Methods | σ1 | μ=0.8 | μ=0.9 | Unknownμ |
[30] | σ1=0.5 | 0.8262 | 0.8215 | - |
Corollary 3.3 | - | - | 0.9976 | |
[30] | σ1=0.75 | 0.9669 | 0.9625 | - |
Corollary 3.3 | - | - | 1.1233 | |
[30] | σ1=1 | 1.1152 | 1.1108 | - |
Corollary 3.3 | - | - | 1.2710 |
This paper studies the mixed passive and H∞ performance for uncertain neural networks with interval discrete and distributed time-varying delays via feedback control. The interval discrete and distributed time-varying delay functions are not assumed to be differentiable. The improved criteria of exponential stability with a mixed passive and H∞ performance are obtained for the uncertain neural networks by constructing a Lyapunov-Krasovskii functional (LKF) comprising single, double, triple, and quadruple integral terms and using a feedback controller. Furthermore, integral inequalities and convex combination technique are applied to achieve the less conservative results for a special case of neural networks. By using the Matlab LMI toolbox, the derived new exponential stability with a mixed passive and H∞ performance criteria is performed in terms of linear matrix inequalities (LMIs) that cover H∞, and passive performance by setting parameters in the general performance index. Numerical examples are shown to demonstrate the benefits and effectiveness of the derived theoretical results. The method given in this paper is less conservative and more general than the others.
Citation: Sunisa Luemsai, Thongchai Botmart, Wajaree Weera, Suphachai Charoensin. Improved results on mixed passive and H∞ performance for uncertain neural networks with mixed interval time-varying delays via feedback control[J]. AIMS Mathematics, 2021, 6(3): 2653-2679. doi: 10.3934/math.2021161
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This paper studies the mixed passive and H∞ performance for uncertain neural networks with interval discrete and distributed time-varying delays via feedback control. The interval discrete and distributed time-varying delay functions are not assumed to be differentiable. The improved criteria of exponential stability with a mixed passive and H∞ performance are obtained for the uncertain neural networks by constructing a Lyapunov-Krasovskii functional (LKF) comprising single, double, triple, and quadruple integral terms and using a feedback controller. Furthermore, integral inequalities and convex combination technique are applied to achieve the less conservative results for a special case of neural networks. By using the Matlab LMI toolbox, the derived new exponential stability with a mixed passive and H∞ performance criteria is performed in terms of linear matrix inequalities (LMIs) that cover H∞, and passive performance by setting parameters in the general performance index. Numerical examples are shown to demonstrate the benefits and effectiveness of the derived theoretical results. The method given in this paper is less conservative and more general than the others.
During the past few decades, many researchers have studied neural networks because of their applications in many fields such as parallel computation, fault diagnosis, image processing, optimization problems, industrial automation, and so on [1,2,3,4,5]. To acquire the above applications, we need to first analyze the theoretical stability for the equilibrium point of neural networks. Further, the important factor affecting system analysis is time delay. It is well known that time delay is a normal phenomenon that appears in neural networks since the neural networks consist of a large number of neurons that connect and communicate with each other into a diversity of axon sizes and lengths. Moreover, the existence of time delay causing poor control performance, divergence, oscillation, and instability to the system [6]. Stability analysis of neural networks with constant, discrete, and distributed time-varying delays has received considerable attentions [7,8,9]. For example, [7], the delay-dependent criterion for exponential stability analysis of neural networks with time-varying delays satisfying 0≤η(t)≤η,˙η(t)≤μ is obtained. In [8], the problem of dissipativity analysis for neural networks with time-varying delays is investigated. However, practically time delay can occur in an irregular fashion such as sometimes the time-varying delays are not differentiable. So, it inspires us to study neural networks without the restriction on the derivative of time-varying delays.
On the other side, since external perturbation, uncertain or slowly varying parameters, an accurate mathematical model does not get easy. Data tends to be uncertain in many applications [10,11,12]. Therefore, it is important to guarantee that the model is stable with respect to the uncertainties. Also, uncertainty in neural networks cannot be avoided. Consequently the problem of robust stability analysis for uncertain neural networks has many studied. For example, Subramanian et al. [13] investigated the robust stabilization of uncertain neural networks with two additive time-varying delays based on Wirtinger-based double integral inequality. In [14], Zeng et al. studied the robust passivity analysis of uncertain neural networks with discrete and distributed delays by constructing an augmented Lyapunov functional and combining a new integral inequality with the reciprocally convex approach.
It is well known that passivity is a special case and a general theory of dissipativeness and it performs an influential part in the designing of linear and nonlinear systems. It is widely applied in many areas such as sliding mode control [15], fuzzy control [16], network control [17], and signal processing [18]. The main property of passivity is that can keep the system internally stable. Recently, the passivity problem has been studied in [14,19,20,21,22]. In addition, the H∞ theory is very important due to the H∞ control design that exposes the control problem as a mathematical optimization problem to find the controller solution. The H∞ approaches are used in control theory to synthesize controllers achieving stabilization with guaranteed performance [23,24]. The problem of mixed H∞ and passivity analysis was first studied in [25,26]. It has received a lot of attention from many researchers. For example, the mixed passive and H∞ synchronization problems of complex dynamical networks have been analyzed in [27,28]. And, the combined H∞ and passivity state estimation of memristive neural networks was studied in [29]. Nevertheless, a mixed passive and H∞ analysis problem for uncertain neural networks with interval discrete and distributed time-varying delays has been few considered which is our motivation.
Inspired by above discussions, the problem of mixed passive and H∞ performance for uncertain neural networks with interval discrete and distributed time-varying delays via feedback control is studied. The main contributions of this paper are three aspects.
∙ In this work, the system consists of the interval discrete and distributed time-varying delays such that does not necessitate being differentiable functions, which mean that a fast interval discrete and distributed time-varying delays is approved. The lower bound of the delays does not restrict to be 0, the activation functions are different, and the output is general.
∙ By using the Lyapunov-Krasovskii stability theory, the new results of the exponential stability with a mixed passive and H∞ performance for the uncertain neural networks are obtained. Based on the weighting parameter, the results are more general such that H∞ performance or passive performance for the uncertain neural networks are included.
∙ Different from the methods in [30,31,32], the Lyapunov-Krasovskii functional comprising single, double, triple, and quadruple integral terms and integral inequalities are employed. Convex combination idea and zero equation are used. The method used in this paper reveals less conservative results when comparing with existing results [30,31,32].
This paper is formed in five sections as follows. In Section 2, network model and preliminaries are provided. Section 3 shows exponential stability analysis with a mixed passive and H∞ performance of the uncertain neural network system, and the stability analysis of a special case neural network. Numerical examples are given in Section 4 and conclusions are addressed in Section 5.
Notations
Throughout this paper, R and Rn represent the set of real numbers and the n-dimensional Euclidean spaces, respectively. M>(≥)0 means that the symmetric matrix M is positive (semi-positive) definite. M<(≤)0 denotes that the symmetric matrix M is negative (semi-negative) definite. MT and M−1 denote the transpose and the inverse of matrix M, respectively. λmax(M) and λmin(M) denote the maximum eigenvalue and the minimum eigenvalue of matrix M, respectively. The symbol ∗ represents the symmetric block in a symmetric matrix. I is the identity matrix with appropriate dimensions. ei represents the unit column vector having one element on its ith row and zeros elsewhere. C([a1,a2],Rn) denotes the set of continuous functions mapping the interval [a1,a2] to Rn. L2[0,∞) represents the space of functions ζ:R+→Rn with the norm ‖ζ‖L2=[∫∞0|ζ(θ)|2dθ]12. For ϑ∈Rn, the norm of ϑ, denoted by ‖ϑ‖, is defined by ‖ϑ‖=[n∑i=1|ϑi|2]1/2; ‖ϑ(t+ν)‖cl =max{sup−max{σ2,δ2}≤ν≤0‖ϑ(t+ν)‖2,sup−max{σ2,δ2}≤ν≤0‖˙ϑ(t+ν)‖2}.
We consider the uncertain neural network model with interval discrete and distributed time-varying delays of the form
˙x(t)=−(A+ΔA(t))x(t)+(B+ΔB(t))f(x(t))+(C+ΔC(t))k(x(t−σ(t)))+(D+ΔD(t))∫t−δ1(t)t−δ2(t)h(x(s))ds+Eω(t)+U(t),z(t)=C1x(t)+C2x(t−σ(t))+C3∫t−δ1(t)t−δ2(t)h(x(s))ds+C4ω(t),x(t)=ϕ(t),t∈[−ϱ,0], | (2.1) |
where x(t)=[x1(t),x2(t),…,xn(t)]T∈Rn is the neuron state vector, f(x(t)),k(x(t)),h(x(t))∈Rn are the neuron activation functions, z(t)∈Rn is the output vector, ω(t)∈Rn is the input vector such that ω(t)∈L2[0,∞), U(t)∈Rn is the control input, A=diag{a1,a2,…,an}>0, B is the connection weight matrix, C is the discretely delayed connection weight matrix, D is the distributively delayed connection weight matrix, E,C1,C2,C3,C4 are given constant matrices, ϕ(t)∈C[[−ϱ,0],Ren] is the initial function. σ(t) is the interval discrete time-varying delay that satisfies 0≤σ1≤σ(t)≤σ2 where σ1,σ2∈R. δi(t) (i=1,2) is the interval distributed time-varying delay that satisfies 0≤δ1≤δ1(t)≤δ2(t)≤δ2 where δ1,δ2∈R. ϱ=max{σ2,δ2} is known real constant, the time-varying uncertainties matrices ΔA(t),ΔB(t),ΔC(t), and ΔD(t) are given by
ΔA(t)=J1S1(t)Σ1,ΔB(t)=J2S2(t)Σ2,ΔC(t)=J3S3(t)Σ3,ΔD(t)=J4S4(t)Σ4, |
and J1,J2,J3,J4,Σ1,Σ2,Σ3 and Σ4 are known constant matrices with appropriate dimensions, S1(t),S2(t),S3(t),S4(t) are unknown uncertain matrices satisfying
ST1(t)S1(t)≤I,ST2(t)S2(t)≤I,ST3(t)S3(t)≤I,ST4(t)S4(t)≤I. |
The neuron activation functions f(x(t)),k(x(t)) and h(x(t)) satisfy the following conditions:
(A1) f is continuous and satisfies
F−i≤fi(α1)−fi(α2)α1−α2≤F+i |
for all α1≠α2, and F−i,F+i∈R, fi(0)=0.
(A2) k is continuous and satisfies
K−i≤ki(α1)−ki(α2)α1−α2≤K+i |
for all α1≠α2, and K−i,K+i∈R, ki(0)=0.
(A3) h is continuous and satisfies
H−i≤hi(α1)−hi(α2)α1−α2≤H+i |
for all α1≠α2, and H−i,H+i∈R, hi(0)=0.
The state feedback is considered with
U(t)=Kx(t). |
Substitute U(t)=Kx(t) into (2.1), we gain
˙x(t)=(K−A−ΔA(t))x(t)+(B+ΔB(t))f(x(t))+(C+ΔC(t))×k(x(t−σ(t)))+(D+ΔD(t))∫t−δ1(t)t−δ2(t)h(x(s))ds+Eω(t),z(t)=C1x(t)+C2x(t−σ(t))+C3∫t−δ1(t)t−δ2(t)h(x(s))ds+C4ω(t),x(t)=ϕ(t),t∈[−ϱ,0]. | (2.2) |
Definition 2.1. [28] The uncertain NNs (2.2) with ω(t)=0 is exponentially stable, if there exist constants b1>0 and b2>0 such that
‖x(t)‖2≤b1e−b2t‖x(ν)‖cl. |
Definition 2.2. [28] For a given scalar υ∈[0,1], the uncertain NNs (2.2) is exponentially stable and meets a predefined passive and H∞ performance index γ, if the following conditions can be ensured simultaneously:
(1) the uncertain NNs (2.2) is exponentially stable in sense of Definition 2.1.
(2) under zero initial condition, there exists a scalar γ>0 such that the following inequality is satisfied:
∫Tp0[−υzT(t)z(t)+2(1−υ)γzT(t)ω(t)]dt≥−γ2∫Tp0[ωT(t)ω(t)]dt, | (2.3) |
for any Tp≥0 and any non-zero ω(t)∈L2[0,∞).
Remark 1. The condition (2.3) includes passive performance index and H∞ performance index. If υ=1, the condition (2.3) reduces to the H∞ performance index; and if υ=0, the condition (2.3) reduces to the passive performance index; when υ takes the value in (0,1), then the condition (2.3) becomes to the mixed passive and H∞ performance index.
Lemma 2.3. [33,34] Suppose 0≤η1<η2 and x(t)∈Rn, for any matrix M>0 the following inequalities hold:
−(η2−η1)∫t−η1t−η2xT(s)Mx(s)ds≤−∫t−η1t−η2xT(s)dsM∫t−η1t−η2x(s)ds,−(η22−η21)2∫−η1−η2∫tt+βxT(s)Mx(s)dsdβ≤−∫−η1−η2∫tt+βxT(s)dsdβ×M∫−η1−η2∫tt+βx(s)dsdβ,−η326∫0−η2∫0β∫tt+λxT(s)Mx(s)dsdλdβ≤−∫0−η2∫0β∫tt+λxT(s)dsdλdβ×M∫0−η2∫0β∫tt+λx(s)dsdλdβ. |
Lemma 2.4. [35] For a matrix M>0, a differentiable function {x(α)|α∈[a1,a2]}, the following inequality holds:
∫a2a1˙xT(α)M˙x(α)dα≥1a2−a1[x(a2)−x(a1)]TM[x(a2)−x(a1)]+3a2−a1×[x(a2)+x(a1)−2a2−a1∫a2a1x(α)dα]TM[x(a2)+x(a1)−2a2−a1∫a2a1x(α)dα]. |
Lemma 2.5. [36] For given matrices P, Q and R with RTR≤I and a scalar α>0, the following inequality holds:
PRQ+(PRQ)T≤αPPT+α−1QTQ. |
Lemma 2.6. [37] Let P,Q,R be given matrices such that R>0, then
[QPTP−R]<0⇔Q+PTR−1P<0. |
In this section, we will firstly find the sufficient conditions which guarantee the neural networks without parameter uncertainties to be exponentially stable with a mixed passive and H∞ performance. That is we consider the following model
˙x(t)=(K−A)x(t)+Bf(x(t))+Ck(x(t−σ(t)))+D∫t−δ1(t)t−δ2(t)h(x(s))ds+Eω(t),z(t)=C1x(t)+C2x(t−σ(t))+C3∫t−δ1(t)t−δ2(t)h(x(s))ds+C4ω(t),x(t)=ϕ(t),t∈[−ϱ,0]. | (3.1) |
In this paper, we define the denotations as follows
˜Fi=max{|F−i|,|F+i|},˜Ki=max{|K−i|,|K+i|},˜Hi=max{|H−i|,|H+i|},F1=diag{F−1F+1,F−2F+2,…,F−nF+n},F2=diag{F−1+F+12,F−2+F+22,…,F−n+F+n2},K1=diag{K−1K+1,K−2K+2,…,K−nK+n},K2=diag{K−1+K+12,K−2+K+22,…,K−n+K+n2},H1=diag{H−1H+1,H−2H+2,…,H−nH+n},H2=diag{H−1+H+12,H−2+H+22,…,H−n+H+n2}, |
ξT(t)=[xT(t),˙xT(t),xT(t−σ1),xT(t−σ2),xT(t−σ(t)),fT(x(t)),kT(x(t−σ(t))),hT(x(t)),1σ1∫tt−σ1xT(s)ds,1σ2∫tt−σ2xT(s)ds,1σ(t)−σ1∫t−σ1t−σ(t)xT(s)ds,1σ2−σ(t)∫t−σ(t)t−σ2xT(s)ds,∫t−δ1(t)t−δ2(t)hT(x(s))ds,∫−σ1−σ(t)∫tt+βxT(s)dsdβ,∫−σ(t)−σ2∫tt+βxT(s)dsdβ,ωT(t)]. |
Theorem 3.1. For given scalars σ1,σ2,δ1,δ2,β1,β2,γ>0, and υ∈[0,1], if there exist eleven n×n matrices P>0,Q1>0,Q2>0,R1>0,R2>0,U>0,L>0,X1>0,X2>0,N>0,Z and three n×n positive diagonal matrices Y1>0,Y2>0,Y3>0 such that the following LMIs hold:
Θ+Θ1<0, | (3.2) |
Θ+Θ2<0, | (3.3) |
wherein,
Θ1=−e15X1eT15,Θ2=−e14X1eT14,Θ=[Θ(1,1)Θ(1,2)∗Θ(2,2)], |
with:
Θ(1,1)=[θ1,1θ1,2−2R1−2R2υCT1C2θ1,6β1NTCH2Y3∗θ2,2000β2NTBβ2NTC0∗∗θ3,30−2U000∗∗∗θ4,4−2U000∗∗∗∗θ5,50K2Y20∗∗∗∗∗−Y100∗∗∗∗∗∗−Y20∗∗∗∗∗∗∗θ8,8], |
Θ(1,2)=[6R16R200θ1,13σ22−σ212X2σ22−σ212X2θ1,160000β2NTD00β2NTE6R106U0000006R206U0000006U6UυCT2C300θ5,16000000000000000000000000], |
Θ(2,2)=[−12R10000000∗−12R2000000∗∗−12U00000∗∗∗−12U0000∗∗∗∗θ13,1300θ13,16∗∗∗∗∗θ14,14−X20∗∗∗∗∗∗θ15,150∗∗∗∗∗∗∗θ16,16], |
in which:
θ1,1=Q1+Q2−4R1−4R2+υCT1C1−F1Y1−H1Y3+2β1Z−2β1NTA+(σ22−σ21)24X1−(σ22−σ21)24X2,θ1,2=P−β1NT+β2ZT−β2NTA,θ1,6=F2Y1+β1NTB,θ1,13=υCT1C3+β1NTD,θ1,16=υCT1C4−(1−υ)γCT1+β1NTE,θ2,2=σ21R1+σ22R2+(σ2−σ1)2U−2β2NT+(σ32−σ31)236X2,θ3,3=−Q1−4R1−4U,θ4,4=−Q2−4R2−4U,θ5,5=−8U−K1Y2+υCT2C2,θ5,16=υCT2C4−(1−υ)γCT2,θ8,8=(δ2−δ1)2L−Y3,θ13,13=−L+υCT3C3,θ13,16=υCT3C4−(1−υ)γCT3,θ14,14=−X1−X2,θ15,15=−X1−X2,θ16,16=υCT4C4−2(1−υ)γCT4−γ2I, |
then, the NNs (3.1) is exponentially stable with a mixed passive and H∞ performance. Moreover, the controller is in the form
K=N−1Z. |
Proof. Consider the model (3.1) with the following Lyapunov-Krasovskii functional
V(x(t),t)=9∑i=1Vi(x(t),t), |
where
V1(x(t),t)=xT(t)Px(t),V2(x(t),t)=∫tt−σ1xT(s)Q1x(s)ds,V3(x(t),t)=∫tt−σ2xT(s)Q2x(s)ds,V4(x(t),t)=σ1∫0−σ1∫tt+s˙xT(τ)R1˙x(τ)dτds,V5(x(t),t)=σ2∫0−σ2∫tt+s˙xT(τ)R2˙x(τ)dτds,V6(x(t),t)=(σ2−σ1)∫−σ1−σ2∫tt+s˙xT(τ)U˙x(τ)dτds,V7(x(t),t)=(δ2−δ1)∫−δ1−δ2∫tt+shT(x(τ))Lh(x(τ))dτds,V8(x(t),t)=(σ22−σ21)2∫−σ1−σ2∫0β∫tt+λxT(s)X1x(s)dsdλdβ,V9(x(t),t)=(σ32−σ31)6∫−σ1−σ2∫0β∫0λ∫tt+φ˙xT(s)X2˙x(s)dsdφdλdβ. | (3.4) |
We find time derivatives of Vi(x(t),t),i=1,2,…,9, along the trajectories of (3.1), we achieve
˙V1(x(t),t)=xT(t)P˙x(t)+˙xT(t)Px(t), | (3.5) |
˙V2(x(t),t)=xT(t)Q1x(t)−xT(t−σ1)Q1x(t−σ1), | (3.6) |
˙V3(x(t),t)=xT(t)Q2x(t)−xT(t−σ2)Q2x(t−σ2), | (3.7) |
˙V4(x(t),t)=σ1∫0−σ1[˙xT(t)R1˙x(t)−˙xT(t+s)R1˙x(t+s)]ds=σ21˙xT(t)R1˙x(t)−σ1∫tt−σ1˙xT(α)R1˙x(α)dα, | (3.8) |
˙V5(x(t),t)=σ2∫0−σ2[˙xT(t)R2˙x(t)−˙xT(t+s)R2˙x(t+s)]ds=σ22˙xT(t)R2˙x(t)−σ2∫tt−σ2˙xT(α)R2˙x(α)dα, | (3.9) |
˙V6(x(t),t)=(σ2−σ1)∫−σ1−σ2[˙xT(t)U˙x(t)−˙xT(t+s)U˙x(t+s)]ds=(σ2−σ1)2˙xT(t)U˙x(t)−(σ2−σ1)∫t−σ1t−σ2˙xT(α)U˙x(α)dα, | (3.10) |
˙V7(x(t),t)=(δ2−δ1)∫−δ1−δ2[hT(x(t))Lh(x(t))−hT(x(t+s))Lh(x(t+s))]ds=(δ2−δ1)2hT(x(t))Lh(x(t))−(δ2−δ1)∫t−δ1t−δ2hT(x(α))Lh(x(α))dα≤(δ2−δ1)2hT(x(t))Lh(x(t))−(δ2(t)−δ1(t))∫t−δ1(t)t−δ2(t)hT(x(α))Lh(x(α))dα, | (3.11) |
˙V8(x(t),t)=(σ22−σ21)2∫−σ1−σ2∫0β[xT(t)X1x(t)−xT(t+λ)X1x(t+λ)]dλdβ=(σ22−σ21)24xT(t)X1x(t)−(σ22−σ21)2∫−σ1−σ2∫tt+βxT(s)X1x(s)dsdβ, | (3.12) |
˙V9(x(t),t)=(σ32−σ31)6×∫−σ1−σ2∫0β∫0λ[˙xT(t)X2˙x(t)−˙xT(t+φ)X2˙x(t+φ)]dφdλdβ=(σ32−σ31)236˙xT(t)X2˙x(t)−(σ32−σ31)6∫−σ1−σ2∫0β∫tt+λ˙xT(s)X2˙x(s)dsdλdβ. | (3.13) |
Utilizing Lemma 2.4., the following inequalities are easily obtained:
−σ1∫tt−σ1˙xT(α)R1˙x(α)dα≤−[x(t)−x(t−σ1)]TR1[x(t)−x(t−σ1)]−3[x(t)+x(t−σ1)−2σ1∫tt−σ1x(α)dα]T×R1[x(t)+x(t−σ1)−2σ1∫tt−σ1x(α)dα], | (3.14) |
−σ2∫tt−σ2˙xT(α)R2˙x(α)dα≤−[x(t)−x(t−σ2)]TR2[x(t)−x(t−σ2)]−3[x(t)+x(t−σ2)−2σ2∫tt−σ2x(α)dα]T×R2[x(t)+x(t−σ2)−2σ2∫tt−σ2x(α)dα], | (3.15) |
−(σ2−σ1)∫t−σ1t−σ2˙xT(α)U˙x(α)dα≤−[x(t−σ(t))−x(t−σ2)]TU[x(t−σ(t))−x(t−σ2)]−3[x(t−σ(t))+x(t−σ2)−2σ2−σ(t)∫t−σ(t)t−σ2x(α)dα]T×U[x(t−σ(t))+x(t−σ2)−2σ2−σ(t)∫t−σ(t)t−σ2x(α)dα]−[x(t−σ1)−x(t−σ(t))]TU[x(t−σ1)−x(t−σ(t))]−3[x(t−σ1)+x(t−σ(t))−2σ(t)−σ1∫t−σ1t−σ(t)x(α)dα]T×U[x(t−σ1)+x(t−σ(t))−2σ(t)−σ1∫t−σ1t−σ(t)x(α)dα]. | (3.16) |
By utilizing Lemma 2.3, we achieve the following inequalities
−(δ2(t)−δ1(t))∫t−δ1(t)t−δ2(t)hT(x(α))Lh(x(α))dα≤−∫t−δ1(t)t−δ2(t)hT(x(α))dαL∫t−δ1(t)t−δ2(t)h(x(α))dα, | (3.17) |
−(σ22−σ21)2∫−σ1−σ2∫tt+βxT(s)X1x(s)dsdβ≤−∫−σ(t)−σ2∫tt+βxT(s)dsdβX1∫−σ(t)−σ2∫tt+βx(s)dsdβ−ε∫−σ(t)−σ2∫tt+βxT(s)dsdβX1∫−σ(t)−σ2∫tt+βx(s)dsdβ−(1−ε)∫−σ1−σ(t)∫tt+βxT(s)dsdβX1∫−σ1−σ(t)∫tt+βx(s)dsdβ−∫−σ1−σ(t)∫tt+βxT(s)dsdβX1∫−σ1−σ(t)∫tt+βx(s)dsdβ, | (3.18) |
where ε=σ2(t)−σ21σ22−σ21.
−(σ32−σ31)6∫−σ1−σ2∫0β∫tt+λ˙xT(s)X2˙x(s)dsdλdβ≤−[σ22−σ212xT(t)−∫−σ(t)−σ2∫tt+βxT(s)dsdβ−∫−σ1−σ(t)∫tt+βxT(s)dsdβ]×X2[σ22−σ212x(t)−∫−σ(t)−σ2∫tt+βx(s)dsdβ−∫−σ1−σ(t)∫tt+βx(s)dsdβ]. | (3.19) |
It follows from (A1) that [fi(xi(t))−F−ixi(t)][fi(xi(t))−F+ixi(t)]≤0 for every i=1,2,…,n, which are equivalent to
[x(t)f(x(t))]T[F−iF+ieieTi−F−i+F+i2eieTi−F−i+F+i2eieTieieTi][x(t)f(x(t))]≤0, |
for every i=1,2,…,n.
Define Y1=diag{y1,y2,…,yn}>0, then
n∑i=1yi[x(t)f(x(t))]T[F−iF+ieieTi−F−i+F+i2eieTi−F−i+F+i2eieTieieTi][x(t)f(x(t))]≤0, |
which is equivalent to
[x(t)f(x(t))]T[−F1Y1F2Y1F2Y1−Y1][x(t)f(x(t))]≥0. | (3.20) |
Similarly, from (A2), (A3) define Y2=diag{˜y1,˜y2,…,˜yn}>0,
Y3=diag{ˆy1,ˆy2,…,ˆyn}>0 we have
[x(t−σ(t))k(x(t−σ(t)))]T[−K1Y2K2Y2K2Y2−Y2][x(t−σ(t))k(x(t−σ(t)))]≥0, | (3.21) |
[x(t)h(x(t))]T[−H1Y3H2Y3H2Y3−Y3][x(t)h(x(t))]≥0. | (3.22) |
We have zero equation as follows
0=2[xT(t)β1NT+˙xT(t)β2NT][−˙x(t)+(N−1Z−A)x(t)+Bf(x(t))+Ck(x(t−σ(t)))+D∫t−δ1(t)t−δ2(t)h(x(s))ds+Eω(t)]. |
Adding above zero equation to ˙V(x(t),t), we obtain the following inequality from (2.3), (3.5)–(3.22)
˙V(x(t),t)+υzT(t)z(t)−2(1−υ)γzT(t)ω(t)−γ2ωT(t)ω(t)≤ξT(t)(εΘ(1)+(1−ε)Θ(2))ξ(t), | (3.23) |
where, Θ(i)=Θ+Θi (i=1,2) with Θ and Θi are defined in (3.2), (3.3).
Since 0≤ε≤1, the term εΘ(1)+(1−ε)Θ(2) is a convex combination of Θ(1) and Θ(2). The combinations are negative definite only if
Θ(1)<0, | (3.24) |
Θ(2)<0. | (3.25) |
So, (3.24) and (3.25) are equivalent to (3.2) and (3.3), respectively.
Hence, we obtain
˙V(x(t),t)+υzT(t)z(t)−2(1−υ)γzT(t)ω(t)−γ2ωT(t)ω(t)<0. | (3.26) |
Under the zero initial condition, for any Tp we find that
∫Tp0υzT(t)z(t)−2(1−υ)γzT(t)ω(t)−γ2ωT(t)ω(t)dt≤∫Tp0˙V(x(t),t)+υzT(t)z(t)−2(1−υ)γzT(t)ω(t)−γ2ωT(t)ω(t)dt<0, |
that is
∫Tp0υzT(t)z(t)−2(1−υ)γzT(t)ω(t)dt≤γ2∫Tp0ωT(t)ω(t)dt. |
In this case, the condition (2.3) is guaranteed for any non-zero ω(t)∈L2[0,∞). If ω(t)=0, in sense of equation (3.26), there exists a scalar υ1>0 such that
˙V(x(t),t)<−υ1xT(t)x(t). | (3.27) |
By the definitions of Vi(x(t),t), it is easy to derive the following inequalities:
V1(x(t),t)≤λmax(P)‖x(t)‖2,V4(x(t),t)≤σ21∫tt−σ1˙xT(α)R1˙x(α)dα,V5(x(t),t)≤σ22∫tt−σ2˙xT(α)R2˙x(α)dα,V6(x(t),t)≤(σ2−σ1)2∫tt−σ2˙xT(τ)U˙x(τ)dτ,V7(x(t),t)≤(δ2−δ1)2∫tt−δ2hT(x(τ))Lh(x(τ))dτ,V8(x(t),t)≤(σ22−σ21)24∫tt−σ2xT(s)X1x(s)ds,V9(x(t),t)≤(σ32−σ31)236∫tt−σ2˙xT(s)X2˙x(s)ds. | (3.28) |
We are now ready to deal with the exponential stability of (3.1). Consider the Lyapunov–Krasovskii functional e2ctV(x(t),t), where c is a constant. Using (3.27), (3.28), we have
ddte2ctV(x(t),t)=e2ct˙V(x(t),t)+2ce2ctV(x(t),t)<e2ct[−υ1+2c(λmax(P)+σ1λmax(Q1)+σ2λmax(Q2)+σ31λmax(R1)+σ32λmax(R2)+σ2(σ2−σ1)2λmax(U)+δ2(δ2−δ1)2λmax(L)maxi∈{1,2,…,n}(˜H2i)+σ2(σ22−σ21)24λmax(X1)+σ2(σ32−σ31)236λmax(X2))]‖x(t+ν)‖cl. | (3.29) |
Let
μ1=λmax(P)+σ1λmax(Q1)+σ2λmax(Q2)+σ31λmax(R1)+σ32λmax(R2)+σ2(σ2−σ1)2λmax(U)+δ2(δ2−δ1)2λmax(L)maxi∈{1,2,…,n}(˜H2i)+σ2(σ22−σ21)24λmax(X1)+σ2(σ32−σ31)236λmax(X2). |
Now, we take c to be a constant satisfying c≤υ12μ1, and then achieve from (3.29) that
ddte2ctV(x(t),t)≤0, | (3.30) |
which, together with (3.4) and (3.28), imply that
e2ctV(x(t),t)≤V(x(0),0)=9∑i=1Vi(x(0),0)≤[λmax(P)‖x(0)‖2+∫0−σ1xT(s)Q1x(s)ds+∫0−σ2xT(s)Q2x(s)ds+σ21∫0−σ1˙xT(τ)R1˙x(τ)dτ+σ22∫0−σ2˙xT(τ)R2˙x(τ)dτ+(σ2−σ1)2∫0−σ2˙xT(τ)U˙x(τ)dτ+(δ2−δ1)2∫0−δ2hT(x(τ))Lh(x(τ))dτ+(σ22−σ21)24∫0−σ2xT(s)X1x(s)ds+(σ32−σ31)236∫0−σ2˙xT(s)X2˙x(s)ds]≤μ0‖x(ν)‖cl, |
where
μ0=λmax(P)+σ1λmax(Q1)+σ2λmax(Q2)+σ31λmax(R1)+σ32λmax(R2)+σ2(σ2−σ1)2λmax(U)+δ2(δ2−δ1)2λmax(L)maxi∈{1,2,…,n}(˜H2i)+σ2(σ22−σ21)24λmax(X1)+σ2(σ32−σ31)236λmax(X2), |
and therefore
V(x(t),t)≤μ0e−2ct‖x(ν)‖cl. |
Noticing λmin(P)‖x(t)‖2≤V(x(t),t), we obtain
‖x(t)‖2≤μ0λmin(P)e−2ct‖x(ν)‖cl. | (3.31) |
Letting b1=μ0λmin(P) and b2=2c, we can rewrite (3.31) as
‖x(t)‖2≤b1e−b2t‖x(ν)‖cl. |
Hence, the NNs (3.1) is exponentially stable with a mixed passive and H∞ performance index γ. The proof is completed.
In the second part, the criteria of exponential stability with a mixed passive and H∞ performance for the uncertain neural networks are obtained by using similar proof of Theorem 3.1 together with Lemma 2.5, 2.6.
Theorem 3.2. For given scalars σ1,σ2,δ1,δ2,β1,β2,γ>0, and υ∈[0,1], if there exist eleven n×n matrices P>0,Q1>0,Q2>0,R1>0,R2>0,U>0,L>0,X1>0,X2>0,N>0,Z, positive diagonal matrices Y1>0,Y2>0,Y3>0 and eight positive constants αi>0(i=1,2,…,8) such that the following LMIs hold:
Ψ+Θ1<0, | (3.32) |
Ψ+Θ2<0, | (3.33) |
wherein,
Θ1=−e15X1eT15,Θ2=−e14X1eT14,ˉΘ=[ˉΘ(1,1)Θ(1,2)∗ˉΘ(2,2)], |
Ψ=[ˉΘNTJ1NTJ1NTJ2NTJ2NTJ3NTJ3NTJ4NTJ4∗−α1I0000000∗∗−α2I000000∗∗∗−α3I00000∗∗∗∗−α4I0000∗∗∗∗∗−α5I000∗∗∗∗∗∗−α6I00∗∗∗∗∗∗∗−α7I0∗∗∗∗∗∗∗∗−α8I], |
with: Θ(1,2) is defined in Theorem 3.1,
ˉΘ(1,1)=[ˉθ1,1θ1,2−2R1−2R2υCT1C2θ1,6β1NTCH2Y3∗ˉθ2,2000β2NTBβ2NTC0∗∗θ3,30−2U000∗∗∗θ4,4−2U000∗∗∗∗θ5,50K2Y20∗∗∗∗∗ˉθ6,600∗∗∗∗∗∗ˉθ7,70∗∗∗∗∗∗∗θ8,8], |
ˉΘ(2,2)=[−12R10000000∗−12R2000000∗∗−12U00000∗∗∗−12U0000∗∗∗∗ˉθ13,1300θ13,16∗∗∗∗∗θ14,14−X20∗∗∗∗∗∗θ15,150∗∗∗∗∗∗∗θ16,16], |
in which:
ˉθ1,1=Q1+Q2−4R1−4R2+υCT1C1−F1Y1−H1Y3+2β1Z−2β1NTA+(σ22−σ21)24X1−(σ22−σ21)24X2+α1β21ΣT1Σ1,ˉθ2,2=σ21R1+σ22R2+(σ2−σ1)2U−2β2NT+(σ32−σ31)236X2+α2β22ΣT1Σ1,ˉθ6,6=−Y1+α3β21ΣT2Σ2+α4β22ΣT2Σ2,ˉθ7,7=−Y2+α5β21ΣT3Σ3+α6β22ΣT3Σ3,ˉθ13,13=−L+υCT3C3+α7β21ΣT4Σ4+α8β22ΣT4Σ4, |
then, the uncertain NNs (2.2) is exponentially stable with a mixed passive and H∞ performance index γ.
Proof. We use the same Lyapunov-Krasovskii functional in Theorem 3.1, such that matrices A,B,C,D are replaced by A+J1S1(t)Σ1,B+J2S2(t)Σ2,C+J3S3(t)Σ3,D+J4S4(t)Σ4, respectively. Then applying Lemma 2.5, we get
xT(t)(−2β1NTΔA(t))x(t)≤α1xT(t)β1ΣT1Σ1β1x(t)+α−11xT(t)NTJ1JT1Nx(t),xT(t)(−β2NTΔA(t))˙x(t)+˙xT(t)(−β2ΔAT(t)N)x(t)≤α2˙xT(t)β2ΣT1Σ1β2˙x(t)+α−12xT(t)NTJ1JT1Nx(t),xT(t)β1NTΔB(t)f(x(t))+fT(x(t))β1ΔBT(t)Nx(t)≤α3fT(x(t))β1ΣT2Σ2β1f(x(t))+α−13xT(t)NTJ2JT2Nx(t),˙xT(t)β2NTΔB(t)f(x(t))+fT(x(t))β2ΔBT(t)N˙x(t)≤α4fT(x(t))β2ΣT2Σ2β2f(x(t))+α−14˙xT(t)NTJ2JT2N˙x(t),xT(t)β1NTΔC(t)k(x(t−σ(t)))+kT(x(t−σ(t)))β1ΔCT(t)Nx(t)≤α5kT(x(t−σ(t)))β1ΣT3Σ3β1k(x(t−σ(t)))+α−15xT(t)NTJ3JT3Nx(t),˙xT(t)β2NTΔC(t)k(x(t−σ(t)))+kT(x(t−σ(t)))β2ΔCT(t)N˙x(t)≤α6kT(x(t−σ(t)))β2ΣT3Σ3β2k(x(t−σ(t)))+α−16˙xT(t)NTJ3JT3N˙x(t),xT(t)β1NTΔD(t)∫t−δ1(t)t−δ2(t)h(x(s))ds+∫t−δ1(t)t−δ2(t)hT(x(s))dsβ1ΔDT(t)Nx(t)≤α7∫t−δ1(t)t−δ2(t)hT(x(s))dsβ1ΣT4Σ4β1∫t−δ1(t)t−δ2(t)h(x(s))ds+α−17xT(t)NTJ4JT4Nx(t),˙xT(t)β2NTΔD(t)∫t−δ1(t)t−δ2(t)h(x(s))ds+∫t−δ1(t)t−δ2(t)hT(x(s))dsβ2ΔDT(t)N˙x(t)≤α8∫t−δ1(t)t−δ2(t)hT(x(s))dsβ2ΣT4Σ4β2∫t−δ1(t)t−δ2(t)h(x(s))ds+α−18˙xT(t)NTJ4JT4N˙x(t). |
Then applying the similar proof of Theorem 3.1 and Lemma 2.6, we have
˙V(t)+υzT(t)z(t)−2(1−υ)γzT(t)ω(t)−γ2ωT(t)ω(t)≤ξT(t)(εΨ(1)+(1−ε)Ψ(2))ξ(t), |
where, Ψ(i)=Ψ+Θi (i=1,2) with Ψ and Θi are defined in (3.32), (3.33).
Since 0≤ε≤1, the term εΨ(1)+(1−ε)Ψ(2) is a convex combination of Ψ(1) and Ψ(2). The combinations are negative definite only if
Ψ(1)<0, | (3.34) |
Ψ(2)<0. | (3.35) |
Therefore, (3.34) and (3.35) are equivalent to (3.32) and (3.33), respectively. This completes the proof.
In the third part, we will investigate the stability of a special model of the neural networks, in order to compare the maximum delay with existing results.
Remark 2. We consider the following neural network model as a special case of the system (2.1)
˙x(t)=−Ax(t)+Bf(x(t))+Ck(x(t−σ(t))). | (3.36) |
Corollary 3.3. For given scalars σ1,σ2,β1 and β2, if there exist nine n×n matrices P>0,Q1>0,Q2>0,R1>0,R2>0,U>0,X1>0,X2>0,N>0 and two n×n positive diagonal matrices Y1>0,Y2>0 such that the following LMIs hold:
Π+Π1<0, | (3.37) |
Π+Π2<0, | (3.38) |
where
Π1=−e13X1eT13,Π2=−e12X1eT12,Π=[θ′(i,j)]13×13, |
with (θ′(i,j))T=θ′(j,i),
θ′(1,1)=Q1+Q2−4R1−4R2−F1Y1−2β1NTA+(σ22−σ21)24X1−(σ22−σ21)24X2,θ′(1,2)=P−β1NT−β2NTA,θ′(1,3)=−2R1,θ′(1,4)=−2R2,θ′(1,6)=F2Y1+β1NTB,θ′(1,7)=β1NTC,θ′(1,8)=6R1,θ′(1,9)=6R2,θ′(1,12)=σ22−σ212X2,θ′(1,13)=σ22−σ212X2,θ′(2,2)=σ21R1+σ22R2+(σ2−σ1)2U−2β2NT+(σ32−σ31)236X2,θ′(2,6)=β2NTB,θ′(2,7)=β2NTC,θ′(3,3)=−Q1−4R1−4U,θ′(3,5)=−2U,θ′(3,8)=6R1,θ′(3,10)=6U,θ′(4,4)=−Q2−4R2−4U,θ′(4,5)=−2U,θ′(4,9)=6R2,θ′(4,11)=6U,θ′(5,5)=−8U−K1Y2,θ′(5,7)=K2Y2,θ′(5,10)=6U,θ′(5,11)=6U,θ′(6,6)=−Y1,θ′(7,7)=−Y2,θ′(8,8)=−12R1,θ′(9,9)=−12R2,θ′(10,10)=−12U,θ′(11,11)=−12U,θ′(12,12)=−X1−X2,θ′(12,13)=−X2,θ′(13,13)=−X1−X2, |
another terms are 0,
then, the NNs (3.36) is exponentially stable.
Proof. We choose the following Lyapunov–Krasovskii functional candidate for the system (3.36) as
V(x(t),t)=8∑i=1Vi(x(t),t), |
where
V1(x(t),t)=xT(t)Px(t),V2(x(t),t)=∫tt−σ1xT(s)Q1x(s)ds,V3(x(t),t)=∫tt−σ2xT(s)Q2x(s)ds,V4(x(t),t)=σ1∫0−σ1∫tt+s˙xT(τ)R1˙x(τ)dτds,V5(x(t),t)=σ2∫0−σ2∫tt+s˙xT(τ)R2˙x(τ)dτds,V6(x(t),t)=(σ2−σ1)∫−σ1−σ2∫tt+s˙xT(τ)U˙x(τ)dτds,V7(x(t),t)=(σ22−σ21)2∫−σ1−σ2∫0β∫tt+λxT(s)X1x(s)dsdλdβ,V8(x(t),t)=(σ32−σ31)6∫−σ1−σ2∫0β∫0λ∫tt+φ˙xT(s)X2˙x(s)dsdφdλdβ. |
By applying similar proof in Theorem 3.1, the system (3.36) is exponentially stable.
Remark 3. Recently, the robust passivity problem of uncertain neural networks with interval discrete and distributed time-varying delays has been studied in [14]. Also, robust reliable H∞ control problem of uncertain neural networks with mixed time delays has been discussed in [23]. However, the problem of mixed passive and H∞ for uncertain neural networks with interval discrete and distributed time-varying delays has not been investigated yet. The results in this paper provide the sufficient conditions to assure that the uncertain neural network is exponentially stable with mixed passive and H∞ index γ. The conditions are obtained by constructing a Lyapunov-Krasovskii functional consisting novel integral terms.
Remark 4. It is well known that time delay is a normal phenomenon that appears in neural networks since the neural networks consist of a large number of neurons that connect with each other into a diversity of axon sizes and lengths. Practically time delay can occur in an irregular fashion such as sometimes the time-varying delays are not differentiable. So, in this work, the interval discrete and distributed time-varying delays do not necessitate being differentiable functions.
Remark 5. It is well known that the H∞ theory is very important in the control problem. Besides, the H∞ approaches are used in control theory to synthesize controllers achieving stabilization with an H∞ norm bound limited to disturbance reduction. The passivity theory is widely used in system synthesis and analysis, as the system with passivity performance can effectively reduce the impact of noise. In fact, the passivity system does not produce energy by itself, but it will use the system's energy. The main property of passivity is that can keep the system internally stable. By the above mentioned, the obtained results are based on mixed passivity and H∞ problem for uncertain neural networks with mixed time-varying delays. In comparison between the design of mixed H∞/passive performance and a single H∞ or passive controller, the control problem under mixed H∞/passive performance consideration is more general than a single H∞ or passive controller for example, a simple actual mixed H∞ and passive performance index is employed in handling with the event-triggered reliable control issue for the fuzzy Markov jump systems (FMJSs), which can achieve the H∞ or passive event-triggered reliable control problem for FMJSs by turning some fixed parameters. Hence, this paper are more general and convenient than the existing individual passive and H∞ problem.
Remark 6. In this work, the Lyapunov-Krasovskii functional consisting single, double, triple, and quadruple integral terms, which full of the information of the delays σ1,σ2,δ1,δ2, and a state variable x(t). Furthermore, more information on activation functions has taken fully into the stability and performance analysis that is F−i≤fi(xi(t))xi(t)≤F+i, K−i≤ki(xi(t−σ(t)))xi(t−σ(t))≤K+i, and H−i≤hi(xi(t))xi(t)≤H+i are addressed in the calculation. Hence, the construction and the technique for computation of the Lyapunov-Krasovskii functional are the main key to improve results of this work. In the proof of Theorems 3.1, 3.2, and Corollary 3.3, integral inequalities and convex combination technique are used to bound the derivative of Lyapunov-Krasovskii functional, which provide tighter than the inequalities in [30,31,32,38]. All of these lead to the improved results in our work as we can see the compared results with some exiting works in numerical examples. However, the complex computation of the Lyapunov-Krasovskii functional leads to the LMI derived in this work which contains many information of the system. It is feasible for NNs with large number of neurons which can be solved by using the Matlab LMI toolbox. Hence, for further work, it is interesting for researchers to improve the technique for a simple Lyapunov-Krasovskii functional and also achieve better results.
In this section, we provided four numerical examples which are illustrated the effectiveness of the proposed results. Moreover, two numerical examples show less conservative results than others.
Example 4.1. We consider the neural networks (3.36) with matrix parameters in [30]:
A=[1001],B=[−10.50.5−1.5],C=[−20.50.5−2],F1=K1=[0000], and F2=K2=[0.2000.4]. |
By taking parameters β1=β2=1 and solving Example 4.1 using LMIs in Corollary 3.3, we obtain maximum allowable values of σ2 for different σ1 without the upper bound of differentiable delay (μ) as shown in Table 1. Table 1 shows that the results derived in this paper are less conservative than the results in [30].
Methods | σ1 | μ=0.8 | μ=0.9 | Unknownμ |
[30] | σ1=0.5 | 0.8262 | 0.8215 | - |
Corollary 3.3 | - | - | 0.9976 | |
[30] | σ1=0.75 | 0.9669 | 0.9625 | - |
Corollary 3.3 | - | - | 1.1233 | |
[30] | σ1=1 | 1.1152 | 1.1108 | - |
Corollary 3.3 | - | - | 1.2710 |
Example 4.2. We consider the neural networks (3.36) with matrix parameters in [31,32,38]:
A=[1.5000.7],B=[0.05030.04540.09870.2075],C=[0.23810.93200.03880.5062],F1=K1=[0000], and F2=K2=[0.15000.4]. |
By taking parameters β1=β2=1 and solving Example 4.2 using LMIs in Corollary 3.3, we get maximum allowable values of σ2 for σ1=0 without the upper bound of differentiable delay (μ) as shown in Table 2. Table 2 illustrates that the results obtained in this paper are less conservative than the results in [31,32,38].
Methods | μ=0.5 | μ=0.55 | Unknownμ |
[38] | 3.0594 | 2.9814 | - |
[31] | 3.3377 | 3.2350 | - |
[32] | 3.4600 | 3.4100 | - |
Corollary 3.3 | - | - | 3.5814 |
Example 4.3. We consider the neural networks (3.1) with σ1=0.5,σ2=1.75,δ1=0.2,δ2=1.0,υ=0.1,β1=0.9,β2=0.2,
A=[1001],B=[0.2−0.1−0.50.1],C=[−0.50−0.3−0.2],D=[0.150.10−0.3],C1=[0.5000.3],I=[1001],C2=C3=C4=0.1I,F1=K1=H1=−0.4I,F2=K2=H2=0.4I,hi(xi)=tanh(xi), and fi(xi)=ki(xi)=0.2(∣xi+1∣−∣xi−1∣). |
LMIs of (3.2), (3.3) in Theorem 3.1 are solved, we obtain
P=[3.6577−0.2200−0.22003.7479],Q1=[4.03380.04600.04603.9284],Q2=[4.16840.05800.05804.0614],R1=[0.1741−0.0295−0.02950.1987],R2=[0.0292−0.0121−0.01210.0394],U=[0.2121−0.0178−0.01780.2033],L=[3.64300.02770.02773.7766],X1=[2.84160.01970.01972.8239],X2=[0.0479−0.0193−0.01930.0642],N=[2.3363−0.2470−0.24322.3840],Z=[−13.6905−0.9328−0.5358−12.9525],Y1=[3.9748003.9748],Y2=[0.4324000.4324],Y3=[4.6765004.6765]. |
The state feedback control is obtained by
U(t)=N−1Zx(t)=[−5.9479−0.9843−0.8316−5.5335]x(t),t≥0. |
The maximum allowable values of σ2 for different values of σ1 are shown in Table 3. Furthermore, we want to find the relation among the scalars σ2, υ, and γ. For three different values of υ, we set υ=0, υ=0.5, and υ=1, respectively, which means the passivity case, passivity and H∞ case, and H∞ case are studied, respectively. Moreover, we choose the values of σ2 from σ2=0.5 to σ2=2 and other parameters are fixed by σ1=0.2, δ1=0.2, δ2=0.8, β1=0.9, β2=0.2. By applying Theorem 3.1 and Matlab LMI toolbox to solve LMIs (3.2) and (3.3), we have the relation among the parameters σ2, υ, and γ, which is presented in Table 4. Figure 1 shows the response solution x(t) in Example 4.3 where ω(t)=0 and the initial condition ϕ(t)=[−0.10.1]T. Figure 2 shows the response solution x(t) in Example 4.3 where ω(t) is Gaussian noise with mean 0 and variance 1 and the initial condition ϕ(t)=[−0.10.1]T.
Method | σ1=0 | σ1=0.5 | σ1=1 | σ1=2 | σ1=3 |
Theorem 3.1 | 2.1176 | 2.3865 | 2.5354 | 3.3564 | 4.1253 |
γmin | σ2=0.5 | σ2=1 | σ2=1.5 | σ2=2 |
υ=0 | 0.5672 | 0.6835 | 0.8135 | 0.9465 |
υ=0.5 | 0.7752 | 0.9683 | 1.1035 | 1.2156 |
υ=1 | 1.2331 | 1.4452 | 1.6862 | 1.7965 |
The numerical simulations are accomplished using the explicit Runge-Kutta-like method (dde45), extrapolation and interpolation by spline of the third order.
Example 4.4. We consider the uncertain neural networks (2.2) with σ1=0.7,σ2=1.5,δ1=0.2,δ2=1,υ=0.1,β1=0.9,β2=0.2,
A=[1001],B=[0.2−0.1−0.50.1],C=[−0.50−0.3−0.2],D=[0.150.10−0.3],C1=[0.5000.3],I=[1001],C2=C3=C4=0.1I,F1=K1=H1=−0.4I,F2=K2=H2=0.4I,J1=J2=J3=J4=0.2I,Σ1=Σ2=Σ3=Σ4=I,hi(xi)=tanh(xi), andfi(xi)=ki(xi)=0.2(∣xi+1∣−∣xi−1∣). |
LMIs of (3.32), (3.33) in Theorem 3.2 are solved, we obtain
P=[2.8510−0.1478−0.14782.7756],Q1=[2.31540.14640.14642.0641],Q2=[2.35340.14020.14022.1063],R1=[0.0523−0.0200−0.02000.0680],R2=[0.0131−0.0077−0.00770.0192],U=[0.2104−0.0062−0.00620.1748],L=[2.26850.01080.01082.3401],X1=[1.33480.01250.01251.3274],X2=[0.0224−0.0103−0.01030.0305],N=[1.6053−0.0435−0.09781.4801],Z=[−10.2677−0.5664−0.5137−9.2901],Y1=[3.0132003.0132],Y2=[0.4054000.4054],Y3=[3.4945003.4945],α1=1.9270,α2=0.6465,α3=1.2926,α4=1.9011,α5=0.0754,α6=0.6677,α7=1.2734,α8=1.8989. |
The state feedback control is obtained by
U(t)=N−1Zx(t)=[−6.4170−0.5239−0.7712−6.3115]x(t),t≥0. |
The maximum allowable values of σ2 for different values of σ1 are shown in Table 5. Furthermore, we want to find the relation among the scalars σ2, υ, and γ. For three different values of υ, we set υ=0, υ=0.5, and υ=1, respectively, which means the passivity case, passivity and H∞ case, and H∞ case are considered, respectively. Moreover, we choose the values of σ2 from σ2=0.5 to σ2=2 and other parameters are fixed by σ1=0.2, δ1=0.2, δ2=0.8, β1=0.9, β2=0.2. By applying Theorem 3.2 and Matlab LMI toolbox to solve LMIs (3.32) and (3.33), we have the relation among the parameters σ2, υ, and γ, which is presented in Table 6. Figure 3 shows the response solution x(t) in Example 4.4 where ω(t)=0 and the initial condition ϕ(t)=[−0.10.1]T. Figure 4 shows the response solution x(t) in Example 4.4 where ω(t) is Gaussian noise with mean 0 and variance 1 and the initial condition ϕ(t)=[−0.10.1]T.
Method | σ1=0 | σ1=0.5 | σ1=1 | σ1=2 | σ1=3 |
Theorem 3.2. | 1.8308 | 2.2056 | 2.4233 | 3.1232 | 3.8142 |
γmin | σ2=0.5 | σ2=1 | σ2=1.5 | σ2=2 |
υ=0 | 0.6354 | 0.7534 | 0.8756 | 0.9869 |
υ=0.5 | 0.8965 | 1.0231 | 1.2231 | 1.4365 |
υ=1 | 1.6352 | 1.7563 | 1.8641 | 1.9634 |
Remark 7. In this work, we choose σ1,σ2,δ1,δ2,β1,β2,γ are real numbers that satisfy 0≤σ1≤σ(t)≤σ2, 0≤δ1≤δ1(t)≤δ2(t)≤δ2, and γ>0. In practice, the designing of these parameters can occur in an appropriate range. Furthermore, the suitable values of σ1,σ2,δ1,δ2,β1,β2 lead to the smallest γ for the mixed passive and H∞ analysis.
Remark 8. The stability criteria of Theorem 3.1 in the form LMIs (3.2) and (3.3) can be easily to examine by using LMI toolbox in MATLAB [39]. The improved stability criteria by using the Lyapunov-Krasovskii functional is based on LMIs and the dimension of the LMIs depends on the number of the neurons in neural networks. Thus, the computational burden problem goes up. This problem is the issue in studying needs of LMI optimization in applied mathematics and the optimization research. Hence, in the further, new techniques should be considered to reduce the conservativeness caused by the time-delays such as the delay-fractioning approach and so on.
Remark 9. In the future work, it is very challenging to apply some lemmas or Lyapunov-Krasovskii functional used in this paper to apply into the quaternion-valued case to get improved stability conditions.
The problem of mixed passive and H∞ analysis for uncertain neural networks with the state feedback control is investigated in this paper. We obtain the new sufficient conditions to guarantee exponential stability with mixed passive and H∞ performance for the uncertain neural networks by using a Lyapunov-Krasovskii functional consisting single, double, triple, and quadruple integral terms with a feedback controller. Furthermore, integral inequalities and convex combination technique are applied to achieve the less conservative results for a special case of neural networks with interval discrete time-varying delays. The new criteria are in terms of linear matrix inequalities (LMIs) that cover H∞, and passive performance by setting parameters in the general performance index. Finally, numerical examples have been given to show the effectiveness of the proposed results and improve over some existing results in the literature. In the future work, the derived results and methods in this paper are expected to be applied to other systems such as fuzzy control systems, complex dynamical networks, quaternion-valued neural networks and so on [16,40,41].
The first author was supported by the Science Achievement Scholarship of Thailand (SAST). The second author was financially supported by Khon Kaen University. The third and the fourth authors were supported by the Program Management Unit for Human Resources & Institutional Development, Research and Innovation (grant number : B05F630095).
The authors declare no conflict of interest.
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Method | σ1=0 | σ1=0.5 | σ1=1 | σ1=2 | σ1=3 |
Theorem 3.1 | 2.1176 | 2.3865 | 2.5354 | 3.3564 | 4.1253 |
γmin | σ2=0.5 | σ2=1 | σ2=1.5 | σ2=2 |
υ=0 | 0.5672 | 0.6835 | 0.8135 | 0.9465 |
υ=0.5 | 0.7752 | 0.9683 | 1.1035 | 1.2156 |
υ=1 | 1.2331 | 1.4452 | 1.6862 | 1.7965 |
Method | σ1=0 | σ1=0.5 | σ1=1 | σ1=2 | σ1=3 |
Theorem 3.2. | 1.8308 | 2.2056 | 2.4233 | 3.1232 | 3.8142 |
γmin | σ2=0.5 | σ2=1 | σ2=1.5 | σ2=2 |
υ=0 | 0.6354 | 0.7534 | 0.8756 | 0.9869 |
υ=0.5 | 0.8965 | 1.0231 | 1.2231 | 1.4365 |
υ=1 | 1.6352 | 1.7563 | 1.8641 | 1.9634 |
Methods | σ1 | μ=0.8 | μ=0.9 | Unknownμ |
[30] | σ1=0.5 | 0.8262 | 0.8215 | - |
Corollary 3.3 | - | - | 0.9976 | |
[30] | σ1=0.75 | 0.9669 | 0.9625 | - |
Corollary 3.3 | - | - | 1.1233 | |
[30] | σ1=1 | 1.1152 | 1.1108 | - |
Corollary 3.3 | - | - | 1.2710 |
Methods | μ=0.5 | μ=0.55 | Unknownμ |
[38] | 3.0594 | 2.9814 | - |
[31] | 3.3377 | 3.2350 | - |
[32] | 3.4600 | 3.4100 | - |
Corollary 3.3 | - | - | 3.5814 |
Method | σ1=0 | σ1=0.5 | σ1=1 | σ1=2 | σ1=3 |
Theorem 3.1 | 2.1176 | 2.3865 | 2.5354 | 3.3564 | 4.1253 |
γmin | σ2=0.5 | σ2=1 | σ2=1.5 | σ2=2 |
υ=0 | 0.5672 | 0.6835 | 0.8135 | 0.9465 |
υ=0.5 | 0.7752 | 0.9683 | 1.1035 | 1.2156 |
υ=1 | 1.2331 | 1.4452 | 1.6862 | 1.7965 |
Method | σ1=0 | σ1=0.5 | σ1=1 | σ1=2 | σ1=3 |
Theorem 3.2. | 1.8308 | 2.2056 | 2.4233 | 3.1232 | 3.8142 |
γmin | σ2=0.5 | σ2=1 | σ2=1.5 | σ2=2 |
υ=0 | 0.6354 | 0.7534 | 0.8756 | 0.9869 |
υ=0.5 | 0.8965 | 1.0231 | 1.2231 | 1.4365 |
υ=1 | 1.6352 | 1.7563 | 1.8641 | 1.9634 |