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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 657839, 9 pages
Exact Solution of Impulse Response to a Class of Fractional Oscillators and Its Stability
1School of Information Science and Technology, East China Normal University, no. 500, Dong-Chuan Road, Shanghai 200241, China
228, Farrer Road, #05-01, Sutton Place, Singapore 268831
3College of Computer Science, Zhejiang University of Technology, Hangzhou 310023, China
Received 18 August 2010; Accepted 15 September 2010
Academic Editor: Cristian Toma
Copyright © 2011 Ming Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Oscillator of single-degree-freedom is a typical model in system analysis. Oscillations resulted from differential equations with fractional order attract the interests of researchers since such a type of oscillations may appear dramatic behaviors in system responses. However, a solution to the impulse response of a class of fractional oscillators studied in this paper remains unknown in the field. In this paper, we propose the solution in the closed form to the impulse response of the class of fractional oscillators. Based on it, we reveal the stability behavior of this class of fractional oscillators as follows. A fractional oscillator in this class may be strictly stable, nonstable, or marginally stable, depending on the ranges of its fractional order.
Fractional systems gain increasing attention in applied sciences, ranging from mechanical engineering to electrical engineering, see, for example, [1–11]. Recall that stability is an essential property of systems, see for example , for the stability of conventional systems of integer order and [13–15] for fractional systems.
One of the typical models used in system analysis is the oscillator of single-degree-freedom . It is given by where is the mass, the damping constant, the stiffness and the forcing function. Let and . Then, we rewrite the above by The parameter is called damping coefficient and is inherent frequency.
Let be the impulse response to the above equation. It is the solution to (1.2) for = (the Dirac delta function) with zero initial conditions and is given by where is angular frequency. Equation (1.3) implies a damped oscillation.
In the case of , that is, the zero damping, (1.2) reduces to The impulse response to the above system is which corresponds to a free oscillation.
Recently, research on fractional oscillators has attracted considerable interests, see for example, [17–22]. In the stability analysis of fractional oscillators, the authors in [17–19] studied a class of fractional oscillators expressed by They concluded that the above oscillator may be strictly stable as if it is a damped oscillator. In this paper, we focus on another class of fractional oscillators that were first introduced by Lim and Muniandy . It satisfies The solutions to (1.7) for being a white noise in time domain and frequency domain are obtained in , which are further extended to the oscillator with two fractional indexes in [22, 24, 25]. However, its stability remains an unsolved issue. This paper shows that this type of fractional oscillator is strictly stable when , nonstable when , and marginally stable for .
The remainder of the paper is organized as follows. In Section 2, the impulse response of the fractional system (1.7) in the closed form is proposed. Stability analysis is given in Section 3. Discussions and conclusions are in Section 4.
2. Impulse Response
For and , denote by the fractional derivative of Caputo type  defined by where Γ is the Gamma function. For simplicity, we write by below. One can regard as a shifted fractional derivative of . By taking the binomial expansion, one gets where is the Pochhammer symbol, that is, The fractional oscillator satisfying has the impulse response function , which is the solution to . Denote the Laplace transform of by . Then, we have where L is the operator of the Laplace transform. Expanding the right side of the above using the binomial series yields Therefore, the impulse response to (1.7) is given by where is the operator of the inverse Laplace transform.
Another form of , which may be more convenient for the stability analysis, is expressed below. Considering the Laplace transform pair given in , we have where is the Bessel function of the first kind of order . Therefore, we have the impulse response given by
3. Stability Analysis
The above discussion allows us to obtain the following results concerning the stability of the fractional oscillator under consideration. Before we discuss these stability properties, we first recall the criteria of stability based on the principle of bounded-input and bounded-output (BIBO). A system is said to be stable if which implies and poles of are located on the left-hand portion of the -plane. A system is said to be nonstable if is increasing, and accordingly poles of are located on the right-hand portion of the s-plane. One says that a system is neutral if poles of are on the complex -axis [28, 29].
Note that  Thus, according to (3.2) for and considering (2.8) for , in (2.8) reduces to the simple case with impulse response corresponds to the free oscillator. That is, This leads to the following remark.
Remark 3.1. reduces to the impulse response to the ordinary oscillator for . Figure 1 indicates the plot of for . The ordinary oscillator is neutral.
One notes that is unbounded if . As a matter of fact, from (3.2), we have Since implies , one immediately sees that both the right side and the left one on the above expression are, respectively, unbounded as . Figure 2 shows the oscillations for various values of β for . Thus, we have the following remark.
Remark 3.2. For , the fractional oscillator (1.7) is non stable.
The other interesting thing is that the becomes an oscillation with decreasing amplitude if . In fact, Thus,
Consequently, we have which implies that the poles of for are located on the left of the -plane. Therefore, the following remark is an obvious consequence. Figure 3 gives plots of for several values of β for .
Remark 3.3. The system (1.7) is strictly stable for .
4. Discussions And Conclusions
As previously noted in , a fractal time series can be considered as a solution to a fractional differential equation driven by a white noise. Thus, there may be research niche for other series, for example, those discussed in [30–43]. In this paper, we have presented two forms, that is, (2.6) and (2.8), of the impulse response to the fractional system expressed by (1.7). Such a type of fractional oscillators has dramatic performances in its stability. We have revealed that the system with (1.7) contains three subclasses of oscillators. Ordinary free oscillators are a special case of (1.7) for β = 1. It corresponds to fractional oscillators of strictly stable for and non stable if . Note that there is no damping term in the discussed oscillators in form. However, in the case of , they are strictly stable, as though they were ordinary oscillators equipped with a certain damping. On the other side, for, they are non stable.
This work was partly supported by the National Natural Science Foundation of China (NSFC) under the Project Grant nos. 60573125, 60873264, 61070214, 60870002, 60703112, and 60974001, the 973 plan under the Project no. 2011CB302800, NCET, and the Science and Technology Department of Zhejiang Province (2009C21008, 2010R10006, 2010C33095, Y1090592). The discussions with Dr. Yang-Quan Chen of the Utah State University are helpful.
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