Sabatier Agrawal Machado Advances in Fractional Calculus Episode 12 doc

6.2 Equivalent integer order system Using function F' defined by 22, it is now possible to assess stability of a fractional system through stability analysis of an equivalent integer sy

Theorem 5: [13] (sufficient condition) Fractional system (7) is t stable if matrix P 0, P M M , exists, such that 0

1 1

A P P A

5.3 Validity of the stability condition

Figure 3 presents stability domain DS of a fractional system characterized using theorem 5 according to fractional order and to argspec A A comparison

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig 3 Stability domain DS ( ) determined by criterion 2 according to the values of and

A simple explanation can be given Systems (7) and (19) have strictly the same behavior However, transformations given by relations (16) to (18) produce

DS’

'

'between Fig 3 and Fig 1 reveals that the entire stability domain is not identi-

whose eigenvalues are outside the left-half complex plane The thus leading to a stable response to nonzero initial conditions Due to fied using theorem 5 It therefore leads to a sufficient but nonnecessary condition

428

In order to analyze such a conservatism, let f be an arguments of an

eigenvalue of matrix A1/ and be the one of system (6) state transition

associates to

f

x x

2,0,0

1 4 3 2, 4 1 2,

0

i

i

A method leading to a necessary and sufficient condition for stability of

fractional systems is therefore necessary

Moze, Sabatier, and Oustaloup

matrix A of system (6) Line D in Fig 4 represents the function F that

'

decays towards, 0, the slope of D increases significantly such that

according to :

6.1 Characterization of the entire stability domain

In order to characterize the entire stability domain DS, it is necessary to define a function that associates every DS with ' belonging to a convex domain of whose characterization is performed through LMI in theorem 4 Such a function can be defined by:

2

12

1,0,0:'

x x

Fig 5. F' as a function of and f , and deduced stability domain ( )

6.2 Equivalent integer order system

Using function F' defined by (22), it is now possible to assess stability of a fractional system through stability analysis of an equivalent integer system whose state transition matrix is to be determined

Let a e j , where j is the complex variable and 0, As

a

one can note that

a F

6 Stability Theorem Based on a Geometric Analysis

of the Stability Domain

the complex plane This convex domain may be the left-half complex plane

'

which is represented by line (D ) in Fig 5

430

2arg

,2

1

a iff

1

A P P A

0 Re

Re Im

Im

Im Im

Re Re

2 1 2

1 2

1 2

1

2 1 2

1 2

1 2

1

A P P A A

P P A

A P P A A

P P A

T T

T T

(25)

6.3 Validity of the method

Figure 6 presents the stability domain DS determined using theorem 6, according to the values of and of

Fig 6. Stability domain DS ( ) determined by theorem 6 according to the values of and

¢



0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

) 2 / ( argspec A rad

DS’’

Moze, Sabatier, and Oustaloup

''

''

7.2 Characterization of the entire unstable domain

u

belongs to Du if and only if it belongs to both Du1andDu2 defined by

21exp(

and

21exp(

Thus belongs to Du if and only if

0)21exp(

Re

0)21exp(

)21exp(

0)21exp(

)21exp(

*

*

j j

j j

which can be rewritten as:

''

When compared with Fig 1, the entire stability domain D is identified

denote the unstable domain as depicted on Fig 2a It is obvious that LetD

7 Stability Criterion Based on Unstability Domain

Characterization

r r

where

2

cos2

r

r r

(32)

As for some spec A , * spec A , and as Du1andDu2are symmetric

in relation to the real axis of the complex plane, condition (32) becomes

1

and fractional system (6) is t stable if and only if

It is now possible to use the following lemma given in [20]

Lemma 1 [20]: There exists a vector p q 0 for some * 0 if and only if pq* qp* 0.

฀

,0

rq

or if and only if

,0

As q Aq, fractional system (6) is t stable if and only if

0

Moze, Sabatier, and Oustaloup

Applied to relation (32), fractional system (6) is thus t stable if and only if

Theorem 7: Fractional system (6) is t stable if and only if there does not

n n such that

02

sin2

cos

2

cos2

sin

T T

T T

QA AQ QA

AQ

QA AQ QA

An analysis of an existing method and two new methods are presented in order

to characterize stability of fractional systems through LMI tools Matignon’s theorem developed for stability analysis of fractional systems is first presented

A new proof of its extension to systems whose fractional order verifies 2

1 is proposed For such derivative orders, stability is granted if all the eigenvalues of its state transition matrix belong to a convex subset of the complex plane, called stability domain A trivial LMI stability condition is thus presented For fractional orders verifying 0 1, stability domain is not a convex subset of the complex plane Three stability conditions involving LMI are however proposed

system have strictly the same behavior, an explanation of the conservatism of the condition is presented

In order to overcome this problem, a third condition is proposed It relies on the fact that instability domain is a convex subset of the complex plane when 1

This work is a first step in fractional system stability analysis using LMI tools towards new conditions and applications

permission to publish this revised contribution of an ASME article

exist any nonnegative rank one matrix Q

The first condition appears in [13] and appears after algebraic mations of the fractional system state-space representation The obtained con- dition is only sufficient Even if the derived system and the original fractional

transfor-The second condition is new and relies on a geometric analysis of the sta- bility domain The resulting LMI stability condition is sufficient and necessarybut is not linear in relation to the state transition matrix of the fractional sys-tem state-space representation, which can limit its applicability

Thanks go to the American Society of Mechanical Engineers (ASME) for the

Acknowledgment

434 Moze, Sabatier, and Oustaloup

4 Chen YQ, Moore KL, Vinagre BM, Podlubny I (2004) Robust PID controller auto tuning with a phase shaper, First IFAC Workwhop on Fractional Derivative and its Application, FDA 04, Bordeaux, France

5 Oustaloup A, Mathieu B (1999) La commande CRONE du scalaire au multivariable Hérmes, Paris

6 Battaglia J-L, Cois O, Puissegur L, Oustaloup A (Juillet 2001) Solving an inverse heat conduction problem using a non-integer identified model, Int J Heat Mass Transf., 44(14) 2671–2680

7 Hotzel R Fliess M (1998) On linear systems with a fractional derivation: Introductory theory and examples, Math Comp Simulation, special issue: Delay Systems, 45:385–395

8 Matignon D (July 1996) Stability results on fractional differential equations with applications to control processing, Comp Eng Syst Appl multiconference, 2:963–968, IMACS, IEEE-SMC

9 Biannic JM (1996) Commande robuste des systèmes à parameters variables, application en aéronautique, PhD Thesis, ENS de l’Aéronautique et de l’Espace

10 Balakrishnan V, Kashyap RL (March 1999) Robust stability and performance analysis of uncertain systems using linear matrix inequalities, J Optim Theory Appl 100(3):457–478

11 Balakrishnan V (August 2002) Linear Matrix Inequalities in Robust Control: A Brief Survey, in Proceedings of the Mathematical Theory of Networks and System Notre Dame, Indiana

12 Boyd S, Vandenberghe L (2004) Convex Optimization, Cambridge University Press

13 Momani S, El-Khazali R (November 19–22, 2001) Stability An alysis of Composite Fractional Systems, in Intelligent Systems and Control, Tampa, Florida

14 Samko AG, Kilbas AA, Marichev OI (1987) Fractional Integrals and Derivatives Gordon and Breach Science, Minsk

15 Miller KS, Ross B (1993) An Introduction to the Fractional Calculus and Fractional Differential Equation Wiley, New York

16 Malti R, Cois O, Aoun M, Levron F, Oustaloup A (July 21–26 2002) Computing impulse response energy of fractional transfer function, in the 15th IFAC World Congress 2002, Barcelona, Spain

17 Boyd S, El Ghaoui L, Feron E, Balakrishnan V (June 1994) Linear matrix inequalities in system and control theory Volume 15 of Studies in Applied Mathematics, Philadelphia

18 Gahinet P, Nemirovski A, Laub AJ, Chilali M (1995) LMI control toolbox user’s guide, The Math Works

19 Tabak D, Kuo BC (1971) Optimal Control by Mathematical Programming Prentice-Hall, New Jersey

20 Ben-Tal A, El Ghaoui L, Nemirovski A (2000) Robustness, in Handbook of Semidefinite Programming: Theory, Algorithms and Applications Kluwer Academic, Boston, pp 68–92

STRUCTURES USING FRACTIONAL

a Laplace transform with an s1/2 or s3/2 term As an example, we present the fractional-order derivatives and integrals of structural responses in the vibration suppression of a thin, light cantilevered beam

1 Introduction

Flexible structures such as large-scale space structures (LSS) have a high active vibration suppression, vibration control approaches based on modal analysis must determine the limits of the spillover instability phenomenon Hence it is necessary to establish a new control methodology that can be applied

to flexible structures Among such novel approaches, the active wave (absorption) control method has attracted attention

It is known that control laws derived from active wave control theory can be expressed using a transfer function including a non-integer order power of the

Tsukuba, Ibaraki 305-8564, Japan; Tel: +81-29-861-7147, Fax: +81-29-861-7098,

E-mail: m-kuroda@aist.go.jp

Keywords

vibration-mode density, even in the low-frequency domain Therefore, to achieve

© 2007 Springer

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

National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1 Namiki,

Recently, active wave control theory has attracted great interest as a novel

Fractional calculus, control, fractional-order transfer function, wave, flexible

in Physics and Engineering, 435–448

structure

435

variable s of the Laplace transform However, there are difficulties implementing

the transfer function due to the non-integer order power In this report we present

a formulation of a feedback-type active wave controller, designed to suppress vibration of a flexible cantilevered beam, described by a transfer function with s ors s and introducing a fractional-order derivative and integral

2 Active Wave Control of a Flexible Structure

Active wave control differs from conventional vibration control in the way it suppresses the vibration modes (standing waves) of a structure The interaction

of progressive and retrogressive waves creates a standing wave, each of which can be treated as a controlled object in the control method developed by von Flotow and Schafer [1]

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Fig 1. Schematic diagram of the active wave control method

As an example, we consider the vibration control of a flexible cantilever (Fig 1)

A sensor and an actuator are placed near the middle of the beam A disturbance

is applied at the free end of the beam The relationship between the progressiveand retrogressive wave vectors generated by the disturbance on the cantilevercan be described in matrix form using boundary conditions on the control point.Backward propagating waves are produced by the reflection of the progressivewave, but are also produced by the control input, allowing control of the back-ward propagating wave The progressive wave vectors can also be controlled However, we note that only one control force can control any one of the wave

In the feedback control of the system, the beam deflection angle is detected and fed to a bending moment actuator, such as a piezoelectric patch, used as

control input Taking a1 and a2 as progressive wave components and b1 and b2 as retrogressive wave components, we obtain a closed-loop relationship between the progressive wave and the retrogressive wave [2]:

M

a a

C p

201

0

01

2 1 2

H -norm of the transfer function of the closed-loop scattering

angle to the bending moment for the control input, M c The resultant controller can be expressed as [2, 3]

s A EI s

s s

3

1 4 3

(2)

This equation shows that the active wave control law includes the term s: the half-order derivative element In other words, the control law can be qualified as velocity feedback with a phase shift of 45°, rather than 90° According to MacMartin and Hall [3], the controller is capable of extracting half of the power input to the structure over the entire frequency range

Active wave control can be performed using the deflection of the beam as the detected value and the shear force as the control input In this case, the transfer function of the controller can be expressed as [4]

s s A EI s

w

s s

4 3 2 1 1

4  (3)

In this equation, the control law includes s s

We note that the controller must be described by a transfer function in the form of a fractional expression of an integer power series Customarily, transfer

functions including non-integer powers of s have been approximated by

introducing a limitation in the frequency range Following the methods of MacMartin and others, the transfer function is substituted by an approximated

minimize the

moment for a control input p = s/C Using Eq (1), we can devise a controller to

= EI/ A, and M is the bending

A is the cross-sectional area of the cantilever, C

matrix at the actuator point, i.e., the transfer function from the beam deflection components The transfer matrix from the progressive wave vector to the retro-gressive wave vector is called the scattering matrix

transfer function with a finite number of poles and zeros located in the exponential positions along the negative real axis, as shown in the following equation [2]:

101010

10

101010

10

3 1 1 3

2 0 2 4

s s s

s

s s s

d

dt

d q t

)()

1(

1)]

Fractional calculus has been applied to control theory and research into this field has flourished Examples of recent publications include a report on PID-controller parameter tuning based on fractional calculus [10], a history of the development of control based on fractional-order derivatives and integrals [11] and reports on discrete-time modelling and numerical simulation of fractional Some implementation techniques in structural dynamics yield fractional-order derivative/integral responses Examples include a special analogue electrical circuit device [9] and a digital filter designed from a discrete approximated form of the definition equation [14] However, obtaining fractional-order derivatives/integrals of dynamical response at a single point on a structure presents technological difficulties

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systems by transfer function representation and state-space representation [12, 13]

In this study we overcome the difficulties due to fractional derivative responses by constructing the responses at the actuation point from a linear combination of multiple signals at several sensing points, rather than from a signal from a single sensor In this method, special sensors with additional signal-conversion functions are not required and existing displacement and velocity sensors can be used

3.1

Figure 2 shows a flexible cantilevered beam of length L with several sensors and

an actuator Without losing generality, four points on the beam can be designated

Calculated response of 1/2-order and 3/2-order derivatives

Fig 2. Active wave control for a flexible cantilevered beam

as sensor points, such as L/10, 4L/10, 7L/10, and L In this example, the actuator

440 Kuroda

is located at the 4L/10 position of the cantilever Displacement (the zeroth-order time-derivative of w i (t), see below) and velocity (the first-order time-derivative

of w i (t)) can be detected at each sensor A linear combination of the

displacement and velocity signals at each sensor point is fed back to the actuator The actuator then supplies shear force or bending moment to the cantilever

t

t w t w t w t w t

w ) 1(), 2(), 3(), 4 ) (9)The equation of motion (EOM) for the system under free vibration can be described as

0)()

()

1 2

2 2

3

(11)Consequently, the fractional-order EOM can be developed as

0)))

)(

)(

0 2 1

0 2

1 2

1

2 1 2

2 2

3

0 2

1 2

2 2

3 2

1

t w D K

t w D

C

t w D D

C D

t w D

D D

M

t w D D

D D

M D

(12)

directly using traditional methods for eigenvalue problems because it includes fractional-order derivatives To overcome the difficulties arising from the fractional-order derivatives, the expanded EOM can be expressed as

0)ˆˆ)(ˆ

2 1

t w K t w M

D , (13)The eigenstructure of the fractional-order EOM of Eq (12) cannot be solved The system response can be characterized by the displacement vectordefined as

00

00

0

000ˆ

C M

C M

M M

M

M K

00

0

00

000

00

4 4 3 3 2 2 1

vector for the system can be constructed The traditional (integer-order) state vector can be described as

respectively The formulation of Eq (13) can be solved using conventional

Fractional-order derivative feedback

right-hand side of Eq (10), where [G

Measured response of 1/2-order and 3/2-order derivatives

Using Eq (16), the traditional state vector and the fractional-order state

442

t

w w w w w w w wt

x ) 1 2 3 4 1 2 3 4 (18)The fractional-order state vector can be given by

t

w w D w w D w w D w w D

t

1 3 2 2 2 3 2 2 3

, ,

, ,

] [

)(

)(

B

A

t y

t x

,

(21)

)()

()

16 , 4 16 , 4 16 , 3 16 , 3 16 , 2 16 , 2 16 , 1 16 , 1

15 , 4 15 , 4 15 , 3 15 , 3 15 , 2 15 , 2 15 , 1 15 , 1

14 , 4 14 , 4 14 , 3 14 , 3 14 , 2 14 , 2 14 , 1 14 , 1

13 , 4 13 , 4 13 , 3 13 , 3 13 , 2 13 , 2 13 , 1 13 , 1

12 , 4 12 , 4 12 , 3 12 , 3 12 , 2 12 , 2 12 , 1 12 , 1

11 , 4 11 , 4 11 , 3 11 , 3 11 , 2 11 , 2 11 , 1 11 , 1

10 , 4 10 , 4 10 , 3 10 , 3 10 , 2 10 , 2 10 , 1 10 , 1

9 , 4 9 , 4 9 , 3 9 , 3 9 , 2 9 , 2 9 , 1 9 , 1

8 , 4 8 , 4 8 , 3 8 , 3 8 , 2 8 , 2 8 , 1 8 , 1

7 , 4 7 , 4 7 , 3 7 , 3 7 , 2 7 , 2 7 , 1 7 , 1

6 , 4 6 , 4 6 , 3 6 , 3 6 , 2 6 , 2 6 , 1 6 , 1

5 , 4 5 , 4 5 , 3 5 , 3 5 , 2 5 , 2 5 , 1 5 , 1

4 , 4 4 , 4 4 , 3 4 , 3 4 , 2 4 , 2 4 , 1 4 , 1

3 , 4 3 , 4 3 , 3 3 , 3 3 , 2 3 , 2 3 , 1 3 , 1

2 , 4 2 , 4 2 , 3 2 , 3 2 , 2 2 , 2 2 , 1 2 , 1

1 , 4 1 , 4 1 , 3 1 , 3 1 , 2 1 , 2 1 , 1 1 , 1

] [

]

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vectors in the matrix [ ] to create the smaller matrix [ ], as illustrated in Eq

where [ ] and [ ] are the matrices consisting of row vectors of [ ] in Eq (17) and which are associated with the state vectors defined respectively by Eq (18) and Eq (19) Hence, Eq (21) yields the final equation [15]

Using these formulae, we can determine fractional-order temporal-derivative terms of the observed value using a linear combination of displacement and velocity signals at each sensor point, as is necessary to implement active wave control

4 The Active Wave Controller and Its Control Effects

In contrast to traditional vibration suppression methods in which the objective is

to control the vibration modes (standing waves) of a structure, the objective of active wave control is to control travelling waves in the structure Consequently, active wave control is equivalent to control of the power flow in the structure Ideally, an active wave controller can extract half of the power flow transmitted

in the structure at all frequencies [3]

Fractional derivatives enable the formulation of the wave control law directly, rather than using the customary method of approximating the wave control transfer function by a function composed of integer-order power-series of the understanding of the physical meaning of wave control

As an example, we consider the wave control of a steel cantilever of length 2.7 m, width 50 mm and thickness 5.8 mm Sensors and an actuator are placed

on the beam as depicted in Fig 2 and a disturbance is applied at the free end of the beam

Figure 3 shows the eigenvalues for the expanded system satisfying the equation

0 ˆ ˆ ˆ ˆ ˆ

j j

The eigenvalues are in complex conjugate pairs and there exist 4 × 2 pairs of eigenvalues for the expanded system The eigenvalues may be mapped onto the Riemann surface for the function ˆ s1 / 2

j , consisting of two Riemann sheets Four pairs of complex conjugate eigenvalues appear on each sheet

The eigenvalues on the principal Riemann sheet and the corresponding eigenvectors illustrate the sinusoidal motion of the structure; they form the mode shapes of the structure The natural frequencies of the original system can be obtained by squaring the eigen pairs The process gives four conjugate pairs The imaginary part of each conjugate pair gives the eigenfrequency of the original system while the real part gives the product of the damping ratio and the natural frequencies of the original system

variable s of the Laplace transform Furthermore, it provides a deeper

under-444

The eigenvalues on the second Riemann sheet represent poles in the system transfer function, which produce a monotonically decreasing response of the structure This monotonically decreasing motion describes the creep and relaxation response of the original system

An advantage of active wave control is that it yields a controller design that depends directly on the dimensions and material properties of the structure without the necessity of carrying out modal analysis of the structure in advance Additionally, the controller provides active damping for all structural vibration modes However, it cannot actively provide strong damping to a specific vibration mode

We carried out a simulation using a combination of deflection-angle sensors show the driving-point compliance and impulse response evaluate vibration suppression effects of wave control for this simulation

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Fig 3. Eigenvalues of the expanded system

4.1 Combination of beam-slope sensor and bending-moment actuator

and a bending-moment actuator, as expressed in Eq (2) Figures 4a and 4b

Fig 4. Active wave control of a flexible cantilevered beam with beam-slope sensors and a

4.2

We carried out a second simulation using a combination of deflection sensors

bending-moment actuator

Combination of beam-deflection sensor and shear-force actuator

and a shear-force actuator, as expressed in Eq (3) Figures 5a and 5b show the wave control for this simulation

446

Fig 5. Active wave control of a flexible cantilevered beam with beam-deflection sensors

The retrogressive wave is eliminated at the control point in both the above simulations, but the results are quite different

Interestingly, it has been reported that viscoelastic materials, such as silicon

s [16] Accordingly, using a reaction surface, a passive control system that supports a cantilever by viscoelasticity may achieve a similar control effect as active wave control for a sensor/actuator colocation

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and a shear-force actuator

gel, also have a frequency response characterized by