A Note on the Sequential Linear Fractional Dynamical Systems from the Control
System Viewpoint and L2 -Theory
Abolhassan Razminia*, Vahid Johari Majd**

* Intelligent Control Systems Laboratory, School of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran
(e-mail: a.razminia@gmail.com ).
** Intelligent Control Systems Laboratory, School of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran (Tel:
+21 8288-3353; e-mail: majd@modares.ac.ir )
Abstract: In this paper, we consider a sequential linear fractional differential equation with constant
coefficients. After deriving a closed analytical solution based on the sequential fractional derivatives, we
propose a theorem that gives some weak conditions under which any arbitrary function in L2 can be
approximated by an output of a linear time-invariant fractional differential equation. More precisely, we
prove that all L2 functions can be represented as the L2 limit of functions that are the outputs of
fractional linear control systems. The analysis is developed in the Hilbert space.
Keywords: Linear fractional differential equations, LTI control system, L2 theory.

spaces. They are sometimes called Lebesgue spaces. One of
the main topics in Lp space is approximation theory [11],
which is employed here for deriving a relationship between
an arbitrary function from L2 space and the output of a linear
time invariant fractional control system. The dynamic of the
control system is stated in the sequential format. Sequential
fractional operator is a special case among other fractional
operators which is discussed in the next section.
1. INTRODUCTION
Fractional calculus is a generalization of ordinary
differentiation and integration to an arbitrary real-valued
order. This subject is as old as the ordinary differential
calculus, and goes back to times when Leibniz and Newton
invented differential calculus. The problem raised by Leibniz
in a letter dated September 30, 1695 for a fractional
derivative has become an ongoing topic for more than
hundreds of years [1-2].
Due to the computing demands of the fractional
differentiation and integration and the need for approximating
some complicated functions which are used in physical
process, we present a novel method in this paper by which
any arbitrary function from L2 space can be approximated by
the output of a linear time invariant time fractional control
system (LTIFCS). Using a rigorous proof, we provide the
construction procedure of LTIFCS whose output
approximated the given function in the sense of L2 -norm.
This subject has attracted the attention of researchers from
different fields in the recent years. While it was developed by
mathematicians few hundred years ago, efforts on its usage in
practical applications have been made only recently. It is
known that many real systems are fractional in nature, thus, it
is more effective to model them by means of fractional order
than integer order systems. Applications such as modeling of
damping behavior of viscoelastic materials [3], cell diffusion
processes [4], transmission of signals through strong
magnetic fields [5], and finance systems [6] are some
examples. Moreover, fractional order dynamic systems have
been used in both design and implementation of control
systems. Studies have shown that a fractional order controller
can provide better performance than an integer order one and
lead to more robust control performance [7].
This paper is organized as follows: after this introduction
in section 1, a historical review is presented in section 2.
Some basic concepts and mathematical preliminaries are
summarized in section 3. Section 4 studies the main result
which is stated in a theorem form. Finally conclusions in
section 5 close the paper.
2. HISTORICAL REVIEW
It is has been reported that specific fractional differential
equations with order less than three, may exhibit complex
dynamical evolution even chaotic dynamics [8-9]; however,
this is not the case for ordinary differential equations due to
the famous Poincaré–Bendixson theorem [10].
L2 space as a special case of Lp space is considered here.
The Lp spaces are function spaces defined using natural
generalizations of p-norms for finite-dimensional vector
How well can functions be approximated on a finite
interval by the output of SISO controllable and observable
fractional linear time-invariant system? This question was
first considered in the framework of constructing the inverse
of a linear system in the integer order system basin [12]. Here
we want to extend this problem for the first time, in a more
comprehensive field, i.e. fractional order systems. Although
in [13], Unser’s method has been extended for fractional
1
systems, the method presented in this paper is completely
different and has a more general form.
for ordinary derivatives. Some of the current definitions for
fractional derivatives are described in [21]. The Riemann–
Liouville definition is the simplest definition to use. Based on
Over the past few decades, the theory of linear control
theory has centered on stability properties that only really
make sense on an infinite interval. However, there are many
problems in which the only thing of interest is a finite
interval. Indeed, this work began with questions of
constructing flyable trajectories for linear systems [14]. Since
the appearance of those initial papers, there has been a
growing interest on the construction of curve approximation
of using linear control theory, which has developed into the
theory of interpolating and smoothing splines.
q th
this definition, the
order fractional integral of
f  L [0, t ] and the terminal value a is given by [22]:
1
(1)
t

1
(t  s ) q 1 f ( s ) ds
( q )
J a q f (t )  f q (t ) 
; q  R
a
where the Euler-Gamma function is defined as follows:

(2)

( q )  e  q u q 1du
; qR
0
The theory of interpolating splines began with the paper
of Schoenberg [15] in 1946 and has begun to be widely
employed since 1960s with the availability of digital
computers. The theory of smoothing splines began with the
work of [16] and received a tremendous impetus with the
publication of her CBMS lectures. A more theoretic problem
can be found in [14].
As another definition for the fractional derivative we
propose the Caputo’s definition [22]:
t


f (m) ( )
1
d


(
m


)

(t   ) 1 m
D f (t )  J a m  D m f (t )  
a
*

dm

f (t )

dt m


Almost all of the literature has been devoted to
polynomial splines and an impressive body of tools has been
developed for this application. A body of literature has
developed around the concept of control theoretic splines.
The initial work was with respect to flight control and is
represented by the works of Crouch and Jackson [11].
m 1  m
(3)
 m
In the same way we can prove the following property for the
Laplace transform of the Caputo derivative [23]:
L{D f (t )}  s L{ f (t )} 
*
m 1

s  k 1 f (k ) (0)
(4)
k 0
As one can see, in the Caputo derivative we need the
integer order derivatives of the initial conditions while in the
Riemann-Liouville we need the fractional derivative of the
initial conditions. This is a merit of Caputo definition.
In the area of smoothing splines Martin and colleagues
have developed an theory of smoothing splines based on
linear control theory [17]. Some modifications in a more
general prospective under weaker conditions and more
general spaces, e.g. Banach spaces, of the problem studied in
[17], were treated in [17] and [18] recently. These works
were motivated by the trajectory planning problem in order to
avoid high accelerations that were observed when it was
required to be exactly at point at a given time. The constraint
of interpolation was relaxed to approximation and a simple
optimal control problem yielded generalized smoothing
splines. This technique was shown to be adaptable to a
number of situations not easily handled with traditional
polynomial smoothing spline techniques.
The question that has been in the background of all of the
control theoretic splines has been its convergence. This
problem was directly attacked in [19]. In particular the paper
[20] gives a fairly comprehensive theory of convergence of
Miller and Ross in [24] introduced the so called
sequential fractional derivative in the following way: MR D


(5)
MR D  D , 0    1
MR D
k 

MR D MR D
(k 1) , k  2, 3,...

where D is the Caputo fractional derivative. Next we define
a sequential fractional differential equation in an operator
format as follows:
n  a
(n 1)  ...  a



(6)
 MR D
n 1 MR D
1 MR D  a0  y(t )  f (t )


without loss of generality we restrict 0    1 .
As in the usual case, it is easy to obtain the connection
between (6) and the corresponding system of linear fractional
differential equations. If we apply the following change of
variables to (6):
(7)
x (t )  y(t ), ( D x )(t )  x (t ), ( j  1, 2, ..., n  1)
2
splines in a L sense. However, it was necessary to impose
some rather restrictive conditions on the linear system, which
although were natural from an approximation view point,
were not natural from the viewpoint of control theory. Beside
these restrictions and developments, we want to extend these
tasks to a more general form, i.e. fractional order control
systems. The new theory might have an important impact on
the interpolation and spline theory. Moreover, this
methodology can be used fairly in the signal processing and
system identification.
1
j
j 1
j
where we have used the Caputo fractional derivative. Using
the above notation we have:
D x1  x2 ,
D x2  x3 ,
(8)

D xn 1  xn ,
D xn  
2. PRELIMINARIES
n 1
a x
k k
 f (t )
k 0
Or in a matrix form with a companion matrix A ,
D x(t )  Ax (t )  f (t )
Fractional calculus as an extension to ordinary calculus
possesses definitions that stem from the definitions existing
2
(9)
This is a standard linear fractional differential equation
(LFDE).
It is easy to see that the vector M is the initial conditions
of x . Also one can show that the particular solution is as
follows [28]:
Definition 1 [25]. Let 1  p   and S , ,   be a measure
space. Consider the set of all measurable functions from S to
C (or R) whose absolute value raised to the p-th power has
finite integral, or equivalently, that:
1
(10)
p

p
f p : 
f d   


t 
xp 
0 k 0



S


k 0
Ak (t   ) (1 k ) 1
f ( )d
[(k  1) ]
(16)
t
Ak
(t   ) ( k 1) 1 f ( )d
[(k  1) ]

0
In this paper we consider a special case with p  2 which is
Therefore the general solution is:
the most important case in the Lp spaces. Consider the
dynamical system:
x(t )  xh (t )  x p (t ) 

k 0


D x(t )  Ax (t )  bu (t )
y  Cx

(11)

k 0
Definition 1 [26]. The system (11) is observable on [t0 , t f ]
t
k

A
(t   ) ( k 1) 1 f ( )d
[(k  1) ]
y (t ) and u (t ) on [t0 , t f ] .
0
D x(t )  Ax (t )  bu (t )
y  Cx
Definition 2 [26]. System (11) is controllable iff for any
( x0 , x1) there exists a control u (t ) defined on [t0 , t f ] which


y  Cx  C.
k 0
3. MAIN RESULTS


As in the integer order case, consider xh and x p are the

k 0
homogeneous and particular solutions of (9), respectively.
0
(13)
uL2 [ 0,T ]
k 0
2
(21)
0
(u ) is dense in L2[0, T ] .
Suppose that (u ) is not dense in L2[0, T ] . Thus there
M
exists a nonzero function v  L2 [0, T ] such that (u ), v  0 for

  Ak (t ) (1 k ) 1 
(t ) (1 k ) 1
D 
M   Ak
D
M


[(k  1) ]
[(k  1) ]
k 0
 k 0


 (m(t )  y(t )) dt  0
Proof. It is enough to establish that the range space of the
all u  L2 [0, T ] . So
To show that this can be the homogenous solution of (9),
based on the differintegrability of xh it is easy to differentiate
(14)[27]:

(20)
T
inf
[(k  1) ]
k 0
Ak (t   ) (1 k ) 1
bu ( )d
[(k  1) ]
Therefore for every m(t )  L2 [0, T ] there exists a u  L2[0, T ]
such that:

So based on this proposition it is sufficient to find the
homogeneous and particular solutions separately. Using a
similar method in finding the homogeneous solution for
integer order differential equations, we can find the
homogeneous solution for LFDE as

(14)
Ak (t ) (1 k ) 1
 Ak

 
 A( xh  x p )  f  Ax  f .


0
t
 D x  D ( xh  x p )  D xh  D x p  Axh  ( Ax p  f )
k 0
(19)
CA
(t   ) ( k 1) 1 bu ( )d
[(k  1) ]
(u )(t )  C
D xh  Axh ,

0
Theorem. Consider the system (18) which is controllable and
observable. Defining a linear operator as follows:

xh (t ) 

t
k
Proof. Inserting xh and x p in (9) we have:

t
Ak
(t   ) ( k 1) 1 bu ( )d
[(k  1) ]
Now we present the main result of the paper:
Proposition 1. For the LFDE (9) we have the general
solution as follows:
(12)
x(t )  xh (t )  x p (t )

(18)
According to the previously developed theory we can write:
drives the initial state x(t0 )  x0 to the final state x(t f )  x1 .

(17)
Now we return to a control system problem. Consider a
fractional control system with zero initial conditions and
companion matrix which is not a restrictive assumption:
iff x (t ) for t [to , t f ] can be deduced from knowledge of
D x p  Ax p  f ,

Ak (t ) (! k ) 1
M
[(k  1) ]

(t ) k 1 ((1  k ) )
M  A.
[(k  1) ](k )


j 0
(15)
A j (t ) (1 j ) 1
M
[( j  1) ]
3
T
(u ), u 
T
 
v(t ) C (t   ) 1
0



k 0



k 0
k 0
0
T
Now introducing the vector valued function ( ) as follows,
we can manipulate the relations easily:
(29)
C  c1 ... cn , ( )  1( ) 2 ( ) ... n ( )'
A k (t   ) k
bu( )d dt
[(k  1) ]
T
 
CA k b
[(k  1) ]
CA k b
[(k  1) ]


v(t ) (t   ) (k 1) 1 u ( )d dt
0
TT

Using these notations we have
(t   ) (k 1) 1 u ( )d v(t )dt u ( )d

(22)
since ( A, b) and ( A, C ) are controllable and observable,
respectively. So we can write:
n
(31)
 u ,  * (v )
 c D ( )  0
where
i
* (v)( ) 
T
 [(k  1) ]  (t   )
k
CA b
k 0
T 

( k 1) 1
k
CA b
( k 1) 1
v(t )dt
Introducing the following state variables:
(23)
D1  i 1, i  1, 2, ..., n  1
v(t )dt
0
1   0
  


D  2    
  
   c0
n   1
 cn

So it is concluded that * (v)( )  0 . Now we use the fact that
(u ) is onto if * (v) is one-to-one [29]. It suffices to show
that v  0 almost everywhere. Let

 k 0
Ak b
(t   )( k 1) 1 v(t )dt
[(k  1) ]
(24)

( ) 
(l  1)

 l 
(l  1   )



   0
0
D( )  D

 k 0
0
1


0
c1
cn

(Q) k  ( k 1) 1
(0)
((k  1) )





(26)
0 

k 0

( A) k
( ) ( k 1) 1 bv( )d  0
((k  1) )






( A) k

D 0 
( )( k 1) 1bv( )d 

((
k

1
)

)


k 0
0



Or briefly
 D0 
(27)


k 0


( A) k
D ( )( k 1) 1bv( )d
((k  1) )

 0  lim D ( )( k 1) 1bv( )  0
of the original system ( A, b) it is obtained that for an initial
value 0 we have
 v( )  0 a.e.

k 0


k 0
( A) k
((k  1) )
   0


(   ) (k 1) 1 bv( )d
(37)
0
So from * (v)( )  0 and using the controllability condition
( A) k  (k 1) 1
0 
((k  1) )
(36)
0
Differintegrating of (36) once we have:
  A( )  bv( )

(35)
and hence that

D( )   A( )  bv( ), (T )  0
(33)
(34)


( A) k  (k 1) 1
( A) k
0 
(   ) (k 1) 1 bv( )d  0
((k  1) )
((k  1) )
k 0
k 0
0
Ak b
(t   ) (k 1) 1 v(t )dt
[(k  1) ]
T 



Ak b

(
k

1
)


1

D (t   )
v(t )dt  bv( )

[(
k

1
)

]


 k  0

T 
A jb
 A
(t   ) ( j 1) 1 v(t )dt  bv( )
[(k  1) ]
 j 0
0

1 
0
  1 
 
 

 1 0   2   Q 2 

  
  1   
 
c1  n 
n 
 

cn 



We deduce that
T 
0
Since (T )  0 we conclude that (0)  0 and consequently
( )  0 . This establishment is based on the assumption that
cn  0 , but this approach can be easily extended to more
general case. Anyway we have:
(25)
2. D K ( , )d  D K ( , )d  lim D 1K ( , )
0

k 0

1
Using (17), we have:
Using the following important facts [30]
1. D  l
(32)
we have:
k 0
T 
i
i 0

  [(k  1) ] (t   )
( ) 
cii ( )  0
i 0
0

(30)
n
0
So this shows that * is one-to-one and hence that  is onto
(28)
a dense subspace of L2[0, T ] .□
0
0
This idea indicates an important point in control theory.
Indeed given a function in L2 space, we can propose a
fractional order transfer function which can approximate that
 C( )  0
4
function in term of L2 norm. One of the main advantages of
this idea is that we can analyze a very complex function in
L2 space by transferring it to a fractional order transfer
function space which may exhibits a more general transfer
function than the usual ones. Transferring it to the second
space we can develop a n analysis using well-known existing
methods such as frequency methods, root-locus methods, and
state space techniques, which are well established and easy to
use.
belongs to L2 can be approximated by an output of a linear
time-invariant fractional differential control system. More
precisely we gave a proof that every L2 function can be
represented as the L2 limit of functions that are the outputs of
fractional linear control systems. Our analysis was developed
in the Hilbert space time-invariant fractional differential
control system. More precisely we gave a proof that every
L2 function can be represented as the L2 limit of functions
that are the outputs of fractional linear control systems. Our
analysis was developed in the Hilbert space.
4. CONCLUSION
In this paper, considering a fractional differential equation
system with constant matrices and using some facts from
functional analysis, we proved that any arbitrary function
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