Control design for partial stabilization of nonlinear mechanical systems with random disturbances
The problem of the partial stabilization of nonlinear control systems de- scribed by the Ito stochastic differential equations is considered. For these systems, we propose a constructive control design method, which provides the partial asymptotic stability in probability of the trivial solution...
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552922311557120 |
|---|---|
| author | Зуєв, О.Л. Васил'єва, І.Г Зуев, А.Л. Васильева, И.Г. Zuyev, A.L. Vasylieva, I.G. |
| author_facet | Зуєв, О.Л. Васил'єва, І.Г Зуев, А.Л. Васильева, И.Г. Zuyev, A.L. Vasylieva, I.G. |
| author_institution_txt_mv | [
{
"author": "О.Л. Зуєв",
"institution": "IM Department 04"
},
{
"author": "І.Г Васил'єва",
"institution": null
}
] |
| author_sort | Зуєв, О.Л. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2020-08-09T19:28:01Z |
| description |
The problem of the partial stabilization of nonlinear control systems de- scribed by the Ito stochastic differential equations is considered. For these systems, we propose a constructive control design method, which provides the partial asymptotic stability in probability of the trivial solution of the closed-loop system with respect to a part of state variables. Mechanical ex- amples are presented to illustrate the eciency of the proposed controllers.
|
| first_indexed | 2026-08-04T01:07:08Z |
| format | Article |
| fulltext |
Збiрник праць Iнституту математики НАН України 2019, т. 16, № 2, 209–223
УДК 531.36, 519.21, 517.977
Control design for partial
stabilization of nonlinear mechanical
systems with random disturbances
⇤
Zuyev A.L. 1,2, Vasylieva I.G. 1
1
Institute of Applied Mathematics and Mechanics, National Academy
of Sciences of Ukraine, Slovyansk; irisna.shurko@gmail.com,
2
Max Planck Institute for Dynamics of Complex Technical Systems,
Magdeburg, Germany; zuyev@mpi-magdeburg.mpg.de.
The problem of the partial stabilization of nonlinear control systems de-
scribed by the Ito stochastic di↵erential equations is considered. For these
systems, we propose a constructive control design method, which provides
the partial asymptotic stability in probability of the trivial solution of the
closed-loop system with respect to a part of state variables. Mechanical ex-
amples are presented to illustrate the e�ciency of the proposed controllers.
У роботi розглянуто задачу стабiлiзацiї щодо частини змiнних для не-
лiнiйних керованих систем, якi описуються стохастичними диферен-
цiальними рiвняннями Iто. Запропоновано конструктивний метод по-
будови функцiй зворотного зв’язку, що забезпечують часткову асим-
птотичну стiйкiсть за ймовiрнiстю тривiального розв’язку вiдповiдної
замкнутої системи. Ефективнiсть одержаних законiв керування про-
iлюстровано на прикладах механiчних систем.
В работе рассматривается задача стабилизации относительно части пе-
ременных для нелинейных управляемых систем, которые описываются
стохастическими дифференциальными уравнениями Ито. Предложен
конструктивный метод построения функций обратной связи, обеспе-
чивающих частичную ассимптотичесую устойчивость по вероятности
тривиального решения соответствующей замкнутой системы. Эффек-
тивность полученных управлений проиллюстрирована на механиче-
ских примерах.
⇤This work was supported in part by the grant of the President of Ukraine (project
F84/56713) and the budget program of NAS of Ukraine (KPKVK 6541230).
c� Zuyev A.L., Vasylieva I.G., 2019
210 Zuyev A.L., Vasylieva I.G.
A Introduction
To construct adequate mathematical models that describe the behav-
ior of real dynamic processes and analyze their stability properties, it is
necessary to take into account the e↵ects of uncertainties and random
disturbances. The latter leads to the need to study systems of di↵eren-
tial equations with random perturbations. Here, qualitative methods for
investigating the asymptotic behavior of solutions of systems of di↵eren-
tial equations with random disturbances are useful. Lyapunov methods
for analyzing the stability of stochastic systems have been developed by
many authors (see, e.g., [1, 2] and references therein). In particular, the
concept of control Lyapunov functions and Artstein’s theorem [3] have
been extended to stochastic di↵erential equations in [4]. In [5], a crite-
rion for stochastic finite-time stability via multiple Lyapunov functions
has been obtained.
Partial stabilization problem arises in tasks when only the stability
with respect to some variables is needed for a desired performance of
the system. This task is also crucial when the system is not stable in
the sense of Lyapunov, but asymptotically stable with respect to a part
of variables [6–9]. Therefore, the problems of partial stability and sta-
bilization of motion are highly important in engineering applications,
cf. [10, 11]. In the paper [9], conditions of partial stability in probability
for the Ito stochastic di↵erential equations have been obtained by Lya-
punov’s direct method. In [12], su�cient conditions for partial stability
of stochastic reaction-di↵usion systems with Markovian switching have
been derived.
In this paper, we consider the problem of stabilization of the Ito-
type stochastic di↵erential equations with respect to a part of variables.
Our goal is to propose an e�cient control design scheme for the above
problem. To achieve this goal, we present an extension of the universal
stabilizing controllers from [13] to the problem of partial stabilization of
stochastic systems in Section 3. Our main theoretical contribution will
be applied to mechanical examples in Sections 4 and 5.
B Notations and Definitions
Throughout this paper, let w(t) 2 Rk (t � 0) be a standard k-dimensional
Wiener process defined on a complete probability space (⌦,F , P ), and
let {Ft}t�0 be the complete right-continuous filtration generated by w.
Control design for partial stabilization of nonlinear mechanical . . . 211
Consider a control system described by the Ito stochastic di↵erential
equations:
dx(t) = (f(x) + g(x)u)dt+
kX
i=1
�i(x)dwi(t), (1)
where x = (x1, ..., xn)T 2 D ✓ Rn is the state and u = (u1, ..., uk)T 2
U = Rk is the control. We assume that 0 2 D, �i(0) = 0 for i = 1, ..., k,
and the maps f : D ! Rn, g : D ! Rn⇥k, �i : D ! Rn satisfy the
Lipschitz condition on every bounded domain X ⇢ D.
For a map h : D ! U , h(0) = 0, we introduce the closed-loop system
for (1) with the feedback law u = h(x):
dx(t) = (f(x) + g(x)h(x))dt+
kX
i=1
�i(x)dwi(t). (2)
If h is Lipschitz continuous on every bounded X ⇢ D, then there exists a
unique strictly Markov process x⇠,s(t) which is a solution of (2) under the
initial condition x⇠,s(s) = ⇠ (see, e.g., [15]). We relate with the control
system (1) the operator
Lu =
nX
i=1
(f(x)+g(x)u)i
@
@xi
+
1
2
nX
i,j=1
cij(x)
@2
@xi@xj
, [cij(x)] = �(x)�T (x).
In the sequel, we will study stability of the trivial solution of (2)
with respect to the variables x1, x2, ..., xm. Denote these variables as y =
(y1, ..., ym)T 2 Rm and the rest as z = (z1, ..., zp)T 2 Rp, m + p = n,
then x = (yT , zT )T , x0 = (yT0 , z
T
0 )
T , and ||x|| = (x2
1 + ... + x2
n)
1/2 =
(||y||2 + ||z||2)1/2.
We assume also that the solutions of (1) are z�extendable in a closed
domain D = DH , where
DH = {x 2 Rn : ||y(t)|| H, z 2 Rp
}, H = const > 0.
It means that if x(t) 2 DH is a maximal solution of system (1) on t 2
(⌧1, ⌧2) with some admissible control u 2 L1(⌧1, ⌧2), then either ||y(t)|| !
H as t ! ⌧2 almost surely or ⌧2 = 1. This kind of z-extendability
assumption is natural in the problems of partial stability [6]; it is usually
satisfied for well-posed mathematical models in physics whose trajectories
do not blow up in finite time with bounded control.
212 Zuyev A.L., Vasylieva I.G.
Let us introduce the standard class of comparison functions K, whose
elements are continuous strictly increasing functions ↵ : R+
! R+ such
that ↵(0) = 0. We will extend the concept of a control Lyapunov func-
tion [3, 4, 13, 14] to the problem of partial stabilization of stochastic sys-
tems as follows.
Definition B.1. A function V 2 C2(DH ;R) is called a y-stochastic con-
trol Lyapunov function (y-SCLF) for system (1), if there exist ↵,�1,�2 2
K such that
�1(||y||) V (x) �2(||y||), inf
u2U
LuV (x) �↵(||y||),
for all x 2 DH .
Throughout the text, B(x; �) denotes the �-neighborhood of a point
x 2 Rn.
Definition B.2. A function V 2 C2(DH ;R) satisfies the small control
property with respect to y if, for any ✏ > 0 and any x0 2 M = {x|y = 0},
there exists a � > 0 such that
x 2 B(x0; �) ) inf
||u||<✏
LuV (x) �↵(||y||).
Definition B.3. [9, 16–18] The solution x = 0 of system (2) is called
y-stable in probability if, for all s � 0, " > 0, � > 0, there exists a � > 0
such that ⇠ 2 B(0; �) implies
P{sup
t�s
||y⇠,s(t)|| > "} < �.
Definition B.4. [17, 18] The solution x = 0 of system (2) is called
asymptotically y-stable in probability if it is y-stable in probability and
P{ lim
t!1
||y⇠,s(t)|| = 0} = 1
for all ⇠ 2 B(0;�) with some constant � > 0.
C Main result
The following result generalizes the constructive proof of Artstein’s the-
orem [13] for the problem of partial stabilization of stochastic systems.
Control design for partial stabilization of nonlinear mechanical . . . 213
Theorem C.1. Let V 2 C2(DH ;R) be a y-SCLF satisfying the small
control property. Then there exists a continuous feedback law h : DH !
Rk, h(0, z) = 0, such that the trivial solution of the corresponding closed-
loop system (2) with u = h(x) is y-asymptotically stable in probability.
The feedback law h(x) is given as follows:
hi(x) =
8
>><
>>:
0, b = 0,
� bi
kbk2 (a+ (a2 + kbk4)
1
2 ), b 6= 0, 2(a2 + kbk4)
1
2 � ↵(kyk),
� bi
2kbk2 (2a+ ↵(kyk)), otherwise,
(3)
where
a(x) =
nX
i=1
fi(x)
@V (x)
@xi
+
1
2
nX
i,j=1
cij(x)
@2V (x)
@xi@xj
,
bi(x) =
nX
j=1
gij(x)
@V (x)
@xj
, b(x) = (b1(x), ..., bk(x)). (4)
Proof. The proof of continuity of h(x) in (3) goes along the same lines
as the proof of Theorem 4 in [14].
Let us evaluate the operator LuV for (1) using the feedback law u =
h(x):
LhV =
8
><
>:
a(x), b = 0,
�(a2(x) + kb(x)k4)
1
2 , b 6= 0, 2(a2 + kbk4)
1
2 � ↵(kyk),
�
1
2↵(kyk), otherwise.
As V (x) is a y-stochastic control Lyapunov function, the following in-
equality holds:
LhV �
1
2
↵(||y||) for all x 2 DH .
Using Grönwall’s inequality, we have:
E||y⇠,s(t)||2 k1(t� s)e
R t
s k2E||y⇠,s(p)||2dp
N1e
N2�
2
,
where E is the expectation in the probability measure P⇠,s, y⇠,s(t) is the
y-component of the solution x⇠,s(t) of (2) with the initial data x⇠,s(s) = ⇠.
Putting � = ln( ✏2✏
2
1
N1
)
1
2N2 , we get
P{supt�t0 ||y(t)|| > ✏1}
E||y(t)||2
✏21
< ✏2.
214 Zuyev A.L., Vasylieva I.G.
Let ⌧" = inf{t : ky⇠,s(t)k > "}, ⌧"(t) = min(⌧", t).
From Dynkin’s lemma [15], it follows that
EV (x⇠,s(⌧"(t)))� V (⇠) = E
Z ⌧"(t)
s
LhV (x⇠,s(u))du.
Since LhV (x) �
1
2↵(||y||), we will get
EV (x⇠,s(⌧"(t))) V (⇠), t � s. (5)
The above inequality can be rewritten as
Z
⌧"<t
↵1(ky
⇠,s(⌧")k)P⇠,s(d!) +
Z
⌧"�t
↵1(ky
⇠,s(t)k)P⇠,s(d!) V (⇠).
Hence, ↵1(")P⇠,s{⌧" < t} V (⇠).
From the last equality, due to the continuity of the function V (x) and
the equality V (0) = 0, it follows that
lim
⇠!0
P⇠,s{⌧" < t} = 0.
So, the equilibrium x = 0 of system (2) is y-stable in probability.
From (5) it follows that the random process V (x⇠,s(⌧"(t))) is a non-
negative supermartingale, and there exists the limit
lim
t!1
V (x⇠,s(⌧"(t))) = ⌘ (6)
with probability 1.
From the set of sample trajectories of the process x⇠,s(t) we take the
subset B of sample trajectories such that for any x⇠,s
i (t) (i = 1, ..., n) the
following equality holds: ⌧"(t) = t, t 2 R+. Then it follow from the above
assumptions that
lim
⇠y!0
P⇠,s{B} = 1, (7)
where ⇠T = (⇠Ty , ⇠
T
z ).
From (6) and (7), we have
lim
t!1
V (x⇠,s(⌧"(t))) = lim
t!1
V (x⇠,s(t)) = ⌘. (8)
Note that V (x) is a y-stochastic control Lyapunov function, so for all
trajectories from the set B, except a set of probability 0, the following
property holds:
lim
t!1
||y⇠,s(t)|| = 0.
Control design for partial stabilization of nonlinear mechanical . . . 215
From the assumption of z�extendability of solutions and (8), we ob-
tain ⌘ = 0.
So, limt!1 ||y⇠,s(t)|| = 0. From this property it follows that the zero
solution of the closed-loop system (2) is y-asymptotically stable in prob-
ability.
D Inverted pendulum with a moving mass
To illustrate possible applications of Theorem 3.1, we consider a mechan-
ical system consisting of an inverted pendulum (carrier body) and a point
mass m moving in the direction perpendicular to the axis of symmetry
of the carrier body (Fig. 1). It is assumed that the mass m is suspended
by a spring with the sti↵ness coe�cient {.
Figure 1. Inverted pendulum with a moving mass.
We will use the following notations: M is the mass of the carrier
body, ' is the angle between the axis of symmetry of the carrier and the
vertical, y is the displacement of the point mass, and ` is the distance
between the fixed point and the suspension of the mass m.
Let us first derive the equations of motion of this mechanical systems
by using the Lagrangian formalism. The kinetic energy of the system is
T =
✓
I
2
+
m(`2 + y2)
2
◆
'̇2 +
m
2
ẏ2 +m`'̇ẏ,
where I is the moment of inertia of the carrier body with respect to its
216 Zuyev A.L., Vasylieva I.G.
fixed point. The potential energy is
U =
M`g
2
cos'+
{
2
y2 +mg(` cos'� y sin').
Then the Lagrangian of the considered system takes the form
L = T � U =
✓
I
2
+
m(`2 + y2)
2
◆
'̇2 +
m
2
ẏ2 +m`'̇ẏ�
�
{
2
y2 �
✓
M
2
+m
◆
`g cos'+mgy sin'.
We now apply Lagrange’s equations in the form
d
dt
⇣
@L
@'̇
⌘
�
@L
@' = 0,
d
dt
⇣
@L
@ẏ
⌘
�
@L
@y = Fu,
where Fu is the control force applied to the mass m.
This leads to the following equations of motion:
'̈ = 1
I+my2 (�2my'̇ẏ �m`y'̇2 + {`y + M`g
2 sin'+
+mgy cos'� `Fu),
ÿ = `
I+my2 (2my'̇ẏ +m`y'̇2
� {`y � M`g
2 sin'+
+`Fu �mgy cos') + I
I+my2 (y'̇2
�
{y
m + g sin')+
+ 1
I+my2 ((
I
m + y2)Fu +my3'̇2 + y3{ + y2mg sin').
By replacing
v =
1
I +my2
(�2my'̇ẏ �m`y'̇2 + {`y + M`g
2
sin'+mgy cos'� `Fu),
we obtain the following equations with respect to the new control v:
'̈ = v,
ÿ = �(`+ I+my2
m` )v + 1
I+my2 (2y3{ + 2m+M
2m ( I
m+
+y2)g sin')� 2yẏ'̇
` + gy cos'
` .
Let us rewrite the above equations of motion in the form ẋ = f(x) +
g(x)v, where
x =
2
664
x1
x2
x3
x4
3
775 =
2
664
'
y
'̇
ẏ
3
775 , f(x) =
2
664
x3
x4
0
q(x)
3
775 , g(x) =
2
664
0
0
1
�`� I+my2
m`
3
775 , (9)
Control design for partial stabilization of nonlinear mechanical . . . 217
q(x) =
1
I +mx2
2
✓
2x3
2{ +
2m+M
2m
✓
I
m
+ x2
2
◆
g sinx1
◆
�
2x2x3x4
`
+
gx2 cosx1
`
.
It is easy to see that system (9) admits the equilibrium x = 0 with
v = 0 (upper equilibrium). We will consider the stabilization of the
upper equilibrium of the carrier body in the sense of partial stabilization
problem with respect to the variables (x1, x3) by applying control to the
point mass.
To take into account random e↵ects, we substitute the stochastic input
v = u+ �x3ẇ(t) formally into system (9), where w(t) is a standard one-
dimensional Wiener process. As a result, we obtain the following system
of stochastic di↵erential equations:
dx1 = x3dt,
dx2 = x4dt,
dx3 = udt+ �x3dw(t),
dx4 =
⇣
(�`� I+my2
m` )u+ q(x)
⌘
dt� (`+ I+my2
m` )�x3dw(t),
(10)
where u is treated as the control.
Since our goal is to steer the variables ' and '̇ (i.e. x1 and x3) to
zero, we propose the following quadratic Lyapunov function candidate:
2V (x) = (k21 + k22 + k2)x
2
3 + 2k1x1x3 + (k2 + 1)2x2
1,
where k1 and k2 are positive constants.
Let us define the functions a(x) and b(x) according to (4):
a(x) =
4X
i=1
fi(x)
@V (x)
@xi
+
1
2
4X
i,j=1
cij(x)
@2V (x)
@xi@xj
=
= (k1x3 + (k2 + 1)x1)x3 + (k22 + k21 + k2)�
2x2
3,
b(x) = (k22 + k21 + k2)x3 + k1x1.
According to Theorem 3.1, we propose the feedback control law for
system (10) in the form (3) with ↵(kyk) = �kyk2, kyk2 = x2
1 + x2
3, � > 0.
So, the equilibrium x = 0 of the corresponding closed-loop system (10),
(3) is asymptotically stable in probability with respect to (x1, x3) by
Theorem 3.1. Simulation results for the closed-loop system (10), (3)
with k1 = 2, k2 = 1 are presented in Fig. 2. These simulations have been
performed in Maple by using the ItoProcess(·) function.
218 Zuyev A.L., Vasylieva I.G.
Figure 2. Components x1 and x3 of a sample path of the closed-loop sys-
tem (10), (3).
E Stabilization of a three-wheeled trolley by a stochas-
tic feedback law
Figure 3. Three-wheeled trolley.
Consider a mathematical model of the three-wheeled trolley whose
position is determined by three coordinates: (x1, x2) are coordinates of
the midpoint between the steering wheels, and x3 is the angle between
the axis of symmetry of the trolley and the x1-axis, cf. [19]. A cylindrical
hinge whose axis is perpendicular to the axis of symmetry of the trolley is
mounted above the point (x1, x2) (Fig. 3). A weightless and inextensible
rod can rotate in this hinge, and a point mass is attached to the other
end of the rod. We denote the angle between the vertical axis and the
Control design for partial stabilization of nonlinear mechanical . . . 219
rod by ↵. The motion of the trolley is described by the rolling without
slipping conditions:
dx1 = (u1 + u2) cosx3dt,
dx2 = (u1 + u2) sinx3dt,
dx3 = (u1 � u2)dt,
(11)
where the vector u = (u1, u2)T 2 R2 is treated as the control.
Following [19], we also write Lagrange’s equation with respect to the
angle ↵:
↵̈� (ẋ1 cosx3 + ẋ2 sinx3 + ẋ3 sin↵)ẋ3 cos↵ = � sin↵. (12)
Note that the considered model belongs to the class of nonholonomic
systems which, as it is well-known, cannot be stabilized in a neighborhood
of the equilibrium position by a deterministic continuous state feedback
law (see, e.g., [20]). In the sequel, we will study the stabilization problem
with respect to a part of variables in the stochastic sense.
Let us denote the relative angular velocity of the rod by ! = ↵̇ and
perform the following change of variables in (11), (12):
z1 := x3,
z2 := x1 cosx3 + x2 sinx3,
z3 := x1 sinx3 � x2 cosx3,
z4 := ↵,
z5 := !,
⌫1 := u1 � u2,
⌫2 := (u1 + u2)� (u1 � u2)z3.
Then the equations of motion take the form:
ż1 = ⌫1,
ż2 = ⌫2,
ż3 = ⌫1z2,
ż4 = z5,
ż5 = (⌫2 + ⌫1z3 + ⌫1 sin z4)⌫1 cos z4 � sin z4.
(13)
We randomize system (13) by designing the control inputs
⌫1 = v1,
⌫2 = v2 + �z2ẇ(t),
220 Zuyev A.L., Vasylieva I.G.
where ẇ(t) is treated formally as the derivative of a standard one-dimensional
Wiener process w(t). Then we rewrite the stochastic control system as
follows:
dz1 = v1dt,
dz2 = v2dt+ �z2dw(t),
dz3 = v1z2dt,
dz4 = z5dt,
dz5 = ((v2 + v1z3 + v1 sin z4)v1 cos z4 � sin z4) dt+
+�z2v1 cos z4dw(t).
(14)
We consider the partial stabilization problem for system (14) with
respect to the variables z1, z2, z3.
To design stabilizing controls v1, v2, we take a control Lyapunov func-
tion candidate of the following form [21]:
V (z) = 2z3 �
1
2
(z21 + z22)(1 + z23) + 2
✓
|z21 + z22 |
2
◆1+
z23
2
.
Then we define the functions a(z), b1(z), b2(z) according to (4):
a(z) =
1
2
5X
i,j=1
cij(z)
@2V (z)
@zi@zj
=
1
2
�2z22
@2V (z)
@z2
2 ,
b1(z) = �z1(z
2
3 +1)+
4
⇣
|z2
1+z2
2 |
2
⌘1+
z23
2
(1 + z2
3
2 )z1
|z21 + z22 |
+ z2
�
2� (z21 + z22)z3+
+z2
0
@2� (z21 + z22)z3 + 2
✓
|z21 + z22 |
2
◆1+
z23
2
z3 ln
✓
|z21 + z22 |
2
◆1
A ,
b2(z) = �z2(z
2
3 + 1) +
4
⇣
|z2
1+z2
2 |
2
⌘1+
z23
2
(1 + z2
3
2 )z2
|z21 + z22 |
,
b(z) = (b1(z), b2(z)).
Thus, the conditions of Theorem 3.1 are satisfied with the above choice
of a(x), b(x), and ↵(kyk) = �kyk2, kyk2 = z21 +z22 +z23 , � > 0. Numerical
simulation results for system (14) with the feedback law (3) are presented
in Figs. 4-5.
Control design for partial stabilization of nonlinear mechanical . . . 221
Figure 4. Components z1, z2, z3 of a sample path of the closed-loop sys-
tem (14), (3).
Figure 5. Components z4, z5 of a sample path of the closed-loop system (14),
(3).
222 Zuyev A.L., Vasylieva I.G.
F Conclusion
A constructive proof of Artstein’s theorem has been extended to the prob-
lem of partial stabilization of the Ito stochastic di↵erential equations.
This construction allows e↵ective computing of stabilizing feedback con-
trols if a control Lyapunov function in the sense of Definitions 2.1-2.2
is known. The control design scheme of Theorem 3.1 is shown to be
applicable to nonlinear systems with stochastic e↵ects that describe the
dynamics of an inverted pendulum with a moving masses and a three-
wheeled trolley with an additional degree of freedom. The simulation
results, presented in Figs. 2 and 4-5, illustrate the required behavior of
sampled paths of the corresponding closed-loop systems.
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|
| id | oai:trim.imath.kiev.ua:article-423 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:07:08Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/0f/8ad16a0519504d98b73df877affba40f.pdf |
| spelling | oai:trim.imath.kiev.ua:article-4232020-08-09T19:28:01Z Control design for partial stabilization of nonlinear mechanical systems with random disturbances Управління для часткової стабілізації нелінійних механічних систем з випадковими збуреннями Управління для часткової стабілізації нелінійних механічних систем з випадковими збуреннями Зуєв, О.Л. Васил'єва, І.Г Зуев, А.Л. Васильева, И.Г. Zuyev, A.L. Vasylieva, I.G. The problem of the partial stabilization of nonlinear control systems de- scribed by the Ito stochastic differential equations is considered. For these systems, we propose a constructive control design method, which provides the partial asymptotic stability in probability of the trivial solution of the closed-loop system with respect to a part of state variables. Mechanical ex- amples are presented to illustrate the eciency of the proposed controllers. В работе рассматривается задача стабилизации относительно части переменных для нелинейных управляемых систем, которые описываются стохастическими дифференциальными уравнениями Ито. Предложен конструктивный метод построения функций обратной связи, обеспе- чивающих частичную ассимптотичесую устойчивость по вероятности тривиального решения соответствующей замкнутой системы. Эффек- тивность полученных управлений проиллюстрирована на механиче- ских примерах. У роботi розглянуто задачу стабiлiзацiї щодо частини змiнних для не- лiнiйних керованих систем, якi описуються стохастичними диференцiальними рiвняннями Iто. Запропоновано конструктивний метод побудови функцiй зворотного зв’язку, що забезпечують часткову асим- птотичну стiйкiсть за ймовiрнiстю тривiального розв’язку вiдповiдної замкнутої системи. Ефективнiсть одержаних законiв керування про- iлюстровано на прикладах механiчних систем. Інститут математики НАН України 2020-08-09 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/423 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 16 No. 2 (2019): Mathematical problems of mechanics and computational mathematics; 209-223 Сборник Трудов Института математики НАН Украины; Том 16 № 2 (2019): Математические проблеми механики и вычислительной математики; 209-223 Збірник Праць Інституту математики НАН України; Том 16 № 2 (2019): Математичні проблеми механіки та обчислювальної математики; 209-223 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/423/414 Авторське право (c) 2020 A.L. Zuyev, І.Г. Васил'єва http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Зуєв, О.Л. Васил'єва, І.Г Зуев, А.Л. Васильева, И.Г. Zuyev, A.L. Vasylieva, I.G. Control design for partial stabilization of nonlinear mechanical systems with random disturbances |
| title | Control design for partial stabilization of nonlinear mechanical systems with random disturbances |
| title_alt | Управління для часткової стабілізації нелінійних механічних систем з випадковими збуреннями Управління для часткової стабілізації нелінійних механічних систем з випадковими збуреннями |
| title_full | Control design for partial stabilization of nonlinear mechanical systems with random disturbances |
| title_fullStr | Control design for partial stabilization of nonlinear mechanical systems with random disturbances |
| title_full_unstemmed | Control design for partial stabilization of nonlinear mechanical systems with random disturbances |
| title_short | Control design for partial stabilization of nonlinear mechanical systems with random disturbances |
| title_sort | control design for partial stabilization of nonlinear mechanical systems with random disturbances |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/423 |
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