Krzysztof Rogowski
General Response Formula for Fractional 2D Continuous-Time Linear Systems Described by the Roesser Model
GENERAL RESPONSE FORMULA FOR FRACTIONAL
2D CONTINUOUS-TIME LINEAR SYSTEMS
DESCRIBED BY THE ROESSER MODEL
Krzysztof ROGOWSKI*
*Phd student, Faculty of Electrical Engineering, Białystok University of Technology, Wiejska 45 D, 15-351 Białystok
[email protected]
Abstract: A new class of fractional two-dimensional (2D) continuous-time linear systems is introduced. The general response formula for the system is derived using a 2D Laplace transform. It is shown that the classical Cayley-Hamilton theorem is valid for such class of systems. Usefulness of the general response formula to obtain a solution of the system is discussed and illustrated by a numerical example.
1. INTRODUCTION
The most popular models of two-dimensional (2D) linear system are the ones introduced by Roesser (1975), Fornasini and Marchesini (1976, 1978) and Kurek (1985).
An overview of 2D linear systems theory is given in (Bose,
1982, 1985; Kaczorek, 1985, 2001; Gałkowski, 2001, Farina and Rinaldi, 2000).
Mathematical fundamentals of fractional calculus and
its applications are given in the monographs (Oldham and
Spanier, 1974; Nashimoto, 1984; Miller and Ross, 1993;
Podlubny, 1999, Ostalczyk, 2008).
The notion of fractional 2D discrete-time linear systems
was introduced by Kaczorek (2008a) and extended
in (Kaczorek, 2008b, 2009, Kaczorek and Rogowski, 2010,
Rogowski, 2011). An overview in state of the art in 1D
and 2D fractional systems is given in the monograph
(Kaczorek, 2011).
In this paper a new 2D continuous-time fractional
Roesser type model will be introduced. The general response formula for the system will be derived using the 2D
Laplace transform method (Section 2). Moreover the classical Cayley-Hamilton theorem will be extended to fractional
2D continuous-time systems in Section 3. In Section 4
usefulness of the general response formula to obtaining
the solution of the system will be discussed and illustrated
by a numerical example. Concluding remarks are given
in Section 5.
To the best knowledge of the author 2D continuous-time
fractional linear systems have not been considered yet.
2. FRACTIONAL 2D STATE EQUATIONS
AND THEIR SOLUTION
Let ௡×௠ be the set of × real matrices and
: = ௡×ଵ . The set of nonnegative integers will be denoted by ା and the × identity matrix will be denoted
by ௡ .
We introduce the following definition of fractional partial derivative of a 2D continuous function (ଵ , ଶ ) of two
independent variables ଵ , ଶ ≥ 0.
௡
112
Definition 1. The ௜ order partial derivative of a 2D continuous function (ଵ , ଶ ) is given by the formula
α
Dt i f ( t1 , t2 ) =
i
=
∂αi
αi
∂ti
f ( t1 , t2 )
ti
ft
( Ni )
(τ i )
(1)
1
i
dτ i ,
Γ ( Ni − αi ) ∫ ( t − τ )α i − Ni +1
0 i
i
where = 1,2; ௜ − 1 < ௜ < ௜ ∈ = {1,2, … }, ௜ ∈ is the order of fractional partial derivative,
∞
Γ( x) = ∫ e−t t x −1dt
(2)
0
for ≥ 0is the gamma function and
 ∂ N1 f (τ1 , t2 )

N

∂τ1 1
(N )
ft i (τ i ) = 
N
i
 ∂ 2 f ( t1 ,τ 2 )

N
∂τ 2 2

for i = 1
(3)
for i = 2.
Consider the fractional 2D continuous-time system described by the state equations
 Dα1 x h ( t , t ) 
1 2
 t1
 =  A11
 α2 v
 A
 Dt2 x ( t1 , t2 )   21
A12   x h ( t1 , t2 )   B1 

+
u (t , t ) ,
A22   x v ( t , t )   B2  1 2
1 2 

(4a)
 x ( t1 , t2 ) 
 + Du ( t1 , t2 ) ,
y ( t1 , t2 ) = C 
 xv ( t1 , t2 ) 
h
(4b)
where ௛ (ଵ , ଶ ) ∈ ௡భ , ௩ (ଵ , ଶ ) ∈ ௡మ ( = ଵ + ଶ )
are the horizontal and vertical state vectors, respectively,
(ଵ , ଶ ) ∈ ௠ is the input vector, (ଵ , ଶ ) ∈ ௣ is the output vector and ௞௟ ∈ ௡ೖ×௟ , ௞ ∈ ௡ೖ×௠ for , = 1,2;
∈ ௣×௡ ; ∈ ௣×௠ .
acta mechanica et automatica, vol.5 no.2(2011)
where
The boundary conditions for (4) are given in the form
h( k )
xt
1
 ∂ k x h ( t1 , t2 ) 



∂t1k
t1 =0
( 0, t2 ) = 
(5a)
for = 0, 1, … , ଵ − 1 and ଶ ≥ 0,
v( l )
xt
2
 ∂ l xv ( t1 , t2 ) 

( t1 , 0 ) = 
∂t2l


Itα f ( t ) =
1
Γ (α ) ∫
(5b)
t2 =0
α −1
f (τ )( t − τ )
dτ ,
(6)
0
where > 0 is the fractional (real) order of the integration.
Similarly, we may define the 2D fractional integral of function
(ଵ , ଶ )
Itα,,tβ f ( t1 , t2 ) = Itα  Itβ f ( t1 , t2 )  = Itβ  Itα f ( t1 , t2 ) 
1 2
1  2

2  1

1
=
×
Γ (α ) Γ ( β )
t1 t2
α −1
∫ ∫ ( t1 − τ1 )
A
T10 =  11
 0
A12 
,
0 
 0
T01 = 
 A21
0 
.
A22 
(10)
Proof. Let (, ଶ ) (ଵ , ) be the Laplace transform
of a 2D continuous function (ଵ , ଶ ) with respect to ଵ (ଶ )
defined by
F ( p, t2 ) = Lt1  f ( t1 , t2 )  =
∞
∫ f ( t1 , t2 )e
− pt1
dt1
0
(11)
∞


 F ( t1 , s ) = Lt  f ( t1 , t2 )  = ∫ f ( t1 , t2 )e − st2 dt2  .
2


0


The 2D Laplace transform of (ଵ , ଶ ) will be denoted
by (, ) and defined by
{
}
{
}
F ( p, s ) = Lt1 Lt2  f ( t1 , t2 )  = Lt2 Lt1  f ( t1 , t2 ) 
= Lt1 ,t2  f ( t1 , t2 ) 
( t2 − τ 2 )β −1 f (τ1 ,τ 2 ) dτ 2 dτ1 ,
=
00
∞∞
∫ ∫ f ( t1 , t2 ) e
− pt1 − st2
(12)
dt1dt2 .
0 0
(7)
where , > 0.
Theorem 1. The solution to the equation (4a) with the boundary conditions (5) is given by
k +iα −1
t1 1
 h( k −1)
jα 2  xt
1
 x ( t1 , t2 )  ∞ ∞


 = ∑ ∑ Tij  ∑
I
v
 x ( t1 , t2 )  i =0 j =1  k =1 Γ ( k + iα1 ) t2
h
(9)
and
for = 0, 1, … , ଶ − 1 and ଵ ≥ 0.
In the following theorem the Riemman-Liouville formula
of fractional integration of a function () will be used (Podlubny, 1999)
t
In
for i = 0, j = 0

Tij = T10Ti −1, j + T01Ti , j −1 for i + j > 0, ( i, j ∈ Z + )

for i < 0 and/or j < 0
0 (zero matrix)
 N1


( 0, t2 ) 


0
 B  ( i +1)α , jα

+  1  It ,t 1 2 u ( t1 , t2 ) 
1 2
0
 

1
k
+
i
−
α
 x h( k −1) ( 0, t ) 
∞
1
 N1 t
2 
 t1
+ ∑ Ti 0  ∑ 1
α
Γ
k
+
i


(
)
1

i =0
0
 k =1


Applying (12) to (1) for i = 1 and taking into account that
(Kaczorek, 2011)
Γ (α + 1)
Lt1 t1α  =
 
pα +1
(13)
and
N
(N)
( k −1)
Lt1  ft ( t1 )  = p N F ( p, t2 ) − ∑ p N − k ft
( 0, t2 ) (14)
1
 1

k =1
for N = 0,1,... ; we obtain
{
}
α
α
Lt1 ,t2  Dt 1 f ( t1 , t2 )  = Lt2 Lt 1  Dt 1 f ( t1 , t2 ) 
 1

 1


 B  ( i +1)α1
+  1  It
u ( t1 , t2 ) 
1

0
∞ ∞
 N 2 t l + jα 2 −1
iα
+ ∑ ∑ Tij ∑ 2
It 1
1
Γ
+
α
l
j
(
)
2
 l =1
i =1 j = 0
0


 v( l −1)

 xt
t1 , 0 ) 
(
 2


0
  iα ,( j +1)α 2

+   I t ,t1
u ( t1 , t2 ) 
1 2
B
 2

l + jα −1
t2 2
0


∞
 N 2
 v( l −1)

+ ∑ T0 j ∑
( t1 , 0 ) 
 l =1 Γ ( l + jα 2 )  xt2
j =0
(N )
 t1

ft 1 (τ1 )
1
 

1
Lt2 Lt 1 ∫
dτ1  
=
 t − τ α1 − N1 +1

Γ ( N1 − α1 )
 0 ( 1 1 )

 
 
{
}
(15)
1
N −α −1
(N )
=
Lt Lt 1 t1 1 1  Lt 1  ft 1 (τ1 ) 


 1

Γ ( N1 − α1 ) 2
N
1
( k −1) 0, s ,
= pα1 F ( p, s ) − ∑ pα1 − k Ft
( )
k =1
1
where
 k

k)
 ∂ f ( t1 , t2 ) 

(

Ft ( 0, s ) = Lt2  

k
1

∂t1
 

=
0
t


1

 0  ( j +1)α 2
+   It
u ( t1 , t2 )  ,
2
 B2 

(8)
(16)
for k = 0,1,... .
113
Krzysztof Rogowski
General Response Formula for Fractional 2D Continuous-Time Linear Systems Described by the Roesser Model
Similarly, for i = 2 in (1) we have
N
2
α
( l −1) p, 0 ,
Lt1 ,t2  Dt 2 f ( t1 , t2 )  = sα 2 F ( p, s ) − ∑ sα 2 −l Ft
( )
 2

2
l =1
(17)
where
 l

l)
 ∂ f ( t1 , t2 ) 

(

Ft ( p, 0 ) = Lt1  

l
2

∂t2
 

t
=
0

2

where T10 and T01 are defined by (10).
Comparing the coefficients at the same powers of and we obtain (9).
Substituting the expansion (22) into (20) we obtain
  −( i +1)α1 s − jα 2 B 
 X h ( p, s )  ∞ ∞
 p
1

 = ∑ ∑ Tij  
U ( p, s )
 X v ( p, s )  i =0 j =0   p −iα1 s −( j +1)α 2 B 
2

 N1 − k −iα − jα h( k −1)

1s
2X
∑ p
( 0, s ) 
t1
 k =1
 
+
 ,
N

 2 −iα1 −l − jα 2 v( l −1)
s
Xt
( p, 0 )  
∑ p
2
 l =1
 
(18)
for ݈ = 0, 1, …
Taking into account (15) and (17) we obtain the 2D Laplace transform of the state equation (4a)
N1
 α h

h( k −1)
0, s ) 
 p 1 X ( p, s ) − ∑ pα1 − k X t
(
1


k =1

=
N2
−
1
v
l
 α2 v
(
)
α 2 −l
Xt
( p, 0 ) 
 s X ( p, s ) − ∑ s
2
l =1


 A11
A
 21
Taking into account (Kaczorek, 2011)
L−1  p −α F ( p )  = I tα f ( t ) ,


(19)
A12   X h ( p, s )   B1 

+
U ( p, s ) .
A22   X v ( p, s )   B2 


(25)
where > 0 and ℒ denotes the inverse Laplace transform,
it is easy to show that
α ,α
Lt−1,t  p −α1 s −α 2 F ( p, s )  = I t ,1t 2 f ( t1 , t2 ) ,

1 2 
1 2
(26)
where , > 0.
Applying the inverse 2D Laplace transform to (24)
and taking into account (26) we obtain the formula (8).
Premultiplying (19) by the matrix
blockdiag  p −α1 I n1 , s −α 2 I n2 


we obtain
3. EXTENSION OF CAYLEY-HAMILTON
THEOREM
 X h ( p, s ) 

 = G −1 ( p , s )
 X v ( p, s ) 
Theorem 2. Let
  N1


h k −1
  ∑ p − k X t ( ) ( 0, s ) 

−α
1
  p 1 B1 
  k =1

 U ( p, s )  ,
× 
 +  −α
N2
 s 2 B2 


−l v( l −1)
( p, 0 )  
  ∑ s X t2


  l =1

(20)
det G ( p, s ) =
=
I n1 − p −α1 A11
− p −α1 A12
− sα 2 A21
I n2 − sα 2 A22
(27)
n1 n2
∑ ∑ an −k ,n −l p −kα s −lα
1
k =0 l = 0
where
 I − p −α1 A
n1
11
G ( p, s ) = 
 − sα 2 A
21

(24)
1
2
2
be the characteristic polynomial of the system (4). Then
the transition matrices satisfy the equality
− p −α1 A12 
.
I n2 − sα 2 A22 
(21)
n1 n2
∑ ∑ ak ,l Tk + m1,l + m2 = 0 ,
(28)
k =0 l =0
Let
G
−1
∞ ∞
( p, s ) = ∑ ∑ Tij p
−iα1 − jα 2
s
.
(22)
i =0 j =0
 n1 n2
AdjG ( p, s ) =  ∑ ∑ an1 − k ,n2 −l p − kα1 s −lα 2

 k =0 l = 0
∞ ∞

 ∑ ∑ Tij p −iα1 s − jα 2 
 i =0 j =0



From
G −1 ( p , s ) G ( p , s ) = G ( p , s ) G −1 ( p , s ) = I n ,
using (21) and (22), it follows that
∞ ∞
∑ ∑ {Tij − T10Ti −1, j − T01Ti, j −1}p −iα s − jα
1
i =0 j =0
2
= In
where , = 0, 1, ….
Proof. From the definition of the inverse matrix, as well as
(22) and (27), we have
(23)
=
n1 n2
∞
∞
∑ ∑ ∑ ∑ aklTi +k , j +l p
k = 0 l = 0 i =− k j =−l
114

×


(29)
−( i + n1 )α1 −( j + n2 )α 2
s
,
acta mechanica et automatica, vol.5 no.2(2011)
where AdjG ( p, s ) denotes the adjoint matrix of (, ).
Comparing the coefficients at the same powers of and for ≥ 0 and ≥ 0we obtain (28) since (, ) is a polynomial matrix of the form
AdjG ( p, s ) =
n1 n2
∑ ∑ Dij p−iα s − jα
1
2
,
(30)
i = 0 j =0
i , j ≠ n1 ,n2
Itα H ( t ) =
tα
.
Γ (1 + α )
(34)
From (34) and (7) it is easy to show that
where ∈ × are some real matrices.
Theorem 2 is an extension of the well-known classical
Cayley-Hamilton theorem to fractional 2D continuous-time
systems.
4. NUMERICAL EXAMPLE
Example 1. Consider fractional 2D system (4) with = 0,7,
= 0,9and matrices
 A11
A
 21
 x h ( t1 , t2 ) 
.
y ( t1 , t2 ) = 
 xv ( t1 , t2 ) 
It is well-known that (Podlubny, 1999)
A12   −0.9 0.7   B1  1
=
,
=
,
A22   0
−0.3  B2  1
1 0 
C=
D = [ 0] .
,
0 1 
(31)
α ,α 2
It ,1t
1 2
α ,α 2
u (t1 , t2 ) = I t ,1t
1 2
α α
H (t1 , t2 ) =
t1 1 t2 2
Γ (1 + α1 ) Γ (1 + α 2 )
. (35)
Using (8) for , = 1 and taking into account (31),
(32) (33) and (35) we obtain
( i +1)α1 t jα 2
 x h ( t1 , t2 )  ∞ ∞
t1
1 
2

 = ∑ ∑ Tij  
 xv ( t1 , t2 )  i =0 j =1  0  Γ 1 + ( i + 1) α1  Γ (1 + jα 2 )
( i +1)α1
∞
t1
1 
+ ∑ Ti 0  
 0  Γ 1 + ( i + 1) α1 
i =0
(36)
iα ( j +1)α 2
∞ ∞
t1 1 t2
0
+ ∑ ∑ Tij  
1  Γ [1 + iα1 ] Γ 1 + ( j + 1) α 2 
i =1 j =0
( j +1)α 2
∞
t2
0
,
+ ∑ T0 j  
1  Γ 1 + ( j + 1) α 2 
j =0
where transition matrices Tij are given by (9).
Fig. 1. State variable ‫ ݔ‬௛ (‫ݐ‬ଵ , ‫ݐ‬ଶ ) of the system
Find a step response of the system (4) with the matrices
(31), i.e. ( , ) for , ≥ 0and
0
u (t1 , t2 ) = H (t1 , t2 ) = 
1
for t1 < 0 and / or t2 < 0
(32)
for t1 , t2 ≥ 0
and zero boundary conditions
x h ( 0, t2 ) = 0,
x v ( t1 ,0 ) = 0.
(33)
Note that in this case from (31) and (4) it follows that
Fig. 2. State variable ‫ ݔ‬௩ (‫ݐ‬ଵ , ‫ݐ‬ଶ ) of the system
Formula (36) describes the step response of the system
(4) with the matrices (31). It is easy to show that
the coefficients 1/Γ(. ) strongly decrease when and increase. Therefore, in numerical analysis we may assume
that and are bounded by some natural numbers and .
The plots of the step response (36) where = 50 and
= 50 are shown on Fig. 1 and 2.
115
Krzysztof Rogowski
General Response Formula for Fractional 2D Continuous-Time Linear Systems Described by the Roesser Model
5. CONCLUDING REMARKS
A new class of fractional 2D continuous-time linear systems described by the Roesser model has been introduced.
The general response formula for such systems has been
derived (Theorem 1) using the 2D Laplace transform.
The classical Cayley-Hamilton theorem has been extended
to fractional 2D continuous-time systems (Theorem 2).
It has been shown that using the general response formula
we are able to obtain the step response of the fractional 2D
continuous-time system. The considerations have been illustrated by a numerical example.
The above considerations can be extended for general
2D model (Kurek, 1985). An open problems are the positivity and stability of fractional 2D continuous-time systems.
116
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