Research Article Robust Nonfragile Controllers...
Transcript of Research Article Robust Nonfragile Controllers...
Research ArticleRobust Nonfragile Controllers Design forFractional Order Large-Scale Uncertain Systems witha Commensurate Order 1 < ๐ผ < 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 ยฉ 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 < ๐ผ < 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 [1โ7].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[11โ13]. 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 < ๐ผ < 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 [16โ20]. The robust stability of fractional orderinterval systems has been investigated in [21, 22]. It is wellknown that Matignonโs stability theorem [16] is the basis forstability analysis of the fractional order system by checkingthe location of eigenvalues in the complex plane. Matignonโstheorem is in fact the pioneering works of stability analysisof the fractional order system. Based onMatignonโs theorem,the stability criteria of fractional order systems have beenproposed in both cases of 1 < ๐ผ < 2 and 0 < ๐ผ < 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 < ๐ผ <
2. Here, it should be also pointed out that [25] only focuseson the case of a fractional order 0 < ๐ผ < 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 ๐ the conju-gate of the complex number ๐. ๐ denotes the imaginary unit.๐ผ denotes the identity matrix with appropriate dimensions.block diag denotes the block diagonal matrix. ๐ ๐ denotesthe ๐-dimensional Euclidean space and ๐ ๐ร๐ is the set of all๐ ร ๐ real matrices. ๐๐ denotes the transpose of ๐ and๐โ denotes the Hermitian transpose of๐. 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 < ๐ผ < 2
composed of๐ fractional order subsystems:
๐๐ผ
๐ฅ๐(๐ก)
๐๐ก๐ผ
= [๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก)
+
๐
โ
๐=1,๐ =๐
[๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก) + ๐ต
๐๐ข๐(๐ก) ,
(1)
where ๐ผ โ ๐ is the fractional commensurate order, ๐, ๐ =
1, 2, . . . ๐, and ๐ฅ๐(๐ก) โ ๐
๐๐ and ๐ข๐(๐ก) โ ๐
๐๐ are the stateand input of the ๐th fractional order subsystem, respectively.Assume that the nominal systems ๐ด
๐๐โ ๐ ๐๐ร๐๐ , ๐ด
๐๐โ ๐ ๐๐ร๐๐ ,
and ๐ต๐โ ๐ ๐๐ร๐๐ are constant and of appropriate dimensions
and the pair (๐ด๐๐,๐ต๐) is controllable. The fractional order
subsystems interact with each other through the intercon-nections โ๐
๐=1,๐ =๐๐ด๐๐๐ฅ๐(๐ก). The main objective of the note is
to find the decentralized local state feedback control law ofthe following form:
๐ข๐(๐ก) = (๐พ
๐+ ฮ๐พ๐) ๐ฅ๐(๐ก) , ๐ = 1, 2, . . . , ๐, (2)
such that the resulting fractional order closed-loop systemis asymptotically stable, where ๐พ
๐โ ๐ ๐๐ร๐๐ is the state
feedback gainmatrix to be designed andฮ๐พ๐= ๐ท๐๐๐น๐๐๐ธ๐๐and
ฮ๐พ๐= ๐ท๐๐๐น๐๐๐ธ๐๐๐พ๐represent 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 ฮ๐ด
๐๐=
๐ท๐๐๐๐น๐๐๐๐ธ๐๐๐, ๐น๐๐๐๐๐น๐๐๐
โค ๐ผ; ฮ๐ด๐๐= ๐ท๐๐๐๐น๐๐๐๐ธ๐๐๐, ๐น๐๐๐๐๐น๐๐๐
โค ๐ผ;ฮ๐พ๐= ๐ท๐๐๐น๐๐๐ธ๐๐, ๐น๐๐๐๐น๐๐โค ๐ผ; ฮ๐พ
๐= ๐ท๐๐๐น๐๐๐ธ๐๐๐พ๐, ๐น๐๐๐๐น๐๐โค ๐ผ,
where the elements are Lebesgue measurable and ๐ท๐๐๐, ๐ท๐๐๐,
๐ท๐๐, ๐ท๐๐, ๐ธ๐๐๐, ๐ธ๐๐๐, ๐ธ๐๐, and ๐ธ
๐๐are 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
๐๐ผ
๐ฅ (๐ก)
๐๐ก๐ผ
= (๐ด + ฮ๐ด) ๐ฅ (๐ก) + ๐ต (๐พ + ฮ๐พ) ๐ฅ (๐ก) , (3)
with the composite matrices ๐ด and๐พ having the structure
๐ด =[[
[
๐ด11
โ โ โ ๐ด1๐
.
.
. d...
๐ด๐1
โ โ โ ๐ด๐๐
]]
]
,
ฮ๐ด =[[
[
ฮ๐ด11
โ โ โ ฮ๐ด1๐
.
.
. d...
ฮ๐ด๐1
โ โ โ ฮ๐ด๐๐
]]
]
,
๐พ = block diag [๐พ1, ๐พ2โ โ โ ๐พ๐] ,
ฮ๐พ = block diag [ฮ๐พ1, ฮ๐พ2โ โ โ ฮ๐พ
๐] .
(4)
Definition 1 (see [26]). For all nonzero real vectors ๐ โ ๐ ๐,
๐ด โ ๐ ๐ร๐ is real matrix; if the inequality ๐๐๐ด๐ < 0 holds,
then ๐ด is said to be negative definite matrix.
Definition 2. The fractional order large-scale uncertain sys-tem can be stabilized via decentralized state feedback ๐ข
๐(๐ก) =
(๐พ๐+ฮ๐พ๐)๐ฅ๐(๐ก) if there exists gainmatrix๐พ
๐โ ๐ ๐๐ร๐๐ such that
the closed-loop fractional order large-scale uncertain system๐๐ผ
๐ฅ (๐ก)
๐๐ก๐ผ
= (๐ด + ฮ๐ด) ๐ฅ (๐ก) + ๐ต (๐พ + ฮ๐พ) ๐ฅ (๐ก) (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 ๐ถ โ ๐ ๐ร๐, ๐ด, ๐ต โ ๐
๐ร๐, ๐ด โฅ ๐ต;then ๐ถ๐๐ด๐ถ โฅ ๐ถ
๐
๐ต๐ถ.
Mathematical Problems in Engineering 3
Lemma 4 (see [27]). For any matrices ๐ and ๐ with appro-priate dimensions and for any ๐ฝ > 0, the following inequalityholds:
๐๐
๐ + ๐๐
๐ < ๐ฝ๐๐
๐ + ๐ฝโ1
๐๐
๐. (6)
Lemma5 (see [28]). The fractional order system๐๐ผ
๐ฅ(๐ก)/๐๐ก๐ผ
=
๐ด๐ฅ(๐ก) with a commensurate order ๐ผ is asymptotically stableif | arg(spec(๐ด))| > ๐ผ(๐/2), where ๐ผ is the order of fractionalorder system and spec(๐ด) is the spectrum of all eigenvalues of๐ด.
Lemma 6 (see [23]). Let 1 < ๐ผ < 2 and ๐ = (๐ผ โ 1)(๐/2),๐ = ๐๐๐. The fractional order system ๐
๐ผ
๐ฅ(๐ก)/๐๐ก๐ผ
= ๐ด๐ฅ(๐ก) witha commensurate order 1 < ๐ผ < 2 is asymptotically stable if andonly if there exist positive definite matrices ๐ = ๐
๐
โ ๐ ๐ร๐,
such that
[
[
(๐ด๐ + ๐๐ด๐
) sin ๐ (๐ด๐ โ ๐๐ด๐
) cos ๐
(๐๐ด๐
โ ๐ด๐) cos ๐ (๐ด๐ + ๐๐ด๐
) sin ๐]
]
< 0, (7)
or equivalently, ๐๐๐ด + ๐๐ด๐
๐ < 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 < ๐ผ < 2 isconvex. By using Linear Matrix Inequalities (LMI) approach,it is obtained as the above LMI. Therefore, it is equivalent to๐๐๐ด + ๐๐ด
๐
๐ < 0.
Lemma 7. A complex Hermitian matrix๐ satisfies๐ < 0 ifand only if the following real LMI inequality holds:
[Re (๐) Im (๐)
โ Im (๐) Re (๐)] < 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 < ๐ผ < 2. Let ๐ =๐๐(๐ผโ1)(๐/2). The fractional order large-scale uncertain systemcan be stabilized via decentralized state feedback ๐ข
๐(๐ก) = (๐พ
๐+
ฮ๐พ๐)๐ฅ๐(๐ก) if there exist positive-definite block diagonal matrices
๐๐= block diag[๐
1, ๐2. . . , ๐๐], matrix ๐
๐, and positive real
scalar constants ๐ผ๐, ๐พ๐๐, ๐ฝ๐, ๐, ๐ = 1, 2, . . . , ๐, such that the
following matrix inequalities hold:
[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
[
๐1๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐๐2๐
0 0 0 0 0 0
๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0 0 0 0 0 0 0 0
๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0 0 0 0 0 0 0 0
๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0 0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0 0 0 0 0 0 0 0
๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0 0 0 0 0 0 0 0
๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ 0 0 0 0 0 0 0
โ๐2๐
0 0 0 0 0 0 ๐1๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐
0 0 0 0 0 0 0 ๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0
0 0 0 0 0 0 0 ๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0
0 0 0 0 0 0 0 ๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0
0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0
0 0 0 0 0 0 0 ๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0
0 0 0 0 0 0 0 ๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ
]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
]
< 0, (9)
where๐๐= ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐+
โ๐
๐=1,๐๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐+ ๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐+ ๐๐ต๐๐๐+ ๐๐๐
๐๐ต๐
๐, ๐๐=
๐โ1
๐, and ๐
1๐and ๐
2๐are the real part and imaginary
part of matrices ๐๐, respectively. Moreover, the stabilization
decentralized state-feedback gain matrix can be calculated asfollows: ๐พ
๐= ๐๐๐๐.
Proof. Under decentralized state-feedback control law (2),the closed-loop fractional order large-scale interconnected
system is obtained as๐๐ผ
๐ฅ๐(๐ก)
๐๐ก๐ผ
= [๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก) +
๐
โ
๐=1,๐ =๐
[๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก)
+ ๐ต๐[๐พ๐+ ฮ๐พ๐] ๐ฅ๐(๐ก) ,
(10)
where ๐,๐ = 1, 2, . . . ๐. Based on Lemma 6, the necessaryand sufficient condition on the asymptotical stability of thefractional order system with order 1 < ๐ผ < 2 is that
4 Mathematical Problems in Engineering
๐๐๐ด + ๐๐ด๐
๐ < 0. According to Definition 1, the sufficientcondition on the stabilization of fractional order large-scale interconnected system satisfies the following quadraticmatrix inequality:
๐๐
{๐๐ [(๐ด + ฮ๐ด) + ๐ต (๐พ + ฮ๐พ)]
+ ๐ [(๐ด + ฮ๐ด) + ๐ต (๐พ + ฮ๐พ)]๐
๐} ๐ < 0,
(11)
๐
โ
๐=1
{
{
{
๐โ
๐[๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐ฮ๐ด๐๐
+ ๐ฮ๐ด๐
๐๐๐๐+ ๐๐๐๐ต๐๐พ๐+ ๐๐พ๐
๐๐ต๐
๐๐๐
+ ๐๐๐๐ต๐ฮ๐พ๐+ ๐๐๐ฮ๐พ๐
๐๐ต๐
๐] ๐๐
+ 2๐๐โ
๐
[
[
๐
โ
๐=1,๐ =๐
(๐ด๐๐+ ฮ๐ด๐๐)]
]
๐๐
}
}
}
< 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
๐๐๐ฮ๐ด๐๐+ ๐ฮ๐ด
๐
๐๐๐๐โค ๐ผ๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐+ ๐ผโ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐. (13)
Meanwhile, based on Lemmas 3 and 4, by means of enlargingthe inequality we have
๐๐๐๐ต๐ฮ๐พ๐+ ๐ฮ๐พ
๐
๐๐ต๐
๐๐๐โค ๐ฝ๐๐๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐๐๐+ ๐ฝโ1
๐๐ธ๐
๐๐๐ธ๐๐,
๐
โ
๐=1
๐
โ
๐=1,๐ =๐
๐โ
๐[๐๐๐ฮ๐ด๐๐+ ๐ฮ๐ด
๐
๐๐๐๐] ๐๐
โค
๐
โ
๐=1
๐
โ
๐=1,๐ =๐
๐โ
๐๐พ๐๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐๐๐
+
๐
โ
๐=1,๐ =๐
๐
โ
๐=1
๐โ
๐๐พโ1
๐๐๐ธ๐
๐๐๐๐ธ๐๐๐๐๐
โค
๐
โ
๐=1
๐
โ
๐=1,๐ =๐
๐โ
๐๐พ๐๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐๐๐
+
๐
โ
๐=1,๐ =๐
๐
โ
๐=1
๐โ
๐๐พโ1
๐๐๐ธ๐
๐๐๐๐ธ๐๐๐๐๐.
(14)
Substituting (11), (12), and (13) into (12) results in thefollowing quadratic matrix inequality:
๐
โ
๐=1
{
{
{
๐โ
๐
[
[
๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐
+ ๐๐๐๐ต๐๐พ๐+ ๐๐พ๐
๐๐ต๐
๐๐๐
+ ๐ผ๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐+
๐
โ
๐=1,๐ =๐
๐พ๐๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐
+ ๐ผโ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐+
๐
โ
๐=1,๐ =๐
๐พโ1
๐๐๐ธ๐
๐๐๐๐ธ๐๐๐
+ ๐ฝโ1
๐๐ธ๐
๐๐๐ธ๐๐+ ๐ฝ๐๐๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐๐๐
]
]
๐๐
}
}
}
< 0,
(15)
๐
โ
๐=1
{
{
{
๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐
+ ๐๐๐๐ต๐๐พ๐+ ๐๐พ๐
๐๐ต๐
๐๐๐
+ ๐ผ๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐+ ๐๐
๐
โ
๐=1,๐ =๐
(๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐) ๐๐
+ ๐ผโ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐+
๐
โ
๐=1,๐ =๐
๐พโ1
๐๐๐ธ๐
๐๐๐๐ธ๐๐๐
+๐ฝ๐๐๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐๐๐+ ๐ฝโ1
๐๐ธ๐
๐๐๐ธ๐๐
}
}
}
< 0.
(16)
Let ๐๐= ๐โ1
๐and ๐
๐= ๐พ๐๐๐. By premultiplying and post-
multiplying ๐โ1๐
onto (16), one has
๐
โ
๐=1
{
{
{
๐๐ด๐๐๐โ1
๐+ ๐๐โ1
๐๐ด๐
๐๐+ ๐๐ด๐๐๐โ1
๐+ ๐๐โ1
๐๐ด๐
๐๐
+ ๐๐ต๐๐พ๐๐โ1
๐+ ๐๐โ1
๐๐พ๐
๐๐ต๐
๐+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐
+
๐
โ
๐=1,๐ =๐
๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐+ ๐ผโ1
๐๐โ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐๐โ1
๐
+
๐
โ
๐=1,๐ =๐
๐พโ1
๐๐๐โ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐๐โ1
๐+ ๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐
+๐ฝโ1
๐๐โ1
๐๐ธ๐
๐๐๐ธ๐๐๐โ1
๐
}
}
}
=
๐
โ
๐=1
{
{
{
๐๐ด๐๐๐๐+ ๐๐๐๐ด๐
๐๐+ ๐๐ด๐๐๐๐+ ๐๐๐๐ด๐
๐๐
+ ๐๐ต๐๐๐+ ๐๐๐
๐๐ต๐
๐+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐
+
๐
โ
๐=1,๐ =๐
๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐+ ๐ผโ1
๐๐๐๐ธ๐
๐๐๐๐ธ๐๐๐๐๐
Mathematical Problems in Engineering 5
+
๐
โ
๐=1,๐ =๐
๐พโ1
๐๐๐๐๐ธ๐
๐๐๐๐ธ๐๐๐๐๐
+ ๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐+ ๐ฝโ1
๐๐๐
๐๐ธ๐
๐๐๐ธ๐๐๐๐
}
}
}
< 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:
[[[[[[[[[[
[
๐๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐
๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0
๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0
๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0
.
.
. 0 0 0 d 0 0
๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0
๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ
]]]]]]]]]]
]
< 0,
(18)
where ๐๐= ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐
+โ๐
๐=1,๐ =๐๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐+ ๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐+ ๐๐ต๐๐๐+ ๐๐๐
๐๐ต๐
๐. In
practice, the feedback matrix๐พ๐has no imaginary part. So let
Im(๐๐) = 0; then ๐
๐= Re(๐
๐). According to the relationship
๐๐= ๐พ๐๐๐, the outputmatrix๐
๐has no imaginary part; that is,
Im(๐๐) = 0; then ๐
๐= Re(๐
๐). Substituting ๐ = cos ๐ + ๐ sin ๐
into๐๐gives
๐๐= cos ๐๐ด
๐๐๐๐+ cos ๐๐
๐๐ด๐
๐๐+ cos ๐๐ด
๐๐๐๐+ cos ๐๐
๐๐ด๐
๐๐
+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐+
๐
โ
๐=1,๐ =๐
๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐+ ๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐
+ cos ๐๐ต๐๐๐+ cos ๐๐๐
๐๐ต๐
๐
+ ๐ (sin ๐๐ด๐๐๐๐โ sin ๐๐
๐๐ด๐
๐๐+ sin ๐๐ด
๐๐๐๐โ sin ๐๐
๐๐ด๐
๐๐
+ sin ๐๐ต๐๐๐โ sin ๐๐๐
๐๐ต๐
๐) .
(19)
Based on Lemma 7, the complex LMI (18) is transformed intothe real LMI. Consider
[[[[[[[[[[[[[[[[[[[[[[[[[
[
๐1๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐๐2๐
0 0 0 0 0 0
๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0 0 0 0 0 0 0 0
๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0 0 0 0 0 0 0 0
๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0 0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0 0 0 0 0 0 0 0
๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0 0 0 0 0 0 0 0
๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ 0 0 0 0 0 0 0
โ๐2๐
0 0 0 0 0 0 ๐1๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐
0 0 0 0 0 0 0 ๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0
0 0 0 0 0 0 0 ๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0
0 0 0 0 0 0 0 ๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0
0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0
0 0 0 0 0 0 0 ๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0
0 0 0 0 0 0 0 ๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ
]]]]]]]]]]]]]]]]]]]]]]]]]
]
< 0, (20)
where๐1๐= Re(๐
๐) and๐
2๐= Im(๐
๐). 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 < ๐ผ < 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 < ๐ผ < 2. Let
6 Mathematical Problems in Engineering
๐ = ๐๐(๐ผโ1)(๐/2). The fractional order large-scale uncertain
system can be stabilized via decentralized state feedback ๐ข๐(๐ก) =
(๐พ๐+ ฮ๐พ
๐)๐ฅ๐(๐ก) if there exist positive-definite block diagonal
matrices ๐๐
= block diag[๐1, ๐2. . . , ๐๐], matrix ๐
๐and
positive number ๐ผ๐, ๐พ๐๐, ๐ฝ๐, ๐, ๐ = 1, 2, . . . , ๐, such that the
following matrix inequalities hold:
[[[[[[[[[[[[[[[[[[[[[[[[[
[
๐1๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐๐2๐
0 0 0 0 0 0
๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0 0 0 0 0 0 0 0
๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0 0 0 0 0 0 0 0
๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0 0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0 0 0 0 0 0 0 0
๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0 0 0 0 0 0 0 0
๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ 0 0 0 0 0 0 0
โ๐2๐
0 0 0 0 0 0 ๐1๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐
0 0 0 0 0 0 0 ๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0
0 0 0 0 0 0 0 ๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0
0 0 0 0 0 0 0 ๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0
0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0
0 0 0 0 0 0 0 ๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0
0 0 0 0 0 0 0 ๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ
]]]]]]]]]]]]]]]]]]]]]]]]]
]
< 0, (21)
where๐๐= ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐+
โ๐
๐=1,๐๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐+ ๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐+ ๐๐ต๐๐๐+ ๐๐๐
๐๐ต๐
๐, ๐๐=
๐โ1
๐, and ๐
1๐and ๐
2๐are the real part and imaginary
part of matrices ๐๐, respectively. Moreover, the stabilization
decentralized state-feedback gain matrix is given by ๐พ๐= ๐๐๐๐.
Proof. By the means of decentralized state-feedback controllaw (2), the closed-loop fractional order large-scale intercon-nected system is obtained as
๐๐ผ
๐ฅ๐(๐ก)
๐๐ก๐ผ
= [๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก) +
๐
โ
๐=1,๐ =๐
[๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก)
+ ๐ต๐[๐พ๐+ ฮ๐พ๐] ๐ฅ๐(๐ก) ,
(22)where ๐, ๐ = 1, 2, . . . ๐, ฮ๐พ
๐= ๐ท๐๐๐น๐๐๐ธ๐๐๐พ๐. Based on
Lemma 6, the necessary and sufficient condition on theasymptotical stability of the fractional order system withorder 1 < ๐ผ < 2 is that ๐๐๐ด + ๐๐ด
๐
๐ < 0.According to Definition 1, the sufficient condition on the
stabilization of fractional order large-scale interconnectedsystem satisfies the following quadratic matrix inequality:๐
โ
๐=1
๐โ
{๐๐ [(๐ด + ฮ๐ด) + ๐ต (๐พ + ฮ๐พ)]
+ ๐[(๐ด + ฮ๐ด) + ๐ต (๐พ + ฮ๐พ)]๐
๐} ๐ < 0,
(23)
๐
โ
๐=1
{
{
{
๐โ
๐[๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐ฮ๐ด๐๐+ ๐ฮ๐ด
๐
๐๐๐๐
+ ๐๐๐๐ต๐๐พ๐+ ๐๐พ๐
๐๐ต๐
๐๐๐+ ๐๐๐๐ต๐ฮ๐พ๐+ ๐๐๐ฮ๐พ๐
๐๐ต๐
๐] ๐๐
+ 2๐๐โ
๐
[
[
๐
โ
๐=1,๐ =๐
(๐ด๐๐+ ฮ๐ด๐๐)]
]
๐๐
}
}
}
< 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
๐๐๐ฮ๐ด๐๐+ ๐ฮ๐ด
๐
๐๐๐๐โค ๐ผ๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐+ ๐ผโ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐. (25)
Likewise, it is obtained that
๐๐๐๐ต๐ฮ๐พ๐+ ๐ฮ๐พ
๐
๐๐ต๐
๐๐๐
= ๐๐๐๐ต๐๐ท๐๐๐น๐๐๐ธ๐๐๐พ๐+ ๐๐พ๐
๐๐ธ๐
๐๐๐น๐
๐๐๐ท๐
๐๐๐ต๐
๐๐๐
โค ๐ฝ๐๐๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐๐๐+ ๐ฝโ1
๐๐พ๐
๐๐ธ๐
๐๐๐ธ๐๐๐พ๐,
(26)
๐
โ
๐=1
๐
โ
๐=1,๐ =๐
๐โ
๐[๐๐๐ฮ๐ด๐๐+ ๐ฮ๐ด
๐
๐๐๐๐] ๐๐
โค
๐
โ
๐=1
๐
โ
๐=1,๐ =๐
๐โ
๐๐พ๐๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐๐๐
+
๐
โ
๐=1,๐ =๐
๐
โ
๐=1
๐โ
๐๐พโ1
๐๐๐ธ๐
๐๐๐๐ธ๐๐๐๐๐
Mathematical Problems in Engineering 7
โค
๐
โ
๐=1
๐
โ
๐=1,๐ =๐
๐โ
๐๐พ๐๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐๐๐
+
๐
โ
๐=1,๐ =๐
๐
โ
๐=1
๐โ
๐๐พโ1
๐๐๐ธ๐
๐๐๐๐ธ๐๐๐๐๐.
(27)
Substituting (24), (25), and (26) into (24) results in the follow-ing quadratic matrix inequality, and we have
๐
โ
๐=1
{
{
{
๐โ
๐
[
[
๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐
+ ๐๐๐๐ต๐๐พ๐+ ๐๐พ๐
๐๐ต๐
๐๐๐+ ๐ผ๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐
+
๐
โ
๐=1,๐ =๐
๐พ๐๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐+ ๐ผโ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐
+
๐
โ
๐=1,๐ =๐
๐พโ1
๐๐๐ธ๐
๐๐๐๐ธ๐๐๐+ ๐ฝโ1
๐๐พ๐
๐๐ธ๐
๐๐๐ธ๐๐๐พ๐
+ ๐ฝ๐๐๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐๐๐
]
]
๐๐
}
}
}
< 0,
๐
โ
๐=1
{
{
{
๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐
+ ๐๐๐๐ต๐๐พ๐+ ๐๐พ๐
๐๐ต๐
๐๐๐
+ ๐ผ๐๐๐๐ท๐๐๐๐ท๐
๐๐๐๐๐+ ๐๐
๐
โ
๐=1,๐ =๐
(๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐) ๐๐
+ ๐ผโ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐+
๐
โ
๐=1,๐ =๐
๐พโ1
๐๐๐ธ๐
๐๐๐๐ธ๐๐๐
+ ๐ฝ๐๐๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐๐๐+ ๐ฝโ1
๐๐พ๐
๐๐ธ๐
๐๐๐ธ๐๐๐พ๐
}
}
}
< 0.
(28)
Let ๐๐= ๐โ1
๐and ๐
๐= ๐พ๐๐๐. By premultiplying and post-
multiplying ๐โ1๐
onto (27), one has
๐
โ
๐=1
{
{
{
๐๐ด๐๐๐โ1
๐+ ๐๐โ1
๐๐ด๐
๐๐+ ๐๐ด๐๐๐โ1
๐+ ๐๐โ1
๐๐ด๐
๐๐
+ ๐๐ต๐๐พ๐๐โ1
๐+ ๐๐โ1
๐๐พ๐
๐๐ต๐
๐
+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐+
๐
โ
๐=1,๐ =๐
๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐
+ ๐ผโ1
๐๐โ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐๐โ1
๐
+
๐
โ
๐=1,๐ =๐
๐พโ1
๐๐๐โ1
๐๐ธ๐
๐๐๐๐ธ๐๐๐๐โ1
๐+ ๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐
+ ๐ฝโ1
๐๐โ1
๐๐พ๐
๐๐ธ๐
๐๐๐ธ๐๐๐พ๐๐โ1
๐
}
}
}
=
๐
โ
๐=1
{
{
{
๐๐ด๐๐๐๐+ ๐๐๐๐ด๐
๐๐+ ๐๐ด๐๐๐๐
+ ๐๐๐๐ด๐
๐๐+ ๐๐ต๐๐๐+ ๐๐๐
๐๐ต๐
๐
+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐+
๐
โ
๐=1,๐ =๐
๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐
+ ๐ผโ1
๐๐๐๐ธ๐
๐๐๐๐ธ๐๐๐๐๐
+
๐
โ
๐=1,๐ =๐
๐พโ1
๐๐๐๐๐ธ๐
๐๐๐๐ธ๐๐๐๐๐
+๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐+ ๐ฝโ1
๐๐๐
๐๐ธ๐
๐๐๐ธ๐๐๐๐
}
}
}
> 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:
[[[[[[[[[[[[
[
๐๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐
๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0
๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0
๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0
.
.
. 0 0 0 d 0 0
๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0
๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ
]]]]]]]]]]]]
]
< 0,
(30)
where๐๐= ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐๐๐๐ด๐๐+ ๐๐ด๐
๐๐๐๐+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐+
โ๐
๐=1,๐ =๐๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐+ ๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐+ ๐๐ต๐๐๐+ ๐๐๐
๐๐ต๐
๐. In
practice, the feedback matrix๐พ๐has no imaginary part. So let
Im(๐๐) = 0; then ๐
๐= Re(๐
๐). According to the relationship
๐๐= ๐พ๐๐๐, the outputmatrix๐
๐has no imaginary part; that is,
8 Mathematical Problems in Engineering
Im(๐๐) = 0; then ๐
๐= Re(๐
๐). Substituting ๐ = cos ๐ + ๐ sin ๐
into๐๐gives
๐๐= cos ๐๐ด
๐๐๐๐+ cos ๐๐
๐๐ด๐
๐๐+ cos ๐๐ด
๐๐๐๐+ cos ๐๐
๐๐ด๐
๐๐
+ ๐ผ๐๐ท๐๐๐๐ท๐
๐๐๐+
๐
โ
๐=1,๐ =๐
๐พ๐๐๐ท๐๐๐๐ท๐
๐๐๐+ ๐ฝ๐๐ต๐๐ท๐๐๐ท๐
๐๐๐ต๐
๐
+ cos ๐๐ต๐๐๐+ cos ๐๐๐
๐๐ต๐
๐
+ ๐ (sin ๐๐ด๐๐๐๐โ sin ๐๐
๐๐ด๐
๐๐+ sin ๐๐ด
๐๐๐๐โ sin ๐๐
๐๐ด๐
๐๐
+ sin ๐๐ต๐๐๐โ sin ๐๐๐
๐๐ต๐
๐) .
(31)
Based on Lemma 7, the complex LMI (30) is transformed intothe real LMI. Consider
[[[[[[[[[[[[[[[[[[[[[[[[[
[
๐1๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐๐2๐
0 0 0 0 0 0
๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0 0 0 0 0 0 0 0
๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0 0 0 0 0 0 0 0
๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0 0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0 0 0 0 0 0 0 0
๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0 0 0 0 0 0 0 0
๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ 0 0 0 0 0 0 0
โ๐2๐
0 0 0 0 0 0 ๐1๐
๐๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐1๐๐๐
๐๐ธ๐
๐2๐โ โ โ ๐
๐
๐๐ธ๐
๐๐๐๐๐
๐๐ธ๐
๐๐
0 0 0 0 0 0 0 ๐ธ๐๐๐๐๐
โ๐ผ๐๐ผ 0 0 0 0 0
0 0 0 0 0 0 0 ๐ธ๐1๐๐๐
0 โ๐พ1๐๐ผ 0 0 0 0
0 0 0 0 0 0 0 ๐ธ๐2๐๐๐
0 0 โ๐พ2๐๐ผ 0 0 0
0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0
0 0 0 0 0 0 0 ๐ธ๐๐๐
๐๐
0 0 0 0 โ๐พ๐๐๐ผ 0
0 0 0 0 0 0 0 ๐ธ๐๐๐๐
0 0 0 0 0 โ๐ฝ๐๐ผ
]]]]]]]]]]]]]]]]]]]]]]]]]
]
< 0, (32)
where๐1๐= Re(๐
๐),๐2๐= Im(๐
๐).
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:
๐๐ผ
๐ฅ (๐ก)
๐๐ก๐ผ
= [๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก) +
๐
โ
๐=1,๐ =๐
[๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก)
+ ๐ต๐[๐พ๐+ ฮ๐พ๐] ๐ฅ๐(๐ก) ,
(33)
where ๐, ๐ = 1, 2, . . . ๐, ๐ผ = 1.5,๐ = 2, ๐ = ๐/3,
๐ด11= [
โ2.5 3.7
1.9 โ2] , ๐ด
12= [
3.5 1.2
1.2 3.5] ,
๐ด21= [
0.2 โ0.1
0.3 0.1] , ๐ด
22= [
1.8 1.2
โ1.2 0.8] ,
๐ต1= [
1
1] , ๐ต
2= [
0 15
15 25] ,
ฮ๐ด11= ๐ท๐11๐น๐11๐ธ๐11
= [0.4 0.2
1 0.3] [
sin (๐) 0
0 sin (๐)] [0.4 0.2
1 0.3] ,
ฮ๐ด22= ๐ท๐22๐น๐22๐ธ๐22
= [0.4 0.2
1 0.3] [
sin (๐) 0
0 sin (๐)] [0.3 0.5
0.1 0.5] ,
Mathematical Problems in Engineering 9
ฮ๐ด12= ๐ท๐12๐น๐12๐ธ๐12
= [0.5
0.3] [
sin (๐) 0
0 sin (๐)] [0.3 0.5
0.1 0.5] ,
ฮ๐ด21= ๐ท๐21๐น๐21๐ธ๐21
= [0.5
0.3] [
sin (๐) 0
0 sin (๐)] [0.4 0.2
1 0.3] .
(34)
Meanwhile, the following additive gain perturbations areconsidered:
ฮ๐พ1= ๐ท๐1๐น๐1๐ธ๐1= [0.5 0.3] [
sin (๐) 0
0 sin (๐)] [0.4 0.2
1 0.3] ,
ฮ๐พ2= ๐ท๐2๐น๐2๐ธ๐2= [
0.5 0
0 0.5] [
sin (๐) 0
0 sin (๐)] [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
๐พ1= [โ58.6565 โ61.7286] ,
๐พ2= [
โ0.2560 โ6.2675
โ1.7459 โ4.9510] .
(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:
๐๐ผ
๐ฅ (๐ก)
๐๐ก๐ผ
= [๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก) +
๐
โ
๐=1,๐ =๐
[๐ด๐๐+ ฮ๐ด๐๐] ๐ฅ๐(๐ก)
+ ๐ต๐[๐พ๐+ ฮ๐พ๐] ๐ฅ๐(๐ก) ,
(37)
where ๐, ๐ = 1, 2, . . . , ๐, ๐ผ = 1.5,๐ = 2, ๐ = ๐/3,
๐ด11= [
โ3.7 3.5
1.9 โ3] , ๐ด
12= [
0.5 0.2
0.2 0.5] ,
๐ด21= [
0.2 โ0.1
0.3 0.1] , ๐ด
22= [
1 0.2
โ0.2 0.8] ,
๐ต1= [
6.5
0] , ๐ต
2= [
0.5 0.5
0 1.5] ,
ฮ๐ด11= ๐ท๐11๐น๐11๐ธ๐11
= [0.4 0.2
1 0.3] [
sin (๐) 0
0 sin (๐)] [0.4 0.2
1 0.3] ,
0 1 2 3 4 5t
x1
x2
x3
x4
1
0.5
0
โ0.5
Figure 1: Example 1: time responses of the closed-loop fractionalorder large-scale interconnected uncertain system.
ฮ๐ด22= ๐ท๐22๐น๐22๐ธ๐22
= [0.4 0.2
1 0.3] [
sin (๐) 0
0 sin (๐)] [0.3 0.5
0.1 0.5] ,
ฮ๐ด12= ๐ท๐12๐น๐12๐ธ๐12
= [0.5
0.3] [
sin (๐) 0
0 sin (๐)] [0.3 0.5
0.1 0.5] ,
ฮ๐ด21= ๐ท๐21๐น๐21๐ธ๐21
= [0.5
0.3] [
sin (๐) 0
0 sin (๐)] [0.4 0.2
1 0.3] .
(38)
Meanwhile, the following multiplicative gain perturbationsare considered:
๐ท๐1= [10.5 10.5] , ๐ธ
๐1= [
2.2
2.2] ,
๐ท๐2= [
0.5 0
0 0.5] , ๐ธ
๐2= [
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
๐พ1= [โ0.0304 โ0.0119] , ๐พ
2= [
โ3.0473 2.8800
2.0568 โ2.2810] .
(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.
10 Mathematical Problems in Engineering
0 20 40 60 80 100t
x1
x2
x3
x4
1
1.5
0.5
0
โ0.5
โ1
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 < ๐ผ < 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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