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1 2 3 1 2 3 Abstract Fractional-order systems, fractional calculus, conjugated-order differintegrals, complex order-distributions. complex-order differintegrals. 1 Introduction CONJUGATED ORDER DIFFERINTEGRALS COMPLEX ORDER-DISTRIBUTIONS USING Jay L. Adams , Tom T. Hartley , and Carl F. Lorenzo Keywords OH 44325-3904; E-mail: JLA36@uakron.edu Department of Electrical and Computer Engineering, The University of Akron, Akron, OH 44325-3904; E-mail: TomHartley@aol.com NASA Glenn Research Center, Cleveland, OH 44135; E-mail: Carl.F.Lorenzo@grc.nasa.gov © 2007 Springer. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications Department of Electrical and Computer Engineering, The University of Akron, Akron, This paper develops the concept of the complex order-distribution. This is a continuum of fractional differintegrals of complex order. Two types of complex order-distributions are considered, uniformly distributed and Gaussian distri- buted. It is shown that these basis distributions can be summed to approxi- mate other complex order-distributions. Conjugated differintegrals, introduced in this paper, are an essential analytical tool applied in this development. Con- jugated-order differintegrals are fractional derivatives whose orders are complex conjugates. These conjugate-order differintegrals allow the use of complex-order differintegrals while still resulting in real time-responses and real transfer-func- tions. An example is presented to demonstrate the complex order-distribution concept. This work enables the generalization of fractional system identification to allow the search for complex order-derivatives that may better describe real- time behaviors. of fractional-order operators. In that discussion, the distribution of order was duced by Hartley and Lorenzo [1,2] as the continuum extension of collections lopment of complex order-distributions. Order distributions have been intro- This paper uses the concept of conjugate-order differintegrals for the deve- required implicitly to be real, but it was able to include any real number. This concept of an order-distribution is expanded to include distributions which have non-real portions, i.e., complex order-distributions. This is done to expand on the system identification technique that used real order-distributions [1] in Physics and Engineering, 347–360. 347 348 Fractional operators of non-integer, but real, order have been the focus of numerous studies. Complex, or even purely imaginary, operators have been studied by a few [2,3]. A motivation in the development of complex operators is limited work in the area of complex-order differintegrals has been done [5]. Both blockwise constant and Gaussian complex order-distributions are presented in the Laplace domain. Approximate complex order-distributions with either the blockwise constant or Gaussian distributions are shown. Finally, the frequency response of a conjugate-symmetric complex order-distribution is compared to that of impulsive distributions in an example. 2 Complex Differintegrals In general, we will consider the complex differintegral acting on a function f(t) to be defined as )()()( 00 tfdtfdtg ivu t q t . (1) uninitialized operator will have the Laplace transform )()()()()}({ )ln( sFessFsssFssGtgL sivuivuivu . (2) Using Euler’s identity, this can be rewritten as )())ln(sin())ln(cos()( sFsvisvssG u . (3) To obtain the impulse response of this operator, the inverse Laplace transform is required. It is defined for 0q as )( 1 1 q t sL q q (4) For our specific case it becomes, with an impulsive input g(t), Adams, Hartley, and Lorenzo to include the possibility of using complex order-distributions. To ensure that only real time-responses are considered, the idea of conjugate-order differinte- grals is utilized. Just as conjugate-differintegrals provide real time-responses, so do complex order-distributions which are conjugate-symmetric. to generalize the idea of derivatives and integrals of distributed order. Very While the physical meaning of a complex function of time is still under dis- cussion, a goal of this paper is the development of complex-order differintegrals which yield purely real time-respsonses. To this end, the concept of conjugate- differintegral is introduced. Following the work of Kober [3], Love [4], and Oustaloup et al. [5], this 349 )( )( 1 )(1 ivu t sLtf ivu ivu , (5) and u and v such that the transform is defined. This can be rewritten as )ln( 11 )(1 )()( )( tiv u iv u ivu e ivu t t ivu t sLtf (6) or by using Euler’s identity as ))ln(sin()ln(cos( )( )( 1 )(1 tvitv ivu t sLtf u ivu . (7) Imaginary time responses have limited physical meaning. However, the functions ))ln(cos( tv and ))ln(sin( tv show up regularly as solutions of special 3 Conjugated-Order Differintegrals The interpretations and inferences of individual complex-order operators are not well understood. However, we can create useful operators by considering the complex-order derivative or integral analogously to a complex eigenvalue of a define the uninitialized conjugated differintegral as )()()()()()( 0000 ),( 0 tfdtfdtfdtfdtfdtg ivu t ivu t q t q t vuq t . (8) Representing this in the Laplace domain gives )()()()( ),( 0 sFsssssFsstfdLtgL ivuivuivuivuvuq t . (9) Rearranging and applying Euler’s identity allows this to be written as )()()( )ln()ln(),( 0 sFeestfdLsG sivsivuvuq t )())ln(sin())ln(cos())ln(sin())ln(cos( sFsvisvsvisvs u )())ln(cos(2 sFsvs u , (10) )()()()()()( 0000 ),( 0 tfdtfdtfdtfdtfdtg ivu t ivu t q t q t vuq t . (11) COMPLEX ORDER-DISTRIBUTIONS USING CONJUGATED ORDER time-varying differential equations known as Cauchy–Euler differential equa- tions. dynamic system, that is, coexisting with its complex-order conjugate. We now which is a purely real operator. Likewise, the complementary conjugated differ- integral is defined as 350 Representing this in the Laplace domain gives )()()()( ),( 0 sFsssssFsstfdLtgL ivuivuivuivuvuq t . (12) Rearranging and using the Euler identity allows us to write )()()( )ln()ln(),( 0 sFesestfdLsG sivusivuvuq t )())ln(sin())ln(cos())ln(sin())ln(cos( sFsvisvssvisvs uu )())ln(sin(2 sFsvis u , (13) which is a purely imaginary operator. It should be noted that a multiplicative operation returns a real operator, )()( 2 sFssFss uivuivu , (14) while a division will yield the imaginary operator )( 2 sFs iv . We note that a real differintegral can always be broken into the product of two complex conjugate derivatives. The conjugated-order fractional integral may be expressed for negative real order as )()()()( 00 ),( 0 tfdtfdtfdtg ivu t ivu t vuq t , (15) with Laplace transform given by )()()()( ),( 0 sFssssFsstfdLtgL ivivuivuivuvuq t . (16) For )(tf a unit impulse, the inverse Laplace transform of the conjugated )()( )( 11 )()(1 ivu t ivu t ssLtg ivuivu ivuivu (17) The presence of the gamma function of complex argument is somewhat problematic, and to move forward we note that the reciprocal gamma function has symmetry about the real axis [6]. Thus we can write )( 1 Im )( 1 Re )( 1 ivu i ivuivu (18) and )( 1 Im )( 1 Re )( 1 ivu i ivuivu . (19) The desired inverse Laplace transform can then be written Adams, Hartley, and Lorenzo integral can also be obtained using the operator inverse of Eq. (5), iviv iviv u t ivu it ivu t ivu it ivu ttg )( 1 Im )( 1 Re )( 1 Im )( 1 Re )( 1 ivivivivu tt ivu itt ivu t )( 1 Im )( 1 Re 1 (20) We can now write )ln(tiviv et and use Euler’s identity to give ))ln(sin())ln(cos())ln(sin())ln(cos( )( 1 Im ))ln(sin())ln(cos())ln(sin())ln(cos( )( 1 Re)( 1 1 tvitvtvitv ivu ti tvitvtvitv ivu ttg u u Thus ))ln(sin( )( 1 Im))ln(cos( )( 1 Re2 ))ln(cos(2)( 1 1)()(1 tv ivu tv ivu t svsLssLtg u uivuivu . (21) When )(tf is not a unit impulse, the time response is given by the convolution of )(tg with )(tf . It should be noted then that the conjugated differintegral has a purely real time response. Similarly, the inverse transform of the complementary conjugated-order derivative of a unit impulse can be found as ))ln(cos( )( 1 Im))ln(sin( )( 1 Re2 ))ln(sin(2)( 1 1)()(1 tv ivu tv ivu ti svsiLssLtg u uivuivu , (22) a purely imaginary time response. The frequency response of a particular conjugated integral is shown in by u. It has superimposed on it a variation that is periodic in log(w), the period of variation that is periodic in log(w). Frequency responses of this form are said to have scale-invariant frequency responses [7], which are fractal in the frequency seen to have a spiral form. Finally the Nichols plane representation is given in approximated to any accuracy using rational transfer functions over any desired range of frequencies [8]. A frequency response of this form is of great use for which is determined by v. The phase-frequency response also rolls off (or up) at COMPLEX ORDER-DISTRIBUTIONS USING CONJUGATED ORDER 351 Fig. 1. The magnitude frequency response rolls off (or up) at a mean rate set an average linear rate, similar to a delay. It also has superimposed on it a domain. The Nyquist plane representation is given in the Fig. 2a. It can be Fig. 2b. Here the plot is a roughly straight line, having the angle from the horizontal determined by v. Frequency domain functions of this form can be the angle and roll-off rate easily defined by ivu , respectively. The CRONE (controller) design [5], contains terms similar to those seen here, however, they are not recognized as being related to conjugated-order differintegrals. In the introduction of conjugated derivatives the weightings of the complex 4.01.04.01.0 ii s . Real coefficients: ))ln(cos(2 )( )(ln()ln( svks sekssskskskssG u sivsivuivivuivuivu (23) ))ln(sin(2 )( )(ln()ln( svksi sekssskskskssG u sivsivuivivuivuivu (24) ))ln(cos(2 )( )(ln()ln( svksi seiksssiksiksikssG u sivsivuivivujvujvu (25) 352 Adams, Hartley, and Lorenzo control-system design as it is roughly a straight line in the Nichol’s plane, with 3.1 Special conjugate derivative forms Fig. 1. Bode (a) magnitude and (b) phase plots for s Imaginary coefficients: derivatives were real and unity. However, complex derivatives can also have complex weightings. Such complex coefficients may lead to real time-responses, so it is important to determine the effects of different combinations. The deter- mination of effects is presented here, with the purely real time-responses boxed. There is also a corresponding impulse response. COMPLEX ORDER-DISTRIBUTIONS USING CONJUGATED ORDER 353 4.01.04.01.0 ii s . ))ln(sin(2 )( )(ln()ln( svks seiksssiksiksikssG u sivsivuivivuivuivu (26) Complex coefficients (4 of the 16 possible): ))ln(cos(2))ln(cos(2 )()()( )(ln()ln()(ln()ln( svbsisvas eeibseeas ssibssasibasibasG uu sivsivusivsivu ivuivuivuivuivuivu (27) ))ln(sin(2))ln(cos(2 )()()( )(ln()ln()(ln()ln( svbssvas eeibseeas ssibssasibasibasG uu sivsivusivsivu ivuivuivuivuivuivu (28) ))ln(sin(2))ln(sin(2 )()()( )(ln()ln()(ln()ln( svbssvasi eeibseeas ssibssasibasibasG uu sivsivusivsivu ivuivuivuivuivuivu (29) Fig. 2. (a) Nyquist and (b) Nichols Ppots for s 354 ))ln(sin(2))ln(sin(2 )()()( )(ln()ln()(ln()ln( svbssvasi eeibseeas ssibssasibasibasG uu sivsivusivsivu ivuivuivuivuivuivu (30) 4 Complex Order-Distribution Definition The conjugated derivative will now be applied to the development of complex defined as dqtfdqkth b a q t )()()( 0 , (31) for q real. We will define the complex order-distribution as dvdutfdvukth ivu t )(),()( 0 . (32) This equation can be Laplace transformed as dvdusFsvuksH ivu )(),()( . (33) We now must consider two complex planes as in [5]. One is the standard Laplace s-plane, and the other is the complex order-plane, or q-plane, where ivuq . It is understood that the order of a given operator is not necessarily an impulse in the q-plane as is usually the case for fractional-order differential equations, )()( qqk . The order will now be considered to be a continuum or distribution in the complex order-plane, a complex generalization of [1]. When the weighting function ),( vuk is complex and it has symmetry about the real order-axis, then the corresponding time response is real. We now consider complex order-distributions that are constant intensity, k, symmetric about the real axis from uu to uu , and from vi to results are presented here. v v u u ivuw v v uu uu ivu dvdwksdvdukssH , where dwduuwu , , Adams, Hartley, and Lorenzo order-distributions. In previous studies [1,7] the real order-distribution was 4.1 Blockwise constant complex order-distribution i v . A detailed derivation is given by Hartley et al. [9] but the idea and COMPLEX ORDER-DISTRIBUTIONS USING CONJUGATED ORDER 355 v v siv u u swu v v u u ivwu dvedweksdvdwssks lnln v v u uw sw u s sv s e ks 0 ln ln lnsin ln 2 )ln( lnsin )ln( lnsinh 4 s sv s su ks u . (34) vv vv uu uu ivu dvduks 1 vv vv iv uu uu u dvsduks v v vri u u uw drsdwks v v ir u u wviu drsdwsks v r u uw sw viu s sr s e ks 0 ln ln lnsin ln 2 svsu s ks viu lnsinlnsinh ln 4 2 Similarly, for constant block order-distributions of intensity k which are centered at viuq [9] vv vv uu uu ivu dvdukssH 1 svsu s ks viu lnsinlnsinh ln 4 2 . (36) conjugated block differintegral as shown below [9]. viuviu sssvsu s k sHsHsH lnsinlnsinh ln 4 2 11 viuviu sssssvsu s k lnsinlnsinh ln 4 2 svsvsu s ke su lncoslnsinlnsinh ln 8 2 ln . (37) at q u iv (off the real-axis), then, as shown by Hartley et al. [9], is For constant block order-distributions of intensity k which are centered H s . (35) Combining these two complex results, Eqs. (35) and (36), give the real Sums of these order-distributions can be used to approximate complex order- distributions that are symmetric with respect to the real-order axis as follows. Assuming the widths of each block are the same and the intensities are nm k , , dvdusksH n uu uu ivu mn vv vv m n n m m 1 , )ln( lnsin ln lnsinh 1 , s sv s su sk n viu mn m mn 1 , )ln( lnsin ln lnsinh 4 n viu mn m mn sk s sv s su . (38) Finally, we consider complex order-distributions that have the form of Gaussians of intensity k centered on, and symmetric about, the real order-axis [9], dvduskesH ivu vuu vu 2 2 2 2 dvseduske iv v u uu vu 2 2 2 2 0 2 2 2 2 dvssedwseks jviv v w w u vu v v v s v u u u s u u s uis Erfie i s us Erfeks v u 2ln 2 2ln 2 2 ln 4 1 2 ln 4 1 22 22 s v s u u vu eeks 2222 ln 4 1 ln 4 1 u s vu sek vu 222 ln 4 1 , (39) 356 Adams, Hartley, and Lorenzo 4.2 Gaussian complex order-distribution [...]... formed in the process of growth of random fractals If these QF having slow logarithmic asymptotic can be applicable for description of different random clusters, including clusters having near-neighboring order then with their usage one can describe 377 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 377–388 © 2007 Springer... differintegrals have been defined in the time-domain, and their Laplace transforms have been determined The use of conjugated-order differintegrals allows the use of complex-order operators while retaining real timeresponses Complex-weighted conjugated differeintegrals have been investigated, showing that particular weightings have real time-responses Expanding collections of conjugated differintegrals... statement has been confirmed in paper [4] and a “universal” decoupling procedure leading to kinetic equations, containing non-integer operators has been recently suggested in [5] In this paper we obtained different types of QF modifying and generalizing the conventional diffusion-limited aggregation (DLA) procedure and procedure and considered the distorted lattices also In all cases considered we confirmed... whole growing process In such way we “ planted” 10 clusters having approximately 10 4 particles in each cluster 3.2 Random rain model This method is obtained from the first model if we switched off respectively the random and friction forces The particles in the process of growing are moving in the arbitrary direction with constant value of the given velocity Other peculiarities are remained the same... in the experiment Keywords Viscoelastic cylindrical column, slow quasi-static phenomenon, rapid dynamic phenomenon, nonlinear fractional derivative model 1 Introduction Fractional calculus is known as a fundamental tool to describe the behavior of weak frequency dependence of viscoelastic materials in a broad frequency range Fractional derivative constitutive models offer many successes in engineering... derivative element, a nonlinear elastic element, and a friction element for a rubber vibration isolator under harmonic displacement 363 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 363–376 © 2007 Springer 364 2 Nasuno, Shimizu, and Fukunaga excitation with static pre-compression Deng et al (2004) presented a fractional derivative... 377–388 © 2007 Springer 378 Nigmatullin and Alekhin describe wide class of disordered media From another side, it could increase possibilities of the application of the fractional calculus for description of relaxation and transport properties in such kind of media, where the non-integer operators of differentiation and integration are appeared in the result of averaging procedure of a smooth function... of nonlinear analytical responses for a type of element xν Dq x(t) are investigated analytically In Chapter 3, the experiments are summarized briefly to extract the nonlinear fractinal derivative models for the slow and the rapid phenomena In Chapters 4, the following type of nonlinear fractional derivative model c(x)Dq x(t) = F (t), (1) is proposed for both in the slowly compressed process and in the... centers located in the point (0,0) and having the radiuses rinit 100 r0 and rout 300r0 , respectively The first circle serves as a source of the generated particles, the second one is used as a particles sink Let us consider the process of the particle movement in detail A particle having the unit mass m 1 is generated on the circle rinit and starts its moving with the value of velocity V 1 in randomly... subscript exp indicates the experimental value 4.2 Nonlinear rapid process The energy dissipation of a viscoelastic body or a damping device per unit cycle is called damping capacity, and is defined by W = cα Dα x(t)dx, (18) where the suffix α indicates the rapid process Substituting x = x0 +y0 sin(ωt) into Eq (18), and neglecting the contribution from higher-order terms, one obtains 2 W = π cα y0 ω α sin πα . Sj¨oberg et al. sists of a linear fractional derivative element, a nonlinear elastic element,and a © 2007 Springer. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments. Carl.F.Lorenzo@grc.nasa.gov © 2007 Springer. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications Department of Electrical and Computer Engineering, The University. distributions in an example. 2 Complex Differintegrals In general, we will consider the complex differintegral acting on a function f(t) to be defined as )()()( 00 tfdtfdtg ivu t q t . (1) uninitialized
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