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Adaptive stabilization of coupled PDE–ODE systems with multiple uncertainties∗

Published online by Cambridge University Press:  14 March 2014

Jian Li  and
Yungang Liu
Jian Li
Affiliation:
School of Control Science and Engineering, Shandong University, Jinan 250061, P.R. China. [email protected] School of Mathematics and Information Science, Yantai University, Yantai 264005, P.R. China
Yungang Liu
Affiliation:
School of Control Science and Engineering, Shandong University, Jinan 250061, P.R. China. [email protected]
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Abstract

The adaptive stabilization is investigated for a class of coupled PDE-ODE systems with multiple uncertainties. The presence of the multiple uncertainties and the interaction between the sub-systems makes the systems to be considered more general and representative, and moreover it may result in the ineffectiveness of the conventional methods on this topic. Motivated by the existing literature, an infinite-dimensional backsteppping transformation with new kernel functions is first introduced to change the original system into a target system, from which the control design and performance analysis of the original system will become quite convenient. Then, by certainty equivalence principle and Lyapunov method, an adaptive stabilizing controller is successfully constructed, which guarantees that all the closed-loop system states are bounded while the original system states converging to zero. A simulation example is provided to validate the proposed method.

Keywords

Coupled PDE-ODE systemsspatially varying coefficientadaptive stabilizationinfinite-dimensional backstepping
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 2 , April 2014 , pp. 488 - 516
Copyright
© EDP Sciences, SMAI, 2014

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References

Krstić, M. and Smyshlyaev, A., Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst. Control Lett. 57 (2008) 750–758. Google Scholar
Pietri, D.B. and Krstić, M., Adaptive trajectory tracking despite unknown input delay and plant parameters. Automatica 45 (2009) 2074–2081. Google Scholar
Krstić, M., Compensating a string PDE in the actuation or sensing path of an unstable ODE. IEEE Trans. Automatic Control 54 (2009) 1362–1368. Google Scholar
Susto, G.A. and Krstić, M., Control of PDE-ODE cascades with Neumann interconnections. J. Franklin Institute 347 (2010) 284–314. Google Scholar
Krstić, M., Compensating actuator and sensor dynamics governed by diffusion PDEs. Syst. Control Lett. 58 (2009) 372–377. Google Scholar
Li, J. and Liu, Y.G., Adaptive control of the ODE systems with uncertain diffusion-dominated actuator dynamics. Internat. J. Control 85 (2012) 868–879. Google Scholar
Tang, S.X. and Xie, C.K., Stabilization for a coupled PDE-ODE control system. J. Franklin Institute 348 (2011) 2142–2155. Google Scholar
Tang, S.X. and Xie, C.K., State and output feedback boundary control for a coupled PDE-ODE system. Syst. Control Lett. 60 (2011) 540–545. Google Scholar
Z.C. Zhou and S.X. Tang, Boundary stabilization of a coupled wave-ODE system. Proc. Chinese Control Conf. Yantai, China (2011) 1048–1052.
A.A. Masoud and S.A. Masoud, A self-organizing, hybrid PDE-ODE structure for motion control in informationally-deprived situations. Proc. IEEE Conf. Decision Control. Tampa, Florida, USA (1998) 2535–2540.
Baicu, C.F., Rahn, C.D. and Dawson, D.M., Backstepping boundary control of flexible-link electrically driven gantry robots. IEEE/ASME Trans. Mechatr. 3 (1998) 60–66. Google Scholar
Morgul, O., Orientation and stabilization of a flexible beam attached to a rigid body: planar motion. IEEE Trans. Autom. Control 36 (1991) 953–962. Google Scholar
Dawson, D.M., Carroll, J.J. and Schneider, M., Integrator backstepping control of a brush DC motor turning a robotic load. IEEE Trans. Control Syst. Technol. 2 (1994) 233–244. Google Scholar
D.B. Chentouf, A note on stabilization of a hybrid PDE-ODE system. Proc. IEEE Conf. Decision Control. Orlando, Florida, USA (2001) 137–142.
d’Andrea-Novel, B. and Coron, J.M., Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach. Automatica 36 (2000) 587–593. Google Scholar
d’Andrea-Novel, B., Boustany, F., Conrad, F. and Rao, B.P., Feedback stabilization of a hybrid PDE-ODE systems: application to an overhead crane. Math. Control, Signals, Syst. 7 (1994) 1–22. Google Scholar
Panjapornpon, C., Limpanachaipornkul, P. and Charinpanitkul, T., Control of coupled PDEs-ODEs using input-output linearization: application to cracking furnace. Chemical Engrg. Sci. 75 (2012) 144–151. Google Scholar
Krstić, M. and Smyshlyaev, A., Adaptive boundary control for unstable parabolic PDEs-part I: Lyapunov design. IEEE Trans. Automatic Control 53 (2008) 1575–1591. Google Scholar
W.Y. Yang, W. Cao, T.S. Chung and J. Morris, Appl. Numer. Methods Using Matlab. John Wiley & Sons, Hoboken, New Jersey (2005).
Do, K.D. and Pan, J., Boundary control of three-dimensional inextensible marine risers. J. Sound and Vibration 327 (2009) 299–321. Google Scholar
Cao, Y.Y. and Lam, J., Robust H∞ control of uncertain markovian jump systems with time-delay. IEEE Trans. Autom. Control 45 (2000) 77–83. Google Scholar