An effective analytical criterion for stability testing of fractional-delay systems

Automatica 47 (2011) 2001–2005

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An effective analytical criterion for stability testing of fractional-delay systems✩ Min Shi, Zaihua Wang ∗ State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China

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Article history: Received 18 August 2010 Received in revised form 23 February 2011 Accepted 27 February 2011 Available online 19 June 2011 Keywords: BIBO stability Fractional-delay system Stability test

abstract This paper investigates the BIBO stability of a class of fractional-delay systems with rational orders, a stability that holds true if all the characteristic roots have negative real parts only. Based on the Argument Principle for complex functions as well as Hassard’s technique for ordinary time-delay systems, an explicit formula is established for calculating the number of characteristic roots lying in the closed right-half complex plane of the first sheet of the Riemann surface, and in turn a sufficient and necessary condition is obtained for testing the BIBO stability of fractional-delay systems. As shown in the illustrative examples, this stability criterion involves easy computation and works effectively. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Fractional-order derivative is a general term for derivatives of non-integer orders. It has been shown to be an adequate tool to describe real physical phenomena including anomalous diffusion and wave propagation (Kilbas, Srivastava, & Trujillo, 2006; Metzler & Klafter, 2000; Podlubny, 1999; Schneider & Wyss, 1989), to model real materials with memory effect for both narrow and broad frequency ranges (Bagley & Torvik, 1984; Eldred, Baker, & Palazotto, 1995; Rossikhin & Shitikova, 2010), to construct new control strategies with stronger robustness and better accuracy than classic controllers (Machado, 1997; Machado, Silva, Barbosa, Jesus, Reis, Marcos, & Galhano, 2010; Podlubny, 1999; Wang & Zheng, 2009). Dynamical systems with elements described by fractional-order derivatives, fractional systems for short, exhibit more complicated dynamics than the conventional ones. For instance, the fractional-order Newton–Leipnik system shows reverse Hopf bifurcation process, i.e., its dynamical behavior mutates from double chaotic attractor to single one as the order decreases (Wang & Wang, 2010). In addition, chaos is found in a fractional-order simplified Lorenz system with a wide range of fractional orders, smaller than 3.0 or greater than 3.0, and the lowest order for this system leading to chaos is found to be 2.62 (Sun, Wang, & Sprott, 2010), while chaos cannot occur in ordinary

✩ The authors wish to thank the financial support of the NSF of China under Grants 10825207 and 11032009. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Maria Elena Valcher under the direction of Editor Roberto Tempo. ∗ Corresponding author. E-mail addresses: [email protected] (M. Shi), [email protected] (Z.H. Wang).

0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.05.018

continuous systems of total order smaller than 3. In Chen and Chen (2008), the fractional damped van der Pol system undergoes chaotic motion when the order of fractional damping exceeds 1, while periodic, quasi-periodic and chaotic motions occur when the order of fractional damping becomes smaller than 1. Complicated dynamics is possible only if some stationary solutions lose their stability. In some cases such as for the equilibria, the stability can be evaluated from the root location of the characteristic functions. The characteristic functions of fractional systems are multi-valued functions, and they are single-valued functions on each sheet of the Riemann surface if the orders of the fractionalorder derivatives are rational numbers. Only the characteristic roots on the first Riemann sheet have physical meaning and are required in stability analysis (Farshad & Masoud, 2008). As shown in (Bonnet & Partington, 2002, 2007; Caponetto, Dongola, Fortuna, & Petras, 2010; Monje, Chen, Vinagre, Xue, & Feliu, 2010), the BIBO (bounded input bounded output) stability of a forced time-invariant fractional dynamic system holds if and only if the characteristic roots on the first Riemann sheet have negative real parts only. This condition guarantees also the asymptotic stability of unforced time-invariant fractional dynamic systems. Usually time delays are inevitable in interconnected real systems or processes, in particular in active controllers and digital filters. When time delays and fractional-order derivative are involved in dynamical systems, we have fractional-delay systems. The characteristic function of a fractional-delay system involves exponential type transcendental terms, so a fractionaldelay system has in general an infinite number of characteristic roots. This makes the stability analysis of fractional-delay systems a challenging task. A number of algorithms for BIBO stability testing are available in the literature. For some special cases, for example, the stability of fractional-delay systems can be completed numerically by using Lambert W function (Chen & More, 2002),

2002

M. Shi, Z.H. Wang / Automatica 47 (2011) 2001–2005

where the rightmost characteristic root on the first Riemann sheet can be expressed explicitly in terms of Lambert W function. In general, the stability test can be carried out numerically by using the algorithms proposed in Hwang and Cheng (2006). In addition, the stability of fractional-delay systems involving one parameter can also be checked by using the method of stability switches (Bonnet & Partington, 2002, 2007); see also Wang and Hu (2010) for the case when the stability domain is a sector in the complex plane, not the open left-half complex plane. The most convenient testing method is probably the method of frequency response plot (Buslowicz, 2008; Ostalczyk, 2001), a graphical testing method firstly reported in Fu, Olbrot, and Polis (1989) for ordinary delay differential equations. The graphical method may cause testing problems when the stability margin is very small. In any case, it is important to develop some easy criteria or algorithms to check the BIBO stability either analytically or numerically. This paper aims at establishing an effective criterion for testing the BIBO stability. The main result will be presented in Section 2, with a proof in Section 3. Three examples will be given in Section 4 to demonstrate the effectiveness of the proposed analytical algorithms. Finally in Section 5 some concluding remarks are drawn from the investigation.

Fig. 1. The contour C . The domain encircled by C becomes the open right-half complex plane when R → +∞.

where C = C1 + C2 is the anti-clockwise closed curve, shown in Fig. 1 and defined by C1 : s = Reiθ ,

as θ varies from −

π

2

to

2

;

2. Statement of the problem and the main result

C2 : s = iω,

In this paper, we are interested in linear fractional-delay systems, described by the following equation

where i2 = −1. Then, using the Argument Principle one obtains the number N of roots of p(s) = 0 with ℜ(s) > 0 (namely, the characteristic roots in the open right-half complex plane) as follows

C α 0 Dt x

l −

(t ) = Ax(t ) +

Bi x(t − hi ),

(1)

where x ∈ Rn¯ , A, Bi ∈ Rn¯ ׯn , 0 < α < 2, hi > 0, and C0 Dαt x(t ), Caputo’s derivative of x(t ), is defined by C α 0 Dt x

(t ) =

x(m) (ξ )

t

∫

Γ (m − α)

0

(t − τ )α−m+1

dξ ,

(2)

where m is the integer satisfying m − 1 ≤ α < m, x(m) (t ) is the mth order derivative of x(t ), and Γ (z ) is the Gamma function satisfying Γ (z + 1) = z Γ (z ) for z > 0. When α = 1, Eq. (1) is the conventional time-delay differential equations. Under the zero initial condition, the Laplace transform gives [L(C0 Dαt x(t ))](s) = sα [L(x(t ))](s), Thus, the characteristic equation of Eq. (1) becomes

 α

det s I − A −

l −

 Bi e

= 0,

−hi s

(3)

i =1

where I is the identity matrix. Assume that α is rational: α = and set n = n¯ · nˆ , then the characteristic function p(s), namely the left-hand side of (3), can be written as nˆ , m



1

p(s) = p0 s m



+

l −

1C arg(p(s)) . 2π

N = lim

R→+∞

i=1

1

as ω varies from R to − R

π



1

pi s m



e−τi s

βi

m

,

(4)

i=1

where 1 ≤ βi ≤ m, and pi (w) are polynomials in w , with deg(p0 ) = n > deg(pi ), (i = 1, 2, . . . , l), the coefficient of the leading term of p0 (w) equals 1. The characteristic function p(s) is a multi-valued complex function, and it is a single-valued function if s stays in the first sheet of the Riemann surface determined by w = s1/m with π π −π < arg(s) ≤ π and − m < arg(w) ≤ m . The BIBO stability is determined by the roots of p(s) in the first sheet of the Riemann surface only (Caponetto et al., 2010; Monje et al., 2010). Assume that p(s) has no roots on the imaginary axis, and let 1C arg(p(s)) denote the change in the argument of p(s) over C ,

(5)

Moreover, we use Hassard’s technique (Hassard, 1997) and introduce M (ω) and N (ω) by using n M (ω) + iN (ω) = e−i 2m π p(iω)

(6)

and present the main result of this paper as follows. Theorem 1. Let M (ω), N (ω) be the real and imaginary parts of n e−i 2m π p(iω), respectively, as defined in (6), let

ρ1 ≥ ρ2 ≥ · · · ≥ ρr > 0 be the positive roots (counted with their multiplicity) of M (ω) = 0. Then, the fractional-delay equation (1) is BIBO stable, namely N = 0, if and only if the following three conditions hold: (i) p(0) ̸= 0, (ii) N (ρj ) ̸= 0, (∀j = 1, 2, . . . , r ), and n (iii) if m ≤ 1, then n/m 2

+

+

n/m 2

(−1)r sgnN (0)

r − (−1)j−1 sgnN (ρj ) = 0,

(7)

j =1

> 1, then   n/m n/m (−1)r sgnN (0) + 1−

or if

n m

2

2

r

+

− (−1)j−1 sgnN (ρj ) = 0.

(8)

j =1

3. Proof of Theorem 1 Obviously, p(iω) = 0 if and only if M (ω) = N (ω) = 0, thus conditions (i)–(ii) in Theorem 1 mean that p(s) has no roots on the imaginary axis. In order to prove Theorem 1, we need to prove

M. Shi, Z.H. Wang / Automatica 47 (2011) 2001–2005

that N coincides with the left-hand side either of (7) or of (8), depending on the value of n/m. To this end, it is required to prove the following two lemmas. Lemma 1. Let M (ω) + iN (ω) = A(ω)eiφ(ω) , where A(ω) > 0, and let 1φ(ω) denote the change of argument of φ(ω) when ω varies from 0 to +∞, then

N =

n/m 2

1φ(ω) . π

−

(9)

2003

Case 1: M (ω) = 0 has no positive roots. In this case, as ω increases from 0 to +∞, according to (6), the curve of {M (ω) + iN (ω): ω ≥ 0}, stays always in the open right-half complex plane, and the starting point M (0) + iN (0) ̸= 0 must fall in one of the following three cases: (a) M (0) > 0, N (0) > 0; (b) M (0) > 0, N (0) < 0; and (c) M (0) > 0, N (0) = 0. From the definition of M (ω) and N (ω), it is easy to see that the leading term of M (ω) is ωn/m , thus for ω ≫ 1, one has M (ω) > 0 and



n

M (ω) − ω m



n

/ω m ≈ 0,

n

N (ω)/ω m ≈ 0.

(12)

Lemma 2. Let ρj , (j = 1, 2, . . . , r ), be the r positive roots (counted with their multiplicity) of M (ω) = 0, let 1φ(ω) denote the change of argument of φ(ω) when ω varies from 0 to +∞, then

It follows that φ(ω) → 0 when ω → +∞, due to

 r − n/m  − (− 1)r sgnN (0) − (−1)j−1 sgnN (ρj ),   2   j = 1   n   ≤1  if 1φ(ω) m  = n/m  π − 1 − (−1)r sgnN (0)    2   r  −  n   > 1. − (−1)j−1 sgnN (ρj ), if 

Then the initial phase angle φ(0) satisfies

− (10)

It follows that

  p(s) − sn/m 1C1 arg(p(s)) = 1C1 arg(sn/m ) + 1C1 arg 1 + sn/m   n = 1C1 arg Rn/m ei m θ   p(s) − sn/m + 1C1 arg 1 + n/ m m 2

−

m

2

+0=

m

π,

(a) N (ρ1 ) > 0;

= −2arg e

→ −21φ(ω),

ω=+R  p(iω)  ω=0

(R → ∞).

Thus, one has

N = lim

R→+∞

1C arg(p(s)) n/m 1φ(ω) = − . 2π 2 π

This completes the proof of Lemma 1.

π · sgnN (0).

n m

(b) N (ρ1 ) < 0;

(c) N (ρ1 ) = 0.

(13)

(14)

2

R −2arg(p(iω))|ω=+ ω=0 n −i 2m π

2

1ω∈[ρ1 ,+∞) φ(ω) = φ(+∞) − φ(ρ1 ) = −φ(ρ1 ) π = − · sgnN (ρ1 ).

R 1C2 arg(p(s)) = arg(p(iω))|ω=− ω=R



n/m

Because φ(ω) → 0 as ω → +∞, one has

(R → ∞).

On the line segment C2 , one has

=

≤ 1 and ω increases from 0 to +∞, the change of argun/m π , n/2m π and 0, corresponding 2

ment φ(+∞) − φ(0) equals to − to (a)–(c), respectively, namely

where 1ω∈[ρi ,ρi−1 ] φ(ω) denotes the change of argument of φ(ω) when ω varies from ρi to ρi−1 . As in Case 1, if ω ∈ [ρ1 , +∞), then M (ρ1 ) + iN (ρ1 ) = 0 + iN (ρ1 ) falls into one of the following three cases:

s

→

.

1φ(ω) = (φ(ρr ) − φ(0)) + (φ(ρr −1 ) − φ(ρr )) + · · · + (φ(+∞) − φ(ρ1 )) = 1ω∈[0,ρr ] φ(ω) + 1ω∈[ρr ,ρr −1 ] φ(ω) + · · · + 1ω∈[ρ1 ,+∞) φ(ω),

(R = |s| → +∞).

n

n m

2

Case 2: M (ω) = 0 has positive roots. If M (ω) = 0 has r (finite) positive roots (counted with their multiplicity) ρ1 ≥ ρ2 ≥ · · · ≥ ρr > 0, then one has

Proof of Lemma 1. In fact, along the curve C1 , one has

n (−π )

Now, if

π

2

m

n π

< φ(0) = arctan(N (0)/M (0)) <

> 1 and ω increases from 0 to +∞, the   n/m change of argument φ(+∞) − φ(0) equals to − 1 − 2 π ,   n/m 1 − 2 π and 0, corresponding to (a)–(c), respectively, namely   n/m 1φ(ω) = φ(+∞) − φ(0) = − 1 − π · sgnN (0).

 r − n/m n/m   + (− 1)r sgnN (0) + (−1)j−1 sgnN (ρj ),   2 2   j =1   n    if ≤ 1 m  N = n/m  n/m  + 1 − (−1)r sgnN (0)    2 2   r  −  n   (−1)j−1 sgnN (ρj ), if > 1.  +

(p(s) − sn/m )/sn/m → 0,

2

On the other hand, for

Theorem 1 follows directly from Lemmas 1 and 2, because

j=1

π

(ω → +∞).

1φ(ω) = φ(+∞) − φ(0) = −

m

j=1

N (ω)/M (ω) → 0,

(11)



Proof of Lemma 2. In the calculation of 1φ(ω), the number of times the curve of {M (ω) + iN (ω): ω ≥ 0} crosses the imaginary axis of the complex plane must be taken into consideration.

For ω ∈ [ρ2 , ρ1 ] in case ρ2 does exist, several cases need to be considered. For example, if N (ρ1 )N (ρ2 ) > 0, one has φ(ρ1 ) − φ(ρ2 ) = ρ +ρ  0. If N (ρ1 ) > 0, N (ρ2 ) < 0 and M 1 2 2 > 0, then the curve of M (ω) + iN (ω) starts from the imaginary axis at 0 + iN (ρ1 ), enters the right-half complex plane, passes again through the imaginary axis, and comes back again to the imaginary axis at 0 + iN (ρ2 ), thus φ(ρ1 ) − φ(ρ2 ) = π /2 − (−π /2) = π . The other cases can be dealt with in a similar way. The results can be expressed in a compact form as follows

1ω∈[ρ2 ,ρ1 ] φ(ω) = φ(ρ1 ) − φ(ρ2 )

=

π 2

(sgnN (ρ1 ) − sgnN (ρ2 ))sgnM



ρ1 + ρ2 2



.

(15)

2004

M. Shi, Z.H. Wang / Automatica 47 (2011) 2001–2005

In general, when ω ∈ [ρj+1 , ρj ] with ρr +1 = 0, one has

N (ρ1 ) = −11.5916, N (ρ2 ) = −12.7551, N (ρ3 ) = 0.0256. Hence, the system is not stable because

1ω∈[ρj+1 ,ρj ] φ(ω) = φ(ρj ) − φ(ρj+1 )

=

π 2

(sgnN (ρj ) − sgnN (ρj+1 ))sgnM



ρj + ρj+1



2

N =

,

(16)

where j = 1, 2, . . . , r, and

 sgnM

ρ j + ρ j +1



2

0,

ρj = ρj+1 ρj+1 < ρj .

 =

(−1)j ,

ρ 

For any ω ∈ [0, ρr ], one has sgnM (ω) = sgnM 2r . The calculation of φ(ρr )−φ(0) depends on the power n/m of the leading term in p(s). n Case (i): m ≤ 1. In this case, if N (ρr )N (0) > 0, then φ(ρr ) −

± n/2m π , where the negative or positive sign coincides with that of M (ω). φ(0) = 0, and if N (ρr )N (0) < 0, one has φ(ρr )−φ(0) =

π

2

Hence, it holds

1ω∈[0,ρr ] φ(ω) = φ(ρr ) − φ(0)   ρ  π n/m r = sgnN (ρr ) − π sgnN (0) sgnM . 2

2

(17)

r − 1φ(ω) n/m (−1)j−1 sgnN (ρj ). =− (−1)r sgnN (0) − π 2 j =1

(18)

n Case (ii): m > 1. In this case, similar to the analysis given in Case 1 as well as in Case (i) above, one has

1ω∈[0,ρr ] φ(ω) = φ(ρr ) − φ(0)     ρ  π n/m r = sgnN (ρr ) − 1 − π sgnN (0) sgnM . 2

(19)

(−1)j−1 sgnN (ρj ).

(20)

j =1

This completes the proof of Lemma 2.



Example 1. Firstly, let us consider a fractional-delay system with the following characteristic function 3

1

p(s) = s 2 − 1.5s − 1.5se−τ s + 4s 2 + 8. In Hwang and Cheng (2006), it has been shown that the system is BIBO stable for τ ∈ (0.99830, 1.57079), and in particular, the system is BIBO stable for τ = 1 and it is unstable for τ = 0.99. We check these results by using Theorem 1, which works for the 3

√ √ N (ω) =

3 2 4

3 2 4

2

2

This implies that the system is BIBO stable. Example 2. Next, let us consider the following fractional-delay characteristic function



5

1

1

p(s) = s 6 + s 2 + s 3 5





1

e− 2 s + e− s .

5

/s 6 → 0, (|s| → +∞), and     1 1 π π 1 5 1 6 2 3 + ω + ω cos + ω M (ω) = ω + ω cos 6 2 4 2  5π + cos +ω , 12     1 1 π π 1 1 N (ω) = −ω 2 sin + ω − ω 3 sin + ω 2 4 2  6  5π − sin +ω . 12

5/6

5/6 + (−1)0 sgnN (0) = 0. 2 2 It follows that the system is BIBO stable. In the case of irrational orders, Theorem 1 actually holds too, because each irrational number can be considered as the limit of a sequence of rational numbers. Example 3. Finally, let us study the stability of the characteristic function p(s) = s2q + (k1 + k + k2 )sq + k1 (k + k2 ) − e−sτ ,

3

π

2

4

ω − ω cos

3

π

2

4

ω + ω sin

M (ω) = ω2q + (k1 + k + k2 )ωq cos

 √ + ωτ − 4 2,

 √ √ + ωτ − 4 ω − 4 2.

(i) At τ = 0.99, the positive roots of M (ω) = 0 are found to be ρ1 = 2.9124, ρ2 = 3.1672 and ρ3 = 6.6695, and accordingly

π  2

q

+ k1 (k + k2 ) cos(π q) − cos(π q + ωτ ), π  N (ω) = −(k1 + k + k2 )ωq sin q + k1 (k + k2 ) sin(π q) + sin(π q + ωτ ),

2

M ′ (ω) = 2qω2q−1 + (k1 + k + k2 )qωq−1 cos

π 

+ τ sin(π q + ωτ ),

3

BIBO stability test because (p(s) − s 2 )/s 2 → 0, (|s| → +∞). In addition, one has 3

(−1)3 (−1) + (−1)1−1 (−1)

arising from a synchronization problem (Deng, Li, & Lü, 2007), where 0 < q < 1, τ > 0, k > 0, and k1 , k2 are control parameters. In this case, one has

4. Illustrative examples

M (ω) = ω 2 −

2



+ (−1)2−1 (−1) + (−1)3−1 = 2. (ii) At τ = 1, the unique positive root of M (ω) = 0 is ρ1 = 6.6157, then N (ρ1 ) = −0.0051. Thus,   3/2 3/2 N = + 1− (−1)1 sgnN (0) + (−1)1−1 sgnN (ρ1 ) = 0.

N =

  1φ(ω) n/m = − 1− (−1)r sgnN (0) π 2 r −

3/2

In this case, N (0) = −0.9659 and M (ω) = 0 has no positive real root, then

Together with (14)–(16), one has

−

+ 1−



By adding together Eqs. (14)–(17), one has

2

2



It holds p(s) − s 6

2

2

3/2

2

q

N (ω) = τ cos (π q + ωτ ) − (k1 + k + k2 )qωq−1 sin ′

(i) If 0 < q ≤

π  2

q .

> 0 and k2 > k11 − k > 0, then M (0) > 0 and N (0) < 0. With a small τ > 0, one has M ′ (ω) > 0, which holds for all positive ω. In this case, M (ω) = 0 has no positive roots, thus N =

2q

+

2q

1 , k1 2

(−1)0 sgnN (0) =

2q

−

2q

2 2 2 2 It means that the system is BIBO stable.

= 0.

M. Shi, Z.H. Wang / Automatica 47 (2011) 2001–2005

< q < 1, k1 > 0 and k2 > k11 − k > 0, then M (0) < 0 and N (0) < 0. For small τ > 0, one has M ′ (ω) > 0, N ′ (ω) < 0 for all positive ω. Thus, M (ω) = 0 has exactly one positive root ρ1 , satisfying N (ρ1 ) < 0. It follows that   2q 2q + 1− (−1)1 sgnN (0) + (−1)1−1 sgnN (ρ1 ) = 0. N = (ii) If

1 2

2

2

Thus, the system is BIBO stable if 0 < q < 1, k1 > 0 and k2 > 1 − k > 0, the same condition obtained in Deng et al. (2007). k 1

5. Conclusion This paper generalizes Hassard’s criterion for the stability of time-delay systems to the BIBO stability testing of fractionaldelay systems. The explicit testing formula calculates the number of unstable roots in the closed right-half complex plane for the characteristic function in a straightforward way, and the BIBO stability is justified if the number of unstable characteristic roots equals to zero. The illustrative examples show that the proposed criterion is very easy to be carried out and works effectively in the BIBO stability analysis for fractional-delay systems. The main advantage of this criterion is that the number of unstable roots can be calculated automatically, and thus it could be useful in the design of feedback control in practical applications. Acknowledgments The authors wish to thank the anonymous reviewers for their helpful comments on improving the presentation of this paper. References Bagley, R. L., & Torvik, P. J. (1984). On the appearance of the fractional derivative in the behavior of real materials. ASME Journal of Applied Mechanics, 51, 294–298. Bonnet, C., & Partington, J. R. (2002). Analysis of fractional delay systems of retarded and neutral type. Automatica, 38, 1133–1138. Bonnet, C., & Partington, J. R. (2007). Stabilization of some fractional delay systems of retarded and neutral type. Automatica, 43, 2047–2053. Buslowicz, M. (2008). Stability of linear continuous-time fractional order systems with delays of the retarded type. Bulletin of the Polish Academy of Sciences: Technical Sciences, 56, 319–324. Caponetto, R., Dongola, G., Fortuna, L., & Petras, I. (2010). Fractional order systems: Modeling and control applications. New Jersey: World Scientific. Chen, J. H., & Chen, W. C. (2008). Chaotic dynamics of the fractionally damped van der Pol equation. Chaos, Solitons and Fractals, 35, 188–198. Chen, Y. Q., & More, K. L. (2002). Analytical stability bound for a class of delayed fractional-order systems. Nonlinear Dynamics, 29, 191–200. Deng, W. H., Li, C. P., & Lü, J. H. (2007). Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynamics, 48, 409–416. Eldred, L. B., Baker, W. P., & Palazotto, A. N. (1995). Kelvin-Voigt vs fractional derivative model as constitutive relations for viscoelastic materials. AIAA Journal, 33, 547–550. Farshad, M.-B., & Masoud, K.-G. (2008). On the essential instabilities caused by fractional-order transfer functions. Mathematical Problems in Engineering, 13. doi:10.1155/2008/419046. Article ID 419046. Fu, M. Y., Olbrot, A. W., & Polis, M. P. (1989). Robust stability for time-delay systems: the edge theorem and graphical tests. IEEE Transactions on Automatic Control, 34, 813–820.

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Min Shi was born in Hubei, China, in 1981. He received his B.S. and M.S. degrees in mathematics from Huanggang Normal College, Huanggang, China and Jiangsu University, Zhenjiang, China, in 2006 and 2008, respectively. Currently he is a Ph.D. student at the State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, China. His research interests include dynamics of fractional-order systems in engineering.

Zaihua Wang was born in Hunan, China, in 1964. He graduated from Xiangtan Teacher’s College, Xiangtan, China, in 1981, received the M.S. degree from China Institute of Atomic Energy, Beijing, China, in 1989, both majoring in Mathematics, and received the Ph.D. degree in Dynamics and Control (Mechanics) from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 2000. He is a professor at the State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics. Dr. Wang is the co-author of ‘‘Dynamics of Controlled Mechanical Systems with Delayed Feedback’’ (Springer-Verlag, 2002). His current research interest is dynamics and control of dynamical systems with memory (timedelay systems and fractional-order dynamical systems).