Boundary Backstepping Control for Multi-Dimensional Wave Equations

Yıl 2018, Cilt: 10 Sayı: 1, 172 - 182, 29.01.2017

Aziz Sezgin

https://doi.org/10.29137/umagd.441671

Öz

In
this study, we consider the problem of the stabilization of chosen unstable
multi-dimensional wave equations by using boundary backstepping control theory.
For this purpose, we design boundary backstepping controllers inspired by the
1-D unstable wave equation stabilization procedure and the target systems are
considered by using Lyapunov stability procedure. We assume that one side of
the boundary is hinged and the other side is controlled for each direction.
Thus, we design two controllers for 2-D domain, three controllers for 3-D
domain and “n” controllers for n-D domain. Generalized Volterra/Fredholm type
transformations are used to map the unstable systems to an exponentially stable
system.

Anahtar Kelimeler

Backstepping Control, Distributed parameter systems, multi dimensional wave equation

Çok Boyutlu Dalga Denklemleri için Geri Adımlamalı Sınır Değer Kontrolü

Öz

Bu
çalışmada seçilen bazı kararsız çok boyutlu dalga denklemleri geri adımlamalı
sınır değer kontrol kuramı ile kararlı hale getirilmiştir. Bu amaçla bir
boyutlu geri adımlamalı kontrol teorisinden yola çıkılarak çok boyutlu
sistemler için kontrolcü tasarımı yapılmış ve hedef sistemler Lyapunov
kararlılık analizi ile incelenmiştir. Her boyut için sistemin bir ucundan
tutturulup diğer ucundan kontrol uygulandığı düşünülmüştür. Böylelikle iki
boyutlu bir sistem için iki adet, üç boyutlu bir sistem için üç adet ve n
boyutlu bir sistem için de n adet kontrolcü tasarlanmıştır. Kararsız sistemin
üstel kararlı bir hedef sisteme dönüştürülebilmesi için Volterra/Felholm tipi
dönüşümler kullanılmıştır.

Anahtar Kelimeler

Geri adımlamalı Kontrol, Dağılmış parametreli sistemler, çok boyutlu dalga denklemi

Kaynakça

• Krstic, M., Smyshlyaev, A. (2003). Explicit State and Output Feedback Boundary Controllers for Partial Differential Equations, Journal of Automatic Control, University of Belgrade, 13(2), 1-9. doi:10.2298/JAC0302001S
• Krstic, M., Smyshlyaev, A. (2008). Boundary Control of PDEs, A Course on Backstepping Designs, Siam.
• Krstic, M., Smyshlyaev, A. (2008). Adaptive control of PDEs. Annual Reviews in Control, 32, 149-160. doi:10.1016/j.arccontrol.2008.05.001
• Krstic, M., Guo, B. Z., Balogh, A., Smyshlyaev, A. (2008). Output-feedback stabilization of an unstable wave equation. Automatica , 44, 63-74. doi:10.1016/j.automatica.2007.05.012
• Krstic, M. (2011). Dead-Time Compensation for Wave/String PDEs. Journal of Dynamic Systems, Measurement, and Control , 133, 031004/1–13, doi:10.1115/1.4003638
• Sezgin A., Krstic, M. (2015). Boundary Backstepping Control of Flow-Induced Vibrations of a Membrane at High Mach Numbers. Journal of Dynamic Systems, Measurement, and Control, 137(8), 081003/1-8 , doi: 10.1115/1.4029468
• Krstic, M., Smyshlyaev, A. (2008). Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays. Systems and Control Letters, 57, 750-758. doi:10.1016/j.sysconle.2008.02.005.
• Bekiaris-Liberis , N., Krstic, M., (2010). Compensating the distributed effect of a wave PDE in the actuation or sensing path of MIMO LTI systems. Systems and Control Letters, 59, 713-719. doi:10.1016/j.sysconle.2008.02.005.
• Cheng, M.B., Radisavljevic, V., Su, W.C. (2011). Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties. Automatica, 47(2), 381-387. doi:10.1016/j.automatica.2010.10.045
• Ng., J., Dubljevic, S., (2012). Optimal boundary control of a diffusionconvection-reaction PDE model with time-dependent spatial domain: Czochralski crystal growth process. Chemical Engineering Science, 67(1), 111-119. doi:10.1016/j.ces.2011.06.050
• Chrysafinos, K., Gunzburger, M.D., Hou, L.S. (2006). Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE. Journal of Mathematical Analysis and Applications, 323(2), 891-912. doi:10.1016/j.jmaa.2005.10.053
• Tang., S., Xie, C., (2011). State and output feedback boundary control for a coupled PDE-ODE system. Systems and Control Letters, 60(8), 540-545. doi:10.1016/j.sysconle.2011.04.011.
• Bekiaris-Liberis , N., Krstic, M., (2011). Compensating the Distributed Effect of Diffusion and Counter-Convection in Multi-Input and Multi-Output LTI Systems. IEEE Transactions on Automatic Control, 56(3), 637-643. doi: 10.1109/CDC.2010.5716993
• Ramirez, J.A., Puebla, H., Ochoa-Tapia, J.A. (2001). Linear boundary control for a class of nonlinear PDE processes. Systems and Control Letters, 44(5), 395-403. doi:10.1016/S0167-6911(01)00159-1
• Cheung, W.S., (2001). Some New Poincare–Type inequalities. Bulletin of the Australian Mathematical Society, 63(2), 321–327,. Doi:10.1017/S0004972700019365
• Krstic, M. (2010). Adaptive Control of an Anti-Stable Wave PDE: Theory and Application Oil Drilling, Dynamics of Continuous, Discrete and Impulsive System, 17, 853-882, doi:10.3182/20130703-3-FR-4038.00154
• Cox, S., Zuazua, E., (1994). The rate at which energy decays in a string damped at one end. Comm. Partial Differential Equations, 19, 213-243.
• Krstic, M., Kanellakopoulos I., Kokotovic, P., (1995). Nonlinear and Adaptive Control Design, Wiley.