable Suspended Loads,” CEAI, Vol. 10, No. 3, 2008, pp. 3-10.

[14]   H. Bouadi and M. Tadjine, “Nonlinear Observer Design and Sliding Mode Control of Four Rotors Helicopter,” Vorld Academy of Science, Engineering and Technology, Vol. 25, 2007, pp. 225-229.

[15]   A. S. I. Zinober, “Variable Structure and Lyapunov Control,” Springer Verlag, London, 1994. doi:10.1007/BFb0033675

[16]   J.-C. Le, and Y.-H. Kuo, “Decoupled Fuzzy Sliding Mode Control,” IEEE Transactions on Fuzzy Systems, 1998, Vol. 6, No. 3, pp.426-435. doi:10.1109/91.705510

[17]   S. H. Jang and S. W. Kim, “A New Sliding Surface Design Method of Linear Systems with Mismatched Uncertainties,” IEICE Transactions on Fundamentals of Electronics, Vol. E88-A, No. 1,2005, pp. 387-391.

[18]   B. Bandyopadhyay and S. Janardhanan, “Discrete-Time Sliding Mode Control,” Springer-Verlag, Berlin, 2005.

[19]   M. Thoma, F. Allg?wer and M. Morari, “Time-Varying Sliding Modes for Second and Third Order Systems,” Springer-Verlag, Berlin, 2009.

[20]   A. J. Koshkouei, “Passivity-Based Sliding Mode Control for Nonlinear Systems,” International Journal of Adaptive Control and Signal Processing, Vol. 22, No. 9, 2008, pp. 859-874. doi:10.1002/acs.1028

[21]   A. J. Koshkouei, K. Burnham and A. S. I. Zinober, “Flatness, Backstepping and Sliding Mode Controllers for Nonlinear Systems,” In: G. Bartolini, L. Fridman, A. Pisano and E. Usai, Eds., Modern Sliding Mode Control Theory, Springer-Verlag, Berlin, 2008, pp. 269-290. doi:10.1007/978-3-540-79016-7_13

[22]   B. Friedland, “Advanced Control System Design,” Prentice-Hall, Englewood Cliffs, 1996.

[23]   A. J. Koshkouei and A. S. I. Zinober, “Sliding Mode Controller-Observer Design for SISO Linear Systems,” International Journal of Systems Science, Vol. 29, No. 12, 1998, pp. 1363-1373. doi:10.1080/00207729808929622

[24]   S. H. ?ak and S. Hui, “On Variable Structure Output Feedback Controllers for Uncertain Dynamic Systems,” IEEE Transactions on Automatic Control, Vol. 38, No. 10, 1993, pp. 1509-1512. doi:10.1109/9.241564

[25]   R. El-Khazali and R. DeCarlo, “Output Feedback Variable Structure Control Design,” Automatica, Vol. 31, No. 6, 1995, pp. 805-816. doi:10.1016/0005-1098(94)00151-8

[26]   C. Edwards and S. K. Spurgeon, “Sliding Mode Control: Theory and Applications,” Taylor & Francies, London, 1998.

[27]   J. E. Rubio, “Control and Optimization; The Linear Treatment of Nonlinear Problems,” Manchester University Press, Manchester, 1986.

[28]   W. Rudin, “Real and Complex Analysis,” 3rd Edition, McGraw-Hill, New York, 1987.

[29]   S. Effati, A. V. Kamyad and R. A. Kamyabi-Gol, “On Infinite-Horizon Optimal Control Problems,” Journal for Analysis and Its Applications, Vol. 19, No. 1, 2000, pp. 269-278.

 
 
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