Solving output control problems using Lyapunov gradient-velocity vector function

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 9, No. 4, August 2019, pp. 2874~2879
ISSN: 2088-8708, DOI: 10.11591/ijece.v9i4.pp2874-2879  2874
Journal homepage: http://iaescore.com/journals/index.php/IJECE
Solving output control problems using Lyapunov
gradient-velocity vector function
М. А. Beisenbi, Zh. О. Basheyeva
Department of Systems Analysis and Control, L.N.Gumilyov Eurasian National University, Kazakhstan
Article Info ABSTRACT
Article history:
Received Sep 20, 2018
Revised Mar 13, 2019
Accepted Mar 21, 2019
This paper describes a controller and observer parameter definition approach
in one input-one output (closed-loop) control systems using Lyapunov
gradient-velocity vector function. Construction of the vector function is
based on the gradient nature of the control systems and the parity of the
vector functions with the potential function from the theory of catastrophe.
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. The approach allows
to select parameters based on the requested characteristics of the system.
Keywords:
Closed-loop control systems
Control systems
Gradient-velocity method
Lyapunov vector function
Оne input-one output control
system
Copyright © 2019 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Zh. О. Basheyeva,
Department of Systems Analysis and Control,
Eurasian National University,
2 Satpayev St. Astana, 010000, Kazakhstan.
Email: zhuldyz.basheyeva@gmail.com
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 [1-4].
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 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 with one input and one output. To research the dynamic
equalizer we use the Lyapunov gradient-velocity vector function [8-11]. Construction of the vector function
is based on the gradient nature of the control systems and the parity of the vector functions with the potential
function 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 [14-16]. 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.

Int J Elec & Comp Eng ISSN: 2088-8708 
Solving output control problems using Lyapunov gradient-velocity vector function (М. А. Beisenbi)
2875
2. RESEARCH
Assume the control system can be described by this set of equations [1-4]:
00 )(),()(),()()( xtxtCxtytButAxtx 
, (1)
)(ˆ)( txKtu  , (2)
00
ˆ)(ˆ),()()(ˆ)()( xtxtLytButxLCAtx 
, (3)
Modify the state (1)-(3). For this we will use the estimation error )(ˆ)()( txtxt  . Then we can
write is as: )()()(ˆ ttxtx  , and Equations (1)-(3) will transform to:
00 )(),()()()( xtxtBKtBKxtAxtx  
, (4)
00 )(),(),()(   ttLCtAt
(5)
For brevity consider the system with one input and one output, hence the system looks like:
nnnnn l
L
b
B
aaaa
A
0
...
0
0
,
0
...
0
0
,
...
0...000
...............
121




nn cccCkkkK ,...,,,,...,, 2121 
The set of (4), (5) will transform into:































nnnnnnnnnn
nn
nnnnnn
nnnnnnnnnn
nn
claclaclacla
kbkbkbkb
xkbaxkbaxkbaxkbax
xx
xx
xx





)(,...,)()()(
...
,...,
)(,...,)()()(
...
133222111
1
32
21
332211
133222111
1
32
21





(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 )(tK
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 [8-11].
From (6) we find the components of the vector gradient for the Lyapunov vector function
:)),(),...,,(),,((),( 221  xVxVxVxV n

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 9, No. 4, August 2019 : 2874 - 2879
2876






























































nnn
n
n
nn
n
nn
n
nn
nnn
n
n
n
n
n
n
nnn
n
n
nn
n
nn
n
n
n
n
cla
xV
cla
xV
cla
xV
xVxV
kb
xV
kb
xV
kb
xV
xkba
x
xV
xkba
x
xV
xkba
x
xV
x
x
xV
x
x
xV
x
x
xV


























)(
),(
,...,)(
),(
,)(
),(
.
;...,
),(
;
),(
),(
,...,
),(
,
),(
,)(
),(
,...,)(
),(
,)(
),(
;
),(
;.
),(
;.
),(
1
2
221
2
2
11
1
2
3
2
2
2
1
1
22
2
11
1
1221
2
11
1
1
3
3
2
2
2
1
(7)
From (6) we find the decomposition of the velocity vector to the coordinates ).,...,,,...,( 11 nnxx 
















































































































,)(,...,)(,)(
...
;;;
,,...,,
,)(,...,)(,)(
,,...,,
122111
1
3
2
2
1
2211
122111
3
2
2
1
21
31
21
21
32
nnn
n
nn
n
nn
n
n
n
nnn
n
n
n
n
n
nnn
x
n
nn
x
n
nn
x
n
n
x
n
xx
cla
dt
d
cla
dt
d
cla
dt
d
dt
d
dt
d
dt
d
kb
dt
dx
kb
dt
dx
kb
dt
dx
xkba
dt
dx
xkba
dt
dx
xkba
dt
dx
x
dt
dx
x
dt
dx
x
dt
dx
n
n
n
n
n
















(8)
To research the stability of the system (6) we use the basics of Lyapunov’s direct method [14-16].
For the system to achieve the asymptotical equilibrium we need to secure the existence of a positive function
),( xV 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):
22
1
2
2
2
21
2
1
22
3
2
2
222
2
2
2
2
22
1
2
1
222
1
2
3
2
32
2
2
2
21
2
1
2
1
22
3
2
2
2
1 11 1
)(,...,)()(,...,
)(,...,)()(
)(,...,
),(),(),(
nnnnnnnnnnn
nnnnnnnnn
nnn
n
ni
n
k
i
k
i
n
i
n
k x
i
k
i
claclaclakb
kbkbxkbaxkbaxkba
xkbaxxx
dt
dxV
dt
dx
x
xV
dt
xdV
kk




























  

(9)
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:

2877
,)1(
2
1
,...,)1(
2
1
)1(
2
1
)(
2
1
)1(
2
1
,...,)1(
2
1
)1(
2
1
)(
2
1
)(
2
1
,...,)(
2
1
)(
2
1
)(
2
1
2
1
,...,
2
1
2
1
2
1
,...,
2
1
2
1
2
1
)(
2
1
,...,)(
2
1
)(
2
1
)(
2
1
2
1
,...,
2
1
2
1
),(
2
1
2
3332
2
2221
2
111
2
1
2
332
2
221
2
11
2
1
2
332
2
221
2
11
22
3
2
2
22
33
2
22
2
11
2
1
2
332
2
221
2
11
22
3
2
2
nnnnnnnnnnn
nnnnnnnnnn
nnnnnnnnnnn
nnnnnnnnnn
nnnnnnn
kbclakbclakbcla
kbclaxkbaxkbaxkba
xkbaсlaclaclacla
kbkbkbkbxkba
xkbaxkbaxkbaxxxxV














(10)
The condition for the positive certainty (10) i.e. existence of Lyapunov function will be defined:














0>1
...
0>1
0>1
0>
1
32
21
1
nn
nn
nn
nn
kba
kba
kba
kba
(11)














0>1
...
0>1
0>1
0>
1
332
221
11
nnnn
nnn
nnn
nnn
kbcla
kbcla
kbcla
kbcla
(12)
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:














nnnnn
nn
xdxdxdxdx
xx
xx
xx
132211
1
32
21
,...,
...




(13)
Explore the system (13) with the given coefficients n),...,1i( id , using the gradient-velocity
function [8]. From (13) we find the components of the gradient-vector function ))(),...,(()( 1 xVxVxV n






























nn
n
n
n
n
n
n
n
n
n
n
n
xd
x
xV
xd
x
xV
xd
x
xV
xd
x
xV
x
x
xV
x
x
xV
x
x
xV
)(
,...,
)(
,
)(
,
)(
;
)(
;....,
)(
;.
)(
32
3
21
2
1
1
1
3
3
2
2
2
1
(14)
From (13) we determine the decomposition of the velocity vector according to the coordinates:



















































n
x
n
n
x
n
n
x
n
n
x
n
n
x
n
xx
xd
dt
dx
xd
dt
dx
xd
dt
dx
xd
dt
dx
x
dt
dx
x
dt
dx
x
dt
dx
n
n
132211
1
3
2
2
1
,...,,,
;;....,;.
321
32
(15)

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 9, No. 4, August 2019 : 2874 - 2879
2878
The total time derivative from the vector function is defined as a scalr product of the gradient vector
(14) and the velocity vector (15):
22
1
2
3
2
2
2
2
2
1
2
1
22
32 ,...,;....,
)(
nnnnn xdxdxdxdxxx
dt
xdV
 
(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:
2
1
2
32
2
21
2
1 )1(
2
1
,...,)1(
2
1
)1(
2
1
2
1
)( nnnn xdxdxdxdxV  
(17)
Now the task is to define the controller coefficients (the elements of the matrix K) so that they have values id
This way exploring the stability of the system with set coefficients id will result in:
0,>1-0,...,>1-0,>1-0,> 121 dddd nnn  (18)
Equalizing the elements of the set of inequalities (11) and (18) we will receive:





















)a-(
1
...
)a-(
1
)a-(
1
)a-(
1
11
2-n23
1-n12
n1
d
b
k
d
b
k
d
b
k
d
b
k
n
n
n
n
n
n
n
n
(19)














0>12
...
0>1d-c2
0>1d-c2
0>2
11
2-n3n2
1-n2n1
1
dcla
la
la
dcla
nn
n
n
nnn
(20)
From the set of inequalities (20) we will receive:
,1,...,2,1,
21d
max,
2d
max
1
i-n
1
n
n 




 



ni
c
a
c
a
l
i
in
i
n
(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 )(tBk in
the equation (4). The speed of convergence of the error )(t can be determined during the synthesis of the
observer from (21).

2879
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.
REFERENCES
[1] Andrievsky B. R. and Fratkov A. L., “Selected chapters of the theory of automatic control with examples in
the language MATLAB – SPb,” Nauka, pp. 475, 2000.
[2] Kvakernah H. and Sivan R., “Linear optimal control systems,” M. Mir, pp. 650, 1986.
[3] Andreev Y. N., “Managing finite-dimensional linear objects,” M. Nauka, pp. 424, 1976.
[4] Rey U., “Techniques for managing technological processes,” M. Mir, pp. 638, 1983.
[5] Kukharenko N. V., “Synthesis of modal regulators with incomplete controllability of objects,” Izvestiya Akademii
Nauk. Russian Academy of Sciences. Technical cybernetics, no. 3.
[6] Gantmakher F. R., “Theory of matrices,” Moscow, Nauka, 1967.
[7] W. Streitz, “The method of the space of states in the theory of discrete linear control systems, Per. with English,”
Moscow, Nauka, 1985.
[8] Beisenbi M. A., “Investigation of robust stability of automatic control systems by the method of functions of AM
Lyapunov,” Astana, pp. 204, 2015.
[9] Beisenbi M. and Uskenbayeva G., “The New Approach of Design Robust Stability for Linear Control System,”
Proc. of the Intl. Conf. on Advances in Electronics and Electrical Technology—AEET, pp. 11-18, 2014.
[10] Beisenbi M. and Yermekbayeva J., “Construction of Lyapunov function to examine Robust Stability for Linear
System,” International Journal of Control, Energy and Electrical Engineering (CEEE), vol. 1, pp. 17-22, 2014.
[11] Beisenbi M., et al., “Robust stability of spacecraft traffic control system using Lyapunov functions,” 16th
International Conference on Control, Automation and System (ICCAS), IEEE, pp. 743-748, 2016.
[12] Gilmore R., “Applied theory of catastrophes,” T.1. Moscow, The World, vol. 2, 1984.
[13] Poston T. and Stuart I., “The theory of catastrophes and its applications,” Moscow, Nauka, no. 6, 2001.
[14] Malkin I. G., “Theory of stability of motion,” Moscow, Nauka, pp. 534, 1966.
[15] E. A. Barbashin, “Introduction to the theory of stability,” Nauka, Moscow, pp. 225, 1967.
[16] Voronov A. A. and Matrosov V. М., “The method of Lyapunov vector functions in stability theory,” M. Nauka, pp.
252, 1987.
BIOGRAPHIES OF AUTHORS
М.А. Beisenbi. Dr. Sci. (Engineering), Professor. Graduate of the Kazakh Polytechnic Institute
V.I. Lenin, specialty "Automation and telemechanics" (1967-1972 gg.). Graduated postgraduate
studies at the Moscow Higher Technical School Bauman (1977 - 1980). In 1982 in the MHTS
Bauman successfully defended the candidate dissertation "Solution of optimal control problems
for objects with distributed parameters in automated control systems", specialty 05.13.02 -
System theory, theory of automatic control and system analysis. In 1998, defended doctoral
thesis "Models and methods of analysis and synthesis of ultimately stable systems", specialty
05.13.01 - Management in technical systems, the Institute of Informatics and Management
Problems of the Ministry of Education and Science of the Republic of Kazakhstan. Since 2002. -
Head of the Department of ENU L.N. Gumilyov. Since 2002. - Member of the Dissertation
Council for the defense of a doctoral dissertation at the L.N. Gumilev ENU. Since 2005. -
Member of the Dissertation Council for the protection of a doctoral dissertation at K.I.Satpaev
KazNTU. Teaching experience - 39 years.
Zh.О. Basheyeva. PhD Doctoral student of the Department of Automated Control Systems,
L.N.Gumilyov Eurasian National University, Аstana.

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