Embedding-Based Sliding Mode Control for Linear Time Varying Systems

Author(s)
Mohammad Reza Zarrabi,
Mohammad Hadi Farahi,
Ali Jafar Koshkouei,
Sohrab Effati,
Keith Burnham

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.

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.

KEYWORDS

Time Optimal Control Problem, Measure Theory, Sliding Mode Control, Sliding Surface Design, Equivalent Control

Time Optimal Control Problem, Measure Theory, Sliding Mode Control, Sliding Surface Design, Equivalent Control

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.

nullM. Zarrabi, M. Farahi, A. Koshkouei, S. Effati and K. Burnham, "Embedding-Based Sliding Mode Control for Linear Time Varying Systems,"

References

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[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.

[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.