fractional linear systems and electrical circuits pdf

Studies in Systems, Decision and Control 13

Tadeusz Kaczorek
Krzysztof Rogowski

Fractional Linear
Systems and
Electrical Circuits


Studies in Systems, Decision and Control
Volume 13

Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
e-mail:

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Tadeusz Kaczorek · Krzysztof Rogowski

Fractional Linear Systems
and Electrical Circuits

ABC
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Tadeusz Kaczorek
Faculty of Electrical Engineering
Białystok University of Technology
Białystok
Poland

ISSN 2198-4182
ISBN 978-3-319-11360-9
DOI 10.1007/978-3-319-11361-6

Krzysztof Rogowski
Faculty of Electrical Engineering
Białystok University of Technology
Białystok
Poland

ISSN 2198-4190 (electronic)
ISBN 978-3-319-11361-6 (eBook)

Library of Congress Control Number: 2014949175
Springer Cham Heidelberg New York Dordrecht London
c Springer International Publishing Switzerland 2015
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Preface

This monograph covers some selected problems of positive and fractional
electrical circuits composed of resistors, coils, capacitors and voltage (current)
sources.
The monograph consists of 8 chapters, 4 appendices and a list of references.
Chapter 1 is devoted to fractional standard and positive continuous-time
and discrete-time linear systems without and with delays. The state equations
and their solutions of linear continuous-time and discrete-time linear systems
are presented. Necessary and sufficient conditions for the internal and external positivity of the linear systems are given. Solutions of the descriptor
standard and fractional linear systems using the Weierstrass–Kronecker decomposition and Drazin inverse matrix method are also presented. In chapter 2 the standard and positive fractional electrical circuits are considered.
The fractional electrical circuits in transient states are analyzed. The reciprocity theorem, equivalent voltage source theorem and equivalent current
source theorem are presented. Descriptor linear electrical circuits and their
properties are investigated in chapter 3. The Weierstrass–Kronecker decomposition method and the shuffle algorithm method are discussed. The regularity, pointwise completeness and pointwise degeneracy of descriptor electrical
circuits is analyzed. The descriptor fractional standard and positive electrical
circuits are also investigated. Chapter 4 is devoted to the stability of fractional
standard and positive linear electrical circuits. It is shown that the electrical
circuits with resistances can be also unstable. The reachability, observability
and recontsructability of fractional positive electrical circuits and their decoupling zeros are analyzed in chapter 5. Necessary and sufficient conditions
for the reachability, observability and reconstructability are established and
illustrated by examples of electrical circuits. The decompositions of the pairs
(A, B) and (A, C) of the electrical circuits are given and the decoupling zeros
of the positive electrical circuits are proposed.
The fractional linear electrical circuits with feedbacks are considered
in chapter 6. The zeroing of the state vector of electrical circuits by state
and output-feedbacks is discussed. In chapter 7 the problem of minimum

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VI

Preface

energy control for standard and fractional systems with and without bounded
inputs has been solved. In chapter 8 the fractional continuous-time 2D linear
systems described by the Roesser type models are investigated. The fractional derivatives and integrals of 2D functions are introduced. The descriptor fractional 2D model is proposed and its solution is derived. The standard
fractional 2D Roesser type models are also investigated.
In Appendix A some basic definitions and theorems on Laplace transforms and Z-transforms are given. The elementary row and column operations on matrices are recalled in Appendix B. In Appendix C some properties
of the nilpotent matrices are given. Definition of Drazin inverse of matrices
and its properties are presented in Appendix D.
The monograph contains some original results of the authors, most of which
have already been published. It is dedicated to scientists and Ph.D. students
from the field of electrical circuits theory and control systems theory.
We would like to express our gratitude to Professors Mikołaj Busłowicz,
Krzysztof Latawiec and Wojciech Mitkowski for their invaluable remarks,
comments and suggestions which helped to improve the monograph.
Białystok, June 2014

Tadeusz Kaczorek
Krzysztof Rogowski

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V

1

1
1
2
3
3
5

Fractional Differential Equations . . . . . . . . . . . . . . . . . . . . . .
1.1 Definition of Euler Gamma Function and Its Properties . . .
1.2 Mittag-Leffler Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Definitions of Fractional Derivative-Integral . . . . . . . . . . . . . .
1.3.1 Riemann-Liouville Definition . . . . . . . . . . . . . . . . . . . . .
1.3.2 Caputo Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Solutions of the Fractional State Equation
of Continuous-Time Linear System . . . . . . . . . . . . . . . . . . . . .
1.5 Positivity of the Fractional Systems . . . . . . . . . . . . . . . . . . . . .
1.6 External Positivity of the Fractional Systems . . . . . . . . . . . .
1.7 Positive Continuous-Time Linear Systems with Delays . . . .
1.8 Positive Linear Systems Consisting of n Subsystems
with Different Fractional Orders . . . . . . . . . . . . . . . . . . . . . . . .
1.8.1 Linear Differential Equations with Different
Fractional Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8.2 Positive Fractional Systems with Different
Fractional Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.9 Descriptor Fractional Continuous-Time Linear Systems . . . .
1.9.1 Solution of the Descriptor Fractional Systems . . . . . .

1.9.2 Drazin Inverse Method for the Solution
of Fractional Descriptor Continuous-Time Linear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.10 Definition of n-Order Difference . . . . . . . . . . . . . . . . . . . . . . . .
1.11 State Equations of the Discrete-Time Fractional Linear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.11.1 Fractional Systems without Delays . . . . . . . . . . . . . . .
1.11.2 Fractional Systems with Delays . . . . . . . . . . . . . . . . . .

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6
11
12
13
15
15
20
21
22

24
28
30
30
31


VIII

Contents

1.12 Solution of the State Equations of the Fractional
Discrete-Time Linear System . . . . . . . . . . . . . . . . . . . . . . . . . .
1.12.1 Fractional Systems with Delays . . . . . . . . . . . . . . . . . .
1.12.2 Fractional Systems with Delays in State Vector . . . .
1.12.3 Fractional Systems without Delays . . . . . . . . . . . . . . .
1.13 Positive Fractional Linear Systems . . . . . . . . . . . . . . . . . . . . . .
1.14 Externally Positive Fractional Systems . . . . . . . . . . . . . . . . . .
1.15 Fractional Different Orders Discrete-Time Linear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.16 Positive Fractional Different Orders Discrete-Time Linear
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.17 Descriptor Fractional Discrete-Time Linear Systems . . . . . .
1.17.1 Solution to the State Equation . . . . . . . . . . . . . . . . . . .
2

3

4

Positive Fractional Electrical Circuits . . . . . . . . . . . . . . . . . .
2.1 Fractional Electrical Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Positive Fractional Electrical Circuits . . . . . . . . . . . . . . . . . . .
2.2.1 Fractional R, C, e Type Electrical Circuits . . . . . . . .
2.2.2 Fractional R, L, e Type Electrical Circuits . . . . . . . .
2.2.3 Fractional R, L, C Type Electrical Circuits . . . . . . . .
2.3 Analysis of the Fractional Electrical Circuits
in Transient States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Reciprocity Theorem for Fractional Circuits . . . . . . . . . . . . .

2.5 Equivalent Voltage Source Theorem and Equivalent
Current Source Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Descriptor Linear Electrical Circuits and Their
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Descriptor Linear Electrical Circuits . . . . . . . . . . . . . . . . . . . .
3.1.1 Regularity of Descriptor Electrical Circuits . . . . . . . .
3.1.2 Pointwise Completeness of Descriptor Electrical
Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Pointwise Degeneracy of Descriptor Electrical
Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Descriptor Fractional Linear Electrical Circuits . . . . . . . . . . .
3.3 Polynomial Approach to Fractional Descriptor Electrical
Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Positive Descriptor Fractional Electrical Circuits . . . . . . . . .
3.4.1 Pointwise Completeness and Pointwise Degeneracy
of Positive Fractional Descriptor Electrical
Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31
31
34
37
38
39
41
44
45
45
49
49

52
53
59
64
73
76
78
81
81
84
94
96
97
100
109

114

Stability of Positive Standard Linear Electrical
Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.1 Stability of Positive Electrical Circuits . . . . . . . . . . . . . . . . . . 117
4.2 Positive Unstable R, L, e Electrical Circuits . . . . . . . . . . . . . 118

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IX

4.3 Positive Unstable G, C, is Electrical Circuit . . . . . . . . . . . . .
4.4 Positive Unstable R, L, C, e Type Electrical Circuits . . . . .

123
126

5

Reachability, Observability and Reconstructability
of Fractional Positive Electrical Circuits
and Their Decoupling Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.1 Decomposition of the Pairs (A, B) and (A, C) of Linear
Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.2 Reachability of Positive Electrical Circuits . . . . . . . . . . . . . . . 141
5.3 Observability of Positive Electrical Circuits . . . . . . . . . . . . . . 148
5.4 Constructability of Positive Electrical Circuits . . . . . . . . . . . 151
5.5 Decomposition of the Positive Pair (A, B) . . . . . . . . . . . . . . . 155
5.6 Decomposition of the Positive Pair (A, C) . . . . . . . . . . . . . . . 157
5.7 Decoupling Zeros of the Positive Electrical Circuits . . . . . . . 159
5.8 Reachability of Positive Fractional Electrical Circuits . . . . . 161
5.9 Observability of Positive Fractional Electrical Circuits . . . . . 167

6

Standard and Fractional Linear Circuits
with Feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
6.1 Linear Dependence on Time of State Variable in Standard
Electrical Circuits with State Feedbacks . . . . . . . . . . . . . . . . . 169
6.2 Zeroing of the State Vector of Standard Circuits by
State-Feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.3 Zeroing of the State Vector of Standard Circuits by
Output-Feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.4 Zeroing of the State Vector of Fractional Electrical
Circuits by State-Feedbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7

Minimum Energy Control of Electrical Circuits . . . . . . . . . 197
7.1 Minimum Energy Control of Positive Standard Electrical
Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.2 Minimum Energy Control of Fractional Positive Electrical
Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.3 Minimum Energy Control of Fractional Positive Electrical
Circuits with Bounded Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8

Fractional 2D Linear Systems Described by the
Standard and Descriptor Roesser Model with
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.1 Fractional Derivatives and Integrals of 2D Functions . . . . . . 209
8.2 Descriptor Fractional 2D Roesser Model and Its
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.3 Fractional-Order Model of the Long Transmission Line . . . . 214
8.4 Standard Fractional 2D Roesser Model and Its Solution . . . 217
8.5 Generalization of Cayley-Hamilton Theorem . . . . . . . . . . . . . 221

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X

Contents

A

Laplace Transforms of Continuous-Time Functions
and Z-Transforms of Discrete-Time Functions . . . . . . . . . . 225
A.1 Convolutions of Continuous-Time and Discrete-Time
Functions and Their Transforms . . . . . . . . . . . . . . . . . . . . . . . . 225
A.2 Laplace Transforms of Derivative-Integrals . . . . . . . . . . . . . . . 226
A.3 Laplace Transforms of Two-Dimensional Fractional
Differintegrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
A.4 Z-Transforms of Discrete-Time Functions . . . . . . . . . . . . . . . 231

B

Elementary Operations on Matrices . . . . . . . . . . . . . . . . . . . 235

C

Nilpotent Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

D

Drazin Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
D.1 Definition and Properties of Drazin Inverse Matrix . . . . . . . . 239
D.2 Procedure for Computation of Drazin Inverse Matrices . . . . 240

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

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List of Symbols

Γ (x)
R
R+
C
N
Z
Z+
Rn×m
Rn = Rn×1
Rn×m
+
Rn+ = Rn×1
+
In
Mn
a∈A
detA
rankA
AdjA
degP (s)

the Euler gamma function
the set of real numbers

the set of real nonnegative numbers
the set of complex numbers
the set of natural numbers
the set of integers
the set of nonnegative integers
the set of n × m real matrices
the set of n-components real vectors
the set of n × m real matrices with nonnegative
entries
the set of n-rows real vectors with nonnegative
components
the n × n identity matrix
the set of n × n Metzler matrices
a is an element of the set A
determinant of the matrix A
rank of the matrix A
adjoint matrix
degree of polynomial (polynomial matrix) P (s)

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XII

A−1
AD
AT
ImA
kerA
n!

diag[a1 , . . . , an ]
(x)
(x)
Eα (z)
Eα,β (z)
n
0 It f (t)
α
0 It f (t)
RL α
0 Dt f (t)
C α
0 Dt f (t)
n
0 Δk xi

L [f (t)] = F (s)
Z [fk ] = F (z)
df (t)
f˙(t) =
dt

List of Symbols

inverse matrix
Drazin inverse matrix
transpose matrix
image of the matrix A
kernel of the matrix A
the factorial of natural number n

diagonal matrix with a1 , . . . , an on diagonal
real part of complex number x
imaginary part of complex number x
one parameter Mittag-Leffler function of z
two parameters Mittag-Leffler function of z
n-multiple integral of the function f (t)
on the interval (0, t)
Riemann-Liouville fractional (α-order) integral
Riemmann-Liouville fractional (α-order)
derivative-integral
Caputo fractional (α-order) derivative-integral
n-order backward difference of xi
on the interval [0, k]
Laplace transform of the function f (t)
Z-transform of the function fk
first order derivative
of continuous-time function f (t)

dn f (t)
n-order derivative of continuous-time function f (t)
dtn
f (t) ∗ g(t)
the convolution of functions f (t) and g(t)
δ(t)
Dirac impulse function
1(t)
unit step function

f (n) (t) =

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Chapter 1

Fractional Differential Equations

1.1

Definition of Euler Gamma Function
and Its Properties

There exist the following two definitions of the Euler gamma function.
Definition 1.1. A function given by the integral [51, 154, 163]
∞

tx−1 e−t dt,

Γ (x) =

(x) > 0

(1.1)

0

is called the Euler gamma function.
The Euler gamma function can be also defined by
n!nx
,

n→∞ x(x + 1) · · · (x + n)

Γ (x) = lim

x ∈ C\{0, −1, −2, . . .},

where C is the field of complex numbers.
We shall show that Γ (x) satisfies the equality
(1.2)

Γ (x + 1) = xΓ (x).
Proof. Using (1.1), we obtain
∞

tx e−t dt = −tx e−t

Γ (x + 1) =
0

∞
0

∞

tx−1 e−t dt = xΓ (x).

+x
0

c Springer International Publishing Switzerland 2015

T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits,
Studies in Systems, Decision and Control 13, DOI: 10.1007/978-3-319-11361-6_1

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1


2

1 Fractional Differential Equations

Example 1.1. From (1.2) we have for:
∞

x=1:

Γ (2) = 1 · Γ (1) = 1,

x=2:
x=3:

Γ (3) = 2 · Γ (2) = 1 · 2 = 2!,
Γ (4) = 3 · Γ (3) = 3 · 2 · Γ (2) = 3!.

since

e−t dt = 1,

Γ (1) =

0

In general case, for x ∈ N we have
Γ (n + 1) = nΓ (n) = n(n − 1)Γ (n − 1) = n(n − 1)(n − 2) · · · (1) = n!.
The gamma function is also well-defined for x being any real (complex)
numbers. For example we have for:
x = 1.5 :
x = −0.5 :

1.2

Γ (2.5) = 1.5 · Γ (1.5) = 1.5 · 0.5Γ (0.5),
Γ (0.5) = −0.5 · Γ (−0.5) = −0.5 · (−1.5)Γ (−1.5).

Mittag-Leffler Function

The Mittag-Leffler function is a generalization of the exponential function esi t
and it plays important role in solution of the fractional differential equations.
Definition 1.2. A function of the complex variable z defined by [51, 154, 163]
∞

Eα (z) =
k=0

zk
Γ (kα + 1)

(1.3)

is called the one parameter Mittag-Leffler function.

Example 1.2. For α = 1 we obtain
∞

E1 (z) =
k=0

zk
=
Γ (k + 1)

∞
k=0

zk
= ez ,
k!

i.e. the classical exponential function.
An extension of the one parameter Mittag-Leffler function is the following
two parameters function.
Definition 1.3. A function of the complex variable z defined by [51, 154, 163]
∞

Eα,β (z) =
k=0

zk
Γ (kα + β)

is called the two parameters Mittag-Leffler function.

For β = 1 from (1.4) we obtain (1.3).

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(1.4)


1.3 Definitions of Fractional Derivative-Integral

1.3
1.3.1

3

Definitions of Fractional Derivative-Integral
Riemann-Liouville Definition

It is well known that to reduce N -multiple integral to 1-tiple integral the following formula
uN −1

x u1
N
0 Ix f (u)

···

=
0

=

f (uN )duN duN −1 · · · du1

0

1
(N − 1)!

0
x

(x − u)N −1 f (u)du

(1.5)

0

can be used, where f (u) is a given function.
Using the equality (N − 1)! = Γ (N ), the formula (1.5) can be extended
for any N ∈ R and we obtain Riemann-Liouville fractional integral
α
0 It f (t)

1
=
Γ (α)

t

(t − τ )α−1 f (τ )dτ,

(1.6)

0

where α ∈ R+ \{0} is the order of integral.
Definition 1.4. The function defined by [51, 154, 163]
RL α
0 Dt f (t)

=

dα
dN
f
(t)
=
dtα
dtN

1
dN
=
Γ (N − α) dtN
where N − 1 ≤ α < N , N
derivative-integral.

(N −α)
f (t)
0 It

t

(1.7)
N −α−1

(t − τ )

f (τ )dτ,

0

∈ N is called Riemann-Liouville fractional

Note, that from (1.7), for α = 0 we obtain
RL 0
0 Dt f (t)

1 d
=
Γ (1) dt

t

f (τ )dτ =
0

d0
f (t) = f (t)
dt0

and for α = 1 we have
RL 1
0 Dt f (t)

=

1 d2
Γ (1) dt2

t

f (τ )dτ =
0

d
f (t).
dt

Therefore, by induction, Definition 1.4 is true for α ∈ N.

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4

1 Fractional Differential Equations

Example 1.3. Consider the unit-step function
f (t) = 1(t) =

t ≥ 0,
t < 0.

1 for
0 for

Using (1.7), we obtain
dα
1
dN
1(t)
=
dtα
Γ (N − α) dtN
1
dN
=
Γ (N − α) dtN

t

(t − τ )N −α−1 dτ
0
t

−1
(t − τ )N −α
N −α

=

0

1
1
dN N −α
t
Γ (N − α) N − α dtN

1
1
t−α
=
(N − α)(N − α − 1) · · · (1 − α)t−α =
.
Γ (N − α) N − α
Γ (1 − α)
Therefore, the α order Riemann-Liouville derivative of unit-step function
is a decreasing in time function.
Theorem 1.1. The Riemann-Liouville derivative-integral operator is a linear operator satisfying the relation
RL α
0 Dt [λf (t)

+ μg(t)] = λRL0 Dtα f (t) + μRL0 Dtα g(t),

λ, μ ∈ R.

Proof.
RL α
0 Dt (λf (t)

1
dN
+ μg(t)) =
Γ (N − α) dtN

t

(t − τ )N −α−1 [λf (τ ) + μg(τ )]dτ
0
t

N

λ
d
=
Γ (N − α) dtN

(t − τ )N −α−1 f (τ )dτ
0

μ
dN
+
Γ (N − α) dtN

t

(t − τ )N −α−1 g(τ )dτ
0

=λRL0 Dtα f (t) + μRL0 Dtα g(t).

Theorem 1.2. The Laplace transform of the derivative-integral (1.7) for
N − 1 < α < N has the form
L

RL α
0 Dt f (t)

α

N

= s F (s) −

sk−1 f (α−k) (0+ ),

k=1

where f (α−k) (0+ ) =

RL α−k
f (t) t=0
0 Dt

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(1.8)

1.3 Definitions of Fractional Derivative-Integral

5

Proof. Using (1.6), (1.7) and (A.5), (A.6) (see Appendix A.1) for
N − 1 < α < N we obtain
⎧
⎡
⎤⎫
t
⎨ dN
⎬
1
N −α−1
⎣
⎦
L RL0 Dtα f (t) =L
(t
−
τ
)
f
(τ
)dτ
⎩ dtN Γ (N − α)
⎭
0

=L

=

dN
dtN

N −α
f (t)
0 It

sN F (s)
−
sN −α

=sα F (s) −

N

sN −k

k=1
N

sN −k

k=1

=sα F (s) −

N

sk−1

k=1

1.3.2

dk−1
dtk−1

N −α
f (t) t=0
0 It

RL k−N −1+α
f (t) t=0
0 Dt

RL α−k
f (t) t=0
0 Dt

.

Caputo Definition

Definition 1.5. The function defined by [51, 154, 163]
C α
0 Dt f (t)

1

=
Γ (N − α)

t

0

f (N ) (τ )
dτ,
(t − τ )α+1−N

f (N ) (τ ) =

dN f (τ )
dτ N

(1.9)

is called the Caputo fractional derivative-integral, where N − 1 ≤ α < N ,
N ∈ N.
Remark 1.1. From Definition 1.5 it follows that the Caputo derivative
of constant is equal to zero.
Theorem 1.3. The Caputo derivative-integral operator is linear satisfying
the relation
C α
0 Dt [λf (t)

α
C α
+ μg(t)] = λC

0 Dt f (t) + μ0 Dt g(t).

Proof. The proof is similar to the proof of Theorem 1.1.
Theorem 1.4. The Laplace transform of the derivative-integral (1.9) for
N − 1 < α < N has the form
L

C α
0 Dt f (t)

= sα F (s) −

N

sα−k f (k−1) (0+ ).

(1.10)

k=1

Proof. Using Definitions 1.5 and
N − 1 < α < N we obtain

A.2, equations

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(A.3),

(A.5) for

6

1 Fractional Differential Equations

⎡
1
α
⎣
L C
0 Dt f (t) = L
Γ (N − α)

⎤

t

(t − τ )N −α−1 f (N ) (τ )dτ ⎦
0

1
=
L tN −α−1 L f (N ) (t)
Γ (N − α)
=

1
Γ (N − α) N
s F (s) −

Γ (N − α) sN −α

= sα F (s) −

N

N

sN −k f (k−1) (0+ )

k=1

sα−k f (k−1) (0+ ).

k=1

1.4

Solutions of the Fractional State Equation
of Continuous-Time Linear System

Consider the continuous-time linear system described by the equations [52]
C α
0 Dt x(t)

=

dα x(t)
= Ax(t) + Bu(t),
dtα

y(t) = Cx(t) + Du(t),

0 < α < 1,

(1.11a)
(1.11b)

where x(t) ∈ Rn , u(t) ∈ Rm , y(t) ∈ Rp are state, input and output vectors
and A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n , D ∈ Rp×m .
Theorem 1.5. The solution of the equation (1.11a) has the form
t

Φ(t − τ )Bu(τ )dτ,

x(t) = Φ0 (t)x0 +

x(0) = x0 ∈ Rn ,

(1.12)

0

where
Φ0 (t) = Eα (Atα ) =
∞

Φ(t) =
k=0

∞

k=0

Ak tkα
,
Γ (kα + 1)

Ak t(k+1)α−1
.
Γ [(k + 1)α]

(1.13)
(1.14)

and Eα (Atα ) is the Mittag-Leffler function and Γ (x) is the Euler gamma
function.
Proof. Applying the Laplace transform to (1.11a) and taking into account
that

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1.4 Solutions of the Fractional State Equation of Continuous-Time System

7

∞

x(t)e−st dt,

X(s) = L[x(t)] =

L
we obtain

C α
0 D x(t)

U (s) = L[u(t)],

0
α

= s X(s) − sα−1 x0

for 0 < α < 1

X(s) = [In sα − A]−1 sα−1 x0 + BU (s) .

It is easy to show that
[In sα − A]−1 =

∞

(1.15)

Ak s−(k+1)α ,

(1.16)

k=0

since
[In sα − A]

∞

Ak s−(k+1)α

= In .

k=0

Substituting of (1.16) to (1.15), we obtain
∞

Ak s−(kα+1) x0 +

X(s) =
k=0

∞

Ak s−(k+1)α BU (s).

(1.17)

k=0

Using the inverse Laplace transform and the convolution theorem (Appendix A.1) to (1.17) we obtain
x(t) =L−1 [X(s)] =

∞

∞

Ak L−1 s−(kα+1) x0 +

k=0

Ak L−1 s−(k+1)α BU (s)

k=0

t

Φ(t − τ )Bu(τ )dτ,

=Φ0 (t)x0 +
0

where
∞

Φ0 (t) =

Ak L−1 s−(kα+1) =

k=0

Φ(t) = L−1 [In sα − A]−1 =

∞

k=0
∞

Ak tkα
,
Γ (kα + 1)

Ak L−1 s−(k+1)α =

k=0

∞
k=0

Ak t(k+1)α−1
.
Γ [(k + 1)α]

Theorem 1.6. The matrix Φ0 (t) defined by (1.13) satisfies the equality
dα Φ0 (t)
= AΦ0 (t).
dtα
Proof. By Theorem 1.5 for u(t) = 0, t ≥ 0, the matrix Φ0 (t) satisfies the
equation (1.11a).

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8

1 Fractional Differential Equations

Remark 1.2. From (1.13) and (1.14) for α = 1 we have
∞

Φ0 (t) = Φ(t) =
k=0

(At)k
= eAt .
Γ (k + 1)

Remark 1.3. From classical Cayley-Hamilton theorem [39, 44, 115] it follows
that if
det [In sα − A] = (sα )n + an−1 (sα )n−1 + · · · + a1 sα + a0 ,
then

An + an−1 An−1 + · · · + a1 A + a0 In = 0.

Example 1.4. Find the solution of the equation (1.11a) for 0 < α < 1 and
A=

01
,
00

B=

0
,
1

x0 =

1
,
1

u(t) = 1(t).

Using (1.13) and (1.14), we obtain
∞

Φ0 (t) =
k=0
∞

Φ(t) =
k=0

Ak tkα
Atα
= I2 +
,
Γ (kα + 1)
Γ (α + 1)
Ak t(k+1)α−1
tα−1

t2α−1
= I2
+A
.
Γ [(k + 1)α]
Γ (α)
Γ (2α)

(1.18)

Substituting (1.18) and u(t) = 1(t) into (1.12), we obtain
Ax0 tα
x(t) =x0 +
+
Γ (α + 1)

t

B
AB
(t − τ )α−1 +
(t − τ )2α−1 dτ
Γ (α)
Γ (2α)

0

α

Ax0 t

Btα
ABt2α
=x0 +
+
+
Γ (α + 1) Γ (α + 1) Γ (2α + 1)
⎡
⎤
tα
t2α
1
+
+
⎢
Γ (α + 1) Γ (2α + 1) ⎥
⎥,
=⎢
⎣
⎦
tα
1+
Γ (α + 1)
since Γ (α + 1) = αΓ (α).
Theorem 1.7. If the Caputo Definition 1.5 is used, then the solution
of the equation (1.11a) for N − 1 < α < N has the form
N

x(t) =

t

(l−1)

Φl (t)x
l=1

+

Φ(t − τ )Bu(τ )dτ,

(0 ) +
0

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(1.19)


1.4 Solutions of the Fractional State Equation of Continuous-Time System

where

∞

∞

Ak tkα+l−1
,
Γ (kα + l)

Φl (t) =

k=0

9

Ak t(k+1)α−1
.
Γ [(k + 1)α]

Φ(t) =
k=0

Proof. Taking into account (A.1), (1.10) we obtain the Laplace transform of
(1.11a)
X(s) = [In sα − A]

N

−1

sα−l x(l−1) (0+ ) + BU (s) ,

U (s) = L[u(t)].

l=1

(1.20)
Substitution of (1.16) into (1.20) yields
∞

X(s) =

Ak s−(k+1)α

k=0
∞

N

sα−l x(l−1) (0+ ) + BU (s)

l=1
N

=

Ak s−(kα+l) x(l−1) (0+ ) +

k=0 l=1

(1.21)

∞

Ak s−(k+1)α BU (s).

k=0

Applying the inverse Laplace transform and the convolution theorem (Appendix A.1) to (1.21), we obtain
∞

N

Ak L−1 s−(kα+l) x(l−1) (0+ ) +

x(t) =
k=0 l=1
n

=

∞

Ak L−1 s−(k+1)α BU (s)

k=0
t

Φl (t)x(l−1) (0+ ) +

l=1

Φ(t − τ )Bu(τ )dτ,
0

where
∞

∞

Ak L−1 s−(kα+l) =

Φl (t) =
k=0
∞

Φ(t) =

Ak L−1 s−(k+1)α =

k=0

k=0
∞
k=0

Ak tkα+l−1
,
Γ (kα + l)
Ak t(k+1)α−1
.
Γ [(k + 1)α]

Theorem 1.8. If the Riemann-Liouville Definition 1.4 is used, then the solution of the equation (1.11a) for N − 1 ≤ α ≤ N has the form
N

x(t) =

t
(α−l)

Φl (t)x
l=1

+

Φ(t − τ )Bu(τ )dτ,

(0 ) +
0

where

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(1.22)


10

1 Fractional Differential Equations
∞

Φl (t) =
k=0

∞

Ak t(k+1)α−l
,
Γ [(k + l)α − l + 1]

Φ(t) =
k=0

Ak t(k+1)α−1
.
Γ [(k + 1)α]

Proof. Taking into account (A.1) and (1.8), from (1.11a) we obtain
N

X(s) = [In sα − A]−1

sl−1 x(α−l) (0+ ) + BU (s) ,

U (s) = L[u(t)].

l=1

(1.23)
Substitution of (1.16) to (1.23) yields
∞

X(s) =

N

Ak s−(k+1)α

k=0

∞

sl−1 x(α−l) (0+ ) + BU (s)

l=1
N

k −(k+1)α+l−1 (α−l)

=

A s

x

(1.24)

∞

+

(0 ) +

k=0 l=1

k −(k+1)α

A s

BU (s).

k=0

Applying the inverse Laplace transform and the convolution theorem (Appendix A.1) to (1.24), we obtain
∞

N

Ak L−1 s−(k+1)α+l−1 x(α−l) (0+ )

x(t) =
k=0 l=1
∞

Ak L−1 s−(k+1)α BU (s)

+
k=0
N

=

t
(α−l)

Φl (t)x

+

Φ(t − τ )Bu(τ )dτ,

(0 ) +

l=1

0

where
∞

Φl (t) =
k=0
∞

Φ(t) =
k=0

Ak L−1 s−(k+1)α+l−1 =
Ak L−1 s−(k+1)α =

∞
k=0

∞
k=0

Ak t(k+1)α−l
,
Γ [(k + l)α − l + 1]

Ak t(k+1)α−1
.
Γ [(k + 1)α]

Remark 1.4. From comparison of (1.19) and (1.22) it follows that the component of the solution corresponding to u(t) is the same.

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1.5 Positivity of the Fractional Systems

1.5

11

Positivity of the Fractional Systems

Definition 1.6. [47, 71] The fractional system (1.11) is called (internally)
positive if the state vector x(t) ∈ Rn+ and the output vector y(t) ∈ Rp+
for t ≥ 0 for all initial conditions x0 ∈ Rn+ and all inputs u(t) ∈ Rm
+ , t ≥ 0.
Definition 1.7. [47, 71] A real square matrix A = [aij ] is called Metzler
matrix if its off diagonal entries are nonnegative, i.e. aij ≥ 0 for i = j.
Lemma 1.1. Let A ∈ Rn×n and 0 < α ≤ 1. Then
∞

Φ0 (t) =
k=0
∞

Φ(t) =
k=0

Ak tkα
∈ Rn×n
+
Γ (kα + 1)

for t ≥ 0,

Ak t(k+1)α−1
∈ Rn×n
+
Γ [(k + 1)α]

for t ≥ 0

(1.25)
(1.26)

if and only if A is a Metzler matrix.
Proof. Necessity. From
Atα
+ ··· ,
Γ (α + 1)
tα−1
t2α−1
Φ(t) =In
+A
+ ···

Γ (α)
Γ (2α)

Φ0 (t) =In +

it follows that Φ0 (t) ∈ Rn×n
and Φ(t) ∈ Rn×n
for small value t > 0 only if A
+
+
is a Metzler matrix.
Sufficiency. It is well-known [47] that
eAt ∈ Rn×n
+

for t ≥ 0

(1.27)

if and only if A is a Metzler matrix.
Using (1.25), we can write
∞

(Atα )k
(Atα )k
−
Γ (kα + 1)
k!

∞

k! − Γ (kα + 1) (Atα )k
·
Γ (kα + 1)
k!
k=0
k=0
(1.28)
α
for t ≥ 0. From (1.27) and (1.28) we have Φ0 (t) ≥ eAt ≥ 0 for t ≥ 0, since
k! ≥ Γ (kα + 1) for 0 < α ≤ 1.
The proof for (1.26) is similar.
α

Φ0 (t) − eAt =

=

Theorem 1.9. The fractional continuous-time system (1.11) is (internally)
positive if and only if
A ∈ Mn ,

B ∈ Rn×m
,
+

C ∈ Rp×n
+ ,

D ∈ Rp×m

.
+

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(1.29)


12

1 Fractional Differential Equations

Proof. Sufficiency. By Theorem 1.5 the solution (1.11a) has the form (1.12)
and x(t) ∈ Rn+ , t ≥ 0, if the condition (1.29) is satisfied, since Φ0 ∈ Rn×n
+ ,
x0 ∈ Rn+ and u(t) ∈ Rm
for
t
≥
0.
+
Necessity. Let u(t) = 0, t ≥ 0 and x0 = ei (i-th column of the identity matrix
In ). The trajectory does not leave the orthant Rn+ only if the derivative of order α, xα (0) = Aei ≥ 0, what implies aij ≥ 0 for i = j. The matrix A is a Metzler matrix. From the same reason for x0 = 0 we have xα (0) = Bu(0) ≥ 0,
what implies B ∈ Rn×m
, since u(0) ∈ Rm
+
+ can be arbitrary. From (1.11b)
n
for u(t) = 0, t ≥ 0 we have y(0) = Cx0 ≥ 0 and C ∈ Rp×n
+ , since x0 ∈ R+

can be arbitrary. In a similar way assuming x0 = 0, we obtain y(0) = Du(0) ≥
0 and D ∈ Rp×m
, since u(0) ∈ Rm
+
+ is arbitrary.

1.6

External Positivity of the Fractional Systems

Definition 1.8. [71] The fractional system (1.11) is called externally positive
if for all u(t) ∈ Rm
+ , t ≥ 0 and zero initial conditions x0 = 0 the output vector
y(t) ∈ Rp+ , t ≥ 0.
Definition 1.9. [71] Output of the fractional SISO (single-input single-output)
system with zero initial conditions for Dirac impulse u(t) = δ(t) is called the impulse response of the system. In a similar way we define the matrix of impulse response of the MIMO (multiple-input multiple-output) fractional system (1.11).
Lemma 1.2. Matrix of the impulse responses g(t) of the fractional system
(1.11) is given by
g(t) = CΦ(t)B + Dδ(t), t ≥ 0,
(1.30)
where δ(t) is the Dirac delta defined by
δ(t) =

∞ for t = 0,
0 for t = 0,

which satisfies the condition
∞

δ(t)dt = 1.

−∞

Proof. Substituting (1.12) into (1.11b) and taking into account x0 = 0,
u(t) = δ(t), y(t) = g(t) we obtain
t

CΦ(t − τ )Bδ(τ )dτ + Dδ(t) = CΦ(t)B + Dδ(t).

g(t) =
0

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