# Synthesis of Decentralized Variable Gain Robust Controllers for Large-Scale Interconnected Systems with Structured Uncertainties

Authors: Shunya Nagai; Hidetoshi Oya
Journal: Journal of Control Science and Engineering
 (2014)

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## Abstract

In this paper, we propose a decentralized variable
gain robust controller which achieves not only robust stability
but also satisfactory transient behavior for a class of uncertain
large-scale interconnected systems. For the uncertain large-scale
interconnected system, the uncertainties and the interactions
satisfy the matching condition. The proposed decentralized robust
controller consists of a fixed feedback gain controller and a
variable gain one determined by a parameter adjustment law. In
this paper, we show that sufficient conditions for the existence
of the proposed decentralized variable gain robust controller are
given in terms of LMIs. Finally, a simple numerical example is
included.


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## Body

## 1. Introduction

To design control systems, it is necessary to derive a mathematical model for controlled systems. However, there always exist some gaps between the mathematical model and the controlled system; that is, uncertainties between actual systems and mathematical models are unavoidable. Hence for dynamical systems with uncertainties, so-called robust controller design methods have been well studied in the past thirty years, and there are a large number of studies for linear uncertain dynamical systems (e.g., see [1, 2] and references therein). In particular for uncertain linear systems, several quadratic stabilizing controllers and H∞ one have been suggested (e.g., [3–5]), and a connection between H∞ control and quadratic stabilization has also been established [6]. In addition, several design methods of variable gain controllers for uncertain continuous-time systems have been shown (e.g., [7–9]). These robust controllers are composed of a fixed gain controller and a variable gain one, and variable gain controllers are tuned by updating laws. Especially, in Oya and Hagino [8] the error signal between the desired trajectory and the actual response is introduced and the variable gain controller is determined so as to reduce the effect of uncertainties.On the other hand, the decentralized control for large-scale interconnected systems has been widely studied, because large-scale interconnected systems can be seen in such diverse fields as economic systems, electrical systems, and so on. A large number of results in decentralized control systems can be seen in Šiljak [10]. A framework for the design of decentralized robust model reference adaptive control for interconnected time-delay systems has been considered in the work of Hua et al. [11] and decentralized fault tolerant control problem has also been studied [12]. Additionally, there are many existing results for decentralized robust control for large-scale interconnected systems with uncertainties (e.g., [13–16]). In Mao and Lin [13] for large-scale interconnected systems with unmodelled interaction, the aggregative derivation is tracked by using a model following technique with online improvement, and a sufficient condition for which the overall system when controlled by the completely decentralized control is asymptotically stable has been established. Gong [15] has proposed decentralized robust controllers which guarantee robust stability with prescribed degree of exponential convergence. Furthermore, some design methods of decentralized guaranteed cost controllers for uncertain large-scale interconnected systems have also been suggested [17, 18].In this paper, on the basis of the existing results [8, 9], we propose a decentralized variable gain robust controller for a class of uncertain large-scale interconnected systems; that is, this study is an extension of the previous studies in this field. For the uncertain large-scale interconnected systems, uncertainties and interactions satisfy the matching condition. The proposed decentralized variable gain robust control input is defined as a state feedback with a fixed feedback gain matrix designed by using nominal subsystem and an error signal feedback with a fixed compensation gain matrix and a variable one determined by a parameter adjustment law. In this paper, LMI-based sufficient conditions for the existence of the proposed decentralized variable gain robust controller are derived. One can see that the crucial difference between the existing results and our new one is that the derived LMI conditions in this paper are always feasible; that is, the proposed decentralized variable gain robust control system can always be designed. Additionally, the number of LMIs needed to be solved equals the number of subsystems and is less than the conventional decentralized robust control with a fixed feedback gain matrix based on Lyapunov stability criterion. Therefore, the proposed decentralized robust control scheme is very useful.This paper is organized as follows. We show notations and useful lemmas which are used in this paper in Section2, and in Section 3, the class of uncertain large-scale interconnected systems under consideration is presented, and uncertain error subsystems and a decentralized variable gain robust control input are introduced. Section 4 contains the main results; that is, LMI-based sufficient conditions for the existence of the proposed decentralized variable gain robust controller are derived. Finally, simple illustrative examples are included to show the effectiveness of the proposed decentralized variable gain robust control system.


## 2. Preliminaries

In this section, notations and useful and well-known lemmas (see [19, 20] for details) which are used in this paper are shown.In the paper, the following notations are used. For a matrixX, the inverse of matrix X and the transpose of one are denoted by X-1 and XT, respectively. Additionally He{X} and In mean X+XT and n-dimensional identity matrix, respectively, and the notation diag⁡(X1,…,XM) represents a block diagonal matrix composed of matrices Xi for i=1,…,M. For real symmetric matrices X and Y, X>Y(resp.,X≥Y) means that X-Y is positive (resp., nonnegative) definite matrix. For a vector α∈Rn, ||α|| denotes standard Euclidian norm and,for a matrix X, ||X|| represents its induced norm. The symbols “≜” and “⋆” mean equality by definition and symmetric blocks in matrix inequalities, respectively.Lemma 1.
For arbitrary vectorsλ and ξ and the matrices G and H which have appropriate dimensions, the following inequality holds: (1)2λTGΔ(t)Hξ≤2∥GTλ∥∥Hξ∥,
where Δ(t) with appropriate dimensions is a time-varying unknown matrix satisfying ∥Δ(t)∥≤1.0.Lemma 2 (Schur complement).
For a given constant real symmetric matrixΞ, the following items are equivalent: (i)
Ξ=(Ξ11Ξ12Ξ12TΞ22)>0,(ii)
Ξ11>0 and Ξ22-Ξ12TΞ11-1Ξ12>0,(iii)
Ξ22>0 and Ξ11-Ξ12Ξ22-1Ξ12T>0.


## 3. Problem Formulation

Consider the uncertain large-scale interconnected system composed ofN subsystems described as
(2)ddtxi(t)=Aii(θii)xi(t)+∑j=1j≠iN‍Aij(θij)xj(t)+Biui(t),
where xi(t)∈Rni and ui(t)∈Rmi(i=1,…,N) are the vectors of the state and the control input for the ith subsystem, respectively, and the vector x(t)=(x1T(t),…,xNT(t))T is the state of the overall system. The matrices Aii(t) and Aij(t) in (2) are given by
(3)Aii(θii)=Aii+∑k=1pii‍θii(k)(t)BiEii(k),Aij(θij)=BiDij+∑k=1pij‍θij(k)(t)BiEij(k);
that is, the uncertainties and the interaction terms satisfy the matching condition. In (2) and (3), the matrices Aii∈Rni×ni,Aij∈Rni×nj, and Bi∈Rni×mi denote the nominal values of the system, and the matrices Dij, Eii(k), and Eij(k) with appropriate dimensions represent the structure of interactions or uncertainties. Besides, θii(k)(t)∈R1 and θij(k)(t)∈R1 denote unknown parameters which belong to the following ellipsoidal sets:
(4)Δii≜{θii∈Rpii∣θiiT(t)Σii-1θii(t)≤1},Δij≜{θij∈Rpij∣θijT(t)Σij-1θij(t)≤1}.
In (4), Σii∈Rpii×pii and Σij∈Rpij×pij represent the size of the ellipsoid and are defined as Σii=diag⁡(σii(1),…,σii(pii)) and Σij=diag⁡(σij(1),…,σij(pij)), respectively, where σii(k) and σij(k) are positive constant; that is, matrices Σii and Σij are symmetric positive definite. Besides, θii(t) and θij(t) are the unknown parameter vectors given by
(5)θii(t)=(θii(1)(t),…,θii(pii)(t))T,θij(t)=(θij(1)(t),…,θij(pij)(t))T.Now, the nominal subsystem, ignoring the unknown parameters and the interactions in (2), is given by
(6)ddtx¯i(t)=Aiix¯i(t)+Biu¯i(t),
where x¯i(t)∈Rni and u¯i(t)∈Rmi are the vectors of the state and the control input for the ith nominal subsystem, respectively.First of all, in order to generate the desired trajectory in time response for the uncertainith subsystem of (2) systematically, we adopt the standard linear quadratic control problem for the ith nominal subsystem of (6). Of course, one can adopt some other design method for deriving the desirable response (e.g., pole placement). It is well known that the optimal control input for the ith nominal subsystem of (6) can be obtained as
(7)u¯i(t)=Kix¯i(t),Ki=-Ri-1BiTXi.
In (7), Xi∈Rni×ni is a symmetric positive define matrix which satisfies the algebraic Riccati equation
(8)He{AiiTXi}-XiBiRi-1BiTXi+Qi=0,
where the weighting matrices Qi∈Rni×ni and Ri∈Rmi×mi are positive definite and are determined in advance so that the desirable transient behavior is achieved.Based on the existing result [8], we introduce an error vector ei(t)≜xi(t)-x¯i(t). Besides, for the ith subsystem of (2), using the feedback gain matrix Ki∈Rmi×ni of (7), we define the following control input:
(9)ui(t)≜Kixi(t)+vi(t),
where vi(t)∈Rmi is the compensation input [8, 9] defined as
(10)vi(t)≜Fiei(t)+Li(xi,ei,t)ei(t).
In (10), Fi∈Rmi×ni and Li(xi,ei,t)∈Rmi×ni are the fixed compensation gain matrix and the variable one for the ith subsystem of (2). From (2), (3), (6), (9), and (10), the uncertain error subsystem is derived as
(11)ddtei(t)=(AKi+BiFi)ei(t)+∑k=1pii‍θi(k)(t)BiEii(k)xi(t)+Bi∑j=1j≠iN‍(Dij+∑k=1pij‍θij(k)(t)Eij(k))xj(t)+BiLi(xi,ei,t)ei(t).
In (11), AKi∈Rni×ni is the stable matrix given by AKi=Aii+BiKi.From the above discussion, our design objective in this paper is to determine the decentralized variable gain robust control of (9) such that the resultant overall system achieves not only robust stability but also satisfactory transient behavior, that is, to design the fixed gain matrix Fi∈Rmi×ni and the variable one Li(xi,ei,t)∈Rmi×ni such that asymptotical stability of the overall error system composed of N error subsystems of (11) is guaranteed.


## 4. Decentralized Variable Gain Controllers

The following theorem shows sufficient conditions for the existence of the proposed decentralized control system.Theorem 3.
Consider the uncertain error subsystem of (11) and the control input of (9).
By using symmetric positive definite matricesYi∈Rni×ni and Si∈Rni×ni, matrices Wi∈Rmi×ni, and positive constants ϵi which satisfy the LMIs
(12)(He{AKiYi+BiWi}Λ(Yi)⋆-Π(ϵi))<0,(13)(He{SiAKi}Ψi⋆-Π(ϵi))<0
the fixed gain matrix Fi∈Rmi×ni and the variable one Li(xi,ei,t)∈Rmi×ni are determined as Fi=WiYi-1 and (14), respectively:(14)Li(xi,ei,t)≜{-ζi(xi,ei,t)+ηi(ei,t)∥BiTPiei(t)∥2BiTPiLi(xi,ei,tϵ).
In (12)–(14), matrices Λ(Yi), Ψi, Π(ϵi), Γ(xi,t), ζ(xi,ei,t), and η(ei,t) are given by
(15)Λ(Yi)≜(YiD1iTYiG1iT⋯YiDi-1iTYiGi-1iT121YiDi+1iTYiGi+1iT⋯YiDNiTYiGNiT),(16)Ψi≜(D1iTG1iT⋯Di-1iTGi-1iT2222Di+1iTGi+1iT⋯DNiTGNiT),(17)Π(ϵi)≜diag⁡(Π1(ϵ1),Π2(ϵ2),…,vvvvv.vΠi-1(ϵi-1),Πi+1(ϵi+1),…,vvvvvvΠN(ϵN))(18)Πk(ϵk)≜diag⁡(ϵkImk,ϵkIqk)(k=1,…,N),(19)Γii(xi,t)≜(Eii(1)xi(t)⋯Eii(pii)xi(t)),(20)ζi(xi,ei,t)=∥Σii1/2ΓiiT(xi,t)BiTPiei(t)∥,(21)ηi(ei,t)=ϵi(1+σij*)(N-1)∥BiTPiei(t)∥2,
where the matrices Gij in (15) and scalars σij* with appropriate dimensions in (18) are given by
(22)σij*≜∑j=1j≠iN‍∑k=1pij‍σij(k),GijTGij≜∑k=1pij(Eij(k))TEij(k).
Note that since the matrix (Eij(k))TEij(k) is symmetric semipositive definite, there always exist the matrices Gij. Besides, tϵ in (14) is given by tϵ=lim⁡ϵ>0,ϵ→0(t-ϵ) [7].
Then the overall error system composed of theN error subsystems of (11) is robustly stable.Proof.
In order to prove Theorem3, let us define the following quadratic function:
(23)V(t)≜∑i=1N‍Vei(t)+∑i=1N‍Vx¯i(t),
where Vei(t) and Vx¯i(t) are given by
(24)Vei(t)≜eiT(t)Piei(t),(25)Vx¯i(t)≜x¯iT(t)Six¯i(t).
Since the state vector xi(t) can be written as xi(t)=ei(t)+x¯i(t), we can obtain the following relation for the time derivative of the quadratic functions Vei(t) of (24):
(26)ddtVei(t)=eiT(t)[He{(AKi+BiFi)TPi}]ei(t)+2eiT(t)Pi(∑k=1pii‍θi(k)(t)BiEii(k)xi(t))+2eiT(t)PiBi{∑j=1j≠iN‍(Dij+∑k=1pij‍θij(k)(t)Eij(k))VVVVVVVVVVVv×(ej(t)+x¯j(t))∑j=1j≠iN‍(Dij+∑k=1pij‍θij(k)(t)Eij(k))∑j=1j≠iN‍}+2eiT(t)PiBiLi(t)ei(t)=eiT(t)[He{(AKi+BiFi)TPi}]ei(t)+2eiT(t)PiBiΓii(xi,t)θii(t)+2eiT(t)PiBi(∑j=1j≠iN‍Dij(ej(t)+x¯j(t)))+2eiT(t)PiBi×{∑j=1j≠iN‍∑k=1pij‍θij(k)(t)Eij(k)(ej(t)+x¯j(t))}+2eiT(t)PiBiLi(t)ei(t).
Besides, by using Lemma 1, assumptions of (4) for the unknown parameters θii(t) and θij(t), and the well-known inequality for any vectors α and β with appropriate dimensions and a positive scalar δ(27)2αTβ≤δαTα+1δβTβ,
we have the following relation:
(28)ddtVei(t)≤eiT(t)[He{(AKi+BiFi)TPi}]ei(t)+2∥Σii1/2ΓiiT(xi,t)BiTPiei(t)∥+2ϵi(N-1)eiT(t)PiBiBiTPiei(t)+2σij*ϵi(N-1)eiT(t)PiBiBTPiei(t)+1ϵi∑j=1j≠iN‍ejT(t)DijTDijej(t)+1ϵi∑j=1j≠iN‍ejT(t){∑k=1pij‍(Eij(k))TEij(k)}ej(t)+1ϵi∑j=1j≠iN‍x¯jT(t)DijTDijx¯j(t)+1ϵi∑j=1j≠iN‍x¯jT(t){∑k=1pij‍(Eij(k))TEij(k)}x¯j(t)+2eiT(t)PiBiLi(t)ei(t).
Therefore from the definition of the matrices Gij in (22), the inequality of (28) can be rewritten as
(29)ddtVei(t)≤eiT(t)[He{(AKi+BiFi)TPi}]ei(t)+2∥Σii1/2ΓiiT(xi,t)BiTPiei(t)∥+2ϵi(N-1)eiT(t)PiBiBiTPiei(t)+2σij*ϵi(N-1)eiT(t)PiBiBTPiei(t)+1ϵi∑j=1j≠iN‍ejT(t)(DijTDij+GijTGij)ej(t)+1ϵi∑j=1j≠iN‍x¯jT(t)(DijTDij+GijTGij)x¯j(t)+2eiT(t)PiBiLi(t)ei(t).
Furthermore, one can easily see that the following relation for the quadratic functionsVx¯i(t) of (25) is satisfied:
(30)ddtVx¯i(t)=x¯iT(t)[He{AKiTSi}]x¯i(t).
Firstly, we consider the case ofBiTPiei(t)≠0. In this case, substituting the variable gain matrix of (14) into (29) and some algebraic manipulations give the inequality
(31)ddtVei(t)≤eiT(t)[He{(AKi+BiFi)TPi}]ei(t)+1ϵi∑j=1j≠iN‍ejT(t)(DijTDij+GijTGij)ej(t)+1ϵi∑j=1j≠iN‍x¯jT(t)(DijTDij+GijTGij)x¯j(t).
Thus, from the definition of the quadratic function V(t) of (23), the following relation can be obtained:
(32)ddtV(t)≤∑i=1N‍eiT(t)[He{(AKi+BiFi)TPi}]ei(t)+∑i=1N‍x¯iT(t)[He{AKiTSi}]x¯i(t)+∑i=1N‍{1ϵi∑j=1j≠iN‍ejT(t)(DijTDij+GijTGij)ej(t)vvvvvvv+1ϵi∑j=1j≠iN‍x¯jT(t)(DijTDij+GijTGij)x¯j(t)}.
The inequality of (32) can be rewritten as
(33)ddtV(t)≤∑i=1N‍eiT(t)[∑i=1N‍∑j=1j≠iN‍He{(AKi+BiFi)TPi}vvvvvvvvv+∑j=1j≠iN‍1ϵj(DjiTDji+GjiTGji)]ei(t)+∑i=1N‍x¯iT(t)[∑i=1N‍∑j=1j≠iN‍He{AKiTSi}vvvvvvvvvvv+∑j=1j≠iN‍1ϵj(DjiTDji+GjiTGji)]x¯i(t).
If the matrix inequalities
(34)He{(AKi+BiFi)TPi}+∑j=1j≠iN‍1ϵj(DjiTDji+GjiTGji)<0,(35)He{AKiTSi}+∑j=1j≠iN‍1ϵj(DjiTDji+GjiTGji)<0
hold, then the following inequality for the time derivative of the quadratic function V(t) of (23) is satisfied:
(36)ddtV(t)<0,∀χ(t)≠0,
where χ(t)≜(e1T(t),…,eNT(t),x¯1T(t),…,x¯NT(t))T.
Next we consider the case ofBiTPiei(t)=0. In this case, one can see from (26) and (30) that the time derivative of the quadratic function V(t) of (23) can be written as
(37)ddtV(t)=∑i=1N‍eiT(t)[He{(AKi+BiFi)TPi}]ei(t)+∑i=1N‍x¯iT(t)[He{AKiTSi}]x¯i(t).
Namely, in the case of BiTPiei(t)=0, the relation of (36) is also satisfied, provided that the inequalities of (34) and (35) hold.
From the above, the overall error system is clearly guaranteed to be stable, because all nominal subsystems are asymptotically stable.
Finally, we consider the inequalities of (34) and (35). By introducing the matrices Yi≜Pi-1 and Wi≜FiPi and pre- and postmultiplying both sides of the matrix inequality of (34) by Yi, we have the inequality
(38)He{AKiYi+BiWi}+∑j=1j≠iN‍1ϵjYi(DjiTDji+GjiTGji)Yi<0.
Thus by applying Lemma 2 (Schur complement) to (35) and (38), we find that the inequalities of (35) and (38) are equivalent to the LMIs of (13) and (12), respectively.
Therefore the proof of Theorem3 is accomplished.Remark 4.
The positive constantsϵi in LMIs of (12) and (13) are included in (2,2)-block only. Namely, the positive constants ϵi are independent of LMI variables Yi and Wi. Thus the positive constants ϵi can arbitrarily be selected and one can see that the LMIs of (12) and (13) are always feasible. Thus by using the solution of the LMIs of (12) and (13) the fixed compensation gain matrix is determined as Fi=WiYi-1 and the variable one is given by (14).Remark 5.
The proposed decentralized robust controller design method is adaptable when some assumptions are satisfied. Namely, if the matching condition for uncertainties and interactions is satisfied, then the LMIs of (12) and (13) are feasible; that is, the proposed decentralized variable gain robust controller is applicable. Besides, in the proposed controller design method, the number of LMIs needed to be solved is N.
On the other hand, the number of the inequalities needed to be solved in the conventional decentralized robust control with fixed gain matrices is “2M”, provided that the number of unknown parameters included in the overall system is M (see the appendix). Therefore, one can see that the proposed controller design method is very useful.


## 5. Numerical Examples

In order to demonstrate the efficiency of the proposed robust controller, we have run a simple example. The control problem considered here is not necessarily practical. However, the simulation results stated below illustrate the distinct feature of the proposed decentralized robust controller.In this example, we consider the uncertain large-scale interconnected systems consisting of three two-dimensional subsystems; that is,N=3. The system parameters are given as follows:
(39)A11=(-1.01.00.01.0),A22=(0.01.0-1.0-1.0),A33=(1.00.0-1.0-3.0),B1=(0.01.0),B2=(1.01.0),B3=(1.00.0),(40)E11(1)T=(1.00.0),E11(2)T=(0.01.0),E22(1)T=(1.00.0),E22(2)T=(0.01.0),E33(1)T=(0.01.0),E33(2)T=(1.00.0),(41)D12T=(1.00.0),D13T=(2.01.0),D21T=(1.02.0),D23T=(0.03.0),D31T=(2.02.0),D32T=(1.01.0),E12(1)T=(1.00.0),E12(2)T=(0.01.0),E13(1)T=(1.00.0),(42)E13(2)T=(0.01.0),E21(1)T=(1.00.0),E21(2)T=(0.01.0),E23(1)T=(1.00.0),E23(2)T=(0.01.0),E31(1)T=(0.01.0),E31(2)T=(1.00.0),E32(1)T=(0.01.0),E32(2)T=(0.01.0).Firstly, we choose the weighting matricesQi∈R2×2 and Ri∈R1×1(i=1,…,3) for the nominal subsystems such as Q1=diag⁡(1.0,2.0), Q2=diag⁡(1.0,1.0×101), Q3=I3, R1=1.0, R2=1.0×101, and R3=1.0×101. Besides, the design parameters Σii and Σij are selected as Σii=diag⁡(3.0,2.0) and Σij=diag⁡(4.0,3.0). By solving the algebraic Riccati equation of (8) the following optimal gain matrices Ki∈R1×2 for the ith nominal subsystem are derived:
(43)K1=(-1.71572×10-1-2.82843),K2=(-4.45866-3.42538)×10-1,K3=(-2.052724.11182×10-3).Next, by using Theorem3 we design the proposed decentralized variable gain robust controller. By solving LMIs of (12) and (13), we have positive definite matrices Yi∈R2×2, matrices Wi∈R1×2, and positive scalars ϵi given by
(44)Y1=(6.1524-2.8845⋆1.1481×101),Y2=(3.77681.5321⋆5.3977),Y3=(7.4690-1.8692⋆7.1578),W1T=(-1.2512×101-1.8741),W2T=(-8.6852-8.9201),W3T=(-1.5089×101-2.1731×101),ϵ1=1.5697×101,ϵ2=1.2076×101,ϵ3=2.8810×101.
Thus the fixed gain matrices Fi∈R1×2 can be computed as
(45)F1=(-2.3919-7.6416×10-1),F2=(-1.8413-1.1300),F3=(-2.9744-3.8127).Note that the number of LMIs needed to be solved in the conventional decentralized robust control with a fixed gain matrix is29=512 (see the appendix). Namely, it is difficult to design the conventional fixed gain decentralized robust controller comparing with the proposed design method of the decentralized variable gain robust control.The result of this example is shown in Figures1, 2, 3, and 4. In this example, initial values of the uncertain large-scale system with system parameters of (39)–(42) and the nominal system are selected as follows:
(46)x(0)=(1.0-1.02.00.01.0-2.0)T,x¯(0)=(1.0-1.52.250.01.5-1.5)T.Figure 1
Time histories ofx1(t) and x¯1(t).Figure 2
Time histories ofx2(t) and x¯2(t).Figure 3
Time histories ofx3(t) and x¯3(t).Figure 4
Time histories ofui(t) and u¯i(t).In these figures,xi(l)(t) and x¯i(l)(t) denote the lth element of xi(t) for ith subsystem and one of the state x¯i(t) for ith nominal subsystem, respectively. Furthermore, unknown parameters are given as
(47)θii(t)=(3.0sin⁡(5.0πt)-2.0cos⁡⁡(2.0πt))T,θij(t)=(2.0cos⁡⁡(3.0πt)3.0sin⁡(4.0πt))T.From these figures, the proposed decentralized variable gain controller stabilizes the uncertain large-scale systems with system parameters of (39)–(42) in spite of uncertainties and interactions. Besides, the proposed variable gain robust control system achieves good transient response close to desired transient behavior generated by the nominal subsystem. Therefore, the effectiveness of the proposed decentralized robust control system has been shown.


## 6. Conclusions

In this paper, on the basis of the work of Oya and Hagino [8], we have suggested a decentralized variable gain robust controller for a large-scale interconnected system with uncertainties. For the uncertain large-scale interconnected systems, we have presented an LMI-based design method of the proposed decentralized variable gain robust controller. In addition, the effectiveness of the proposed decentralized robust controller has been shown by simple numerical examples. One can easily find that since the parameters ϵi in LMIs of (12) and (13) can arbitrarily be selected, these LMIs can be easily solved. Namely, the proposed design approach is very useful comparing with the conventional decentralized robust control.In the future research, we will extend the proposed controller synthesis to such a broad class of systems as large-scale systems with general uncertainties, uncertain large-scale systems with time delays, and so on.



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