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Analysis of a Nonlinear Control Law with Cubic Nonlinearity
Melnikov Vitaly
1,2 a
, Melnikov Gennady
2 b
and Dudarenko Natalia
2 c
1
Department of Mechanics, Saint Petersburg Mining University, 2, 21st Line, Saint-Petersburg, 199106, Russia
2
Department of Control Systems and Industrial Robotics, ITMO University,
49 Kronverksky Pr., Saint-Petersburg, 197101, Russia
Keywords:
Nonlinear Control, Cubic Nonlinearity, Polynomial Transformation.
Abstract:
The paper is considered the problem of improving the stabilization of nonlinear controlled systems. It is
proposed to solve the problem by introducing cubic components into the control law. Using the method of
polynomial transformation, a comparative analysis of the influence of cubic components on the dynamics of a
controlled systems is presented. As a result, some conclusions about the choice of the structure and parameters
of the nonlinear control law are presented and recommendations are given.
1 INTRODUCTION
The presence of nonlinearity in the model description
of any controlled rigid body can have a negative im-
pact on the quality of control system processes. This
problem is relevant for many controlled robotic sys-
tems. There are many approaches to reduce the influ-
ence of nonlinearity with correction devices and pro-
vide a control system with required dynamic quality
(Popov, 1989; Frank et al., 2004; Ivanov et al., 2014).
For example, correction methods for nonlinear sys-
tems can be realized with the changing of the struc-
ture and parameters of a linear part of the system, ad-
ditional feedbacks or elements. There are modes of a
control system when it is reasonable to do the correc-
tion of a controlled system with nonlinear control law
(Fang et al., 2003; Ilyukhin et al., 2015; Reichensdor-
fer et al., 2018).
The choice of parameters of the nonlinear con-
trol law makes it possible to control the regulation
time of the process and its oscillation, which im-
proves the dynamic qualities of a wide class of sys-
tems, including aircraft stabilization systems, robotic
and mechatronic complexes and its applications (Ser-
aji, 1998; Ansarieshlaghi and Eberhard, 2020; Alek-
sandrov et al., 2018; Qi et al., 2018). The case of non-
linear control law with cubic nonlinearity is consid-
ered in the paper and the influence of the cubic com-
ponents of the nonlinear control law on the system
a
https://orcid.org/0000-0002-2114-7891
b
https://orcid.org/0000-0003-2606-7572
c
https://orcid.org/0000-0002-3553-0584
state variables is analysed. The results of the paper
can be useful for the design of robotic applications
with nonlinear control law including a cubic nonlin-
earity.
The paper is laid out as follows. Firstly, the prob-
lem of nonlinear systems correction with a nonlin-
ear control law with cubic nonlinearity is discussed.
Then, a methodology of reducing a first-order aperi-
odic controller to an ideal controller and integrating
a simplified equation is proposed. Thereafter, expres-
sions for the choice of characteristic coefficients and
control parameters are presented. The analysis results
are discussed and the paper is finished with some con-
cluding remarks about the choice of the parameters
for a cubic control law.
2 PROBLEM DESCRIPTION
Let the motion of the controlled system be described
by a system of differential equations of the third order.
¨
θ = −aα + cθ,
˙
α = f (σ),
(1)
where a, c are constants and f (σ) is the known non-
linearity of the control device, represented in the form
of the following odd cubic polynomial
f (σ) = k
∗
σ − k
∗
σ
3
, (2)
where k
∗
, k
∗
are constants σ is a control, a linear or
non-linear function of α, θ,
˙
θ. The function f (σ) can
also be an odd piecewise linear function admitting a
Vitaly, M., Gennady, M. and Natalia, D.
Analysis of a Nonlinear Control Law with Cubic Nonlinearity.
DOI: 10.5220/0010604407010706
In Proceedings of the 18th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2021), pages 701-706
ISBN: 978-989-758-522-7
Copyright
c
2021 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
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