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 systems, spatially varying coefficient, adaptive stabilization, infinite-dimensional backstepping

@article{COCV_2014__20_2_488_0, author = {Li, Jian and Liu, Yungang}, title = {Adaptive stabilization of coupled {PDE-ODE} systems with multiple uncertainties}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {488--516}, publisher = {EDP-Sciences}, volume = {20}, number = {2}, year = {2014}, doi = {10.1051/cocv/2013072}, mrnumber = {3264213}, zbl = {1285.93084}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2013072/} }

TY - JOUR AU - Li, Jian AU - Liu, Yungang TI - Adaptive stabilization of coupled PDE-ODE systems with multiple uncertainties JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 488 EP - 516 VL - 20 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2013072/ DO - 10.1051/cocv/2013072 LA - en ID - COCV_2014__20_2_488_0 ER -

%0 Journal Article %A Li, Jian %A Liu, Yungang %T Adaptive stabilization of coupled PDE-ODE systems with multiple uncertainties %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 488-516 %V 20 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2013072/ %R 10.1051/cocv/2013072 %G en %F COCV_2014__20_2_488_0

Li, Jian; Liu, Yungang. Adaptive stabilization of coupled PDE-ODE systems with multiple uncertainties. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 2, pp. 488-516. doi : 10.1051/cocv/2013072. http://archive.numdam.org/articles/10.1051/cocv/2013072/

[1] Backstepping boundary control for first order hyperbolic PDEs and application to systems with actuator and sensor delays. Syst. Control Lett. 57 (2008) 750-758. | MR | Zbl

and ,[2] Adaptive trajectory tracking despite unknown input delay and plant parameters. Automatica 45 (2009) 2074-2081. | MR | Zbl

and ,[3] Compensating a string PDE in the actuation or sensing path of an unstable ODE. IEEE Trans. Automatic Control 54 (2009) 1362-1368. | MR

,[4] Control of PDE-ODE cascades with Neumann interconnections. J. Franklin Institute 347 (2010) 284-314. | MR | Zbl

and ,[5] Compensating actuator and sensor dynamics governed by diffusion PDEs. Syst. Control Lett. 58 (2009) 372-377. | MR | Zbl

,[6] Adaptive control of the ODE systems with uncertain diffusion-dominated actuator dynamics. Internat. J. Control 85 (2012) 868-879. | MR | Zbl

and ,[7] Stabilization for a coupled PDE-ODE control system. J. Franklin Institute 348 (2011) 2142-2155. | MR | Zbl

and ,[8] State and output feedback boundary control for a coupled PDE-ODE system. Syst. Control Lett. 60 (2011) 540-545. | MR | Zbl

and ,[9] Boundary stabilization of a coupled wave-ODE system. Proc. Chinese Control Conf. Yantai, China (2011) 1048-1052.

and ,[10] 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.

and ,[11] Backstepping boundary control of flexible-link electrically driven gantry robots. IEEE/ASME Trans. Mechatr. 3 (1998) 60-66.

, and ,[12] Orientation and stabilization of a flexible beam attached to a rigid body: planar motion. IEEE Trans. Autom. Control 36 (1991) 953-962. | MR | Zbl

,[13] Integrator backstepping control of a brush DC motor turning a robotic load. IEEE Trans. Control Syst. Technol. 2 (1994) 233-244.

, and ,[14] A note on stabilization of a hybrid PDE-ODE system. Proc. IEEE Conf. Decision Control. Orlando, Florida, USA (2001) 137-142.

,[15] Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach. Automatica 36 (2000) 587-593. | MR | Zbl

and ,[16] Feedback stabilization of a hybrid PDE-ODE systems: application to an overhead crane. Math. Control, Signals, Syst. 7 (1994) 1-22. | MR | Zbl

, , and ,[17] Control of coupled PDEs-ODEs using input-output linearization: application to cracking furnace. Chemical Engrg. Sci. 75 (2012) 144-151.

, and ,[18] Adaptive boundary control for unstable parabolic PDEs-part I: Lyapunov design. IEEE Trans. Automatic Control 53 (2008) 1575-1591. | MR | Zbl

and ,[19] Appl. Numer. Methods Using Matlab. John Wiley & Sons, Hoboken, New Jersey (2005). | MR

, , and ,[20] Boundary control of three-dimensional inextensible marine risers. J. Sound and Vibration 327 (2009) 299-321.

and ,[21] Robust H∞ control of uncertain markovian jump systems with time-delay. IEEE Trans. Autom. Control 45 (2000) 77-83. | MR | Zbl

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