American Institute of Mathematical Sciences

Advanced
  • 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

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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

[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.

[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.

[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.

[1]

Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks and Heterogeneous Media, 2009, 4 (1) : 19-34. doi: 10.3934/nhm.2009.4.19
[2]

Meng Zhang, Kaiyuan Liu, Lansun Chen, Zeyu Li. State feedback impulsive control of computer worm and virus with saturated incidence. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1465-1478. doi: 10.3934/mbe.2018067
[3]

Rohit Gupta, Farhad Jafari, Robert J. Kipka, Boris S. Mordukhovich. Linear openness and feedback stabilization of nonlinear control systems. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1103-1119. doi: 10.3934/dcdss.2018063
[4]

Lorena Bociu, Steven Derochers, Daniel Toundykov. Feedback stabilization of a linear hydro-elastic system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1107-1132. doi: 10.3934/dcdsb.2018144
[5]

Sanmei Zhu, Jun-e Feng, Jianli Zhao. State feedback for set stabilization of Markovian jump Boolean control networks. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1591-1605. doi: 10.3934/dcdss.2020413
[6]

Zhiling Guo, Shugen Chai. Exponential stabilization of the problem of transmission of wave equation with linear dynamical feedback control. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2022001
[7]

Fabio S. Priuli. State constrained patchy feedback stabilization. Mathematical Control and Related Fields, 2015, 5 (1) : 141-163. doi: 10.3934/mcrf.2015.5.141
[8]

Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial and Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061
[9]

Ruofeng Rao, Shouming Zhong. Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1375-1393. doi: 10.3934/dcdss.2020280
[10]

Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014
[11]

A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289
[12]

Elena Braverman, Alexandra Rodkina. Stabilization of difference equations with noisy proportional feedback control. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2067-2088. doi: 10.3934/dcdsb.2017085
[13]

Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Erratum and addendum to "Feedback stabilization of a coupled string-beam system" by K. Ammari, M. Jellouli and M. Mehrenberger; N. H. M: 4 (2009), 19--34. Networks and Heterogeneous Media, 2011, 6 (4) : 783-784. doi: 10.3934/nhm.2011.6.783
[14]

Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167
[15]

Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783
[16]

Cătălin-George Lefter, Elena-Alexandra Melnig. Feedback stabilization with one simultaneous control for systems of parabolic equations. Mathematical Control and Related Fields, 2018, 8 (3&4) : 777-787. doi: 10.3934/mcrf.2018034
[17]

Xiaorui Wang, Genqi Xu. Uniform stabilization of a wave equation with partial Dirichlet delayed control. Evolution Equations and Control Theory, 2020, 9 (2) : 509-533. doi: 10.3934/eect.2020022
[18]

Vanessa Baumgärtner, Simone Göttlich, Stephan Knapp. Feedback stabilization for a coupled PDE-ODE production system. Mathematical Control and Related Fields, 2020, 10 (2) : 405-424. doi: 10.3934/mcrf.2020003
[19]

H. W. J. Lee, Y. C. E. Lee, Kar Hung Wong. Differential equation approximation and enhancing control method for finding the PID gain of a quarter-car suspension model with state-dependent ODE. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2305-2330. doi: 10.3934/jimo.2019055
[20]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (118)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy