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January  2016, 12(1): 17-29. doi: 10.3934/jimo.2016.12.17

Global stabilization for ball-and-beam systems via state and partial state feedback

Huawen Ye 1, and  Honglei Xu 2,

1. 

School of Information Science and Engineering, Central South University, Changsha, Hunan 410083, China
2. 

Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845

Received  June 2014 Revised  October 2014 Published  April 2015

In this paper, we present new state and partial state feedback laws as global stabilizers of the well-known frictionless ball and beam system. Dealing with nonlinear terms in the manner different from the ones in the literature, we have achieved a new, simple state-dependent saturation control law. The key technique is to assign a suitable state-dependent saturation level function and jointly use the computation techniques of linear gains. Then, combining such a state feedback law with a homogeneous observer, we again obtain a new partial state feedback design.
Keywords: linear gain, partial state feedback., saturated control, Stabilization, ball and beam system.
Mathematics Subject Classification: Primary: 93C10, 93D15; Secondary: 93D0.
Citation: Huawen Ye, Honglei Xu. Global stabilization for ball-and-beam systems via state and partial state feedback. Journal of Industrial & Management Optimization, 2016, 12 (1) : 17-29. doi: 10.3934/jimo.2016.12.17
References:
[1]

D. Angeli and E. D. Sontag, Forward completeness, unboundedness observability, and their Lyapunov characterizations, Systems and Control Letters, 38 (1999), 209-217. doi: 10.1016/S0167-6911(99)00055-9.  Google Scholar

[2]

C. Barbu, R. Sepulchre, P. V. Kokotovic and W. Lin, Global asymptotic stabilization of the ball-and-beam model, in IEEE Conf. Decision Contr., San Diego, USA, (1997), 2351-2355. Google Scholar

[3]

Y. H. Chang, C. W. Chang, C. W. Tao, H. W. Lin and J. S. Taur, Fuzzy sliding-mode controlfor ball and beam system with fuzzy ant colony optimization, Expert Systems with Applications, 39 (2012), 3624-3633. Google Scholar

[4]

A. Chen, J. Cao and L. Huang, An estimation of upper bound of delays for global asymptotic stability of delayed Hopfield neural networks, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 49 (2002), 1028-1032. doi: 10.1109/TCSI.2002.800841.  Google Scholar

[5]

T. L. Chien, C. C. Chen, M. C. Tsai and Y. C. Chen, Control of AMIRA's ball and beam system via improved fuzzy feedback linearization approach, Applied Mathematical Modelling, 34 (2010), 3791-3804. doi: 10.1016/j.apm.2010.03.020.  Google Scholar

[6]

J. Hauser, S. Sastry and P. V. Kokotovic, Nonlinear control via approximate input-output linearization: The ball and beam example, IEEE Transactions on Automatic Control, 37 (1992), 392-398. doi: 10.1109/9.119645.  Google Scholar

[7]

Y. Hong, J. Huang and Y. Xu, On an output feedback finite-time stabilization problem, IEEE Transactions on Automatic Control, 46 (2001), 305-309. doi: 10.1109/9.905699.  Google Scholar

[8]

H. K. Khalil, Nonlinear Systems, Prentice Hall, New Jersey, 2002. Google Scholar

[9]

X. Li and W. Yu, Synchronization of ball and beam systems with neural compensation, International Journal of Control, Automation, and Systems, 8 (2010), 491-496. Google Scholar

[10]

W. Lin and X. Li, Synthesis of upper-triangular nonlinear systems with marginally unstable free dynamics using state-dependent saturation, Int. J. Control, 72 (1999), 1078-1086. doi: 10.1080/002071799220434.  Google Scholar

[11]

R. Ortega, M. W. Spong, F. Gomez-Estern and G. Blankenstein, Stabilization of a class of under-actuated mechanical system via interconnection and damping assignment, IEEE Transactions on Automatic Control, 47 (2002), 1218-1233. doi: 10.1109/TAC.2002.800770.  Google Scholar

[12]

C. Qian, Global output feedback stabilization of a class of upper-triangular nonlinear systems, in American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA, (2009), 3995-4000. Google Scholar

[13]

S. Sastry, Nonlinear Systems: Analysis, Stability and Control, Springer, New York, 1999. doi: 10.1007/978-1-4757-3108-8.  Google Scholar

[14]

R. Sepulchre, Slow peaking and low-gain designs for global stabilization of nonlinear systems, IEEE Transactions on Automatic Control, 45 (2000), 453-461. doi: 10.1109/9.847724.  Google Scholar

[15]

R. Sepulchre, M. Jankovic and P. V. Kokotovic, Constructive Nonlinear Control, Springer, London, 1997. doi: 10.1007/978-1-4471-0967-9.  Google Scholar

[16]

E. D. Sontag, Remarks on stabilization and input-to-state stability, in IEEE Conf. Decision and Control, Tempa, FL, (1989), 1376-1378.  Google Scholar

[17]

A. R. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls, Systems and Control Letters, 18 (1992), 165-171. doi: 10.1016/0167-6911(92)90001-9.  Google Scholar

[18]

A. R. Teel, Semi-global stabilization of the ball and beam using output feedback, in American Control Conference, San Diego, CA, (1993), 2577-2581. Google Scholar

[19]

H. Ye, H. Wang and H. B. Wang, Stabilization of a PVTOL aircraft and an inertia wheel pendulum using saturation technique, IEEE Transactions on Control Systems Technology, 15 ( 2007), 1143-1150. Google Scholar

show all references

References:
[1]

D. Angeli and E. D. Sontag, Forward completeness, unboundedness observability, and their Lyapunov characterizations, Systems and Control Letters, 38 (1999), 209-217. doi: 10.1016/S0167-6911(99)00055-9.  Google Scholar

[2]

C. Barbu, R. Sepulchre, P. V. Kokotovic and W. Lin, Global asymptotic stabilization of the ball-and-beam model, in IEEE Conf. Decision Contr., San Diego, USA, (1997), 2351-2355. Google Scholar

[3]

Y. H. Chang, C. W. Chang, C. W. Tao, H. W. Lin and J. S. Taur, Fuzzy sliding-mode controlfor ball and beam system with fuzzy ant colony optimization, Expert Systems with Applications, 39 (2012), 3624-3633. Google Scholar

[4]

A. Chen, J. Cao and L. Huang, An estimation of upper bound of delays for global asymptotic stability of delayed Hopfield neural networks, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 49 (2002), 1028-1032. doi: 10.1109/TCSI.2002.800841.  Google Scholar

[5]

T. L. Chien, C. C. Chen, M. C. Tsai and Y. C. Chen, Control of AMIRA's ball and beam system via improved fuzzy feedback linearization approach, Applied Mathematical Modelling, 34 (2010), 3791-3804. doi: 10.1016/j.apm.2010.03.020.  Google Scholar

[6]

J. Hauser, S. Sastry and P. V. Kokotovic, Nonlinear control via approximate input-output linearization: The ball and beam example, IEEE Transactions on Automatic Control, 37 (1992), 392-398. doi: 10.1109/9.119645.  Google Scholar

[7]

Y. Hong, J. Huang and Y. Xu, On an output feedback finite-time stabilization problem, IEEE Transactions on Automatic Control, 46 (2001), 305-309. doi: 10.1109/9.905699.  Google Scholar

[8]

H. K. Khalil, Nonlinear Systems, Prentice Hall, New Jersey, 2002. Google Scholar

[9]

X. Li and W. Yu, Synchronization of ball and beam systems with neural compensation, International Journal of Control, Automation, and Systems, 8 (2010), 491-496. Google Scholar

[10]

W. Lin and X. Li, Synthesis of upper-triangular nonlinear systems with marginally unstable free dynamics using state-dependent saturation, Int. J. Control, 72 (1999), 1078-1086. doi: 10.1080/002071799220434.  Google Scholar

[11]

R. Ortega, M. W. Spong, F. Gomez-Estern and G. Blankenstein, Stabilization of a class of under-actuated mechanical system via interconnection and damping assignment, IEEE Transactions on Automatic Control, 47 (2002), 1218-1233. doi: 10.1109/TAC.2002.800770.  Google Scholar

[12]

C. Qian, Global output feedback stabilization of a class of upper-triangular nonlinear systems, in American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA, (2009), 3995-4000. Google Scholar

[13]

S. Sastry, Nonlinear Systems: Analysis, Stability and Control, Springer, New York, 1999. doi: 10.1007/978-1-4757-3108-8.  Google Scholar

[14]

R. Sepulchre, Slow peaking and low-gain designs for global stabilization of nonlinear systems, IEEE Transactions on Automatic Control, 45 (2000), 453-461. doi: 10.1109/9.847724.  Google Scholar

[15]

R. Sepulchre, M. Jankovic and P. V. Kokotovic, Constructive Nonlinear Control, Springer, London, 1997. doi: 10.1007/978-1-4471-0967-9.  Google Scholar

[16]

E. D. Sontag, Remarks on stabilization and input-to-state stability, in IEEE Conf. Decision and Control, Tempa, FL, (1989), 1376-1378.  Google Scholar

[17]

A. R. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls, Systems and Control Letters, 18 (1992), 165-171. doi: 10.1016/0167-6911(92)90001-9.  Google Scholar

[18]

A. R. Teel, Semi-global stabilization of the ball and beam using output feedback, in American Control Conference, San Diego, CA, (1993), 2577-2581. Google Scholar

[19]

H. Ye, H. Wang and H. B. Wang, Stabilization of a PVTOL aircraft and an inertia wheel pendulum using saturation technique, IEEE Transactions on Control Systems Technology, 15 ( 2007), 1143-1150. Google Scholar

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