Parameter estimation and stabilization for a wave equation with boundary output harmonic disturbance and non-collocated control
Corresponding Author
Wei Guo
School of Information Technology and Management, University of International Business and Economics, Beijing 100029, People's Republic of China
School of Information Technology and Management, University of International Business and Economics, Beijing 100029, People's Republic of China===Search for more papers by this authorBao-Zhu Guo
Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, People's Republic of China
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, People's Republic of China
School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
Search for more papers by this authorZhi-Chao Shao
School of Information Technology and Management, University of International Business and Economics, Beijing 100029, People's Republic of China
Search for more papers by this authorCorresponding Author
Wei Guo
School of Information Technology and Management, University of International Business and Economics, Beijing 100029, People's Republic of China
School of Information Technology and Management, University of International Business and Economics, Beijing 100029, People's Republic of China===Search for more papers by this authorBao-Zhu Guo
Academy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, People's Republic of China
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, People's Republic of China
School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa
Search for more papers by this authorZhi-Chao Shao
School of Information Technology and Management, University of International Business and Economics, Beijing 100029, People's Republic of China
Search for more papers by this authorAbstract
This paper is concerned with the parameter estimation and stabilization of a one-dimensional wave equation with harmonic disturbance suffered by boundary observation at one end and the non-collocated control at the other end. An adaptive observer is designed in terms of measured velocity corrupted by harmonic disturbance with unknown magnitude. The backstepping method for infinite-dimensional system is adopted in the design of the feedback law. It is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity. Copyright © 2010 John Wiley & Sons, Ltd.
REFERENCES
- 1 Logemann H, Zwart H. Some remarks on adaptive stabilization of infinite-dimensional systems. Systems and Control Letters 1991; 16: 199–207.
- 2 Townley S. Simple adaptive stabilization of output feedback stabilizable distributed parameter systems. Dynamic Control 1995; 5: 107–123.
- 3 Dahleh MW, Hopkins Jr W. Adaptive stabilization of single-input single-output delay systems. IEEE Transactions on Automatic Control 1986; 31: 577–579.
- 4 Kobayashi T. Global adaptive stabilization of infinite-dimensional systems. Systems and Control Letters 1987; 9: 215–223.
- 5 Kobayashi T. Adaptive stabilization and regulator design for distributed-parameter systems in the case of collocated actuators and sensors. IMA Journal of Mathematical Control and Information 1999; 16: 363–367.
10.1093/imamci/16.4.363 Google Scholar
- 6 Guo BZ, Luo ZH. Intial-boundary value problem and exponential decay for a flexible-beam vibration with gain adaptive direct strain feedback control. Nonlinear Analysis, TMA 1996; 27: 353–365.
- 7 Logemann H, Townley S. Adaptive control of infinite-dimensional systems without parameter estimation: an overview. IMA Journal of Mathematical Control and Information 1997; 14: 175–206.
10.1093/imamci/14.2.175 Google Scholar
- 8 Krstic M. Systematization of approaches to adaptive boundary stabilization of PDEs. Journal of Robust and Nonlinear Control 2006; 16: 801–818.
- 9 Krstic M, Smyshlyaev A. Adaptive boundary control for unstable parabolic PDEs—part I: Lyapunov design. IEEE Transactions on Automatic Control 2008; 53: 1575–1591.
- 10 Smyshlyaev A, Krstic M. Adaptive boundary control for unstable parabolic PDEs—part II: estimation-based designs. Automatica 2008; 43: 1543–1556.
- 11 Smyshlyaev A, Krstic M. Adaptive boundary control for unstable parabolic PDEs—part III: output-feedback examples with swapping identifiers. Automatica 2007; 43: 1557–1564.
- 12 Kobayashi T, Sakamoto T, Oya M. Adaptive regulator design for undamped second-order hyperbolic systems with output disturbances. IMA Journal of Mathematical Control and Information 2008; 25: 205–219.
- 13 Kobayashi T. Adaptive regulator design of a viscous burgers' system by boundary control. IMA Journal of Mathematical Control and Information 2001; 18: 513–527.
10.1093/imamci/18.3.427 Google Scholar
- 14 Liu W, Krstic M. Adaptive control of Burger's equation with unknown viscosity. International Journal of Adaptive Control and Signal Processing 2001; 15: 745–766.
- 15 Guo BZ, Guo W. Adaptive stabilization for a Kirchhoff-type nonlinear beam under boundary output feedback control. Nonlinear Analysis, TMA 2007; 66: 427–441.
- 16 Chodavarapu PA, Spong MW. On noncollocated Control of a single flexible link. Proceedings of the IEEE International Conference on Robotics and Automation, Minneapolis, MN, 1996; 1101–1106.
- 17 Liu LY, Yuan K. Noncollocated passivity-based control of a single-link flexible robot with revolute joint. IEEE Transactions on Automatic Control 1997; 42: 53–56.
10.1109/9.553687 Google Scholar
- 18 Cannon RH, Schmitz E. Initial experiments on the end-point control of a flexible one-link robot. International Journal of Robotics Research 1984; 3: 62–75.
- 19 Lacarbonara W, Yabuno H. Closed-loop non-linear control of an initally imperfect beam with non-colocated input. Journal of Sound and Vibration 2004; 273: 695–711.
- 20 de Querioz MS, Rhan CD. Boundary control and noise in distributed parameter systems: an overview. Mechanical Systems and Signal Processing 2002; 16: 19–38.
- 21 Spector VA, Flashner H. Modelling and design implications of non-collocated control in flexible systems. ASME Journal of Dynamic Systems, Measurement and Control 1990; 112: 186–193.
- 22 Ryu JH, Kwon DS, Hannsford B. Control of a flexible manipulator with non-collocated feedback: time-domain passivity approach. In Problems in Robotics, A Bicchi, HI Christensen, D Prattichizzo (eds), Springer Tracts in Advanced Robotics Series, vol. 4. Spring: Berlin, 2003; 121–134.
10.1007/3-540-36224-X_8 Google Scholar
- 23 Udwadia FE. Noncollocated point control of nondispersive distributed-parameters systems using time delays. Applied Mathematics and Computations 1991; 42: 23–63.
- 24 Wu ST. Virtual passive control of flexible arms with collocated and noncollocated feedback. Journal of Robotic Systems 2001; 18: 645–655.
- 25 Yang B, Mote Jr CD. On time delay in noncollocated control of flexible mechanical systems. ASME Journal of Dynamic Systems, Measurement, and Control 1992; 114: 409–415.
- 26 Yuan K, Liu LY. On the stability of zero dynamics of a single-link flexible manipulator for a class of parameter outputs. Journal of Robotic Systems 2003; 20: 581–586.
- 27 Deguenon AJ, Geguenon, Sallet G, Xu CZ. A Kalman observer for infinte-dimensional skew-symmentric symmetric systems with application to an elastic beam. Proceedings of the Second International Symposium on Communications, Control and Signal Processing, March 2006, Marrakech, Morocco.
- 28 Guo BZ, Xu CZ. The stabilization of a one-dimensional wave equation by boundary feedback with non-collocated observation. IEEE Transactions on Automatic Control 2007; 52: 371–377.
- 29 Krstic M, Guo BZ, Balogh A, Smyshlyaev A. Output-feedback stabilization of an unstable wave equation. Automatica 2008; 44: 63–74.
- 30 Smyshlyaev A, Kristic M. Backstepping observers for a class of parabolic PDEs. System Control Letters 2005; 54: 613–625.
- 31 Guo BZ, Guo W. The strong stabilization of a one-dimensional wave equation by non-collocated dynamic boundary feedback control. Automatica 2009; 45: 790–797.
- 32 Luo ZH, Guo BZ, Morgul O. Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer: London, 1999.
10.1007/978-1-4471-0419-3 Google Scholar
- 33 Guo BZ, Yu R. The Riesz basis property of discrete operators and application to a Euler–Bernoulli beam equation with boundary linear feedback control. IMA Journal of Mathematical Control and Information 2001; 18: 241–251.
- 34 Guo BZ, Zhang X. The regularity of the wave equation with partial Dirichlet control and colocated observation. SIAM Journal on Control and Optimization 2005; 44: 1598–1613.
- 35 Weiss G. Admissible observation operators for linear semigroups. Israel Journal of Mathematics 1989; 65: 17–43.
- 36 Trefethen LN. Spectral Methods in Matlab. SIAM: Philadelphia, PA, 2000.
10.1137/1.9780898719598 Google Scholar
