research article robust nonfragile controllers...

Research ArticleRobust Nonfragile Controllers Design forFractional Order Large-Scale Uncertain Systems witha Commensurate Order 1 < 𝛼 < 2

Jianyu Lin1,2

1Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China2Department of Communication, Shanghai University of Electric Power, Shanghai 200090, China

Correspondence should be addressed to Jianyu Lin; [email protected]

Received 12 July 2014; Accepted 10 September 2014

Academic Editor: Dan Ye

Copyright © 2015 Jianyu Lin. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The paper concerns the problem of stabilization of large-scale fractional order uncertain systems with a commensurate order1 < 𝛼 < 2 under controller gain uncertainties. The uncertainties are of norm-bounded type. Based on the stability criterion offractional order system, sufficient conditions on the decentralized stabilization of fractional order large-scale uncertain systems inboth cases of additive and multiplicative gain perturbations are established by using the complex Lyapunov inequality. Moreover,the decentralized nonfragile controllers are designed. Finally, some numerical examples are given to validate the proposedmethod.

1. Introduction

In the past decades, a great deal of attention has been paidto the stability and stabilization of large-scale systems [1–7].This is due to the fact that there exist a large numberof large-scale interconnected dynamical systems in manypractical physical systems, such as process control systems,computer communication networks, transportation systems,and economic systems. Meanwhile, nonfragile controllershave been nominated by resilient and the fragility of thePID controllers has been analyzed in [8]. The controllergain perturbations can commonly be modeled as uncertaingains which are dependent on uncertain parameters in theliterature [9, 10]. The robust nonfragile control problem foruncertain integer order large-scale system has been studied[11–13]. In recent years, the nonfragile control problem hasbeen an attractive topic in theory analysis and practicalimplement, because of perturbations often appearing in thecontroller gain, whichmay result in either the actuator degra-dations or the requirements for readjustment of controllergains. The problem of reliable dissipative control withinnonfragile control framework has been investigated in [14,15]. The nonfragile control idea is how to design a feedbackcontrol that will be insensitive to perturbations in gains of

feedback control.The robust resilient stabilization problem isto design a nonfragile state feedback controller such that theuncertain fractional order large-scale interconnected closed-loop systemwith a commensurate order 1 < 𝛼 < 2 is robustlystable for all admissible parameter uncertainties.

On the other hand, pioneering works in stability analysisand stabilization of fractional order control systems can befound in [16–20]. The robust stability of fractional orderinterval systems has been investigated in [21, 22]. It is wellknown that Matignon’s stability theorem [16] is the basis forstability analysis of the fractional order system by checkingthe location of eigenvalues in the complex plane. Matignon’stheorem is in fact the pioneering works of stability analysisof the fractional order system. Based onMatignon’s theorem,the stability criteria of fractional order systems have beenproposed in both cases of 1 < 𝛼 < 2 and 0 < 𝛼 < 1

in [23, 24]. The necessary and sufficient LMI conditions forstability analysis of a commensurate fractional order systemhave been established in [23, 24], in which complex Lyapunovinequality holds. However, very few studies provide LMIconditions for the stability analysis of the fractional orderlarge-scale interconnected system in the literature. Our studyis mainly motivated by the works [23, 24]. The importantfeature is that the proposed method can be implemented to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 206908, 11 pageshttp://dx.doi.org/10.1155/2015/206908

2 Mathematical Problems in Engineering

the fractional order large-scale interconnected system. Theobjective of the paper is to design a nonfragile controllerwhich is robust to system uncertainties and resilient tocontroller gain variations for the fractional order large-scaleinterconnected systems with a commensurate order 1 < 𝛼 <

2. Here, it should be also pointed out that [25] only focuseson the case of a fractional order 0 < 𝛼 < 1. This paper isorganized as follows. Some preliminaries and the problemstatement are given in Section 2. The main results of thesufficient condition of stabilization of the fractional ordersystem under additive gain perturbations are presented inSection 3. Furthermore, the decentralized stabilization statefeedback controller are designed. Meanwhile, the LMI resultsof the sufficient condition of stabilization of the fractionalorder system under multiplicative gain perturbations arepresented in Section 4. The examples are given in Section 5to illustrate the effectiveness of our LMI-based results forchecking the stabilization of the fractional order large-scaleinterconnected system. Finally, a brief conclusion is drawn inSection 6.

Notations. Throughout the paper, we denote by 𝑟 the conju-gate of the complex number 𝑟. 𝑗 denotes the imaginary unit.𝐼 denotes the identity matrix with appropriate dimensions.block diag denotes the block diagonal matrix. 𝑅𝑛 denotesthe 𝑛-dimensional Euclidean space and 𝑅𝑛×𝑚 is the set of all𝑛 × 𝑚 real matrices. 𝑀𝑇 denotes the transpose of 𝑀 and𝑀∗ denotes the Hermitian transpose of𝑀. Re() and Im() are

corresponding to the real and imaginary parts of the matrix,respectively.

2. Preliminaries and Problem Formulation

Let us consider a fractional order large-scale interconnecteduncertain system with a commensurate order 1 < 𝛼 < 2

composed of𝑁 fractional order subsystems:

𝑑𝛼

𝑥𝑖(𝑡)

𝑑𝑡𝛼

= [𝐴𝑖𝑖+ Δ𝐴𝑖𝑖] 𝑥𝑖(𝑡)

+

𝑁

∑

𝑗=1,𝑗 =𝑖

[𝐴𝑖𝑗+ Δ𝐴𝑖𝑗] 𝑥𝑗(𝑡) + 𝐵

𝑖𝑢𝑖(𝑡) ,

(1)

where 𝛼 ∈ 𝑅 is the fractional commensurate order, 𝑖, 𝑗 =

1, 2, . . . 𝑁, and 𝑥𝑖(𝑡) ∈ 𝑅

𝑛𝑖 and 𝑢𝑖(𝑡) ∈ 𝑅

𝑚𝑖 are the stateand input of the 𝑖th fractional order subsystem, respectively.Assume that the nominal systems 𝐴

𝑖𝑖∈ 𝑅𝑛𝑖×𝑛𝑖 , 𝐴

𝑖𝑗∈ 𝑅𝑛𝑖×𝑛𝑖 ,

and 𝐵𝑖∈ 𝑅𝑛𝑖×𝑚𝑖 are constant and of appropriate dimensions

and the pair (𝐴𝑖𝑖,𝐵𝑖) is controllable. The fractional order

subsystems interact with each other through the intercon-nections ∑𝑁

𝑗=1,𝑗 =𝑖𝐴𝑖𝑗𝑥𝑗(𝑡). The main objective of the note is

to find the decentralized local state feedback control law ofthe following form:

𝑢𝑖(𝑡) = (𝐾

𝑖+ Δ𝐾𝑖) 𝑥𝑖(𝑡) , 𝑖 = 1, 2, . . . , 𝑁, (2)

such that the resulting fractional order closed-loop systemis asymptotically stable, where 𝐾

𝑖∈ 𝑅𝑚𝑖×𝑛𝑖 is the state

feedback gainmatrix to be designed andΔ𝐾𝑖= 𝐷𝑏𝑖𝐹𝑏𝑖𝐸𝑏𝑖and

Δ𝐾𝑖= 𝐷𝑑𝑖𝐹𝑑𝑖𝐸𝑑𝑖𝐾𝑖represent the additive and multiplicative

gain perturbations, respectively. In this note, the uncertaintyis bounded as follows. The parameter uncertainties consid-ered here are norm-bounded and are of the forms Δ𝐴

𝑖𝑖=

𝐷𝑎𝑖𝑖𝐹𝑎𝑖𝑖𝐸𝑎𝑖𝑖, 𝐹𝑇𝑎𝑖𝑖𝐹𝑎𝑖𝑖

≤ 𝐼; Δ𝐴𝑖𝑗= 𝐷𝑎𝑖𝑗𝐹𝑎𝑖𝑗𝐸𝑎𝑖𝑗, 𝐹𝑇𝑎𝑖𝑗𝐹𝑎𝑖𝑗

≤ 𝐼;Δ𝐾𝑖= 𝐷𝑏𝑖𝐹𝑏𝑖𝐸𝑏𝑖, 𝐹𝑇𝑏𝑖𝐹𝑏𝑖≤ 𝐼; Δ𝐾

𝑖= 𝐷𝑑𝑖𝐹𝑑𝑖𝐸𝑑𝑖𝐾𝑖, 𝐹𝑇𝑑𝑖𝐹𝑑𝑖≤ 𝐼,

where the elements are Lebesgue measurable and 𝐷𝑎𝑖𝑖, 𝐷𝑎𝑖𝑗,

𝐷𝑏𝑖, 𝐷𝑑𝑖, 𝐸𝑎𝑖𝑖, 𝐸𝑎𝑖𝑗, 𝐸𝑏𝑖, and 𝐸

𝑑𝑖are known real matrices

of appropriate dimensions which characterize the structureof the uncertainty. The overall system is described by thecomposite fractional order large-scale state equations

𝑑𝛼

𝑥 (𝑡)

𝑑𝑡𝛼

= (𝐴 + Δ𝐴) 𝑥 (𝑡) + 𝐵 (𝐾 + Δ𝐾) 𝑥 (𝑡) , (3)

with the composite matrices 𝐴 and𝐾 having the structure

𝐴 =[[

[

𝐴11

⋅ ⋅ ⋅ 𝐴1𝑁

.

.

. d...

𝐴𝑁1

⋅ ⋅ ⋅ 𝐴𝑁𝑁

]]

]

,

Δ𝐴 =[[

[

Δ𝐴11

⋅ ⋅ ⋅ Δ𝐴1𝑁

.

.

. d...

Δ𝐴𝑁1

⋅ ⋅ ⋅ Δ𝐴𝑁𝑁

]]

]

,

𝐾 = block diag [𝐾1, 𝐾2⋅ ⋅ ⋅ 𝐾𝑁] ,

Δ𝐾 = block diag [Δ𝐾1, Δ𝐾2⋅ ⋅ ⋅ Δ𝐾

𝑁] .

(4)

Definition 1 (see [26]). For all nonzero real vectors 𝑋 ∈ 𝑅𝑛,

𝐴 ∈ 𝑅𝑛×𝑛 is real matrix; if the inequality 𝑋𝑇𝐴𝑋 < 0 holds,

then 𝐴 is said to be negative definite matrix.

Definition 2. The fractional order large-scale uncertain sys-tem can be stabilized via decentralized state feedback 𝑢

𝑖(𝑡) =

(𝐾𝑖+Δ𝐾𝑖)𝑥𝑖(𝑡) if there exists gainmatrix𝐾

𝑖∈ 𝑅𝑚𝑖×𝑛𝑖 such that

the closed-loop fractional order large-scale uncertain system𝑑𝛼

𝑥 (𝑡)

𝑑𝑡𝛼

= (𝐴 + Δ𝐴) 𝑥 (𝑡) + 𝐵 (𝐾 + Δ𝐾) 𝑥 (𝑡) (5)

is asymptotically stable.

3. Nonfragile Controller Design ofFractional Order Large-Scale System withAdditive Gain Perturbations

In this section, the resilient controller synthesis problem isformulated for the fractional order large-scale interconnectedsystem under additive gain perturbations. Sufficient condi-tions are firstly derived for the decentralized stabilizationof fractional order large-scale interconnected system withnorm-bounded uncertainties given by (1). Before proceedingfurther, we will state the following well-known lemmas.We will use the lemmas and theorems to establish suffi-cient conditions on decentralized stabilization of fractionalorder large-scale interconnected system with norm-boundeduncertainties under additive gain perturbations.

Lemma 3 (see [27]). For all 𝐶 ∈ 𝑅𝑛×𝑘, 𝐴, 𝐵 ∈ 𝑅

𝑛×𝑛, 𝐴 ≥ 𝐵;then 𝐶𝑇𝐴𝐶 ≥ 𝐶

𝑇

𝐵𝐶.

Mathematical Problems in Engineering 3

Lemma 4 (see [27]). For any matrices 𝑋 and 𝑌 with appro-priate dimensions and for any 𝛽 > 0, the following inequalityholds:

𝑋𝑇

𝑌 + 𝑌𝑇

𝑋 < 𝛽𝑋𝑇

𝑋 + 𝛽−1

𝑌𝑇

𝑌. (6)

Lemma5 (see [28]). The fractional order system𝑑𝛼

𝑥(𝑡)/𝑑𝑡𝛼

=

𝐴𝑥(𝑡) with a commensurate order 𝛼 is asymptotically stableif | arg(spec(𝐴))| > 𝛼(𝜋/2), where 𝛼 is the order of fractionalorder system and spec(𝐴) is the spectrum of all eigenvalues of𝐴.

Lemma 6 (see [23]). Let 1 < 𝛼 < 2 and 𝜃 = (𝛼 − 1)(𝜋/2),𝑟 = 𝑒𝑗𝜃. The fractional order system 𝑑

𝛼

𝑥(𝑡)/𝑑𝑡𝛼

= 𝐴𝑥(𝑡) witha commensurate order 1 < 𝛼 < 2 is asymptotically stable if andonly if there exist positive definite matrices 𝑃 = 𝑃

𝑇

∈ 𝑅𝑛×𝑛,

such that

[

[

(𝐴𝑃 + 𝑃𝐴𝑇

) sin 𝜃 (𝐴𝑃 − 𝑃𝐴𝑇

) cos 𝜃

(𝑃𝐴𝑇

− 𝐴𝑃) cos 𝜃 (𝐴𝑃 + 𝑃𝐴𝑇

) sin 𝜃]

]

< 0, (7)

or equivalently, 𝑟𝑃𝐴 + 𝑟𝐴𝑇

𝑃 < 0.

Proof. The idea is mainly based on the geometric analysisof a fractional system stability domain. Based on Lemma 5,the stability domain for a fractional order 1 < 𝛼 < 2 isconvex. By using Linear Matrix Inequalities (LMI) approach,it is obtained as the above LMI. Therefore, it is equivalent to𝑟𝑃𝐴 + 𝑟𝐴

𝑇

𝑃 < 0.

Lemma 7. A complex Hermitian matrix𝑀 satisfies𝑀 < 0 ifand only if the following real LMI inequality holds:

[Re (𝑀) Im (𝑀)

− Im (𝑀) Re (𝑀)] < 0. (8)

Under commensurate order hypothesis, our finding is summa-rized in the following theorem.

Theorem8. Consider the fractional order large-scale intercon-nected system (1)with a commensurate order 1 < 𝛼 < 2. Let 𝑟 =𝑒𝑗(𝛼−1)(𝜋/2). The fractional order large-scale uncertain systemcan be stabilized via decentralized state feedback 𝑢

𝑖(𝑡) = (𝐾

𝑖+

Δ𝐾𝑖)𝑥𝑖(𝑡) if there exist positive-definite block diagonal matrices

𝑃𝑖= block diag[𝑃

1, 𝑃2. . . , 𝑃𝑁], matrix 𝑌

𝑖, and positive real

scalar constants 𝛼𝑖, 𝛾𝑖𝑗, 𝛽𝑖, 𝑖, 𝑗 = 1, 2, . . . , 𝑁, such that the

following matrix inequalities hold:

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

𝑀1𝑖

𝑋𝑇

𝑖𝐸𝑇

𝑎𝑖𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇

𝑎𝑁𝑖𝑋𝑇

𝑖𝐸𝑇

𝑏𝑖𝑀2𝑖

0 0 0 0 0 0

𝐸𝑎𝑖𝑖𝑋𝑖

−𝛼𝑖𝐼 0 0 0 0 0 0 0 0 0 0 0 0

𝐸𝑎1𝑖𝑋𝑖

0 −𝛾1𝑖𝐼 0 0 0 0 0 0 0 0 0 0 0


0 0 −𝛾2𝑖𝐼 0 0 0 0 0 0 0 0 0 0

.

.

. 0 0 0 d 0 0 0 0 0 0 0 0 0

𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0 0 0 0 0 0 0 0

𝐸𝑏𝑖𝑋𝑖

0 0 0 0 0 −𝛽𝑖𝐼 0 0 0 0 0 0 0

−𝑀2𝑖

0 0 0 0 0 0 𝑀1𝑖

𝑋𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑏𝑖

0 0 0 0 0 0 0 𝐸𝑎𝑖𝑖𝑋𝑖

−𝛼𝑖𝐼 0 0 0 0 0

0 0 0 0 0 0 0 𝐸𝑎1𝑖𝑋𝑖

0 −𝛾1𝑖𝐼 0 0 0 0

0 0 0 0 0 0 0 𝐸𝑎2𝑖𝑋𝑖

0 0 −𝛾2𝑖𝐼 0 0 0

0 0 0 0 0 0 0

.

.

. 0 0 0 d 0 0

0 0 0 0 0 0 0 𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0

0 0 0 0 0 0 0 𝐸𝑏𝑖𝑋𝑖

0 0 0 0 0 −𝛽𝑖𝐼

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

< 0, (9)

where𝑀𝑖= 𝑟𝑃𝑖𝐴𝑖𝑖+ 𝑟𝐴𝑇

𝑖𝑖𝑃𝑖+ 𝑟𝑃𝑖𝐴𝑖𝑗+ 𝑟𝐴𝑇

𝑖𝑗𝑃𝑖+ 𝛼𝑖𝐷𝑎𝑖𝑖𝐷𝑇

𝑎𝑖𝑖+

∑𝑁

𝑗=1,𝑗𝛾𝑖𝑗𝐷𝑎𝑖𝑗𝐷𝑇

𝑎𝑖𝑗+ 𝛽𝑖𝐵𝑖𝐷𝑏𝑖𝐷𝑇

𝑏𝑖𝐵𝑇

𝑖+ 𝑟𝐵𝑖𝑌𝑖+ 𝑟𝑌𝑇

𝑖𝐵𝑇

𝑖, 𝑋𝑖=

𝑃−1

𝑖, and 𝑀

1𝑖and 𝑀

2𝑖are the real part and imaginary

part of matrices 𝑀𝑖, respectively. Moreover, the stabilization

decentralized state-feedback gain matrix can be calculated asfollows: 𝐾

𝑖= 𝑌𝑖𝑃𝑖.

Proof. Under decentralized state-feedback control law (2),the closed-loop fractional order large-scale interconnected

system is obtained as𝑑𝛼

𝑥𝑖(𝑡)

𝑑𝑡𝛼

= [𝐴𝑖𝑖+ Δ𝐴𝑖𝑖] 𝑥𝑖(𝑡) +

𝑁

∑

𝑗=1,𝑗 =𝑖

[𝐴𝑖𝑗+ Δ𝐴𝑖𝑗] 𝑥𝑗(𝑡)

+ 𝐵𝑖[𝐾𝑖+ Δ𝐾𝑖] 𝑥𝑖(𝑡) ,

(10)

where 𝑖,𝑗 = 1, 2, . . . 𝑁. Based on Lemma 6, the necessaryand sufficient condition on the asymptotical stability of thefractional order system with order 1 < 𝛼 < 2 is that


𝑟𝑃𝐴 + 𝑟𝐴𝑇

𝑃 < 0. According to Definition 1, the sufficientcondition on the stabilization of fractional order large-scale interconnected system satisfies the following quadraticmatrix inequality:

𝜁𝑇

{𝑟𝑃 [(𝐴 + Δ𝐴) + 𝐵 (𝐾 + Δ𝐾)]

+ 𝑟 [(𝐴 + Δ𝐴) + 𝐵 (𝐾 + Δ𝐾)]𝑇

𝑃} 𝜁 < 0,

(11)

𝑁

∑

𝑖=1

{

{

{

𝜁∗

𝑖[𝑟𝑃𝑖𝐴𝑖𝑖+ 𝑟𝐴𝑇

𝑖𝑖𝑃𝑖+ 𝑟𝑃𝑖Δ𝐴𝑖𝑖

+ 𝑟Δ𝐴𝑇

𝑖𝑖𝑃𝑖+ 𝑟𝑃𝑖𝐵𝑖𝐾𝑖+ 𝑟𝐾𝑇

𝑖𝐵𝑇

𝑖𝑃𝑖

+ 𝑟𝑃𝑖𝐵𝑖Δ𝐾𝑖+ 𝑟𝑃𝑖Δ𝐾𝑇

𝑖𝐵𝑇

𝑖] 𝜁𝑖

+ 2𝑟𝜁∗

𝑖

[

[

𝑁

∑

𝑗=1,𝑗 =𝑖

(𝐴𝑖𝑗+ Δ𝐴𝑖𝑗)]

]

𝜁𝑗

}

}

}

< 0.

(12)

Consequently, the sufficient condition on the decentralizedstabilization of fractional order large-scale interconnectedsystem is that quadratic matrix inequality (11) holds.

Based on Lemmas 3 and 4, by means of enlarging theinequality, it yields

𝑟𝑃𝑖Δ𝐴𝑖𝑖+ 𝑟Δ𝐴

𝑇

𝑖𝑖𝑃𝑖≤ 𝛼𝑖𝑃𝑖𝐷𝑎𝑖𝑖𝐷𝑇

𝑎𝑖𝑖𝑃𝑖+ 𝛼−1

𝑖𝐸𝑇

𝑎𝑖𝑖𝐸𝑎𝑖𝑖. (13)

Meanwhile, based on Lemmas 3 and 4, by means of enlargingthe inequality we have

𝑟𝑃𝑖𝐵𝑖Δ𝐾𝑖+ 𝑟Δ𝐾

𝑇

𝑖𝐵𝑇

𝑖𝑃𝑖≤ 𝛽𝑖𝑃𝑖𝐵𝑖𝐷𝑏𝑖𝐷𝑇

𝑏𝑖𝐵𝑇

𝑖𝑃𝑖+ 𝛽−1

𝑖𝐸𝑇

𝑏𝑖𝐸𝑏𝑖,

𝑁

∑

𝑖=1

𝑁

∑

𝑗=1,𝑗 =𝑖

𝜁∗

𝑖[𝑟𝑃𝑖Δ𝐴𝑖𝑗+ 𝑟Δ𝐴

𝑇

𝑖𝑗𝑃𝑖] 𝜁𝑗

≤

𝑁

∑

𝑖=1

𝑁

∑

𝑗=1,𝑗 =𝑖

𝜁∗

𝑖𝛾𝑖𝑗𝑃𝑖𝐷𝑎𝑖𝑗𝐷𝑇

𝑎𝑖𝑗𝑃𝑖𝜁𝑖

+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝑁

∑

𝑖=1

𝜁∗

𝑗𝛾−1

𝑖𝑗𝐸𝑇

𝑎𝑖𝑗𝐸𝑎𝑖𝑗𝜁𝑗

≤

𝑁

∑

𝑖=1

𝑁

∑

𝑗=1,𝑗 =𝑖

𝜁∗



+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝑁

∑

𝑖=1

𝜁∗

𝑖𝛾−1

𝑗𝑖𝐸𝑇

𝑎𝑗𝑖𝐸𝑎𝑗𝑖𝜁𝑖.

(14)

Substituting (11), (12), and (13) into (12) results in thefollowing quadratic matrix inequality:

𝑁

∑

𝑖=1

{

{

{

𝜁∗

𝑖

[

[

𝑟𝑃𝑖𝐴𝑖𝑖+ 𝑟𝐴𝑇


𝑖𝑗𝑃𝑖

+ 𝑟𝑃𝑖𝐵𝑖𝐾𝑖+ 𝑟𝐾𝑇

𝑖𝐵𝑇

𝑖𝑃𝑖

+ 𝛼𝑖𝑃𝑖𝐷𝑎𝑖𝑖𝐷𝑇

𝑎𝑖𝑖𝑃𝑖+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾𝑖𝑗𝑃𝑖𝐷𝑎𝑖𝑗𝐷𝑇

𝑎𝑖𝑗𝑃𝑖

+ 𝛼−1

𝑖𝐸𝑇

𝑎𝑖𝑖𝐸𝑎𝑖𝑖+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾−1

𝑖𝑗𝐸𝑇

𝑎𝑖𝑗𝐸𝑎𝑖𝑗

+ 𝛽−1

𝑖𝐸𝑇

𝑏𝑖𝐸𝑏𝑖+ 𝛽𝑖𝑃𝑖𝐵𝑖𝐷𝑏𝑖𝐷𝑇

𝑏𝑖𝐵𝑇

𝑖𝑃𝑖

]

]

𝜁𝑖

}

}

}

< 0,

(15)

𝑁

∑

𝑖=1

{

{

{



𝑖𝑗𝑃𝑖


𝑖𝐵𝑇

𝑖𝑃𝑖


𝑎𝑖𝑖𝑃𝑖+ 𝑃𝑖

𝑁

∑

𝑗=1,𝑗 =𝑖

(𝛾𝑖𝑗𝐷𝑎𝑖𝑗𝐷𝑇

𝑎𝑖𝑗) 𝑃𝑖

+ 𝛼−1

𝑖𝐸𝑇


𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾−1

𝑖𝑗𝐸𝑇


+𝛽𝑖𝑃𝑖𝐵𝑖𝐷𝑏𝑖𝐷𝑇

𝑏𝑖𝐵𝑇


𝑖𝐸𝑇

𝑏𝑖𝐸𝑏𝑖

}

}

}

< 0.

(16)

Let 𝑋𝑖= 𝑃−1

𝑖and 𝑌

𝑖= 𝐾𝑖𝑋𝑖. By premultiplying and post-

multiplying 𝑃−1𝑖

onto (16), one has

𝑁

∑

𝑖=1

{

{

{

𝑟𝐴𝑖𝑖𝑃−1

𝑖+ 𝑟𝑃−1

𝑖𝐴𝑇

𝑖𝑖+ 𝑟𝐴𝑖𝑗𝑃−1

𝑖+ 𝑟𝑃−1

𝑖𝐴𝑇

𝑖𝑗

+ 𝑟𝐵𝑖𝐾𝑖𝑃−1

𝑖+ 𝑟𝑃−1

𝑖𝐾𝑇

𝑖𝐵𝑇

𝑖+ 𝛼𝑖𝐷𝑎𝑖𝑖𝐷𝑇

𝑎𝑖𝑖

+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾𝑖𝑗𝐷𝑎𝑖𝑗𝐷𝑇

𝑎𝑖𝑗+ 𝛼−1

𝑖𝑃−1

𝑖𝐸𝑇

𝑎𝑖𝑖𝐸𝑎𝑖𝑖𝑃−1

𝑖

+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾−1

𝑖𝑗𝑃−1

𝑖𝐸𝑇

𝑎𝑖𝑗𝐸𝑎𝑖𝑗𝑃−1

𝑖+ 𝛽𝑖𝐵𝑖𝐷𝑏𝑖𝐷𝑇

𝑏𝑖𝐵𝑇

𝑖

+𝛽−1

𝑖𝑃−1

𝑖𝐸𝑇

𝑏𝑖𝐸𝑏𝑖𝑃−1

𝑖

}

}

}

=

𝑁

∑

𝑖=1

{

{

{

𝑟𝐴𝑖𝑖𝑋𝑖+ 𝑟𝑋𝑖𝐴𝑇

𝑖𝑖+ 𝑟𝐴𝑖𝑗𝑋𝑖+ 𝑟𝑋𝑖𝐴𝑇

𝑖𝑗

+ 𝑟𝐵𝑖𝑌𝑖+ 𝑟𝑌𝑇

𝑖𝐵𝑇

𝑖+ 𝛼𝑖𝐷𝑎𝑖𝑖𝐷𝑇

𝑎𝑖𝑖

+

𝑁

∑

𝑗=1,𝑗 =𝑖


𝑎𝑖𝑗+ 𝛼−1

𝑖𝑋𝑖𝐸𝑇

𝑎𝑖𝑖𝐸𝑎𝑖𝑖𝑋𝑖


+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾−1

𝑖𝑗𝑋𝑖𝐸𝑇

𝑎𝑖𝑗𝐸𝑎𝑖𝑗𝑋𝑖

+ 𝛽𝑖𝐵𝑖𝐷𝑏𝑖𝐷𝑇

𝑏𝑖𝐵𝑇

𝑖+ 𝛽−1

𝑖𝑋𝑇

𝑖𝐸𝑇

𝑏𝑖𝐸𝑏𝑖𝑋𝑖

}

}

}

< 0.

(17)

If the quadratic matrix inequality holds, then the frac-tional order large-scale interconnected system is asymptoti-cally stable.

By applying Schur complement, the above matrixinequality is equivalent to the following complex LMI:

[[[[[[[[[[

[

𝑀𝑖

𝑋𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑏𝑖


−𝛼𝑖𝐼 0 0 0 0 0


0 −𝛾1𝑖𝐼 0 0 0 0


0 0 −𝛾2𝑖𝐼 0 0 0

.

.

. 0 0 0 d 0 0

𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0


0 0 0 0 0 −𝛽𝑖𝐼

]]]]]]]]]]

]

< 0,

(18)

where 𝑀𝑖= 𝑟𝑃𝑖𝐴𝑖𝑖+ 𝑟𝐴𝑇



𝑎𝑖𝑖

+∑𝑁

𝑗=1,𝑗 =𝑖𝛾𝑖𝑗𝐷𝑎𝑖𝑗𝐷𝑇


𝑏𝑖𝐵𝑇


𝑖𝐵𝑇

𝑖. In

practice, the feedback matrix𝐾𝑖has no imaginary part. So let

Im(𝑋𝑖) = 0; then 𝑋

𝑖= Re(𝑋

𝑖). According to the relationship

𝑌𝑖= 𝐾𝑖𝑋𝑖, the outputmatrix𝑌

𝑖has no imaginary part; that is,

Im(𝑌𝑖) = 0; then 𝑌

𝑖= Re(𝑌

𝑖). Substituting 𝑟 = cos 𝜃 + 𝑗 sin 𝜃

into𝑀𝑖gives

𝑀𝑖= cos 𝜃𝐴

𝑖𝑖𝑋𝑖+ cos 𝜃𝑋

𝑖𝐴𝑇

𝑖𝑖+ cos 𝜃𝐴

𝑖𝑗𝑋𝑖+ cos 𝜃𝑋

𝑖𝐴𝑇

𝑖𝑗

+ 𝛼𝑖𝐷𝑎𝑖𝑖𝐷𝑇

𝑎𝑖𝑖+

𝑁

∑

𝑗=1,𝑗 =𝑖



𝑏𝑖𝐵𝑇

𝑖

+ cos 𝜃𝐵𝑖𝑌𝑖+ cos 𝜃𝑌𝑇

𝑖𝐵𝑇

𝑖

+ 𝑗 (sin 𝜃𝐴𝑖𝑖𝑋𝑖− sin 𝜃𝑋

𝑖𝐴𝑇

𝑖𝑖+ sin 𝜃𝐴

𝑖𝑗𝑋𝑖− sin 𝜃𝑋

𝑖𝐴𝑇

𝑖𝑗

+ sin 𝜃𝐵𝑖𝑌𝑖− sin 𝜃𝑌𝑇

𝑖𝐵𝑇

𝑖) .

(19)

Based on Lemma 7, the complex LMI (18) is transformed intothe real LMI. Consider

[[[[[[[[[[[[[[[[[[[[[[[[[

[

𝑀1𝑖

𝑋𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑏𝑖𝑀2𝑖

0 0 0 0 0 0


−𝛼𝑖𝐼 0 0 0 0 0 0 0 0 0 0 0 0


0 −𝛾1𝑖𝐼 0 0 0 0 0 0 0 0 0 0 0


0 0 −𝛾2𝑖𝐼 0 0 0 0 0 0 0 0 0 0

.

.

. 0 0 0 d 0 0 0 0 0 0 0 0 0

𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0 0 0 0 0 0 0 0


0 0 0 0 0 −𝛽𝑖𝐼 0 0 0 0 0 0 0

−𝑀2𝑖

0 0 0 0 0 0 𝑀1𝑖

𝑋𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑏𝑖

0 0 0 0 0 0 0 𝐸𝑎𝑖𝑖𝑋𝑖

−𝛼𝑖𝐼 0 0 0 0 0

0 0 0 0 0 0 0 𝐸𝑎1𝑖𝑋𝑖

0 −𝛾1𝑖𝐼 0 0 0 0

0 0 0 0 0 0 0 𝐸𝑎2𝑖𝑋𝑖

0 0 −𝛾2𝑖𝐼 0 0 0

0 0 0 0 0 0 0

.

.

. 0 0 0 d 0 0

0 0 0 0 0 0 0 𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0

0 0 0 0 0 0 0 𝐸𝑏𝑖𝑋𝑖

0 0 0 0 0 −𝛽𝑖𝐼

]]]]]]]]]]]]]]]]]]]]]]]]]

]

< 0, (20)

where𝑀1𝑖= Re(𝑀

𝑖) and𝑀

2𝑖= Im(𝑀

𝑖). This completes the

proof.

Therefore, the sufficient condition for decentralizedrobust stabilization of fractional order large-scale inter-connected system with norm-bounded uncertainties underadditive gain perturbations is derived. Furthermore, thiscondition is transformed into the solvability problem oflinear matrix inequalities. In summary, by solving the LMI(18), we derive the sufficient condition on stabilizability viadecentralized state feedback of the fractional order uncertainsystem with order 1 < 𝛼 < 2.

4. Nonfragile Controller Design ofFractional Order Large-Scale System withMultiplicative Gain Perturbations

In this section, the nonfragile controller design problem isformulated for the fractional order large-scale interconnectedsystem under multiplicative gain perturbations. Sufficientconditions are established for the decentralized stabilizationof fractional order large-scale interconnected system withnorm-bounded uncertainties under multiplicative gain per-turbations. We are in a position to present our main result.

Theorem 9. Consider the fractional order large-scale uncer-tain system (1) with a commensurate order 1 < 𝛼 < 2. Let


𝑟 = 𝑒𝑗(𝛼−1)(𝜋/2). The fractional order large-scale uncertain

system can be stabilized via decentralized state feedback 𝑢𝑖(𝑡) =

(𝐾𝑖+ Δ𝐾

𝑖)𝑥𝑖(𝑡) if there exist positive-definite block diagonal

matrices 𝑃𝑖

= block diag[𝑃1, 𝑃2. . . , 𝑃𝑁], matrix 𝑌

𝑖and

positive number 𝛼𝑖, 𝛾𝑖𝑗, 𝛽𝑖, 𝑖, 𝑗 = 1, 2, . . . , 𝑁, such that the

following matrix inequalities hold:

[[[[[[[[[[[[[[[[[[[[[[[[[

[

𝑀1𝑖

𝑋𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇

𝑎𝑁𝑖𝑌𝑇

𝑖𝐸𝑇

𝑑𝑖𝑀2𝑖

0 0 0 0 0 0


−𝛼𝑖𝐼 0 0 0 0 0 0 0 0 0 0 0 0


0 −𝛾1𝑖𝐼 0 0 0 0 0 0 0 0 0 0 0


0 0 −𝛾2𝑖𝐼 0 0 0 0 0 0 0 0 0 0

.

.

. 0 0 0 d 0 0 0 0 0 0 0 0 0

𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0 0 0 0 0 0 0 0

𝐸𝑑𝑖𝑌𝑖

0 0 0 0 0 −𝛽𝑖𝐼 0 0 0 0 0 0 0

−𝑀2𝑖

0 0 0 0 0 0 𝑀1𝑖

𝑋𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑑𝑖

0 0 0 0 0 0 0 𝐸𝑎𝑖𝑖𝑋𝑖

−𝛼𝑖𝐼 0 0 0 0 0

0 0 0 0 0 0 0 𝐸𝑎1𝑖𝑋𝑖

0 −𝛾1𝑖𝐼 0 0 0 0

0 0 0 0 0 0 0 𝐸𝑎2𝑖𝑋𝑖

0 0 −𝛾2𝑖𝐼 0 0 0

0 0 0 0 0 0 0

.

.

. 0 0 0 d 0 0

0 0 0 0 0 0 0 𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0

0 0 0 0 0 0 0 𝐸𝑑𝑖𝑌𝑖

0 0 0 0 0 −𝛽𝑖𝐼

]]]]]]]]]]]]]]]]]]]]]]]]]

]

< 0, (21)




𝑎𝑖𝑖+

∑𝑁

𝑗=1,𝑗𝛾𝑖𝑗𝐷𝑎𝑖𝑗𝐷𝑇

𝑎𝑖𝑗+ 𝛽𝑖𝐵𝑖𝐷𝑑𝑖𝐷𝑇

𝑑𝑖𝐵𝑇


𝑖𝐵𝑇

𝑖, 𝑋𝑖=

𝑃−1

𝑖, and 𝑀

1𝑖and 𝑀

2𝑖are the real part and imaginary

part of matrices 𝑀𝑖, respectively. Moreover, the stabilization

decentralized state-feedback gain matrix is given by 𝐾𝑖= 𝑌𝑖𝑃𝑖.

Proof. By the means of decentralized state-feedback controllaw (2), the closed-loop fractional order large-scale intercon-nected system is obtained as

𝑑𝛼

𝑥𝑖(𝑡)

𝑑𝑡𝛼


𝑁

∑

𝑗=1,𝑗 =𝑖



(22)where 𝑖, 𝑗 = 1, 2, . . . 𝑁, Δ𝐾

𝑖= 𝐷𝑏𝑖𝐹𝑏𝑖𝐸𝑏𝑖𝐾𝑖. Based on

Lemma 6, the necessary and sufficient condition on theasymptotical stability of the fractional order system withorder 1 < 𝛼 < 2 is that 𝑟𝑃𝐴 + 𝑟𝐴

𝑇

𝑃 < 0.According to Definition 1, the sufficient condition on the

stabilization of fractional order large-scale interconnectedsystem satisfies the following quadratic matrix inequality:𝑁

∑

𝑖=1

𝜁∗

{𝑟𝑃 [(𝐴 + Δ𝐴) + 𝐵 (𝐾 + Δ𝐾)]

+ 𝑟[(𝐴 + Δ𝐴) + 𝐵 (𝐾 + Δ𝐾)]𝑇

𝑃} 𝜁 < 0,

(23)

𝑁

∑

𝑖=1

{

{

{

𝜁∗

𝑖[𝑟𝑃𝑖𝐴𝑖𝑖+ 𝑟𝐴𝑇

𝑖𝑖𝑃𝑖+ 𝑟𝑃𝑖Δ𝐴𝑖𝑖+ 𝑟Δ𝐴

𝑇

𝑖𝑖𝑃𝑖


𝑖𝐵𝑇

𝑖𝑃𝑖+ 𝑟𝑃𝑖𝐵𝑖Δ𝐾𝑖+ 𝑟𝑃𝑖Δ𝐾𝑇

𝑖𝐵𝑇

𝑖] 𝜁𝑖

+ 2𝑟𝜁∗

𝑖

[

[

𝑁

∑

𝑗=1,𝑗 =𝑖

(𝐴𝑖𝑗+ Δ𝐴𝑖𝑗)]

]

𝜁𝑗

}

}

}

< 0.

(24)

Consequently, the sufficient condition on the decentralizedstabilization of fractional order large-scale interconnectedsystem is that quadratic matrix inequality (23) holds.

Based on Lemmas 3 and 4, by means of enlarging theinequality we have

𝑟𝑃𝑖Δ𝐴𝑖𝑖+ 𝑟Δ𝐴

𝑇

𝑖𝑖𝑃𝑖≤ 𝛼𝑖𝑃𝑖𝐷𝑎𝑖𝑖𝐷𝑇

𝑎𝑖𝑖𝑃𝑖+ 𝛼−1

𝑖𝐸𝑇

𝑎𝑖𝑖𝐸𝑎𝑖𝑖. (25)

Likewise, it is obtained that

𝑟𝑃𝑖𝐵𝑖Δ𝐾𝑖+ 𝑟Δ𝐾

𝑇

𝑖𝐵𝑇

𝑖𝑃𝑖

= 𝑟𝑃𝑖𝐵𝑖𝐷𝑑𝑖𝐹𝑑𝑖𝐸𝑑𝑖𝐾𝑖+ 𝑟𝐾𝑇

𝑖𝐸𝑇

𝑑𝑖𝐹𝑇

𝑑𝑖𝐷𝑇

𝑑𝑖𝐵𝑇

𝑖𝑃𝑖

≤ 𝛽𝑖𝑃𝑖𝐵𝑖𝐷𝑑𝑖𝐷𝑇

𝑑𝑖𝐵𝑇


𝑖𝐾𝑇

𝑖𝐸𝑇

𝑑𝑖𝐸𝑑𝑖𝐾𝑖,

(26)

𝑁

∑

𝑖=1

𝑁

∑

𝑗=1,𝑗 =𝑖

𝜁∗

𝑖[𝑟𝑃𝑖Δ𝐴𝑖𝑗+ 𝑟Δ𝐴

𝑇

𝑖𝑗𝑃𝑖] 𝜁𝑗

≤

𝑁

∑

𝑖=1

𝑁

∑

𝑗=1,𝑗 =𝑖

𝜁∗



+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝑁

∑

𝑖=1

𝜁∗

𝑗𝛾−1

𝑖𝑗𝐸𝑇

𝑎𝑖𝑗𝐸𝑎𝑖𝑗𝜁𝑗


≤

𝑁

∑

𝑖=1

𝑁

∑

𝑗=1,𝑗 =𝑖

𝜁∗



+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝑁

∑

𝑖=1

𝜁∗

𝑖𝛾−1

𝑗𝑖𝐸𝑇

𝑎𝑗𝑖𝐸𝑎𝑗𝑖𝜁𝑖.

(27)

Substituting (24), (25), and (26) into (24) results in the follow-ing quadratic matrix inequality, and we have

𝑁

∑

𝑖=1

{

{

{

𝜁∗

𝑖

[

[



𝑖𝑗𝑃𝑖


𝑖𝐵𝑇

𝑖𝑃𝑖+ 𝛼𝑖𝑃𝑖𝐷𝑎𝑖𝑖𝐷𝑇

𝑎𝑖𝑖𝑃𝑖

+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾𝑖𝑗𝑃𝑖𝐷𝑎𝑖𝑗𝐷𝑇

𝑎𝑖𝑗𝑃𝑖+ 𝛼−1

𝑖𝐸𝑇

𝑎𝑖𝑖𝐸𝑎𝑖𝑖

+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾−1

𝑖𝑗𝐸𝑇

𝑎𝑖𝑗𝐸𝑎𝑖𝑗+ 𝛽−1

𝑖𝐾𝑇

𝑖𝐸𝑇

𝑏𝑖𝐸𝑏𝑖𝐾𝑖

+ 𝛽𝑖𝑃𝑖𝐵𝑖𝐷𝑏𝑖𝐷𝑇

𝑏𝑖𝐵𝑇

𝑖𝑃𝑖

]

]

𝜁𝑖

}

}

}

< 0,

𝑁

∑

𝑖=1

{

{

{



𝑖𝑗𝑃𝑖


𝑖𝐵𝑇

𝑖𝑃𝑖


𝑎𝑖𝑖𝑃𝑖+ 𝑃𝑖

𝑁

∑

𝑗=1,𝑗 =𝑖

(𝛾𝑖𝑗𝐷𝑎𝑖𝑗𝐷𝑇

𝑎𝑖𝑗) 𝑃𝑖

+ 𝛼−1

𝑖𝐸𝑇


𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾−1

𝑖𝑗𝐸𝑇


+ 𝛽𝑖𝑃𝑖𝐵𝑖𝐷𝑑𝑖𝐷𝑇

𝑑𝑖𝐵𝑇


𝑖𝐾𝑇

𝑖𝐸𝑇

𝑑𝑖𝐸𝑑𝑖𝐾𝑖

}

}

}

< 0.

(28)

Let 𝑋𝑖= 𝑃−1

𝑖and 𝑌

𝑖= 𝐾𝑖𝑋𝑖. By premultiplying and post-

multiplying 𝑃−1𝑖

onto (27), one has

𝑁

∑

𝑖=1

{

{

{

𝑟𝐴𝑖𝑖𝑃−1

𝑖+ 𝑟𝑃−1

𝑖𝐴𝑇

𝑖𝑖+ 𝑟𝐴𝑖𝑗𝑃−1

𝑖+ 𝑟𝑃−1

𝑖𝐴𝑇

𝑖𝑗

+ 𝑟𝐵𝑖𝐾𝑖𝑃−1

𝑖+ 𝑟𝑃−1

𝑖𝐾𝑇

𝑖𝐵𝑇

𝑖


𝑎𝑖𝑖+

𝑁

∑

𝑗=1,𝑗 =𝑖


𝑎𝑖𝑗

+ 𝛼−1

𝑖𝑃−1

𝑖𝐸𝑇

𝑎𝑖𝑖𝐸𝑎𝑖𝑖𝑃−1

𝑖

+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾−1

𝑖𝑗𝑃−1

𝑖𝐸𝑇

𝑎𝑖𝑗𝐸𝑎𝑖𝑗𝑃−1

𝑖+ 𝛽𝑖𝐵𝑖𝐷𝑑𝑖𝐷𝑇

𝑑𝑖𝐵𝑇

𝑖

+ 𝛽−1

𝑖𝑃−1

𝑖𝐾𝑇

𝑖𝐸𝑇

𝑑𝑖𝐸𝑑𝑖𝐾𝑖𝑃−1

𝑖

}

}

}

=

𝑁

∑

𝑖=1

{

{

{

𝑟𝐴𝑖𝑖𝑋𝑖+ 𝑟𝑋𝑖𝐴𝑇

𝑖𝑖+ 𝑟𝐴𝑖𝑗𝑋𝑖

+ 𝑟𝑋𝑖𝐴𝑇

𝑖𝑗+ 𝑟𝐵𝑖𝑌𝑖+ 𝑟𝑌𝑇

𝑖𝐵𝑇

𝑖


𝑎𝑖𝑖+

𝑁

∑

𝑗=1,𝑗 =𝑖


𝑎𝑖𝑗

+ 𝛼−1

𝑖𝑋𝑖𝐸𝑇

𝑎𝑖𝑖𝐸𝑎𝑖𝑖𝑋𝑖

+

𝑁

∑

𝑗=1,𝑗 =𝑖

𝛾−1

𝑖𝑗𝑋𝑖𝐸𝑇

𝑎𝑖𝑗𝐸𝑎𝑖𝑗𝑋𝑖

+𝛽𝑖𝐵𝑖𝐷𝑑𝑖𝐷𝑇

𝑑𝑖𝐵𝑇

𝑖+ 𝛽−1

𝑖𝑌𝑇

𝑖𝐸𝑇

𝑑𝑖𝐸𝑑𝑖𝑌𝑖

}

}

}

> 0.

(29)

If the quadratic matrix inequality holds, then the fractionalorder large-scale interconnected system is asymptoticallystable. By applying Schur complement, the above matrixinequality is equivalent to the following complex LMI:

[[[[[[[[[[[[

[

𝑀𝑖

𝑋𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑑𝑖


−𝛼𝑖𝐼 0 0 0 0 0


0 −𝛾1𝑖𝐼 0 0 0 0


0 0 −𝛾2𝑖𝐼 0 0 0

.

.

. 0 0 0 d 0 0

𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0


0 0 0 0 0 −𝛽𝑖𝐼

]]]]]]]]]]]]

]

< 0,

(30)




𝑎𝑖𝑖+

∑𝑁

𝑗=1,𝑗 =𝑖𝛾𝑖𝑗𝐷𝑎𝑖𝑗𝐷𝑇


𝑑𝑖𝐵𝑇


𝑖𝐵𝑇

𝑖. In

practice, the feedback matrix𝐾𝑖has no imaginary part. So let

Im(𝑋𝑖) = 0; then 𝑋

𝑖= Re(𝑋

𝑖). According to the relationship

𝑌𝑖= 𝐾𝑖𝑋𝑖, the outputmatrix𝑌

𝑖has no imaginary part; that is,


Im(𝑌𝑖) = 0; then 𝑌

𝑖= Re(𝑌

𝑖). Substituting 𝑟 = cos 𝜃 + 𝑗 sin 𝜃

into𝑀𝑖gives

𝑀𝑖= cos 𝜃𝐴

𝑖𝑖𝑋𝑖+ cos 𝜃𝑋

𝑖𝐴𝑇

𝑖𝑖+ cos 𝜃𝐴

𝑖𝑗𝑋𝑖+ cos 𝜃𝑋

𝑖𝐴𝑇

𝑖𝑗


𝑎𝑖𝑖+

𝑁

∑

𝑗=1,𝑗 =𝑖



𝑑𝑖𝐵𝑇

𝑖

+ cos 𝜃𝐵𝑖𝑌𝑖+ cos 𝜃𝑌𝑇

𝑖𝐵𝑇

𝑖

+ 𝑗 (sin 𝜃𝐴𝑖𝑖𝑋𝑖− sin 𝜃𝑋

𝑖𝐴𝑇

𝑖𝑖+ sin 𝜃𝐴

𝑖𝑗𝑋𝑖− sin 𝜃𝑋

𝑖𝐴𝑇

𝑖𝑗

+ sin 𝜃𝐵𝑖𝑌𝑖− sin 𝜃𝑌𝑇

𝑖𝐵𝑇

𝑖) .

(31)

Based on Lemma 7, the complex LMI (30) is transformed intothe real LMI. Consider

[[[[[[[[[[[[[[[[[[[[[[[[[

[

𝑀1𝑖

𝑋𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑑𝑖𝑀2𝑖

0 0 0 0 0 0


−𝛼𝑖𝐼 0 0 0 0 0 0 0 0 0 0 0 0


0 −𝛾1𝑖𝐼 0 0 0 0 0 0 0 0 0 0 0


0 0 −𝛾2𝑖𝐼 0 0 0 0 0 0 0 0 0 0

.

.

. 0 0 0 d 0 0 0 0 0 0 0 0 0

𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0 0 0 0 0 0 0 0


0 0 0 0 0 −𝛽𝑖𝐼 0 0 0 0 0 0 0

−𝑀2𝑖

0 0 0 0 0 0 𝑀1𝑖

𝑋𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑎1𝑖𝑋𝑇

𝑖𝐸𝑇

𝑎2𝑖⋅ ⋅ ⋅ 𝑋

𝑇

𝑖𝐸𝑇


𝑖𝐸𝑇

𝑑𝑖

0 0 0 0 0 0 0 𝐸𝑎𝑖𝑖𝑋𝑖

−𝛼𝑖𝐼 0 0 0 0 0

0 0 0 0 0 0 0 𝐸𝑎1𝑖𝑋𝑖

0 −𝛾1𝑖𝐼 0 0 0 0

0 0 0 0 0 0 0 𝐸𝑎2𝑖𝑋𝑖

0 0 −𝛾2𝑖𝐼 0 0 0

0 0 0 0 0 0 0

.

.

. 0 0 0 d 0 0

0 0 0 0 0 0 0 𝐸𝑎𝑁𝑖

𝑋𝑖

0 0 0 0 −𝛾𝑁𝑖𝐼 0

0 0 0 0 0 0 0 𝐸𝑑𝑖𝑌𝑖

0 0 0 0 0 −𝛽𝑖𝐼

]]]]]]]]]]]]]]]]]]]]]]]]]

]

< 0, (32)

where𝑀1𝑖= Re(𝑀

𝑖),𝑀2𝑖= Im(𝑀

𝑖).

This completes the proof.

Therefore, the sufficient condition for decentralizedrobust stabilization of fractional order large-scale inter-connected system with norm-bounded uncertainties undermultiplicative gain perturbations is obtained. Furthermore,this condition is transformed into the solvability problem oflinear matrix inequalities. In summary, by solving the LMI(27), we derive the sufficient conditions on stabilizability viadecentralized state feedback of the uncertain fractional ordersystem under multiplicative gain perturbations.

5. Numerical Examples

In this section, to verify and demonstrate the effective-ness of the proposed method, two numerical examples areinvestigated. The fractional order large-scale interconnecteduncertain system under controller gain perturbations isstabilized by the decentralized state feedback controllers.TheAdams-type predictor-corrector method [29] is used for thenumerical solution of fractional differential equations duringthe simulation.

Example 1. Consider the stabilization problem of fractionalorder large-scale interconnected uncertain system underadditive gain perturbations:

𝑑𝛼

𝑥 (𝑡)

𝑑𝑡𝛼


𝑁

∑

𝑗=1,𝑗 =𝑖



(33)

where 𝑖, 𝑗 = 1, 2, . . . 𝑁, 𝛼 = 1.5,𝑁 = 2, 𝜙 = 𝜋/3,

𝐴11= [

−2.5 3.7

1.9 −2] , 𝐴

12= [

3.5 1.2

1.2 3.5] ,

𝐴21= [

0.2 −0.1

0.3 0.1] , 𝐴

22= [

1.8 1.2

−1.2 0.8] ,

𝐵1= [

1

1] , 𝐵

2= [

0 15

15 25] ,

Δ𝐴11= 𝐷𝑎11𝐹𝑎11𝐸𝑎11

= [0.4 0.2

1 0.3] [

sin (𝜙) 0

0 sin (𝜙)] [0.4 0.2

1 0.3] ,


= [0.4 0.2

1 0.3] [

sin (𝜙) 0

0 sin (𝜙)] [0.3 0.5

0.1 0.5] ,



= [0.5

0.3] [

sin (𝜙) 0

0 sin (𝜙)] [0.3 0.5

0.1 0.5] ,


= [0.5

0.3] [

sin (𝜙) 0

0 sin (𝜙)] [0.4 0.2

1 0.3] .

(34)

Meanwhile, the following additive gain perturbations areconsidered:

Δ𝐾1= 𝐷𝑏1𝐹𝑏1𝐸𝑏1= [0.5 0.3] [

sin (𝜙) 0

0 sin (𝜙)] [0.4 0.2

1 0.3] ,

Δ𝐾2= 𝐷𝑏2𝐹𝑏2𝐸𝑏2= [

0.5 0

0 0.5] [

sin (𝜙) 0

0 sin (𝜙)] [0.5 0

0 0.5] .

(35)

By using the LMI technique, it is verified that the matrixinequalities are feasible in view of Theorem 8. So the decen-tralized local state feedback gain matrix is obtained as

𝐾1= [−58.6565 −61.7286] ,

𝐾2= [

−0.2560 −6.2675

−1.7459 −4.9510] .

(36)

The time responses of system are shown in Figure 1. It isobserved that its four states all converge to zero. It can beconcluded that fractional order large-scale interconnectedsystem with additive gain perturbations can be stabilized bythe nonfragile controller.

Example 2. Consider the stabilization problem of fractionalorder large-scale interconnected uncertain system with mul-tiplicative gain perturbations:

𝑑𝛼

𝑥 (𝑡)

𝑑𝑡𝛼


𝑁

∑

𝑗=1,𝑗 =𝑖



(37)

where 𝑖, 𝑗 = 1, 2, . . . , 𝑁, 𝛼 = 1.5,𝑁 = 2, 𝜙 = 𝜋/3,

𝐴11= [

−3.7 3.5

1.9 −3] , 𝐴

12= [

0.5 0.2

0.2 0.5] ,

𝐴21= [

0.2 −0.1

0.3 0.1] , 𝐴

22= [

1 0.2

−0.2 0.8] ,

𝐵1= [

6.5

0] , 𝐵

2= [

0.5 0.5

0 1.5] ,


= [0.4 0.2

1 0.3] [

sin (𝜙) 0

0 sin (𝜙)] [0.4 0.2

1 0.3] ,

0 1 2 3 4 5t

x1

x2

x3

x4

1

0.5

0

−0.5

Figure 1: Example 1: time responses of the closed-loop fractionalorder large-scale interconnected uncertain system.


= [0.4 0.2

1 0.3] [

sin (𝜙) 0

0 sin (𝜙)] [0.3 0.5

0.1 0.5] ,


= [0.5

0.3] [

sin (𝜙) 0

0 sin (𝜙)] [0.3 0.5

0.1 0.5] ,


= [0.5

0.3] [

sin (𝜙) 0

0 sin (𝜙)] [0.4 0.2

1 0.3] .

(38)

Meanwhile, the following multiplicative gain perturbationsare considered:

𝐷𝑑1= [10.5 10.5] , 𝐸

𝑑1= [

2.2

2.2] ,

𝐷𝑑2= [

0.5 0

0 0.5] , 𝐸

𝑑2= [

0.5 0

0 0.5] .

(39)

By using the LMI technique, it is verified that the matrixinequalities are feasible in view of Theorem 9. So the decen-tralized local state feedback gain matrix is obtained as

𝐾1= [−0.0304 −0.0119] , 𝐾

2= [

−3.0473 2.8800

2.0568 −2.2810] .

(40)

The time responses of system are shown in Figure 2. It isshown that its four states all converge to zero. It can beconcluded that fractional order large-scale interconnectedsystem with multiplicative gain perturbations can be stabi-lized by the nonfragile controller.


0 20 40 60 80 100t

x1

x2

x3

x4

1

1.5

0.5

0

−0.5

−1

Figure 2: Example 2: time responses of the closed-loop fractionalorder large-scale interconnected uncertain system.

6. Conclusions

In this paper, sufficient conditions have been derived on thestabilization of fractional order large-scale interconnecteduncertain system with a commensurate order 1 < 𝛼 < 2

under two kinds of controller gain perturbations, that is,additive and multiplicative gain perturbations. The proposedmethod is based on the stability criterion of fractional ordersystem by using the complex Lyapunov inequality. Moreover,the nonfragile controllers are designed. Simulation resultshave demonstrated the effectiveness of the proposedmethod.

Conflict of Interests

The author declares that there is no conflict of interestsregarding to the publication of this paper.

Acknowledgment

This work was supported in part by a Zhiyuan Professorshipat Shanghai Jiao Tong University.

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