Abstract
The paper is concerned with the stabilization of a class of coupled PDE-ODE systems with spatially varying coefficient, via state-feedback or output-feedback. The system is more general than that of the related literature due to the presence of the spatially varying coefficient which makes the problem more difficult to solve. By infinite-dimensional backstepping method, both state-feedback and output-feedback stabilizing controllers are explicitly constructed, which guarantee that the closed-loop system is exponentially stable in the sense of certain norm. It is worthwhile pointing out that, in the case of output-feedback, by appropriately choosing the state observer gains, the severe restriction on the ODE sub-system in the existing results is completely removed. A simulation example is presented to illustrate the effectiveness of the proposed method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Krstić M, Compensating actuator and sensor dynamics governed by diffusion PDEs, Systems and Control Letters, 2009, 58(5): 372–377.
Li J and Liu Y G, Adaptive control of the ODE systems with uncertain diffusion-dominated actuator dynamics, International Journal of Control, 2012, 85(7): 868–879.
Tang S X and Xie C K, State and output feedback boundary control for a coupled PDE-ODE system, Systems and Control Letters, 2011, 60(8): 540–545.
Tang S X and Xie C K, Stabilization for a coupled PDE-ODE control system, Journal of the Franklin Institute, 2011, 348(8): 2142–2155.
Zhou Z C and Tang S X, Boundary stabilization of a coupled wave-ODE system, Proceedings of the Chinese Control Conference, Yantai, China, 2011.
Lynch A F and Wang D, Flatness-based control of a flexible beam in a gravitational field, Proceedings of American Control Conference, Boston, Massachusetts, USA, 2004.
Rawlings J B, Witkowski W R, and Eaton J W, Modelling and control of crystallizers, Powder Technology, 1992, 69(1): 3–9.
Masoud A A and Masoud S A, A self-organizing, hybrid PDE-ODE structure for motion control in informationally-deprived situations, Proceedings of the IEEE Conference on Decision and Control, Tampa, Florida, USA, 1998.
Baicu C F, Rahn C D, and Dawson D M, Backstepping boundary control of flexible-link electrically driven gantry robots, IEEE/ASME Transactions on Mechatronics, 1998, 3(1): 60–66.
Morgül Ö, Orientation and stabilization of a flexible beam attached to a rigid body: Planar motion, IEEE Transactions on Automatic Control, 1991, 36(8): 953–962.
Dawson D M, Carroll J J, and Schneider M, Integrator backstepping control of a brush DC motor turning a robotic load, IEEE Transactions on Control Systems Technology, 1994, 2(3): 233–244.
Chentouf D B, A note on stabilization of a hybrid PDE-ODE system, Proceedings of the IEEE Conference on Decision and Control, Orlando, Florida, USA, 2011.
d’Andrea-Novel B and Coron J M, Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach, Automatica, 2000, 36(4): 587–593.
d’Andrea-Novel B, Boustany F, Conrad F, and Rao B P, Feedback stabilization of a hybrid PDEODE systems: Application to an overhead crane, Mathematics of Control, Signals, and Systems, 1994, 7(1): 1–22.
d’Andrea-Novel B, Boustany F, and Conrad F, Control of an overhead crane: Stabilization of flexibles, Lecture Notes in Control and Information Sciences, 1992, 178: 1–26.
Moghadam A A, Aksikas I, Dubljevic S, and Forbes J F, LQ control of coupled hyperbolic PDEs and ODEs: Application to a CSTR-PFR system, Proceedings of the Ninth International Symposium on Dynamics and Control of Process Systems, Leuven, Belgium, 2010, 713–718.
Panjapornpon C, Limpanachaipornkul P, and Charinpanitkul T, Control of coupled PDEs-ODEs using input-output linearization: Application to cracking furnace, Chemical Engineering Science, 2012, 75(16): 144–151.
Miletic M and Arnold A, Euler-Bernoulli beam with boundary control: Stability and FEM, Proceedings in Applied Mathematics and Mechanics, 2011, 11(1): 681–682.
Yang W Y, Cao W, Chung T S, and Morris J, Applied Numerical Methods Using Matlab, New Jersey, John Wiley & Sons, Inc., Hoboken, 2005.
Do K D and Pan J, Boundary control of three-dimensional inextensible marine risers, Journal of Sound and Vibration, 2009, 327(3): 299–321.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research is supported by the National Natural Science Foundations of China under Grant Nos. 60974003, 61143011, 61273084, and 61233014, the Natural Science Foundation for Distinguished Young Scholar of Shandong Province of China under Grant No. JQ200919, and the Independent Innovation Foundation of Shandong University under Grant No. 2012JC014.
This paper was recommended for publication by Editor ZHANG Bingyu.
Rights and permissions
About this article
Cite this article
Li, J., Liu, Y. Stabilization of coupled pde-ode systems with spatially varying coefficient. J Syst Sci Complex 26, 151–174 (2013). https://doi.org/10.1007/s11424-013-2070-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-013-2070-0