Research Article Exact Solution of Impulse Response to a Class of Fractional Oscillators and Its Stability

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1 Mahemaical Problems in Engineering Volume 2, Aricle ID , 9 pages doi:.55/2/ Research Aricle Exac Soluion of Impulse Response o a Class of Fracional Oscillaors and Is Sabiliy Ming Li, S. C. Lim, 2 and Shengyong Chen 3 School of Informaion Science and Technology, Eas China Normal Universiy, no. 5, Dong-Chuan Road, Shanghai 224, China 2 28 Farrer Road, #5-, Suon Place, Singapore College of Compuer Science, Zhejiang Universiy of Technology, Hangzhou 323, China Correspondence should be addressed o Ming Li, ming lihk@yahoo.com Received 8 Augus 2; Acceped 5 Sepember 2 Academic Edior: Crisian Toma Copyrigh q 2 Ming Li e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. Oscillaor of single-degree-freedom is a ypical model in sysem analysis. Oscillaions resuled from differenial equaions wih fracional order arac he ineress of researchers since such a ype of oscillaions may appear dramaic behaviors in sysem responses. However, a soluion o he impulse response of a class of fracional oscillaors sudied in his paper remains unknown in he field. In his paper, we propose he soluion in he closed form o he impulse response of he class of fracional oscillaors. Based on i, we reveal he sabiliy behavior of his class of fracional oscillaors as follows. A fracional oscillaor in his class may be sricly sable, nonsable, or marginally sable, depending on he ranges of is fracional order.. Inroducion Fracional sysems gain increasing aenion in applied sciences, ranging from mechanical engineering o elecrical engineering, see, for example,. Recall ha sabiliy is an essenial propery of sysems, see for example 2, for he sabiliy of convenional sysems of ineger order and 3 5 for fracional sysems. One of he ypical models used in sysem analysis is he oscillaor of single-degreefreedom 6. I is given by m d2 Y d 2 c dy d ky e,.

2 2 Mahemaical Problems in Engineering where m> is he mass, c he damping consan, k>hesiffness and e he forcing funcion. Le 2b c/m andω k/m. Then, we rewrie he above by d 2 Y d 2 2b dy d ω 2 Y e..2 The parameer b is called damping coefficien and ω is inheren frequency. Le g be he impulse response o he above equaion. I is he soluion o.2 for e δ he Dirac dela funcion wih zero iniial condiions and is given by g ω e b sin ω,.3 where ω ω 2 b2 is angular frequency. Equaion.3 implies a damped oscillaion. In he case of b, ha is, he zero damping,.2 reduces o ( ) d 2 d 2 ω2 Y e, ω >..4 The impulse response o he above sysem is g ω sin ω,.5 which corresponds o a free oscillaion. Recenly, research on fracional oscillaors has araced considerable ineress, see for example, In he sabiliy analysis of fracional oscillaors, he auhors in 7 9 sudied a class of fracional oscillaors expressed by d 2 ε d 2 ε ω2 x e, <ε<..6 They concluded ha he above oscillaor may be sricly sable as if i is a damped oscillaor. In his paper, we focus on anoher class of fracional oscillaors ha were firs inroduced by Lim and Muniandy 23.Isaisfies ( d 2 d 2 ω2 ) β x e, β >..7 The soluions o.7 for e being a whie noise in ime domain and frequency domain are obained in 2, which are furher exended o he oscillaor wih wo fracional indexes in 22, 24, 25. However, is sabiliy remains an unsolved issue. This paper shows ha his ype of fracional oscillaor is sricly sable when <β<, nonsable when β>, and marginally sable for β.

3 Mahemaical Problems in Engineering 3 The remainder of he paper is organized as follows. In Secion 2, he impulse response of he fracional sysem.7 in he closed form is proposed. Sabiliy analysis is given in Secion 3. Discussions and conclusions are in Secion Impulse Response For >andv>, denoe by D v he fracional derivaive of Capuo ype 26 defined by D ν f Γ n ν f n u du, n α n, 2. ν n u where Γ is he Gamma funcion. For simpliciy, we wrie D v by D v below. One can regard D 2 ω 2 β as a shifed fracional derivaive of D 2ν. By aking he binomial expansion, one ges ( ) β D 2 ω 2 ( β )n ω2n D2 β n, 2.2 n where β m is he Pochhammer symbol, ha is, ( β )m β! ( β m )! 2.3 The fracional oscillaor saisfying D 2 ω 2 β x e, β > has he impulse response funcion h, which is he soluion o D 2 ω 2 β h δ. Denoe he Laplace ransform of h by H s. Then, we have H s L h ( s2 ω 2 ) β ω 2β ( s2 /ω) 2 β, 2.4 wherelisheoperaorofhelaplaceransform.expandingherighsideofheaboveusing he binomial series yields ω 2β ( s2 /ω 2 ) β ω 2β ( ) β m m s m, m! m s <. ω 2.5 Therefore, he impulse response o.7 is given by h ω 2β [ ( ) ] β L m m s m m! m ω 2β ( ) β m m δ m,, 2.6 m! m where L is he operaor of he inverse Laplace ransform.

4 4 Mahemaical Problems in Engineering Anoher form of h, which may be more convenien for he sabiliy analysis, is expressed below. Considering he Laplace ransform pair given in 27, wehave L ( s2 ω) 2 β π Γ ( β ) 2ω β /2 β /2 J β /2 ω, 2.7 where J β /2 ω is he Bessel funcion of he firs kind of order β /2. Therefore, we have he impulse response given by h, h ( ; β ) π Γ ( β ) 2ω β /2 β /2 J β /2 ω, β >, Sabiliy Analysis The above discussion allows us o obain he following resuls concerning he sabiliy of he fracional oscillaor under consideraion. Before we discuss hese sabiliy properies, we firs recall he crieria of sabiliy based on he principle of bounded-inpu and bounded-oupu BIBO. A sysem is said o be sable if h d consan, 3. which implies lim h andpolesofl h are locaed on he lef-hand porion of he s-plane. A sysem is said o be nonsable if h is increasing, and accordingly poles of L h are locaed on he righ-hand porion of he s-plane. One says ha a sysem is neural if poles of L h are on he complex jω-axis 28, 29. Noe ha 27 J v /2 v ( u 2) v /2 cos u du, Re v> / Γ v /2 Γ /2 Thus, according o 3.2 for v /2 and considering 2.8 for β, h ; β in 2.8 reduces o he simple case wih impulse response corresponds o he free oscillaor. Tha is, h ( ; β ) β sin ω ω. 3.3 This leads o he following remark. Remark 3.. h ; β reduces o he impulse response o he ordinary oscillaor for β. Figure indicaes he plo of h ; for ω. The ordinary oscillaor is neural.

5 Mahemaical Problems in Engineering 5.5 h ; Figure : Plo of he impulse response of he ordinary oscillaor. One noes ha h ; β is unbounded if β>. As a maer of fac, from 3.2,wehave /2 v ( u 2) v /2 /2 v ( du Jv u 2) v /2 du. 3.4 Γ v /2 Γ /2 Γ v /2 Γ /2 Since β>implies v>/2, one immediaely sees ha boh he righ side and he lef one on he above expression are, respecively, unbounded as. Figure 2 shows he oscillaions for various values of β for w. Thus, we have he following remark. Remark 3.2. For β>, he fracional oscillaor.7 is non sable. The oher ineresing hing is ha he h ; β becomes an oscillaion wih decreasing ampliude if <β<. In fac, J v m ( ) 2m v. 3.5 m!γ m v 2 m Thus, J v for. 3.6 Consequenly, we have lim h( ; β ) π lim Γ ( β ) 2ω β /2 β /2 J β /2 ω, <β<, 3.7 which implies ha he poles of L h ; β for <β< are locaed on he lef of he s-plane. Therefore, he following remark is an obvious consequence. Figure 3 gives plos of h ; β for several values of β for ω. Remark 3.3. The sysem.7 is sricly sable for <β<. The sabiliy of he fracional oscillaors expressed by.7 is summarized in Table.

6 6 Mahemaical Problems in Engineering h ;.2 2 h ; a b h ;.6 h ; c d h ; e Figure 2: Increasing oscillaions. a β.2. b β.4. c β.6. d β.8. e β.2. h ;.3.5 h ; h ; a c h ; b d Figure 3: Oscillaions. a β.3 b β.5 c β.7 d β.9.

7 Mahemaical Problems in Engineering 7 Table : Sabiliy performances of.7. Value of β Type of sabiliy <β< Sricly sable β> Non sable β Neural 4. Discussions And Conclusions As previously noed in 25, a fracal ime series can be considered as a soluion o a fracional differenial equaion driven by a whie noise. Thus, here may be research niche for oher series, for example, hose discussed in In his paper, we have presened wo forms, ha is, 2.6 and 2.8, of he impulse response o he fracional sysem expressed by.7. Such a ype of fracional oscillaors has dramaic performances in is sabiliy. We have revealed ha he sysem wih.7 conains hree subclasses of oscillaors. Ordinary free oscillaors are a special case of.7 for β. I corresponds o fracional oscillaors of sricly sable for <β< and non sable if β>. Noe ha here is no damping erm in he discussed oscillaors in form. However, in he case of <β<, hey are sricly sable, as hough hey were ordinary oscillaors equipped wih a cerain damping. On he oher side, for β>, hey are non sable. Acknowledgmens This work was parly suppored by he Naional Naural Science Foundaion of China NSFC under he Projec Gran nos , , 6724, 6872, 6732, and 6974, he 973 plan under he Projec no. 2CB328, NCET, and he Science and Technology Deparmen of Zhejiang Province 29C28, 2R6, 2C3395, Y9592. The discussions wih Dr. Yang-Quan Chen of he Uah Sae Universiy are helpful. References M. D. Origueira and A. G. Baisa, On he relaion beween he fracional Brownian moion and he fracional derivaives, Physics Leers A, vol. 372, no. 7, pp , M. D. Origueira, An inroducion o he fracional coninuous-ime linear sysems: he 2s cenury sysems, IEEE Circuis and Sysems Magazine, vol. 8, no. 3, pp. 9 26, Y. Luo and Y. Chen, Fracional order proporional derivaive conroller for a class of fracional order sysems, Auomaica, vol. 45, no., pp , Y. Q. Chen and K. L. Moore, Discreizaion schemes for fracional-order differeniaors and inegraors, IEEE Transacions on Circuis and Sysems I, vol. 49, no. 3, pp , J. A. Tenreiro Machado, M. F. Silva, R. S. Barbosa e al., Some applicaions of fracional calculus in engineering, Mahemaical Problems in Engineering, vol. 2, Aricle ID 6398, 34 pages, 2. 6 O. P. Agrawal, Soluion for a fracional diffusion-wave equaion defined in a bounded domain, Nonlinear Dynamics, vol. 29, no. 4, pp , I. Podlubny, I. Peráš, B. M. Vinagre, P. O Leary, and L. Dorčák, Analogue realizaions of fracionalorder conrollers, Nonlinear Dynamics, vol. 29, no. 4, pp , C. H. Eab and S. C. Lim, Pah inegral represenaion of fracional harmonic oscillaor, Physica A, vol. 37, no. 2, pp , C. H. Eab and S. C. Lim, Fracional generalized Langevin equaion approach o single-file diffusion, Physica A, vol. 389, no. 3, pp , 2.

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