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Nonlinear Feedback Control and Artificial Intelligence Computational
Methods applied to a Dissipative Dynamic Contact Problem
Daniela Danciu
1
, Andaluzia Cristina Matei
2
, Sorin Micu
2,3
and Ionel Rovent¸a
2
1
Department of Automation, Electronics and Mechatronics, University of Craiova, 200585, Craiova, Romania
2
Department of Mathematics, University of Craiova, 200585, Craiova, Romania
3
Institute of Mathematical Statistics and Applied Mathematics, 70700, Bucharest, Romania
Keywords:
Hyperbolic partial differential equation, Contact problem, Method of Lines, Galerkin method, Cellular Neural
Networks, Computational methods, Neuromathematics.
Abstract:
In this paper we consider a vibrational percussion system described by a one-dimensional hyperbolic partial
differential equation with boundary dissipation at one extremity and a normal compliance contact condition at
the other extremity. Firstly, we obtain the mathematical model using the Calculus of variations and we prove
the existence of weak solutions. Secondly, we focus on the numerical approximation of solutions by using a
neuromathematics approach – a well-structured numerical technique which combines a specific approach of
Method of Lines with the paradigm of Cellular Neural Networks. Our technique ensures from the beginning
the requirements for convergence and stability preservation of the initial problem and, exploiting the local con-
nectivity of the approximating system, leads to a low computational effort. A comprehensive set of numerical
simulations, considering both contact and non-contact cases, ends the contribution.
1 INTRODUCTION
The purpose of the paper is twofold. Firstly, we
verify, on a new mathematical model from Con-
tact Mechanics, a well-structured technique for solv-
ing hyperbolic partial differential equations (hPDEs),
by using Method of Lines (MoL) combined with
the features and flexibility of Cellular Neural Net-
works (CNNs) – nonlinear processing devices having
a large amount of applications from image processing
to numerical solving of differential equations. This
is a neuromathematics approach, i.e., according to
(Galushkin, 2010), an approach used for solving both
non-formalized (or weakly formalized) and formal-
ized problems using the neural networks paradigm.
Our approach addresses the formalized-type prob-
lems where the structure of the neural network is
based only on the “natural parallelism” of the problem
itself and does not need a learning process based on
experimental data. We have already successfully ap-
plied our technique for the overhead crane with non-
homogeneous cable (Danciu, 2013a), the torsional
stick slip oscillations in oilwell drillstrings (Danciu,
2013b) and the controlled flexible arm of an ocean
vessel riser (Danciu and R˘asvan, 2014).
It is assumed that our method is even more effec-
tive for high dimensions, where the well-organized
technique may have the following advantages: a) ac-
curacy (for example, fewer rounding errors), b) ro-
bustness to ill numerical conditioning and c) use of
existing high-performance software for solving ordi-
nary differential equations (ODEs).
Secondly, we study a particular robotic system of
type vibrational percussion system, which is mod-
elled by a one-dimensional hPDE with a dissipative
boundary condition at one extremity and a contact
nonlinear phenomenon at the other extremity. The
study concerns mathematical modelling of the sys-
tem, existence of the weak solutions for the hPDE
problem, numerical approximation and asymptotic
behavior. The problem can be viewed as a structure
with nonlinear feedback and reference signal. From
the mathematical point of view, the main difficulties
reside in the nonlinearity of the problem, the lack of a
regularization effect of solutions, possibly the effects
of discretization on the dissipative mechanism.
The rest of paper is organized as follows. Sec-
tion 2 deals with mathematical modelling of the sys-
tem via Calculus of variation methods. Section 3 fo-
cuses on the study of the existence of weak solutions
for the hPDE problem, whereas Section 4 is devoted
to numerical approximation of the solution. We im-
528
Danciu D., Matei A., Micu S. and Roven¸ta I..
Nonlinear Feedback Control and Artificial Intelligence Computational Methods applied to a Dissipative Dynamic Contact Problem.
DOI: 10.5220/0005055005280539
In Proceedings of the 11th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2014), pages 528-539
ISBN: 978-989-758-039-0
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)