International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 07, July 2019, pp. 120-130, Article ID: IJCIET_10_07_014
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=7
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication
RESEARCH AND SSYNTHESIS OF CONTROL
SYSTEMS BY OUTPUT OF THE OBJECT BY
SPEED- GRADIENT METHOD OF
A.M. LYAPUNOV VECTOR FUNCTIONS
М. А. Beisenbi, Zh. О. Basheyeva
Department of Systems Analysis and Control,
L.N. Gumilyov Eurasian National University, Nur-Sultan, Kazakhstan
ABSTRACT
This paper describes one method of research and synthesis of control systems for
the output of the object by the gradient-speed method of Lyapunov vector functions.
The task of the synthesis of the regulator and the observer is considered as a system
that can provide the specified (desired) transition characteristics of a closed system.
The gradient-speed method of Lyapunov vector functions allows one to solve the
problem of control systems directly using the elements of the matrix of the controller
object and the observer.
The approach to determine the range of changes in the object parameters, a
regulator, and an observer that provides the specified (desired) transient
characteristics of a closed system.
Key words: Control systems. Closed-loop control systems. Lyapunov vector function.
Gradient-velocity method.
Cite this Article: М. А. Beisenbi, Zh. О. Basheyeva, Research and Ssynthesis of
Control Systems by Output of the Object by Speed - Gradient Method of
A.M.Lyapunov Vector Functions. International Journal of Civil Engineering and
Technology 10(7), 2019, pp. 120-130.
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1. INTRODUCTION
In practice the state vector is available for manipulation less often than the plant output . This
leads to the use of the state value in the control law instead of the state variables received by
the observers [1,2,3,4]. This, in return, requires that the dynamic properties of the system
change accordingly. We aim to observe how the replacement of the state variables by state
values affects the properties of the system. In the modal control [1,5] case characteristic
polynomials are found through output.
The characteristic polynomial [6] of the closed-loop system with a controller that uses
state values with an observer requires that the roots of the polynomial with a modal control be
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Research and Ssynthesis of Control Systems by Output of the Object by Speed - Gradient Method
of A.M.Lyapunov Vector Functions
combined with the observer’s own number [1,3,6]. This way synthesizing the modal
controller with the observer becomes a challenging task.
The known iterative algorithms [1,3] of separate own value control are based on the
preliminary matrix triangularization or block-diagonalization. Notice that the control matrix is
used as the nonsingular transition matrix in this canonical transformation, and it is defined by
its own vectors in complicated unequivocal ways described here [6,7].
The paper considers the control systems for the output of an object. To research the
dynamic equalizer we use the Lyapunov gradient-velocity vector function [8,9,10,11].
Construction of the vector function is based on the gradient nature of the control systems from
the theory of catastrophe [12,13]. Investigation of the closed-loop control system’s stability
and solution of the problem of controller (determining the coefficient of magnitude matrix)
and observer (calculation of the matrix elements of the observing equipment) synthesis is
based on the direct methods of Lyapunov [14,15,16]. The approach offered in this paper can
be considered as a way of determining the parameters of the controller and observer for a
closed-loop with certain transitional characteristics.
2. RESEARCH
In the present study tsunami forecast stations were selected for output of tsunami simulation
along the coast of India, Pakistan, Oman and Iran. Most of the tsunami forecast stations were
selected in such a way that sea depth is less than 10.0m to better
Assume the control system can be described by this set of equations [1,2,3,4]:
x (t )  Ax(t )  Bu (t ), y(t )  Cx (t ), x(t 0 )  x0
,
(1)
(2)
u (t )   Kxˆ (t ) ,
x (t )  ( A  LC ) xˆ (t )  Bu (t )  Ly (t ), xˆ (t 0 )  xˆ 0
,
(3)
Modify the state equation (1)-(3). For this we will use the estimation error  (t )  x(t )  xˆ(t ) .
Then we can write is as: xˆ(t )  x(t )   (t ) , and equations (1)-(3) will transform to:
x (t )  Ax(t )  BKx (t )  BK (t ), x(t 0 )  x0
,
(4)
(t )  A (t ), LC (t ),  (t 0 )   0
(5)
consider the system with one input and one output, hence the system looks like:
A
0
1
0
...
0
0
0
0
0
1
...
0
0
0
...
...
...
...
...
, B  ... , L  ...
0
0
0
...
0
0
0
 an
 a n 1
 a n2
...
 a1
bn
ln
K  k1 , k 2 ,..., k n , C  c1 , c2 ,..., cn
The set of (4) , (5) will transform into:
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М. А. Beisenbi, Zh. О. Basheyeva
 x1  x 2
 x  x
3
 2
...

 x n 1  x n
 x n  (a n  bn k1 ) x1  (a n 1  bn k 2 ) x 2  (a n  2  bn k 3 ) x3 ,..., (a1  bn k n ) x n 


 bn k1 1  bn k 2  2  bn k 3 3 ,..., bn k n  n
  
2
 1
 2   3

...
 n 1   n


n  (a n  l n c1 ) 1  (a n 1  l n c 2 ) 2  (a n  2  l n c3 ) 3 ,..., (a1  l n c n ) n
(6)
Notice that in the absence of the external impact, the process in the set (4), (5) must
asymptotically approach the processes of a system with a controller, as if the closed-loop
system according to a state vector, was affected by the impact of the convergent disturbance
waves. These disturbances are caused by the K (t ) polynom in the equation (5). The error
must converge and the speed of convergence is defined during the synthesis of the observer.
The main property of the set (4) and (5) lies in the asymptomatical stability. This way we
found the requirement for the asymptotical stability of the system using the gradient-velocity
method of the Lyapunov functions [8,9,10,11].
From (6) we find the components of the vector gradient for the Lyapunov vector function
V ( x,  )  (V1 ( x,  ),V2 ( x,  ),..., V2 n ( x,  )) :
 V1 ( x,  )
V ( x,  )
V ( x,  )
  x 2 ;. 2
  x3 ;. n 1
  xn ;

x3
x n
 x 2
 Vn ( x,  )
V ( x,  )
V ( x,  )
 (a n  bn k1 ) x1 , n
 (a n 1  bn k 2 ) x 2 ,..., n
 (a1  bn k n ) x n ,


x

x
x n
1
2

V ( x,  )
V ( x,  )
 Vn ( x,  )
 bn k1 1 , n
 bn k 2  2 ,..., n
 bn k n  n

 2
 n
  1
 Vn 1 ( x,  )
V ( x,  )
  2 ; n  2
  3 ;...,

 1
 2

 V ( x,  )
V ( x,  )
V ( x,  )
. 2 n
 (a n  l n c1 ) 1 , 2 n
 (a n 1  l n c 2 ) 2 ,..., 2 n
 (a1  l n c n ) n

 1
 2
 n
(7)
From (6) we find the decomposition of the velocity vector to the coordinates
( x1 ,..., xn ,  1 ,...,  n ).
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Research and Ssynthesis of Control Systems by Output of the Object by Speed - Gradient Method
of A.M.Lyapunov Vector Functions
 dx1 


 dt  x2
 dx 
 n 
 dt  x1

 dxn 

 dt  1

 d 1 
 dt  
1

...

 d n 

 dt  1
 dx 
 dx 
 x 2 ,  2   x3 ,...,  n   x n ,
dt

 x3
 dt  xn
 dx 
 dx 
 (a n  bn k1 ) x1 ,  n   (a n 1  bn k 2 ) x 2 ,...,  n   (a1  bn k n ) x n ,
 dt  x2
 dt  xn
 dx 
 dx 
 bn k1 1 ,  n   bn k 2  2 ,...,  n   bn k n  n ,
 dt   2
 dt   n
(8)
 d 
 d 
  2 ;  2    3 ;  n 1    n ;
 dt   3
 dt   n
 d 
 d 
 (a n  l n c1 ) 1 ,  n   (a n 1  l n c 2 ) 2 ,...,  n   (a1  l n c n ) n ,
dt

 2
 dt   n
To research the stability of the system (6) we use the basics of Lyapunov’s direct method
[14,15,16]. For the system to achieve the asymptotical equilibrium we need to secure the
existence of a positive function V ( x,  ) so that its total derivative on the time axis along the
state function (6) is a negative function. The total derivative from Lyapunov function with
regard to the state equation (6) is defined as a scalar product of the gradient (7) from
Lyapunov and the velocity vector. (8):
2n n
V ( x,  )  d i 
dV ( x,  ) n n Vi ( x,  )  dxi 
 

   i

 
dt
x k  dt  xk i  n 1k 1  k  dt   k
i 1 k 1
(9)
  x 22  x 32 ,...,  x n2  (a n  bn k1 ) 2 x12 
 (a n 1  bn k 2 ) 2 x 22  (a n  2  bn k 3 ) 2 x 32 ,..., (a1  bn k n ) 2 x n2  bn2 k12  12  bn2 k 22  22 
,..., bn2 k n2  n2   22   32  (a n  l n c n ) 2  12  (a n 1  l n c 2 ) 2  22 ,..., (a1  l n c n ) 2  n2
From (9) we derive that the total time derivative of the vector function will be negative.
Lyapunov function from (7) can be represented in the scalar view:
1 2 1 2
1
1
1
1
x 2  x 3 ,...,  x n2  (a n  bn k1 ) x12  (a n 1  bn k 2 ) x 22   (a n  2  bn k 3 ) x 32 ,..., 
2
2
2
2
2
2
1
1
1
1
1
1 2 1 2
1
2
2
2
2
2
(a1  bn k n ) x n  bn k1 1  bn k 2  2  bn k 3 3 ,...,  bn k n  n   2   3 ,...,   n2 
2
2
2
2
2
2
2
2
(10)
1
1
1
1
1
2
2
2
2
2
(a n  l n c1 ) 1  (a n 1  l n c 2 ) 2  ( a n  2  l n c 3 ) 3 ,..., (a1  l n с n ) n  ( a n  bn k1 ) x1 
2
2
2
2
2
1
1
1
1
2
2
2
(a n 1  bn k 2  1) x 2  (a n  2  bn k 3  1) x 3 ,...,  (a1  bn k n  1) x n  (a n  l n c1  bn k1 ) 12 
2
2
2
2
1
1
1
(a n 1  l n c 2  bn k 2  1) 22  (a n  2  l n c 3  bn k 3  1) 32 ,...,  (a1  l n c n  bn k n  1) n2 ,
2
2
2
V ( x,  )  




The condition for the positive certainty (10) i.e. existence of Lyapunov function will be
defined:
a n  bn k1 > 0
a  b k  1 > 0
n 1
n 2


 a n  2  bn k 3  1 > 0
...


a1  bn k n  1 > 0
(11)
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a n  l n c1  bn k1 > 0
a  l c  b k  1 > 0
n 2
 n 1 n 2
 a n  2  l n c 3  bn k 3  1 > 0
...

a1  l n c n  bn k n  1 > 0
123
(12)
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The quality and the stability of the control system is dictated by the elements of the matrix
of the closed- system. That is determining the target values of coefficients in a closed-loop
system will prodivide smooth transitional processes in a system and result in higher quality
control.
The set of inequalities (11) and (12) serve as the necessary condition for the robust
dynamic equalizer. The condition (11) allows for the stability in the state vector.
Imagine a control system with a set of desired transition processes with one input and one
output:
 x1  x 2
 x  x
2
3


...
 x  x
n
 n 1


x


d
n x1  d n 1 x 2  d n  2 x 3 ,...,  d 1 x n
 n
(13)
Explore the system (13) with the given coefficients d i (i  1,..., n) , using the gradient-velocity
function [8].
From (13) we find the components of the gradient-vector function V ( x)  (V1 ( x),..., Vn ( x))
Vn 1 ( x)
V2 ( x)
 V1 ( x)
  xn ;
 x   x 2 ;. x   x3 ;...., x

2
3
n

 Vn ( x)  d x , Vn ( x)  d x , Vn ( x)  d x ,..., Vn ( x)  d x
n 1
n 1 2
n2 3
n n

x 2
x3
x n
 x1
(14)
From (13) we determine the decomposition of the velocity vector according to the
coordinates:
 dx1 
 dx 
  x 2 ;. 2 

 dt  x3
 dt  x2

 dxn   d x ,  dxn
n 1
 dt  x
 dt
1

 dx 
 x3 ;....,  n 1   x n ;
 dt  xn


 x2
(15)
 dx 
 dx 
 d n 1 x 2 ,  n   d n  2 x3 ,...,  n   d1 x n
 dt  x3
 dt  xn
The total time derivative from the vector function is defined as a scalr product of the
gradient vector (14) and the velocity vector (15):
dV ( x)
  x 2  x3 ;....,  x n2  d n2 x12  d n21 x 22  d n2 2 x32 ,..., d12 x n2
dt
(16)
From (16) follows that the total time derivative of the vector-gradient function is negative.
The vector function in the scalar view from (14) can be represented as:
V ( x) 
1
1
1
1
d n x12  (d n 1  1) x 22  (d n  2  1) x32 ,...,  (d1  1) x n2
2
2
2
2
(17)
Now the task is to define the controller coefficients (the elements of the matrix K) so that
they have values d i .
This way exploring the stability of the system with set coefficients d i will result in:
(18)
d n > 0, d n1 - 1 > 0, d n2 - 1 > 0,..., d1 - 1 > 0,
Equalizing the elements of the set of inequalities (11) and (18) we will receive:
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Research and Ssynthesis of Control Systems by Output of the Object by Speed - Gradient Method
of A.M.Lyapunov Vector Functions
1

k 1  b ( d n - a n )
n

1

k 2  b (d n 1 - a n -1 )
n


1
k 3  ( d n  2 - a n - 2 )
bn

...

k  1 ( d - a )
1
1
 n b
n


(19)
2a n  l n c1  d n > 0
2 a  l c - d  1 > 0
 n 1 n 2 n -1
2 a n  2  l n c 3 - d n - 2  1 > 0
...

2a1  l n c n  d1  1 > 0
(20)
From the set of inequalities (20) we will receive:
 d  2a n
d  1  2a n  i
l n  max  n
, max i n -i
ci 1
 c1

, i  1,2,..., n  1,

(21)
This way the equations (4) and (5) allow us to conclude that in the absence of the external
impact, the process in the system asymptotically approaches those in the system with a
controller which state vector was affected by convergent disturbances. The role of the
disturbances is played by the compound Bk (t ) in the equation (4).
The speed of convergence of the error  (t ) can be determined during the synthesis of the
observer from (21).
A system with m-input and n-output is considered, respectively, the output control system of
the object has a matrix of the form
а11
а12
а13
 а1n
b11
b12
b13
a 21
a 22
a 23  a 2 n
b21
b22
b23  b2 m
A 

 
 ,B  

 
a n1
an2
a n 3  a nn
bn 2
bn 3  bnm
C
K
bn1
c11
c12
c13
... c1n
c 21
c 22
c 23
... c 2 n
...
...
...
... ...
cl1
cl 2
cl 3
... c ln
k11
k12
k13
... k1n
k 21
k 22
k 23
... k 2 n
...
...
...
... ...
k m1
k m2
k m3
... k mn
 b1m
x1 (t )
x(t ) 
x 2 (t )
x n (t )
,
L
,
 (t ) 
...
 ,
 1 (t )
 2 (t )
...
 n (t )
,
l11
l12
l13
... l1l
l 21
l 22
l 23
... l 2l
...
...
...
... ...
l n1
ln2
l n3
... l nl
,
Us (4) and (5) can be expanded as
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М. А. Beisenbi, Zh. О. Basheyeva
m
m
m


x

(

a

b
k
)
x

(

a

b
k
)
x

(

a

bij k j 3 ) x3 



1
i
1
ij
j
1
1
12
ij
j
2
2
13

j 1
j 1
j 1

m
m
m
m
m

 ...  (a1n   bij k jn ) x n  ( bij k j1 ) 1  ( bij k j 2 ) 2 ( bij k ij ) 3  ...  ( bij k jn ) n , i  1,..., n; (22)

j 1
j 1
j 1
j 1
j 1

l
l
l
   (a  l c )  (a  l c )  (a  l c ) 



i1
ij j1
1
12
ij j 2
2
13
ij j 3
3
 1
j 1
j 1
j 1

l


...

(

a

l ij c jn ) n , i  1,..., n

1
n

i 1

We find the condition of robust asiptotic stability of the system (22) by the gradientvelocity method of Lyapunov vector functions [8,9].
From (22) we find the components of the gradient vector for Lyapunov vector functions.
m
 Vi ( x,  )
 (a ik   bij k jk ) x k , i  1,..., n; k  1,..., n

j 1
  k
m
 Vi ( x,  )
 ( bij k jk ) k , i  1,..., n; k  1,..., n

j 1
  k
l
 V ( x,  )
 (a ik   l ij c jk ) k , i  1,..., n; k  1,..., n
 n 1
  k
j 1
(23)
(23) we decompose the velocity vector to coordinates ( x1 ,..., xn ,  1 ,... n )
m
 dxi
( ) xk  ( aik   bij k jk ) xk , i  1,..., n; k  1,..., n
j 1
 dt
m
 dx
i
( )  k  ( bij k jk ) k , i  1,..., n; k  1,..., n
dt
j 1

l
 d
( i )  k  ( aik   lij c jk ) k , i  1,..., n; k  1,..., n
 dt
j 1
(24)
The total time derivative of Lyapunov functions is defined as the scalar product of the
gradient vector (23) and the velocity vector (24)
n
n
n
n
V ( x,  )  dxi 
V ( x,  )  d i 
dV ( x,  ) n n Vi ( x,  )  dxi 
 

   i

    i

 
dt
x k  dt  xk i 1 k 1 x k  dt   k i 1 k 1
x k
 dt   k (25)
i 1 k 1
n
n
m
n
n
m
n
n
l
  (a ik   bij k jk ) 2 x k2  ( bij k jk ) 2  k2  (a ik   l ij c jk ) 2  k2 ,
i 1 k 1
j 1
i 1 k 1
j 1
i 1 k 1
j 1
From (25) it follows that the total time derivative of Lyapunov vector functions is a signnegative function. From (23), the Lyapunov function can be represented as:
V ( x,  ) 
l
m
1 n n m
1 n n
2
(
b
k

a
)
x

[(
l
c

a
)

bij k jk )] k2 , (26)
  ij jk ik k 2 


ij jk
ik
2 i 1 k 1 j 1
i 1 k 1
j 1
i 1
The condition for the existence of a Lyapunov function is defined by the inequalities:
n
b k
j 1
ij
jk
 aik > 0, i  1,..., n; k  1,..., n
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n
m
j 1
j 1
 lij c jk  aik -  bij k jk  aik > 0, i  1,..., n; k  1,..., n
(28)
The system of inequalities (27) and (28) is a condition for the robust stability of the
dynamic compensator. Condition (27) allows for robust stability of the system when
controlled by a state vector. Suppose we have a closed control system of desired transients
with m-inputs and n-outputs with matrix:
 d11
 d12
 d13   d1n
 d 21
 d 22
 d 23   d 2 n
G 
 d n1

 d n2



 d n 3   d nn
The task is to determine the coefficient of the regulator (elements of the matrix K) such
that the elements of the matrix of a closed system have the specified values.
We study the stability of a system with given values of the coefficients by the gradientspeed method of Lyapunov vector functions.
For a system with given parameters, write the equations of state in expanded form:
 x1   d11 x1  d12 x2  d13 x3 ,...,  d1n xn
 x   d x  d x  d x ,...,  d x
 2
21 1
22 2
23 3
2n n

...
 x n   d n1 x1  d n 2 x2  d n 3 x3 ,...,  d nn xn
(29)
From (29) we find the components of the gradient vector of Lyapunov vector functions
V ( x)  (V1 ( x),..., Vn ( x))
V1 ( x)
V1 ( x)
V1 ( x)
 V1 ( x)
 x  d11 x1 , x  d12 x2 , x  d13 x3 ,..., x  d1n xn
1
2
3
n

V ( x)
V ( x)
V ( x)
 V2 ( x)
 d 21 x1 , 2
 d 22 x2 , 2
 d 23 x3 ,..., 2
 d 2 n xn

x2
x3
xn
 x1
...

Vn ( x)
Vn ( x)
Vn ( x)
 Vn ( x)
 x  d n1 x1 , x  d n 2 x2 , x  d n 3 x3 ,..., x  d nn xn
1
2
3
n

(30)
The total derivative of the Lyapunov vector-function with respect to time is determined by
the expression:
V ( x)
2 2
2 2
2 2
 d112 x12  d122 x22  d132 x32 ,..., d12n xn2  d 21
x1  d 22
x2  d 23
x3 ,..., d 22n xn2 ,..., 
(31)
t
2 2
2
2
2 2
2 2
 d n1 x1  d n 2 x2  d n3 x3 ,..., d nn xn
From (31) it follows that the total time derivative of Lyapunov vector functions is a signnegative function.
From (30), the Lyapunov function can be represented in scalar form:
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М. А. Beisenbi, Zh. О. Basheyeva
1
1
(d11  d 21 ,...,  d n1 ) x12  (d12  d 22 ,...,  d n 2 ) x22 ,..., 
2
2
V ( x) 

1
(d1n  d 2 n ,...,  d nn ) xn2
2
(32)
The condition for the existence of Lyapunov vector functions for system (29) is described
by the system of inequalities:
d11  d 21 ,...,  d n1  0
d  d ,...,  d  0
22
n2
 12
d

d

,...,

d
 13
23
n3  0
...

d1n  d 2 n ,...,  d nn  0
(33)
We can equate inequalities (22) for system (27) and (33) to have the specified properties
and from there find the required values of the coefficients of the matrix k:
n
b k
ij
j 1
n
b k
ij
j 1
jk
jk
 aik  d ik , i  1,..., n; k  1,..., n
(34)
 aik  d ik  0, i  1,..., n; k  1,..., n
(35)
From this n-system of algebraic equations (35) we can find the values of the elements of
the matrix k (kij ; i  1,..., n; j  1,..., n) .
System (28) can be rewritten in the form:
n
l с
j 1
ij
jk
 2aik  d ik  0, i  1,..., n; k  1,..., n
(36)
Solving the system (36) with respect to the coefficients lij ; j  1,..., n; i  1,..., l ) , we can find the
boundary values for the coefficients of the observing device.
Suppose, for simplicity, we assume that the matrices B, L and C have dimension,
respectively n  1, l  1, l  1 , and the matrices A and D are given a controlled canonical form.
Then the equations can be rewritten in the form:
n
k
j 1
j

1
(ai  d i )  0, i  1,..., n;
bi
(37)
From (37), the boundary values of the coefficients
will be determined:
li 
2a i  d i
n
k
j 1
, i  1,..., n
(li ; i  1,..., l )
of the observability matrix
(38)
j
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Research and Ssynthesis of Control Systems by Output of the Object by Speed - Gradient Method
of A.M.Lyapunov Vector Functions
Thus, the problem of ontrol for the output of the object is solved by the gradient-speed
method of Lyapunov vector functions completely. The calculation of the coefficients of the
controller and the observing device according to the desired characteristic of the system (29)
is presented as a solution of the system of algebraic equations (35) and (36).
The calculation of the matrix elements of the regulator and the observer requires certain
regulated transformations - complex and ambiguous calculations of the roots of the
characteristic equation of a closed system. The roots of the characteristic polynomial of a
closed system are obtained by combining the roots of the system with a model controller and
the eigenvalues of the state observer.
3. CONCLUSION
The known methods of synthesis of the systems with controllers that use the state value and
the observer are based on the combination of the roots of the characteristic polynomial with a
modal control with the observer’s own number. This process requires substantial calculations
and requires preliminary matrix block-diagonalization. This nonsingular matrix in the
canonical transformation is defined by its own vectors in.
Researching the closed-loop control system using the gradient-velocity method of
Lyapunov function gave us the opportunity to develop an approach for controller and observer
parametrization that provides us with a system of our desire without extraneous calculations.
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