Some recent developments in theory of fractional positive and cone linear systems
The overview of some recent developments and new results in the theory of fractional positive and cone one-dimensional (1D) and two-dimensional (2D) linear systems are presented in the article. The state equations and their solution for the fractional continuous-time and discrete-time linear systems...
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nasplib_isofts_kiev_ua-123456789-618722025-02-09T09:56:53Z Some recent developments in theory of fractional positive and cone linear systems Kaczorek, T. Теоретична електротехніка та електрофізика The overview of some recent developments and new results in the theory of fractional positive and cone one-dimensional (1D) and two-dimensional (2D) linear systems are presented in the article. The state equations and their solution for the fractional continuous-time and discrete-time linear systems are given. Necessary and sufficient conditions for the internal and external positivity of the fractional linear systems are established. The reachability of the fractional linear systems is addressed. A new notation of the cone systems is introduced and methods for computation of such systems are proposed. Positive fractional 2D linear systems are intoduced. Necessary and sufficient conditions for the positivity and reachability are established. The considerations are illustrated with many examples of 1D and 2D linear systems. Запропоновано огляд та деякі нові результати у теорії фракційних додатніх та конусних одновимірних (1D) та двовимірних (2D) лінійних систем. Подано рівняння стану та їхнє розв’язання для фракційних неперервних та дискретних лінійних систем. Встановлено необхідні та достатні умови для внутрішньої та зовнішньої додатності фракційних лінійних систем. Показано їхню досяжність. Запропоновано нову форму запису конусних систем та методи, придатні для комп’ютерного розрахунку таких систем. Представлено додатні фракційні 2D лінійні системи. Встановлено необхідні та достатні умови для додатності та досяжності. Теоретичні виклади проілюстровано чисельними прикладами 1D та 2D лінійних систем. Предложены анализ состояния вопроса и некоторые новые результаты, полученные в теории фракционных положительных и конусных одномерных (1D) и двумерных (2D) линейных систем. Приведены уравнения состояния и их решения для фракционных непрерывных и дискретных линейных систем. Выявлены необходимые и достаточные условия для внутренней и внешней положительности фракционных линейных систем. Показана их достижимость. Предложена новая форма записи конусных систем, а также методы их компьютерного расчета. Представлены положительные фракционные 2D линейные системы. Установлены необходимые и достаточные условия для положительности и достижимости. Теоретические выкладки иллюстрированы численными примерами 1D и 1D линейных систем. This work was supported by Ministry of Science and Higher Education in Poland under work No NN514 1939 33. 2010 Article Some recent developments in theory of fractional positive and cone linear systems / T. Kaczorek // Технічна електродинаміка. — 2010. — № 2. — С. 20-32. — Бібліогр.: 26 назв. — англ. 0204-3599 https://nasplib.isofts.kiev.ua/handle/123456789/61872 621.3.011.74.005 en application/pdf Інститут електродинаміки НАН України |
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Теоретична електротехніка та електрофізика Теоретична електротехніка та електрофізика |
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Теоретична електротехніка та електрофізика Теоретична електротехніка та електрофізика Kaczorek, T. Some recent developments in theory of fractional positive and cone linear systems |
| description |
The overview of some recent developments and new results in the theory of fractional positive and cone one-dimensional (1D) and two-dimensional (2D) linear systems are presented in the article. The state equations and their solution for the fractional continuous-time and discrete-time linear systems are given. Necessary and sufficient conditions for the internal and external positivity of the fractional linear systems are established. The reachability of the fractional linear systems is addressed. A new notation of the cone systems is introduced and methods for computation of such systems are proposed. Positive fractional 2D linear systems are intoduced. Necessary and sufficient conditions for the positivity and reachability are established. The considerations are illustrated with many examples of 1D and 2D linear systems. |
| format |
Article |
| author |
Kaczorek, T. |
| author_facet |
Kaczorek, T. |
| author_sort |
Kaczorek, T. |
| title |
Some recent developments in theory of fractional positive and cone linear systems |
| title_short |
Some recent developments in theory of fractional positive and cone linear systems |
| title_full |
Some recent developments in theory of fractional positive and cone linear systems |
| title_fullStr |
Some recent developments in theory of fractional positive and cone linear systems |
| title_full_unstemmed |
Some recent developments in theory of fractional positive and cone linear systems |
| title_sort |
some recent developments in theory of fractional positive and cone linear systems |
| publisher |
Інститут електродинаміки НАН України |
| publishDate |
2010 |
| topic_facet |
Теоретична електротехніка та електрофізика |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/61872 |
| citation_txt |
Some recent developments in theory of fractional positive and cone linear systems / T. Kaczorek // Технічна електродинаміка. — 2010. — № 2. — С. 20-32. — Бібліогр.: 26 назв. — англ. |
| work_keys_str_mv |
AT kaczorekt somerecentdevelopmentsintheoryoffractionalpositiveandconelinearsystems |
| first_indexed |
2025-11-25T15:15:29Z |
| last_indexed |
2025-11-25T15:15:29Z |
| _version_ |
1849775868058009600 |
| fulltext |
20 ISSN 0204-3599. . . 2010. 2
621.3.011.74.005
T. Kaczorek (Warsaw University of Technology, Poland)
SOME RECENT DEVELOPMENTS IN THEORY OF FRACTIONAL POSITIVE AND
CONE LINEAR SYSTEMS
(1D)
(2D) .
. -
. .
, .
2D . .
1D 2D .
,
(1D) (2D) . -
.
. -
. ,
. 2D .
.
1D 1D .
Introduction. In positive systems inputs, state variables and outputs take only non-negative values.
Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and
distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A
variety of models having positive linear systems behaviour can be found in engineering, management
science, economics, social sciences, biology and medicine, etc.
Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of
positive systems is more complicated and less advanced. An overview of state of the art in positive systems
is given in the monographs [2, 5]. An extension of positive systems are the cone systems. The notion of cone
systems was introduced in [6]. Roughly speaking cone system is a system obtained from positive one by
substitution of the positive orthants of states, inputs and outputs by suitable arbitrary cones. The realization
problem for cone systems has been addressed in [6].
The first definition of the fractional derivative was introduced by Liouville and Riemann at the end
of the 19th century [21, 24, 26]. This idea has been used by engineers for modeling different processes in the
late 1960s [1, 3, 25-27]. Mathematical fundamentals of fractional calculus are given in [21, 23, 24, 26]. A
generalization of the Kalman filter for fractional order systems has been proposed in [27]. Fractional
polynomials and nD systems have been investigated in [4]. The positive controllability of positive systems
and approximate constrained controllability of mechanical systems have been investigated in [20, 21].
The aim of this paper is to give an overview of some recent developments and new results in the
theory of fractional positive and cone 1D and 2D linear systems.
The paper is organized as follows. The standard and positive fractional continuous-time linear
systems are addressed in section 2. Necessary and sufficient conditions for the positivity of the system are
established. Similar problem for the discrete-time linear systems are considered in section 3. Section 4 is
devoted to the reachability of positive fractional discrete-time linear systems. The realization problem for
positive fractional continuous-time systems is addressed in section 5. The cone fractional discrete-time linear
systems and their reachability are considered in section 6. Positive fractional 2D systems and their
reachability are addressed in section 7. Concluding remarks and open problems are given in section 8.
The following notation will be used in this paper.
© Kaczorek T., 2010
ISSN 0204-3599. . . 2010. 2 21
Let n m be the set of n m real matrices and 1: .n n The set of m n matrices with
nonnegative entries will be denoted by m n and 1: .n n The set of nonnegative integers will be
denoted by Z and the n n identity matrix by .nI
Continuous-time linear systems. 1. Continuous-time fractional linear systems. In this paper the
following Caputo definition of the fractional derivative will be used [21, 24]
( )
1
0
1( ) ( ) ,
( ) ( )
t n
n
d fD f t f t d
dt n t
1 {1, 2,...}n n N , (1)
where is the order of fractional derivative and ( ) ( )( ) .
n
n
n
d ff
d
Consider the continuous-time fractional linear system described by the state equations
( ) ( ) ( ), 0 1D x t Ax t Bu t ( ) ( ) ( )y t Cx t Du t , (2 ,b)
where ( ) ,nx t ( ) ,mu t ( ) py t are the state, input and output vectors and ,n nA ,n mB
,p nC .p mD
Theorem 1. [10] The solution of equation (2a) is given by
0 0 0
0
( ) ( ) ( ) ( ) , (0)
t
x t t x t Bu d x x , (3)
where 0
0
( ) ( )
( 1)
k k
k
A tt E At
k
,
( 1) 1
0
( )
[( 1) ]
k k
k
A tt
k
(4,5)
and ( )E At is the Mittage-Leffler matrix function, 1
0
( ) t xx e t dt is the gamma function.
Remarks. 1. From (4) and (5) for 1 we have 0
0
( )( ) ( )
( 1)
k
At
k
Att t e
k
. (6)
2. From the Cayley-Hamilton theorem we have. If 1
1 1 0det[ ] ( ) ( ) ...n n
n nI s A s a s a s a , (7)
then 1
1 1 0... 0n n
nA a A a A a I . (8)
Example. Find the solution of eq.(2a) for 10 , 0 1 0, ,0 0 1A B 0
1 ,1x 1, for 0( ) 0, for 0
tu t t . (9)
Using (4) and (5) we obtain 0 2
0
( )
( 1) ( 1)
k k
k
A t Att I
k
1 2 1
2( )
( ) (2 )
t tt I A , (10a,b)
since 0 1 0 0 for 2,3,...0 0 0 0
k
kA k .
Substitution of (10) and ( ) 1u t into (3) yields
1 2 10
0 0 0
0 0
2
2
0
0
( ) ( ) ( ) ( ) ( ) ( )
( 1) ( ) (2 )
1
( 1) (2 1)
( 1) ( 1) (2 1)
1
( 1)
t tAx t B ABx t t x t Bu d x t t d
t t
Ax t Bt ABtx
t
, (11)
since ( 1) ( ).
2. Positivity of continuous-time fractional systems.
Definition 1. The fractional system (2) is called the internally positive fractional system if and only
if ( ) nx t and ( ) py t for 0t for any initial conditions 0
nx and all inputs ( ) ,mu t 0.t
22 ISSN 0204-3599. . . 2010. 2
A square real matrix [ ]ijA a is called the Metzler matrix if its off-diagonal entries are nonnegative,
i.e. 0ija for i j [2, 5].
Lemma 1. Let n nA and 0 1 . Then
0
0
( ) for 0
( 1)
k k
n n
k
A tt t
k
,
( 1) 1
0
( ) for 0
[( 1) ]
k k
n n
k
A tt t
k
, (12,13)
if and only if A is a Metzler matrix. Proof is given in [10].
Theorem 2. The continuous-time fractional system (2) is internally positive if and only if
, , ,n m p n p m
nA M B C D (14)
nM is the set of Metzler matrices. Proof is given in [10].
Definition 2. The fractional system (2) is called externally positive if and only if ( ) , 0py t t
for every input ( ) , 0mu t t and 0 0.x
The impulse response ( )g t of single-input single-output system is called its output for the input
equal to the Dirac impulse ( )t with zero initial conditions. Assuming successively that only one input is
equal to ( )t and the remaining inputs and initial conditions are zero we may define the impulse response
matrix ( ) p mg t of the system (2). The impulse response matrix of the system (2) is given by
( ) ( ) ( ) for 0g t C t B D t t . (15)
Substitution of (3) into (2b) for 0 0x yields
0
( ) ( ) ( ) ( ), 0
t
y t C t Bu d Du t t . (16)
The formula (15) follows from (16) for ( ) ( ).u t t
Theorem 3. The continuous-time fractional system (2) is externally positive if and only if its
impulse response matrix (15) is nonnegative, i.e. ( ) for 0p mg t t . (17)
Proof. The necessity of the condition (17) follows immediately from Definition 2. The output ( )y t
of the system (2) with zero initial conditions for any input ( )u t is given by the formula
0
( ) ( ) ( )
t
y t g t u d (18)
which can be obtained by substitution of (15) into (16). If the condition (17) is met and ( ) ,mu t then
from (18) we have ( ) for 0.py t t From (15) and (13) it follows that if A is a Metzler matrix and
(14) holds then the impulse response matrix (15) is nonnegative. Therefore, we have the following two
corollaries: 1. The impulse response matrix (15) of the internally positive system (2) is nonnegative.
2. Every continuous-time fractional internally positive system (2) is also externally positive.
A example of electrical circuit composed of a resistance R, capacitance C and voltage source
described by fractional differential equation is given in [18].
Discrete-time linear systems. 1. Discrete-time fractional systems. The following definition of the
fractional difference will be used
0
( 1)
k
j
k k j
j
x xj , 0 1 , (19)
where R is the order of the fractional difference, and
1, for 0
( 1) ( 1) , for 1, 2,...
!
j
j j j
j
(20)
Consider the fractional discrete-time linear system, described by the state-space equations
1 ,k k kx Ax Bu u Z k k ky Cx Du , (21a,b)
ISSN 0204-3599. . . 2010. 2 23
where ,n
kx ,m
ku p
ky are the state, input and output vectors and ,n nA
,n mB ,p nC .p mD
Using (19) we may write the equations (21) in the form
1
1 1
1
( 1) ,
k
j
k k j k k
j
x x Ax Bu k Z
j k k ky Cx Du . (22a,b)
Definition 3. The system (22) is called the (internally) positive fractional system if and only if n
kx and
,p
ky k Z for any initial conditions 0
nx and all input sequences ,m
ku .k Z
Theorem 4. [8] The solution of equation (22a) is given by
1
0 1
0
k
k k k i i
i
x x Bu , (23)
where k is determined by the equation
1
1
1 1
2
( ) ( 1)
k
i
k n k k i
i
A I
i
with 0 .nI (24)
Theorem 5. Let 1 1
0
det ( )
M
i
n M i
i
I z Az a z (25)
be the characteristic polynomial of the system (22). Then the matrices 0 1, ,..., M satisfy the equation
0
0
M
i i
i
a . (26)
Proof. From definition of inverse matrix and (25) we have
11 1
0 0
Adj ( )
M
i i
n i m i
i i
I z Az z a z , (27)
where AdjF denotes the adjoint matrix of F.
Comparison of the coefficients at the same power Mz of the equality (27) yields (26) since degree
of 1 1Adj[ ( ) ]nI z Az less than M.
Theorem 5 is an extension of the well-known classical Cayley-Hamilton theorem for the fractional
system (20). Note that the degree M of the characteristic polynomial (25) depends on k and it increases to
infinity for .k In practical problems it is assumed that k is bounded by some natural number L. If k L
then ( 1).M N L
2. Positivity of discrete-time fractional systems. The following two lemmas are used in the proof of
the positivity of the fractional system (23).
Lemma 2. [8] If 0 1 , then 1( 1) 0i
i for 1, 2,...i (28,29)
Lemma 3. [8] If (29) holds and
n n
nA I , then
n n
k for 1, 2,...k (30,31)
Theorem 6. Let (28) be satisfied. Then the fractional system (28) is positive if and only if
,n n
nA I ,n mB ,p nC p mD . (32)
Proof is given in [8, 18].
Example 2. Consider the fractional system (22) for 0 1 with 1 0 0, , ( 2)0 1A B n . (33)
The fractional system is positive since
2 21 0
0 0nA I .
Using (24) for 0,1,...k we obtain diagonal matrices of the forms
24 ISSN 0204-3599. . . 2010. 2
1 0
2
2 1 0
1 0( ) ,0 0
1 5 2 0( ) ,2 0 (1 )2
n
n
A I
A I
3 2 1 0
2
( )
2 3
3( 5 2)( 1) ( 1)(2 5) 01
6 0 (1 )( 2)
nA I
. (34)
From (23) and (24) we have
1
0 1
0
0
1
k
k k k i i
i
x x u , (35)
where k is given by (34).
Definition 4. The discrete-time fractional system (22) is called externally positive if
,p
ky k Z for every input sequence ,m
ku k Z and 0 0.x
Theorem 7. The discrete-time fractional system (22) is externally positive if and only if its response
matrix 1
, for 0
, for 1,2,...kk
D kg CA B k is nonnegative, i.e. p m
kg for k Z . (36,37)
The proof is similar to the proof of Theorem 3.
Remark 4. The impulse response matrix (36) of the internally positive system (22) is nonnegative
and every discrete-time fractional internally positive system is also externally positive.
Reachability of positive fractional discrete-time systems. Consider the positive fractional discrete-
time linear system (22).
Definition 5. A state n
fx of the positive fractional system (22) is called reachable in q steps if
there exist an input sequence m
ku , 0,1, , 1k q which steers the state of the system from zero
0( 0)x to the final state fx , i.e. q fx x .
Let ,ie ni ,...,1 be the i-th column of the identity matrix .nI A column iae for 0a is called a
monomial column.
Theorem 8. The positive fractional system (22) is reachable in q steps if and only if the reachability
matrix 1 1: [ , ,..., ]q qR B B B (38)
contains n linearly independent monomial columns.
Proof. Using (22) for k q and 0 0x , obtain
1 1
21
0 0
q q
qf q q i i q
i
u
ux x Bu R
u
. (39)
From Definition 5 and (39) it follows that for every n
fx there exists an input sequence ,m
iu
0,1,..., 1i q if and only if the matrix (38) contains n linearly independent monomial columns.
Example 3. Consider the positive fractional systems (22) for 0 1 with (33).Using (24) and (38)
we obtain
2 1 3 1 2
0 0 0 0 0[ , ] , [ , , ]1 0 1 0 0,5(1 )R B B R B B B
’
4
0 0 0 0
(1 ) (1 )( 2) ,...1 0
2 6
R (40)
Note that the matrices (40) contain only one linearly independent monomial column. Therefore, by
Theorem 8 the system (22) with (33) is unreachable.
ISSN 0204-3599. . . 2010. 2 25
Example 4. Consider the fractional systems (22) for 10 with 0 1, , ( 2)1 2 0A B n . (41)
The system is positive since 2 20 0
1 2nA I R . Using (24) for 0k we obtain
1 0( )nA I 0 0
1 2 . The reachability matrix (38) for 2q has the form
1
1 0[ , ] 0 1qR B B .
It contains two linearly independent monomial columns. Therefore, the positive fractional system
with (41) is reachable in two steps.
Remark 5. From (24) and (38) it follows that the positive fractional system (22) is reachable only if
the matrix [ , ]nB A I (42)
contain n linearly independent monomial columns.
The controllability of positive fractional discrete-time linear systems has been considered in [8] and
the reachability of positive fractional continuous-time linear systems in [10].
Realisation problem for positive fractional continuous-time linear systems.
1. Problem formulation. Using the Laplace transform it is easy to show that the transfer matrix of the
systems is given by the formula 1( ) [ ]nT s C I s A B D . (43)
The transfer matrix is called proper if and only if lim ( ) p m
s
T s K R and it is called strictly proper if
and only if 0K . From (43) we have lim ( )
s
T s D , since 1lim[ ] 0ns
I s A . (44)
Definition 6. Matrices (14) are called a positive fractional realization of given transfer matrix ( )T s
if they satisfy the equality (43). A realization is called minimal if the dimension of A is minimal among all
realizations of ( )T s .
The positive realization problem can be stated as follows. Given a proper transfer matrix ( )T s , find its
positive realizations (14). In this section sufficient conditions for the existence of positive fractional realizations will
be established and procedure for computation of the positive fractional realizations will be proposed.
Problem Solution. The realization problem will be solved for single-input single-output (SISO)
linear fractional systems with the proper transfer function
1
1 1 0
1
1 1 0
( ) ( ) ...
( )
( ) ( ) ...
n n
n n
n n
n
b s b s b s bT s
s a s a s a
. (45)
Using (44) we obtain lim ( ) ns
D T s b (46)
and the strictly proper transfer function has the form
1 2
1 2 1 0
1
1 1 0
( ) ( ) ...
( ) ( )
( ) ( ) ...
n n
n n
sp n n
n
b s b s b s bT s T s D
s a s a s a
, (47)
where , 0,1,..., 1k k k nb b a b k n . (48)
From (48) it follows that if 0ka and 0kb for 0,1,...,k n then also 0kb for 0,1,..., 1k n .
Theorem 9. There exist positive fractional minimal realizations of the forms
0 1 2 1
0 1 1
0 1 0 ... 0
0 0 1 ... 0
,
0 0 0 ... 1
...
0
, ... ,0
1
n
n n
A
a a a a
B C b b b D b
2 0
2
12
11
0 0 ... 0
1 0 ... 0
, ,0 1 ... 0
...
0 0 ... 1
0 ... 0 1 ,
nn
n
a ba bA Ba
ba
C D b
(49a,b)
26 ISSN 0204-3599. . . 2010. 2
1 2 1 0
1 1 0
... 11 0 ... 0 0 0, ,0 1 ... 0 0
00 0 ... 1 0
... ,
n n
n n
a a a a
A B
C b b b D b
1 1
2
11
00
1 0 ... 0
0 1 ... 0
, ,
0 0 ... 1
0 0 ... 0
1 0 ... 0 ,
n n
n
n
a ba
A B ba ba
C D b
(49c,d)
of the transfer function (45) if a) 0kb for 0,1,...,k n , b) 0ka for 0,1,..., 2k n and
1 1 0n n nb a b .
Proof. Taking into account that for (49) 1
1 1 0det[ ] ( ) ( ) ...n n
n nI s A s a s a s a and
1Adj[ ] [1 ... ( ) ]n
nI s A B s s it is easy to verify that
1 2
1 1 2 1 0
1
1 1 0
Adj[ ] ( ) ( ) ...
[ ]
det[ ] ( ) ( ) ...
n n
n n n
n n n
n n
C I s A B b s b s b s b
C I s A B
I s A s a s a s a
.
The matrix A is Metzler matrix if and only if 0ka for 0,1,..., 2k n and arbitrary 1na . Note that the
coefficient of matrices C and D are nonnegative if the conditions a) is met and 1 1 1 0n n n nb b a b . The
proof for (49b), (49c) and (49b) are similar (dual). The matrices (49) are minimal realizations if and only if
the transfer function (45) is irreducible. If the conditions of Theorem 9 are satisfied then the positive minimal
realizations (49) of the transfer function (45) can be computed by use of the following procedure.
Procedure. Step 1. Knowing ( )T s and using (46) find D and the strictly proper function (47). Step 2.
Using (49) find the desired realizations.
Example 5. Find the positive minimal fractional realizations (49) of the irreducible transfer function
2
2
2( ) 5 1( )
( ) 2 3
s sT s
s s
. (50)
Using Procedure and (50) we obtain the following. Step 1. From (46) and (50) we obtain
2
2
2( ) 5 1lim 2
( ) 2 3s
s sD
s s
and 2
7( ) ( )
( ) 2 3sp
sT s T s D
s s
. (51,52)
Step 2. Taking into that in this case 0 17, 1b b and using (49) we obtain the desired positive
minimal fractional realizations
0 1 0, , [7 1], 23 2 1A B C D ; 0 3 7, , [0 1], 21 2 1A B C D ; (53a,b)
2 3 1
, , [1 7], 2
1 0 0
A B C D ;
2 1 1
, , [1 0], 2
3 0 7
A B C D . (53c,d)
An extension of this method for multi-input multi-output positive fractional continuous-time linear
systems has been given in [11]. The presented method can be easily extended for positive fractional discrete-
time linear systems.
Cone Fractional Discrete Time Systems And Their Reachibility.
Definition 7. Let
1 n n
n
p
P
p
be nonsingular and kp be the k-th ( 1,...,k n ) its row.
The set
1
: : 0
n
n
k
k
x p xP (54)
is called a linear cone generated by the matrix P.
In a similar way we may define for the inputs u the linear cone
1
: : 0
m
m
k
k
u q uQ (55)
ISSN 0204-3599. . . 2010. 2 27
generated by the nonsingular matrix 1 m m
m
q
Q
q
and for the outputs y , the linear cone
1
: : 0
p
p
k
k
y v yV (56)
generated by the nonsingular matrix
1 p p
p
v
V
v
.
Definition 8. The fractional system (22) is called (P Q V, , ) cone fractional system if kx P and
ky V , k Z for every 0x P , ku Q , .k Z
The (P Q V, , ) cone fractional system (4) will be shortly called the cone fractional system. Note
that if nP , ,mQ pV then the ),,( pmn cone system is equivalent to the classical
positive system [2, 5].
Theorem 10. The fractional system (22) is (P Q V, , ) cone fractional system if and only if
,1 nnPAPA ,1 mnPBQB ,1 npVCPC mpVDQD 1 (57)
Proof. Let ,kk Pxx kk Quu and kk Vyy , Zk . (58)
From Definition 8 it follows that if kx P then ,n
kx if ku Q then m
ku and if
ky V then .p
ky From (22) and (58) we have
1 1
1 1
1 1 1 1
1 1
( 1) ( 1) ,
k k
j j
k k j k k j k k k k k k
j j
x x Px Px PAx PBu PAP x PBQ u Ax Bu k Z
j j
and 1 1 ,k k k k k k k ky Vy VCx VDu VCP x VDQ u Cx Du k Z . (59a,b)
It is well-known [5] that the system (59) is the positive one if and only if the conditions (57) are satisfied.
Definition 9. A state fx P of the cone fractional system (22) is called reachable in q steps if there
exists an input sequence ku Q , 0,1,..., 1k q which steers the state of the system from zero initial
state 0( 0)x to the desired state fx , i.e. .q fx x If every state fx P is reachable in q steps then the
cone fractional system is called reachable in q steps. If for every state fx P there exists a natural number
q such that the state is reachable in q steps then the cone fractional system is called reachable.
Theorem 11. The cone fractional system (22) is reachable in q steps if and only if the matrix
],...,,[ 1
1
1
1
1 BQPBQPPBQR qq (60)
contains n linearly independent monomial columns.
Proof. From the relations (58) it follows that if kx P then n
k kx Px and if ku Q then
m
k ku Qu for .k Z Hence by Definition 8 and 9 the cone fractional system (22) is reachable in q
steps if and only if the positive fractional system (59) is reachable in q steps.
Using (24) and (57) it is easy to show that k of the system (59) with k of he system (22) are
related by 1PPk for 0,1,...k . (61)
Taking into account that 1 1 1, 1, 2,..., 1k k kB P P PBQ P BQ k q (62)
28 ISSN 0204-3599. . . 2010. 2
we may write
1 1
1 1 1
1 1
[ , ,..., ]
[ , ,..., ]
q q
q
R B B B
PBQ P BQ P BQ
. (63)
By Theorem 8 the positive fractional system (59) is reachable in q steps if and only if the matrix (60)
contain n linearly independent monomial columns.
Example 6. Consider the cone fractional system (22)
for
1 1 , [1],1 1
, 0, 0 11 1
P Q
aA aa
(64)
and for the following two forms of the matrix B
0,, 21 b
b
b
B
b
b
B .
The P -cone generated by the matrix P is shown in
Fig.. In case 1 we shall show that the cone fractional
system is not reachable.
Using (57) and (64) we obtain
1
1
1 1
11 1 0 1 1 1( ) , 01 1 1 1 1 1 1 02
1 1 2 , 01 1 0
d n
a a aA P A I P aa
b bB PB Q bb
. (65)
The system (59) with matrices (65) is a positive fractional system. Using (60) for 2q , (64) and
taking into account that 1 dA we obtain
1 1
2 1 1 1 1 1
1 1 2 2( 1)[ , ] [ , ] 0 21 1 ( 2)d
b ab b a bR PB Q P B Q P B A B bb a b . (66)
The matrix (66) contains only one (the first) monomial column. Thus by Theorem 11 the cone
fractional system is unreachable.
In case 2 we have 0,
2
0
11
111
22 b
bb
b
QPBB . (67)
The system (59) with matrices dA and 2B given by (65) and (67) is also a positive fractional
system. Using (60) and (64) we obtain the matrix
2 2 2
1 1 0 2[ , ] , 0, 01 1 2 0d
b ab abR P B A B a bb ab b , (68)
which contains two linearly independent monomial columns. Therefore, by Theorem 11 the cone fractional
system is reachable.
The controllability to zero of the cone fractional discrete-time linear systems has been considered in [9].
Positive Fractional 2d Linear Systems And Their Reachibility And Their Reachibility.
1. Fractional 2D linear systems. The positive fractional 2D linear systems have been introduced in
[15, 16] and the positive 2D hybrid linear systems in [17].
Definition 10. The ( , ) orders fractional difference of and 2D function ijx is defined by the formula
,
,
0 0
( , ) ,
ji
ij i k j l
k l
x c k l x , 1 , 1 , {1, 2,...}n n n n n N , (69)
where ,
ij i j ijx x and
,
1 for 0 or/and 0
( 1)...( 1 ) ( 1)...( 1 )( , ) ( 1)
! !
for 0
k l
k l
k lc k l
k l
k l
. (70)
x2
x1
x2 = -x1
x2 = x1
0
ISSN 0204-3599. . . 2010. 2 29
The justification of Definition 10 is given in [15].
Consider the ( , ) order 2D fractional linear system, described by the state equations
,
1, 1 0 1 1, 2 , 1 0 1 1, 2 , 1i j ij i j i j ij i j i jx A x A x A x B u B u B u ij ij ijy Cx Du , (71a,b)
where , ,n m p
ij ij ijx u y are the state, input and output vectors and ,n n
kA
, 0,1, 2,n m
kB k , .p n p mC D
Using Definition 10 we may write the equation (71a) in the form
11
1, 1 0 1 1, 2 , 1 1, 1 0 1 1, 2 , 1
0 0
0
( , )
ji
i j ij i j i j i k j l ij i j i j
k l
k l
x A x A x A x c k l x B u B u B u , (72)
where 0 0 nA A I , 1 1 nA A I , 2 2 nA A I .
From (69) it follows that the coefficients (70) in (69) strongly decrease when k and l increase.
Therefore, in practical problems it is assumed that i and j are bounded by some natural numbers L1 and L2. In
this case (72) takes the form
1 21 1
1, 1 0 1 1, 2 , 1 1, 1 0 1 1, 2 , 1
0 0
( , )
L L k
i j ij i j i j i k j l ij i j i j
k l
x A x A x A x c k l x B u B u B u . (73)
Note that the fractional systems are 2D linear systems with delays increasing with i and j.
The boundary conditions for the equation (72) and (73) are given in the form
0 0, and ,i jx i Z x j Z . (74)
Theorem 12. The solution of equation (72) with boundary conditions (74) is given by
1
, 1 1 0 1 0 1, 2 0 2 0 1, 1 0 0
1 1 1
1 11
1, 1 0 0 1, 1 0 00 1, 1 0 1, 1 1 , 1 2
1 0 0 0 0
( ) ( )
( )
ji i
ij i p j p p i j q q q i p j p
p q p
j j ji i
i j q q i j i p j q pq i p j q i p j q pq
q p q p q
x T A x B u T A x B u T A x
T A x T A u T B u T B T B u
(75)
where the transition matrices pqT are defined by the formula
0 1, 1 1 , 1 2 1, ,
0 0
2
for 0
( , ) for 0
0 (zero matrix) for 0 or/and 0
n
p q
pq p q p q p q kl
k l
k l p q
I p q
T A T AT A T c p k q l T p q
p q
. (76)
Proof is given in [16]. Let
1 21 1
1 1 1 1
1 2 1 2 0 2 1 2 2 1
0 0
( , ) ( , )
L L
k l
n n
k l
G z z I I c k l z z A z z A z A z (77)
and
1 1
1 21 2 , 1 2
0 0
det ( , )
N N
k l
N k N l
k l
G z z a z z . (78)
It is assumed that i and j are bounded by some natural numbers 1 2,L L which determined the degrees 1 2,N N .
Theorem 13. Let (78) be the characteristic polynomial of the system (71). Then the matrices klT
satisfy the equation
1 2
0 0
0
N N
kl kl
k l
a T . (79)
Proof is given in [16].
Theorem 13 is an extension of the well-known classical Cayley-Hamilton theorem for the 2D
fractional system (71).
30 ISSN 0204-3599. . . 2010. 2
2. Positivity of the fractional 2D systems
Lemma 4. [16]
a) If 0 1 and 1 2 then ( , ) 0 for 1, 2,...; 2,3,...c k l k l (80a)
b) If 1 2 and 0 1 then ( , ) 0 for 2,3,...; 1, 2,...c k l k l . (80b)
Lemma 5. [16] If (80) is met and n n
kA for 0,1, 2,k (81)
then n n
pqT for ,p q Z . (82)
Definition 11. The system (71) is called the (internally) positive 2D fractional system if and only if
n
ijx and , ,p
ijy i j Z for any boundary conditions Zix n
i ,0 Zix n
j ,0 and all
input sequences .,, Zjiu m
ij
Theorem 14. The 2D fractional system (71) for 10 and 21 is positive if and only if
, , 0,1, 2, ,n n n m p n p m
k kA B k C D . (83)
Proof is given in [16].
Remark 6. From (70) and (71) it follows that if , 0 1 then ( , ) 0 for 1, 2,...c k l k l
and the fractional 2D system (71) is not positive.
3. Reachability of the positive fractional 2D systems.
Definition 12. The positive 2D fractional system (71) is called reachable at the point
( , )h k Z Z if and only if for zero boundary conditions (74) 0 0( 0, , , )i jx i Z x j Z and every
vector n
fx there exists a sequence of inputs m
iju for
( , ) {( , ) : 0 ,
0 , }
hki j D i j Z Z i h
j k i j h k
(84)
such that .fhk xx A vector is called monomial if and only if its one component is positive and the remaining
components are zero.
Theorem 15. The positive 2D fractional system (71) is reachable at the point ),( kh if and only if
the reachability matrix
1 1 2 2
0 1 2 1 11 1 21[ , ,..., , ,..., , ,..., , ,..., ]hk h k k hkR M M M M M M M M M
1
0 1, 1 0 , 1 1 1, 1 0
2
1, 1 2 1, 1 0
1, 1 0 , 1 1 1, 1 2
, , 1,...,
, 1,...,
, 1,..., , 1,...,
h k i h i k h i k
j h k h k j
ij h i k j h i k j h i k
M T B M T B T B i h
M T B T B j k
M T B T B T B i h j k
(85,86)
contains n linearly independent monomial columns.
Proof. Using the solution (75) for kjhi , and zero boundary conditions we obtain
( , )f hkx R u h k , (87)
where 00 10 0 01 0 11 1 21 ,( , ) [ , ,..., , ,..., , ,..., , ,..., ]T T T T T T T T T T
h k k h ku h k u u u u u u u u u (88)
and T denotes the transpose.
For the positive 2D fractional system (71) from (86) and (85) we have 1
0 , ,n m n m
iM M
2 , , 1,..., , 1,...,n m n m
j ijM M i h j h and [( 1)( 1) 1] .n h k m
hkR From (87) it follows that there
exists a sequence m
iju for ( , ) hki j D for every n
fx if and only if the matrix (85) contains n
linearly independent monomial columns.
The following theorem gives sufficient conditions for the reachability of the positive 2D fractional system (71).
Theorem 16. The positive 2D fractional system (71) is reachable at the point ( , )h k if rank hkR n
and the right inverse r
hkR of the matrix (85) has nonnegative entries
ISSN 0204-3599. . . 2010. 2 31
1 [( 1)( 1) 1][ ]r T T h k m n
hk hk hk hkR R R R . (89)
Proof. If rank hkR n then there exists the right inverse r
hkR of the matrix .hkR If the condition
(89) is met then from (87) we obtain [( 1)( 1) 1]( , ) r h k m
hk fu h k R x for every .n
fx
Example 7. Consider the positive 2D fractional system (71) with
0 1 2 0 1 2
0 1 1 0 1 0 1 0 1, , , , , ,1 0 0 1 1 1 0 1 1A A A B B B . (90)
To check the reachability at the point ( , ) (1,1)h k of the system we use Theorem 15. From (86) and
(85) we obtain
1 2
0 0 1 1 1 2
1 0 1, , , 00 1 1 ijM B M B M B M for 1, 1i j ,
1 2
11 0 1 1
1 0 1[ , , ] 0 1 1R M M M . (91)
The first two columns of (91) are linearly independent monomial columns and by Theorem 15 the
positive 2D fractional system (71) with (90) is reachable at the point (1,1). The sequence of inputs steering
the state of the system from zero boundary conditions to an arbitrary state 2
fx at the point (1,1) is
given by 00
10 f
u xu and 01 0.u Using (89) and (91) we obtain
1
1
1 0 2 112 1[ ] 0 1 1 21 2 31 1 1 1
r T T
hk hk hk hkR R R R . (92)
From (92) it follows that the condition (89) is not satisfied in spite of the fact that the system is
reachable at the point (1,1). Note that the system is reachable at the point (1,1) for any fractional order
( , ) 0 1 , 1 2 (or 1 2 , 0 1 ) and any matrices , 0,1, 2.kA k
Necessary and sufficient conditions for the controllability to zero of positive fractional 2D linear
systems have been established in [16].
Concluding remarks and open problems. An overview of some resent developments and new
results in the theory of fractional positive and cone 1D and 2D linear system have been given. The state
equations and their solutions for fractional continuous-time and discrete-time linear systems have been
proposed. Necessary and sufficient conditions for the internal and external positivity and reachability of the
systems have been established. The realization problem for positive fractional continuous-time linear
systems has been formulated and solved. A new class of cone fractional discrete-time linear systems has
been introduced. The positive fractional 2D linear systems have been also introduced and their reachability
has been investigated. From the long list of the open problems in the fractional systems theory the following
are the natural steps:
- 1D and 2D fractional linear systems with delays,
- Positive 1D and 2D fractional linear systems with delays,
- Positive fractional 2D hybrid systems with and without delays,
- Standard and positive 2D fractional continuous-time systems,
- Standard and positive 1D and 2D nonlinear systems.
This work was supported by Ministry of Science and Higher Education in Poland under work No NN514
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