Robust Finite -Time Stability Analysis of Fractional Order

Time Delay Systems: New Results

MIHAILO LAZAREVIĆ*, ALEKSANDAR OBRADOVIĆ,VASILIJE VASIĆ Department of Mechanics, Faculty of Mechanical Engineering

University of Belgrade st. Kraljice Marije 16, 11020 Belgrade 35

*mlazarevic@mas.bg.ac.rs ; http://www28.brinkster.com/mlazarevic (http://www.mas.bg.ac.rs)

SERBIA

Abstract: In this paper, a stability test procedure is proposed for perturbed nonlinear nonhomogeneous fractional order systems with a pure time delay. Sufficient conditions of this kind of stability are derived for particular class of fractional time-delay systems using generalized Gronwall inequality. A numerical example is given to illustrate the validity of the proposed procedure. Key-Words: Stability criteria, Analysis, Fractional calculus, Time delay, Robust, Perturbed systems,Nonlinear

1 Introduction The question of stability is of main interest in control theory. Also, the problem of investigation of time delay system has been exploited over many years,[1]. A numerous reports have been published on this matter, with particular emphasis on the application of Lyapunov`s second method, or on using idea of matrix measure, [2,3]. Here, one other approach is presented, i.e system stability from the non-Lyapunov point of view (finite and practical stability) is studied,[4-6]. Also, analysis of the linear time-delay systems in the context of finite and practical stability was introduced and considered,[7,8]. Recently there have been some advances in control theory of fractional (non-integer order) dynamical systems for stability questions such as robust stability, bounded input–bounded output stability, internal stability,finite time stability, practical stability, root-locus, robust controllability, robust observability, etc. For example, regarding linear fractional differential systems of finite dimensions in state-space form, both internal and external stabilities are investigated by Matignon,[9].Some properties and (robust) stability results for linear, continuous, (uncertain) fractional order state-space systems are presented and discussed [10,11].However, we can not directly use an algebraic tools as for example Routh-Hurwitz criteria for the fractional order system because we do not have a characteristic polynomial but pseudopolynomial with rational power-multivalued function. An analytical approach was suggested by Chen and Moore,[12], who considered the analytical stability bound using Lambert function W. Further, analysis and stabilization of fractional (exponential) delay systems of retarded/neutral type are considered [13,14], and BIBO stability [15].

Whereas Lyapunov methods have been developed for stability analysis and control law synthesis of integer linear systems and have been extended to stability of fractional systems, only few studies deal with non-Lyapunov stability of fractional systems. Recently, for the first time, finite-time stability analysis of fractional time delay systems is presented and reported on papers [16,17]. The main contribution of this paper is to propose finite time stability test procedure perturbed nonlinear (non)autonomous time-invariant delay fractional order systems. Here, a Bellman-Gronwall`s approach is proposed, using a recently obtained generalized Gronwall inequality reported in [18] as a starting point. The problem of sufficient conditions that enable system trajectories to stay within the a priori given sets for the particular class of nonlinear (non)autonomous fractional order time-delay systems has been examined.

2 Motivation and Preliminaries

2.1 Preliminaries on integer time-delay systems A linear, multivariable time-delay system can be represented by the following differential equation:

0 1 0( )

( ) ( ) ( ),d t

A t A t B tdt

τ= + − +x

x x u (1)

and with the associated function of the initial state: ( ) ( ), 0.xt t tτ= − ≤ ≤x ψψψψ (2)

In equation (1), ( ) nt ∈x ℝ is a state vector, ( ) mt ∈u ℝ is a

control vector, 0 1 0, ,A A B are constant system matrices of

appropriate dimensions, and τ is a pure time delay,

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constτ = . ( 0τ > ). Moreover, here we are interested in a class of non-linear perturbed system with time delay described by the state space equation: (3)

( ) ( )

( )

0 0 1 1

0 0

( )( ) ( )

( ) ,

d tt t

dt

t

τ= + ∆ + + ∆ − +

+

xA A x A A x

B u(t) + f x

Vector function 0f present nonlinear parameter

perturbation of system in respect to ( )tx and matrices

0 1,∆ ∆A A present perturbations of system, too. Also, it

is introduced next assumption:

)0 0( , 0,(t)) c (t) t ≤ ∈ ∞f x x (4)

where 0+∈c R is known real positive number.

Dynamical behaviour of system (1), with a given initial function is defined over time interval

[ ], ,o oJ t t T J R= + ⊂ where quantity T may be either a

positive real number or symbol +∞ , so finite time stability and practical stability can be treated simultaneously. Time invariant sets, used as bounds of system trajectories, are assumed to be open, connected and bounded. Let index " "ε stands for the set of all allowable states of the system and index " "δ for the set of all initial states of the system, such that the set S Sδ ε⊆ .In general, one may write:

[ ]2: ( ) , ,

QS tρ ρ ρ δ ε= < ∈x x , where Q will be

assumed to be a symmetric, positive definite, real matrix.

uSα denotes the set of all allowable control actions. Let

( ).x be any vector norm (e.g., . 1,2,= ∞ ) and (.) the

matrix norm induced by this vector. The initial function can be written in its general form as:

[ ]( ) ( ), 0, ( ) ,0o x xt Cθ θ τ θ θ τ+ = − ≤ ≤ ∈ −x ψ ψψ ψψ ψψ ψ ,where

0t is the initial time of observation of the system (1) and

[ ],0C −τ is a Banach space of continuous functions over

a time interval of length τ , mapping the interval

[ ],t t− τ into nR with the norm defined in the following

manner: 0

max ( )C τ θ

θ− ≤ ≤

=ψ ψψ ψψ ψψ ψ . It is assumed that the

usual smoothness condition is present so that there is no difficulty with questions of existence, uniqueness, and continuity of solutions with respect to initial data. Definition 2.1: The system given by homogenous state equation (1) (when ( ) 0,t t≡ ∀u ), satisfying initial

condition ( ) ( ), 0xt t tτ= − ≤ ≤x ψψψψ is finite stable w.r.t

, , , , ,ot Jδ ε δ ε< if and only if

C

δ<ψψψψ (5)

imply: ( ) ,t t Jε< ∀ ∈x (6)

where 0t denotes the initial time of observation of the

system and J denotes time interval

[ ], ,o oJ t t T J R= + ⊂ .

Definition 2.2: System given by (1) satisfying initial condition ( ) ( ), 0xt t tτ= − ≤ ≤x ψψψψ is finite stable w.r.t

, , , , , ,u ot Jδ ε α δ ε< if and only if:

Cδ<ψψψψ

(7)

and ( ) ,ut t Jα< ∀ ∈u (8) imply: ( ) ,t t Jε< ∀ ∈x (9)

2.2. Fundamentals of fractional calculus

The fractional integro-differential operators-(fractional calculus–(FC)) is a generalization of integration and derivation to non-integer order (fractional) operators. The idea of FC has been known since the development of the regular calculus, with the first reference probably being associated with Leibniz and Marquis de l’ Hopitalˆ in 1695. The modern epoch started in 1974 when a consistent formalism of the fractional calculus has been developed by K.B. Oldham and J. Spanier [19]. Applications of FC are very wide nowadays in rheology, viscoelasticity, acoustics, optics, chemical physics, robotics, control theory of dynamical systems, electrical engineering, bioengineering and so on, [19-22]. Recently, [23] choosing a relevant wavelet, we may obtained a new representation for fractional differintegral and new the wavelet transform which is based on fractional B-splines,[24]. The main reason for the success of applications FC is that these new fractional-order models are more accurate than integer-order models and fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a ”memory” term in a model. Three definitions are generally used for the fractional differintegral. First is the Grunwald definition,given as [19,20]:

( ) ( )[ ]( ) /

00

1( ) lim 1

t a hj

ah

j

D f t f t jhjh

αα

α−

→=

= − −

∑ , (10)

where a, t are the limits of operator and [x] means the integer part of x . The Riemann-Liouville (RL) definition of fractional derivative and integral are: Definition 2.3 ([21]) Let [ ] ( ), ,a b a bΩ = −∞ < < < ∞ be a

finite interval on the real axis ℝ and f(t) be a continuous function defined on [ ],a b . The left-sided Riemann-

Liouville fractional derivative of order ( )( )0ℜ ≥α ,

,∈ℂα is:

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( )

( )( )

1

1 ( )( ) ,

( )

1,

tn

a n n

a

d fD f t d

n dt t

n t a

αατ

τα τ

α

+ − +=

Γ − −

= ℜ + >

∫ (11)

Definition 2.4 ([21]) Let f(t) be a continuous function defined on [ ],a b . The left-sided Riemann-Liouville

fractional integral of order ( )( ), 0∈ ℜ >ℂα α , is: (12)

( ) ( )( )11( ) ( ) , , 0

( )

t

a

a

I f t t f d t aαα τ τ τ α

α+

−= − > ℜ >Γ ∫

Without loss of generality we denote ( )aD f t+

α by

( )aD f tα

and ( )aI f t+α

by ( )aI f tα

, respectively. Here, Γ(.)

is the Euler's gamma function which is defined by:

1

0

( ) ,t zz e t dt z

∞− −Γ = ∈∫ ℂ (13)

For this function the reduction formula holds: ( 1) ( ),z z zΓ + = Γ (14)

Lemma2.1([21]) If ( )( )0ℜ >α and ( ) ( ),pf t L a b∈

( )1 p≤ ≤ ∞ , (where ( ),pL a b are set of those Lebesgue

complex-valued measurable functions f on [ ] ( ), ,a b a bΩ = −∞ < < < ∞ for which

pf < ∞ ) then the

following equality:

( )( ) ( ),a aD I f t f tα α

+ + = (15)

holds almost everywhere on [ ],a b . Also, left-sided

Caputo fractional derivative of order ( )( ), 0∈ ℜ ≥ℂα α

on [ ],a b are defined via the above Riemann-Liouville

fractional derivative by, [22] (16)

( )1 ( )

0

( )( ) ( ) ( ),

!

n kkC

a ak

f aD f t D f s s a t

k

α α+ +

−

=

= − −

∑

where

( ) 0

0

1 0,1,2...

,

forn

for

ℜ α + α∉ = = α α∈

ℕ

ℕ

(17)

The Caputo and Riemann-Liouville formulation coincide when the initial conditions are zero. In particular, Caputo fractional derivative is defined for function ( )f t which

belongs to the space of absolutely continuous functions

[ ] [ ] [ ] 1 1, : , ( ) / ,n n nAC a b f a b and d f t dt AC a b− −= → ∈ℂ ,

: 1, 2,3,..n∈ =ℕ ,[22].Thus the following statement

holds. Theorem 2.1 [22] Let ( )( )0ℜ ≥α and let n be given by

(14). If ( ) [ ],n

f t AC a b∈ then the Caputo fractional

derivative ( )C

aD f t+

α exist almost everywhere on [ ],a b .

a) If 0α∉ℕ , then

( ) ( )

( )

1

1 ( )( )

tn

C

na

a

f sD f t ds

n t s+

αα− +

=Γ −α −∫ (18)

b) If 0nα = ∈ℕ , then

( ) ( )( ) ( )

nC n

na

d f tD f t f t

dt+

α = = (19)

Theorem 2.2 ([18] Generalized Gronwall inequality) Suppose ( ) , ( )x t a t are nonnegative and local integrable

on 0 ,t T some T≤ < ≤ +∞ ,and ( )g t is a nonnegative,

nondecreasing continuous function defined on 0 ,t T≤ < ( ) .g t M const≤ = , 0α > with

( ) 1

0

( ) ( ) ( ) ( )

t

x t a t g t t s x s dsα−

≤ + −∫ (20)

on this interval.Then

( )( )( )

( ) 1

10

( )( ) ( ) ( ) ,

0

nt

n

n

g tx t a t t s a s ds

n

t T

∞α−

=

Γ α ≤ + −

Γ α

≤ <

∑∫ (21)

Corollary 2.1 of (Theorem 2.2) [18] Under the hypothesis of Theorem 2.2, let ( )a t be a nondecreasing

function on [ )0,T . Then holds:

( ) ( )( )( ) ( )x t a t E g t tαα≤ Γ α (22)

where Eα is the Mittag-Leffler function defined by

( )0

( ) / 1k

k

E z z k

∞

α

=

= Γ α +∑ (23)

2.3 Previous results related to time-delay

systems (non)integer order Recently, there have been some developments in the control theory of FDS (with time-delay) for the stability questions [13-17]. Particularly, authors in [20] using short memory principle and taking into account initial function one can obtain correct initial function where it is assumed that is no difficulty with questions of continuity of solutions with respect to initial data (function). Also, in in signal and image processing time delay appears where wavelet transform has growing importance may be generalized by association with both the wavelet transform and the fractional Fourier transform, and fractional B splines. Also, it is shown that PDα control of Newcastle robot can be presented by the linear time delay fractional order of differential equation in the state space form, [16]:

0 1( )

( ) ( )d t

A t A tdt

α

α τ= + −x

x x (24)

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and with associated function of initial state: ( ) ( ), 0xt t tψ τ= − ≤ ≤x , for (0 1)α< < .

Further more, for the first time, finite time stability analysis of (non)linear (non)autonomous fractional order systems with delayed state are presented by Lazarević et al.[17,23,24]. Also, more generally, non-homogenous, linear multivariable time-delay system can be represented by the following differential equation:

0 1 0( )

( ) ( ) ( )d t

A t A t B tdt

α

α τ= + − +x

x x u (25)

and with associated function of the initial state (2).

Theorem 2.3. ([25]).Autonomous system ( ( ) 0t ≡u ,(25))

satisfying initial condition (2) is finite time stable w.r.t. , , , , ,ot Jδ ε δ ε< , if the following condition is

satisfied:

( )( )

( )max 0( )

1max 01 / , .1

A t tA t t

e t J

ασαασ

ε δα

−Γ +

− + ⋅ ≤ ∀ ∈ Γ +

(26)

where (.)maxσ being the largest singular value of matrix

(.), where ( ) ( )max 01 max 0 max 1A Aσ σ σ= + .

Remark 2.1: If 1α = , see (1,24), one can obtain same conditions which related to integer order time delay systems (1) as follows [7]: (27)

( )1

max 0( )1max 0 11 / , , (2) 1

1

A t tA t te t J

σσ

ε δ− − + ⋅ ≤ ∀ ∈ Γ =

Theorem 2.4.([26]) Nonautonomous system given by (25) satisfying initial condition (2) is finite time stable w.r.t. 0, , , , , , ,u ot Jδ ε α α δ ε< , if the following

condition is satisfied: (28)

( )( )

( )( )

max 0( )

1max 0 0( )1 / , .

1 1

A t tA t t t t

e t J

ασα αασ

γ ε δα α

−Γ +

− − + ⋅ + ≤ ∀ ∈ Γ + Γ +

i

where 0 0 0/ ,ub B bγ α δ= =i .

Remark 2.2: If 1α = , see (28), one can obtain same conditions which related to integer order time delay systems (3) as follows [8]: (29)

( )1

max 0( )1 1max 0 01 ( )

1 / , ,1 1

At tA t t t t

e t J

σσ

γ ε δ− − − + ⋅ + ≤ ∀ ∈

i

Theorem 2.5 ([17]) The linear nonautonomous system given by (25) satisfying initial condition

( ) ( ), 0xt t tτ= − ≤ ≤x ψψψψ is finite time stable w.r.t.

0, , , , ,u Jδ ε α δ ε< , where 0 0 / .u ubγ α δ• = if the

following condition is satisfied: (30)

( ) ( ) ( )

max 01 0max 01

0

1 / ,1 1

0,

ut tE t

t J T

α αα

ασ γ

σ ε δα α

• + + ≤ Γ + Γ +

∀ ∈ =

,

Remark 2.3. If 1α = , see (1), one can obtain same conditions which related to integer order time delay systems (1) as follows [8]:

( )1

max 0( )1 1max 0 01

1

( )1 / ,

1 1

, (2) 1, ( )

At tA

z

t t t te

t J E z e

σ

α

σγ ε δ

−

=

− − + ⋅ + ≤

∀ ∈ Γ = =

i

(31)

3 Main Result In this paper, we discuss the case 1, 0 1n α= < < . Here, we examine the problem of sufficient conditions that enable system trajectories to stay within the a priori given sets for the particular class of (non)linear perturbed (non)autonomous fractional order time-delay system which is presented in the state space form:

( ) ( )

( )

0 0 1 1

0 0

( )( ) ( )

( ) ,

d tt t

dt

t

α

ατ= + ∆ + + ∆ − +

+

xA A x A A x

B u(t) + f x

(32)

with the initial functions (2) of the system and vector functions 0f satisfied (4).

Theorem 3.1 The nonlinear nonautonomous system given by (32) satisfying initial condition

( ) ( ), 0xt t tτ= − ≤ ≤x ψψψψ is finite time stable w.r.t.

0, , , , ,u Jδ ε α δ ε< , if the following condition is

satisfied:

( ) ( ) ( )

0

0

1 /1 1

0,

p up

tE t t

t J T

αα α

αµ γ

µ ε δα α

• + + ≤ Γ + Γ +

∀ ∈ =

(33)

where 0

1

0

1 1 1

,

,

Aoco Ao A

A A A p Aoco A

cµ σ γ

σ σ γ µ µ σ∆

∆ ∆ ∆

= + +

= + = + (34)

0 0 / ,u ubγ α δ• = and (.)σ being the largest singular value of matrix (.) Proof of Theorem 3.1. In accordance with the property of the fractional order 0 1α< < , one can obtain a solution in the form of the equivalent Volterra integral equation, where is 0 0t = : (35)

( ) ( )

( )( ) ( )

( )

0

0 0 1 11

0 00

( ) ( )1( )

( )

t

t t

t tt s ds

t

α τ

α−

= +

+∆ + +∆ − ++ −

Γ + ∫

x x

A A x A A x

B u(t) +f x

Applying the norm (.) on equation (35) and using

appropriate property of the norm, it yields: (36)

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( ) ( )

( ) ( )

0

0 0 11

1 0 00

( ) ( ) ( )1( )

( ) ( ) ( )

t

t t

s s st s ds

s B s f s

α τ

τα−

≤ +

+ ∆ + − ++ −

+∆ − + +Γ ∫

x x

A x A x A x

A x u x

Also, applying the norm (.) on equation (32) and taking into account assumption (4), one can obtain: (37)

( ) ( )( )

( ) ( )( )

max 0 max 0 0

max 1 max 1 0

( )( )

( ) ( )

d tc t

dt

t B t

α

ασ σ

σ σ τ

≤ + ∆ + +

+ + ∆ − +

xA A x

A A x u

,

where A denotes the induced norm of a matrix A, as

well as,

( ) sup ( )t t t

t tτ

τ•

•

− ≤ ≤− ≤x x , (38)

Applying this inequality, equation (37) can be presented in the following manner:

1

0

( )( ) sup ( )

sup ( ) ( ) ,

Aoco At t t

p ot t t

d tt t

dt

t b t t t

α

ατ

τ

µ σ

µ τ

•

•

•∆

− ≤ ≤

•

− ≤ ≤

≤ +

≤ + > +

xx x

x u

(39)

where:

0 1 1 1

1 1 1

0 , ,

, , ,

Aoco Ao A A A A

p Aoco A Ao Ao A A

cµ σ γ σ σ γ

µ µ σ σ γ σ γ∆ ∆ ∆

∆ ∆ ∆ ∆ ∆

= + + = +

= + ≤ ≤ (40)

or (41)

( )0 1 0

1 0 0

0

( ) ( ) ( )

( ) ( )

sup ( ) ( )p x oCt t t

t t t

t t

t b t t tτ

τ

τ

µ +•

•

− ≤ ≤

+ − + ∆ +

∆ − +

≤ + + >

A x A x A x

A x B u(t) + f x

x uψψψψ

After combining (41) with (36), it yields: (42)

( )( )

( )( )

( )1

00

11

1sup ( ) ( )

1

px C

tp

ut t t

tt

t s t ds b t

α

α α

τ

µ

α

µα

α α•

− •

− ≤ ≤

≤ + +

Γ +

+ − +Γ Γ +∫

x

x

ψψψψ

,

Obviously, one can introduce nondecreasing function

( )tβ such as:

( ) ( )1

1

px C

tt

αµβ

α

= + Γ +

ψψψψ , (43)

Now, applying generalized Gronwall inequality,[18], here, Corollary 2.1 of (Theorem 2.2) (22). Moreover, it is easy to show:

( ) ( ) ( )sup ( ) pt t t

t t t E tαατ

β µ•

•

− ≤ ≤≤ ≤x x , (44)

and (45)

( )( ) ( ) ( ) 0

11 ( )

1 1p

p u

tt E t b t

αα α

αµ

δ µ αα α

≤ + + Γ + Γ +

x

Hence, using the basic condition of Theorem 4.1, relation (33) yields:

0( ) , .t t Jε< ∀ ∈x (46)

This is a proof of the theorem. Remark 3.1: If we have no perturbed system

0 1 00, 0, ( (t)) 0∆ = ∆ = =A A f x one can obtain same conditions which related to Theorem 2.5.

Remark 3.2. If 1α = , see (22), one can obtain same conditions which related to integer order time delay systems (3) as follows [8]: (47)

11 1

111 / , (2) 1, ( )

1 1

pt

p zt t

e E z e

µ

αµ

γ ε δ =

+ ⋅ + ≤ Γ = =

i

3.1. An Illustrative Example A perturbed nonlinear nonhomogenous fractional, time-delay system is considered in the following state space description: (48)

( ) ( )

( )

1/ 20 0 1 1

0 0

( )( ) ( ) ( )

( ) ,

t

d tD t t t

dt

f t

α

α τ= = + ∆ + + ∆ − +

+

xx A A x A A x

B u(t) + x

where are

0 1 0

1 0

0 1 0 0 0.3 0, , ,

2 0 1 1 0.2 0.1

0.1 0.2 0,

0 0.1 1

A A A

A B

= = ∆ = −

∆ = =

(49)

with an associated function of the initial state: ( ) ( ) 0, 0,xt t tτ= = − ≤ ≤x ψψψψ (50)

Also, there is the task of checking the finite time stability wrt 0 0, 0,2 , 0.1, 50, 0.1, 1 ,ut J δ ε τ α= = = = = =

where 0( ) 0, 0.1,0 , 0.1x t t cψ = ∀ ∈ − = .

From the initial data and the equations (43) and (22) one can obtain: (51)

( ) ( )

( ) ( )max 0 max 1

max 0 max 1

( ) 0.1, 2, 2

0.838, 0.2414

x Ct σ σ

σ σ

< = =

∆ = ∆ =

A A

A A

ψψψψ

( ) ( )

( ) ( )

max 0 max 0 0

max 1 max 1

2 0.838 0.1 2.938

2.938 1.41 0.2414 4.592

Aoco

p Aoco

cµ σ σ

µ µ σ σ

= ∆ + ∆ +

= + + =

= + + ∆ =

= + + =

A A

A A

Applying the condition of the Theorem 3.1, (33) one can get: (52)

( )0.5 0.5

0.50.5

4.592 101 4.592 50/ 0.1 0.54 .

0.886 0.886e e

e eT T

E T T s ⋅ + ⋅ + ≤ ⇒ ≈

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eT being “estimated time” of finite time stability.

4. Conclusion In this paper, robust finite-time stability analysis for a class of perturbed nonlinear fractional order systems with time invariant delay was considered. New stability criteria for this class of fractional order systems were derived by applying generalized Gronwall inequality where sufficient conditions of this kind of stability are obtained. In that way, one can check system stability over a finite time. Finally,an illustrative example for this class of system was presented. Acknowledgment This work is supported by EUREKA program- E!4930 AWAST and partially supported by the Ministry of Science and Environmental Protection of Republic of Serbia, No.142034.

References

[1] Zavarei M.,Jamshidi M., Time-Delay Systems: Analysis,Optimization and Applications, North-Holland, Amsterdam, 1987.

[2] Lee, T. N., S. Diant, Stability of Time Delay Systems, IEEE Trans. Automat. Control AC 31 (3) 951-953,1981.

[3] Chen, J., D. Xu, B. Shafai, On Sufficient Conditions for Stability Independent of Delay, IEEE Trans. Auto.Cont.AC-40(9)1675-1680, 1995.

[4] Weiss, L., F. Infante, On the Stability of Systems Defined over Finite Time Interval. Proc. National Acad. Science 54(1). 44-48,1965.

[5] Grujić, Lj. T., Non-Lyapunov Stability Analysis of Large-Scale Systems on Time-Varying Sets, Int. J. Control 21(3),401-405,1975a.

[6] Lashirer,M.,C.Story,Final-Stability with Applica-tions, J. Inst. Math. Appl.,9,379-410,1972.

[7] Lazarević M, Debeljković D., et.al., Finite Time Stability of Time Delay Systems,IMA Journal of Math. Cont. and Inf., 17,101-109,2000.

[8] Debeljković,D., Lazarević M. et.al., Further Results On Non-Lyapunov Stability of the Linear Nonautonomous Systems with Delayed State, Journal Facta Uversitatis,Vol.3,No11,231-241, Niš, Serbia,Yugoslavia,2001.

[9] Matignon, D., Stability properties for generalized fractional differential systems, ESAIM:

Proceedings, 5: 145–158, December,1998. [10] Vinagre, M.,C.Monje and A. Calderon, Fractional

order systems and fractional order control actions. In: Lecture 3 of the IEEE CDC02: Fractional

Calculus Applications in Automatic Control and

Robotics, 2002.

[11] Chen Q, Ahn H., Podlubny I.,Robust stability check of fractional order linear time invariant systems with interval uncertainties,Signal Processing 86 (2006) 2611–2618.

[12] Chen, Y. Q., Moore K.L., Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems, Nonlinear Dynamics, Vol.29: 191-200,2002.

[13] Bonnet C., J.R Partington,Stabilization of fractional exponential systems including delays,Kybernetika, 37,pp.345-353,2001.

[14] Bonnet C., J.R Partington,Analysis of fractional delay systems of retarded and neutral type, Automatica, 38, pp.1133-1138, 2002.

[15] Hotzel, R., Fliess M.,On linear systems with a fractional derivation: Introductory theory and examples, Mathematics and Computers in

Simulations, 45 385-395,1998. [16] Lazarević,M., Finite time stability analysis of PDα

fractional control of robotic time-delay systems, Mechanics Research Communications,33,(2006) 269–279.

[17] Lazarević M., A.Spasić, Finite-Time Stability Analysis of Fractional Order Time Delay Systems: Gronwall`s Approach, Mathematical and Computer Modelling,vol 49,(3-4),2009,475-481.

[18] H. Ye., J.Gao., Y. Ding.,A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328 (2007),1075–1081.

[19] Oldham K. B.,J Spanier, The Fractional Calculus, Academic Press, New York,1974

[20] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego,1999.

[21] Kilbas A., H. Srivastava, J.Trujillo,Theory and applications of fractional differential equations, Elsevier, Amsterdam,2006.

[22] Samko, S.G., Kilbas,A.A., and Marichev,O.I., Fractional Integrals and derivatives: Theory and

Applications,Gordon&Breach,Sci.Pub.,Switz.1993. [23] Rubin B., D.Ryabogin and E. Shamir, Fractional

integrals and wavelet transforms, Transform

Methods&Spec.Func,Proc.IIInt.Workshop,Varna, 1996, 100-105.

[24] M. Unser, T. Blu, Fractional splines and wavelets, SIAM Rev. 42 (1) ,2000, 43–67.

[25] Lazarević M.,D.Debeljković, Finite Time Stability Analysis of Linear Autonomous Fractional Order Systems with Delayed State,Asian Journal of Control,Vol.7,No.4, 2005,440-447.

[26] Lazarević,M., Further results on finite time stability of nonautonomous fractional order time delay systems, J.Problems of nonlinear analysis in engineering systems,no.1(27),Vol.13,2007,123-148.

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