Boundary Backstepping Control for Multi-Dimensional Wave Equations

Year 2018, Volume: 10 Issue: 1, 172 - 182, 29.01.2017

Aziz Sezgin

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

Abstract

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.

Keywords

Backstepping Control, Distributed parameter systems, multi dimensional wave equation

Çok Boyutlu Dalga Denklemleri için Geri Adımlamalı Sınır Değer Kontrolü

Abstract

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.

Keywords

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

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