计算方法：分数阶滞延系统稳定区间的时间延迟

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Automatica 50 (2014) 1611–1616

Contents lists available at ScienceDirect

Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

A computing method on stability intervals of time-delay for

fractional-order retarded systems with commensurate time-delays

✩

Zhe Gao

College of Light Industry, Liaoning University, Shenyang, 110036, PR China

a r t i c l e i n f o

Article history:

Received 8 April 2013

Received in revised form

22 February 2014

Accepted 24 February 2014

Available online 6 May 2014

Keywords:

Fractional-order systems

Time-delays

Stability intervals

Crossing frequencies

a b s t r a c t

This paper investigates the stability intervals of time-delays for fractional-order retarded time-delay

systems. By the Orlando formula, the existence of the crossing frequencies is brought to verify the

stability related to the commensurate time-delay. For each crossing frequency, the corresponding critical

time-delays are determined by the generalized eigenvalues of two matrices constructed by the crossing

frequency, the commensurate fractional-order and the coefficients of the characteristic function. The root

tendency (RT) is defined to provide a method to analyze the number of the unstable roots for a given

crossing frequency and critical time-delay. Based on the RT values and the number of the unstable roots

for fractional-order systems with no time-delay, a computing method on the stability intervals of time-

delay is proposed in this paper. Finally, a numerical example is offered to validate the effectiveness of this

method.

1. Introduction

Recently, the investigations on fractional-order systems have

attracted much attention, due to its more accurate descriptions for

real-world systems, especially for systems with the dynamic char-

acteristics of viscoelasticity and diffusion (Krishna, 2011; Machado,

Kiryakova, & Mainardi, 2011). Meanwhile, fractional-order con-

trollers have been applied to many engineering applications, by

introducing the flexibility in control systems (Efe, 2011). The

essential requirement of controller design is to achieve of the sta-

bility of any control system. For the fractional-order systems repre-

sented by transfer function, the condition that all the characteristic

roots of the fractional-order characteristic equation locate at the

left half-plane is the stability criterion (Bonnet & Partington, 2002;

Matignon, 1998). Based on this criterion, the stability of fractional-

order systems with interval uncertainties was discussed by Gao

and Liao (2013) and Moornani and Haeri (2010). For state-space

description of fractional-order systems with no time-delay, the sta-

bility criteria were offered for the commensurate fractional-order

✩

This work was supported by the National Natural Science Foundation of

China (Grant No. 61304094). The material in this paper was not presented at

any conference. This paper was recommended for publication in revised form by

Associate Editor Hitay Ozbay under the direction of Editor Miroslav Krstic.

E-mail address: gaozhe83@gmail.com.

Tel.: +86 24 62202139; fax: +86 24 62202139.

α belonging to 0 < α < 1 and 1 < α < 2 by Lu and Chen (2010),

Farges, Moze, and Sabatier (2010) and Lan and Zhou (2011).

In practical plants including fractional-order systems, the time-

delay is a common phenomenon such as in the heating process of

the aluminum rod (Victor, Malti, Garnier, & Oustaloup, 2013). Al-

though the requirements of the fractional-order Lyapunov func-

tions were proposed by Li, Chen, and Podlubny (2009) and Baleanu,

Ranjbar, Sadati, Delavari, and Abdeljawad (2011), it is not straight-

forward to establish a specific Lyapunov function, such as the

quadratic function for integer-order systems. Thereby, most of the

investigations on the stability of time-delay fractional-order sys-

tems are based on transfer function models. Bonnet and Parting-

ton (2002) and Moornani and Haeri (2011) extended the stability

criteria for fractional-order retarded and neutral systems with

time-delays in the frequency domain, requiring that all the roots

of the characteristic equations lie at the left half-plane. Since the

exponential type transcendental term is involved in the charac-

teristic function, an infinite number of characteristic roots exist,

leading to the complexity in the stability analysis. To overcome

this obstacle, a number of criteria have been presented. Hwang

and Cheng (2006) presented a numerical algorithm for the BIBO-

stability of fractional-order time-delay systems based on using

Cauchy’s integral theorem and solving an initial-value problem. Shi

and Wang (2011) proposed an analytical criterion for the BIBO-

stability of fractional-order time-delay systems by argument for-

mula for the complex function. Yu and Wang (2011) proposed a

graphical method of the BIBO-stability on fractional-order systems

http://dx.doi.org/10.1016/j.automatica.2014.03.019

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