# Observer-Based Decentralized Control for Uncertain Interconnected Systems of Neutral Type

Authors: Heli Hu; Dan Zhao; Qingling Zhang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The observer-based decentralized control problem is investigated for a class of uncertain interconnected systems of neutral type. Using the singular value decomposition approach, a full-order observer is designed to guarantee the asymptotic stability of the error dynamic system. A novel mathematical technique is developed to solve this design problem. Sufficient condition for uncertain interconnected systems of neutral type to be asymptotic stable is established based on the singular value decomposition method. Furthermore, the desired gains of observer and controller are obtained by the explicit expressions in terms of some free parameters. Finally, an illustrative example is used to demonstrate the proposed approach, and the corresponding simulation results are given to elucidate the effectiveness.


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## Body

## 1. Introduction

Nowadays the systems have become more and more large, and for the interconnected systems decentralized control has obvious advantage that overcomes the limitations of the traditional centralized control requiring sufficiently large communication bandwidth to exchange information between the subsystems. Therefore, the decentralized control scheme [1–11] only making use of local information available is very popular among the researchers and the engineers. In [1], an algorithm formulated within the convex optimization framework is proposed to investigate the strict dissipativity of the linear interconnected systems. A decentralized structure of dissipative state-feedback controllers is designed. In [2, 3], a decentralized adaptive output-feedback stabilizer and a decentralized L1 adaptive controller are proposed to stabilize a class of large-scale nonlinear systems, respectively. Different from the constant delays involved in the considered system without a priori knowledge of subsystem high-frequency-gain signs in [2], the interconnected nonlinearities and unmodeled dynamics are included in the considered systems in [3]. Based on the concepts of dynamic graphs and dynamic adjacency matrix, a modeling method of a complex dynamic interconnected system is considered in [4]. In [5], a decentralized dynamic output feedback based on linear controller is proposed to robust stabilize a class of nonlinear interconnected systems coupled by nonlinear interconnections that are unknown and quadratically bounded. However, neither constant delay nor time-varying delays are considered in [4, 5].On the other hand, the neutral system is the general form of delay system that contains the same highest order derivatives for the state vectorx(t), at both time t and past time(s) ts≤t. Many models of practical systems can be described by functional differential equation of neutral type [12]. Physical examples for neutral system have distributed networks [13], population ecology [14], heat exchangers, robots in contact with rigid environments [15], and so forth. In recent years, the stability analysis and robust control problems of neutral delay systems have been considered extensively (see, e.g., [16–23]). In [16–18], the stability problems of neutral systems are investigated. The difference among them is that Balasubramaniam et al. [16] focus on the stability of the neutral systems with both constant and time-varying delays using the method of nonuniformly dividing the whole delay interval into multiple segments, Rakkiyappan et al. [17] study the stability of the neutral systems with interval time-varying delays and nonlinear perturbations using the method of a new Lyapunov functional approach, and Nian et al. [18] deal with the stability of neutral systems with only a constant delay using the method of state matrix decomposition. However, the design problems of the control law have not been considered in them.Furthermore, Ma et al. [19] develop a control method for neutral systems with a single input and some restrictions on the system matrices using a differential-difference inequality and the transformation technique. Han et al. [20] utilize a discretized Lyapunov-Krasovskii functional approach to investigate the stability of the linear neutral systems with small and nonsmall discrete delays. But its discrete application to control design yields nonlinear conditions, which may not be easily computable. In [21], the robust adaptive stabilization problem is investigated for neutral time-delay systems with uncertainties, and an adaptive scheme is introduced to estimate the bounds on uncertainties. But the matched condition is required for the disturbance vector of the considered system. It is worth pointing out that many researches on the neutral systems are often restricted to the stability analysis without controller or static state feedback control schemes.Nowadays, observer control is also an attractive topic [24–28]. In [24], two controllers based on state feedback and observer output feedback are designed for networked systems with discrete and distributed delays subject to quantization and packet dropout. Also, a compensation scheme is proposed to deal with the effect of random packet dropout through communication network. In [25], Su et al. introduce a new model transformation for considered discrete-time T-S fuzzy systems to realize the design of dynamic output-feedback controller. Utilizing an approximation for time-varying delay state, a new comparison model is proposed. Furthermore, the l2–l∞ filtering problem for a class of discrete-time T-S fuzzy systems with time-varying delays is studied in [26]. The anticipated full- and reduced-order filter design is cast into a convex optimization problem, which can be efficiently solved by standard numerical algorithms. In [27], an improved fuzzy observer design that has some advantages, such as the less conservatism and the satisfactory multiple performance required by the fault detection, is presented. Moreover, Messaoud et al. [28] propose a new design of functional unknown input observer for nonlinear systems. In particular, necessary and sufficient conditions for the existence of the anticipated observer are presented.To the best of the authors’ knowledge, the observer-based decentralized control problems of uncertain interconnected systems with neutral type have not yet been investigated, which motivates the present study. In this paper, the asymptotic stabilization of a class of uncertain neutral interconnected systems with time-varying delays in state, control input, and interconnections is made. In framework of linear matrix equalities, the design of the observer and controller is formulated. Sufficiently the influence of interconnections on the system performance is taken into account, and interconnections are dealt with by flexible techniques. These strategies allow one to obtain less conservative stabilization conditions. A numerical example and the corresponding simulation results are given to illustrate the effectiveness of the proposed method of decentralized controller based on observer.The remainder of the paper is organized as follows. The observer-based decentralized control problem formulation is described in Section2. In Section 3, the desired gains of observer and controller are obtained by the explicit expressions in terms of some free parameters. A numerical example and the corresponding simulation results are presented in Section 4. The conclusion is provided in Section 5.


## 2. Problem Formulation

Consider the following uncertain neutral interconnected systems composed ofN subsystems:
(1)x˙i(t)-Aiηix˙i(t-ηi(t))=[Ai+ΔAi(t)]xi(t)+[Aiσi+ΔAiσi(t)]xi(t-σi(t))+[Bi+ΔBi(t)]ui(t)+∑j=1,j≠iN‍[Aij+ΔAij]xj(t-τij(t)),yi(t)=Cixi(t)+Diui(t),xi(t)=ϕi(t),t∈[-l,0],i=1,2,…,N,
where xi(t)∈ℜni, ui(t)∈ℜmi, and yi(t)∈ℜpi are the state, control input, and measurement output of the ith subsystem, respectively. Ai, Aiσi, Aiηi, Bi, Aij, Ci, and Di are known constant matrices of appropriate dimensions. ϕi(t) is the initial condition. σi(t), ηi(t), and τij(t) are the time-varying delays. Assume that there exist constants fi0, gi0, li0, fi, gi, li, and l satisfying
(2)0≤σi(t)≤fi0,0≤ηi(t)≤gi0,0≤τij(t)≤li0,σ˙i(t)≤fi<1,η˙i(t)≤gi<1,τ˙ij(t)≤li<1,l=max{fi0,gi0,li0},i,j=1,2…,N,j≠i.Time-varying parametric uncertaintiesΔAi(t), ΔAiσi(t), ΔBi(t), and ΔAij(t) are assumed to satisfy
(3)[ΔAi(t)ΔAiσi(t)ΔBi(t)ΔAij(t)]=MiFi(t)[Ni1NiσiNi2Eij],
where matrices Mi, Ni1, Niσi, Ni2, and Eij are constant matrices of appropriate dimensions and Fi(t) is the unknown matrix function satisfying
(4)FiT(t)Fi(t)≤I,∀t≥0.The following assumptions are made on the considered system (1).Assumption 1.
Suppose that the matrixCi has full row rank (i.e., rank(Ci)=pi). Then the singular value decomposition of Ci presents as follows:
(5)Ci=Ui[Si0]ViT,
where Si∈ℜpi×pi is a diagonal matrix with positive elements in a decreasing order, 0∈ℜpi×(ni-pi) is a zero matrix, and Ui∈ℜpi×pi and Vi∈ℜni×ni are unitary matrices.Assumption 2.
The matrixAiηi≠0 and ∥Aiηi∥<1.Consider the following observer-based decentralized control for system (1):
(6)x^˙i(t)-Aiηix^˙i(t-ηi(t))=Aix^i(t)+Biui(t)+Aiσix^i(t-σi(t))+∑j=1,j≠iN‍Aijx^j(t-τij(t))+Li(yi(t)-y^i(t)),y^i(t)=Cix^i(t)+Diui(t),ui(t)=-Kix^i(t),
where x^i(t)∈ℜni and y^i(t)∈ℜpi are the state and output vectors of the observer and Ki∈ℜmi×ni and Li∈ℜni×pi are the controller gain and observer gain to be designed. System (1) with the observer-based control (6) can be rewritten as
(7)[x^˙i(t)e˙i(t)]=[Ai-BiKiLiCi0Ai-LiCi][x^i(t)ei(t)]+[Aiσi00Aiσi][x^i(t-σi(t))ei(t-σi(t))]+[Aiηi00Aiηi][x^˙i(t-ηi(t))e˙i(t-ηi(t))]+[∑j=1,j≠iN‍Aijx^j(t-τij(t))∑j=1,j≠iN‍Aijej(t-τij(t))]+[0Mi]Δi(t),where the signal ei(t)=xi(t)-x^i(t) is defined as the estimated error of system, Δi(t)=Fi(t)[(Ni1-Ni2Ki)x^i(t)+Ni1ei+Niσix^iσi+Niσieiσi+∑j=1,j≠iNEijx^j(t-τij(t))+∑j=1,j≠iNEijej(t-τij(t))], and the uncertainty Δi(t) satisfies the following quadratic inequality:(8)ΔiT(t)Δi(t)≤ξiT(t)ΘiTΘiξi(t),
where
(9)ξiT(t)=[x^iT(t)eiT(t)x^iT(t-σi(t))eiT(t-σi(t))nnnnnx^˙iT(t-ηi(t))e˙iT(t-ηi(t))x^1T(t-τi1(t))nnnnn⋯x^i-1T(t-τii-1(t))x^i+1T(t-τii+1(t))nnnnn⋯x^NT(t-τiN(t))e1T(t-τi1(t))nnnnn⋯ei-1T(t-τii-1(t))ei+1T(t-τii+1(t))nnnnn⋯eNT(t-τiN(t))],Θi=[(Ni1-Ni2Ki)Ni1NiσiNiσi00Ei1nnnn⋯Eii-1Eii+1⋯EiNEi1nnnn⋯Eii-1Eii+1⋯EiN].Definition 3.
Consider system (1) with the observer-based control (6). System (1) is said to be robustly stabilizable by the observer-based control (6), if the closed-loop system (7) satisfying Assumptions 1 and 2 is asymptotically stable.Lemma 4 (see [29]).
For a givenCi∈ℜpi×ni with rank(Ci)=pi, assume that Qi2∈ℜni×ni is a symmetric matrix; then there exists a matrix Q^i2∈ℜpi×pi such that CiQi2=Q^i2Ci if and only if(10)Qi2=Vi[X^i100X^i2]ViT,
whereX^i1∈ℜpi×pi and X^i2∈ℜ(ni-pi)×(ni-pi).


## 3. Main Result

Theorem 5.
Consider uncertain neutral interconnected systems (1) with (2), (3), and (6). If there exists a solution Qik>0(k=1,3,4,5,6), X^i1>0, X^i2>0, W¯ji>0, G¯ji>0, Wij>0, Gij>0, Ri1, and Ri2 such that the following inequalities hold:
(11)Ωi<0,
then system (1) is robustly stabilizable by the observer-based control (6) with Ki=Ri1Qi1-1 and Li=Ri2UiSiX^i1-1Si-1UiT, where(12)
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According to Schur complement, inequality (11) is equivalent to the following inequality:(13)Γi1=[Ω11iΩ12iΩ13i0Ω15i0Ω17i00*Ω22i0Ω24i0Ω26i0Ω28iMi**-Qi3000000***-Qi400000****-Qi50000*****-Qi6000******Ω77i00*******Ω88i0********-I]+Υi1T(11-fiQi3-1)Υi1+Υi2T(11-fiQi4-1)Υi2+Υi3T(11-giQi5-1)Υi3+Υi4T(11-giQi6-1)Υi4+Υi5TΥi5<0,
where(14)Υi1=[(Qi1)T00000000],Υi2=[0(Ω211i)T0000000],Υi3=[(Ω112i)T(Ω212i)T(Ω312i)T0(Ω512i)T0(Ω712i)T00],Υi4=[0(Ω213i)T0(Ω413i)T0(Ω613i)T0(Ω813i)TMi],Υi5=[(Ω114i)T(Ω214i)T(Ω314i)T(Ω414i)T00(Ω714i)T(Ω814i)T0].
By Lemma4, the conditions Qi2=Vi[X^i100X^i2]ViT and Q^i2=UiSiX^i1Si-1UiT imply the condition CiQi2=Q^i2Ci. In view of Ki=Ri1Qi1-1, Li=Ri2UiSiX^i1-1Si-1UiT, W¯ji=Qi1WjiQi1, and G¯ji=Qi1GjiQi1, the matrix inequality (13) is equivalent to
(15)Γi2=[Ξ11iΞ12iΞ13i0Ξ15i0Ω17i00*Ξ22i0Ξ24i0Ξ26i0Ω18iMi**-Qi3000000***-Qi400000****-Qi50000*****-Qi6000******Ω77i00*******Ω88i0********-I]+ΛiT(11-giQi5-1)Λi+Λ^iT(11-giQi6-1)Λ^i+Λ¯iTΛ¯i+[Qi100000000](11-fiQi3-1)[Qi100000000]T+[0Qi20000000](11-fiQi4-1)[0Qi20000000]T<0,
where(16)Λi=[(Ai-BiKi)Qi1LiCiQi2AiσiQi30AiηiQi50Ω17i00],Λ^i=[0(Ai-LiCi)Qi20AiσiQi40AiηiQi60Ω17iMi],Λ¯i=[(Ni1-Ni2Ki)Qi1Ni1Qi2NiσiQi3NiσiQi400(Ω714i)T(Ω814i)T0],Ξ11i=Qi1(Ai-BiKi)T+(Ai-BiKi)Qi1+∑j=1,j≠iN‍11-ljW¯ji,Ξ12i=LiCiQi2,Ξ13i=AiσiQi3,Ξ15i=AiηiQi5,Ξ22i=Qi2(Ai-LiCi)T+(Ai-LiCi)Qi2+∑j=1,j≠iN‍11-ljG¯ji,Ξ24i=AiσiQi4,Ξ26i=AiηiQi6.
Pre- and postmultiplying matrixΓi2 in (15) by ΠiT and Πi, where
(17)Πi=diag(Qi1-1,Qi2-1,Qi3-1,Qi4-1,Qi5-1,Qi6-1,I,I,I)=diag(Pi1,Pi2,Pi3,Pi4,Pi5,Pi6,I,I,I)>0,
we obtain
(18)Γi3=[Ψ11iΨ12iΨ13i0Ψ15i0Ψ17i00*Ψ22i0Ψ24i0Ψ26i0Ψ28iPi2Mi**-Qi3000000***-Qi400000****-Qi50000*****-Qi6000******Ψ77i00*******Ψ88i0********-I]+ΦiT(11-giPi5)Φi+Φ^iT(11-giPi6)Φ^i+Φ¯iTΦ¯i<0,
where
(19)Φi=[Ai-BiKiLiCiAiσi0Aiηi0Ω17i00],Φ^i=[0Ai-LiCi0Aiσi0Aiηi0Ω17iMi],Φ¯i=[Ni1-Ni2KiNi1NiσiNiσi00(Φ714i)T(Φ814i)T0],Ψ11i=(Ai-BiKi)TPi1+Pi1(Ai-BiKi)nnnn+∑j=1,j≠iN‍11-ljW¯ji+11-fiPi3,Ψ12i=Pi1LiCi,Ψ13i=Pi1Aiσi,Ψ15i=Pi1Aiηi,Ψ17i=Pi1Ω17i,Ψ24i=Pi2Aiσi,Ψ26i=Pi2Aiηi,Ψ22i=(Ai-LiCi)TPi2+Pi2(Ai-LiCi)nnnn+∑j=1,j≠iN‍11-ljG¯ji+11-fiPi4,Ψ28i=Pi2Ω17i.
Construct the following Lyapunov functional candidate:(20)V(x^t,et)=∑i=1N{∑j=1,j≠iN‍∫t-τij(t)t‍x^iT(t)Pi1x^i(t)+eiT(t)Pi2ei(t)‍nnnnnn+11-finnnnnn×∫t-σi(t)t‍[x^iT(s)Pi3x^i(s)+eiT(s)Pi4ei(s)]dsnnnnnn+11-ginnnnnn×∫t-ηi(t)t‍[x^˙iT(s)Pi5x^˙i(s)+e˙iT(s)Pi6e˙i(s)]dsnnnnnn+11-linnnnnn×∑j=1,j≠iN‍∫t-τij(t)t‍[x^jT(s)Wijx^j(s)nnnnnnnnnnnnnnnnnnnnnn+ejT(s)Gijej(s)]ds∑j=1,j≠iN‍∫t-τij(t)t‍}.
For the system (1), the following structural identity holds:
(21)∑i=1N‍11-li∑j=1,j≠iN‍xjT(t)Wijxj(t)=∑i=1N‍xiT(t)∑j=1,j≠iN‍11-ljWjixi(t).
By some simple derivations, the time derivative ofV(x^t,et) along the trajectories of (7) satisfies the following inequality:
(22)V˙(x^t,et)≤∑i=1N‍ζiT(t)Γi3ζi(t),
where(23)ζiT(t)=[ξiT(t)ΔiT(t)].
According to conditions (18) and (22), one can obtain that, for all ζi≠0,
(24)V˙(x^t,et)<0.
In addition, Assumption2 guarantees system (1) is Lipschitzian in the term x˙i(t-ηi(t)) with Lipschitz constant less than 1 [12]. Therefore, by Definition 3 with conditions (20) and (24), system (7) is asymptotically stable and system (1) is robustly stabilizable by observer-based control (6). This completes the proof.Remark 6.
One of the distinctive features of this paper is that theN inequalities are included in (11), and only one is LMI due to the existence of the interconnected matrix variables  W-ji=Qi1WjiQi1, Wij, G¯ji=Qi2GjiQi2, and Gij among N inequalities. When i=1, the corresponding equality Ω1<0 in (11) is an LMI. Under the solvable condition of Ω1<0, we can apply Schur complement formula to the second inequality Ω2<0 in order to obtain the solvable LMI. The process is repeated until the last inequality. Thus, the controller gain Ki and observer gain Li can be obtained by finding feasible set to Ωi<0 with feasp in [30].


## 4. Illustrative Example

Consider system (1) composed of two three-order subsystems with the following parameters:
(25)A1=[-7.11.22.5-3.8-2.10.42.1-3.1-5.4],A1σ1=[-0.2-0.1-0.20.20.10.10.1-0.2-0.1],B1=[-2.9-1.3-0.32.52.7-2.2],A1η1=[-0.20.10.20.2-0.30.20.1-0.2-0.1],M1=[-0.2-0.1-0.40.6-0.20.2-0.1-0.30.3],N12=[-0.10.30.2-0.50.10.2],A12=[-2.2-1.51.1-1.11.30.20.31.1-0.5],N11=[0.20.70.30.2-0.5-0.20.5-0.1-0.1],D1=[-1.40.41.1-1.1],N1σ1=[0.1-0.1-0.30.2-0.8-0.2-0.50.10.1],E12=[0.30.40.2-0.3-0.3-0.1-0.10.20.1],σ1(t)=0.1(2+sin(t)),C1=[-2.33.91.20.22.3-2.1],η1(t)=0.2(1+cos(t)),τ12(t)=0.1(1+cos(t)),A2=[-7.62.11.3-2.1-1.5-2.11.2-0.5-5.3],A2σ2=[0.1-0.1-0.2-0.20.30.10.20.30.2],B2=[-2.11.8-4.10.22.1-2.1],A2η2=[0.10.1-0.10.30.20.40.10.20.1],A21=[-0.10.3-0.10.1-0.20.20.1-0.10.1],M2=[0.5-0.10.3-0.2-0.10.1-0.20.30.1],N21=[-0.10.20.1-0.1-0.30.5-0.20.1-0.5],N2σ2=[0.2-0.10.10.30.20.2-0.20.1-0.1],N22=[0.30.1-0.2-0.20.10.3],E21=[-0.20.2-0.10.10.10.2-0.1-0.20.1],C2=[-4.50.10.10.10.3-0.2],D2=[0.170.31-0.120.23],σ2(t)=0.2(1+cos(t)),η2(t)=0.1(2+cos(t)),τ21(t)=0.2(2+sin(t)).Applying Matlab toolbox to solving inequality (11), we obtain the following results:
(26)Q11=[1.48650.79550.42310.79552.2754-1.51400.4231-1.51405.2053],Q13=[1.43240.26460.86370.26461.1728-1.35990.8637-1.35995.0693],Q14=[1.42850.24130.65130.24131.0183-1.17820.6513-1.17824.2542],Q15=[47.92356.6463-2.81486.646344.10281.5746-2.81481.574643.3414],Q16=[40.19029.0430-2.84929.043039.54472.1458-2.84922.145841.1207],K1=[1.4531-2.1903-0.7468-1.45702.45540.6993],Q^12=[0.57470.2102-0.74203.6344],X^11=[0.8267-0.7426-0.74263.3824],X^12=2.0431,W¯21=[6.62560.8468-2.81660.84683.7842-1.9443-2.8166-1.94437.7792],W21=[8.0860-4.9568-2.7975-4.95683.91971.9259-2.79751.92591.2428],G¯21=[3.63291.1769-1.72761.17693.4642-1.9414-1.7276-1.94147.6914],G21=[2.2006-2.2199-1.0134-2.21994.20631.6203-1.01341.62031.4176],R11=[-5.33503.3676-0.9899-1.58434.02580.3958],R12=[-1.2497-0.63241.44330.67110.6835-0.8005],W12=[32.5716-0.15600.7872-0.156033.06801.22480.78721.224832.6264],G12=[33.16990.4384-0.01190.438433.46290.6883-0.01190.688333.2668],L1=[-2.2325-0.04492.55880.03660.8420-0.2690],Q21=[0.2476-0.0193-0.1246-0.01930.1848-0.0060-0.1246-0.00600.2346],Q^22=[0.1239-0.0000-0.00000.1239],X^21=[0.1239-0.0005-0.00050.1197],Q23=[1.36380.0541-2.65500.05412.8900-0.9487-2.6550-0.948711.4203],R22=[-0.03361.59120.07975.74630.0145-2.5460],Q24=[1.80541.0651-3.05581.06511.7394-3.5831-3.0558-3.58319.2685],K2=[0.5773-8.9195-3.60703.8724-6.5449-12.5895],Q25=[416.8897-302.9290-158.6713-302.9290231.3825118.1184-158.6713118.118479.4590],X^22=0.1013,Q26=[139.5613-92.2506-33.9039-92.250666.792618.6639-33.903918.663912.5276],R21=[0.7642-1.6381-0.86472.6539-1.2087-3.3972],L2=[-0.271212.84240.643446.37660.1172-20.5476].Under the following initial condition(27)ϕ1(t)=[-0.2e2t0.10.4t-0.05],ϕ2(t)=[0.2et0.2t-0.10.05],the simulation results are shown in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. With the observer-based control applied, the state responses of the closed-loop system are depicted in Figures 1–2. The control signals based on the observer are rather smooth in Figures 3 and 4. The output signals of the closed-loop system are shown in Figures 5 and 6. The state responses and the output signals of the observer system (6) are presented in Figures 7–10, respectively. The state responds of the error dynamic system (7) are shown in Figures 11 and 12.Figure 1
State responses of the first closed-loop subsystem.Figure 2
State responses of the second closed-loop subsystem.Figure 3
The control signals of the first subsystem.Figure 4
The control signals of the second subsystem.Figure 5
Output responses of the first subsystem.Figure 6
Output responses of the second subsystem.Figure 7
State responses of the first observer subsystem.Figure 8
State responses of the second observer subsystem.Figure 9
Output responses of the first observer subsystem.Figure 10
Output responses of the second observer subsystem.Figure 11
State responses of the first error subsystem.Figure 12
State responses of the second error subsystem.The simulations results indicate that the designed observer can stabilize the error dynamic system and estimate the states of the interconnected systems of neutral type.Remark 7.
For this example,N=2. When i=1, we can obtain Q11, Q12, W¯21, W12, G¯21, and G12 by solving LMI Ω1<0 in (11). Furthermore, according to W¯21=Q11W21Q11 and G¯21=Q12G21Q12, we have W21 and G21. When i=2, inequality Ω2<0 in (11) is not LMI because of the existence of the interconnection matrix W¯12=Q21W12Q21 and G¯12=Q22G12Q22 (here, W12 and G12 are constant matrices coming from Ω1<0). However, the transformed inequality is an LMI after applying Schur complement to Ω2<0. Hence, the controller gain K2 and observer gain L2 can be obtained by finding feasible set to the transformed inequality of Ω2<0.


## 5. Conclusion

The observer-based decentralized control problem of uncertain interconnected systems of neutral type is complex and challenging. In framework of Lyapunov stability theory, a novel mathematical technique to deal with the parametric disturbances is developed to obtain the sufficient conditions of existing anticipated controller and observer. The sufficient conditions are the coupled LMIs and depend on not only the sizes of delays, but also the information of derivatives. The numerical example and the corresponding simulation results elucidate that the results obtained in this paper are effective.



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*Source: 134935-2013-07-09.xml*