Adaptive boundary stabilization is investigated for a class of PDE-ODE cascade systems with general uncertain harmonic disturbances. The essential difference between this paper and the existing related literature is the presence of the uncertain disturbances belonging to an unknown interval, which makes the problem unsolved so far. Motivated by the existing related literature, the paper develops the adaptive boundary stabilization for the PDE-ODE cascade system in question. First, an adaptive boundary feedback controller is constructed in two steps by adaptive and Lyapunov techniques. Then, it is shown that the resulting closed-loop system is well-posed and asymptotically stable, by the semigroup approach and LaSalle’s invariance principle, respectively. Moreover, the parameter estimates involved in the designed controller are shown to ultimately converge to their own real values. Finally, the effectiveness of the proposed method is illustrated by a simulation example.

Accepted:

DOI: 10.1051/cocv/2016001

Keywords: PDE-ODE cascade systems, uncertain harmonic disturbances, boundary stabilization, adaptive technique, semigroup approach

^{1}; Liu, Yungang

^{1}; Li, Jian

^{2}

@article{COCV_2017__23_2_497_0, author = {Xu, Zaihua and Liu, Yungang and Li, Jian}, title = {Adaptive stabilization for a class of {PDE-ODE} cascade systems with uncertain harmonic disturbances}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {497--515}, publisher = {EDP-Sciences}, volume = {23}, number = {2}, year = {2017}, doi = {10.1051/cocv/2016001}, mrnumber = {3608091}, zbl = {1358.93095}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2016001/} }

TY - JOUR AU - Xu, Zaihua AU - Liu, Yungang AU - Li, Jian TI - Adaptive stabilization for a class of PDE-ODE cascade systems with uncertain harmonic disturbances JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 497 EP - 515 VL - 23 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016001/ DO - 10.1051/cocv/2016001 LA - en ID - COCV_2017__23_2_497_0 ER -

%0 Journal Article %A Xu, Zaihua %A Liu, Yungang %A Li, Jian %T Adaptive stabilization for a class of PDE-ODE cascade systems with uncertain harmonic disturbances %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 497-515 %V 23 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016001/ %R 10.1051/cocv/2016001 %G en %F COCV_2017__23_2_497_0

Xu, Zaihua; Liu, Yungang; Li, Jian. Adaptive stabilization for a class of PDE-ODE cascade systems with uncertain harmonic disturbances. ESAIM: Control, Optimisation and Calculus of Variations, Volume 23 (2017) no. 2, pp. 497-515. doi : 10.1051/cocv/2016001. http://archive.numdam.org/articles/10.1051/cocv/2016001/

Disturbance rejection in $2\times 2$ linear hyperbolic systems. IEEE Trans. Automat. Control 58 (2013) 1095–1106. | DOI | MR | Zbl

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

and ,Stabilization of an overhead crane with a variable length flexible cable. Comput. Appl. Math. 21 (2002) 101–134. | MR | Zbl

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

, , and ,V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, Boston (1993). | MR | Zbl

Compensation of wave actuator dynamics for nonlinear systems. IEEE Trans. Automat. Control 59 (2014) 1555–1570. | DOI | MR | Zbl

and ,R.F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York (1995). | MR | Zbl

Control design of an overhead crane system from the perspective of stabilizing undesired oscillations. IMA J. Math. Control Inform. 28 (2011) 267–278. | DOI | MR | Zbl

,Force control and exponential stabilisation of one-link flexible arm. Int. J. Control 87 (2014) 1794–1807. | DOI | MR | Zbl

, and ,A. Hasan, Disturbance Attenuation of $n+1$ Coupled Hyperbolic PDEs. Proceedings of the IEEE Conference on Decision and Control. Los Angeles, USA (2014).

A. Hasan, Adaptive Boundary Control and Observer of Linear Hyperbolic Systems with Application to Managed Pressure Drilling. Proceedings of the ASME Dynamic Systems and Control Conference. San Antonio, USA (2014).

Robust adaptive boundary control of a flexible marine riser with vessel dynamics. Automatica 47 (2011) 722–732. | DOI | MR | Zbl

, , , and ,Boundary control of a flexible riser with the application to marine installation. IEEE Trans. Ind. Electron. 60 (2013) 5802–5810. | DOI

, and ,Adaptive control of a flexible crane system with the boundary output constraint. IEEE Trans. Ind. Electron. 61 (2014) 4126–4133. | DOI

, and ,Control of coupled vessel, crane, cable, and payload dynamics for subsea installation operations. IEEE Trans. Control Syst. Technol. 19 (2011) 208–220. | DOI

, and ,Adaptive output feedback stabilization for one-dimensional wave equation with corrupted observation by harmonic disturbance. SIAM J. Control Optim. 51 (2013) 1679–1706. | DOI | MR | Zbl

and ,Parameter estimation and non-collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance. IEEE Trans. Automat. Control 58 (2013) 1631–1643. | DOI | MR | Zbl

and ,Stabilization and regulator design for a one-dimensional unstable wave equation with input harmonic disturbance. Int. J. Robust Nonlin. Control 23 (2013) 514–533. | DOI | MR | Zbl

and ,Parameter estimation and stabilisation for a one-dimensional wave equation with boundary output constant disturbance and non-collocated control. Int. J. Control 84 (2011) 381–395. | DOI | MR | Zbl

and ,C.W. Kim, K.S. Hong and G. Lodewijks, Anti-sway control of container cranes: an active mass-damper approach. Proceedings of the SICE Annual Conference. Sapporo, Japan (2004).

Compensating a string PDE in the actuation or sensing path of an unstable ODE. IEEE Trans. Automat. Control 54 (2009) 1362–1368. | DOI | MR | Zbl

,M. Krstić, I. Kanellakopoulos and P. Kokotović, Nonlinear and adaptive control design. John Wiley $$ Sons, New York (1995).

Adaptive stabilization of coupled PDE-ODE systems with multiple uncertainties. ESAIM: COCV 20 (2014) 488–516. | Numdam | MR | Zbl

and ,Adaptive stabilization for ODE systems via boundary measurement of uncertain diffusion-dominated actuator dynamics. Int. J. Robust Nonlin. Control 24 (2014) 3214–3238. | DOI | MR | Zbl

and ,Z.H. Luo, B.Z. Guo and O. Morgul, Stability and Stabilization of Infinite Dimensional Systems with Applications. Springer, London (1999). | MR | Zbl

Boundary optimal (LQ) control of coupled hyperbolic PDEs and ODEs. Automatica 49 (2013) 526–533. | DOI | MR | Zbl

, , and ,A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983). | MR | Zbl

Propagation, delays and stabilization I. J. Control Eng. Appl. Inform. 10 (2008) 11–17.

,Boundary stabilization of hyperbolic systems related to overhead cranes. IMA J. Math. Control Inform. 25 (2008) 353–366. | DOI | MR | Zbl

,A. Smyshlyaev and M. Krstić, Adaptive control of parabolic PDEs. Princeton University Press, New Jersey (2010). | MR | Zbl

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

and ,W.Y. Yang, W. Cao, T.S. Chung and J. Morris, Applied Numerical Methods Using MATLAB. John Wiley & Sons, New Jersey (2005). | MR | Zbl

E. Zeidler, Nonlinear Functional Analysis and its Applications III: Variational Methods and Optimization. Springer, New York (1985). | MR | Zbl

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