Sabatier Agrawal Machado Advances in Fractional Calculus Episode 3 pps

Axtell M, Bise EM 1990 Fractional calculus applications in control systems.. 1 where {akx}n−1 k=0 are continuous real functions defined in [a, b] ⊂ R and Dα a+= Dα a+ Dkα a+= Dα α-Keyword

PSEUDO-RATIONAL APPROXIMATIONS TO FO-LTI SYSTEMS 69

controller for the optimally reduced model Gr(s) and let us see if the designedcontroller still works for the original system

The integer order PID controller to be designed is in the following form:

The optimum ITAE criterion-based PID tuning formula [43] can be used

Based on this tuning algorithm, a PID controller can be designed for Gr(s)

tion is simple and effective It is also demonstrated

approximation method is effective in designing integer order controllers forFO-LTI systems in general form

Finally, we would like to remark that the so-called pseudo-rational proximation is essentially by cascading irrational transfer function (a timedelay) and a rational transfer function Since a delay element is also infinitesystem involving time delay Although it might not fully make physical sense,the pseudo-rational approximation proposed in this paper will find its prac-systems, as illustrated in Example 3

Xue and Chen

Finally, the step response of the original FO-LTI with the above -designed

used for integer-order controller design for general FO-LTI systems

Fig 4.Step response of fractional-order plant model under the PID controller

to arbitrary FO-LTI

pseudo-rational that this suboptimum

approxima-dimensional, it makes sense to approximate a general fractional-order LTI

tical applications in designing an integer-order controller for fractional-order

systems with suboptimummation

In this paper, we presented a procedure to achieve pseudo-rational

approxi-PSEUDO-RATIONAL APPROXIMATIONS TO FO-LTI SYSTEMS 71

We acknowledge that this paper is a modified version of a paper published inthe Proceedings of IDETC/CIE 2005 (Paper# DETC2005-84743) We wouldlike to thank the ASME for granting us permission in written form to pub-lish a modified version of IDETC/CIE 2005 (Paper# DETC2005-84743) as

Professors Machado, Sabatier, and Agrawal (Springer)

fractional model reduction

a chapter in the book entitled

if ierr==1, y=10*ff0; else, ff0=y; end

• get2h.m internal function to evaluate the H2norm of a rational transferfunction model

function [v,ierr]=geth2(G)

G=tf(G); num=G.num{1}; den=G.den{1}; ierr=0; n=length(den);

if abs(num(1))>eps

disp(’System not strictly proper’); ierr=1; return

else, a1=den; b1=num(2:end); end

1 Oldham KB, Spanier J (1974) The Fractional Calculus Academic Press, New York

2 Samko SG, Kilbas AA, Marichev OI (1987) Fractional integrals and derivatives and

some of their applications Nauka i technika, Minsk, Russia

3 Miller KS, Ross B (1993) An Introduction to the Fractional Calculus and Fractional

Differential Equations Wiley, New York

PSEUDO-RATIONAL APPROXIMATIONS TO FO-LTI SYSTEMS 73

4 Podlubny I (1994) Fractional-order systems and fractional-order controllers Tech Rep UEF-03-94, Slovak Academy of Sciences Institute of Experimental Physics, Department of Control Engineering Faculty of Mining, University of Technology Kosice, Slovakia, November

5 Lurie BJ (1994) ‘Three-parameter tunable tilt-integral-derivative (TID) controller’

12 Oustaloup A (1981) Fractional-order sinusoidal oscilators: optimization and their use

in highly linear FM modulators IEEE Trans Circ Syst 28(10):1007–1009

13 Axtell M, Bise EM (1990) Fractional calculus applications in control systems In: Proceedings of the IEEE 1990 National Aerospace and Electronics Conference, pp 563–566

14 Vinagre BM, Chen Y (2002) Lecture notes on fractional calculus applications in automatic control and robotics In the 41st IEEE CDC2002 Tutorial Workshop # 2, B

M Vinagre and Y Chen, eds., pp 1–310 [Online] http://mechatronics.ece.usu.edu /foc/cdc02_tw2_1n.pdf

15 JATMG (2002) Special issue on fractional calculus and applications Nonlinear Dynamics, 29(March):1–385

16 Ortigueira MD, JATMG (eds) (2003) Special issue on fractional signal processing and applications Signal Processing, 83(11) November: 2285–2480

17 Oustaloup A (1995) La Dérivation non Entière HERMES, Paris

18 Oustaloup A, Sabatier J, Lanusse P (1999) From fractal robustness to CRONE control Fract Calc Appl Anal 2(1):1–30

19

20 Petrás I (1999) The fractional-order controllers: methods for their synthesis and application J Electr Eng 50(9–10):284–288

21 Oustaloup A, Levron F, Nanot F, Mathieu B (2000) Frequency band complex non

Fundamental Theory and Applications, 47(1) January: 25–40

22 Ichise M, Nagayanagi Y, Kojima T (1971) An analog simulation of non-integer order transfer functions for analysis of electrode processes J electroanal chem 33:253–

265

Oustaloup A, Mathieu B (1999) La Commande CRONE: du Scalaire au Multi-

variable HERMES, Paris

integer differentiator: characterization and synthesis IEEE Trans Circ Syst I:

74 Xue and Chen

23 Oldham KB (1973) Semiintegral electroanalysis: analog implementation Anal Chem 45(1):39–47

24 Sugi M, Hirano Y, Miura YF, Saito K (1999) Simulation of fractal immittance by analog circuits: an approach to optimized circuits IEICE Trans Fundam (8):1627–

1635

25 Podlubny I, Petráš I, Vinagre BM, O’leary P, Dorcak L (2002) Analogue Realizations

of Fractional-Order Controllers Nonlinear Dynamics, 29(1-4) July: 281–296

26 Machado JAT (1997) Analysis and design of fractional-order digital control systems

J Syst Anal.-Model.-Simul 27:107–122

27 Vinagre BM, Petráš I, Merchan P, Dorcak L (2001) Two digital realisation of fractional controllers: application to temperature control of a solid In: Proceedings of the European Control Conference (ECC2001), pp 1764–1767

28 Vinagre BM, Podlubny I, Hernandez A, Feliu V (2000) On realization of order controllers In: Proceedings of the Conference Internationale Francophone d’Automatique

fractional-29 Vinagre BM, Podlubny I, Hernandez A, Feliu V (2000) Some approximations of fractional-order operators used in control theory and applications Fract Calc Appl Anal 3(3):231–248

30 Chen Y, Moore KL (2002) Discretization schemes for fractional-order differentiators

and integrators IEEE Trans Circ Syst I: Fundamental Theory and Applications,

34 Al-Alaoui MA (1995) A class of second-order integrators and low-pass differentiators

IEEE Trans Cir Syst I: Fundamental Theory and Applications, 42(4):220–223

35 Al-Alaoui MA (1997) Filling the gap between the bilinear and the back-ward difference transforms: an interactive design approach Int J Electr Eng Educ 34(4):331–337

36 Tseng C, Pei S, Hsia S (2000) Computation of fractional derivatives using Fourier transform and digital FIR differentiator Signal Processing, 80:151–159

37 Tseng C (2001) Design of fractional order digital FIR differentiator IEEE Signal Proc Lett 8(3):77–79

38 Chen Y, Vinagre BM, Podlubny I (2003) A new discretization method for fractional order differentiators via continued fraction expansion In Proceedings of The First Symposium on Fractional Derivatives and Their Applications at The 19th Biennial Conference on Mechanical Vibration and Noise, the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (ASME DETC2003), pp 1–8, DETC2003/VIB

39 Oustaloup A, Melchoir P, Lanusse P, Cois C, Dancla F (2000) The CRONE toolbox for Matlab In: Proceedings of the 11th IEEE International Symposium on Computer Aided Control System Design - CACSD

40 Xue D, Atherton DP (1994) A suboptimal reduction algorithm for linear systems with

a time delay Int J Control, 60(2):181–196

PSEUDO-RATIONAL APPROXIMATIONS TO FO-LTI SYSTEMS 75

41 Åström KJ (1970) Introduction to Stochastic Control Theory Academic Press,

London

42 Press WH, Flannery BP, Teukolsky SA (1986) Numerical Recipes, the Art of Scientific

Computing Cambridge University Press, Cambridge

43 Wang FS, Juang WS, Chan CT (1995) Optimal tuning of PID controllers for single and cascade control loops Chem Eng Comm 132:15–34

LINEAR DIFFERENTIAL EQUATIONS

OF FRACTIONAL ORDER

Blanca Bonilla1, Margarita Rivero2, and Juan J Trujillo1

1 Departamento de An´alisis Matem´atico, Universidad de la Laguna

2 Departamento de Matem´atica Fundamental, Universidad de la Laguna

ak(x)Dkαa+

(y) = y(nα+

n−1

k=0

ak(x)y(kα= f (x) (1)

where {ak(x)}n−1

k=0 are continuous real functions defined in [a, b] ⊂ R and

Dα a+= Dα a+

Dkα a+= Dα

α-Keywords

1 Introduction

Questions as to what we mean by, and where we could apply, the fractionalcalculus operators have fascinated us all ever since 1695 when the so-calledfractional calculus was conceptualised in connection with the infinitesimal

38271 La Laguna-Tenerife, Spain; E-mail: BBonilla@ull.es;JTrujill@ull.es

Spain; MRivero@ull.es

ferential equations, involving the well known Riemann Liouville fractional operators,−

We then introduce the Mittag-Leffler-type function e

Fractional differential equations, Caputo, Riemann Liouville, linear.−

© 2007 Springer

77

in Physics and Engineering, 77– 91

J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications

E-mail:

ex-of pollution in the atmosphere, and many more.

In most of the above-mentioned cases, this kind of anomalous process has

a complex macroscopic behaviour, the dynamics of which cannot be terised by classical derivative models Nevertheless, a heuristic solution to thecorresponding models of some of those processes can be frequently obtainedusing tools from statistical physics For such an explanation, one must usesome generalised concepts from classical physics such as fractional Brownianmotion, the continuous time random walk (CTRW) method involving L´evystable distributions (instead of Gaussian distributions), the generalised cen-tral limit theorem (instead of the classical central limit theorem), and non-Markovian distributions which means non-local distributions (instead of theclassical Markovian ones) From this approach it is also important to note thatthe anomalous behaviour of many complex processes includes multi-scaling inthe time and space variables

charac-The above-mentioned tools have been used extensively during last 30 years.But the connection between these statistical models and certain fractionaldifferential equations involving the fractional integral and derivative operatorsbeen formally established during the last 15 years; (see, for instance, [10], [9][19], [14])

We could ask what are the useful properties of these processes? From the point of view of the authors and from known experi-mental results, most of the processes associated with complex systems have

frac-Perhaps this is one of the reasons why these fractional calculus operators losethe above-mentioned useful properties of the ordinary derivative D

This manuscript is organised as follows Sections 2 and 3 presents somefractional operators and their main properties and introduce some types of

new direct method for solving the homogeneous and non-homogeneous casewith constant coefficients, using the α-exponential function and certain frac-tional Green functions, including some illustrative examples

Bonilla, Rivero, and Trujillo

(Riemann Liouville, Caputo, Liouville or Weyl, Riesz, etc.; see [17]) has only−

ourselves,tional calculus operators, which help in the modelling of so many anomalous

non-local dynamics involving long memory in time, and the fractional gral and fractional-derivative operators do have some of those characteristics

inte-Mittag-Leffler functions In section 4 we develop a general theory for tial linear fractional differential equations, while in section 5 we introduce a

sequen-LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 79

used for the case where the fractional derivative involved in the fractionaldifferential equation is Dα

0+, and never when we want to use the more generalfractional derivative Dα

a+(a < 0), as must be done when the initial conditions

of the corresponding model are given in the origin On the other hand, it isclear that those mentioned integral transforms are not utile when the probleminvolve a distributional delta function as a initial condition

2 Fractional operators

derivatives See [17] and [1]

Let α ∈ R (α > 0), m − 1 < α ≤ m, m ∈ N, [a, b] ⊂ R and f be a

dx is the ordinary derivative

Let us remember that, in general, when α, β ∈ R+, the operators Daα+β+

and Dα

a +Daβ+ are different Also, as usual, we will use AC([a, b]) to refer tothe set of absolutely continuous functions in [a, b], and ACn([a, b]) (n ∈ N),for the set of functions f , such that there exist (Dn)(f ) = f(n in [a, b] and

almost everywhere in [a, b]

The following Property holds from the rule for the parametric derivationunder the integral sign (see [14])

Property 2 Let 0 < η ≤ 1,Dηa+K

∈ L1(a, b) with a suitable f (for example,

f ∈ C([a, b])) Then we have

We must point out that the Laplace or Fourier transform can only be

We will consider here the so-called sequential Riemann Liouville and Caputo−

−

and the corresponding Riemann Liouville fractional derivative by−

measurable function, that is f ∈ L (a, b) Then the Riemann Liouville integral

In 1993 Miller Ross [11] introduced sequential

Dα in the following way

Dα= Dα, (0 < α ≤ 1)

Dkα= DαD(k−1)α, (k = 2, 3, ), (8)where Dα is a fractional derivative

A sequential fractional differential equation of order nα has the followingrelationship

F (x, y(x), (Dαy)(x), (D2αy)(x), , (Dnαy)(x)) = g(x) (9)Let Dα= Dα

a+be the Riemann Liouville fractional derivative Then, ing into account Property 1, we can obtain the relation between Dnα

Γ (α)



On the other hand, if α = n

p (n, p ∈ N) and y(x) is a continuous realfunction defined in [a, b], that is y ∈ C([a, b]), we can deduce from Property 1the important property:

(Dny)(t) = (Da+pαy)(t), (t > a) (11)

In this paper we study the linear sequential fractional differential equations

of order nα which can be written in the following normalised form

n−1

k=0

ak(x)y(kα= f (x), (12)

where {ak(x)}n−1k=0 are continuous real functions defined in an interval [a, b] ⊂

Rand f (x) ∈ C([a, b]) or f(x) ∈ C((a, b])

different kinds of functional spaces We present below two of the theoremswhich will be used in this paper

Bonilla, Rivero, and Trujillo

−

The existence and uniqueness of solutions to the Cauchy-type problemfor fractional differential Eq (12) was established in [4], [6], and [7] for

LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 81Theorem 1 Let x0∈ (a, b) ⊂ R and {yk}n−1k=0∈ Rn Let f (x) and {ak(x)}n−1k=0

be continuous real functions in [a, b] Then there exists a unique continuous



Dkα

a+y(x0) = y(kα(x0) = yk (k = 0, 1, , n − 1), (14)Moreover, this solution y(x) satisfies

In particular C0([a, b]) = C([a, b])

Theorem 2 Let {ak(x)}n−1k=0 be continuous functions in [a, b], f ∈ C1−α([a, b])and {bk}n−1k=0 ∈ Rn Then there exists a unique continuous function y(x) de-fined in (a, b] which is a solution to the linear sequential fractional differentialequation of order nα

For the particular case f (x) = 0 we have the following

Corollary 1 Let x0∈ (a, b], (or x0= a) Let {ak(x)}n−1k=0 be continuous realfunctions defined in (a, b] and such that (x − a)1−αak(x)|x=a < ∞, ∀k =

1, 2, , n The homogeneous linear sequential fractional differential equation

has y(x) = 0 as the unique solution in (a, b], satisfying the initial conditions

y(jα(x0) = 0 (o [(x − a)1−αy(kα(x)]x=a+= 0) (k = 0, 1, , n − 1)).function y(x) defined in (a, b], which is a solution to the Cauchy-type problem

82

3 α-Exponential functions

In this section we introduce two special functions of the Mittag-Leffler type,

Definition 1 Let λ, ν ∈ C, α ∈ R+ and a ∈ R We will call α-exponentialfunction eλ(x−a)α

Proposition 1 Under the restrictions of definition 1, it is easy to prove thefollowing properties

i)

Dαa+eλ(x−a)α = λeλ(x−a)α (23)ii)

eλ(x−a)α = (x − a)α−1Eα,α(λ(x − a)α) (24)where Eβ,η(x − a) is the Mittag-Leffler function

where L denotes the Laplace transform

Definition 2 Let α ∈ R+, l ∈ N0, a ∈ R and λ = b + ic ∈ C We will call

i)

∂l

∂λleλ(x−a)α = (x − a)lαEα,lλ(x−a) (27)ii)

Eα,lλx= l!xα−1Eα,(l+1)αl (λxα) (28)iii)

LxαlEα,lλx

(sα− λ)l+1 (|s|α< |λ|) (29)

Bonilla, Rivero, and Trujillo

the Mittag-Leffler-type function

the Mittag-Leffler-type function

which will be used in the next sections See, for instance, [16], [12], [2], [5],and [3]

LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 83

In this section we study the solutions to a homogeneous linear sequential

(y) = y(nα+

is the sequential Riemann Liouville fractional derivative

Definition 3 As usual, a fundamental set of solutions to equation (30) insome interval V ⊂ [a, b] is a set of n functions linearly independent in V ,which are solutions to (30)

Definition 4 The α-Wronskian of the n functions {uk(x)}n

1, which admititerated fractional derivatives up to order (n −1)α in some interval V ⊂ (a, b],refers to the following determinant

To simplify the notation, this will be represented by |Wα(x)|

= |Wα(u1, , un)(x)| We will use Wα(x) for the corresponding Wronskianmatrix

Theorem 3 Let {uk(x)}n

k=1 be a family of functions with sequential tional derivatives up to order (n − 1)α in (a, b] and such that, if j = 1, 2, , nand k = 0, 1, , n − 1

frac-lim

x→a+[(x − a)1−αu(kαj (x)] < ∞ (32)

If the functions {(x−a)1−αuj(x)}n

j=1are linearly dependent in [a, b], it followsthat for all x ∈ [a, b]

in (a, b] which satisfies

4 General Theory for Linear Fractional Differential Equations

−

be a solution family of functions to Eq (30)fractional-differential equation

V ⊂ [a, b].

Usually, the general solution to a non-homogeneous linear sequential tional differential equation

will be given as in the following proposition:

Proposition 3 If yp(x) is a particular solution to (35) and yh(x) is a generalsolution to the corresponding homogeneous equation

Bonilla, Rivero, and Trujillo

dinary case, a fundamental set of solutions for Eq (30) in some interval

(36), then a general solution to the non-homogeneous Eq (35) is

−

LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 85

In this section we present a direct method for obtaining the explicit generalsolution to a linear sequential fractional differential equation with constantcoefficients, such as

method At the end, we will introduce a fractional Green function to obtain

As in the ordinary case, if we try to find solutions to (42) of the type y(x) =

eλ(x−a)α , it follows that

In the following it will be assumed that λ ∈ C

By the use of the properties of the α-exponential function, we obtain thefollowing result

Lemma 1 If λ is a root of characteristic polynomial (44), then



(45)and

Cauchy-an explicit particular solution to the non-homogeneous Eq (41)

Let us consider now the corresponding homogeneous Eq to (41)

is referred to as the characteristic polynomial associated with Eq (42)

with Constant Coefficients

of multiplicity {σj}pj=1, respectively, of (44) Then the union set of the sets

k

&

m=1

'(x − a)lαEλm (x−a)

Bonilla, Rivero, and Trujillo

Eq (42), then, in accordance with Proposition 4, to obtain the explicit

LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 87First of all we will obtain the general solution to the simpler equation



Da+α yp

(x) = Dαa+

 x a

eλ(x−ξ)

α f (ξ)dξ + f (x) = λyp+ f (x),which concludes the proof

f (x) ∈ C1−α([a, b]) Moreover

I1−αGα

(a+) = 0

Proposition 4 Let f ∈ L (a, b) ∩ C[(a, b)] Then Eq (53) admits

Theorem 6 A particular solution to Eq (41) is given by

instead of the α-exponential function eλ(x−a)α

Example 2 Let us consider the equation

sin∗α[λ(x − a)] = Im{Eα(λ(x − a)α)} (63)

We point out here that the sin∗

α(x) and cos∗

α(x) functions are a new alisation of the usual cos(x) and sin(x) functions, which, like the sinα(x) andcosα(x) functions, could play a fundamental role, for instance, in the develop-which are solutions to elementary fractional differential equations

gener-Liouville non-sequential linear fractional differential equations It is easy toprove the following:

Corollary 2 Let f ∈ C1−α([a, b]) and a0, a1∈ R Then equation

D2α a+y + a1Dα

a+y + a0y = f (x) (0 < α ≤ 1) (64)has the general solution

y(x) = C1z1(x) + C2z2(x) + zp(x) − C

Γ (α)(x − a)α−1, (65)

Bonilla, Rivero, and Trujillo

ated with non-homogeneous Eq (41), analogous to the usual case, this

−

−

ment of a fractional Fourier theory, or of Weierstrass-type fractal functions,

In addition, the results previously presented may be applied to Riemann−