review article stability of fractional order...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 356215 14 pageshttpdxdoiorg1011552013356215

Review ArticleStability of Fractional Order Systems

Margarita Rivero1 Sergei V Rogosin2 Joseacute A Tenreiro Machado3 and Juan J Trujillo4

1 Departamento de Matematica Fundamental University of La Laguna Tenerife 38271 La Laguna Spain2Department of Economics Belarusian State University Nezavisimosti Avenue 4 220030 Minsk Belarus3 Department of Electrical Engineering Institute of Engineering Polytechnic of Porto 4200-072 Porto Portugal4Departamento de Analisis Matematico University of La Laguna Tenerife 38271 La Laguna Spain

Correspondence should be addressed to Jose A Tenreiro Machado jtmisepipppt

Received 26 December 2012 Revised 8 April 2013 Accepted 11 April 2013

Academic Editor Clara Ionescu

Copyright copy 2013 Margarita Rivero et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The theory and applications of fractional calculus (FC) had a considerable progress during the last years Dynamical systems andcontrol are one of the most active areas and several authors focused on the stability of fractional order systems Nevertheless dueto the multitude of efforts in a short period of time contributions are scattered along the literature and it becomes difficult forresearchers to have a complete and systematic picture of the present day knowledge This paper is an attempt to overcome thissituation by reviewing the state of the art and putting this topic in a systematic form While the problem is formulated with rigourfrom the mathematical point of view the exposition intends to be easy to read by the applied researchers Different types of systemsare considered namely linearnonlinear positive with delay distributed and continuousdiscrete Several possible routes of futureprogress that emerge are also tackled

1 Classical Stability Analysis

The study of stability of polynomial and related questions fordifferential equations goes back to XIX century Hurwitz (orRouth-Hurwitz) criterion [1] is a necessary and sufficientcondition for all the roots of a polynomial

119875 (119911) = 119886119899119911119899

+ 119886119899minus1

119911119899minus1

+ sdot sdot sdot + 1198860 (119911 isin C) (1)

with real coefficients and 119886119899

gt 0 to have negative real partsIt consists of the following all principal minors Δ

119896 119896 =

1 2 119899 of the Hurwitz matrix 119867 are positiveHere119867 is a matrix of order 119899whose 119895th row is of the form

1198862119899minus119895

1198862119899minus2minus119895

1198864minus119895

1198862minus119895

(2)

where 119886119896

= 0 if 119896 gt 119899 or 119896 lt 0 Polynomial 119875(119911) satisfyingthe Hurwitz condition is called a Hurwitz polynomial or inapplications of the Routh-Hurwitz criterion in the stabilitytheory of oscillating systems a stable polynomial Exactand approximate methods of Hurwitz factorization weredeveloped intensively (see eg [2 3])

Among other criteria concerning zeros distribution ofpolynomials we have tomentionMikhailov stability criterion[4] It states that all roots of a polynomial

119875 (119911) = 119911119899

+ 119886119899minus1

119911119899minus1

+ sdot sdot sdot + 1198860 (119911 isin C) (3)

with real coefficients have strictly negative real part if and onlyif the complex-valued function 120577 = 119875(119894120596) 119894 = radicminus1 of areal variable 120596 isin [0infin) describes a curve (the Mikhailovhodograph) in the complex 120577-plane which starts on the positivereal semiaxis and does not cross the origin and successivelygenerates an anticlockwise motion through 119899 quadrants

An equivalent condition is as follows the radius vector119875(119894120596) as 120596 increases from 0 to +infin never vanishes andmonotonically rotates in a positive direction through an angle1198991205872

The Mikhailov criterion gives a necessary and sufficientcondition for the asymptotic stability of a linear differentialequation of order 119899

119909(119899)

+ 119886119899minus1

119909(119899minus1)

+ sdot sdot sdot + 1198860119909 = 0 (4)

2 Mathematical Problems in Engineering

with constant coefficients or of a linear system (where theprime symbol denotes first derivative with respect to time)

1198831015840

= 119860119883 119883 isin R119899

(5)

with a constant matrix 119860 the characteristic polynomial ofwhich is 119875(119911)

A very general result on the zero location problem forthe polynomial is the Hermite-Biehler theorem [5] whichstates that the roots of 119892(119911) + 119894ℎ(119911) are all on the same sideof the real axis when 119892(119911) and ℎ(119911) are polynomials withreal coefficients if and only if the zeros of 119892(119911) and ℎ(119911)

are real and alternateTheHermite-Biehler theorem providesnecessary and sufficient conditions for Hurwitz stability ofreal polynomials in terms of an interlacing (alternating)property [6ndash8] Notice that if a given real polynomial is nota Hurwitz one then the Hermite-Biehler theorem does notprovide information on its roots distribution

During the last years several surveys addressed this topicsee for instance [9ndash12]This paper without being exhaustiveis complementary to the contents of such other works In ourpaper we introduced formally a selected set of methods tocharacterize the stability of fractional systems It is intendedto form a comprehensive text so that readers can follow easilythe concepts Furthermore the limitations of the knownmethods are also pointed out giving readers the opportunityto consider the open problems

Bearing these ideas in mind this paper is organized asfollows Sections 2 and 3 introduce fundamental aspectsnamely the concepts of quasi-polynomials and fractionalquasi-polynomials respectively Section 4 addresses themaincore of the paper the stability of fractional order systemsand is divided into eight subsections The section starts bypresenting general fractional order systems In the next sub-sections issues associated with linear time-invariant systemsare discussed in more details such as controllers positivesystems systems with delay and distributed discrete-timeand nonlinear systems Stability of closed-loop linear controlsystems becomes a major motivation of the paper Thereforefour examples in Section 5 deal with this topic FinallySection 6 outlines several techniques that are presentlyemerging and draws the main conclusions

2 Quasi-Polynomials

Pontryagin [13] gave a generalization of the Hermite-BiehlerTheorem which appeared to be very relevant formal tool forthe mathematical analysis of stability of quasi-polynomialsthat is of the functions of the following type

119865 (119911) =

119899

sum

119896=0

119891119896(119911) 119890

120582119896119911 (6)

where 119891119896(119911) are polynomials in 119911 with constant coefficients

and 120582119896 119896 = 0 119899 are real (or complex) numbers By

other words 119865(119911) is a sum where the terms are the productof and exponential and polynomial function with constantcoefficients In control theory such exponentials correspondto delays If 120582

119896are commensurable real numbers that is

120582119896

= 120582sdot119896 119896 = 0 119899 and 120582 gt 0 then the quasi-polynomial(6) can be written in the form

120575 (119911) = 119875 (119911 119890119911

) (7)

with

119875 (119911 119904) =

119898

sum

119895=0

119899

sum

119896=0

119886119895119896

119911119895

119904120582119896

119911 119904 isin C (8)

where 119875(119911 119904) is a polynomial function of two variables andthen if 119904 = 119890

119911 we get 120575(119911)Thus from this point of view the determination of the

zeros of a quasi-polynomial (7) by means of Pontryagintheorem can be considered to be a mathematical methodfor analysis of stabilization of a class of linear time invariantsystems with time delay (see eg [14])

119898

sum

119895=0

119899

sum

119896=0

119886119895119896

119909(119895)

(119905 + 120582 sdot 119896) = 0 (9)

where 119886119895119896are constant coefficients

PontryaginTheorem (see [13]) Let 120575(119911) = 119875(119911 119890119911

) be aquasi-polynomial of the type (7) where 119875(119911 119904) is a polynomialfunction in two variables with real coefficients Suppose thatthe ldquooldestrdquo coefficient 119886

119898119899= 0 Let 120575(119894120596) be the restriction of

the quasi-polynomial 120575(119911) to imaginary axis One can express120575(119894120596) = 119891(120596) + 119894119892(120596) where the real functions (of a realvariable) 119891(120596) and 119892(120596) are the real and imaginary parts of120575(119894120596) respectively Let one denote by 120596

119903and 120596

119894 respectively

the zeros of the functions 119891(120596) and 119892(120596) If all the zeros ofthe quasi-polynomial 120575(119911) lie to the left side of the imaginaryaxis then the zeros of the functions 119891(120596) and 119892(120596) are realalternating and

1198921015840

(120596) 119891 (120596) minus 119892 (120596) 1198911015840

(120596) gt 0 (10)

for each 120596 isin RReciprocally let one of the following conditions be satisfied

(1) all the zeros of the functions119891(120596) and 119892(120596) are real andalternate and the inequality (10) is satisfied for at leastone value 120596

(2) all the zeros of the function 119891(120596) are real and for eachzero of 119891(120596) the inequality (10) is satisfied that is119892(120596

119903)119891

1015840

(120596119903) lt 0

(3) all the zeros of the function 119892(120596) are real and for eachzero of 119892(120596) the inequality (10) is satisfied that is1198921015840

(120596119894)119891(120596

119894) gt 0

Then all the zeros of the quasi-polynomial 120575(119911) lie to the leftside of the imaginary axis

In [15] quasi-polynomial of the type

119865 (119911) = 119860 (119911) + 119861 (119911) 119890minus120591119911 (11)

is studied where119860(119911) and119861(119911) are polynomials with constantcoefficients given by

119860 (119911) =

119898

sum

119896=0

119886119898minus119896

119911119896

119861 (119911) =

119899

sum

119896=0

119887119899minus119896

119911119896

(12)

Mathematical Problems in Engineering 3

A notion of the principal term closely connected with thestability problem is used in this study namely the principalterm of quasi-polynomial (11) after premultiplying it by 119890

120591119911 isthe term 119888

119896119911119896

119890120591119911 in which the argument of the power 119911 and

120591 has the highest value for some 119896 = 0 1 119899 From thePontryagin criterion follows that a quasi-polynomial with noprincipal term has infinitely many roots with arbitrary largepositive real parts Hence the presence of principal term in aquasi-polynomial is a necessary condition for its stability In[15] the following formula related zeros of quasi-polynomials119911119895and it coefficients are proved

infin

sum

119895=1

1

119911119895

=1

2[

1198651015840

(119911)

119865(119911)]

119911=infin

minus [119865

1015840

(119911)

119865(119911)]

119911=0

(13)

Stability of systems of differential equations with delay(or in other words systems of differential-difference equa-tions)

1199091015840

(119905) = 119860119909 (119905 minus 120591) 119909 isin R119899 (14)

with a constant matrix 119860 was also investigated by usingproperties of quasi-polynomials

Thus in [16] the following criterion was proved let 119860 bea 2 times 2 matrix of real constants Then all zeros of the quasi-polynomial Δ(119911) = 119911

2

1198902119911

minus tr(119860)119911119890119911

+ |119860| have negative realparts if and only if

(120587

2)

2

+120587

2tr (119860) + |119860| gt 0

0 lt |119860| lt 1205772

lt (120587

2)

2

(15)

where 120577 is the smallest positive root of the equation 119910 sin119910 =

minus(12) tr(119860)Distribution of zeros of quasi-polynomials related to the

coupled renewal-differential system

119908 (119905) = 119891 (119905) + int

119905

0

119860 (119905 minus 120591)119908 (120591) 119889120591 + 119887119910 (119905)

1199101015840

(119905) = 119862119908 (119905) + 119863119910 (119905) 119910 (0) = 1199100

(16)

is discussed in [17]) A numerical method for calculation ofzeros of quasi-polynomials is proposed for example in [18]

Special attention was paid in the last years to (finite andinfinite) Dirichlet series In [19] it was proved that if

infin

sum

119896=0

119875119896(119911) 119890

minus120582119896119911

119911 = 120590 + 119894119905 120582119896

isin C

119875119896(119911) isin C [119911] Re 120582

119896uarr infin

(17)

is a convergent Taylor-Dirichlet series where as usualC[119911] means the set of polynomial with constant complexcoefficients The symbol 120582

119896uarr infin denotes 120582

1lt 120582

2lt sdot sdot sdot lt

120582119899

lt sdot sdot sdot rarr infin when 119899 rarr infin and satisfies an algebraicdifferential-difference equation

119866(119909 119891(1198981) (119909 + ℎ

1) 119891

(119898119903) (119909 + ℎ119903)) = 0

119866 (119909 1199091 119909

119903) = sum119862

1198961 1198961199031199091198961 sdot sdot sdot 119909

119896119903

(18)

where 1198621198961 119896119903

are constant coefficients and then the set of itsexponents 120582

119896infin

119896=0has a finite linear integral basis

Different questions related to the series of polynomial ofexponents were discussed in [20] (see also references thereinand [21]) In [22] the large 119899 asymptotic of zeros of sections ofa generic exponential series are derived where the 119899th sectionof the mentioned series mean the the sum of the first 119899 termsof it In [23] it is given an answer on the question when thereciprocal to the product of Gamma-functions coincide witha quasi-polynomial of type (6)

3 Fractional Quasi-Polynomials

Recently an attention is paid to the study of linear fractionalsystems with delays described by the transfer function

119875 (119911) =

1199020(119911) + sum

1198982

119895=1119902119895(119911) exp (minus119911

119903

120574119895)

1199010(119911) + sum

1198981

119895=1119901119895(119911) exp (minus119911119903120575

119895)

=119873 (119911)

119863 (119911) (19)

where 119903 is such a real number (0 lt 119903 le 1) and thefractional degree nontrivial polynomials 119901

119895(119911) and 119902

119895(119911)with

real coefficients have the forms

119901119895(119911) =

119899

sum

119896=0

119886119895119896

119911120572119896 119895 = 0 1 119898

1

119902119895(119911) =

119898

sum

119896=0

119887119895119896

119911120573119896 119895 = 0 1 119898

2

(20)

where 120572119896 120573

119896are real nonnegative numbers and 119886

0119899= 0

1198870119898

= 0The fractional degree characteristic quasi-polynomial of

the system (19) has the form

119863 (119911) = 1199010(119911) +

1198981

sum

119895=1

119901119895(119911) exp (minus119911

119903

120575119895) (21)

In the case of a system with delays of a fractional commen-surate order (ie when 120572

119896= 120572 sdot 119896 (119896 = 0 1 119899) 120573

119896=

120572 sdot 119896 (119896 = 0 1 119898)) one can consider the natural degreequasi-polynomial

119863 (120582) = 1199010(120582) +

1198981

sum

119895=1

119901119895(120582) exp (minus120582

119903120572

120575119895) 120582 = 119911

120572

(22)

associated with the characteristic quasi-polynomial (21) of afractional order

In [24] new frequency domain methods for stabilityanalysis of linear continuous-time fractional order systemswith delays of the retarded type are proposed The methods


are obtained by generalization to the class of fractional ordersystems with delays of the Mikhailov stability criterion andthe modified Mikhailov stability criterion known from thetheory of natural order systems without and with delays

The following results concerning stability of the consid-ered system are proved in [24]

(1) The fractional quasi-polynomial (21) of commensu-rate degree satisfies the condition 119863(119911) = 0 Re 119911 ge 0 if andonly if all the zeros of the associated natural degree quasi-polynomial (22) satisfy the condition

1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

120572120587

2 (23)

where arg(120582) means the principal branch of the multivaluedfunction Arg(120582) 120582 isin C that is arg(120582) isin (minus120587 120587]

(2) The fractional quasi-polynomial (21) of commensu-rate degree is not stable for any 120572 gt 1

(3) The fractional characteristic quasi-polynomial (21) ofcommensurate degree is stable if and only if

Δ0le120596lt+infin

arg (119863 (119894120596)) =119899120587

2 (24)

which means that the plot of 119863(119894120596) with 120596 increasing from0 to +infin runs in the positive direction by 119899 quadrants of thecomplex plane missing the origin of this plane

(4)The fractional characteristic quasi-polynomial (21) (ofcommensurate or noncommensurate degree) is stable if andonly if

Δminusinfinlt120596lt+infin

arg (120595 (119894120596)) = 0 (25)

where

120595 (119911) =119863 (119911)

119908119903(119911)

(26)

and 119908119903(119911) can be chosen for example as

119908119903(119911) = 119886

0119899(119911 + 119888)

120572119899 119888 gt 0 (27)

Remark 1 Stability of fractional polynomials is related to thestability of ordinary quasi-polynomials due to the followingrelation

1198860+ 119886

11199111205721 + sdot sdot sdot + 119886

119899119911120572119899 = 119886

0+ 119886

1sdot 119890

1205721119908 + sdot sdot sdot

+ 119886119899sdot 119890

120572119899119908

(28)

where119908 = log 119911 Application of formula (28) needs to be verycareful since119908 is now the point on the Riemann surface of thelogarithmic function (see eg [22])

An approach describing the stability of fractional quasi-polynomials in terms of zeros distribution of these polyno-mials on certain Riemann surfaces is widely used now (seeeg survey paper [10] and references therein) Sometimes it iscalled ldquoroot-locusmethodrdquoThe characteristic result obtainedwith the application of this method is the following [25]

(5) The fractional order system with characteristic poly-nomial 119908(119911) is stable if and only if 119908(119911) has no zeros in theclosed right-half of the Riemann surface that is

119908 (119911) = 0 forallRe 119911 ge 0 (29)

The fractional order polynomial 119908(119911) is a multivaluedfunction whose domain is a Riemann surface In generalthis surface has an infinite number of sheets and thus thefractional polynomial has (in general) an infinite number ofzerosWe are interested only in those zeros which are situatedon the main sheet of the Riemann surface which can be fixedin the following way minus120587 lt arg(119911) lt 120587

Recently the notion of robust stability was introducedfor systems with characteristic polynomials dependent onuncertainty parameter (see [26ndash29]) In [25] this notion isapplied to the convex combination of two fractional degreespolynomials

119882(119911 119902) = 119908 (119911 119902) 119902 isin 119876 = [0 1] (30)

119908 (119911 119902) = (1 minus 119902)119908119886(119911) + 119902119908

119887(119911) (31)

where 119902 is uncertainty parameter and119908119886(119911) 119908

119887(119911) are frac-

tional degree polynomialsThe family (30) of fractional degree polynomials is called

robust stable if polynomial 119908(119911 119902) is stable for all 119902 isin 119876Generalization of the Mikhailov-type criterion (see [24])

to this case has the following form [25](6) Let the nominal polynomial 119908

119886(119911) be stable The

family of polynomials (30) is robust stable if and only if theplot of the function

120599 (119895120596) =119908

119887(119895120596)

119908119886(119895120596)

120596 isin Ω = [0infin) (32)

does not cross the nonpositive part (minusinfin 0] of the real axis inthe complex plane

4 Stability of Fractional Order Systems

Several applications of the results on stability of the fractionalpolynomials to the systems describing different processes andphenomena are presented for example [30 31] Some ofthese results are closely related to the recent achievementsand the theory and applications of fractional calculus andfractional differential equations (see [32ndash34]) The results inthis area need to be systematized as from the point of theideas technical point of view

41 General Fractional Order Systems A general fractionalorder system can be described by a fractional differentialequation of the form

119886119899119863

120572119899119910 (119905) + 119886119899minus1

119863120572119899minus1119910 (119905) + sdot sdot sdot + 119886

0119863

1205720119910 (119905)

= 119887119898119863

120573119898119910 (119905) + 119887119898minus1


0119863

1205730119910 (119905)

(33)

where 119863120574

=0119863

120574

119905denotes the Riemann-Liouville RL

0119863

120574

119905or

Caputo 119862

0119863

120574

119905fractional derivative [32 33] Another form of

general fractional order system is due to properties of theLaplace transform [32 34] This form represents the systemin terms of corresponding transfer function

119866 (119904) =119887119898119904120573119898 + 119887

119898minus1119904120573119898minus1 + sdot sdot sdot + 119887

01199041205730

119886119899119904120572119899 + 119886


01199041205720

=119876 (119904)

119875 (119904) (34)


where 119904 is the Laplace variable Here 119886119899 119886

0 119887

119898 119887

0are

given real constants and 120572119899 120572

0 120573

119898 120573

0are given real

numbers (usually positive) Without loss of generality thesesets of parameters can be ordered as 120572

119899gt sdot sdot sdot gt 120572

0 120573

119898gt

sdot sdot sdot gt 1205730

If both sets 120572-119904 and 120573-119904 constitute an arithmetical pro-gression with the same difference that is 120572

119896= 119896120572 119896 =

0 119899 120573119896

= 119896120572 119896 = 0 119898 then system (33) is calledcommensurate order system Usually it is supposed that par-ameter 120572 satisfies the inequality 0 lt 120572 lt 1 In all other casessystem (33) is called incommensurate order system Anywayif parameters 120572 and 120573 are rational numbers then this casecan be considered as commensurate one with 120572 = 1119873

being a least common multiple of denominators of fractions120572119899 120572

0 120573

119898 120573

0[35]

For commensurate order system its transfer function canbe thought as certain branch of the following multivaluedfunction

119866 (119904) =sum

119898

119896=0119887119896(119904

120572

)119896

sum119899

119896=0119886119896(119904

120572)119896

=119876 (119904

120572

)

(119904120572)

(35)

Since the right hand-side of this relation is a rational functionof 119904120572 then one can represent 119866(119904) in the form of generalizedsimple fractions The most descriptive representation of sucha type is that for 119899 gt 119898

119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904120572 + 120582119894)119895

(36)

where minus120582119894is a root of polynomial 119875(119911) of multiplicity 119903

119894 In

particular if all roots are simple then the representation (36)has the most simple form

119866 (119904) =

119899

sum

119894=1

119861119894

119904120572 + 120582119894

(37)

In this case an analytic solution to system (33) is given by theformula

119910 (119905) = Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

sdot (L119906) (119904)

= Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

lowast 119906 (119905)

= (

119899

sum

119894=1

119861119894119905120572

119864120572120572

(minus120582119894119905120572

)) lowast 119906 (119905)

(38)

where the symbol ldquolowastrdquo means the Laplace-type convolutionand 119864

120583] is the two-parametric Mittag-Leffler function [36]

119864120583] (119911) =

infin

sum

119896=0

119911119896

Γ (120583119896 + ]) (39)

In the case of homogeneous fractional order system

119886119899119863

120572119899119910 (119905) + 119886119899minus1


0119863

1205720119910 (119905) = 0 (40)

the analytical solution is given by the following formula (seeeg [10 37])

119910 (119905)

=1

119886119899

infin

sum

119896=0

(minus1)119896

119896sum

1198960+sdotsdotsdot+119896119899minus2=119896

1198960ge0119896119899minus2ge0

(119896 1198960 119896

119899minus2)

times

119899minus2

prod

119894=0

(119886119894

119886119899

)

119896119894

E119896

times (119905 minus119886119899minus1

119886119899

119886119899minus 119886

119899minus1 119886

119899+

119899minus2

sum

119895=0

(119886119899minus1

minus 119886119895) 119896

119895+ 1)

(41)

where (119896 1198960 119896

119899minus2) are the multinomial coefficients and

E119896(119905 119910 120583 ]) is defined by the formula [33]

E119896(119905 119910 120583 ]) = 119905

120583119896+]minus1119864

(119896)

120583] (119910119905120583

) (119896 = 0 1 2 )

119864(119896)

120583] (119911) =

infin

sum

119895=0

(119895 + 119896)119911119895

119895Γ (120583119895 + 120583119896 + ])(119896 = 0 1 2 )

(42)

is the 119896th derivative of two-parametric Mittag-Leffler func-tion [32 38]

119864120583] (119911) =

infin

sum

119895=0

119911119895

Γ (120583119895 + ])(119896 = 0 1 2 ) (43)

The stability analysis of the fractional order system givesthe following results (see [10 24 39 40])

Theorem 2 A commensurate order system with transferfunction (37) is stable if and only if

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 gt 120572

120587

2forall119894 = 1 119899 (44)

with minus120582119894being the 119894th root of the generalized polynomial 119875(119904

120572

)

To formulate the result in incommensurate case we usethe concept of bounded input-bounded output (BIBO) orexternal stability (see [10 39])

Theorem 3 Let the transfer function of an incommensurateorder system be represented in the form

119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904119902119894 + 120582119894)119895

(45)

for some complex numbers 119860119894119895 120582

119894 positive 119902

119894 and positive

integer 119903119894

Such system is BIBO stable if and only if parameters 119902119894and

arguments of numbers 120582119894satisfy the following inequality

0 lt 119902119894lt 2

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 lt 120587 (1 minus

119902119894

2) forall119894 = 1 119901 (46)

The result of Theorem 3 was obtained by using thestability results given in [40 41]


42 Fractional Order Linear Time-Invariant Systems Besidesthe conception of stability for fractional order linear time-invariant systems

0119863

q119905x (119905) = Ax (119905) + Bu (119905)

y (119905) = Cx (119905)

(47)

the conceptions of controllability and observability (knownas linear and nonlinear differential systems [42 43]) areintroduced too

In (47) x isin R119899 is an unknown state vector and u isin

R119903 y isin R119901 are the control vector and output vector res-pectively Given (constant) matrices A B and C are of thefollowing size A isin R119899times119899

B isin R119899times119903

and C isin R119901times119899 Positivevector q = [119902

1 119902

119899]119879 denotes the (fractional) order of

system (47) If 1199021

= sdot sdot sdot = 119902119899

= 119902 then system (47) is called acommensurate order system

As in case of ordinary linear differential time-invari-ant systems controllability and observability conditions [44]are represented in terms of controllability 119862

119886= [119861|119860119861

|1198602

119861| sdot sdot sdot |119860119899minus1

119861] and observability 119874119886

= [119862|119862119860|1198621198602

| sdot sdot sdot

|119862119860119899minus1

] matrices respectivelyWe have to mention also the stability criterion for the

system (47) (see [40 45ndash48])

Theorem 4 Commensurate system (47) is stable if the follow-ing conditions are satisfied

1003816100381610038161003816arg (eig (A))1003816100381610038161003816 gt 119902

120587

2 0 lt 119902 lt 2 (48)

for all eigenvalues eig(A) of the matrix A

Several new results on stability controllability andobservability of system (47) are presented in the recentmonograph [49] (see also [37 50ndash52]) is proposed We cansay that control theory for fractional order systems becomes aspecial branch of fractional order systems andmention in thisconnection several important papers developing this theory[53ndash67] A number of applications of the fractional ordersystems are presented in [30 31 68]

43 Fractional Order Controllers The fractional-order con-troller (FOC) PI120582D120575 (also known as PI120582D120583 controller) wasproposed in [33] as a generalization of the PID controller withintegrator of real order120582 anddifferentiator of real order120575Thetransfer function of such controller in the Laplace domain hasthis form

119862 (119904) =119880 (119904)

119864 (119904)= 119870

119901+ 119879

119894119904minus120582

+ 119879119889119904120575

(120582 120575 gt 0) (49)

where 119870119901is the proportional constant 119879

119894is the integration

constant and 119879119889is the differentiation constant

In [69] a classification of different modifications of thefractional PI120582D120575 controllers (see also [49 70ndash72])

(i) CRONE controller (1st generation) characterized bythe bandlimited lead effect

119862 (119904) = 1198620

(1 + 119904120596119887)119903

(1 + 119904120596ℎ)119903minus1

(50)

There are a number of real-life applications of threegenerations of the CRONE controller [73]

(ii) Fractional lead-lag compensator [49] which is givenby

119862 (119904) = 119896119888(

119904 + 1120582

119904 + 1120582)

119903

(51)

where 0 lt 119904 lt 1 120582 isin R and 119903 isin R(iii) Noninteger integral and its application to control as a

reference function [74 75] Bode suggested an idealshape of the loop transfer function in his work ondesign of feedback amplifiers in 1945 Ideal looptransfer function has the form

119871 (119904) = (119904

120596119892119888

)

120572

(120572 lt 0) (52)

where 120596119892119888is desired crossover frequency and 120572 is the

slope of the ideal cut-off characteristic The Nyquistcurve for ideal Bode transfer function is simply astraight line through the origin with arg(119871(119895120596)) =

1205721205872(iv) TID compensator [76] which has structure similar to

a PID controller but the proportional component isreplaced with a tilted component having a transferfunction 119904 to the power of (minus1119899) The resultingtransfer function of the TID controller has the form

119862 (119904) =119879

1199041119899+

119868

119904+ 119863119904 (53)

where 119879 119868 and 119863 are the controller constants and 119899

is a non-zero real number preferably between 2 and3 The transfer function of TID compensator moreclosely approximates an optimal transfer functionand an overall response is achieved which is closer tothe theoretical optimal response determined by Bode[74]

Different methods for determination of PI120582D120575 controllerparameters satisfying the given requirements are proposed(see eg [69] and references therein)

44 Positive Fractional Order Systems A new concept(notion) of the practical stability of the positive fractional2D linear systems is proposed in [77] Necessary and suf-ficient conditions for the practical stability of the positivefractional 2D systems are established It is shown that thepositive fractional 2D systems are practically unstable (1) if acorresponding positive 2D system is asymptotically unstableand (2) if some matrices of the 2D system are nonnegative

Simple necessary and sufficient conditions for practicalstability independent of the length of practical implementa-tion are established in [78] It is shown that practical stabilityof the system is equivalent to asymptotic stability of thecorresponding standard positive discrete-time systems of thesame order


45 Fractional Order Systems with Delay Fractional ordersystems with delay meet a number of important applications(see eg [31 49]) There are several important works aboutstability of closed-loop fractional order systemscontrollerswith time delays Some relevant examples can be found in[79ndash81]

To describe simplest fractional order systems with delaylet us introduce some notations (see [82])

Let C([119886 119887]R119899

) be the set of continuous functionsmapping the interval [119886 119887] to R119899

) One may wish to identifya maximum time delay 119903 of a system In this case weare interested in the set of continuous function mapping[minus119903 0] to R119899

) for which we simplify the notation to C =

C([minus119903 0]R119899

) For any119860 gt 0 and any continuous function oftime x isin C([119905

0minus 119903 119905

0+119860]R119899

) 1199050le 119905

0+119860 let x

119905(120579) isin C be a

segment of function defined as x119905(120579) = x(119905 + 120579) minus119903 le 120579 le 0

Let the fractional nonlinear time-delay system be thesystem of the following type

119862

1199050

119863q119905x (119905) = f (119905 x

119905(119905)) (54)

where x isin C([1199050

minus 119903 1199050

+ 119860]R119899

) for any 119860 gt 0 q =

(119902 119902) 0 lt 119902 lt 1 and 119891 R times C rarr R119899 To determinethe future evolution of the state it is necessary to specify theinitial state variables x(119905) in a time interval of length 119903 sayfrom 119905

0minus 119903 to 119905

0 that is

x (1199050) = 120593 (55)

where 120593 isin C is given In other words x(1199050)(120579) = 120593(120579) minus119903 le

120579 le 0Several stability results for fractional order systems with

delay were obtained in [82ndash84] In particular in [84] thelinear and time-invariant differential-functional Caputo frac-tional differential systems of order 120572 are considered

119862

119863120572

0119905x (119905) =

1

Γ (119896 minus 120572)int

119905

0

x(119896)

(120591)

(119905 minus 120591)120572+1minus119896

119889120591

=

119901

sum

119894=0

A119894x (119905 minus ℎ

119894) + Bu (119905)

(56)

119896 minus 1 lt 120572 le 119896 119896 minus 1 isin Z0+ 0 lt ℎ

0lt ℎ

1lt sdot sdot sdot lt ℎ

119901= ℎ lt infin

A0 A

119894isin R119899times119899 are matrices of dynamics for each delay ℎ

119894

and B isin R119899times119898 is the control matrix Under standard initialconditions the solution to this problem is represented viaMittag-Leffler function and the dependence of the differentdelay parameters is studied

46 Distributed Order Fractional Systems An example of dis-tributed order fractional systems is the systemof the followingtype (see eg [85]) containing the so-called distributed orderfractional derivative

119862

119889119900119863

120572

119905119909 (119905) = 119860

119862

119889119900119863

120573

119905119909 (119905) + 119861119906 (119905) 119909 (0) = 119909

0 (57)

where 0 lt 120573 lt 120572 le 1

119889119900119863

120572

119905(sdot) = int

120574

119905

119887 (120572)119889120572

(sdot)

119889119905120572119889120572 120574 gt 119897 ge 0 119887 (120572) ge 0 (58)

and119904119900119863

120572

119905(sdot) = 119889

120572

(sdot)119889119905120572 is a standard (single order) fractional

derivativeApplication of such systems to the description of the

ultraslow diffusion is given in series of articles by Kochubei(see eg [86])

47 Discrete-Time Fractional Systems There are differentdefinitions of the fractional derivative (see eg [34 87])The Grunwald-Letnikov definition which is the discreteapproximation of the fractional order derivative is used hereTheGrunwald-Letnikov fractional order derivative of a givenfunction 119891(119905) is given by

GL119886

119863120572

119905119891 (119905) = lim

ℎrarr0

119886Δ

120572

ℎ119891 (119905)

ℎ120572 (59)

where the real number 120572 denotes the order of the derivative119886 is the initial time and ℎ is a sampling time The differenceoperator Δ is given by

119886Δ

120572

ℎ119891 (119905) =

[(119905minus119886)ℎ]

sum

119895=0

(minus1)119895

(120572

119895)119891 (119905 minus 119895ℎ) (60)

where (120572

119895 ) is the Pochhammer symbol and [sdot] denotes aninteger part of a number

Traditional discrete-time state-space model of integerorder that is when 120572 is equal to unity has the form

x (119896 + 1) = Ax (119896) + Bu (119896) 119909 (0) = 1199090

y (119896) = Cx (119896) + Du (119896)

(61)

where u(119896) isin R119901 and y(119896) isin R119902 are respectively the inputand the output vectors and x(119896) = [119909

1(119896) 119909

119899(119896)] isin

R119899 is the state vector Its initial value is denoted by 1199090

=

119909(0) and can be set equal to zero without loss of generalityA B C and D are the conventional state space matriceswith appropriate dimensions

The generalization of the integer-order difference to anoninteger order (or fractional-order) difference has beenaddressed in [88] where the discrete fractional-order differ-ence operator with the initial time taken equal to zero isdefined as follows

Δ120572x (119896) =

1

ℎ120572

119896

sum

119895=0

(minus1)119895

(120572

119895) x (119896 minus 119895) (62)

In the sequel the sampling time ℎ is taken equal to 1These results conducted to conceive the linear discrete-timefractional-order state-space model using

Δ120572x (119896 + 1) = A

119889x (119896) + Bu (119896) x (0) = x

0 (63)

The discrete-time fractional order system is represented bythe following state space model

x (119896 + 1) =

119896

sum

119895=0

A119895x (119896 minus 119895) + Bu (119896) x (0) = x

0

y (119896) = Cx (119896) + Du (119896)

(64)


where A0

= A119889

minus 1198881I119899and A

119895= minus119888

119895+1119868119899for 119895 ge 2 with 119888

119895=

minus(minus1)119895

(120572

119895 ) This description can be extended to the case ofnoncommensurate fractional-order systems modeled in [88]by introducing the following vector difference operator

ΔΥx (119896 + 1) = A

119889x (119896) + Bu (119896)

x (119896 + 1) = ΔΥx (119896 + 1) +

119896+1

sum

119895=1

A119895x (119896 minus 119895 + 1)

ΔΥx (119896 + 1) =

[[[[

[

Δ1205721119909

1(119896 + 1)

Δ1205722119909

2(119896 + 1)

Δ

120572119899119909119899(119896 + 1)

]]]]

]

(65)

Stability analysis of such system is performed for example[89] (see also [90 91])

The paper [92] is devoted to controllability analysis ofdiscrete-time fractional systems

48 Fractional Nonlinear Systems The simplest fractionalnonlinear system in the incommensurate case is the systemof the type

0119863

q119905x (119905) = f (x (119905) 119905)

x (0) = c(66)

It has been mentioned in [39] that exponential stabilitycannot be used to characterize the asymptotic stability offractional order systems A new definition of power lawstability was introduced [93]

Definition 5 Trajectory x(119905) = 0 of the system (66) is powerlaw 119905

minus119902 asymptotically stable if there exists a positive 119902 gt 0

such that

forall x (119905) with 119905 le 1199050 exist119873 = 119873 (x (sdot))

such that forall119905 ge 1199050997904rArr x (119905) le 119873119905

minus119902

(67)

Power law 119905minus119902 asymptotic stability is a special case of the

Mittag-Leffler stability [94] which has the following form

Definition 6 (definition of the Mittag-Leffler stability) Thesolution of the nonlinear problem

1199050119863

q119905x (119905) = 119891 (x (119905) 119905)

x (1199050) = c

(68)

is said to be Mittag-Leffler stable if

119909 (119905) le 119898 [119909 (1199050)] 119864

119902(minus120582(119905 minus 119905

0)119902

)119887

(69)

where 119864119902(119906) is the classical Mittag-Leffler function q =

(119902 119902) 119902 isin (0 1) 120582 gt 0 119887 gt 0 and 119898(0) = 0 119898(x) ge

0 119898 is locally Lipschitz on x isin B sube R119899 with Lipschitzconstant 119898

0

Among the many methods that have been proposed forthe study of ldquodifferent kinds of stability definitionsrdquo of non-linear fractional order systems (66) wemention perturbationanalysis

Thus in [95] it is investigates the qualitative behaviourof a perturbed fractional order differential equations withCaputo derivatives that differs in initial position and initialtime with respect to the unperturbed fractional order dif-ferential equation with Caputo derivatives In [96] the sta-bility of 119899-dimensional linear fractional differential systemswith commensurate order and the corresponding perturbedsystems is investigated By using the Laplace transformthe asymptotic expansion of the Mittag-Leffler functionand the Gronwallrsquos inequality some conditions on stabilityand asymptotic stability are given In [97] the stabilityof nonlinear fractional differential systems with Caputoderivatives by utilizing a Lyapunov-type function is studiedTaking into account the relation between asymptotic stabilityand generalized Mittag-Leffler stability the condition onLyapunov-type function is weakened

5 Some Examples

In this section we analyze the stability of the four systems bymeans of the root locus and the polar diagram For calculatingthe root-locus the algorithmproposed in [98] is adoptedTheclosed-loop system is constituted by a controller and a plantwith transfer functions 119862(119904) and 119866(119904) respectively and unitfeedback

For studying the stability the following classical criteriaare applied

(i) ultimate (or critical) gain 119870119906 1 + 119870

119906119862(119904)119866(119904) = 0

Re(119904) = 0(ii) phasemargin PM |119862119866(119894120596

1)| = 1 PM = arg119862119866(119894120596

1)

+120587(iii) gain margin GM arg119862119866(119894120596

120587) = minus120587 GM =

|119862119866(119894120596120587)|

minus1

51 Example 1 This example was discussed in [10] In thiscase we have 119862(119904) = 119870(6447 + 1246119904) and 119866(119904) = 1(0598 +

3996119904125

) Figure 1 depicts the root locus where the whitecircles represent the roots for119870 = 1 119904

12= minus10788plusmn119894 06064

The system is always stableFigure 2 shows the polar diagram for 119870 = 1 The corre-

sponding phase margin is MF = 14720 rad for 1205961

= 15195Varying the gain we verify again that the system is alwaysstable

52 Example 2 This example was discussed in [99] In thiscase we have 119862(119904) = 119870(1 + 11694(1119904

11011

) minus 0151711990401855

)

and 119866(119904) = 119890minus05119904

(1 + 11990405

) Figure 3 depicts the root locuswhere the white circles represent the roots for 119870 = 14098namely 119904

12= minus06620 plusmn 119894 04552 119904

34= minus20323 plusmn 119894 41818

The limit of stability occurs for 119870119906

= 35549 11990412

= 0 plusmn

119894 45124Figure 4 shows the polar diagram for 119870 = 14098 The

corresponding phase margin is PM = 11854 rad for 1205961

=


1

05

00

minus05

minus1

minus1minus2minus3minus4

minus5minus6

Re(119904)

Im(119904

)Figure 1 Root locus for 119862(119904) = 119870(6447 + 1246119904) and 119866(119904) =

1(0598 + 3996119904125

)

minus20 00

20 40 60 80 100 120

minus50

50

minus100

100

Re

Im

Figure 2 Polar diagram for 119862(119904) = 119870(6447 + 1246119904) 119866(119904) =

1(0598 + 3996119904125

) and 119870 = 1

10478 and gain margin GM = 25201 for 120596120587

= 45062 radwhich leads to 119870

119906= 35549 as the ultimate gain

53 Example 3 This example analyzes the system 119870119904

(119904 + 1)120572

(119904 + 2) for 120572 isin 0 12 1 32 2 52 3 Since theinteger-order cases are trivial in Figures 5 6 and 7 areonly depicted the fractional-order cases Table 1 shows thecorresponding gain in the limit of stability 119870

119906 the phase

margin PM and gain margin GM for several values of 120572

54 Example 4 This example analyzes the nonminimumphase system 119870((119904 + 2)

1205722(119904 minus 1)1205721) for 120572

1= 33 120572

2= 23

Figure 8 depicts the root locus The limit of stability occursfor 119870

119906= 43358 119904

12= 0 plusmn 119894 48493

Figure 9 shows the polar diagram for 119870 = 10 Thecorresponding gain margin is GM = 43358 for 120596

120587=

48493 rad which leads to 119870119906

= 43358 as the ultimate gain

00

minus1

minus2

minus2minus3

minus4

minus6

2

4

6

Re(119904)

Im(119904

)

Figure 3 Root locus for 119862(119904) = 119870(1 + 11694(111990411011

) minus

0151711990401855

) and 119866(119904) = 119890minus05119904

(1 + 11990405

)

00

minus1minus1

minus2

minus2

minus3

minus4

minus15minus05 05

1

2

3

4

Re

Im

Figure 4 Polar diagram for 119862(119904) = 119870(1 + 11694(111990411011

) minus

0151711990401855

) 119866(119904) = 119890minus05119904

(1 + 11990405

) and 119870 = 14098

6 Future Directions of Research andConclusions

In the previous discussion one can outline themainmethodsapplied at the study of stability of ordinary and fractionalorder systems These directions are as follows

(i) complex analytic methods related to the properties ofsingle- and multivalued analytic functions

(ii) methods of geometric functions theory describing thebehaviour of polynomials or systems in geometricalterms

(iii) methods of linear algebra (mainly matrix analysis)

(iv) methods of stochastic analysis

(v) methods of fuzzy data analysis

(vi) perturbation analysis of nonlinear systems

(vii) methods of differential equations of fractional order


00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5

minus05 minus01minus04 minus03 minus02

00

1

2

3

Re

Im

(b)

Figure 5 Root locus and polar diagram for 119870119904(119904 + 1)120572

(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5

minus08 minus06 minus04minus02 02

00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00

minus1minus16 minus14 minus12 minus1 minus08 minus06 minus04 minus02 02

minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25


Table 1 Stability indices of 119870(119904(119904 + 1)120572

(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945

minus10 minus8 minus6

minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)

Figure 8 Root locus for 119870((119904 + 2)1205722(119904 minus 1)

1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im

Figure 9 Polar diagram for119870((119904 + 2)1205722(119904 minus 1)

1205721 ) for 1205721= 33 120572

2=

23

We can also mention the possibility to apply the study offractional order stability methods of Pade (or Hermite-Pade)approximation (see eg [100ndash102])

In conclusion this paper reviewed themain contributionsthat were proposed during the last years for analysing thestability of fractional order systems Different problems were

addressed such as control systems including a delay or witha distributed nature as well as discrete-time and nonlinearsystemsThe paper presented in a comprehensive and conciseway many details that are scattered in the literature andprovide researchers a reference text for work in this area

Acknowledgments

The authors thank the anonymous referees for their valuablesuggestions that helped to improve the paper The work wasdone in the framework of cooperation between BelarusianState University and University La LagunaThe work is partlysupported by ProjectMTM2010-16499 from theGovernmentof Spain

References

[1] A Hurwitz ldquoUeber die Bedingungen unter welchen eine Gle-ichung nur Wurzeln mit negativen reellen Theilen besitztrdquoMathematische Annalen vol 46 no 2 pp 273ndash284 1895

[2] F J MacWilliams ldquoAn iterative method for the direct hurwitz-factorization of a polynomialrdquo IRE Transactions on CircuitTheory vol 5 no 4 pp 347ndash352 1958

[3] MWunchHurwitz-Faktorisierung von Polynomen undGanzenFunktionen Giessen University Selbstverlag des Mathematis-chen Instituts 1993

[4] A V Mikhailov ldquoMethods for harmonic analysis in automaticcontrol systemsrdquo Avtomat i Telemekh vol 3 pp 27ndash81 1938(Russian)

[5] N G Cebotarev andN NMeıman ldquoThe Routh-Hurwitz prob-lem for polynomials and entire functionsrdquo Trudy Matematich-eskogo Instituta imeni V A Steklova vol 26 p 331 1949

[6] R Caponetto and G Dongola ldquoA numerical approach forcomputing stability region of FO-PID controllerrdquo Journal of theFranklin Institute vol 350 no 4 pp 871ndash889 2013

[7] A Datta M Ho and S Bhattacharyya Structure and Synthesisof PID Controllers Springer London UK 2000

[8] M-T Ho A Datta and S P Bhattacharyya ldquoGeneralizations ofthe Hermite-Biehler theoremrdquo Linear Algebra and its Applica-tions vol 302303 pp 135ndash153 1999 Special issue dedicated toHans Schneider (Madison WI 1998)

[9] C P Li and F R Zhang ldquoA survey on the stability of fractionaldifferential equationsrdquo European Physical Journal vol 193 no 1pp 27ndash47 2011

[10] I Petras ldquoStability of fractional-order systems with rationalorders a surveyrdquo Fractional Calculus amp Applied Analysis vol12 no 3 pp 269ndash298 2009


[11] I Petras Fractional-Order Nonlinear Systems Modelling Analy-sis and Simulation Springer Dordrecht The Netherlands 2011

[12] J C Trigeassou N Maamri J Sabatier and A Oustaloup ldquoALyapunov approach to the stability of fractional differentialequationsrdquo Signal Processing vol 91 no 3 pp 437ndash445 2011

[13] L S Pontryagin ldquoOn the zeros of some elementary transcen-dental functionsrdquo American Mathematical Society Translationsvol 1 no 2 pp 95ndash110 1955

[14] L Cossi ldquoIntroduction to stability of quasipolynomialsrdquo inTime-Dealay Systems D Debeljkovich Ed pp 1ndash14 InTech2001

[15] H Gorecki and S Bialas ldquoRelations between roots and coef-ficients of the transcendental equationsrdquo Bulletin of the PolishAcademy of Sciences vol 58 no 4 pp 631ndash634 2010

[16] T Hara and J Sugie ldquoStability region for systems of differential-difference equationsrdquo Funkcialaj Ekvacioj vol 39 no 1 pp 69ndash86 1996

[17] R P Rhoten and J K Aggarwal ldquoThe location of zeros of finiteexponential seriesrdquo SIAM Journal on Applied Mathematics vol20 pp 44ndash50 1971

[18] E O Movsesyan ldquoCalculation of zeros of a quasipolynomialby the method of the derivative of the argumentrdquo ZhurnalVychislitelrsquonoi Matematiki i Matematicheskoi Fiziki vol 26 no8 pp 1132ndash1140 1986 (Russian)

[19] F Wadleigh ldquoTaylor-Dirichlet series and algebraic differential-difference equationsrdquoProceedings of theAmericanMathematicalSociety vol 80 no 1 pp 83ndash89 1980

[20] A F Leontrsquoev Posledovatelnosti Polinomov iz Eksponent NaukaMoscow Russia 1980

[21] A F Leontrsquoev ldquoSeries and sequences of polynomials of exponen-tialsrdquoTrudyMatematicheskogo Instituta imeni VA Steklova vol176 pp 308ndash325 1987 (Russian)

[22] P Bleher and R Mallison Jr ldquoZeros of sections of exponentialsumsrdquo International Mathematics Research Notices vol 2006Article ID 38937 49 pages 2006

[23] V V Napalkov and S G Merzlyakov ldquoEntire functions ofexponential type represented by using the gamma functionrdquoDoklady Mathematics vol 74 no 2 pp 748ndash750 2006

[24] M Busłowicz ldquoStability of linear continuous-time fractionalorder systems with delays of the retarded typerdquo Bulletin of thePolish Academy of Sciences vol 56 no 4 pp 319ndash324 2008

[25] M Buslowicz ldquoRobust stability of convex combination of twofractional degree characteristic polynomialsrdquo Acta Mechanicaet Automatica vol 2 no 2 pp 5ndash10 2008

[26] M Busłowicz ldquoRobust stability of positive continuous-timelinear systems with delaysrdquo International Journal of AppliedMathematics and Computer Science vol 20 no 4 pp 665ndash6702010

[27] M Busłowicz ldquoSimple conditions for robust stability of positivediscrete-time linear systems with delaysrdquoControl and Cybernet-ics vol 39 no 4 pp 1159ndash1171 2010

[28] T Kaczorek Selected Problems of Fractional Systems TheorySpringer Berlin Germany 2011

[29] T Kaczorek andM Busłowicz ldquoMinimal realization for positivemultivariable linear systems with delayrdquo International Journal ofApplied Mathematics and Computer Science vol 14 no 2 pp181ndash187 2004

[30] J Sabatier O P Agraval and J A T Machado Eds Advancesin Fractional Calculus Springer Dordrecht The Netherlands2007

[31] D Baleanu Z B Guvenc and J A T Tenreiro MachadoEds New Trends in Nanotechnology and Fractional CalculusApplications Springer Dordrecht The Netherlands 2010

[32] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations vol 204 ofNorth-Holland Mathematics Studies Elsevier Amsterdam TheNetherlands 2006

[33] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[34] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers New York NY USA 1993

[35] F Merrikh-Bayat and M Afshar ldquoExtending the root-locusmethod to fractional-order systemsrdquo Journal of Applied Math-ematics vol 2008 Article ID 528934 13 pages 2008

[36] R Gorenflo and F Mainardi ldquoFractional calculus integraland differential equations of fractional orderrdquo in Fractals andFractional Calculus in Continuum Mechanics vol 378 of CISMCourses and Lectures pp 223ndash276 Springer Berlin Germany1997

[37] R Caponetto G Doongola L Fortuna and I Petras FractionalOrder SystemsModeling and Control ApplicationsWorld Scien-tific Singapore 2010

[38] A M Mathai and H J Haubold Special Functions for AppliedScientists Springer New York NY USA 2008

[39] D Matignon ldquoStability result on fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the International Meeting on Automated Compliance Systemsand the International Conference on Systems Man and Cyber-netics (IMACS-SMC rsquo96) pp 963ndash968 Lille France 1996

[40] D Matignon ldquoStability properties for generalized fractionaldifferential systemsrdquo in Proceedings of the Fractional DifferentialSystems Models Methods and Applications pp 145ndash158 1998

[41] H Akcay and R Malti ldquoOn the completeness problem for frac-tional rationals with incommensurable differentiation ordersrdquoin Proceedings of the 17th World Congress International Feder-ation of Automatic Control (IFAC rsquo08) pp 15367ndash15371 SoulKorea July 2008

[42] L S Pontryagin V G Boltyanskii R V Gamkrelidze and EF Mishchenko The Mathematical Theory of Optimal ProcessesJohn Wiley amp Sons New York NY USA 1962

[43] Y Y Liu J J Slotine and A L Barabasi ldquoControllability ofcomplex networksrdquo Nature vol 473 pp 167ndash173 2011

[44] D Matignon and B DrsquoAndrea-Novel ldquoSome results on control-lability and observability of nite-dimensional fractional differ-ential systemsrdquo in Proceedings of Computational Engineering inSystems Applications pp 952ndash956 Lille France 1996

[45] M Aoun R Malti F Levron and A Oustaloup ldquoNumericalsimulations of fractional systems an overview of existingmeth-ods and improvementsrdquo Nonlinear Dynamics vol 38 no 1ndash4pp 117ndash131 2004

[46] M S Tavazoei and M Haeri ldquoUnreliability of frequency-domain approximation in recognising chaos in fractional-ordersystemsrdquo IET Signal Processing vol 1 no 4 pp 171ndash181 2007

[47] M S Tavazoei andM Haeri ldquoA necessary condition for doublescroll attractor existence in fractional-order systemsrdquo PhysicsLetters A vol 367 no 1-2 pp 102ndash113 2007

[48] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications A vol 69no 4 pp 1299ndash1320 2008


[49] C A Monje Y Chen B M Vinagre D Xue and V FeliuFractional-Order Systems and Controls Fundamentals andApplications Springer New York NY USA 2010

[50] S Das Functional Fractional Calculus for System Identificationand Controls Springer Berlin Germany 2008

[51] S Das and I Pan Fractional Order Signal Processing Introduc-tory Concepts and Applications Springer Heidelberg Germany2012

[52] W M Haddad and V Chellaboina Nonlinear DynamicalSystems and Control A Lyapunov-Based Approach PrincetonUniversity Press Princeton NJ USA 2008

[53] K Balachandran J Y Park and J J Trujillo ldquoControllabilityof nonlinear fractional dynamical systemsrdquo Nonlinear AnalysisTheory Methods amp Applications A vol 75 no 4 pp 1919ndash19262012

[54] A Debbouche and D Baleanu ldquoControllability of fractionalevolution nonlocal impulsive quasilinear delay integro-differ-ential systemsrdquo Computers amp Mathematics with Applicationsvol 62 no 3 pp 1442ndash1450 2011

[55] R Dorville G M Mophou and V S Valmorin ldquoOptimalcontrol of a nonhomogeneous Dirichlet boundary fractionaldiffusion equationrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1472ndash1481 2011

[56] M Hazi and M Bragdi ldquoControllability of fractional integrod-ifferential systems via semigroup theory in Banach spacesrdquoMathematical Journal of Okayama University vol 54 pp 133ndash143 2012

[57] V Kavitha and M M Arjunan ldquoControllability of impulsivequasi-linear fractional mixed Volterra-Fredholm-type inte-grodifferential equations inBanach spacesrdquo Journal ofNonlinearScience and its Applications vol 4 no 2 pp 152ndash169 2011

[58] G M Mophou ldquoOptimal control of fractional diffusion equa-tionrdquo Computers amp Mathematics with Applications vol 61 no1 pp 68ndash78 2011

[59] D Mozyrska and D F M Torres ldquoMinimal modified energycontrol for fractional linear control systems with the Caputoderivativerdquo Carpathian Journal of Mathematics vol 26 no 2pp 210ndash221 2010

[60] R Sakthivel Y Ren and N I Mahmudov ldquoOn the approximatecontrollability of semilinear fractional differential systemsrdquoComputers amp Mathematics with Applications vol 62 no 3 pp1451ndash1459 2011

[61] J Si and W Jiang ldquoThe optimal control for a class of fractionaldifferential equationsrdquo International Journal of Biomathematicsvol 26 no 1 pp 17ndash24 2011

[62] N Sukavanam and S Kumar ldquoApproximate controllability offractional order semilinear delay systemsrdquo Journal of Optimiza-tion Theory and Applications vol 151 no 2 pp 373ndash384 2011

[63] Z Tai ldquoControllability of fractional impulsive neutral integrod-ifferential systems with a nonlocal Cauchy condition in Banachspacesrdquo Applied Mathematics Letters vol 24 no 12 pp 2158ndash2161 2011

[64] J Wang and Y Zhou ldquoExistence and controllability results forfractional semilinear differential inclusionsrdquo Nonlinear Analy-sis Real World Applications vol 12 no 6 pp 3642ndash3653 2011

[65] J Wang and Y Zhou ldquoAnalysis of nonlinear fractional controlsystems in Banach spacesrdquoNonlinear Analysis Theory Methodsamp Applications A vol 74 no 17 pp 5929ndash5942 2011

[66] J Wang Y Zhou W Wei and H Xu ldquoNonlocal problems forfractional integrodifferential equations via fractional operatorsand optimal controlsrdquo Computers amp Mathematics with Applica-tions vol 62 no 3 pp 1427ndash1441 2011

[67] C Yeroglu andN Tan ldquoAbsolute stability of uncertain fractionalorder control systemsrdquo Journal of Applied Functional Analysisvol 6 no 3 pp 211ndash218 2011

[68] V E Tarasov Fractional Dynamics Applications of FractionalCalculus to Dynamics of Particles Fields and Media SpringerNew York NY USA 2010

[69] I Petras ldquoTuning and implementation methods for fractional-order controllersrdquo Fractional Calculus and Applied Analysis vol15 no 2 pp 282ndash303 2012

[70] Y Luo and Y Q Chen ldquoFractional-order [proportional deriva-tive] controller for robustmotion control tuning procedure andvalidationrdquo in Proceedings of the American Control Conference(ACC rsquo09) pp 1412ndash1417 Hyatt Regency Riverfront St LouisMo USA June 2009

[71] C A Monje B M Vinagre V Feliu and Y Chen ldquoTuningand auto-tuning of fractional order controllers for industryapplicationsrdquo Control Engineering Practice vol 16 no 7 pp798ndash812 2008

[72] B M Vinagre C A Monje A J Caldern and J I Surez ldquoFrac-tional PID controllers for industry application a brief introduc-tionrdquo Journal of Vibration andControl vol 13 no 9-10 pp 1419ndash1429 2007

[73] A Oustaloup La Derivation Non Entiere Theorie Synthese etApplications Hermes Paris France 1995

[74] H W Bode Network Analysis and Feedback Amplifier DesignTung Hwa Book Company 1949

[75] S Manabe ldquoThe non-integer integral and its application tocontrol systemsrdquo ETJ of Japan vol 6 no 3-4 pp 83ndash87 1961

[76] B J Lurie ldquoThree-parameter tunable Tilt-Integral-Derivative(TID) controllerrdquo United States Patent 5371670 1994

[77] T Kaczorek ldquoPractical stability of positive fractional 2D linearsystemsrdquo Multidimensional Systems and Signal Processing vol21 no 3 pp 231ndash238 2010

[78] M Busłowicz and T Kaczorek ldquoSimple conditions for practicalstability of positive fractional discrete-time linear systemsrdquoInternational Journal of Applied Mathematics and ComputerScience vol 19 no 2 pp 263ndash269 2009

[79] S E Hamamci ldquoAn algorithm for stabilization of fractional-order time delay systems using fractional-order PID con-trollersrdquo IEEE Transactions on Automatic Control vol 52 no10 pp 1964ndash1968 2007

[80] S E Hamamci and M Koksal ldquoCalculation of all stabilizingfractional-order PD controllers for integrating time delay sys-temsrdquo Computers amp Mathematics with Applications vol 59 no5 pp 1621ndash1629 2010

[81] T Liang J Chen and C Lei ldquoAlgorithm of robust stabilityregion for interval plant with time delay using fractional orderPI120582D120583 controllerrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 2 pp 979ndash991 2012

[82] D Baleanu S J Sadati R Ghaderi A Ranjbar T Abdeljawadand F Jarad ldquoRazumikhin stability theorem for fractionalsystems with delayrdquo Abstract and Applied Analysis vol 2010Article ID 124812 9 pages 2010

[83] D Baleanu A Ranjbar N S J Sadati R H Delavari T Abdel-jawad and V Gejji ldquoLyapunov-Krasovskii stability theorem forfractional systems with delayrdquo Romanian Journal of Physics vol56 no 5-6 pp 636ndash643 2011

[84] M De la Sen ldquoPositivity and stability of the solutions ofCaputo fractional linear time-invariant systems of any orderwith internal point delaysrdquo Abstract and Applied Analysis vol2011 Article ID 161246 25 pages 2011


[85] H S Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[86] A N Kochubei ldquoDistributed order calculus and equationsof ultraslow diffusionrdquo Journal of Mathematical Analysis andApplications vol 340 no 1 pp 252ndash281 2008

[87] A V Letnikov and V A Chernykh The Foundations of Frac-tional Calculus Neftegaz Moscow Russia 2011 (Russian)

[88] A Dzielinski and D Sierociuk ldquoAdaptive feedback control offractional order discrete state-space systemsrdquo in Proceedingsof the International Conference on Computational Intelligencefor Modelling Control and Automation and the InternationalConference on Intelligent Agents Web Technologies and Inter-net Commerce (CIMCA-IAWTICrsquo05) pp 804ndash809 November2005

[89] S Guermah S Djennoune and M Bettayeb ldquoA new approachfor stability analysis of linear discrete-time fractional-ordersystemsrdquo in New Trends in Nanotechnology and FractionalCalculus Applications D Baleanu Z B Guvenc and J ATenreiro Machado Eds pp 151ndash162 Springer Dordrecht TheNetherlands 2010

[90] F Chen and Z Liu ldquoAsymptotic stability results for nonlinearfractional difference equationsrdquo Journal of AppliedMathematicsvol 2012 Article ID 879657 14 pages 2012

[91] P Ostalczyk ldquoStability analysis of a discrete-time system withvariablefractional-order controlrdquo Bulletin of the Polish Academyof Sciences vol 58 no 4 pp 613ndash619 2010

[92] J Klamka ldquoControllability and minimum energy control prob-lem of fractional discrete-time systemsrdquo in New Trends in Nan-otechnology and Fractional Calculus Applications D BaleanuZ B Guvenc and J A Tenreiro Machado Eds pp 503ndash509Springer New York NY USA 2010

[93] A Oustaloup J Sabatier P Lanusse R Malti P Melchior andMMoze ldquoAn overview of the CRONE approach in system anal-ysis modeling and identification observation and controlrdquo inProceedings of the 17th World Congress International Federationof Automatic Control (IFAC rsquo08) pp 14254ndash14265 Soul KoreaJuly 2008

[94] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[95] C Yakar ldquoFractional differential equations in terms of compar-ison results and Lyapunov stability with initial time differencerdquoAbstract and Applied Analysis vol 2010 Article ID 762857 16pages 2010

[96] F Zhang and C Li ldquoStability analysis of fractional differentialsystems with order lying in (12)rdquo Advances in DifferenceEquations vol 2011 Article ID 213485 2011

[97] F Zhang C Li andYChen ldquoAsymptotical stability of nonlinearfractional differential system with Caputo derivativerdquo Interna-tional Journal of Differential Equations vol 2011 Article ID635165 12 pages 2011

[98] J A Tenreiro Machado ldquoRoot locus of fractional linear sys-temsrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 10 pp 3855ndash3862 2011

[99] D Valerio and J S da Costa ldquoTuning of fractional PIDcontrollers with Ziegler-Nichols-type rulesrdquo Signal Processingvol 86 no 10 pp 2771ndash2784 2006

[100] A I Aptekarev and W Van Assche ldquoScalar and matrixRiemann-Hilbert approach to the strong asymptotics of Pade

approximants and complex orthogonal polynomials with vary-ing weightrdquo Journal of ApproximationTheory vol 129 no 2 pp129ndash166 2004

[101] A I Aptekarev ldquoAsymptotics of Hermite-Pade approximantsfor a pair of functions with branch pointsrdquo Doklady Mathemat-ics vol 78 no 2 pp 717ndash719 2008

[102] A I Aptekarev A Kuijlaars and W Van Assche ldquoAsymptoticsof Hermite-Pade rational approximants for two analytic func-tions with separated pairs of branch points (case of genus 0)rdquoInternationalMathematics Research Papers vol 2008 Article IDrpm007 p 128 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


with constant coefficients or of a linear system (where theprime symbol denotes first derivative with respect to time)

1198831015840

= 119860119883 119883 isin R119899

(5)

with a constant matrix 119860 the characteristic polynomial ofwhich is 119875(119911)

A very general result on the zero location problem forthe polynomial is the Hermite-Biehler theorem [5] whichstates that the roots of 119892(119911) + 119894ℎ(119911) are all on the same sideof the real axis when 119892(119911) and ℎ(119911) are polynomials withreal coefficients if and only if the zeros of 119892(119911) and ℎ(119911)

are real and alternateTheHermite-Biehler theorem providesnecessary and sufficient conditions for Hurwitz stability ofreal polynomials in terms of an interlacing (alternating)property [6ndash8] Notice that if a given real polynomial is nota Hurwitz one then the Hermite-Biehler theorem does notprovide information on its roots distribution

During the last years several surveys addressed this topicsee for instance [9ndash12]This paper without being exhaustiveis complementary to the contents of such other works In ourpaper we introduced formally a selected set of methods tocharacterize the stability of fractional systems It is intendedto form a comprehensive text so that readers can follow easilythe concepts Furthermore the limitations of the knownmethods are also pointed out giving readers the opportunityto consider the open problems

Bearing these ideas in mind this paper is organized asfollows Sections 2 and 3 introduce fundamental aspectsnamely the concepts of quasi-polynomials and fractionalquasi-polynomials respectively Section 4 addresses themaincore of the paper the stability of fractional order systemsand is divided into eight subsections The section starts bypresenting general fractional order systems In the next sub-sections issues associated with linear time-invariant systemsare discussed in more details such as controllers positivesystems systems with delay and distributed discrete-timeand nonlinear systems Stability of closed-loop linear controlsystems becomes a major motivation of the paper Thereforefour examples in Section 5 deal with this topic FinallySection 6 outlines several techniques that are presentlyemerging and draws the main conclusions

2 Quasi-Polynomials

Pontryagin [13] gave a generalization of the Hermite-BiehlerTheorem which appeared to be very relevant formal tool forthe mathematical analysis of stability of quasi-polynomialsthat is of the functions of the following type

119865 (119911) =

119899

sum

119896=0

119891119896(119911) 119890

120582119896119911 (6)

where 119891119896(119911) are polynomials in 119911 with constant coefficients

and 120582119896 119896 = 0 119899 are real (or complex) numbers By

other words 119865(119911) is a sum where the terms are the productof and exponential and polynomial function with constantcoefficients In control theory such exponentials correspondto delays If 120582

119896are commensurable real numbers that is

120582119896

= 120582sdot119896 119896 = 0 119899 and 120582 gt 0 then the quasi-polynomial(6) can be written in the form

120575 (119911) = 119875 (119911 119890119911

) (7)

with

119875 (119911 119904) =

119898

sum

119895=0

119899

sum

119896=0

119886119895119896

119911119895

119904120582119896

119911 119904 isin C (8)

where 119875(119911 119904) is a polynomial function of two variables andthen if 119904 = 119890

119911 we get 120575(119911)Thus from this point of view the determination of the

zeros of a quasi-polynomial (7) by means of Pontryagintheorem can be considered to be a mathematical methodfor analysis of stabilization of a class of linear time invariantsystems with time delay (see eg [14])

119898

sum

119895=0

119899

sum

119896=0

119886119895119896

119909(119895)

(119905 + 120582 sdot 119896) = 0 (9)

where 119886119895119896are constant coefficients

PontryaginTheorem (see [13]) Let 120575(119911) = 119875(119911 119890119911

) be aquasi-polynomial of the type (7) where 119875(119911 119904) is a polynomialfunction in two variables with real coefficients Suppose thatthe ldquooldestrdquo coefficient 119886

119898119899= 0 Let 120575(119894120596) be the restriction of

the quasi-polynomial 120575(119911) to imaginary axis One can express120575(119894120596) = 119891(120596) + 119894119892(120596) where the real functions (of a realvariable) 119891(120596) and 119892(120596) are the real and imaginary parts of120575(119894120596) respectively Let one denote by 120596

119903and 120596

119894 respectively

the zeros of the functions 119891(120596) and 119892(120596) If all the zeros ofthe quasi-polynomial 120575(119911) lie to the left side of the imaginaryaxis then the zeros of the functions 119891(120596) and 119892(120596) are realalternating and

1198921015840

(120596) 119891 (120596) minus 119892 (120596) 1198911015840

(120596) gt 0 (10)

for each 120596 isin RReciprocally let one of the following conditions be satisfied

(1) all the zeros of the functions119891(120596) and 119892(120596) are real andalternate and the inequality (10) is satisfied for at leastone value 120596

(2) all the zeros of the function 119891(120596) are real and for eachzero of 119891(120596) the inequality (10) is satisfied that is119892(120596

119903)119891

1015840

(120596119903) lt 0

(3) all the zeros of the function 119892(120596) are real and for eachzero of 119892(120596) the inequality (10) is satisfied that is1198921015840

(120596119894)119891(120596

119894) gt 0

Then all the zeros of the quasi-polynomial 120575(119911) lie to the leftside of the imaginary axis

In [15] quasi-polynomial of the type

119865 (119911) = 119860 (119911) + 119861 (119911) 119890minus120591119911 (11)

is studied where119860(119911) and119861(119911) are polynomials with constantcoefficients given by

119860 (119911) =

119898

sum

119896=0

119886119898minus119896

119911119896

119861 (119911) =

119899

sum

119896=0

119887119899minus119896

119911119896

(12)



120591119911 isthe term 119888

119896119911119896



infin

sum

119895=1

1

119911119895

=1

2[

1198651015840

(119911)

119865(119911)]

119911=infin

minus [119865

1015840

(119911)

119865(119911)]

119911=0

(13)


1199091015840

(119905) = 119860119909 (119905 minus 120591) 119909 isin R119899 (14)



2

1198902119911

minus tr(119860)119911119890119911


(120587

2)

2

+120587

2tr (119860) + |119860| gt 0

0 lt |119860| lt 1205772

lt (120587

2)

2

(15)




119908 (119905) = 119891 (119905) + int

119905

0

119860 (119905 minus 120591)119908 (120591) 119889120591 + 119887119910 (119905)

1199101015840

(119905) = 119862119908 (119905) + 119863119910 (119905) 119910 (0) = 1199100

(16)



infin

sum

119896=0

119875119896(119911) 119890

minus120582119896119911

119911 = 120590 + 119894119905 120582119896

isin C

119875119896(119911) isin C [119911] Re 120582

119896uarr infin

(17)



1lt 120582


120582119899


119866(119909 119891(1198981) (119909 + ℎ

1) 119891

(119898119903) (119909 + ℎ119903)) = 0

119866 (119909 1199091 119909

119903) = sum119862

1198961 1198961199031199091198961 sdot sdot sdot 119909

119896119903

(18)

where 1198621198961 119896119903


119896infin





119875 (119911) =

1199020(119911) + sum

1198982

119895=1119902119895(119911) exp (minus119911

119903

120574119895)

1199010(119911) + sum

1198981

119895=1119901119895(119911) exp (minus119911119903120575

119895)

=119873 (119911)

119863 (119911) (19)


119895(119911) and 119902

119895(119911)with


119901119895(119911) =

119899

sum

119896=0

119886119895119896

119911120572119896 119895 = 0 1 119898

1

119902119895(119911) =

119898

sum

119896=0

119887119895119896

119911120573119896 119895 = 0 1 119898

2

(20)

where 120572119896 120573


0119899= 0

1198870119898



119863 (119911) = 1199010(119911) +

1198981

sum

119895=1

119901119895(119911) exp (minus119911

119903

120575119895) (21)


119896= 120572 sdot 119896 (119896 = 0 1 119899) 120573

119896=


119863 (120582) = 1199010(120582) +

1198981

sum

119895=1

119901119895(120582) exp (minus120582

119903120572

120575119895) 120582 = 119911

120572

(22)







1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

120572120587

2 (23)




Δ0le120596lt+infin

arg (119863 (119894120596)) =119899120587

2 (24)




arg (120595 (119894120596)) = 0 (25)

where

120595 (119911) =119863 (119911)

119908119903(119911)

(26)


119908119903(119911) = 119886

0119899(119911 + 119888)

120572119899 119888 gt 0 (27)


1198860+ 119886

11199111205721 + sdot sdot sdot + 119886

119899119911120572119899 = 119886

0+ 119886

1sdot 119890


+ 119886119899sdot 119890

120572119899119908

(28)




119908 (119911) = 0 forallRe 119911 ge 0 (29)



119882(119911 119902) = 119908 (119911 119902) 119902 isin 119876 = [0 1] (30)

119908 (119911 119902) = (1 minus 119902)119908119886(119911) + 119902119908

119887(119911) (31)


119887(119911) are frac-




119886(119911) be stable The


120599 (119895120596) =119908

119887(119895120596)

119908119886(119895120596)

120596 isin Ω = [0infin) (32)





119886119899119863

120572119899119910 (119905) + 119886119899minus1


0119863

1205720119910 (119905)

= 119887119898119863

120573119898119910 (119905) + 119887119898minus1


0119863

1205730119910 (119905)

(33)

where 119863120574

=0119863

120574


0119863

120574

119905or

Caputo 119862

0119863

120574



119866 (119904) =119887119898119904120573119898 + 119887


01199041205730

119886119899119904120572119899 + 119886


01199041205720

=119876 (119904)

119875 (119904) (34)



0 119887

119898 119887

0are


0 120573

119898 120573

0are given real



0 120573

119898gt



119896= 119896120572 119896 =

0 119899 120573119896



0 120573

119898 120573

0[35]


119866 (119904) =sum

119898

119896=0119887119896(119904

120572

)119896

sum119899

119896=0119886119896(119904

120572)119896

=119876 (119904

120572

)

(119904120572)

(35)


119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904120572 + 120582119894)119895

(36)


119894 In


119866 (119904) =

119899

sum

119894=1

119861119894

119904120572 + 120582119894

(37)


119910 (119905) = Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

sdot (L119906) (119904)

= Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

lowast 119906 (119905)

= (

119899

sum

119894=1

119861119894119905120572

119864120572120572

(minus120582119894119905120572

)) lowast 119906 (119905)

(38)



119864120583] (119911) =

infin

sum

119896=0

119911119896

Γ (120583119896 + ]) (39)


119886119899119863

120572119899119910 (119905) + 119886119899minus1


0119863

1205720119910 (119905) = 0 (40)


119910 (119905)

=1

119886119899

infin

sum

119896=0

(minus1)119896

119896sum


1198960ge0119896119899minus2ge0

(119896 1198960 119896

119899minus2)

times

119899minus2

prod

119894=0

(119886119894

119886119899

)

119896119894

E119896


119886119899

119886119899minus 119886

119899minus1 119886

119899+

119899minus2

sum

119895=0

(119886119899minus1

minus 119886119895) 119896

119895+ 1)

(41)

where (119896 1198960 119896



E119896(119905 119910 120583 ]) = 119905

120583119896+]minus1119864

(119896)

120583] (119910119905120583

) (119896 = 0 1 2 )

119864(119896)

120583] (119911) =

infin

sum

119895=0

(119895 + 119896)119911119895

119895Γ (120583119895 + 120583119896 + ])(119896 = 0 1 2 )

(42)


119864120583] (119911) =

infin

sum

119895=0

119911119895

Γ (120583119895 + ])(119896 = 0 1 2 ) (43)



1003816100381610038161003816arg (120582119894)1003816100381610038161003816 gt 120572

120587

2forall119894 = 1 119899 (44)


120572

)



119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904119902119894 + 120582119894)119895

(45)


119894 positive 119902

119894 and positive

integer 119903119894



0 lt 119902119894lt 2

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 lt 120587 (1 minus

119902119894

2) forall119894 = 1 119901 (46)




0119863

q119905x (119905) = Ax (119905) + Bu (119905)

y (119905) = Cx (119905)

(47)






1 119902


system (47) If 1199021

= sdot sdot sdot = 119902119899



119886= [119861|119860119861

|1198602



= [119862|119862119860|1198621198602

| sdot sdot sdot

|119862119860119899minus1




1003816100381610038161003816arg (eig (A))1003816100381610038161003816 gt 119902

120587

2 0 lt 119902 lt 2 (48)




119862 (119904) =119880 (119904)

119864 (119904)= 119870

119901+ 119879

119894119904minus120582

+ 119879119889119904120575

(120582 120575 gt 0) (49)






119862 (119904) = 1198620

(1 + 119904120596119887)119903

(1 + 119904120596ℎ)119903minus1

(50)



119862 (119904) = 119896119888(

119904 + 1120582

119904 + 1120582)

119903

(51)



119871 (119904) = (119904

120596119892119888

)

120572

(120572 lt 0) (52)





119862 (119904) =119879

1199041119899+

119868

119904+ 119863119904 (53)









Let C([119886 119887]R119899




C([minus119903 0]R119899


0minus 119903 119905

0+119860]R119899

) 1199050le 119905

0+119860 let x

119905(120579) isin C be a



119862

1199050

119863q119905x (119905) = f (119905 x

119905(119905)) (54)


minus 119903 1199050

+ 119860]R119899

) for any 119860 gt 0 q =


0minus 119903 to 119905

0 that is

x (1199050) = 120593 (55)




119862

119863120572

0119905x (119905) =

1

Γ (119896 minus 120572)int

119905

0

x(119896)

(120591)

(119905 minus 120591)120572+1minus119896

119889120591

=

119901

sum

119894=0

A119894x (119905 minus ℎ

119894) + Bu (119905)

(56)


0lt ℎ



A0 A


119894



119862

119889119900119863

120572

119905119909 (119905) = 119860

119862

119889119900119863

120573

119905119909 (119905) + 119861119906 (119905) 119909 (0) = 119909

0 (57)

where 0 lt 120573 lt 120572 le 1

119889119900119863

120572

119905(sdot) = int

120574

119905

119887 (120572)119889120572

(sdot)

119889119905120572119889120572 120574 gt 119897 ge 0 119887 (120572) ge 0 (58)

and119904119900119863

120572

119905(sdot) = 119889

120572





GL119886

119863120572

119905119891 (119905) = lim

ℎrarr0

119886Δ

120572

ℎ119891 (119905)

ℎ120572 (59)


119886Δ

120572

ℎ119891 (119905) =

[(119905minus119886)ℎ]

sum

119895=0

(minus1)119895

(120572

119895)119891 (119905 minus 119895ℎ) (60)

where (120572



x (119896 + 1) = Ax (119896) + Bu (119896) 119909 (0) = 1199090

y (119896) = Cx (119896) + Du (119896)

(61)


1(119896) 119909

119899(119896)] isin


=



Δ120572x (119896) =

1

ℎ120572

119896

sum

119895=0

(minus1)119895

(120572

119895) x (119896 minus 119895) (62)


Δ120572x (119896 + 1) = A

119889x (119896) + Bu (119896) x (0) = x

0 (63)


x (119896 + 1) =

119896

sum

119895=0

A119895x (119896 minus 119895) + Bu (119896) x (0) = x

0

y (119896) = Cx (119896) + Du (119896)

(64)


where A0

= A119889


119895= minus119888

119895+1119868119899for 119895 ge 2 with 119888

119895=

minus(minus1)119895

(120572


ΔΥx (119896 + 1) = A

119889x (119896) + Bu (119896)

x (119896 + 1) = ΔΥx (119896 + 1) +

119896+1

sum

119895=1

A119895x (119896 minus 119895 + 1)

ΔΥx (119896 + 1) =

[[[[

[

Δ1205721119909

1(119896 + 1)

Δ1205722119909

2(119896 + 1)

Δ

120572119899119909119899(119896 + 1)

]]]]

]

(65)




0119863

q119905x (119905) = f (x (119905) 119905)

x (0) = c(66)




such that



minus119902

(67)




1199050119863

q119905x (119905) = 119891 (x (119905) 119905)

x (1199050) = c

(68)


119909 (119905) le 119898 [119909 (1199050)] 119864

119902(minus120582(119905 minus 119905

0)119902

)119887

(69)


(119902 119902) 119902 isin (0 1) 120582 gt 0 119887 gt 0 and 119898(0) = 0 119898(x) ge


0



5 Some Examples




119906119862(119904)119866(119904) = 0


1)| = 1 PM = arg119862119866(119894120596

1)


120587) = minus120587 GM =

|119862119866(119894120596120587)|

minus1


3996119904125


12= minus10788plusmn119894 06064





11011

) minus 0151711990401855

)

and 119866(119904) = 119890minus05119904

(1 + 11990405


12= minus06620 plusmn 119894 04552 119904

34= minus20323 plusmn 119894 41818


= 35549 11990412

= 0 plusmn



=


1

05

00

minus05

minus1


minus5minus6

Re(119904)

Im(119904


1(0598 + 3996119904125

)

minus20 00

20 40 60 80 100 120

minus50

50

minus100

100

Re

Im


1(0598 + 3996119904125

) and 119870 = 1





(119904 + 1)120572


119906 the phase



1205722(119904 minus 1)1205721) for 120572

1= 33 120572

2= 23


119906= 43358 119904

12= 0 plusmn 119894 48493


120587=



00

minus1

minus2

minus2minus3

minus4

minus6

2

4

6

Re(119904)

Im(119904

)


) minus

0151711990401855

) and 119866(119904) = 119890minus05119904

(1 + 11990405

)

00

minus1minus1

minus2

minus2

minus3

minus4

minus15minus05 05

1

2

3

4

Re

Im


) minus

0151711990401855

) 119866(119904) = 119890minus05119904

(1 + 11990405

) and 119870 = 14098











00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00


minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25



(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120591119911 isthe term 119888

119896119911119896



infin

sum

119895=1

1

119911119895

=1

2[

1198651015840

(119911)

119865(119911)]

119911=infin

minus [119865

1015840

(119911)

119865(119911)]

119911=0

(13)


1199091015840

(119905) = 119860119909 (119905 minus 120591) 119909 isin R119899 (14)



2

1198902119911

minus tr(119860)119911119890119911


(120587

2)

2

+120587

2tr (119860) + |119860| gt 0

0 lt |119860| lt 1205772

lt (120587

2)

2

(15)




119908 (119905) = 119891 (119905) + int

119905

0

119860 (119905 minus 120591)119908 (120591) 119889120591 + 119887119910 (119905)

1199101015840

(119905) = 119862119908 (119905) + 119863119910 (119905) 119910 (0) = 1199100

(16)



infin

sum

119896=0

119875119896(119911) 119890

minus120582119896119911

119911 = 120590 + 119894119905 120582119896

isin C

119875119896(119911) isin C [119911] Re 120582

119896uarr infin

(17)



1lt 120582


120582119899


119866(119909 119891(1198981) (119909 + ℎ

1) 119891

(119898119903) (119909 + ℎ119903)) = 0

119866 (119909 1199091 119909

119903) = sum119862

1198961 1198961199031199091198961 sdot sdot sdot 119909

119896119903

(18)

where 1198621198961 119896119903


119896infin





119875 (119911) =

1199020(119911) + sum

1198982

119895=1119902119895(119911) exp (minus119911

119903

120574119895)

1199010(119911) + sum

1198981

119895=1119901119895(119911) exp (minus119911119903120575

119895)

=119873 (119911)

119863 (119911) (19)


119895(119911) and 119902

119895(119911)with


119901119895(119911) =

119899

sum

119896=0

119886119895119896

119911120572119896 119895 = 0 1 119898

1

119902119895(119911) =

119898

sum

119896=0

119887119895119896

119911120573119896 119895 = 0 1 119898

2

(20)

where 120572119896 120573


0119899= 0

1198870119898



119863 (119911) = 1199010(119911) +

1198981

sum

119895=1

119901119895(119911) exp (minus119911

119903

120575119895) (21)


119896= 120572 sdot 119896 (119896 = 0 1 119899) 120573

119896=


119863 (120582) = 1199010(120582) +

1198981

sum

119895=1

119901119895(120582) exp (minus120582

119903120572

120575119895) 120582 = 119911

120572

(22)







1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

120572120587

2 (23)




Δ0le120596lt+infin

arg (119863 (119894120596)) =119899120587

2 (24)




arg (120595 (119894120596)) = 0 (25)

where

120595 (119911) =119863 (119911)

119908119903(119911)

(26)


119908119903(119911) = 119886

0119899(119911 + 119888)

120572119899 119888 gt 0 (27)


1198860+ 119886

11199111205721 + sdot sdot sdot + 119886

119899119911120572119899 = 119886

0+ 119886

1sdot 119890


+ 119886119899sdot 119890

120572119899119908

(28)




119908 (119911) = 0 forallRe 119911 ge 0 (29)



119882(119911 119902) = 119908 (119911 119902) 119902 isin 119876 = [0 1] (30)

119908 (119911 119902) = (1 minus 119902)119908119886(119911) + 119902119908

119887(119911) (31)


119887(119911) are frac-




119886(119911) be stable The


120599 (119895120596) =119908

119887(119895120596)

119908119886(119895120596)

120596 isin Ω = [0infin) (32)





119886119899119863

120572119899119910 (119905) + 119886119899minus1


0119863

1205720119910 (119905)

= 119887119898119863

120573119898119910 (119905) + 119887119898minus1


0119863

1205730119910 (119905)

(33)

where 119863120574

=0119863

120574


0119863

120574

119905or

Caputo 119862

0119863

120574



119866 (119904) =119887119898119904120573119898 + 119887


01199041205730

119886119899119904120572119899 + 119886


01199041205720

=119876 (119904)

119875 (119904) (34)



0 119887

119898 119887

0are


0 120573

119898 120573

0are given real



0 120573

119898gt



119896= 119896120572 119896 =

0 119899 120573119896



0 120573

119898 120573

0[35]


119866 (119904) =sum

119898

119896=0119887119896(119904

120572

)119896

sum119899

119896=0119886119896(119904

120572)119896

=119876 (119904

120572

)

(119904120572)

(35)


119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904120572 + 120582119894)119895

(36)


119894 In


119866 (119904) =

119899

sum

119894=1

119861119894

119904120572 + 120582119894

(37)


119910 (119905) = Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

sdot (L119906) (119904)

= Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

lowast 119906 (119905)

= (

119899

sum

119894=1

119861119894119905120572

119864120572120572

(minus120582119894119905120572

)) lowast 119906 (119905)

(38)



119864120583] (119911) =

infin

sum

119896=0

119911119896

Γ (120583119896 + ]) (39)


119886119899119863

120572119899119910 (119905) + 119886119899minus1


0119863

1205720119910 (119905) = 0 (40)


119910 (119905)

=1

119886119899

infin

sum

119896=0

(minus1)119896

119896sum


1198960ge0119896119899minus2ge0

(119896 1198960 119896

119899minus2)

times

119899minus2

prod

119894=0

(119886119894

119886119899

)

119896119894

E119896


119886119899

119886119899minus 119886

119899minus1 119886

119899+

119899minus2

sum

119895=0

(119886119899minus1

minus 119886119895) 119896

119895+ 1)

(41)

where (119896 1198960 119896



E119896(119905 119910 120583 ]) = 119905

120583119896+]minus1119864

(119896)

120583] (119910119905120583

) (119896 = 0 1 2 )

119864(119896)

120583] (119911) =

infin

sum

119895=0

(119895 + 119896)119911119895

119895Γ (120583119895 + 120583119896 + ])(119896 = 0 1 2 )

(42)


119864120583] (119911) =

infin

sum

119895=0

119911119895

Γ (120583119895 + ])(119896 = 0 1 2 ) (43)



1003816100381610038161003816arg (120582119894)1003816100381610038161003816 gt 120572

120587

2forall119894 = 1 119899 (44)


120572

)



119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904119902119894 + 120582119894)119895

(45)


119894 positive 119902

119894 and positive

integer 119903119894



0 lt 119902119894lt 2

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 lt 120587 (1 minus

119902119894

2) forall119894 = 1 119901 (46)




0119863

q119905x (119905) = Ax (119905) + Bu (119905)

y (119905) = Cx (119905)

(47)






1 119902


system (47) If 1199021

= sdot sdot sdot = 119902119899



119886= [119861|119860119861

|1198602



= [119862|119862119860|1198621198602

| sdot sdot sdot

|119862119860119899minus1




1003816100381610038161003816arg (eig (A))1003816100381610038161003816 gt 119902

120587

2 0 lt 119902 lt 2 (48)




119862 (119904) =119880 (119904)

119864 (119904)= 119870

119901+ 119879

119894119904minus120582

+ 119879119889119904120575

(120582 120575 gt 0) (49)






119862 (119904) = 1198620

(1 + 119904120596119887)119903

(1 + 119904120596ℎ)119903minus1

(50)



119862 (119904) = 119896119888(

119904 + 1120582

119904 + 1120582)

119903

(51)



119871 (119904) = (119904

120596119892119888

)

120572

(120572 lt 0) (52)





119862 (119904) =119879

1199041119899+

119868

119904+ 119863119904 (53)









Let C([119886 119887]R119899




C([minus119903 0]R119899


0minus 119903 119905

0+119860]R119899

) 1199050le 119905

0+119860 let x

119905(120579) isin C be a



119862

1199050

119863q119905x (119905) = f (119905 x

119905(119905)) (54)


minus 119903 1199050

+ 119860]R119899

) for any 119860 gt 0 q =


0minus 119903 to 119905

0 that is

x (1199050) = 120593 (55)




119862

119863120572

0119905x (119905) =

1

Γ (119896 minus 120572)int

119905

0

x(119896)

(120591)

(119905 minus 120591)120572+1minus119896

119889120591

=

119901

sum

119894=0

A119894x (119905 minus ℎ

119894) + Bu (119905)

(56)


0lt ℎ



A0 A


119894



119862

119889119900119863

120572

119905119909 (119905) = 119860

119862

119889119900119863

120573

119905119909 (119905) + 119861119906 (119905) 119909 (0) = 119909

0 (57)

where 0 lt 120573 lt 120572 le 1

119889119900119863

120572

119905(sdot) = int

120574

119905

119887 (120572)119889120572

(sdot)

119889119905120572119889120572 120574 gt 119897 ge 0 119887 (120572) ge 0 (58)

and119904119900119863

120572

119905(sdot) = 119889

120572





GL119886

119863120572

119905119891 (119905) = lim

ℎrarr0

119886Δ

120572

ℎ119891 (119905)

ℎ120572 (59)


119886Δ

120572

ℎ119891 (119905) =

[(119905minus119886)ℎ]

sum

119895=0

(minus1)119895

(120572

119895)119891 (119905 minus 119895ℎ) (60)

where (120572



x (119896 + 1) = Ax (119896) + Bu (119896) 119909 (0) = 1199090

y (119896) = Cx (119896) + Du (119896)

(61)


1(119896) 119909

119899(119896)] isin


=



Δ120572x (119896) =

1

ℎ120572

119896

sum

119895=0

(minus1)119895

(120572

119895) x (119896 minus 119895) (62)


Δ120572x (119896 + 1) = A

119889x (119896) + Bu (119896) x (0) = x

0 (63)


x (119896 + 1) =

119896

sum

119895=0

A119895x (119896 minus 119895) + Bu (119896) x (0) = x

0

y (119896) = Cx (119896) + Du (119896)

(64)


where A0

= A119889


119895= minus119888

119895+1119868119899for 119895 ge 2 with 119888

119895=

minus(minus1)119895

(120572


ΔΥx (119896 + 1) = A

119889x (119896) + Bu (119896)

x (119896 + 1) = ΔΥx (119896 + 1) +

119896+1

sum

119895=1

A119895x (119896 minus 119895 + 1)

ΔΥx (119896 + 1) =

[[[[

[

Δ1205721119909

1(119896 + 1)

Δ1205722119909

2(119896 + 1)

Δ

120572119899119909119899(119896 + 1)

]]]]

]

(65)




0119863

q119905x (119905) = f (x (119905) 119905)

x (0) = c(66)




such that



minus119902

(67)




1199050119863

q119905x (119905) = 119891 (x (119905) 119905)

x (1199050) = c

(68)


119909 (119905) le 119898 [119909 (1199050)] 119864

119902(minus120582(119905 minus 119905

0)119902

)119887

(69)


(119902 119902) 119902 isin (0 1) 120582 gt 0 119887 gt 0 and 119898(0) = 0 119898(x) ge


0



5 Some Examples




119906119862(119904)119866(119904) = 0


1)| = 1 PM = arg119862119866(119894120596

1)


120587) = minus120587 GM =

|119862119866(119894120596120587)|

minus1


3996119904125


12= minus10788plusmn119894 06064





11011

) minus 0151711990401855

)

and 119866(119904) = 119890minus05119904

(1 + 11990405


12= minus06620 plusmn 119894 04552 119904

34= minus20323 plusmn 119894 41818


= 35549 11990412

= 0 plusmn



=


1

05

00

minus05

minus1


minus5minus6

Re(119904)

Im(119904


1(0598 + 3996119904125

)

minus20 00

20 40 60 80 100 120

minus50

50

minus100

100

Re

Im


1(0598 + 3996119904125

) and 119870 = 1





(119904 + 1)120572


119906 the phase



1205722(119904 minus 1)1205721) for 120572

1= 33 120572

2= 23


119906= 43358 119904

12= 0 plusmn 119894 48493


120587=



00

minus1

minus2

minus2minus3

minus4

minus6

2

4

6

Re(119904)

Im(119904

)


) minus

0151711990401855

) and 119866(119904) = 119890minus05119904

(1 + 11990405

)

00

minus1minus1

minus2

minus2

minus3

minus4

minus15minus05 05

1

2

3

4

Re

Im


) minus

0151711990401855

) 119866(119904) = 119890minus05119904

(1 + 11990405

) and 119870 = 14098











00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00


minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25



(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1003816100381610038161003816arg (120582)1003816100381610038161003816 gt

120572120587

2 (23)




Δ0le120596lt+infin

arg (119863 (119894120596)) =119899120587

2 (24)




arg (120595 (119894120596)) = 0 (25)

where

120595 (119911) =119863 (119911)

119908119903(119911)

(26)


119908119903(119911) = 119886

0119899(119911 + 119888)

120572119899 119888 gt 0 (27)


1198860+ 119886

11199111205721 + sdot sdot sdot + 119886

119899119911120572119899 = 119886

0+ 119886

1sdot 119890


+ 119886119899sdot 119890

120572119899119908

(28)




119908 (119911) = 0 forallRe 119911 ge 0 (29)



119882(119911 119902) = 119908 (119911 119902) 119902 isin 119876 = [0 1] (30)

119908 (119911 119902) = (1 minus 119902)119908119886(119911) + 119902119908

119887(119911) (31)


119887(119911) are frac-




119886(119911) be stable The


120599 (119895120596) =119908

119887(119895120596)

119908119886(119895120596)

120596 isin Ω = [0infin) (32)





119886119899119863

120572119899119910 (119905) + 119886119899minus1


0119863

1205720119910 (119905)

= 119887119898119863

120573119898119910 (119905) + 119887119898minus1


0119863

1205730119910 (119905)

(33)

where 119863120574

=0119863

120574


0119863

120574

119905or

Caputo 119862

0119863

120574



119866 (119904) =119887119898119904120573119898 + 119887


01199041205730

119886119899119904120572119899 + 119886


01199041205720

=119876 (119904)

119875 (119904) (34)



0 119887

119898 119887

0are


0 120573

119898 120573

0are given real



0 120573

119898gt



119896= 119896120572 119896 =

0 119899 120573119896



0 120573

119898 120573

0[35]


119866 (119904) =sum

119898

119896=0119887119896(119904

120572

)119896

sum119899

119896=0119886119896(119904

120572)119896

=119876 (119904

120572

)

(119904120572)

(35)


119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904120572 + 120582119894)119895

(36)


119894 In


119866 (119904) =

119899

sum

119894=1

119861119894

119904120572 + 120582119894

(37)


119910 (119905) = Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

sdot (L119906) (119904)

= Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

lowast 119906 (119905)

= (

119899

sum

119894=1

119861119894119905120572

119864120572120572

(minus120582119894119905120572

)) lowast 119906 (119905)

(38)



119864120583] (119911) =

infin

sum

119896=0

119911119896

Γ (120583119896 + ]) (39)


119886119899119863

120572119899119910 (119905) + 119886119899minus1


0119863

1205720119910 (119905) = 0 (40)


119910 (119905)

=1

119886119899

infin

sum

119896=0

(minus1)119896

119896sum


1198960ge0119896119899minus2ge0

(119896 1198960 119896

119899minus2)

times

119899minus2

prod

119894=0

(119886119894

119886119899

)

119896119894

E119896


119886119899

119886119899minus 119886

119899minus1 119886

119899+

119899minus2

sum

119895=0

(119886119899minus1

minus 119886119895) 119896

119895+ 1)

(41)

where (119896 1198960 119896



E119896(119905 119910 120583 ]) = 119905

120583119896+]minus1119864

(119896)

120583] (119910119905120583

) (119896 = 0 1 2 )

119864(119896)

120583] (119911) =

infin

sum

119895=0

(119895 + 119896)119911119895

119895Γ (120583119895 + 120583119896 + ])(119896 = 0 1 2 )

(42)


119864120583] (119911) =

infin

sum

119895=0

119911119895

Γ (120583119895 + ])(119896 = 0 1 2 ) (43)



1003816100381610038161003816arg (120582119894)1003816100381610038161003816 gt 120572

120587

2forall119894 = 1 119899 (44)


120572

)



119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904119902119894 + 120582119894)119895

(45)


119894 positive 119902

119894 and positive

integer 119903119894



0 lt 119902119894lt 2

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 lt 120587 (1 minus

119902119894

2) forall119894 = 1 119901 (46)




0119863

q119905x (119905) = Ax (119905) + Bu (119905)

y (119905) = Cx (119905)

(47)






1 119902


system (47) If 1199021

= sdot sdot sdot = 119902119899



119886= [119861|119860119861

|1198602



= [119862|119862119860|1198621198602

| sdot sdot sdot

|119862119860119899minus1




1003816100381610038161003816arg (eig (A))1003816100381610038161003816 gt 119902

120587

2 0 lt 119902 lt 2 (48)




119862 (119904) =119880 (119904)

119864 (119904)= 119870

119901+ 119879

119894119904minus120582

+ 119879119889119904120575

(120582 120575 gt 0) (49)






119862 (119904) = 1198620

(1 + 119904120596119887)119903

(1 + 119904120596ℎ)119903minus1

(50)



119862 (119904) = 119896119888(

119904 + 1120582

119904 + 1120582)

119903

(51)



119871 (119904) = (119904

120596119892119888

)

120572

(120572 lt 0) (52)





119862 (119904) =119879

1199041119899+

119868

119904+ 119863119904 (53)









Let C([119886 119887]R119899




C([minus119903 0]R119899


0minus 119903 119905

0+119860]R119899

) 1199050le 119905

0+119860 let x

119905(120579) isin C be a



119862

1199050

119863q119905x (119905) = f (119905 x

119905(119905)) (54)


minus 119903 1199050

+ 119860]R119899

) for any 119860 gt 0 q =


0minus 119903 to 119905

0 that is

x (1199050) = 120593 (55)




119862

119863120572

0119905x (119905) =

1

Γ (119896 minus 120572)int

119905

0

x(119896)

(120591)

(119905 minus 120591)120572+1minus119896

119889120591

=

119901

sum

119894=0

A119894x (119905 minus ℎ

119894) + Bu (119905)

(56)


0lt ℎ



A0 A


119894



119862

119889119900119863

120572

119905119909 (119905) = 119860

119862

119889119900119863

120573

119905119909 (119905) + 119861119906 (119905) 119909 (0) = 119909

0 (57)

where 0 lt 120573 lt 120572 le 1

119889119900119863

120572

119905(sdot) = int

120574

119905

119887 (120572)119889120572

(sdot)

119889119905120572119889120572 120574 gt 119897 ge 0 119887 (120572) ge 0 (58)

and119904119900119863

120572

119905(sdot) = 119889

120572





GL119886

119863120572

119905119891 (119905) = lim

ℎrarr0

119886Δ

120572

ℎ119891 (119905)

ℎ120572 (59)


119886Δ

120572

ℎ119891 (119905) =

[(119905minus119886)ℎ]

sum

119895=0

(minus1)119895

(120572

119895)119891 (119905 minus 119895ℎ) (60)

where (120572



x (119896 + 1) = Ax (119896) + Bu (119896) 119909 (0) = 1199090

y (119896) = Cx (119896) + Du (119896)

(61)


1(119896) 119909

119899(119896)] isin


=



Δ120572x (119896) =

1

ℎ120572

119896

sum

119895=0

(minus1)119895

(120572

119895) x (119896 minus 119895) (62)


Δ120572x (119896 + 1) = A

119889x (119896) + Bu (119896) x (0) = x

0 (63)


x (119896 + 1) =

119896

sum

119895=0

A119895x (119896 minus 119895) + Bu (119896) x (0) = x

0

y (119896) = Cx (119896) + Du (119896)

(64)


where A0

= A119889


119895= minus119888

119895+1119868119899for 119895 ge 2 with 119888

119895=

minus(minus1)119895

(120572


ΔΥx (119896 + 1) = A

119889x (119896) + Bu (119896)

x (119896 + 1) = ΔΥx (119896 + 1) +

119896+1

sum

119895=1

A119895x (119896 minus 119895 + 1)

ΔΥx (119896 + 1) =

[[[[

[

Δ1205721119909

1(119896 + 1)

Δ1205722119909

2(119896 + 1)

Δ

120572119899119909119899(119896 + 1)

]]]]

]

(65)




0119863

q119905x (119905) = f (x (119905) 119905)

x (0) = c(66)




such that



minus119902

(67)




1199050119863

q119905x (119905) = 119891 (x (119905) 119905)

x (1199050) = c

(68)


119909 (119905) le 119898 [119909 (1199050)] 119864

119902(minus120582(119905 minus 119905

0)119902

)119887

(69)


(119902 119902) 119902 isin (0 1) 120582 gt 0 119887 gt 0 and 119898(0) = 0 119898(x) ge


0



5 Some Examples




119906119862(119904)119866(119904) = 0


1)| = 1 PM = arg119862119866(119894120596

1)


120587) = minus120587 GM =

|119862119866(119894120596120587)|

minus1


3996119904125


12= minus10788plusmn119894 06064





11011

) minus 0151711990401855

)

and 119866(119904) = 119890minus05119904

(1 + 11990405


12= minus06620 plusmn 119894 04552 119904

34= minus20323 plusmn 119894 41818


= 35549 11990412

= 0 plusmn



=


1

05

00

minus05

minus1


minus5minus6

Re(119904)

Im(119904


1(0598 + 3996119904125

)

minus20 00

20 40 60 80 100 120

minus50

50

minus100

100

Re

Im


1(0598 + 3996119904125

) and 119870 = 1





(119904 + 1)120572


119906 the phase



1205722(119904 minus 1)1205721) for 120572

1= 33 120572

2= 23


119906= 43358 119904

12= 0 plusmn 119894 48493


120587=



00

minus1

minus2

minus2minus3

minus4

minus6

2

4

6

Re(119904)

Im(119904

)


) minus

0151711990401855

) and 119866(119904) = 119890minus05119904

(1 + 11990405

)

00

minus1minus1

minus2

minus2

minus3

minus4

minus15minus05 05

1

2

3

4

Re

Im


) minus

0151711990401855

) 119866(119904) = 119890minus05119904

(1 + 11990405

) and 119870 = 14098











00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00


minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25



(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 119887

119898 119887

0are


0 120573

119898 120573

0are given real



0 120573

119898gt



119896= 119896120572 119896 =

0 119899 120573119896



0 120573

119898 120573

0[35]


119866 (119904) =sum

119898

119896=0119887119896(119904

120572

)119896

sum119899

119896=0119886119896(119904

120572)119896

=119876 (119904

120572

)

(119904120572)

(35)


119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904120572 + 120582119894)119895

(36)


119894 In


119866 (119904) =

119899

sum

119894=1

119861119894

119904120572 + 120582119894

(37)


119910 (119905) = Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

sdot (L119906) (119904)

= Lminus1

119899

sum

119894=1

119861119894

119904120572 + 120582119894

lowast 119906 (119905)

= (

119899

sum

119894=1

119861119894119905120572

119864120572120572

(minus120582119894119905120572

)) lowast 119906 (119905)

(38)



119864120583] (119911) =

infin

sum

119896=0

119911119896

Γ (120583119896 + ]) (39)


119886119899119863

120572119899119910 (119905) + 119886119899minus1


0119863

1205720119910 (119905) = 0 (40)


119910 (119905)

=1

119886119899

infin

sum

119896=0

(minus1)119896

119896sum


1198960ge0119896119899minus2ge0

(119896 1198960 119896

119899minus2)

times

119899minus2

prod

119894=0

(119886119894

119886119899

)

119896119894

E119896


119886119899

119886119899minus 119886

119899minus1 119886

119899+

119899minus2

sum

119895=0

(119886119899minus1

minus 119886119895) 119896

119895+ 1)

(41)

where (119896 1198960 119896



E119896(119905 119910 120583 ]) = 119905

120583119896+]minus1119864

(119896)

120583] (119910119905120583

) (119896 = 0 1 2 )

119864(119896)

120583] (119911) =

infin

sum

119895=0

(119895 + 119896)119911119895

119895Γ (120583119895 + 120583119896 + ])(119896 = 0 1 2 )

(42)


119864120583] (119911) =

infin

sum

119895=0

119911119895

Γ (120583119895 + ])(119896 = 0 1 2 ) (43)



1003816100381610038161003816arg (120582119894)1003816100381610038161003816 gt 120572

120587

2forall119894 = 1 119899 (44)


120572

)



119866 (119904) =

119901

sum

119894=1

119903119894

sum

119895=1

119860119894119895

(119904119902119894 + 120582119894)119895

(45)


119894 positive 119902

119894 and positive

integer 119903119894



0 lt 119902119894lt 2

1003816100381610038161003816arg (120582119894)1003816100381610038161003816 lt 120587 (1 minus

119902119894

2) forall119894 = 1 119901 (46)




0119863

q119905x (119905) = Ax (119905) + Bu (119905)

y (119905) = Cx (119905)

(47)






1 119902


system (47) If 1199021

= sdot sdot sdot = 119902119899



119886= [119861|119860119861

|1198602



= [119862|119862119860|1198621198602

| sdot sdot sdot

|119862119860119899minus1




1003816100381610038161003816arg (eig (A))1003816100381610038161003816 gt 119902

120587

2 0 lt 119902 lt 2 (48)




119862 (119904) =119880 (119904)

119864 (119904)= 119870

119901+ 119879

119894119904minus120582

+ 119879119889119904120575

(120582 120575 gt 0) (49)






119862 (119904) = 1198620

(1 + 119904120596119887)119903

(1 + 119904120596ℎ)119903minus1

(50)



119862 (119904) = 119896119888(

119904 + 1120582

119904 + 1120582)

119903

(51)



119871 (119904) = (119904

120596119892119888

)

120572

(120572 lt 0) (52)





119862 (119904) =119879

1199041119899+

119868

119904+ 119863119904 (53)









Let C([119886 119887]R119899




C([minus119903 0]R119899


0minus 119903 119905

0+119860]R119899

) 1199050le 119905

0+119860 let x

119905(120579) isin C be a



119862

1199050

119863q119905x (119905) = f (119905 x

119905(119905)) (54)


minus 119903 1199050

+ 119860]R119899

) for any 119860 gt 0 q =


0minus 119903 to 119905

0 that is

x (1199050) = 120593 (55)




119862

119863120572

0119905x (119905) =

1

Γ (119896 minus 120572)int

119905

0

x(119896)

(120591)

(119905 minus 120591)120572+1minus119896

119889120591

=

119901

sum

119894=0

A119894x (119905 minus ℎ

119894) + Bu (119905)

(56)


0lt ℎ



A0 A


119894



119862

119889119900119863

120572

119905119909 (119905) = 119860

119862

119889119900119863

120573

119905119909 (119905) + 119861119906 (119905) 119909 (0) = 119909

0 (57)

where 0 lt 120573 lt 120572 le 1

119889119900119863

120572

119905(sdot) = int

120574

119905

119887 (120572)119889120572

(sdot)

119889119905120572119889120572 120574 gt 119897 ge 0 119887 (120572) ge 0 (58)

and119904119900119863

120572

119905(sdot) = 119889

120572





GL119886

119863120572

119905119891 (119905) = lim

ℎrarr0

119886Δ

120572

ℎ119891 (119905)

ℎ120572 (59)


119886Δ

120572

ℎ119891 (119905) =

[(119905minus119886)ℎ]

sum

119895=0

(minus1)119895

(120572

119895)119891 (119905 minus 119895ℎ) (60)

where (120572



x (119896 + 1) = Ax (119896) + Bu (119896) 119909 (0) = 1199090

y (119896) = Cx (119896) + Du (119896)

(61)


1(119896) 119909

119899(119896)] isin


=



Δ120572x (119896) =

1

ℎ120572

119896

sum

119895=0

(minus1)119895

(120572

119895) x (119896 minus 119895) (62)


Δ120572x (119896 + 1) = A

119889x (119896) + Bu (119896) x (0) = x

0 (63)


x (119896 + 1) =

119896

sum

119895=0

A119895x (119896 minus 119895) + Bu (119896) x (0) = x

0

y (119896) = Cx (119896) + Du (119896)

(64)


where A0

= A119889


119895= minus119888

119895+1119868119899for 119895 ge 2 with 119888

119895=

minus(minus1)119895

(120572


ΔΥx (119896 + 1) = A

119889x (119896) + Bu (119896)

x (119896 + 1) = ΔΥx (119896 + 1) +

119896+1

sum

119895=1

A119895x (119896 minus 119895 + 1)

ΔΥx (119896 + 1) =

[[[[

[

Δ1205721119909

1(119896 + 1)

Δ1205722119909

2(119896 + 1)

Δ

120572119899119909119899(119896 + 1)

]]]]

]

(65)




0119863

q119905x (119905) = f (x (119905) 119905)

x (0) = c(66)




such that



minus119902

(67)




1199050119863

q119905x (119905) = 119891 (x (119905) 119905)

x (1199050) = c

(68)


119909 (119905) le 119898 [119909 (1199050)] 119864

119902(minus120582(119905 minus 119905

0)119902

)119887

(69)


(119902 119902) 119902 isin (0 1) 120582 gt 0 119887 gt 0 and 119898(0) = 0 119898(x) ge


0



5 Some Examples




119906119862(119904)119866(119904) = 0


1)| = 1 PM = arg119862119866(119894120596

1)


120587) = minus120587 GM =

|119862119866(119894120596120587)|

minus1


3996119904125


12= minus10788plusmn119894 06064





11011

) minus 0151711990401855

)

and 119866(119904) = 119890minus05119904

(1 + 11990405


12= minus06620 plusmn 119894 04552 119904

34= minus20323 plusmn 119894 41818


= 35549 11990412

= 0 plusmn



=


1

05

00

minus05

minus1


minus5minus6

Re(119904)

Im(119904


1(0598 + 3996119904125

)

minus20 00

20 40 60 80 100 120

minus50

50

minus100

100

Re

Im


1(0598 + 3996119904125

) and 119870 = 1





(119904 + 1)120572


119906 the phase



1205722(119904 minus 1)1205721) for 120572

1= 33 120572

2= 23


119906= 43358 119904

12= 0 plusmn 119894 48493


120587=



00

minus1

minus2

minus2minus3

minus4

minus6

2

4

6

Re(119904)

Im(119904

)


) minus

0151711990401855

) and 119866(119904) = 119890minus05119904

(1 + 11990405

)

00

minus1minus1

minus2

minus2

minus3

minus4

minus15minus05 05

1

2

3

4

Re

Im


) minus

0151711990401855

) 119866(119904) = 119890minus05119904

(1 + 11990405

) and 119870 = 14098











00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00


minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25



(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0119863

q119905x (119905) = Ax (119905) + Bu (119905)

y (119905) = Cx (119905)

(47)






1 119902


system (47) If 1199021

= sdot sdot sdot = 119902119899



119886= [119861|119860119861

|1198602



= [119862|119862119860|1198621198602

| sdot sdot sdot

|119862119860119899minus1




1003816100381610038161003816arg (eig (A))1003816100381610038161003816 gt 119902

120587

2 0 lt 119902 lt 2 (48)




119862 (119904) =119880 (119904)

119864 (119904)= 119870

119901+ 119879

119894119904minus120582

+ 119879119889119904120575

(120582 120575 gt 0) (49)






119862 (119904) = 1198620

(1 + 119904120596119887)119903

(1 + 119904120596ℎ)119903minus1

(50)



119862 (119904) = 119896119888(

119904 + 1120582

119904 + 1120582)

119903

(51)



119871 (119904) = (119904

120596119892119888

)

120572

(120572 lt 0) (52)





119862 (119904) =119879

1199041119899+

119868

119904+ 119863119904 (53)









Let C([119886 119887]R119899




C([minus119903 0]R119899


0minus 119903 119905

0+119860]R119899

) 1199050le 119905

0+119860 let x

119905(120579) isin C be a



119862

1199050

119863q119905x (119905) = f (119905 x

119905(119905)) (54)


minus 119903 1199050

+ 119860]R119899

) for any 119860 gt 0 q =


0minus 119903 to 119905

0 that is

x (1199050) = 120593 (55)




119862

119863120572

0119905x (119905) =

1

Γ (119896 minus 120572)int

119905

0

x(119896)

(120591)

(119905 minus 120591)120572+1minus119896

119889120591

=

119901

sum

119894=0

A119894x (119905 minus ℎ

119894) + Bu (119905)

(56)


0lt ℎ



A0 A


119894



119862

119889119900119863

120572

119905119909 (119905) = 119860

119862

119889119900119863

120573

119905119909 (119905) + 119861119906 (119905) 119909 (0) = 119909

0 (57)

where 0 lt 120573 lt 120572 le 1

119889119900119863

120572

119905(sdot) = int

120574

119905

119887 (120572)119889120572

(sdot)

119889119905120572119889120572 120574 gt 119897 ge 0 119887 (120572) ge 0 (58)

and119904119900119863

120572

119905(sdot) = 119889

120572





GL119886

119863120572

119905119891 (119905) = lim

ℎrarr0

119886Δ

120572

ℎ119891 (119905)

ℎ120572 (59)


119886Δ

120572

ℎ119891 (119905) =

[(119905minus119886)ℎ]

sum

119895=0

(minus1)119895

(120572

119895)119891 (119905 minus 119895ℎ) (60)

where (120572



x (119896 + 1) = Ax (119896) + Bu (119896) 119909 (0) = 1199090

y (119896) = Cx (119896) + Du (119896)

(61)


1(119896) 119909

119899(119896)] isin


=



Δ120572x (119896) =

1

ℎ120572

119896

sum

119895=0

(minus1)119895

(120572

119895) x (119896 minus 119895) (62)


Δ120572x (119896 + 1) = A

119889x (119896) + Bu (119896) x (0) = x

0 (63)


x (119896 + 1) =

119896

sum

119895=0

A119895x (119896 minus 119895) + Bu (119896) x (0) = x

0

y (119896) = Cx (119896) + Du (119896)

(64)


where A0

= A119889


119895= minus119888

119895+1119868119899for 119895 ge 2 with 119888

119895=

minus(minus1)119895

(120572


ΔΥx (119896 + 1) = A

119889x (119896) + Bu (119896)

x (119896 + 1) = ΔΥx (119896 + 1) +

119896+1

sum

119895=1

A119895x (119896 minus 119895 + 1)

ΔΥx (119896 + 1) =

[[[[

[

Δ1205721119909

1(119896 + 1)

Δ1205722119909

2(119896 + 1)

Δ

120572119899119909119899(119896 + 1)

]]]]

]

(65)




0119863

q119905x (119905) = f (x (119905) 119905)

x (0) = c(66)




such that



minus119902

(67)




1199050119863

q119905x (119905) = 119891 (x (119905) 119905)

x (1199050) = c

(68)


119909 (119905) le 119898 [119909 (1199050)] 119864

119902(minus120582(119905 minus 119905

0)119902

)119887

(69)


(119902 119902) 119902 isin (0 1) 120582 gt 0 119887 gt 0 and 119898(0) = 0 119898(x) ge


0



5 Some Examples




119906119862(119904)119866(119904) = 0


1)| = 1 PM = arg119862119866(119894120596

1)


120587) = minus120587 GM =

|119862119866(119894120596120587)|

minus1


3996119904125


12= minus10788plusmn119894 06064





11011

) minus 0151711990401855

)

and 119866(119904) = 119890minus05119904

(1 + 11990405


12= minus06620 plusmn 119894 04552 119904

34= minus20323 plusmn 119894 41818


= 35549 11990412

= 0 plusmn



=


1

05

00

minus05

minus1


minus5minus6

Re(119904)

Im(119904


1(0598 + 3996119904125

)

minus20 00

20 40 60 80 100 120

minus50

50

minus100

100

Re

Im


1(0598 + 3996119904125

) and 119870 = 1





(119904 + 1)120572


119906 the phase



1205722(119904 minus 1)1205721) for 120572

1= 33 120572

2= 23


119906= 43358 119904

12= 0 plusmn 119894 48493


120587=



00

minus1

minus2

minus2minus3

minus4

minus6

2

4

6

Re(119904)

Im(119904

)


) minus

0151711990401855

) and 119866(119904) = 119890minus05119904

(1 + 11990405

)

00

minus1minus1

minus2

minus2

minus3

minus4

minus15minus05 05

1

2

3

4

Re

Im


) minus

0151711990401855

) 119866(119904) = 119890minus05119904

(1 + 11990405

) and 119870 = 14098











00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00


minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25



(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Let C([119886 119887]R119899




C([minus119903 0]R119899


0minus 119903 119905

0+119860]R119899

) 1199050le 119905

0+119860 let x

119905(120579) isin C be a



119862

1199050

119863q119905x (119905) = f (119905 x

119905(119905)) (54)


minus 119903 1199050

+ 119860]R119899

) for any 119860 gt 0 q =


0minus 119903 to 119905

0 that is

x (1199050) = 120593 (55)




119862

119863120572

0119905x (119905) =

1

Γ (119896 minus 120572)int

119905

0

x(119896)

(120591)

(119905 minus 120591)120572+1minus119896

119889120591

=

119901

sum

119894=0

A119894x (119905 minus ℎ

119894) + Bu (119905)

(56)


0lt ℎ



A0 A


119894



119862

119889119900119863

120572

119905119909 (119905) = 119860

119862

119889119900119863

120573

119905119909 (119905) + 119861119906 (119905) 119909 (0) = 119909

0 (57)

where 0 lt 120573 lt 120572 le 1

119889119900119863

120572

119905(sdot) = int

120574

119905

119887 (120572)119889120572

(sdot)

119889119905120572119889120572 120574 gt 119897 ge 0 119887 (120572) ge 0 (58)

and119904119900119863

120572

119905(sdot) = 119889

120572





GL119886

119863120572

119905119891 (119905) = lim

ℎrarr0

119886Δ

120572

ℎ119891 (119905)

ℎ120572 (59)


119886Δ

120572

ℎ119891 (119905) =

[(119905minus119886)ℎ]

sum

119895=0

(minus1)119895

(120572

119895)119891 (119905 minus 119895ℎ) (60)

where (120572



x (119896 + 1) = Ax (119896) + Bu (119896) 119909 (0) = 1199090

y (119896) = Cx (119896) + Du (119896)

(61)


1(119896) 119909

119899(119896)] isin


=



Δ120572x (119896) =

1

ℎ120572

119896

sum

119895=0

(minus1)119895

(120572

119895) x (119896 minus 119895) (62)


Δ120572x (119896 + 1) = A

119889x (119896) + Bu (119896) x (0) = x

0 (63)


x (119896 + 1) =

119896

sum

119895=0

A119895x (119896 minus 119895) + Bu (119896) x (0) = x

0

y (119896) = Cx (119896) + Du (119896)

(64)


where A0

= A119889


119895= minus119888

119895+1119868119899for 119895 ge 2 with 119888

119895=

minus(minus1)119895

(120572


ΔΥx (119896 + 1) = A

119889x (119896) + Bu (119896)

x (119896 + 1) = ΔΥx (119896 + 1) +

119896+1

sum

119895=1

A119895x (119896 minus 119895 + 1)

ΔΥx (119896 + 1) =

[[[[

[

Δ1205721119909

1(119896 + 1)

Δ1205722119909

2(119896 + 1)

Δ

120572119899119909119899(119896 + 1)

]]]]

]

(65)




0119863

q119905x (119905) = f (x (119905) 119905)

x (0) = c(66)




such that



minus119902

(67)




1199050119863

q119905x (119905) = 119891 (x (119905) 119905)

x (1199050) = c

(68)


119909 (119905) le 119898 [119909 (1199050)] 119864

119902(minus120582(119905 minus 119905

0)119902

)119887

(69)


(119902 119902) 119902 isin (0 1) 120582 gt 0 119887 gt 0 and 119898(0) = 0 119898(x) ge


0



5 Some Examples




119906119862(119904)119866(119904) = 0


1)| = 1 PM = arg119862119866(119894120596

1)


120587) = minus120587 GM =

|119862119866(119894120596120587)|

minus1


3996119904125


12= minus10788plusmn119894 06064





11011

) minus 0151711990401855

)

and 119866(119904) = 119890minus05119904

(1 + 11990405


12= minus06620 plusmn 119894 04552 119904

34= minus20323 plusmn 119894 41818


= 35549 11990412

= 0 plusmn



=


1

05

00

minus05

minus1


minus5minus6

Re(119904)

Im(119904


1(0598 + 3996119904125

)

minus20 00

20 40 60 80 100 120

minus50

50

minus100

100

Re

Im


1(0598 + 3996119904125

) and 119870 = 1





(119904 + 1)120572


119906 the phase



1205722(119904 minus 1)1205721) for 120572

1= 33 120572

2= 23


119906= 43358 119904

12= 0 plusmn 119894 48493


120587=



00

minus1

minus2

minus2minus3

minus4

minus6

2

4

6

Re(119904)

Im(119904

)


) minus

0151711990401855

) and 119866(119904) = 119890minus05119904

(1 + 11990405

)

00

minus1minus1

minus2

minus2

minus3

minus4

minus15minus05 05

1

2

3

4

Re

Im


) minus

0151711990401855

) 119866(119904) = 119890minus05119904

(1 + 11990405

) and 119870 = 14098











00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00


minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25



(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where A0

= A119889


119895= minus119888

119895+1119868119899for 119895 ge 2 with 119888

119895=

minus(minus1)119895

(120572


ΔΥx (119896 + 1) = A

119889x (119896) + Bu (119896)

x (119896 + 1) = ΔΥx (119896 + 1) +

119896+1

sum

119895=1

A119895x (119896 minus 119895 + 1)

ΔΥx (119896 + 1) =

[[[[

[

Δ1205721119909

1(119896 + 1)

Δ1205722119909

2(119896 + 1)

Δ

120572119899119909119899(119896 + 1)

]]]]

]

(65)




0119863

q119905x (119905) = f (x (119905) 119905)

x (0) = c(66)




such that



minus119902

(67)




1199050119863

q119905x (119905) = 119891 (x (119905) 119905)

x (1199050) = c

(68)


119909 (119905) le 119898 [119909 (1199050)] 119864

119902(minus120582(119905 minus 119905

0)119902

)119887

(69)


(119902 119902) 119902 isin (0 1) 120582 gt 0 119887 gt 0 and 119898(0) = 0 119898(x) ge


0



5 Some Examples




119906119862(119904)119866(119904) = 0


1)| = 1 PM = arg119862119866(119894120596

1)


120587) = minus120587 GM =

|119862119866(119894120596120587)|

minus1


3996119904125


12= minus10788plusmn119894 06064





11011

) minus 0151711990401855

)

and 119866(119904) = 119890minus05119904

(1 + 11990405


12= minus06620 plusmn 119894 04552 119904

34= minus20323 plusmn 119894 41818


= 35549 11990412

= 0 plusmn



=


1

05

00

minus05

minus1


minus5minus6

Re(119904)

Im(119904


1(0598 + 3996119904125

)

minus20 00

20 40 60 80 100 120

minus50

50

minus100

100

Re

Im


1(0598 + 3996119904125

) and 119870 = 1





(119904 + 1)120572


119906 the phase



1205722(119904 minus 1)1205721) for 120572

1= 33 120572

2= 23


119906= 43358 119904

12= 0 plusmn 119894 48493


120587=



00

minus1

minus2

minus2minus3

minus4

minus6

2

4

6

Re(119904)

Im(119904

)


) minus

0151711990401855

) and 119866(119904) = 119890minus05119904

(1 + 11990405

)

00

minus1minus1

minus2

minus2

minus3

minus4

minus15minus05 05

1

2

3

4

Re

Im


) minus

0151711990401855

) 119866(119904) = 119890minus05119904

(1 + 11990405

) and 119870 = 14098











00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00


minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25



(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

05

00

minus05

minus1


minus5minus6

Re(119904)

Im(119904


1(0598 + 3996119904125

)

minus20 00

20 40 60 80 100 120

minus50

50

minus100

100

Re

Im


1(0598 + 3996119904125

) and 119870 = 1





(119904 + 1)120572


119906 the phase



1205722(119904 minus 1)1205721) for 120572

1= 33 120572

2= 23


119906= 43358 119904

12= 0 plusmn 119894 48493


120587=



00

minus1

minus2

minus2minus3

minus4

minus6

2

4

6

Re(119904)

Im(119904

)


) minus

0151711990401855

) and 119866(119904) = 119890minus05119904

(1 + 11990405

)

00

minus1minus1

minus2

minus2

minus3

minus4

minus15minus05 05

1

2

3

4

Re

Im


) minus

0151711990401855

) 119866(119904) = 119890minus05119904

(1 + 11990405

) and 119870 = 14098











00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00


minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25



(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 05

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

minus1minus1

minus2

minus3

minus4

4

minus5

5


00

1

2

3

Re

Im

(b)


(119904 + 2) 120572 = 15

00

minus1

minus1

minus2

minus2

minus3

minus3

1

1

2

3

Re(119904)

Im(119904

)

(a)

Re

Im

00


minus2

minus3

minus4minus5

1

2

5

4

3

(b)


(119904 + 2) 120572 = 25



(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119904 + 2)) for 120572 isin 0 12 1 32 2 52 3

120572 119870119906

PM 1205961

GM 120596120587

0 1332478866 0485868267705 170542240216776 1125696835 04639038816 1697056274 28284271241 6 09321940595 04457479715 6 141421356215 33537212820799 07492952635 04303313791 3338364465 093976381612 225 05751435644 04169759154 225 0707106781125 169381413351661 04083841882 04052247852 1680695653 056914295843 1347 02479905647 03947560116 133607116 04774914945


minus6

minus4

minus4

minus2 2

4

6

00

minus2

2

Re(119904)

Im(119904

)


1205721 ) for 1205721

= 33 1205722

=

23

minus5

minus4

minus4

minus3

minus3 minus2 2100

minus1minus1

minus2

4

3

1

2

Re

Im


1205721 ) for 1205721= 33 120572

2=

23




Acknowledgments


References


















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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