1. science
2. mathematics
3. algebra

I R M A
I V A N O V I E N Ė
INVESTIGATION OF
DYNAMICS OF CONTROL
SYSTEMS USING
LAMBERT FUNCTION
S U M M A R Y O F D O C T O R A L
D I S S E R T A T I O N
P H Y S I C A L S C I E N C E S ,
I N F O R M A T I C S ( 0 9 P )
Kaunas
2015
KAUNAS UNIVERSITY OF TECHNOLOGY
IRMA IVANOVIENĖ
INVESTIGATION OF DYNAMICS OF
CONTROL SYSTEMS USING LAMBERT
FUNCTION
Summary of Doctoral Dissertation
Physical sciences, Informatics (09P)
KAUNAS, 2015
The research was accomplished during the period of 2009 to 2014 at
Kaunas University of Technology, Department of Applied Mathematics, Faculty
of Mathematics and Natural Sciences. It was supported by the Lithuanian
Research Council.
Scientific supervisor:
Prof. Dr. Jonas RIMAS (Kaunas University of Technology, Physical
Sciences, Informatics – 09P).
Scientific consultant:
Prof. Dr. Habil. Henrikas PRANEVIČIUS (Kaunas University of
Technology, Physical Sciences, Informatics – 09P).
Dissertation Defence Board of Informatics Science Field:
Chairman
Prof. Dr. Habil. Rimantas BARAUSKAS (Kaunas University of
Technology, Physical Sciences, Informatics – 09P).
Members:
Prof. Dr. Habil. Romas BARONAS (Vilnius University, Physical Sciences,
Informatics – 09P),
Prof. Dr. Habil. Vytautas KAMINSKAS (Vytautas Magnus University,
Physical Sciences, Informatics – 09 P),
Prof. Dr. Gintaras PALUBECKIS (Kaunas University of Technology,
Physical Sciences, Informatics – 09P),
Prof. Dr. Habil. Rimvydas SIMUTIS (Kaunas University of Technology,
Physical Sciences, Informatics – 09P).
Opponents:
Prof. Dr. Habil. Feliksas IVANAUSKAS (Vilnius University, Physical
Sciences, Informatics – 09P),
Prof. Dr. Habil. Minvydas RAGULSKIS (Kaunas University of
Technology, Physical Sciences, Informatics – 09P).
The official defense of the dissertation will be held at 1 p.m. on June 17,
2015 at the Board of Informatics Sciences Field public meeting in the
Dissertation Defense Hall at the Central Building of Kaunas University of
Technology.
Address: K. Donelaičio str. 73-403, LT-44249 Kaunas, Lithuania.
The summary of the doctoral dissertation is sent on 15 May 2015.
The Dissertation is available on the internet (http://ktu.edu) and at the
library of Kaunas University of Technology (K. Donelaičio St. 20, Kaunas).
KAUNO TECHNOLOGIJOS UNIVERSITETAS
IRMA IVANOVIENĖ
VALDYMO SISTEMŲ DINAMIKOS
TYRIMAS, TAIKANT LAMBERTO
FUNKCIJĄ
Daktaro disertacijos santrauka
Fiziniai mokslai, Informatika (09P)
KAUNAS, 2015
Disertacija rengta 2009-2014 metais Kauno Technologijos Universiteto
Taikomosios matematikos katedroje, Matematikos ir gamtos mokslų fakultete,
remiant Lietuvos mokslo tarybai.
Mokslinis vadovas:
Prof. dr. Jonas RIMAS (Kauno technologijos universitetas, fiziniai mokslai,
informatika – 09P).
Mokslinis konsultantas:
Prof. habil. dr. Henrikas PRANEVIČIUS
universitetas, fiziniai mokslai, informatika – 09P).
(Kauno
technologijos
Disertacija ginama informatikos mokslo krypties taryboje:
Pirmininkas
Prof. habil. dr. Rimantas BARAUSKAS (Kauno technologijos
universitetas, fiziniai mokslai, informatika – 09P).
Nariai:
Prof. habil. dr. Romas BARONAS (Vilniaus universitetas, fiziniai mokslai,
informatika – 09P),
Prof. habil. dr. Vytautas KAMINSKAS (Vytauto didžiojo universitetas,
fiziniai mokslai, informatika – 09 P),
Prof. dr. Gintaras PALUBECKIS (Kauno technologijos universitetas,
fiziniai mokslai, informatika – 09P),
Prof. habil. dr. Rimvydas SIMUTIS (Kauno technologijos universitetas,
fiziniai mokslai, informatika – 09P).
Oponentai:
Prof. habil. dr. Feliksas IVANAUSKAS (Vilniaus universitetas, fiziniai
mokslai, informatika – 09P),
Prof. habil. dr. Minvydas RAGULSKIS (Kauno technologijos universitetas,
fiziniai mokslai, informatika – 09P).
Disertacija bus ginama viešame informatikos mokslo krypties tarybos posėdyje
2015 m. birželio mėn. 17 d. 13 val. Kauno Technologijos Universiteto centrinių
rūmų disertacijų gynimo salėje.
Adresas: K. Donelaičio g. 73-403 , LT-44249 Kaunas, Lietuva.
Disertacijos santrauka išsiųsta 2015 m. gegužės mėn. 15 d.
Disertaciją galima peržiūrėti internete (http://ktu.edu) ir Kauno Technologijos
Universiteto bibliotekoje (K. Donelaičio g. 20, Kaunas).
INTRODUCTION
In the first chapter there are presented: research subject, the main aim, its
problems and methods applied in the given analysis.
Research subject: multidimensional linear time-delay systems.
Actuality of the research: automatic control systems are widely used in
the present industrial technologies, cosmic technological devices, aviation,
military industry, energetics, electronics and electrical appliance. When
managing technological processes the signal delay, which influences system's
dynamic features, is often encountered. Multidimensional control systems with
delays are not fully investigated within control theory.
The mathematic model of the multidimensional linear time-delay system is
the matrix differential equation with delayed argument. This type of equations is
being studied two decades intensively and a wide number of monographs and
scientific works were published. Differential equations with delays are often
solved using numerical methods, applying diverse approximations or
analytically. Many numerical methods are implemented
into engineering
packages, such as Matlab, Maple, Mathematica. When the approximate results
received by numerical methods do not satisfy, the exact methods of research are
being sought. The main difficulty when solving the linear differential equations
with delayed argument, is that a transcendental characteristic equation
corresponding to them has an infinite number of roots.
Research aim and problems: The main aim of the work is to propose the
method for investigation of multidimensional linear time delay systems
described by linear matrix differential equations with delayed argument, to use
the proposed method for computation of step responses matrices of concrete
control systems, to examine transients in the systems and to perform analysis of
their stability.
In order to reach the main aim the following tasks are formulated:
1. To perform the analysis of the method of investigation of control systems with
delays, which is based on the use of the Lambert W function, and to test the
possibility of the use of the method for finding solution of various differential
equations with delayed argument: scalar differential equations and matrix
differential equations, when coefficient matrices of the equation are commuting
and not commuting.
2. To propose a new modification of the method of finding of the solution of
linear matrix differential equation with delayed argument and non commuting
coefficient matrices, which is based on the use of the Lambert W function and on
the use of a special auxiliary matrix.
3. To propose a new method of finding of the solution of linear matrix
differential equation with delayed argument, applicable in the case when the
argument of the Lambert W function (which is a matrix) has zero eigenvalues.
5
4. To make the analysis of the concrete multidimensional control systems,
described by the linear matrix differential equation with delayed argument,
applying the offered technique (to find matrices of step responses, to investigate
transients, to carry out the stability analysis).
Research methods and means. The Linear matrix differential equation,
which describes control system with delays, is solved using the method based on
Lambert's W function application, using a special auxiliary matrix, and an
application of a correction for Lambert W function values when its argument is
equal to 0. Lambert's W matrix function is defined relying on the definition of
a standard matrix function used in the matrix analysis.
Represented research results.
1. A new solution method for linear matrix differential equations with delayed
argument and for non- commuting coefficient matrices based on Lambert's W
function application and on the use of special auxiliary matrix. This matrix is
found using iteration process (applying fsolve function given in the Matlab
program). The advantage of the suggested method is a simpler equation,
defining the auxiliary matrix (in comparison with the known method), and the
fact that the initial matrix in the iteration process is described by the same
expression ( independently on the structure of coefficient matrices of the matrix
differential equation with delayed argument).
2. A new method for finding solution of linear matrix differential equation with
delayed argument, applicable in the case, when the argument of the Lambert W
function (the matrix) has zero eigenvalues.
3. The results of analytical investigation of concrete multidimensional control
systems with delays using the Lambert W function
Scientific newness and practical importance. Investigation of control
system with delays is actual. This fact is indicated by abundance of publications
on this subject. There are widely used diverse numerical methods for control
systems' investigations, too. In the case of approximated results, obtained
applying numerical methods, there are pursued different means that would allow
an exact or close to exact results. This is possible to achieve applying consequent
integration method which gives an exact solution by finite sum of functions in
any finite interval, and a method, based on Lambert's W function, that gives an
exact result in the shape of infinite functional series.
In doctoral dissertation control systems with delays are described by a
linear matrix differential equation with delayed argument and by non-commuting
coefficient matrices. They are investigated using a method offered in dissertation
work. The main points of this method are following:
1. An original procedure of finding of an auxiliary matrix which enables to
investigate systems having non- commuting coefficient matrices.
2. A new method for finding the solution, applicable in the case when Lambert
W function's argument has zero eigenvalues.
6
There were analyzed diverse multidimensional control systems with delays
applying the method suggested in dissertation work. The investigation of some
control systems using known method was impossible.
The suggested means could be used in analysis of mathematical models
(described by linear matrix differential equation with delayed argument) that can
be met in different areas such as physics, biochemistry, economics, etc.)
Approval of the work results, publications
9 scientific papers published on the topic of the doctoral dissertation:
• Papers in journals referred in the databases of Institute of Scientific
Information:
Included in the ISI Master journal list: 2
Included in database of ISI Proceedings: 3
• Papers in journals referred in the databases included in the list
approved by the Science Council of Lithuania: 4
10 presentations in conferences:
• Presentations in the national conferences: 4
• Presentations in the international conferences: 6
Work structure
The doctoral dissertation is composed of an introduction, 5 main chapters,
conclusion, references, list of publications. Its volume: 148 pages, 53 pictures
and 94 entries in the references list.
7
2.
OVERVIEW
Lambert's function and its use in diverse control system investigations are
introduced in the second chapter. There is presented a short overview of
mathematical models that describe systems – differential equations with delayed
argument – solution using Lambert function method when we have: linear
homogenous and non-homogenous, scalar and matrix differential equation with
delayed argument. There are also presented consequent integration and
numerical methods, pseudo inverse matrix calculation method. Also control
systems investigation examples, employing Lambert's function method, are
presented.
2.1. Method of research of control systems using Lambert W function.
There we will define a scalar and matrix Lambert function.
1. The scalar Lambert W function: The inverse function of
z = ψ (w) = wew
(2.1)
(here z and w are complex numbers) is called the Lambert W function and is
denoted (Chen, Horng and Yang, 2010) :
W (z ) = w = ψ −1 (z ).
W function
Wk ( z ), k = 0,±1,±2,K
Lambert
has
an
(2.2)
infinite
number
of
2. Matrix Lambert W function. In order to calculate the value
branches:
Wk (H ) of
the k–th branch of the matrix Lambert function W (H ) of argument H , firstly
λi , i = 1, n
found. When matrix H
eigenvalues
and eigenvectors
Vi , i = 1, n of matrix H have to be
is diagonalizable, we will have (Horn, 1986)
0
Wk (λ1 )
 0
W k (λ 2 )
Wk ( H ) = V 
 M
M

0
 0

L
0  −1
V ,
O
M 

L Wk ( λ n ) 
L
k = −∞ ,K ,−1,0,1,K ,+∞;
8
0
(2.3)
here V = [V1 V2 K Vn ]. Otherwise i.e. when H is not diagonalizable,
this expression will be more complicated (Jarlebring and Damm, 2007).
Application of Lambert function allows to write transcendental equation
solutions. This function is very practical because its values Wk H at different
( )
arguments H and for different branches k , can be calculated using Matlab,
Mathematica, Maple packages (Corless, Gonnet, Hare, Jeffrey and Knuth, 1993).
In this work the control system’s research is made using the scalar and
matrix Lambert functions. There will be presented only mathematical models
that describe the systems and general solutions obtained applying Lambert
function (a detailed research is given in the dissertation work).
2.1.1. Application of Lambert W function for solution of linear scalar
homogenous differential equation with delayed argument
Let we have a linear homogenous matrix differential equation with delayed
argument
x' (t ) + ax(t − τ ) + bx(t ) = 0
x(t ) = φ (t ), t ∈ [− τ ,0];
here
τ
(τ > 0), t ≥ 0,
is a fixed time delay, x (t ) – desired function,
(2.4)
φ (t ) is an initial function
defined on the interval [ −τ ,0] , a and b are constants.
The general solution of the equation (2.4), using a the scalar Lambert W
function, is written as (Asl and Ulsoy, 2000):
x(t ) =
∞
∑C e
1

bτ
 Wk ( − aτe ) −b  t
τ

k
;
(2.5)
k = −∞
here
Ck ( k = 0,±1,±2, K ) is a coeficient.
2.1.2. Application of Lambert W function for solution of linear matrix
homogenous differential equation with delayed argument
Let we have a linear homogenous matrix differential equation with delayed
argument:
x' (t ) + B1 x(t ) + B2 x(t − τ ) = 0,
x(t ) = φ (t ), t ∈ [−τ ,0];
(2.6)
9
here
B1
and
B2
are
n× n
numerical coefficient matrices, x (t ) is a vector
τ is a constant time delay, φ (t ) is an initial function defined on the
interval [− τ ;0] . The solution of the homogeneous matrix differential equation
function,
(
)
(2.6 on the interval 0,+∞ obtained
follows (Asl and Ulsoy, 2003):
x(t ) =
∞
∑ e Skt Ck = lim
N →∞
k = −∞
using Lambert function, is presented as
N
∑e
Sk t
Ck ,
j = 1, n, t ∈ (τ ,+∞);
(2.7)
k =− N
Ck ∈ M n×1 (C ) ( M n×m (C ) – is a set of all n × m matrices),
S k ∈ M n×n (C ) . When coefficient matrices of a differential equation (2.6) are
here
commuting (Asl and Ulsoy, 2003),
(
)
1
S k = Wk − B2τe B1τ − B1 , k = 0,±1,±2,K
τ
(2.8)
and if non-commuting (Yi, Ulsoy and Nelson, 2007),
1
S k = Wk (− B2τQk ) − B1 .
(2.9)
τ
2.1.3. Application of Lambert W function for solution of linear scalar nonhomogenous differential equation with delayed argument
Let we have a linear scalar non-homogeneous differential equation with delayed
argument:
x' (t ) + ax(t ) + bx(t − τ ) = u(t ),
x(t ) = φ (t ), t ∈ [−τ ,0];
(2.10)
()
here a, b are coefficients, φ (t ) is an initial function, u t is a free term
(continuous vector-function depending on the initial conditions). The general
solution of non-homogenous differential equation (2.10) on the interval
can be written as follows (Yi and Ulsoy, 2006):
x (t ) =
∞
∞
e sk t C k + ∫ ∑ e sk (t −ζ )C k′ u (ζ ) dζ ;
∑
k = −∞
k = −∞
1424
3 01
44424443
homogenous
10
t
(0,+∞)
non - homogenous
(2.11)
here
(
)
1
s k = Wk − bτe aτ − a.
τ
(2.12)
2.1.4. Application of Lambert W function for solution of linear matrix nonhomogenous differential equation with delayed argument
Let we have a linear non-homogenous matrix differential equation with delayed
argument:
x' (t ) + B1 x(t ) + B2 x(t − τ ) = u (t ),
x(t ) = φ (t ), t ∈ [−τ ,0];
here
B1
function,
and
τ
B2
are
n× n
(2.13)
numerical coefficient matrices, x (t ) is vector
is constant time delay,
φ (t ) is initial vector function on the interval
[− τ ;0] . The general solution of non-homogenous differential equation (2.13)
on the interval (0,+∞) , using Lambert W function, is expressed so (Yi and
Ulsoy, 2006):
x (t ) =
∞
t
∞
e S k t C k + ∫ ∑ e S k (t −ζ )C k′ u (ζ ) dζ ;
∑
k = −∞
k = −∞
1424
3 01
44424443
homogenous
here
(2.14)
non - homogenous
Ck′ = lim (η −1 (T , N ) × (σ − δ )) k , Ck′ ∈ M n×n (C ) and Sk are
N →∞
expressed by (2.8), when coefficient matrices of the equation (2.13) are
commuting, and by (2.9), when they are non-commuting.
In the general solutions (2.5), (2.7), (2.11) and (2.14) the coefficients
Ck ∈ M n×1 C and Ck′ ∈ M n×n C are used. The coefficient Ck is written
( )
( )
so (Yi, Ulsoy and Nelson, 2010):
{
}
Ck = lim X −1 (τ , N )Φ(τ , N ) k ;
N →∞
(2.15)
here
11
 ϕ (0) 
 e S− N ⋅0
  τ 
 S− N ⋅ − τ 

ϕ  −
e  2 N 
  2 N 
 2τ 

Φ (τ , N ) =   2τ ; Ω(τ , N ) =  S− N ⋅ − 2 N 
e
ϕ −
  2 N 

M



M


 e S− N ⋅(−τ )
 ϕ (− v ) 
{
Ck = C− N
L
L
L
O
L
e S N ⋅0



e
 2τ  
S− N ⋅ −

;
e  2N  

M

e SN ⋅(−τ ) 
 τ 
S− N ⋅ −

 2N 
(2.16)
}
T
C− N +1 C− N +2 L C N .
The expression of the coefficient
Ck′ is following (Yi, Ulsoy and Nelson, 2010):
{
}
Ck′ = lim Π −1 (τ , N )Γ(τ , N ) ;
N →∞
(2.17)
here
 e B1 (τ ) 
 e S − N ⋅(τ )
τ


 S − N ⋅ τ − τ 
 B1  τ −  
e  2 N 
e  2 N  
2
2τ 
τ






Γ(τ , N ) =  B1  τ − 2 N  ; Π (τ , N ) =  S − N ⋅ τ − 2 N 
e
e



M
M



 e S − N ⋅(0 )
 e B1 ( 0 ) 
[
C k′ = C −′ N
Γ(τ , N ) ,
C −′ N +1
C −′ N + 2 L C N′
C k′ ∈ M n ( 2 N +1)×n (C ) ,
L
L
L
O
L
];
e S N ⋅(τ )



e
2τ  

S − N ⋅ τ −

;
e  2N  

M

S N ⋅( 0 )

e
τ 

S − N ⋅ τ −

 2N 
(2.18)
T
Π (τ , N ) ∈ M n ( 2 N +1)× n ( 2 N +1) (C ) .
The
coefficient C k can be found using Laplace transform (Yi, Ulsoy and Nelson,
2006):
d
(s − λk1 )(s − λk 2 )  λ + b1 + b2 e− sτ − b1 + b2 e− sτ 
12
12
ds
 ki 1 22 2 22− sτ
×
lim
1
2 − sτ 
 − b +b e
s → λ ki
d
λ
+
b
+
b
e
21
21
ki
11
11


∆s
ds
d 012 
 λki − d 011
C , i = 1,2.
× {x0 − B2G(λki )} = 
λki − d 022  k
 d 021
(
12
)
(
)
(2.19)
The coefficient
Ck′ is found as follows:
d
(s − λ k1 )(s − λk 2 )  λ + b1 + b 2 e − sτ
 ki 1 22 2 22− sτ
lim ds
 − b +b e
s →λ ki
d
21
21

∆s
ds
d 012 
 λ − d 011
C ′ , i = 1,2.
=  ki
λ ki − d 022  k
 d 021
(
)
(
)
1
− b12
+ b122 e − sτ 
=
λ ki + b111 + b112 e − sτ 
(2.20)
The given expressions (2.19) and (2.20) are obtained for the control system
described by the second order matrix differential equation with delayed
argument.
In the matrix equations (2.15) and (2.17) the matrices
X (τ , N ) and
Π(τ , N ) are not singular. When matrices are close to the singular ones or they
are singular, the expressions (2.15) and (2.17) for the search of coefficients
cannot be applied. In this case the matrix equation can be solved using a pseudo
inverse matrix (Barata and Hussein, 2012). The pseudo inverse matrix for the
matrix A is found as it is shown below (Barata and Hussein, 2012):
A + = VS +U T ;
(2.21)
U is eigenvector’s matrix of AA* ( A* is the conjugate transpose matrix),
V is eigenvector’s matrix of A* A , S is a diagonal matrix which contains the
here
eigenvalues of A in its main diagonal. The pseudo inverse matrix of the matrix
S is found as follows:
1
 ,1≤ i ≤ r
+
+
S (i, i ) = σ i
, S (i, j ) = 0, i ≠ j;
0, r + 1 ≤ i ≤ n

(2.22)
r - is a range of the matrix A , σ i are non-zero elements in the main
diagonal of matrix S . Following condition must be fulfilled for pseudo inverse
matrix:
(
)
A+ = AT A
−1
AT
(2.23)
13
2.2. Finding of the coefficients
Ck and Ck′ using the pseudo inverse
matrix
Let we have a matrix equation in which
singular:
X (τ , N )
is singular or close to
X (τ , N )Ck = Φ(τ , N ).
Then the equation (2.24) is multiplied by
X (τ , N )T :
X (τ , N )T X (τ , N )C k = X (τ , N )T Φ (τ , N ).
From the equation (2.25)
(2.24)
(2.25)
Ck is expressed if the product X (τ , N )T X (τ , N ) is
not singular matrix:
(
Ck = X (τ , N )T X (τ , N )
)
−1
X (τ , N )T Φ (τ , N ).
Applying the expression (2.23), we have
Ck = X (τ , N ) + Φ (τ , N ).
If the product X (τ , N )
methods must be used.
T
X (τ , N ) is a singular matrix, then the other
2.3. Investigation of control systems with delays using consequent
integration method
In the given dissertation there are presented solutions of linear matrix
differential equations with delayed argument, describing multidimensional
control systems with various structures of internal links, found applying
consequent integration method.
()
1. The step response matrix h t of the mutual synchronization system . Let
we have a mathematical model of the multidimensional control system
presented by a linear matrix differential equation:
x' (t ) + B1 x(t ) + B2 x(t − τ ) = z (t ),
x(t ) = φ (t ), t ∈ [−τ ,0].
The response of a component
(2.26)
xi (t ) of a vector x(t ) to the unit jump in
the component x j (t ) will be referred as the step response hij (t ) of the system,
14
described by the (2.26) equation. The matrix
h(t ) = (hij (t )) i, j = 1, n we
shall call the step response matrix of the system. On the base of (2.26) equation,
the corresponding systems of differential equations for step responses hij (t )
(i, j = 1, n ) can be written (Rimas,1986). There we shall present expressions for
( (
))(
)
L = 0,1,2,K for few
the solution of this system on the interval 0; L + 1 τ
special cases.
A. Structure of the internal links of the system is "complete graph" or
"star". Coefficient matrices are defined by B1 = κE ( κ –constant, E –
identity matrix) ,
B2 =
κ
n −1
B ; here
0, i = j ,
{B}ij = 
(2.27)
1, i ≠ j
for „complete graph“ structure and
1 1 L 1
 0


 n −1 0 0 O M 
B =  n −1 O O O 0


O 0 0 0
 M
 n −1 L 0 0 0


(2.28)
for „star“ structure. Using the consequent integration method and Laplace
transform the step responses hij (t ) can be presented as follows (Rimas,1986):
 κ  (t − lτ ) −κ (t −lτ ) l
e
B ij 1(t − lτ ),


hij (t ) = ∑  n − 1 
l!
l =0
0 < t < (L + 1)τ .
l
l
{ }
L
(2.29)
B. Structure of the internal links of the system is "chain" or "ring".
Coefficient matrices are defined by the expressions: B1 = κE ( κ – constant,
κ
E – identity matrix) , B2 = − B ;
2
here
15
0 2 0

1 0 1
B = 0 1 0

M O O
0 L 0

L 0

L 0
L 0

O M
2 0 
(2.30)
L 1

L M
L 0

O M
1 0 
(2.31)
for „chain“ structure and
0 1 0

1 0 1
B = 0 1 0

M O O
1 L 0

for „ring“ structure. The step responses can be given by
 κ  (t − lτ ) −κ ( t −lτ ) l
e
B ij 1(t − lτ ),
 
hij (t ) = ∑  2 
l!
l =0
0 < t < (L + 1)τ .
l
L
l
{ }
(2.32)
2. The analytical expressions of the elements hij (t ) of the step responses
matrix h(t ) for forced synchronization system with „chain“ form structure of
the internal links and odd number of oscillators in the system (Leonaitė and
Rimas, 2006). In this case the system is described by the equation (2.26), in
which coefficient matrices are defined as follows:
B2 =
κ
2
16
B; here
B1 = diag(0, κ ,−κ ,K, κ ),
0

1
0
B=
M
0

0

( B1 , B2 , B ∈ M n×n
(R ),
0
0
0
1
0
0
K
1
0
1
K
M
M
O
M
0
K
1
0
0
K
0
2
n - is odd). For this
K
0

0
0
.
M
1 
0 
system the step responses hij (t )
can be written by following expressions (Leonaitė and Rimas, 2006):
e −κt 1(t ) + u ij (t ), i = j > 1,

hij (t ) = u ij (t ), 1 < i ≠ j > 1,
v (t ), i > 1 ir j = 1;
 ij
(2.33)
here
L
 1  κ (t − kτ ) −κ (t − kτ ) k
u ij (t ) = ∑  
e
B
k!
k =1  2 
k
k
{ } 1(t − kτ ) ,
ij
i > 1, j > 1, 0 < t < (L + 1)τ ,
L
1
vij = ∑  
k =1  2 
k
 k −1 κ l (t − kτ )l − κ  t − kτ  

1 −
e 
B k ij 1(t − kτ ) ,
 ∑

l!
l =0


{ }
i > 1, j = 1, 0 < t < (L + 1)τ .
2.4.
Investigation of control systems applying numerical methods
When control system’s transient processes and stability are analyzed,
precise results are not always requested and the use of numerical methods is
sufficient. In dissertation work there are presented some of: widely used RungeKutta method (Tsitouras, 2007), which is described as dde23 function in the
Matlab package (Shampine and Thompson, 2001), and the one based on
Chebyshev polynomials application, which is used for the investigation of
stability (Butcher, Ma, Bueler, Averina and Szabo, 2004).
17
2.4.1. dde23 function described in the Matlab package
This function uses the Runge-Kutta method (its formulas of the second and
third order are obtained choosing Teilor's series components respectively
including the first or the second order derivatives. Runge-Kutta method is used
for finding of solution of ordinary differential equation. It is modified so as it
could be applied for the solution of differential equations with delayed
argument (Shampine and Thompson, 2001). An example which presents the
application of this functions is given in dissertation work.
2.4.2. Method of investigation of stability based on application of
Chebyshev's polynomials (Butcher, Ma, Bueler, Averina and Szabo, 2004)
Stability of the system, described by the linear matrix differential equation with
delayed argument, can be analyzed using Chebyshev polynomials. A detailed
information about this method is presented in dissertation work wherein are also
given program codes for the stability analysis of any system.
In the second chapter application of fsolve function, integrated in Matlab
package, is described, too (fsolve function's purpose is to find solution X = A
( )
for non-linear matrix equation F X = 0 , which would satisfy the equation
(Geletu, 2007)). There also results of analysis of various control systems using
Lambert function method are presented.
3.
NEW COMPLEMENTS FOR METHOD OF SOLUTION OF
MATRIX DIFFERENTIAL EQUATION WITH DELAYED
ARGUMENT BASED ON APPLICATION OF LAMBERT W
FUNCTION
In the third section the modified Lambert W function method and the
method of finding of solution in the case when the argument (the matrix) of
Lambert function has zero eigenvalues, proposed in the dissertation work, are
presented. As well, there is presented the method of finding the step responses
matrix of control system with delays.
3.1. Modified Lambert function method and method of finding solution in
the case when Lambert function’s argument has zero eigenvalue
When analyzing control systems, described by the linear matrix
differential equation with delayed argument with non-commuting coefficient
matrices using Lambert function method, some problems are met. For their
solution the modified Lambert function method is created in the dissertation
work.
18
3.1.1. Modified Lambert function method
Using the method presented in the overview and solution (2.9), we
introduce the substitution Dk = Wk − B2τQ . Then the equation (2.9) can be
(
)
written as follows:
Sk =
1
τ
Dk − B1.
We calculate the unknown matrix
(3.1)
Dk using fsolve function:
Dk e Dk − B1τ = − B2τ .
(3.2)
The solution of the equation (2.6) is written as:
x(t ) =
∞
∑e
Sk t
Ck .
(3.3)
k =−∞
Using fsolve function in the Matlab program we realize an algorithm:
Bτ
(0)
1. Firstly, we choose an initial matrix Dk = Wk − B2τe 1 , which will be
(
)
different for different branches of the Lambert function.
(n)
Rk(n ) = Dk( n ) e Dk −B1τ
2. Then we put
into
D k( 0 )
( )
− B2τ ,
here
D k(n +1) = D k(n ) + ϕ R k(n ) , n (n = 0,1,2, K) – is an iteration’s number. Using
(n +1)
fsolve function, iterations are presented till Rk
(n)
3. D k
Sk =
1
τ
< Rk(n ) .
, obtained by iteration method, is introduced instead of
Dk into
Dk − B1 .
This method is better than the one, based on the matrix
Qk search using
iteration method, because:
1. The initial matrix, using fsolve function, is always selected using the same
Bτ
(0 )
expression Dk = Wk − B2τe 1 (when the branch of the Lambert function
(
)
changes, the initial matrix changes, as well, thus we avoid the choice of different
(0 )
initial matrix every time). While the initial matrix Qk not always can be
determined so that the condition S k − B1 − B2 e
− S kτ
and would be fitting changing Lambert function branch
< ε would be satisfied
k and delay τ .
19
Qk from the equation
2. Thus we avoid a complicated search of
Wk (− B2τQk )eWk (− B2τQk )− B1τ = B2τ using iteration method. Applying the
method proposed in the dissertation work, the matrix
Dk is found using a more
simple equation (besides there is no necessity to calculate values of the matrix
Lambert W function):
Dk eDk e− B1τ = −B2τ .
In control systems’ analysis, presented in the dissertation work and fulfilled
by the modified Lambert function method, obtained results are compared with
the exact results, got using consequent integration method, and results, obtained
by numerical methods.
In the case of non-homogenous differential equation with delayed argument
(2.13), the general solution on the interval
method, can be written as it is shown below:
x(t ) =
∞
∑
(
e
1
)C
Dk − B1 t
t
∞
(
e
1
(0, ∞ ) , got by Lambert function
)
Dk − B1 (t −ζ
)
+∫ ∑
C kN u (ζ )dζ .
k = −∞
k = −∞
1442443 01
4444
4244444
3
τ
homogenous
k
τ
(3.4)
non - homogenous
When investigating control systems by Lambert function’s method there
are cases when we need to find the value of Lambert function at zero argument
(it is known that the Lambert function values at zero argument are equal to
minus infinity for all branches, except the zero branch). In dissertation work
there was proposed the method of finding solution applicable in the case, when
matrix Lambert function’s argument has zero eigenvalue.
3.1.2. Method of finding solution in the case when Lambert function’s
argument has zero eigenvalue
If at least one eigenvalue of the matrix
H = − B2τe B1τ is zero (λi = 0 )
we meet the problem in computing values of Lambert function (this problem is
described in the chapter 2). The value of the k-th branch Wk λ of the Lambert
( )
W (λ ) at argument λ = 0 is equal to minus infinity for all k
(k = ±1,±2,K) except k = 0 (Corless, Gonnet, Hare, Jeffrey and Knuth,
function
1996):
− ∞, k ≠ 0,
Wk (0) = 
k = 0.
0,
20
(3.5)
We will assume that the value of the Lambert function
branch
W (λ ) for k-th
Wk (λ ) at zero argument λ = 0 is equal to zero, as well:
0,
Wk (0) = 
0,
k ≠ 0,
k = 0.
We make this assumption on the base of the
function, W k ( z ) e
Wk ( z )
= z , according which (k ≠ 0)
(3.6)
definition of Lambert
x
=
x →−∞
x → −∞ e − x
x'
1
1
= lim
= lim
= − lim − x = −0,
−
x
′ x →−∞ − e
x → −∞
x →−∞ e
e −x
Wk (0)eWk (0 ) = lim xe x = lim
(3.7)
( )
if we use the expression (3.5), and
Wk (0)eWk (0) = 0e 0 = 0,
(3.8)
if we use the expression (3.6). This assumption is justified by an example
presented in dissertation work.
3.1.3. Modified Lambert function’s method and method of finding of
solution in the case when argument (matrix) of the Lambert function has
zero eigenvalue
If we have non-commuting coefficient matrices of the equations (2.6) and
(2.13) and we use the modified Lambert function’s method then, in order to find
the auxiliary matrix D k , we use the fsolve function and proposed new method of
finding of solution in the case when argument (matrix) of the Lambert function
Bτ
(0 )
and
has zero eigenvalue. Then the initial matrix is Dk = Wk − B2τe 1
(
)
Wk (0) = 0 , ∀k . The details of application of this method is presented in the
dissertation work.
3.2. The step responses matrix of control system, described by linear nonhomogenous matrix differential equation with delayed argument, and its
application
The mathematical model of analyzed control system with delays is the
matrix differential equation with delayed argument given by (2.26) expression.
21
In the case when (2.26) describes multidimensional control system – the
synchronization system of communication network - then the symbol xi t
()
means the phase of the oscillation of i-th oscillator. In the dissertation work the
transients arising in the synchronization system, when the increment of the unit
(
)
step form is delivered to the phase of i-th oscillator i = 1, n , are investigated.
To this end, the step response matrix of the system is found.
The matrix h(t ) = hij (t ) is called the step response matrix of the
(
)
synchronization system. The element hij (t ) of this matrix is the response of
xi (t ) of a vector x(t ) to a unit jump in the component x j (t ) .
There is found the matrix h(t ) .
component
When the unit phase jump affects the phase of oscillation of the j-th
oscillator then the free term z (t ) of the equation (2.26) assumes the increment:
∆ z (t ) = δ (t ) I ( j ) ;
(3.9)
here I ( j ) is a column vector which has all its elements equal to zero, except j-th
element which is equal to 1, δ (t ) is Dirac delta function. Considering the
equation (2.26) we can write a differential equation for vector of step responses
function
h j (t )
( j = 1, n):
h' j (t ) + B1 h j (t ) + B2 h j (t − τ ) = δ (t ) I ( j ) ,
h j (t ) = 0, t ∈ [− τ ,0];
here
j = 1, n,
(3.10)
h j (t ) = (h1 j (t ), h2 j (t ),K, hnj (t )) is j-th column of the step response
T
matrix h(t ) , matrices B1 and B2 are numerical matrices that depend on a the
structure of the internal links of the system.
Firstly, we will write the solution of the matrix differential equation (3.10)
on the interval [ 0, τ ] . Due to the initial conditions (3.10), vector h j (t − τ ) is a
zero vector on the interval [ 0, τ ] . Considering this, the differential equation
(3.10) on the interval [ 0, τ ] is written as a simple differential equation (without
delayed argument):
22
δ (t ) I ( j ) , i = j
h' j (t ) + κh j (t ) = 
0, i ≠ j.
h j (0 ) = 0.
(3.11)
Solution of the equation (3.11) has following form:
[
]
h j (t ) = h1 j (t ) h2 j (t ) K hnj (t ) , t ∈ [0,τ ];
(3.12)
0, i ≠ j
hij (t ) =  −κt
e 1(t ), i = j ,
(3.13)
T
here
1(t ) – unit Heaviside function.
Using the solution (3.12) the differential equation (3.10) on the interval
be written as the linear homogenous matrix differential equation
with delayed argument
(τ , +∞ ) can
h' j (t ) + B1 h j (t ) + B2 h j (t − τ ) = 0,
j = 1, n, t > τ ,
h j (t ) = ϕ j (t ), t ∈ [0,τ ];
here
(3.14)
φ j (t ) = h j (t ) is an initial vector function defined by the expression (3.12)
When the Lambert function method, described in the chapter (2), is used the
approximate solution of the equation (2.26) on the interval (τ , +∞ ) is written as
it is shown below:
h j (t ) =
N
∑e
Sk t
Ck ,
j = 1, n, t ∈ (τ ,+∞);
(3.15)
k =− N
here
N is a sufficiently large number, the matrix S k is defined by equation
(3.1), when coefficient matrices of the differential equation with delayed
argument are non-commuting, and by equation (2.8), when they are commuting.
In the third chapter there also is presented analysis of the control system
made applying the modified Lambert W function method and the proposed
method of finding of solution in the case when the argument (matrix) of the
Lambert function has zero eigenvalues. For every example there are given
23
expressions of the step responses got using consequent integration method and
Laplace transform.
The results are compared, as well, with results obtained by numerical
method which uses dde23 function described in the Matlab's package. There are
given pseudo codes of all algorithms, used for calculations within Matlab
environment. The codes can be applied to generate programs for computing the
step responses of the system by three methods: Lambert function, dde23
numerical and consequent integration.
4.
INVESTIGATION OF SYNCHRONIZATION SYSTEM OF THE
COMMUNICATION NETWORK
In the fourth chapter there are analyzed synchronization systems with
different structures of the internal links and described by the linear matrix nonhomogenous differential equations with delayed argument and with commuting
coefficient matrices.
4.1. The step responses matrix of the synchronization system in the case
of different structures of the internal links
For the analysis of step responses of such systems the Lambert function’s
method, presented in the chapter (2), is applied. Analyzed systems have
following structures of the internal links: ring, star, full graph, chain. For the
analysis of these systems the described method not always is sufficient. When
the argument (matrix) of the Lambert function has zero eigenvalues, we shall
apply the new method of finding of the solution, proposed in the dissertation
(discussed in the third chapter). Looking for a general solution’s coefficients Ck
(
)
, we need to find a inverse matrix of the matrix X τ , N . When this matrix is
singular
a pseudo inverse matrix finding procedure is applied:
X (τ , N ) = VD +V −1 .
+
Derived results are compared with the ones derived using consequent
integration and numerical (using dde23 function described in Matlab package)
methods. For comparing the results a relative error at the fixed point t is
calculated using the following formula:
σ=
xLamberto (t ) − xtikslus (t )
.
xtikslus (t )
4.2. Investigations of transients of phase differences of oscillations of the
oscillators
The phase differences of oscillations of the oscillators are investigated by two
methods: Lambert function method and the consequent integration method (exact
24
method). The mathematical model of the analyzed system is the linear nonhomogenous matrix differential equation with delayed argument and commuting
coefficient matrices. The obtained results are compared, as well, with the results,
obtained by the numerical method, which uses the dde23 function, described in
the Matlab package.
The mathematical model of the multidimensional control system is matrix
differential equation with delayed argument, given by (2.26). The symbol xi t
()
in this equation denote the phase of i-th oscillators.
The phase xi t can be expressed by the following formula:
()
t
 f i (ξ )dξ + x0i , t > 0,
xi (t ) = ∫0
 f t + x , t ≤ 0;
0i
 0i
(4.1)
x0i = xi (0) ( i = 1, n ) is initial phase of the i-th oscillator, f 0i is own
frequency of i-th oscillator when control signal is disconnected (shut off), f i (t )
here
is a frequency of the i-th oscillator. The meaning of the formula (4.1) is
following: control signal for the i-th oscillator is connected at the moment t = 0 .
Till then the oscillator worked at own frequency
f 0i . Taking into account (4.1),
expressions of the initial function and of the free term in the differential
equation (2.26) can be written as it is shown below:
φ (t ) = (φ1 (t ),φ2 (t ),K,φn (t ))T
(4.2)
φi (t ) = f 0i t + x0i , i = 1, n, t ∈ [− τ ,0],
(4.3)
z (t ) = ( f 01 , f 02 ,K, f 0 n ) .
(4.4)
T
4.3. Derivation of expressions of generalized (suited for any finite number
of oscillators n ∈ N ) phase differences for different structures of internal
links when consequent integration method is applied
In the fourth chapter there are derived expressions of phase differences
when synchronization systems have structure of the internal links s of the
following forms: ring, full graph, chain. An exact solution of the differential
equation (2.26), found applying consequent integration method and Laplace
25
transformation, when synchronization system has ring form structure of the
internal links is written as it is shown below:
xi (κt ) − x j (κt ) = σ ij (κt ) + Sij (κt ), 0 < κt < (L + 1)κτ ; i, j = 1, n,
(4.5)
čia
σ ij (κt ) = σ ij ,1 (κt ) + σ ij , 2 (κt ) + σ ij ,3 (κt ) − σ ij ,3 (κt − κτ ) − σ ij , 4 (κt − κτ ),
σ ij ,1 (κt ) = α ij ,1e−κt 1(κt ),
σ ij ,2 (κt ) = βij (t )(1 − e−κt )1(κt ),
σ ij ,3 (κt ) = α ij , 4 (κt − 1 + e −κt )1(κt ),
σ ij , 4 (κt ) = α ij ,3 (1 − e −κt )1(κt ),
α ij ,1 = ∑ ϕ0 m (σ im − σ jm ),
n
m =1
n
α ij , 2 = ∑
m =1
f0m
κ
(σ
im
− σ jm ),
α ij ,3 = ∑ϕm −1, m +1 (σ im − σ jm ),
n
m =1
n
α ij , 4 = ∑
m =1
f m −1, m +1
κ
(σ
im
− σ jm ),
β ij (t ) = α ij , 2 + α ij ,3 − κtα ij , 4
Sij (κt ) = Sij ,1 (κt ) + Sij , 2 (κt ) + Sij ,3 (κt ) − Sij ,3 (κt − κτ ) − Sij , 4 (κt − κτ ),
1
(κt − kκτ ) e− (κt − kκτ )1(κt − kκτ ),
Sij ,1 (κt ) = ∑ k aij ,1 (k )
k!
k =1 2
k
L
k

(
1
κt − kκτ )r −(κt −kκτ ) 

Sij , 2 (κt ) = ∑ k bij (t , k )1 − ∑
e
1(κt − kκτ ),
r!
k =1 2
 r =0

L
26
k
r

(κt − kκτ )s e−(κt −kκτ ) 1(κt − kκτ ),
1

a
(
k
)
(
κ
t
−
k
κτ
)
−
(
k
+
1
)
+
∑∑
k ij , 4


s!
k =1 2
r =0 s = 0


L
Sij ,3 (κt ) = ∑
k

(κt − kκτ )r e −(κt − kκτ ) 1(κt − kκτ ),
1

(
)
a
k
1
−
ij , 3
k
 ∑

r!
k =1 2
 r =0

L
S ij , 4 (κt ) = ∑
aij ,1 (k ) = ∑ ϕ 0 m (d im (k ) − d jm (k )),
n
m =1
n
aij , 2 (k ) = ∑
m =1
f 0m
κ
(d (k ) − d (k )),
im
jm
aij ,3 (k ) = ∑ ϕ m−1,m +1 (d im (k ) − d jm (k )),
n
m =1
n
aij , 4 (k ) = ∑
m =1
f m−1,m+1
κ
(d (k ) − d (k )),
im
jm
bij (t , k ) = aij , 2 + aij ,3 − κtaij , 4 ,
[ ]
dij (k ) = B k ij , i, j = 1, n.
There were used following notations:
ϕ i −1,i − n
 f 0i −1 + f 0i +1
ϕ 0i −1 + ϕ 0i +1
, i ≠ 1,
, i ≠ 1,


2
2


i ≠ n,
i ≠ n,




; f i −1,i − n =  f 0 n + f 02
;
= ϕ 0 n + ϕ 02
,
1
,
i
,
i
=
1
,
=


2
2


 f 0 n −1 + f 01
ϕ 0 n −1 + ϕ 01
, i = n.
, i = n.


2
2
;
here
ϕ 0i
(
is an initial phase of the i-th oscillator i = 1, n
frequency of i-th oscillator
B ∈ M n (R ) is given by (2.31).
(i = 1, n)l.
),
f0i is own
The expression of the matrix
27
When the synchronization system has complete graph structure of the
internal links then in the expressions (4.5) of phase differences the following
changing must be made:
1. In the formulas S ij ,1 (κt ) , S ij , 2 (κt ) , S ij ,3 (κt ) and S ij , 4 (κt )
1
1
to
.
k
2
(n − 1)k
2. The expressions ϕ i −1,i − n and f i −1, i − n are defined as follows:
change
ϕ i −1,i −n =
1 n
∑ ϕ 0l ,
n − 1 l =1
l ≠i
3.
The matrix
f i −1,i − n =
1 n
∑ f 0l .
n − 1 l =1
l ≠i
B ∈ M n (R ) must be defined by the expression
(2.27).
When synchronization system has chain form structure of the internal links,
the following changing in the phase difference expression (4.5) must be made:
1. Marks ϕ i −1,i − n and f i −1, i − n are defined thus:
ϕ0i −1 + ϕ0i +1
, i ≠ 1, i ≠ n,

2

;
ϕi −1, i − n = ϕ02 , i = 1,
ϕ , i = n.
 0n

 f 0i −1 + f 0i +1
, i ≠ 1, i ≠ n,

2

.
f i −1,i − n =  f 02 , i = 1,
 f , i = n.
 0n

2.
28
The matrix
B ∈ M n (R ) is presented by the expression (2.30).
4.4. Calculation of phase differences applying Lambert W function
method and the method of finding the solution in the case when argument
(matrix) of the Lambert function has zero eigenvalues
The solution of non-homogenous differential
argument on the interval
expressed as follows:
∞
∞
x(t ) = lim ∑ e k C k + ∫  lim ∑ e
N →+∞ k = −∞
0  N →+∞ k = −∞
∞
t
S t
t
Sk t
∞
= ∑ e Ck + ∫ ∑ e
k = −∞
here
equation with delayed
(0;+∞) , obtained by Lambert function method, is
S k ( t −ζ )
0 k =−∞
Sk ( t −ζ )
C k′ bu(ζ ) dζ =

(4.7)
Ck′ bu(ζ )dζ ;
Ck is n × 1 vector of the coefficients calculated using the initial function
x (t ) = φ (t ) given on the interval [− τ ;0] . C kN is n × n coefficient matrix
calculated using the coefficient matrix B1 . In the calculation an approximate
formula, derived from (4.7) when
N is fixed, is used:
t
x(t ) = ∑ e k C k + ∫ ∑ e
N
S t
k =N
N
0 k=N
S k ( t −ζ )
C k′ bu (ζ )dζ .
(4.8)
The phase difference xi (t ) − x j (t ) is found on the base of the general
solution
x(t ) = ( x1 (t ) K xn (t )) (see (4.8)).
T
Results, obtained in the dissertation work using the Lambert function
method, are compared with the results, obtained applying numerical method,
which uses dde23 function described in Matlab's package, and applying the
consequent integration method. In order to compare results there is used a
relative error, found as follows: within the chosen interval of κτ variation the
such κt value is found at which the distance between graphs, got by Lambert
function method and the exact method, is biggest. This value is denoted
(κt )max .
Then the relative error of Lambert function method is calculated as it is shown
below
σ max =
x Lambert (κt )max − xtikslus (κt )max
xtikslus (κt )max
.
29
Pseudo codes presented in dissertation work (for Matlab) can be used to
create calculation programs for phase differences when any of these three
methods is applied: Lambert function, the dde23 numerical and consequent
integration.
5.
INVESTIGATION OF CONTROL SYSTEM STABILITY APPLYING
LAMBERT W FUNCTION
In the fifth chapter there is analyzed the method of finding roots of
characteristic transcendental equation, corresponding to the linear scalar and
matrix differential equation with delayed argument. This method is based on
application of Lambert W function. When coefficient matrices of matrix
differential equation with delayed argument are non-commuting a modified
Lambert function method is applied and obtained results are compared with the
ones obtained applying numerical (when Chebyshev polynomial is used) and the
exact method ( sk1, 2
2  1

= Wk  ± τ α  used in the article (Jarlebring and
τ  2

Damm, 2007)).
5.1. System described by linear scalar differential equation with delayed
argument
There is analyzed a characteristic transcendental equation of the linear
differential equation (2.4)
s + b + ae − sτ = 0 ,
(5.1)
using Lambert function method. Since Lambert W function has infinite number
of branches, linear scalar characteristic transcendental equation ( 5.1) will have
infinite number of roots. These roots can be expressed as follows:
1
s k = Wk ( − aτeτβ ) − β , k = 0,±1,±2, K.
τ
(5.2)
The solution of the differential equation with delayed argument (2.4) will
be asymptotically stable if all complex numbers sk , k = 0,±1,±2,K will have
negative real parts.
Using expression (5.2) in dissertation work the dependence of location of
points, representing eigenvalues on the complex plane, on system parameters is
analyzed. The transients in the system are presented, too.
30
5.2. System described by linear matrix differential equation with delayed
argument when coefficient matrices are commuting
There is analyzed characteristic transcendental equation, corresponding to linear
matrix differential equation, applying Lambert function method. The matrix
transcendental characteristic equation will have infinite number of roots that are
written as it is shown below:
(
)
1
S k = Wk − B2τe B1τ − B1 ;
τ
(5.3)
Solution of differential equation with delayed argument (2.6) will be
asymptotically stable if all real parts of eigenvalues of all complex matrices
Sk , k = 0,±1,±2,K will be negative numbers.
Using expression (5.3), in the work there is analyzed dependence of
location of points, representing eigenvalues of the matrices S k , on the complex
plane on system parameters. The transients in the system are presented, too.
5.3. System described by linear matrix differential equation with delayed
argument when coefficient matrices are non-commuting
Let we have control system described by linear matrix differential equation
(2.6) and let its coefficient matrices are non-commuting. In the work a
transcendental characteristic equation, corresponding to linear matrix differential
equation, is solved using modified Lambert function method. The matrix
transcendental characteristic equation will have infinite number of roots:
Sk =
1
τ
Dk − B1 ,
(k = 0,±1,±2,K) .
(5.4)
Solution of differential equation with delayed argument will be asymptotically
stable if all eigenvalues of complex matrices S k , k = 0,±1,±2,K will have
negative real parts.
31
CONCLUSIONS
1. There is proposed a new modification of the method of solution of linear
matrix differential equation with delayed argument in the case, when the
coefficient matrices of the equation are non-commuting. This modification is
based on the use of Lambert W function and on application of special auxiliary
matrix which is found using fsolve function, given in Matlab. The new modified
method is better than the known one because the initial matrix in the iteration
process of fsolve function is described by the same expression, fits for all
branches of Lambert W function and is derived from a simpler expression.
2. There is proposed a new method of finding of the solution of linear matrix
differential equation with delayed argument, applicable in the case when
argument of matrix Lambert W function (matrix) has zero eigenvalues.
(
)
According to this method, the value of the k-th branch k ≠ 0 of the Lambert
function at zero argument is equated to zero (in fact this value is equal to − ∞).
This replacement, as it is shown by example, does not changes all set of
eigenvalues (it influences only their numbering).
3. For computing coefficient matrices, involved in the expression of general
solution of the linear matrix differential equation with delayed argument there is
used a pseudo inverse matrix or procedure of finding pseudo inverse, in the case
when coefficient matrix of corresponding linear matrix differential equation is
singular.
4. Dependence of accuracy of the solution, obtained by the Lambert W
function‘s method, on number of branches, used in calculation of solution, is
investigated. For this purpose the exact solutions, got for some systems using
consequent integration method, are applied. The proposed method has advantage
comparing with consequent integration method when coefficient matrices are of
more than third order or their have not zeros elements, except the case, when
B1 is diagonal matrix (synchronization systems).
32
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Asl, F. M. and Ulsoy, A. G. "Analytical solution of a system of
homogeneous delay differential equations via the Lambert function," in
American Control Conference, 2000. Proceedings of the 2000, 2000,
pp. 2496-2500 vol.4.
Asl, F. M. and Ulsoy, A. G., "Analysis of a system of linear delay
differential equations," Journal of Dynamic Systems, Measurement, and
Control, vol. 125, pp. 215-223, 2003.
Barata, J. and Hussein, M. "The Moore–Penrose Pseudoinverse: A
Tutorial Review of the Theory," Brazilian Journal of Physics, vol. 42,
pp. 146-165, 2012/04/01 2012.
Butcher, A. E., Ma, H., Bueler, E., Averina, V. and Szabo, Z.
"Stability of linear time‐periodic delay‐differential equations via
Chebyshev polynomials." International Journal for Numerical Methods
in Engineering 59.7 (2004): 895-922.
Chen, H. C., Horng, T. and Yang, Y. "Reflexive decompositions for
solving Poisson equation by Chebyshev pseudospectral method,"
Proceedings of Neural, Parallel, and Scientific Computations, vol. 4,
pp. 98-103, 2010.
Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J. and Knuth, D.
"On the LambertW function," Advances in Computational Mathematics,
vol. 5, pp. 329-359, 1996/12/01 1996.
Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J. and Knuth, D.
"Lambert’s W function in Maple," Maple Technical Newsletter, vol. 9,
pp. 12-22, 1993.
Geletu, A. "Solving Optimization Problems using the Matlab
Optimization Toolbox-a Tutorial," TU-Ilmenau, Fakultät für
Mathematik und Naturwissenschaften, 2007.
Horn, R. A. Topics in matrix analysis: Cambridge University Press,
1986.
Yi, S., Ulsoy, A. G. and Nelson, P. W. "Delay differential equations via
the matrix Lambert W function and bifurcation analysis: application to
machine tool chatter," Mathematical biosciences and engineering :
MBE, vol. 4, pp. 355-368, 04/ 2007.
Yi, S., Ulsoy, A. G. and Nelson, P. W. Time-Delay Systems: Analysis
and Control Using the Lambert W Function: World Scientific, 2010.
Yi, S. and Ulsoy, A. G. "Solution of a system of linear delay differential
equations using the matrix Lambert function," in American Control
Conference, 2006, 2006, p. 6 pp.
Yi, S., Ulsoy, A. G. and Nelson, P. W. "Solution of systems of linear
delay differential equations via Laplace transformation," in Decision
and Control, 2006 45th IEEE Conference on, 2006, pp. 2535-2540.
33
14. Jarlebring, E. and Damm, T. "The Lambert W function and the
spectrum of some multidimensional time-delay systems," Automatica,
vol. 43, pp. 2124-2128, 12// 2007.
15. Leonaite, G. and Rimas, J. Investigation of a Multidimensional
Automatic Control System with Delays and Chain Form
Structure. Information
Technology
and
Control,
Kaunas,
Technologija, 2006, Vol. 35, No. 1, 65 – 70.
16. Rimas, J. Analysis of multidimensional delay system: Heverlee :
Katholieke Universiteit Leuven, 2004.
17. Shampine, L. F. and Thompson, S. "Solving DDEs in Matlab," Applied
Numerical Mathematics, vol. 37, pp. 441-458, 6// 2001.
18. Tsitouras, C. "Runge–Kutta interpolants for high precision
computations," Numerical Algorithms, vol. 44, pp. 291-307, 2007.
INFORMATION ABOUT THE AUTHOR OF THE DISSERTATION
Personal Information:
Date of Birth: 2th July, 1984.
Place of Birth: Kaunas.
Education:
2003 – 2007 Bachelor in Sciences of Applied mathematics, Faculty of
Fundamental science, Kaunas University of Technology.
2007 – 2009 Master in Sciences of Applied mathematics, Faculty of
Fundamental science, Kaunas University of Technology.
2009 – 2014 Doctoral studies in Physical Sciences (Informatic 09P),
Department of Applied Mathematics, Faculty of Mathematics and Natural
Sciences, Kaunas University of Technology.
Teaching experience:
Since 2014 Assistant, Department of Applied Mathematics, Faculty of
Mathematics and Natural Sciences, Kaunas University of Technology.
Research interests:
Research of multidimensional time delay systems and their dynamics;
Analysis of the mathematical models of the linear time-delay systems using
Lambert function and numerical methods.
LIST OF PUBLICATIONS
Papers in Master List Journals of the Institute of Scientific
Information (ISI)
1.
34
Ivanovienė, Irma; Rimas, Jonas. The use of the Lambert function
method for analysis of a control system with delays // Informacinės
2.
technologijos ir valdymas = Information technology and control /
Kauno technologijos universitetas. Kaunas : KTU. ISSN 1392-124X.
2013, T. 42, nr. 4, p. 325-332. [Science Citation Index Expanded (Web
of Science); INSPEC]. [IF: 0,667, AIF: 1,688 (E, 2012)].
Ivanovienė, Irma; Rimas, Jonas. Complement to method of analysis of
time delay systems via Lambert W function// Automatica, ISSN 00051098,
Vol
54,
2015,
p.
25-28,
DOI:
http://dx.doi.org/10.1016/j.automatica.2015.01.039.
Papers in Proceedings List of the Institute of Scientific Information (ISI)
1.
2.
3.
Ivanovienė, Irma; Rimas, Jonas. Analysis of control system with delay
using the Lambert function // Information and Software Technologies :
19th international conference, ICIST 2013, Kaunas, Lithuania, October
10-11, 2013 : proceedings / Edited by Tomas Skersys, Rimantas
Butleris, Rita Butkiene. Berlin, Heidelberg : Springer, 2013.
(Communications in computer and information science, Vol. 403, ISSN
1865-0929). ISBN 9783642419461. p. [1-10]. [Conference Proceedings
Citation Index]. [M.kr. 09P].
Ivanovienė, Irma; Rimas, Jonas. Analysis of a multidimensional control
system with delays // Electrical and Control Technologies : proceedings
of the 6th international conference on Electrical and Control
Technologies ECT 2011 / Kaunas University of Technology, IFAC
Committee of National Lithuanian Organisation, Lithuanian Electricity
Association. Kaunas : Technologija. ISSN 1822-5934. 2011, p. 124129. [Conference Proceedings Citation Index]. [M.kr. 01P].
Ivanovienė, Irma; Rimas, Jonas. Investigation of the multidimensional
automatic control system, having strucrure of the chain, applying
Lambert W function // Electrical and Control Technologies :
proceedings of the 7th International Conference on Electrical and
Control Technologies ECT 2012 / Kaunas University of Technology,
IFAC Committee of National Lithuanian Organisation, Lithuanian
Electricity Association. Kaunas : Technologija. ISSN 1822-5934. 2012,
p. 82-86. [M.kr. 01T].
Papers in Journals Referred in the Databases, Included in the List
Approved by the Science Council of Lithuania
1.
Ivanovienė, Irma. The use of Lambert function method for analysis of a
multidimensional control system with delays and with structure of the
complete graph // Lietuvos matematikos rinkinys : Lietuvos matematikų
35
2.
3.
4.
draugijos darbai. Serija B / Lietuvos matematikų draugija, Vilniaus
universitetas. Vilnius : Vilniaus universitetas. ISSN 0132-2818. 2012, t.
53, p. 30-35. [M.kr. 01P].
Ivanovienė, Irma; Rimas, Jonas. Finding of roots of the transcendental
characteristic equation by a method of function Lambert W //
Matematika ir matematinis modeliavimas = Mathematics and
mathematical modelling / Kauno technologijos universitetas. Kaunas :
Technologija. ISSN 1822-2757. 2012, T. 7-8, p. 7-14. [M.kr. 01P].
Ivanovienė, Irma; Rimas, Jonas. The use of the Lambert Function
method for analysis of a multidimensional control system with delays
and with structure having form of the star // International Journal of
Information and Education Technology (IJIET). Singapore : IACSIT.
ISSN 2010-3689. 2012, Vol. 2, no. 4, p. 360-363. [M.kr. 09P].
Ivanovienė, Irma; Rimas, Jonas. Finding of roots of the matrix
transcendental characteristic equation using Lambert W function //
Lietuvos matematikos rinkinys : Lietuvos matematikų draugijos darbai /
Lietuvos matematikų draugija, Vilniaus universitetas. Vilnius : Vilniaus
universitetas. ISSN 0132-2818. 2011, t. 52, p. 189-194. [M.kr. 09P].
VALDYMO SISTEMŲ DINAMIKOS TYRIMAS, TAIKANT
LAMBERTO FUNKCIJĄ
Reziumė
Tyrimo objektas. Daugiamatės tiesinės valdymo sistemos su vėlavimais.
Temos aktualumas. Automatinės valdymo sistemos plačiai yra
naudojamos šiuolaikinės pramonės technologijose, kosminės technikos
įrenginiuose, aviacijoje, karinėje pramonėje, transporte, energetikoje,
elektronikoje ir buitinėje technikoje. Valdant technologinius procesus dažnai
susiduriama su signalų vėlavimu, kuris įtakoja sistemos dinamines savybes.
Daugiamatės valdymo sistemos su vėlavimais yra nepilnai ištirtos valdymo
teorijoje.
Daugiamatės valdymo sistemos su vėlavimais matematinis modelis yra
matricinė difrencialinė lygtis su vėluojančiu argumentu. Tokios lygtys jau keli
dešimtmečiai yra intensyviai studijuojamos. Tai rodo monografijų ir publikacijų
mokslo žurnaluose gausa. Diferencialinės lygtys su vėluojančiu argumentu
dažnai sprendžiamos naudojant skaitinius metodus, taikant įvairias
aproksimacijas, bei analiziškai. Daugelis skaitinių metodų yra realizuota
inžineriniuose paketuose, tokiuose kaip Matlab, Maple, Mathematica. Tačiau
nevisada pakanka realiuose tyrimuose turėti apytikslius rezultatus, tad ieškoma
tikslių tokių sistemų tyrimo metodų. Pagrindinis sunkumas sprendžiant tiesines
diferencialines lygtis su vėluojančiu argumentu yra tai, kad jas atitinkanti
transendentinė charakteristinė lygtis turi be galo daug šaknų.
36
Tyrimo tikslas ir uždaviniai. Pagrindinis disertacinio darbo tikslas –
daugiamatėms valdymo sistemoms (aprašomoms tiesinėmis matricinėmis
diferencialinėmis lygtimis su vėluojančiu argumentu) pasiūlyti tyrimo metodiką
ir pritaikyti šią metodiką konkrečių valdymo sistemų pereinamųjų funkcijų
matricų skaičiavimui, pereinamųjų procesų tyrimui bei stabilumo analizei.
Darbo tikslą įgyvendinti formuluojami šie uždaviniai:
1.
Išanalizuoti valdymo sistemų su vėlavimais tyrimo metodą, paremtą
Lamberto W funkcijos taikymu. Išnagrinėti šio metodo panaudojimo galimybes,
sprendžiant įvairias tiesines diferencialines lygtis su vėluojančiu argumentu:
skaliarines, matricines, kai lygties koeficientų matricos yra komutuojančios, ir
matricines, kai lygties koeficientų matricos yra nekomutuojančios.
2.
Pasiūlyti naują tiesinių diferencialinių lygčių su vėluojančiu argumentu ir
nekomutuojančiomis koeficientų matricomis sprendimo metodą, paremtą
Lamberto W funkcijos taikymu bei specialios pagalbinės matricos panaudojimu.
3.
Pasiūlyti tiesinių matricinių diferencialinių lygčių su vėluojančiu
argumentu sprendinio radimo metodą tuo atveju, kai matricinės Lamberto W
funkcijos argumentas (matrica) turi nulinių tikrinių reikšmių.
4.
Atlikti konkrečių daugiamačių valdymo sistemų (tame tarpe ir ryšio tinklo
sinchronizacijos sistemų), aprašomų tiesine matricine diferencialine lygtimi su
vėluojančiu argumentu, analizinį tyrimą, taikant pasiūlytą metodiką (rasti
pereinamųjų funkcijų matricas, ištirti pereinamuosius procesus sistemoje bei
atlikti stabilumo analizę).
Tyrimo metodai ir priemonės. Valdymo sistemą su vėlavimais aprašanti
tiesinė matricinė diferencialinė lygtis sprendžiama metodu, paremtu Lamberto W
funkcijos taikymu, specialios pagalbinės matricos panaudojimu bei Lamberto W
funkcijos reikšmės, kai argumentas yra lygus nuliui, koregavimu. Matricinė
Lamberto W funkcija apibrėžiama remiantis matricinėje analizėje naudojamu
standartiniu matricinės funkcijos apibrėžimu.
Pateikiami ginti darbo rezultatai.
1.
Naujas tiesinių matricinių diferencialinių lygčių su vėluojančiu argumentu
ir nekomutuojančiomis koeficientų matricomis sprendimo metodas, paremtas
Lamberto W funkcijos taikymu bei specialios pagalbinės matricos panaudojimu.
Ši matrica randama naudojant iteracinį procesą (taikant fsolve funkciją Matlab
programoje). Pasiūlyto metodo privalumas tas, kad pagalbinę matricą
apibrėžianti lygtis paprastesnė (lyginant su žinomu metodu), o pradinė matrica
iteraciniame procese nusakoma ta pačia išraiška (nepriklausomai nuo matricinės
diferencialinės lygties su vėluojančiu argumentu koeficientų matricų struktūros).
2.
Naujas tiesinių matricinių diferencialinių lygčių su vėluojančiu argumentu
sprendinio radimo metodas, taikomas tuo atveju, kai matricinės Lamberto W
funkcijos argumentas (matrica) turi nulinių tikrinių reikšmių.
3.
Daugiamačių valdymo sistemų su vėlavimais, aprašomų tiesine matricine
diferencialine lygtimi su vėluojančiu argumentu, analizinio tyrimo (naudojant
37
Lamberto W funkciją) metodika bei konkrečių daugiamačių valdymo sistemų
(tame tarpe ir ryšio tinklo sinchronizacijos sistemų su skirtingomis vidinių ryšių
struktūromis) analizinio tyrimo rezultatai.
Mokslinis naujumas ir praktinė svarba. Valdymo sistemų su vėlavimais
tyrimas yra aktualus. Tai rodo publikacijų šia tematika gausa. Valdymo
sistemoms tirti plačiai naudojami įvairūs skaitiniai metodai. Kai netenkina
apytiksliai rezultatai, gauti skaitiniais metodais, siekiama naudoti metodus, kurie
pateikia tikslius arba kiek norima artimus tiksliems rezultatus. Tokie yra
nuoseklaus integravimo metodas, kuris bet kuriame baigtiniame intervale tikslų
sprendinį pateikia baigtine funkcijų suma, ir metodas, pagrįstas Lamberto W
funkcijos taikymu, kuris bet kuriame intervale tikslų sprendinį pateikia begalinės
funkcijų eilutės pavidalu.
Disertaciniame darbe valdymo sistemos su vėlavimais, aprašomos tiesine
matricine
diferencialine
lygtimi
su
vėluojančiu
argumentu
ir
nekomutuojančiomis koeficientų matricomis, tiriamos naudojant darbe pasiūlytą
naują metodiką. Šios metodikos pagrindiniai akcentai yra šie:
1.
Orginali pagalbinės matricos radimo procedūra, įgalinanti tirti sistemas su
nekomutuojančiomis koeficientų matricomis, turinčiomis bet kokias struktūras.
2.
Naujas sprendinio radimo metodas, taikytinas tada, kai matricinės
Lamberto W funkcijos argumentas (matrica) turi nulinių tikrinių reikšmių.
Taikant darbe pasiūlytą metodiką, buvo išnagrinėtos įvairios daugiamatės
valdymo sistemos su vėlavimais. Daugelio iš jų tyrimas, taikant žinomą
metodiką, nebuvo galimas.
Pasiūlyta metodika gali būti panaudota analizuojant matematinius
modelius, aprašomus tiesinėmis matricinėmis diferencialinėmis lygtimis su
vėluojančiu argumentu, su kuriais susiduriama fizikoje, biochemijoje,
ekonomikoje ir kitur.
Darbo rezultatų aprobavimas, publikacijos
Disertacijos tema paskelbti 9 moksliniai straipsniai, iš jų 2 straipsniai
leidinyje, įrašytame į Mokslinės informacijos instituto (ISI) pagrindinių žurnalų
sąrašą, 3 straipsniai leidiniuose, įrašytuose į Mokslinės informacijos instituto ISI
Proceedings leidinių sąrašą, 2 straipsniai Lietuvos mokslo tarybos patvirtinto
sąrašo tarptautinėse duomenų bazėse referuojamose leidiniuose, 2 straipsniai
recenzuojamoje konferencijų pranešimų medžiagoje. Pagrindiniai darbo
rezultatai pristatyti 3 respublikinėse ir 6 tarptautinėse konferencijose.
Darbo struktūra
Daktaro disertaciją sudaro įvadas, 5 pagrindiniai skyriai, išvados,
literatūros sąrašas, publikacijų sąrašas. Disertacijos apimtis 148 puslapiai, yra 53
paveikslai ir 94 šaltinių cituojamos literatūros aprašas.
Pirmajame skyriuje pateikiamas tyrimo objektas, pagrindinis darbo tikslas
ir darbo tikslo realizavimo uždaviniai bei metodika, taikoma tyrimuose.
38
Antrajame skyriuje supažindinama su Lamberto funkcija, jos naudojimu
įvairių valdymo sistemų tyrimuose; apžvelgiama sistemas aprašančių
matematinių modelių – diferencialinių lygčių su vėluojančiu argumentu –
sprendimas Lamberto funkcijos metodu, kai turime tiesines homogenines ir
nehomogenines, skaliarines ir matricines diferencialines lygtis su vėluojančiu
argumentu. Pateikiami skaitiniai tokių lygčių sprendimo metodai, pseudo
atvirkštinės matricos skaičiavimo metodika (ji bus naudojama tolimesniuose
skyriuose). Apžvelgiama valdymo sistemų tyrimo, taikant Lamberto funkcijos
metodą, pavyzdžiai.
Trečiame skyriuje pateiktas modifikuotas Lamberto funkcijos metodas
įvairioms daugiamatėms valdymo sistemoms tirti, kai jas aprašantys
matematiniai modeliai – matricinės diferencialinės lygtys su vėluojančiu
argumentu – turi nekomutuojančias koeficientų matricas. Pateikiami metodo
taikymą iliustruojantys pavyzdžiai, kai sistemas aprašančių diferencialinių lygčių
su vėluojančiu argumentu tyrime būtina koreguoti Lamberto funkcijos reikšmę
prie nulinio argumento (naudojamas naujas sprendinio radimo metodas, kai
Lamberto W funkcijos argumentas (matrica) turi nulinių tikrinių reikšmių).
Pateikiama valdymo sistemų su vėlavimais pereinamųjų funkcijų skaičiavimo
metodika, paremta nuoseklaus integravimo metodo taikymu.
Ketvirtajame skyriuje ištirtos daugiamatės valdymo sistemos (ryšio tinklo
sinchronizacijos sistemos), aprašomos
tiesinėmis nehomogeninėmis
matricinėmis diferencialinėmis lygtimis su vėluojančiu argumentu ir
komutuojančiomis koeficientų matricomis. Pereinamųjų funkcijų radimui
taikomas Lamberto funkcijos metodas: pirmajame etape tiesinė nehomogeninė
diferencialinė lygtis su vėluojančiu argumentu pertvarkoma į pereinamųjų
funkcijų vektoriaus atžvilgiu tiesinę homogeninę diferencialinę lygtį ir
tolimesniam tyrimui taikoma Lamberto funkcija. Tiriamos sinchronizacijos
sistemos su įvairiomis vidinių ryšių struktūromis. Sinchronizacijos sistemų,
turinčių tam tikrą vidinių ryšių struktūrą, analizei naudojamas naujas sprendinio
radimo metodas, kai matricinės Lamberto funkcijos argumentas (matrica) turi
nulinių tikrinių reikšmių. Bendrojo sprendinio koeficientų paskaičiavimui
taikoma pseudo atvirkštinė matrica. Rezultatai lyginami su rezultatais, gautais
nuoseklaus integravimo („žingsnių“) metodu (tiksliu metodu) ir skaitiniu
metodu, paremtu dde23 funkcijos, aprašytos Matlab pakete, taikymu. Taip pat
analizuoti sinchronizacijos sistemų su įvairiomis vidinių ryšių struktūromis
generatorių fazių skirtumai, naudojant tris metodus: Lamberto funkcijos, skaitinį
– dde23 funkcijos ir nuoseklaus integravimo. Įvertintos paklaidos, gaunamos
naudojant Lamberto funkcijos metodą ir skaitinį metodą, paremtą dde23
funkcijos taikymu. Tiriamos sistemos, aprašomos nehomogeninėmis
diferencialinėmis lygtimis su vėluojančiu argumentu ir komutuojančiomis
koeficientų matricomis.
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Penktajame skyriuje pateiktas stabilumo tyrimas, atliktas naudojant
modifikuotą Lamberto funkcijos metodą, kai sistemos koeficientų matricos yra
nekomutuojančios, ir paprastą Lamberto funkcijos metodą, kai sistemos
koeficientų matricos yra komutuojančios. Stabilumo tyrimo rezultatai lyginami
su rezultatais, gautais skaitiniu metodu.
Toliau pateikiamos bendrosios išvados, literatūros sąrašas, disertacijos
tema skelbtų publikacijų sąrašas.
UDK 517.935 + 519.711 (043.3)
SL344. 2015-05-05, 2,5 leidyb. apsk. l. Tiražas 70 egz. Užsakymas 150157.
Išleido leidykla „Technologija“, Studentų g. 54, 51424 Kaunas
Spausdino leidyklos „Technologija“ spaustuvė, Studentų g. 54, 51424
Kaunas
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