Solution of the State-Space Linear Fractional Order Systems
Djamel Boucherma(1), Abdelfatah Charef(2) , and Hassene Nezzari(3)
(1) (3)
Unité de Recherche en Technologie Industriel URTI/CSC BP1037, Chaiba, Annaba 23000, Algeria
(2) Département d’Electronique, Université Mentouri -Constantine- Constantine 25000, Algeria (afcharef@yahoo.com)
djamelboucherma25@yahoo.com, afcharef@yahoo.com, nezzarih@yahoo.fr
Abstract--A useful representation of fractional order systems is the state space representation. For the linear fractional systems of commensurate order, the state space representation is defined as for regular integer state space representation with the state vector differentiated to a real order. This paper presents a solution to the state space linear fractional systems of commensurate order. The solution is obtained using a technique based on functions of square matrices and the Cayley-Hamilton Theorem. The solution is calculated in the form of a linear combination of suitable fundamental functions. The basic ideas and the derived formulations of the technique are presented. An example is also given to demonstrate the effectiveness of the presented analytical approach.
Key-Words--Cayley-Hamilton theorem, fractional order differential equations, fractional state space, functions of matrices, rational function
I. INTRODUCTION
The theory of fractional order systems has gained some importance during the last decades [6, 10-13]. These
fractional concepts have been generally used to model physical systems, leading to the formulation of fractional differential equations. So, dynamic systems described by linear fractional differential equations are called linear fractional order systems. These types of systems have more lately been under study from several different points of view, in physics, mathematics, engineering, and many other fields [1-2, 7, 9-10, 13].
In this paper, we present an analytical solution of the state-space linear fractional systems of commensurate order given as:
t Ax t Be(t) x
Dm (1) where Dmx t is the Caputo fractional derivative of order m such that 0 < m < 1, x(t) is the n-state vector, A is the (nxn) state matrix and e(t) is the input.
The solution is obtained using a technique based on functions of square matrices and the Cayley-Hamilton Theorem. The basic ideas and the derived formulations of the technique are presented. An example is also reported to illustrate the method. The objective of this paper is to treat the linear fractional order systems as it is done with usual systems in order to establish a linear fractional order system theory. In this work the eigenvalues of the state- space matrix A are all complexes with null real part.
II. FUNDAMENTAL FUNCTIONS
In this context, the fundamental functions are defined as the functions which will be used in the solution of the state-space linear fractional order systems of equation (1).
Let is complex pole with null real part and m real number such that 0 < m < 1.
we will call gsin(t, , m) and gcos(t, , m) the generalized sine and generalized cosine fundamental functions, its Laplace transform is such that, for a pair of complex poles with null real part:
] s
m) [ λ, gsin(t, L s
H 2
m 2
2
S
(2)
] s
[ m) s λ, gcos(t, L s
H 2
m 2
m
C
(3)
In order to study the dynamical behavior of the solutions of the state-space linear fractional order systems of equation (1), the irrational functions HS(s) and HC(s) of (2) and (3) have to be approximated by rational ones. To do so, we will consider three cases based on the fractional order derivative m.
Case 1: 0 < m < 0.5
For this case, in [3,8], the approximation of the irrational functions of (2) and (3) respectively by a rational function are presented in full details.
Case 2: m = 0.5
For this case, in [3], the approximation of the irrational function of (3) by a rational function is presented in full details.
Case 3: 0.5 < m < 1
For this case, in [3, 8], the approximation of the irrational functions of (2) and (3) respectively by a rational function are presented in full details.
We will introduce a second function hsin(t,, m) and hcos(t,, m) that will also be used in the solution of the state-space linear fractional order systems of equation (1).
The functions hsin(t,, m) and hcos(t,, m) are a simple convolution of the generalized sine gsin(t, , m) and generalized cosine gcos(t, , m) with the unity step function u(t). Hence, it is given as:
gsin(t,λ,m)
u(t) λ,m)hsin(t, (4)
gcos(t,λ,m)
u(t) λ,m)hcos(t, (5) Using Laplace transform, we will have:
2m 2
2
s u(t) s m) λ, gsin(t, L m) λ, (t, in s h
L
(6)
2m 2
m
s s u(t) s m) λ, (t, cos g L m) λ, (t, cos h
L
(7)
the irrational functions of (6) and (7) have to be approximated by rational ones. To do so, we will consider three cases based on the fractional order derivative m. 0 <
m < 0.5, m=0.5 and 0.5 < m < 1. In previous works [10- 12], the fundamental functions defined in (4) and (5) have been approximated by rational ones in order to represent them by linear time-invariant system models.
III. SOLUTION OF THE STATE-SPACE LINEAR
FRACTIONAL ORDER SYSTEM a. Introduction
Let A be an nxn matrix whose characteristic polynomial is written as:
I A nan1n1 a1a0
(8)
Then the corresponding matrix polynomial is:
A An an1An1a1Aa0I
(9)
Cayley-Hamilton Theorem [5]:
Every matrix satisfies its own characteristic equation; that is
A
0 (10)
Functions of square matrices [5]:
Let P() be a function and let A be an nxn matrix whose eigenvalues i (i=1, 2…n) are complexes with null real part. Then P() has a power series representation as:
0 k
k k 0
k k
k P A A
P (11)
It is possible to regroup the infinites series for P() so that:
A A A R A P
R P
0 k
k k 0 k
k k
(12)
The reminder R() will have degree ≤ (n-1) because () has degree n; therefore:
1 n 1 n 2
2 1 0
1 n 1 n 2
2 1 0
A A
A I A R R
(13) For λ = λi (i=1, 2… n), we have Δ(λi) = 0 and Δ(A) = 0, hence, from (12) and (13), we will have :
1 n 1 n 2
2 1 0
) 1 n ( i 1 n 2
i 2 i 1 0 i i
A A
A I A R ) A ( P
R ) ( P
(14)
If the n eigenvalues i (i=1, 2… n) of the matrix A are complexes with null real part, the n coefficients αj(s) (j=0, 1… (n-1)) can be calculated using the following n equations (i=1, 2… n):
i R i 0 1 i 2 i2 n1 (in1) P (15) b. Solution
Consider the state-space linear fractional order system of equation (1):
t Ax t Be(t) x
Dm (16) where Dmx t is the Caputo fractional derivative of order m such that 0 < m < 1, x(t) is the n-state vector, e(t) is the input and A is the (nxn) state matrix whose eigenvalues i
(i=1, 2, …, n) are complexes with null real part. Taking the Laplace transform of equation (16), we obtain:
s
s I A
s x 0
s I A
BE(s) X m 1 (m1) m 1 (17) where x(0) is the initial state. The expression of the state vector x(t) is determined by taking the inverse Laplace transform of equation (17); we will then have:
s I A
*Be(t)L
0 x A I s s L s X L t x
m 1 1
m 1 ) 1 m ( 1 1
(18)
We can easily see that finding the nxn matrix function
smIA
1leads to solution of the state-space linear fractional order system of (16).1. Calculation of the nxn matrix function
smIA
1:The eigenvalues i (i=1, 2… n) of the state matrix A are considered to be complexes with null real part. From subsection 3.1, the function P(A) of the square matrix A is considered to be :
smI A1 0 s I 1 s A n1 s An1 )
A (
P (19)
The functions αj(s) (j=0, 1… (n-1)) can be calculated using the following equations; that is for i=1, 2…n:
i) 1 n (
0 j
j i j
m 1
i s s
) (
P
(20)The above equation can be rewritten in matrix form as:
s s s s
1 1 1 1
s s
s s
1 n
2 n
1 0
1 n n 2
n n
1 n
1 n 2
1 n 1 n
1 n 2 2
2 2
1 n 1 2
1 1
1 n m
1 1 n m
1 2 m
1 1 m
(21)
such that
1 n n 2
n n
1 n
1 n 2
1 n 1 n
1 n 2 2
2 2
1 n 1 2
1 1
1 1 1 1
V
With 21 ,4 3nn1and n is number pair, Then:
s s s s
1 1 1 1
s s s s
1 n
2 n
1 0
1 n 1 n 2
1 n 1 n
1 n
1 n 2
1 n 1 n
1 n 1 2
1 1
1 n 1 2
1 1
1 1 n m
1 1 n m
1 1 m
1 1 m
(22)
The matrix V is non-singular, hence the n coefficients αj(s) (j=0, 1… (n-1)) are obtained from equation (21) as:
1 1 n m
1 1 n m
1 1 m
1 1 m
nn 2
n 1 n
n 1 n 2
1 n 1 1 n
n 2 22
21
n 1 12
11
1 1 n m
1 1 n m
1 1 m
1 1 m
1
1 n
2 n
1 0
s s s s
. s
s s s
V
s s s s
(23)
We notice that:
ij are real numbers if index I is odd number i=1, 3, …, n- 1, j=1, 2, …, n and ijij1
ij are complexes numbers if the index I is even number i=2, 4…n, j=1, 2…n and ij ij1
n
pair k , 2 k
2 k m 2 nk n
impair k , 1 k
2 k m 2 m k 1 n n
pair k , 2 k
2 k m 2 k 2 n
impair k , 1 k
2 k m 2 m k 1
n
1 i
1 i m ni n
1 i
1 i m i 1 n n
1 i
1 i m i 2 n
1 i
1 i m i 1
1 n
2 n
1 0
s L
s s L
s L
s s L
s s s s
s s s s
(24)
The nxn matrix function
smIA
1is then obtained as:
1 n
impair p , 1 p
p n
pair k , 2 k
2 k m k 2 1 p 1
n
pair p , 0 p
p n
impair k , 1 k
2 k m 2
m k 1 p 1
n
0 p
p p m 1
s A L 1
s A L s A
t A
I s
(25)
So, the solution x(t) of the state-space linear fractional order system of equation (18) is given as:
s
A *Be(t)L 1 L
) t ( Be
* s A
L s L
0 x s A
L s L
0 x s A
s L s L t
x
1 n
impair p , 1 p
p n
pair k , 2 k
2 k m 2 1 k 1 p 1
n
pair p , 0 p
p n
impair k , 1 k
2 k m 2
m 1 k 1 p 1
n
impair p , 1 p
p n
pair k , 2 k
2 k m 2
) 1 m ( 1 k 1 p 1
n
pair p , 0 p
p n
impair k , 1 k
2 k m 2
m ) 1 m ( 1 k 1 p
(26)
The functions
k 2
1
m 2
1 s
L and
k 2
1
m 2 m
1 s s
L
are generalized sine and generalized cosine fundamental functions of equation (2) and (3).and are obtained as follows:
s
1 gsint, ,mL 2 k
k 2 1
k m 2
1
(27)
s s
1 gcos(t,λ ,m)L k
k 2 1
k m 2 m 1
(28)
the functions
k 2
1
m 2 1 m
1 s s
L and
k 2 1
m 2 m 1 m
1 s s s
L are obtained as follows:
s s
1 hcos(t,λ ,m)L k
k 2 1
k m 2 1 m 1
(29)
2 k m 2 2 1
k 1 2 k m 2
m 1 m 1
s s L 1 s
L 1 s
s L s
(30)
From equation (7), we can write that:
s s
u t hsint, ,mL1 2m1 2m k 2 1 k (31) Therefore, the solution x(t) of equation (26) is given as:
