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
701

sufficiently accurate approximation of the form (2)
using the Chebyshev polynomials (Melnikov, 2005;
Melnikov, 2010; Mason and Handscomb, 2002). For
example, the function f (cosϕ) is approximated by
two terms of the Fourier series, then it is assumed that
cosϕ = σ
f (σ) = f (cos ϕ) ≈ A
1
cosϕ+A
3
cos3ϕ =
= (A
1
− 3A
3
)cosϕ + 4A
3
cos
3
ϕ =
= (A
1
− 3A
3
)σ + 4A
3
σ
3
,
(3)
where A
k
=
2
π
π
R
0
f (cos ϕ)cos kϕdϕ. Another example
of f (σ):
f (σ) =

σ , 0 ≤ σ ≤ 0, 4
0.4, 0.4 ≤ σ ≤ 1

≈ 0, 90σ − 0, 56σ
3
.
(4)
Let θ,
˙
θ can take values from some elliptic neigh-
borhood of zero values θ,
˙
θ determined on the phase
plane by an inequality of the form
θ
2
+ a
2
1
˙
θ
2
≤ ε
2
0
, (5)
where a
2
1
is a given coefficient, ε
0
is the main param-
eter characterizing the size of the area. This area will
be called the ”large ellipse ε
0
”, from which we will
extract the ”small ellipse” by the following condition
θ
2
+ a
2
1
˙
θ
2
≤ (
ε
0
2
)
2
. (6)
Definition 1. We will say that a controlled system
has a performance defined as (h
1
, h
2
, h
3
, h
4
) if for any
initial perturbations from the large ellipse ε
0
the per-
turbations remain in some ”overshoot ellipse” h
1
and
by the time t = T they contract inside the ellipse of
”residual perturbations” h
2
and if for any initial per-
turbations from the small perturbation ellipse
ε
0
2
, re-
maining in ellipse h
3
contract over time into an ellipse
h
4
, where ε
0
≥ h
1
≥ h
2
≥ h
3
≥ h
4
, ε
0
 h
2
, ε
0
 h
4
.
Definition 2. Let us consider two systems with per-
formance (h
1
, h
2
, h
3
, h
4
) and (h
0
1
, h
0
2
, h
0
3
, h
0
4
), respec-
tively. We will say that the second system has a higher
performance than the first system if the following in-
equalities are satisfied:
h
1
≥ h
0
1
, h
2
≥ h
2
0
, h
3
≥ h
3
0
, h
4
≥ h
4
0
, (7)
where at least one of the four inequalities is strict, for
example h
1
> h
1
0
.
Let the linear control law have the form
σ ≡
˜
σ ≡
1
k
∗
(−kα + k
1
˙
θ + k
2
θ), (8)
and it provides stabilization of the system with a cer-
tain performance (h
1
, h
2
, h
3
, h
4
) for given ε
0
, T .
Let the parameters of the linear part of system (1),
(2), (8) satisfy the condition that the characteristic
equation has a pair of complex roots λ
1,2
with nega-
tive real parts and one real negative root λ
3
with large
modulus:
λ
3
+ kλ
2
+ (ak
1
− c)λ + ak
2
− ck = 0;
λ
1,2
= −χ ± µi; µ ≥ χ; λ
3
= −p.
(9)
Let the performance be good enough for small
perturbations
ε
0
2
, but insufficient for large perturba-
tions ε
0
. That is, we believe that characteristics h
3
,
h
4
are satisfactory, and h
1
, h
2
are unsatisfactory, and
that the linear law (8) with the selected parameters
is considered the best in the sense that no other lin-
ear law of the form (8) can significantly improve the
performance of the system.
Let us introduce into the control law (8) such non-
linear terms that, without significantly changing the
performance at small perturbations, would improve
the performance at large perturbations. We add to (8)
a set of cubic terms with constant coefficients
σ =
˜
σ +
1
k
∗
∑
ν
1
+ν
2
+ν
3
=3
p
(ν
1
,ν
2
,ν
3
)
α
ν
1
˙
θ
ν
2
θ
ν
3
, (10)
where (ν
1
, ν
2
, ν
3
) is a vector index. If the coefficients
p
(ν
1
,ν
2
,ν
3
)
are small in comparison with the charac-
teristic coefficients of the linear part of the system,
then for small values of the parameters α,
˙
θ, θ the cu-
bic terms in (11) are negligible and we can assume
that σ ≈
˜
σ. This means that at small perturbations the
regulation is performed approximately as according
to the linear law (8), therefore the parameters h
3
0
, h
4
0
of the new system are approximately equal to the pa-
rameters h
3
, h
4
of the system controlled according to
the law (8). For large perturbations from the ellipse
ε
0
, cubic terms can be as large as linear ones, so their
influence becomes significant, and the action of some
terms is to some extent equivalent to a change in the
coefficients k, k
1
, k
2
in the linear law (8). For exam-
ple, the presence of a term p
(0,0,3)
θ
3
is equivalent to a
change in the coefficient at large values of θ, since in
expression (11) two terms can be combined and for
them approximately write
(k
2
+ p
(0,0,3)
θ
2
)θ ≈
≈

k
2
θ ,
|
θ
|
≤
ε
0
2
(k
2
+ p
(0,0,3)
θ
2
av
)θ ,
ε
0
2
≤ θ ≤ ε
0
.
(11)
The main feature of the effect of cubic terms, in
comparison with linear terms, is that they have almost
no effect on the behaviour of the system if the system
is near a stationary state θ =
˙
θ = 0 (for example, it
does not accelerate the system to the position θ = 0,
which would lead to subsequent overshoot in θ = 0).
Consequently, cubic terms can be introduced in order
to improve the performance of the system at signifi-
cant (θ,
˙
θ) without impairing the performance at small
θ,
˙
θ.
ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics
702

Let the control law be given in the form (10),
where cubic terms are to be determined Let’s find out
the influence of each of the cubic terms on the sta-
bilization of the system, determine which of the cu-
bic terms improve the performance of the system, and
which terms are inappropriate to leave in expression
(10). Note that if in the process of synthesizing a lin-
ear system it is more convenient to specify the param-
eters χ, µ, p instead of specifying the coefficients k ,
k
1
; k
2
, then we will use the following formulas.
k = p + 2χ, k
1
=
1
a
(c + χ
2
+ µ
2
+ 2χp),
k
2
=
1
a
(cp + 2cχ + p(χ
2
+ µ
2
)).
(12)
3 REDUCING A FIRST-ORDER
APERIODIC CONTROLLER TO
AN IDEAL CONTROLLER
Let us approximately decrease the order of system (1)
by one using the condition p  χ. This can be inter-
preted as replacing the first order aperiodic controller
with an ideal controller. Let us rewrite system (1) up
to terms of the fifth order under conditions (2), (8),
(10):
˙
α = −kα + k
1
ω + k
2
θ+
+
∑
ν
1
+ν
2
+ν
3
=3
p
1
(ν
1
,ν
2
,ν
3
)
α
ν
1
ω
ν
2
θ
ν
3
,
˙
θ = ω,
˙
ω = −aα + cθ
(13)
where cubic terms are defined by rearranging the ex-
pression
∑
ν
1
+ν
2
+ν
3
=3
p
(ν
1
,ν
2
,ν
3
)
α
ν
1
ω
ν
2
θ
ν
3
−
−
k
∗
k
3
∗
(−kα + k
1
ω + k
2
θ)
3
≡
≡
∑
ν
1
+ν
2
+ν
3
=3
p
1
(ν
1
,ν
2
,ν
3
)
α
ν
1
ω
ν
2
θ
ν
3
. (14)
Instead of α, we introduce a variable z in two stages.
First we put
U = α − Aω − Bθ, (15)
where A and B are determined from the condition that
the new variable satisfies the diagonal equation
˙
U = −pu +
∑
ν
1
+ν
2
+ν
3
=3
Q
(ν
1
,ν
2
,ν
3
)
u
ν
1
ω
ν
2
θ
ν
3
. (16)
We have
A =
k − p
a
=
2χ
a
, B = k
1
−
2pχ
a
. (17)
Then we perform a polynomial transformation
z = u −
∑
ν
1
+ν
2
+ν
3
=3
A
(ν
2
,ν
3
)
ω
ν
2
θ
ν
3
, (18)
we determine the coefficients from the condition that
z satisfies an equation of the form
˙z =
(
−p +
∑
ν
1
+ν
2
+ν
3
=3
a
(ν
1
,ν
2
,ν
3
)
z
ν
1
ω
ν
2
θ
ν
3
)
z. (19)
Substituting expressions (13), (15), (16), (18) into
(19), then reducing each term in the resulting equality
to the form of the Q
(...)
u
ν
1
ω
ν
2
θ
ν
3
, renaming the super-
scripts and equating the coefficients at the u
ν
1
ω
ν
2
θ
ν
3
,
we obtain
[(ν
2
+ 1)p − ν
2
k]A
(ν
2
,ν
3
)
+ (ν
2
+ 1)rA
(ν
2
+1,ν
3
−1)
+
+ (ν
3
+ 1)A
(ν
2
−1,ν
3
+1)
= Q
(0,ν
2
,ν
3
)
, (20)
a
(ν
1
−1,ν
2,
ν
3
)
= Q
(ν
1
,ν
2,
ν
3
)
− (ν
2
+ 1)a
2
A
(ν
2
+1,ν
3
)
,
(21)
where r = c −k
1
a −(p −k)p, v
2
= 0, 1, 2, 3, v
3
= 3 −
v
2
, v
2
+ v
3
= 2 − v
1
, v
1
≥ 1.
From here we can find all the coefficients, and
they are small for a sufficiently large ρ. Let equa-
tion (19) be defined. Then it follows from it that z
tends to zero approximately according to the law of
z = z
0
exp(−pt), since ρ is relatively large, and after
a short period of time [0,t
0
] we can assume that for
t ≥ t
0
we have
z ≈ 0
α ≈ Aω + BQ +
∑
ν
1
+ν
2
=3
A
(ν
1
,ν
2
)
˙
θ
ν
1
θ
ν
2
. (22)
We replace the last equation of system (1) with rela-
tion (22). This relation defines an ideal regulator, in
a sense equivalent to the original regulator.
As a result, instead of the original problem, we
can consider the problem of the angular motion of an
object controlled by an ideal controller, which is de-
scribed by a second-order nonlinear differential equa-
tion
¨
θ + 2χ
˙
θ + (χ
2
+ µ
2
)θ +
∑
ν
1
+ν
2
=3
A
(ν
1
,ν
2
)
˙
θ
ν
1
θ
ν
2
= 0.
(23)
Equation (19) shows that the controlled aircraft ro-
tates according to the law of rotation around the fixed
axis of a rigid body with a unit moment of inertia un-
der the action of the following damping and restoring
moments:
M
1
= −(2χ + aA
(3,0)
˙
θ
2
+ A
(1,2)
θ
2
)
˙
θ,
M
2
= −(χ
2
+ µ
2
+ aA
(2,1)
˙
θ
2
+ aA
(0,3)
θ
2
)θ.
(24)
As a result, we can also conclude that by substituting
relation (18) into the cubic terms of expression (11),
cubic regulation with respect to α,
˙
θ, θ can be reduced
to regulation with respect to the
˙
θ, θ and therefore, it
makes no sense to introduce cubic terms containing α
Analysis of a Nonlinear Control Law with Cubic Nonlinearity
703

in ( (11)), it is enough to introduce regulation accord-
ing to the following law:
σ =
1
k
∗
(−kα + k
1
˙
θ + k
2
θ) +
1
k
∗
(
∑
ν
1
+ν
3
=3
p
(ν
1
,ν
2
)
˙
θ
ν
1
θ
ν
2
).
(25)
4 INTEGRATING A SIMPLIFIED
EQUATION
We assume that the imaginary part of the roots ex-
ceeds the real part in absolute value µ > k. We intro-
duce a variable
ξ = Θ + (χ + µi)Θ, (26)
satisfying the diagonal equation
˙
ξ = λ
1
ξ+
∑
ν
1
+ν
2
=3
Q
(ν
1
,ν
2
)
ξ
ν
1
ξ
−ν
2
, λ
(1,2)
= −χ+µi.
(27)
By a polynomial substitution
ξ = z −
∑
ν
1
+ν
2
=3
B
(ν
1
,ν
2
)
z
ν
1
z
−ν
2
for B
(2,1)
= 0, (28)
B
(ν
1
,ν
2
)
=
1
(1−ν
1
)λ
1
−ν
2
λ
2
Q
(ν
1
,ν
2
)
,
ν
1
= 0, 1, 3; ν
2
= 3 − ν,
(29)
we reduce equation (27) to the form
˙z = (λ
1
+ gr
2
)z for g = Q
(2,1)
≡ g
1
+ ig
2
, r = |z|.
(30)
Equation (30) is equivalent to two equations
˙r = χr + g
1
r
3
,
˙
ϕ = µ + g
2
r
2
, where z = re
iϕ
. (31)
Integrating the last equations, we find
r = r
0
[(−1 −
g
1
r
2
0
χ
)e
−2χt
+
g
1
r
2
0
χ
]
−
1
2
,
ϕ = (µ +
χg
2
g
1
)t +
g
2
g
1
ln(
r
r
0
).
(32)
It follows from equations (31) or from solution (32)
that the constants chi,g characterize the rate of de-
crease in r, i.e. the rate of damping of the disturbed
motion, while mu, g
2
determine the oscillation of the
motion. The influence of nonlinear terms in the ex-
pression for the control function (25) on the stabiliza-
tion of the object is mainly characterized by a com-
plex coefficient Q
(2,1)
= g
1
+ ig
2
having two param-
eters g
1
, g
2
. A decrease in g
1
and g
2
in the negative
direction, respectively, increases the rate of decay of
the process and reduces the oscillation of the process.
5 CHARACTERISTIC
COEFFICIENT AND CONTROL
PARAMETERS
The coefficient Q
(2,1)
can be expressed directly
through the coefficients p
(ν
1
,ν
2
)
. Differentiating the
first of equations (1) and subtracting from it the sec-
ond equation multiplied by a, we obtain one third-
order equation equivalent to system (1)
θ
(3)
+ (2χ + p)
¨
θ + (χ
2
+ µ
2
+ 2χp)
˙
θ+
+ (χ
2
+ µ
2
)pθ = −a
∑
ν
1
+ν
2
=3
˜p
(ν
1
,ν
2
)
˙
θ
ν
1
θ
ν
2
(33)
From (2), (8), (10), (25) we obtain the coefficients
of the cubic terms
˜p
(ν
1
,ν
2
)
= p
(ν
1
,ν
2
)
+
k
∗
k
3
∗
C
ν
2
3
N
ν
1
1
N
ν
2
2
, (34)
where N
1
= kA − k
1
, N
2
= kB − k
2
, C
ν
2
3
- number of
combinations from 3 to v
2
. By grouping the linear
terms of equation (33), we represent it in three forms
˙z
s
− λ
s
z
s
+ a
∑
ν
1
+ν
2
=3
˜p
(ν
1
,ν
2
)
˙
θ
ν
1
θ
ν
2
= 0, (s = 1, 2, 3),
(35)
where
z
1,2
=
¨
θ + (p + χ ± µi)
˙
θ + p(χ ± µi)θ;
z
3
=
¨
θ + 2χ
˙
θ + (χ
2
± µ
2
)θ.
(36)
We will express the cubic terms in equations (35) in
terms of z
1
, z
2
, z
3
. Substituting into the first of equa-
tions (1) and comparing the obtained expression with
(18), (15), we conclude that z
3
= −
1
a
z + O
(3)
(
˙
θ, θ)
and since z ≈ 0 at t ≥ t
0
, then z
3
= O
(3)
(
˙
θ, θ) at t ≥ t
0
.
Therefore, when substituted into cubic terms, we can
assume z
3
≈ 0 and from system (36) we obtain
θ =
1
2µi∆
(K
1
z
1
+ K
2
z
2
),
˙
θ =
1
2µi∆
(K
1
λ
1
z
1
+ K
2
λ
2
z
2
),
(37)
where ∆ = (p − χ)
2
+ µ
2
, K
1
= p − χ − µi, K
2
=
−[(p − χ) + µi], λ
1,2
= −χ ± µi.
Substituting (37) into (35), we obtain a system of
two equations in the canonical form
˙z
s
− λ
s
z
s
+
a
i
(2µ∆)
3
S = 0, (38)
where
S =
∑
v
1
+v
2
˜p
(v
1
,v
2
)
(K
1
λ
1
z
1
+ K
2
λ
2
z
2
)
v
1
(K
1
z
1
+ K
2
z
2
)
v
2
.
The real and imaginary parts of the coefficient at
the term z
2
1
z
2
are determined by the formulas
Q
(2,1)
= −
∂
˜
f
∂z
2
|
z
2
=0
z
1
=1
=
=
a
i
8µ
3
∆
2
∑
v
1
+v
2
=3
˜p
(v
1
,v
2
)
(p + λ
2
)(v
1
λ
2
+ v
2
λ
1
)λ
1
v
1
−1
,
(39)
ICINCO 2021 - 18th International Conference on Informatics in Control, Automation and Robotics
704

where
˜
f denotes the entire cubic term of equations
(38)
g
s
= L
2
∑
v
1
+v
2
=3
L
3
(v
1
,v
2
)
˜p
(v
1
,v
2
)
, (s = 1, 2), (40)
where
L
(0,3)
1
= 3µ, L
(0,3)
2
= 3(p − χ), L
2
=
a
8µ
3
∆
2
> 0,
L
(1,2)
1
= (p + 2χ)µ, L
(1,2)
2
= µ
2
− 3χ(p − χ),
L
(2,1)
1
= (χ
2
+ µ
2
+ 2χp)µ,
L
(2,1)
2
= 3χ
2
(p − χ) + µ
2
(p − 3χ),
L
(3,0)
1
= −3(χ
2
+ µ
2
)pµ,
L
(3,0)
2
= −3(χ
2
+ µ
2
)[−µ
2
+ (p − χ)].
(41)
Note that according to (34) the coefficients at σ
(v
1
,v
2
)
differ from p
(v
1
,v
2
)
only by additional terms; there-
fore, g
1
, g
2
depends linearly on p
(v
1
,v
2
)
, and the coef-
ficients at p
(v
1
,v
2
)
are determined by formulas (41).
6 RECOMMENDATIONS ABOUT
THE CHOICE OF
PARAMETERS FOR THE
CUBIC CONTROL LAW
Now, some recommendations about the choice of pa-
rameters for the cubic control law are given. As noted
in (4), the rate of damping of the disturbed motion
will increase if g
1
is reduced to the negative side
by choosing the parameters p
(ν
1
,ν
2
)
; fluctuation de-
creases with decreasing g
2
. Each of p
(ν
1
,ν
2
)
affects
the characteristics of g
1
and g
2
, namely, from expres-
sions (40), (41) it follows that:
1) The inclusion of a term p
(0,3)
θ
3
with a negative
coefficient p
(0,3)
in the control function (25) helps to
weaken the disturbed motion (since L
2
L
(0,3)
1
> 0), and
also helps to reduce the oscillation of the movement
(more precisely, to a decrease of g
2
).
2) The inclusion of p
(1,2)
< 0 contributes to the
damping of motion in about the same way as the in-
clusion of the coefficient p
(0,3)
. The coefficient p
(0,3)
,
is related to p
(1,2)
by the formula related to p
(1,2)
ac-
cording to the formula
p
(0,3)
=
1
3
(p + 2χ)p
(1,2)
< 0. (42)
At the same time, p
(1,2)
decreases the vibrational
value less than p
(0,3)
(42) if the following condition
is satisfied
L
(1,2)
2
(L
(1,2)
1
)
−1
< L
(0,3)
2
(L
(0,3)
1
)
−1
⇒
⇒
µ
2
p−χ
< p + 2χ.
(43)
In the opposite case, the inclusion of p
(1,2)
is prefer-
able.
3) The coefficient p
(2,1)
< 0 contributes to the
damping of motion as p
(0,3)
since
p
(0,3)
=
1
3
(χ
2
+ µ
2
+ 2χp)p
(2,1)
< 0. (44)
But the inclusion of the term p
(0,3)
is preferable, since
it provides less oscillation due to the fulfillment of the
condition
χ −
µ
2
p − χ
< p. (45)
4) The coefficient p
(3,0)
< 0 contributes to the
damping of motion as p
(0,3)
since
p
(0,3)
= −p(χ
2
+ µ
2
)p
(3,0)
. (46)
But taking into account condition (45), the coefficient
p
(0,3)
is preferable.
Thus, the inclusion of a term p
(0,3)
θ
3
with a neg-
ative coefficient into the control function does not al-
low the system to be accelerated unnecessarily to a
stationary state at relatively large values of θ.
If L
(1,2)
2
> L
(0,3)
2
, then it is preferable to include in
the control the term p
(1,2)
˙
θθ
2
in addition to p
(0,3)
θ
3
.
This term at relatively small χ (more precisely, at
L
(1,2)
2
> 0) does not lead to an increase in the oscil-
lation of the system.
If L
(2,1)
1
or L
(3,0)
1
are much larger than L
(0,3)
1
, L
(1,2)
1
,
then regulation with the help of
˙
θ
2
θ or
˙
θ
3
can also be
introduced. Note that these terms can adversely affect
the oscillation of the movement.
7 CONCLUSIONS
The paper presents the analysis results of the influ-
ence of the cubic components of the nonlinear control
law on a system state variables. The choice of param-
eters of the nonlinear control law makes it possible to
control the regulation time of the process and its os-
cillations. Expressions for the choice of characteristic
coefficients and control parameters are obtained in the
paper. The relationship between the parameters of the
nonlinear control law with the cubic components and
the quality of the system is described.
ACKNOWLEDGEMENTS
This work was financially supported by Government
of Russian Federation, Grant 08-08.
Analysis of a Nonlinear Control Law with Cubic Nonlinearity
705

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