• Previous Article
    Asymptotics for random-time ruin probability in a time-dependent renewal risk model with subexponential claims
  • JIMO Home
  • This Issue
  • Next Article
    Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems
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

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.
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.  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., (1997), 2351.   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.   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.  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.  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.  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.  doi: 10.1109/9.905699.  Google Scholar

[8]

H. K. Khalil, Nonlinear Systems,, Prentice Hall, (2002).   Google Scholar

[9]

X. Li and W. Yu, Synchronization of ball and beam systems with neural compensation,, International Journal of Control, 8 (2010), 491.   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.  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.  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, (2009), 3995.   Google Scholar

[13]

S. Sastry, Nonlinear Systems: Analysis, Stability and Control,, Springer, (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.  doi: 10.1109/9.847724.  Google Scholar

[15]

R. Sepulchre, M. Jankovic and P. V. Kokotovic, Constructive Nonlinear Control,, Springer, (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, (1989), 1376.   Google Scholar

[17]

A. R. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls,, Systems and Control Letters, 18 (1992), 165.  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, (1993), 2577.   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.   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.  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., (1997), 2351.   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.   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.  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.  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.  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.  doi: 10.1109/9.905699.  Google Scholar

[8]

H. K. Khalil, Nonlinear Systems,, Prentice Hall, (2002).   Google Scholar

[9]

X. Li and W. Yu, Synchronization of ball and beam systems with neural compensation,, International Journal of Control, 8 (2010), 491.   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.  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.  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, (2009), 3995.   Google Scholar

[13]

S. Sastry, Nonlinear Systems: Analysis, Stability and Control,, Springer, (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.  doi: 10.1109/9.847724.  Google Scholar

[15]

R. Sepulchre, M. Jankovic and P. V. Kokotovic, Constructive Nonlinear Control,, Springer, (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, (1989), 1376.   Google Scholar

[17]

A. R. Teel, Global stabilization and restricted tracking for multiple integrators with bounded controls,, Systems and Control Letters, 18 (1992), 165.  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, (1993), 2577.   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.   Google Scholar

[1]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133

[2]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[3]

Hyung-Chun Lee. Efficient computations for linear feedback control problems for target velocity matching of Navier-Stokes flows via POD and LSTM-ROM. Electronic Research Archive, , () : -. doi: 10.3934/era.2020128

[4]

Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030

[5]

Yubiao Liu, Chunguo Zhang, Tehuan Chen. Stabilization of 2-d Mindlin-Timoshenko plates with localized acoustic boundary feedback. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021006

[6]

Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2021, 17 (1) : 67-79. doi: 10.3934/jimo.2019099

[7]

Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027

[8]

Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025

[9]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

[10]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[11]

Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031

[12]

Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022

[13]

Biao Zeng. Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021001

[14]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[15]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[16]

Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399

[17]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033

[18]

Xianwei Chen, Xiangling Fu, Zhujun Jing. Chaos control in a special pendulum system for ultra-subharmonic resonance. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 847-860. doi: 10.3934/dcdsb.2020144

[19]

Mikhail I. Belishev, Sergey A. Simonov. A canonical model of the one-dimensional dynamical Dirac system with boundary control. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021003

[20]

Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]