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Research ArticleRobust Nonfragile Controllers Design forFractional Order Large-Scale Uncertain Systems witha Commensurate Order 1 < πΌ < 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 Β© 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 < πΌ < 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 [1β7].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[11β13]. 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 < πΌ < 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 [16β20]. The robust stability of fractional orderinterval systems has been investigated in [21, 22]. It is wellknown that Matignonβs stability theorem [16] is the basis forstability analysis of the fractional order system by checkingthe location of eigenvalues in the complex plane. Matignonβstheorem is in fact the pioneering works of stability analysisof the fractional order system. Based onMatignonβs theorem,the stability criteria of fractional order systems have beenproposed in both cases of 1 < πΌ < 2 and 0 < πΌ < 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 < πΌ <
2. Here, it should be also pointed out that [25] only focuseson the case of a fractional order 0 < πΌ < 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 π the conju-gate of the complex number π. π denotes the imaginary unit.πΌ denotes the identity matrix with appropriate dimensions.block diag denotes the block diagonal matrix. π π denotesthe π-dimensional Euclidean space and π πΓπ is the set of allπ Γ π real matrices. ππ denotes the transpose of π andπβ denotes the Hermitian transpose ofπ. 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 < πΌ < 2
composed ofπ fractional order subsystems:
ππΌ
π₯π(π‘)
ππ‘πΌ
= [π΄ππ+ Ξπ΄ππ] π₯π(π‘)
+
π
β
π=1,π =π
[π΄ππ+ Ξπ΄ππ] π₯π(π‘) + π΅
ππ’π(π‘) ,
(1)
where πΌ β π is the fractional commensurate order, π, π =
1, 2, . . . π, and π₯π(π‘) β π
ππ and π’π(π‘) β π
ππ are the stateand input of the πth fractional order subsystem, respectively.Assume that the nominal systems π΄
ππβ π ππΓππ , π΄
ππβ π ππΓππ ,
and π΅πβ π ππΓππ are constant and of appropriate dimensions
and the pair (π΄ππ,π΅π) is controllable. The fractional order
subsystems interact with each other through the intercon-nections βπ
π=1,π =ππ΄πππ₯π(π‘). The main objective of the note is
to find the decentralized local state feedback control law ofthe following form:
π’π(π‘) = (πΎ
π+ ΞπΎπ) π₯π(π‘) , π = 1, 2, . . . , π, (2)
such that the resulting fractional order closed-loop systemis asymptotically stable, where πΎ
πβ π ππΓππ is the state
feedback gainmatrix to be designed andΞπΎπ= π·πππΉπππΈππand
ΞπΎπ= π·πππΉπππΈπππΎπrepresent 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 Ξπ΄
ππ=
π·ππππΉππππΈπππ, πΉπππππΉπππ
β€ πΌ; Ξπ΄ππ= π·ππππΉππππΈπππ, πΉπππππΉπππ
β€ πΌ;ΞπΎπ= π·πππΉπππΈππ, πΉππππΉππβ€ πΌ; ΞπΎ
π= π·πππΉπππΈπππΎπ, πΉππππΉππβ€ πΌ,
where the elements are Lebesgue measurable and π·πππ, π·πππ,
π·ππ, π·ππ, πΈπππ, πΈπππ, πΈππ, and πΈ
ππare 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
ππΌ
π₯ (π‘)
ππ‘πΌ
= (π΄ + Ξπ΄) π₯ (π‘) + π΅ (πΎ + ΞπΎ) π₯ (π‘) , (3)
with the composite matrices π΄ andπΎ having the structure
π΄ =[[
[
π΄11
β β β π΄1π
.
.
. d...
π΄π1
β β β π΄ππ
]]
]
,
Ξπ΄ =[[
[
Ξπ΄11
β β β Ξπ΄1π
.
.
. d...
Ξπ΄π1
β β β Ξπ΄ππ
]]
]
,
πΎ = block diag [πΎ1, πΎ2β β β πΎπ] ,
ΞπΎ = block diag [ΞπΎ1, ΞπΎ2β β β ΞπΎ
π] .
(4)
Definition 1 (see [26]). For all nonzero real vectors π β π π,
π΄ β π πΓπ is real matrix; if the inequality πππ΄π < 0 holds,
then π΄ is said to be negative definite matrix.
Definition 2. The fractional order large-scale uncertain sys-tem can be stabilized via decentralized state feedback π’
π(π‘) =
(πΎπ+ΞπΎπ)π₯π(π‘) if there exists gainmatrixπΎ
πβ π ππΓππ such that
the closed-loop fractional order large-scale uncertain systemππΌ
π₯ (π‘)
ππ‘πΌ
= (π΄ + Ξπ΄) π₯ (π‘) + π΅ (πΎ + ΞπΎ) π₯ (π‘) (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 πΆ β π πΓπ, π΄, π΅ β π
πΓπ, π΄ β₯ π΅;then πΆππ΄πΆ β₯ πΆ
π
π΅πΆ.
Mathematical Problems in Engineering 3
Lemma 4 (see [27]). For any matrices π and π with appro-priate dimensions and for any π½ > 0, the following inequalityholds:
ππ
π + ππ
π < π½ππ
π + π½β1
ππ
π. (6)
Lemma5 (see [28]). The fractional order systemππΌ
π₯(π‘)/ππ‘πΌ
=
π΄π₯(π‘) with a commensurate order πΌ is asymptotically stableif | arg(spec(π΄))| > πΌ(π/2), where πΌ is the order of fractionalorder system and spec(π΄) is the spectrum of all eigenvalues ofπ΄.
Lemma 6 (see [23]). Let 1 < πΌ < 2 and π = (πΌ β 1)(π/2),π = πππ. The fractional order system π
πΌ
π₯(π‘)/ππ‘πΌ
= π΄π₯(π‘) witha commensurate order 1 < πΌ < 2 is asymptotically stable if andonly if there exist positive definite matrices π = π
π
β π πΓπ,
such that
[
[
(π΄π + ππ΄π
) sin π (π΄π β ππ΄π
) cos π
(ππ΄π
β π΄π) cos π (π΄π + ππ΄π
) sin π]
]
< 0, (7)
or equivalently, πππ΄ + ππ΄π
π < 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 < πΌ < 2 isconvex. By using Linear Matrix Inequalities (LMI) approach,it is obtained as the above LMI. Therefore, it is equivalent toπππ΄ + ππ΄
π
π < 0.
Lemma 7. A complex Hermitian matrixπ satisfiesπ < 0 ifand only if the following real LMI inequality holds:
[Re (π) Im (π)
β Im (π) Re (π)] < 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 < πΌ < 2. Let π =ππ(πΌβ1)(π/2). The fractional order large-scale uncertain systemcan be stabilized via decentralized state feedback π’
π(π‘) = (πΎ
π+
ΞπΎπ)π₯π(π‘) if there exist positive-definite block diagonal matrices
ππ= block diag[π
1, π2. . . , ππ], matrix π
π, and positive real
scalar constants πΌπ, πΎππ, π½π, π, π = 1, 2, . . . , π, such that the
following matrix inequalities hold:
[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[
[
π1π
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
πππ2π
0 0 0 0 0 0
πΈπππππ
βπΌππΌ 0 0 0 0 0 0 0 0 0 0 0 0
πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0 0 0 0 0 0 0 0
πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0 0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0 0 0 0 0 0 0 0
πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0 0 0 0 0 0 0 0
πΈππππ
0 0 0 0 0 βπ½ππΌ 0 0 0 0 0 0 0
βπ2π
0 0 0 0 0 0 π1π
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
ππ
0 0 0 0 0 0 0 πΈπππππ
βπΌππΌ 0 0 0 0 0
0 0 0 0 0 0 0 πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0
0 0 0 0 0 0 0 πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0
0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0
0 0 0 0 0 0 0 πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0
0 0 0 0 0 0 0 πΈππππ
0 0 0 0 0 βπ½ππΌ
]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]
]
< 0, (9)
whereππ= ππππ΄ππ+ ππ΄π
ππππ+ ππππ΄ππ+ ππ΄π
ππππ+ πΌππ·ππππ·π
πππ+
βπ
π=1,ππΎπππ·ππππ·π
πππ+ π½ππ΅ππ·πππ·π
πππ΅π
π+ ππ΅πππ+ πππ
ππ΅π
π, ππ=
πβ1
π, and π
1πand π
2πare the real part and imaginary
part of matrices ππ, respectively. Moreover, the stabilization
decentralized state-feedback gain matrix can be calculated asfollows: πΎ
π= ππππ.
Proof. Under decentralized state-feedback control law (2),the closed-loop fractional order large-scale interconnected
system is obtained asππΌ
π₯π(π‘)
ππ‘πΌ
= [π΄ππ+ Ξπ΄ππ] π₯π(π‘) +
π
β
π=1,π =π
[π΄ππ+ Ξπ΄ππ] π₯π(π‘)
+ π΅π[πΎπ+ ΞπΎπ] π₯π(π‘) ,
(10)
where π,π = 1, 2, . . . π. Based on Lemma 6, the necessaryand sufficient condition on the asymptotical stability of thefractional order system with order 1 < πΌ < 2 is that
4 Mathematical Problems in Engineering
πππ΄ + ππ΄π
π < 0. According to Definition 1, the sufficientcondition on the stabilization of fractional order large-scale interconnected system satisfies the following quadraticmatrix inequality:
ππ
{ππ [(π΄ + Ξπ΄) + π΅ (πΎ + ΞπΎ)]
+ π [(π΄ + Ξπ΄) + π΅ (πΎ + ΞπΎ)]π
π} π < 0,
(11)
π
β
π=1
{
{
{
πβ
π[ππππ΄ππ+ ππ΄π
ππππ+ πππΞπ΄ππ
+ πΞπ΄π
ππππ+ ππππ΅ππΎπ+ ππΎπ
ππ΅π
πππ
+ ππππ΅πΞπΎπ+ πππΞπΎπ
ππ΅π
π] ππ
+ 2ππβ
π
[
[
π
β
π=1,π =π
(π΄ππ+ Ξπ΄ππ)]
]
ππ
}
}
}
< 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
πππΞπ΄ππ+ πΞπ΄
π
ππππβ€ πΌππππ·ππππ·π
πππππ+ πΌβ1
ππΈπ
ππππΈπππ. (13)
Meanwhile, based on Lemmas 3 and 4, by means of enlargingthe inequality we have
ππππ΅πΞπΎπ+ πΞπΎ
π
ππ΅π
πππβ€ π½ππππ΅ππ·πππ·π
πππ΅π
πππ+ π½β1
ππΈπ
πππΈππ,
π
β
π=1
π
β
π=1,π =π
πβ
π[πππΞπ΄ππ+ πΞπ΄
π
ππππ] ππ
β€
π
β
π=1
π
β
π=1,π =π
πβ
ππΎπππππ·ππππ·π
πππππππ
+
π
β
π=1,π =π
π
β
π=1
πβ
ππΎβ1
πππΈπ
ππππΈπππππ
β€
π
β
π=1
π
β
π=1,π =π
πβ
ππΎπππππ·ππππ·π
πππππππ
+
π
β
π=1,π =π
π
β
π=1
πβ
ππΎβ1
πππΈπ
ππππΈπππππ.
(14)
Substituting (11), (12), and (13) into (12) results in thefollowing quadratic matrix inequality:
π
β
π=1
{
{
{
πβ
π
[
[
ππππ΄ππ+ ππ΄π
ππππ+ ππππ΄ππ+ ππ΄π
ππππ
+ ππππ΅ππΎπ+ ππΎπ
ππ΅π
πππ
+ πΌππππ·ππππ·π
πππππ+
π
β
π=1,π =π
πΎπππππ·ππππ·π
πππππ
+ πΌβ1
ππΈπ
ππππΈπππ+
π
β
π=1,π =π
πΎβ1
πππΈπ
ππππΈπππ
+ π½β1
ππΈπ
πππΈππ+ π½ππππ΅ππ·πππ·π
πππ΅π
πππ
]
]
ππ
}
}
}
< 0,
(15)
π
β
π=1
{
{
{
ππππ΄ππ+ ππ΄π
ππππ+ ππππ΄ππ+ ππ΄π
ππππ
+ ππππ΅ππΎπ+ ππΎπ
ππ΅π
πππ
+ πΌππππ·ππππ·π
πππππ+ ππ
π
β
π=1,π =π
(πΎπππ·ππππ·π
πππ) ππ
+ πΌβ1
ππΈπ
ππππΈπππ+
π
β
π=1,π =π
πΎβ1
πππΈπ
ππππΈπππ
+π½ππππ΅ππ·πππ·π
πππ΅π
πππ+ π½β1
ππΈπ
πππΈππ
}
}
}
< 0.
(16)
Let ππ= πβ1
πand π
π= πΎπππ. By premultiplying and post-
multiplying πβ1π
onto (16), one has
π
β
π=1
{
{
{
ππ΄πππβ1
π+ ππβ1
ππ΄π
ππ+ ππ΄πππβ1
π+ ππβ1
ππ΄π
ππ
+ ππ΅ππΎππβ1
π+ ππβ1
ππΎπ
ππ΅π
π+ πΌππ·ππππ·π
πππ
+
π
β
π=1,π =π
πΎπππ·ππππ·π
πππ+ πΌβ1
ππβ1
ππΈπ
ππππΈππππβ1
π
+
π
β
π=1,π =π
πΎβ1
πππβ1
ππΈπ
ππππΈππππβ1
π+ π½ππ΅ππ·πππ·π
πππ΅π
π
+π½β1
ππβ1
ππΈπ
πππΈπππβ1
π
}
}
}
=
π
β
π=1
{
{
{
ππ΄ππππ+ ππππ΄π
ππ+ ππ΄ππππ+ ππππ΄π
ππ
+ ππ΅πππ+ πππ
ππ΅π
π+ πΌππ·ππππ·π
πππ
+
π
β
π=1,π =π
πΎπππ·ππππ·π
πππ+ πΌβ1
ππππΈπ
ππππΈπππππ
Mathematical Problems in Engineering 5
+
π
β
π=1,π =π
πΎβ1
πππππΈπ
ππππΈπππππ
+ π½ππ΅ππ·πππ·π
πππ΅π
π+ π½β1
πππ
ππΈπ
πππΈππππ
}
}
}
< 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:
[[[[[[[[[[
[
ππ
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
ππ
πΈπππππ
βπΌππΌ 0 0 0 0 0
πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0
πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0
.
.
. 0 0 0 d 0 0
πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0
πΈππππ
0 0 0 0 0 βπ½ππΌ
]]]]]]]]]]
]
< 0,
(18)
where ππ= ππππ΄ππ+ ππ΄π
ππππ+ ππππ΄ππ+ ππ΄π
ππππ+ πΌππ·ππππ·π
πππ
+βπ
π=1,π =ππΎπππ·ππππ·π
πππ+ π½ππ΅ππ·πππ·π
πππ΅π
π+ ππ΅πππ+ πππ
ππ΅π
π. In
practice, the feedback matrixπΎπhas no imaginary part. So let
Im(ππ) = 0; then π
π= Re(π
π). According to the relationship
ππ= πΎπππ, the outputmatrixπ
πhas no imaginary part; that is,
Im(ππ) = 0; then π
π= Re(π
π). Substituting π = cos π + π sin π
intoππgives
ππ= cos ππ΄
ππππ+ cos ππ
ππ΄π
ππ+ cos ππ΄
ππππ+ cos ππ
ππ΄π
ππ
+ πΌππ·ππππ·π
πππ+
π
β
π=1,π =π
πΎπππ·ππππ·π
πππ+ π½ππ΅ππ·πππ·π
πππ΅π
π
+ cos ππ΅πππ+ cos πππ
ππ΅π
π
+ π (sin ππ΄ππππβ sin ππ
ππ΄π
ππ+ sin ππ΄
ππππβ sin ππ
ππ΄π
ππ
+ sin ππ΅πππβ sin πππ
ππ΅π
π) .
(19)
Based on Lemma 7, the complex LMI (18) is transformed intothe real LMI. Consider
[[[[[[[[[[[[[[[[[[[[[[[[[
[
π1π
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
πππ2π
0 0 0 0 0 0
πΈπππππ
βπΌππΌ 0 0 0 0 0 0 0 0 0 0 0 0
πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0 0 0 0 0 0 0 0
πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0 0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0 0 0 0 0 0 0 0
πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0 0 0 0 0 0 0 0
πΈππππ
0 0 0 0 0 βπ½ππΌ 0 0 0 0 0 0 0
βπ2π
0 0 0 0 0 0 π1π
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
ππ
0 0 0 0 0 0 0 πΈπππππ
βπΌππΌ 0 0 0 0 0
0 0 0 0 0 0 0 πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0
0 0 0 0 0 0 0 πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0
0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0
0 0 0 0 0 0 0 πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0
0 0 0 0 0 0 0 πΈππππ
0 0 0 0 0 βπ½ππΌ
]]]]]]]]]]]]]]]]]]]]]]]]]
]
< 0, (20)
whereπ1π= Re(π
π) andπ
2π= Im(π
π). 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 < πΌ < 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 < πΌ < 2. Let
6 Mathematical Problems in Engineering
π = ππ(πΌβ1)(π/2). The fractional order large-scale uncertain
system can be stabilized via decentralized state feedback π’π(π‘) =
(πΎπ+ ΞπΎ
π)π₯π(π‘) if there exist positive-definite block diagonal
matrices ππ
= block diag[π1, π2. . . , ππ], matrix π
πand
positive number πΌπ, πΎππ, π½π, π, π = 1, 2, . . . , π, such that the
following matrix inequalities hold:
[[[[[[[[[[[[[[[[[[[[[[[[[
[
π1π
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
πππ2π
0 0 0 0 0 0
πΈπππππ
βπΌππΌ 0 0 0 0 0 0 0 0 0 0 0 0
πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0 0 0 0 0 0 0 0
πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0 0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0 0 0 0 0 0 0 0
πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0 0 0 0 0 0 0 0
πΈππππ
0 0 0 0 0 βπ½ππΌ 0 0 0 0 0 0 0
βπ2π
0 0 0 0 0 0 π1π
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
ππ
0 0 0 0 0 0 0 πΈπππππ
βπΌππΌ 0 0 0 0 0
0 0 0 0 0 0 0 πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0
0 0 0 0 0 0 0 πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0
0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0
0 0 0 0 0 0 0 πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0
0 0 0 0 0 0 0 πΈππππ
0 0 0 0 0 βπ½ππΌ
]]]]]]]]]]]]]]]]]]]]]]]]]
]
< 0, (21)
whereππ= ππππ΄ππ+ ππ΄π
ππππ+ ππππ΄ππ+ ππ΄π
ππππ+ πΌππ·ππππ·π
πππ+
βπ
π=1,ππΎπππ·ππππ·π
πππ+ π½ππ΅ππ·πππ·π
πππ΅π
π+ ππ΅πππ+ πππ
ππ΅π
π, ππ=
πβ1
π, and π
1πand π
2πare the real part and imaginary
part of matrices ππ, respectively. Moreover, the stabilization
decentralized state-feedback gain matrix is given by πΎπ= ππππ.
Proof. By the means of decentralized state-feedback controllaw (2), the closed-loop fractional order large-scale intercon-nected system is obtained as
ππΌ
π₯π(π‘)
ππ‘πΌ
= [π΄ππ+ Ξπ΄ππ] π₯π(π‘) +
π
β
π=1,π =π
[π΄ππ+ Ξπ΄ππ] π₯π(π‘)
+ π΅π[πΎπ+ ΞπΎπ] π₯π(π‘) ,
(22)where π, π = 1, 2, . . . π, ΞπΎ
π= π·πππΉπππΈπππΎπ. Based on
Lemma 6, the necessary and sufficient condition on theasymptotical stability of the fractional order system withorder 1 < πΌ < 2 is that πππ΄ + ππ΄
π
π < 0.According to Definition 1, the sufficient condition on the
stabilization of fractional order large-scale interconnectedsystem satisfies the following quadratic matrix inequality:π
β
π=1
πβ
{ππ [(π΄ + Ξπ΄) + π΅ (πΎ + ΞπΎ)]
+ π[(π΄ + Ξπ΄) + π΅ (πΎ + ΞπΎ)]π
π} π < 0,
(23)
π
β
π=1
{
{
{
πβ
π[ππππ΄ππ+ ππ΄π
ππππ+ πππΞπ΄ππ+ πΞπ΄
π
ππππ
+ ππππ΅ππΎπ+ ππΎπ
ππ΅π
πππ+ ππππ΅πΞπΎπ+ πππΞπΎπ
ππ΅π
π] ππ
+ 2ππβ
π
[
[
π
β
π=1,π =π
(π΄ππ+ Ξπ΄ππ)]
]
ππ
}
}
}
< 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
πππΞπ΄ππ+ πΞπ΄
π
ππππβ€ πΌππππ·ππππ·π
πππππ+ πΌβ1
ππΈπ
ππππΈπππ. (25)
Likewise, it is obtained that
ππππ΅πΞπΎπ+ πΞπΎ
π
ππ΅π
πππ
= ππππ΅ππ·πππΉπππΈπππΎπ+ ππΎπ
ππΈπ
πππΉπ
πππ·π
πππ΅π
πππ
β€ π½ππππ΅ππ·πππ·π
πππ΅π
πππ+ π½β1
ππΎπ
ππΈπ
πππΈπππΎπ,
(26)
π
β
π=1
π
β
π=1,π =π
πβ
π[πππΞπ΄ππ+ πΞπ΄
π
ππππ] ππ
β€
π
β
π=1
π
β
π=1,π =π
πβ
ππΎπππππ·ππππ·π
πππππππ
+
π
β
π=1,π =π
π
β
π=1
πβ
ππΎβ1
πππΈπ
ππππΈπππππ
Mathematical Problems in Engineering 7
β€
π
β
π=1
π
β
π=1,π =π
πβ
ππΎπππππ·ππππ·π
πππππππ
+
π
β
π=1,π =π
π
β
π=1
πβ
ππΎβ1
πππΈπ
ππππΈπππππ.
(27)
Substituting (24), (25), and (26) into (24) results in the follow-ing quadratic matrix inequality, and we have
π
β
π=1
{
{
{
πβ
π
[
[
ππππ΄ππ+ ππ΄π
ππππ+ ππππ΄ππ+ ππ΄π
ππππ
+ ππππ΅ππΎπ+ ππΎπ
ππ΅π
πππ+ πΌππππ·ππππ·π
πππππ
+
π
β
π=1,π =π
πΎπππππ·ππππ·π
πππππ+ πΌβ1
ππΈπ
ππππΈπππ
+
π
β
π=1,π =π
πΎβ1
πππΈπ
ππππΈπππ+ π½β1
ππΎπ
ππΈπ
πππΈπππΎπ
+ π½ππππ΅ππ·πππ·π
πππ΅π
πππ
]
]
ππ
}
}
}
< 0,
π
β
π=1
{
{
{
ππππ΄ππ+ ππ΄π
ππππ+ ππππ΄ππ+ ππ΄π
ππππ
+ ππππ΅ππΎπ+ ππΎπ
ππ΅π
πππ
+ πΌππππ·ππππ·π
πππππ+ ππ
π
β
π=1,π =π
(πΎπππ·ππππ·π
πππ) ππ
+ πΌβ1
ππΈπ
ππππΈπππ+
π
β
π=1,π =π
πΎβ1
πππΈπ
ππππΈπππ
+ π½ππππ΅ππ·πππ·π
πππ΅π
πππ+ π½β1
ππΎπ
ππΈπ
πππΈπππΎπ
}
}
}
< 0.
(28)
Let ππ= πβ1
πand π
π= πΎπππ. By premultiplying and post-
multiplying πβ1π
onto (27), one has
π
β
π=1
{
{
{
ππ΄πππβ1
π+ ππβ1
ππ΄π
ππ+ ππ΄πππβ1
π+ ππβ1
ππ΄π
ππ
+ ππ΅ππΎππβ1
π+ ππβ1
ππΎπ
ππ΅π
π
+ πΌππ·ππππ·π
πππ+
π
β
π=1,π =π
πΎπππ·ππππ·π
πππ
+ πΌβ1
ππβ1
ππΈπ
ππππΈππππβ1
π
+
π
β
π=1,π =π
πΎβ1
πππβ1
ππΈπ
ππππΈππππβ1
π+ π½ππ΅ππ·πππ·π
πππ΅π
π
+ π½β1
ππβ1
ππΎπ
ππΈπ
πππΈπππΎππβ1
π
}
}
}
=
π
β
π=1
{
{
{
ππ΄ππππ+ ππππ΄π
ππ+ ππ΄ππππ
+ ππππ΄π
ππ+ ππ΅πππ+ πππ
ππ΅π
π
+ πΌππ·ππππ·π
πππ+
π
β
π=1,π =π
πΎπππ·ππππ·π
πππ
+ πΌβ1
ππππΈπ
ππππΈπππππ
+
π
β
π=1,π =π
πΎβ1
πππππΈπ
ππππΈπππππ
+π½ππ΅ππ·πππ·π
πππ΅π
π+ π½β1
πππ
ππΈπ
πππΈππππ
}
}
}
> 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:
[[[[[[[[[[[[
[
ππ
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
ππ
πΈπππππ
βπΌππΌ 0 0 0 0 0
πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0
πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0
.
.
. 0 0 0 d 0 0
πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0
πΈππππ
0 0 0 0 0 βπ½ππΌ
]]]]]]]]]]]]
]
< 0,
(30)
whereππ= ππππ΄ππ+ ππ΄π
ππππ+ ππππ΄ππ+ ππ΄π
ππππ+ πΌππ·ππππ·π
πππ+
βπ
π=1,π =ππΎπππ·ππππ·π
πππ+ π½ππ΅ππ·πππ·π
πππ΅π
π+ ππ΅πππ+ πππ
ππ΅π
π. In
practice, the feedback matrixπΎπhas no imaginary part. So let
Im(ππ) = 0; then π
π= Re(π
π). According to the relationship
ππ= πΎπππ, the outputmatrixπ
πhas no imaginary part; that is,
8 Mathematical Problems in Engineering
Im(ππ) = 0; then π
π= Re(π
π). Substituting π = cos π + π sin π
intoππgives
ππ= cos ππ΄
ππππ+ cos ππ
ππ΄π
ππ+ cos ππ΄
ππππ+ cos ππ
ππ΄π
ππ
+ πΌππ·ππππ·π
πππ+
π
β
π=1,π =π
πΎπππ·ππππ·π
πππ+ π½ππ΅ππ·πππ·π
πππ΅π
π
+ cos ππ΅πππ+ cos πππ
ππ΅π
π
+ π (sin ππ΄ππππβ sin ππ
ππ΄π
ππ+ sin ππ΄
ππππβ sin ππ
ππ΄π
ππ
+ sin ππ΅πππβ sin πππ
ππ΅π
π) .
(31)
Based on Lemma 7, the complex LMI (30) is transformed intothe real LMI. Consider
[[[[[[[[[[[[[[[[[[[[[[[[[
[
π1π
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
πππ2π
0 0 0 0 0 0
πΈπππππ
βπΌππΌ 0 0 0 0 0 0 0 0 0 0 0 0
πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0 0 0 0 0 0 0 0
πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0 0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0 0 0 0 0 0 0 0
πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0 0 0 0 0 0 0 0
πΈππππ
0 0 0 0 0 βπ½ππΌ 0 0 0 0 0 0 0
βπ2π
0 0 0 0 0 0 π1π
ππ
ππΈπ
πππππ
ππΈπ
π1πππ
ππΈπ
π2πβ β β π
π
ππΈπ
πππππ
ππΈπ
ππ
0 0 0 0 0 0 0 πΈπππππ
βπΌππΌ 0 0 0 0 0
0 0 0 0 0 0 0 πΈπ1πππ
0 βπΎ1ππΌ 0 0 0 0
0 0 0 0 0 0 0 πΈπ2πππ
0 0 βπΎ2ππΌ 0 0 0
0 0 0 0 0 0 0
.
.
. 0 0 0 d 0 0
0 0 0 0 0 0 0 πΈπππ
ππ
0 0 0 0 βπΎπππΌ 0
0 0 0 0 0 0 0 πΈππππ
0 0 0 0 0 βπ½ππΌ
]]]]]]]]]]]]]]]]]]]]]]]]]
]
< 0, (32)
whereπ1π= Re(π
π),π2π= Im(π
π).
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:
ππΌ
π₯ (π‘)
ππ‘πΌ
= [π΄ππ+ Ξπ΄ππ] π₯π(π‘) +
π
β
π=1,π =π
[π΄ππ+ Ξπ΄ππ] π₯π(π‘)
+ π΅π[πΎπ+ ΞπΎπ] π₯π(π‘) ,
(33)
where π, π = 1, 2, . . . π, πΌ = 1.5,π = 2, π = π/3,
π΄11= [
β2.5 3.7
1.9 β2] , π΄
12= [
3.5 1.2
1.2 3.5] ,
π΄21= [
0.2 β0.1
0.3 0.1] , π΄
22= [
1.8 1.2
β1.2 0.8] ,
π΅1= [
1
1] , π΅
2= [
0 15
15 25] ,
Ξπ΄11= π·π11πΉπ11πΈπ11
= [0.4 0.2
1 0.3] [
sin (π) 0
0 sin (π)] [0.4 0.2
1 0.3] ,
Ξπ΄22= π·π22πΉπ22πΈπ22
= [0.4 0.2
1 0.3] [
sin (π) 0
0 sin (π)] [0.3 0.5
0.1 0.5] ,
Mathematical Problems in Engineering 9
Ξπ΄12= π·π12πΉπ12πΈπ12
= [0.5
0.3] [
sin (π) 0
0 sin (π)] [0.3 0.5
0.1 0.5] ,
Ξπ΄21= π·π21πΉπ21πΈπ21
= [0.5
0.3] [
sin (π) 0
0 sin (π)] [0.4 0.2
1 0.3] .
(34)
Meanwhile, the following additive gain perturbations areconsidered:
ΞπΎ1= π·π1πΉπ1πΈπ1= [0.5 0.3] [
sin (π) 0
0 sin (π)] [0.4 0.2
1 0.3] ,
ΞπΎ2= π·π2πΉπ2πΈπ2= [
0.5 0
0 0.5] [
sin (π) 0
0 sin (π)] [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
πΎ1= [β58.6565 β61.7286] ,
πΎ2= [
β0.2560 β6.2675
β1.7459 β4.9510] .
(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:
ππΌ
π₯ (π‘)
ππ‘πΌ
= [π΄ππ+ Ξπ΄ππ] π₯π(π‘) +
π
β
π=1,π =π
[π΄ππ+ Ξπ΄ππ] π₯π(π‘)
+ π΅π[πΎπ+ ΞπΎπ] π₯π(π‘) ,
(37)
where π, π = 1, 2, . . . , π, πΌ = 1.5,π = 2, π = π/3,
π΄11= [
β3.7 3.5
1.9 β3] , π΄
12= [
0.5 0.2
0.2 0.5] ,
π΄21= [
0.2 β0.1
0.3 0.1] , π΄
22= [
1 0.2
β0.2 0.8] ,
π΅1= [
6.5
0] , π΅
2= [
0.5 0.5
0 1.5] ,
Ξπ΄11= π·π11πΉπ11πΈπ11
= [0.4 0.2
1 0.3] [
sin (π) 0
0 sin (π)] [0.4 0.2
1 0.3] ,
0 1 2 3 4 5t
x1
x2
x3
x4
1
0.5
0
β0.5
Figure 1: Example 1: time responses of the closed-loop fractionalorder large-scale interconnected uncertain system.
Ξπ΄22= π·π22πΉπ22πΈπ22
= [0.4 0.2
1 0.3] [
sin (π) 0
0 sin (π)] [0.3 0.5
0.1 0.5] ,
Ξπ΄12= π·π12πΉπ12πΈπ12
= [0.5
0.3] [
sin (π) 0
0 sin (π)] [0.3 0.5
0.1 0.5] ,
Ξπ΄21= π·π21πΉπ21πΈπ21
= [0.5
0.3] [
sin (π) 0
0 sin (π)] [0.4 0.2
1 0.3] .
(38)
Meanwhile, the following multiplicative gain perturbationsare considered:
π·π1= [10.5 10.5] , πΈ
π1= [
2.2
2.2] ,
π·π2= [
0.5 0
0 0.5] , πΈ
π2= [
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
πΎ1= [β0.0304 β0.0119] , πΎ
2= [
β3.0473 2.8800
2.0568 β2.2810] .
(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
β0.5
β1
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 < πΌ < 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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