Computer algorithms for solving optimization problems
for discrete-time fractional systems
Andrzej Dzieli´ nski and Przemyslaw M. Czyronis
Abstract— Dynamic programming and discrete-time calculus
of variations optimization problems for fractional discrete-time
systems with quadratic performance index have been formu-
lated and solved. A new methods for numerical computation of
optimization problems have been presented. The efficiency of
the methods have been demonstrated on numerical example and
illustrated by graphs. Graphs also show the differences between
the fractional and classical (standard) systems theory. Results
for both methods have been obtained through a computer
algorithms written for this purpose.
I. INTRODUCTION
Dynamic optimization problems for integer (not fractional)
order systems have been widely considered in literature (see
e.g. [1]–[3]). Mathematical fundamentals of the fractional
calculus are given in the monographes [4]–[6] and fractional
calculus of variations is given in [7]. Fractional differential
equations and their applications have been addressed in [8],
[9]. The numerical simulation of the fractional order control
systems has been investigated in [10]. One of the fractional
discretization method has been presented in [11]. Some
optimal problems for fractional order systems have been
investigated in [12]–[15]. Dynamic Programming problem
for discrete-time fractional systems has been formulated and
solved in [16]. Variational calculus for discrete-time frac-
tional systems have been investigated in [17]–[22]. Fractional
Kalman filter and its application have been addressed in [23],
[24]. Some recent interesting results in fractional systems
theory and its applications for standard and positive systems
can be found in [25], [26].
In this paper optimization problems for fractional discrete-
time systems with quadratic performance index will be
formulated. A general solutions for dynamic programming
and discrete-time calculus of variations problems will be
presented. A new methods for numerical computation of
optimization problems will be presented. The efficiency of
the methods will be demonstrated on numerical example
and illustrated by graphs. Graphs also show the differences
between the fractional and classical (standard) systems the-
ory. Results for both methods will be obtained through a
computer algorithm written for this purpose. Computer al-
gorithms will be discussed in details and the block diagrams
will be presented.
The paper is organized as follows. In section II some
preliminaries are recalled and the optimization problem for
This work was not supported by any organization
The authors are with the Faculty of Electrical Engineering - The
Institute of Control and Industrial Electronics, Warsaw University of
Technology, Pl. Politechniki 1, 00-661 Warszawa, Poland; email: an-
drzej.dzielinski@ee.pw.edu.pl, przemyslaw.czyronis@ee.pw.edu.pl
dynamic programming method will be formulated. The gen-
eral solutions of the above problem are presented also in that
section. In section III the optimization problem for discrete-
time calculus of variations method will be formulated. The
general solutions of the above problem are presented also in
that section. In section IV a computer algorithms for com-
putation of the optimal control are proposed and numerical
examples are presented. Conclusions of the paper are given
in section V.
II. DYNAMIC PROGRAMMING
A. Problem Formulation
Consider a fractional discrete-time system, obtained by use
of Grunwald-Letnikov’a (shifted) approximation, described
by equations
x
k+1
=
k
j=0
d
j
x
k-j
+ Bu
k
, k ∈ Z
+
, (1a)
where x ∈ R
n
, u ∈ R
m
are respectively the state and control
vectors, A ∈ R
n×n
, B ∈ R
n×m
and
d
0
= A
α
= A + αI
n
, 0 <α< 1 , (1b)
d
j
= (-1)
j
α
j +1
I
n
, j =1,...,k . (1c)
Consider a performance index of the form
J
i
(u)= G(x
N
)+
N-1
k=i
F
k
(x
k
,u
k
)
= x
T
N
Sx
N
+
N-1
k=i
(
x
T
k
Qx
k
+ u
T
k
Ru
k
)
,
(2)
where R ∈ R
m×m
, Q ∈ R
n×n
, S ∈ R
n×n
and S ≥ 0,
Q ≥ 0 and R> 0.
Optimal trajectory starting at the point x
0
and ending
at the point x
k
has been divided into N elementary time
intervals [0,N ]. It is desired to find optimal control sequence
u
0
,u
1
,...,u
N-1
, u ∈ U, which minimizes the performance
index (2) and satisfies the differential equation (1). The
solution of this task by searching for a conditional minimum
of the performance index (2) requires the solution of N
equations with N unknown variables of the form
∂J (u)
∂u
k
=0 , (k =0,...,N - 1) ,
where J (u) is the performance index (2) after substituting
(1) for k =1, 2,...,N - 1.
2013 European Control Conference (ECC)
July 17-19, 2013, Zürich, Switzerland.
978-3-952-41734-8/©2013 EUCA 4009