# ℋ∞ Stability Conditions for Fuzzy Neural Networks

Authors: Choon Ki Ahn
Journal: Advances in Fuzzy Systems
 (2012)

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## Abstract

This paper presents a novel approach to assess the stability of fuzzy neural
networks. First, we propose a new condition for theℋ∞ stability of fuzzy neural
networks. Second, a new ℋ∞ stability condition based on linear matrix inequality
(LMI) is presented for fuzzy neural networks. These conditions also ensure asymptotic
stability without external input.


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## Body

## 1. Introduction

In this paper, we consider the following Takagi-Sugeno (T-S) fuzzy Hopfield neural network:(1)FuzzyRulei:IFω1isμi1and…ωsisμisTHENẋ(t)=Aix(t)+Wiϕ(x(t))+J(t),
where x(t)=[x1(t)⋯xn(t)]T∈Rn is the state vector, Ai=diag{-a(i,1),…,-a(i,n)}∈Rn×n(a(i,k)>0,k=1,…,n) is the self-feedback matrix, Wi∈Rn×n is the connection weight matrix, ϕ(x(t))=[ϕ1(x(t))⋯ϕn(x(t))]T:Rn→Rn is the nonlinear function vector satisfying the global Lipschitz condition with Lipschitz constant Lϕ>0, J(t)∈Rn is an external input vector, ωj(j=1,…,s) is the premise variable, μij(i=1,…,r,j=1,…,s) is the fuzzy set that is characterized by membership function, r is the number of the IF-THEN rules, and s is the number of the premise variables. Using a singleton fuzzifier, product fuzzy inference, and weighted average defuzzifier, the system (1) is inferred as follows:
(2)ẋ(t)=∑i=1rhi(ω)[Aix(t)+Wiϕ(x(t))+J(t)],
where ω=[ω1,…,ωs], hi(ω)=wi(ω)/∑j=1rwj(ω), and wi:Rs→[0,1](i=1,…,r) is the membership function of the system with respect to the fuzzy rule i. hi can be regarded as the normalized weight of each IF-THEN rule, and it satisfies hi(ω)≥0,∑i=1rhi(ω)=1. The T-S fuzzy models have been of great importance in academic research and industrial applications. Essentially, these models are based on the use of a set of fuzzy rules to describe nonlinear systems in terms of a set of local linear models that are smoothly connected by fuzzy membership functions [1]. The T-S fuzzy models can be also used to represent several complex nonlinear systems by having a set of neural networks as its consequent parts. Some stability problems for neural networks based on T-S fuzzy models have been investigated [2–8]. In [2, 3], some linear matrix inequality (LMI) conditions for the stochastic stability of fuzzy neural networks were derived. The authors in [4] investigated the robust stability of uncertain fuzzy neural networks and proposed a delay-dependent condition such that these neural networks are asymptotically stable. In [5], Ahn presented a new delay-dependent state estimation method for fuzzy neural networks. In [6], Mathiyalagan et al. further dealt with the stability problem of fuzzy neural networks with various activation functions. Recently, some conditions for the passivity analysis of fuzzy neural networks were proposed in [7, 8]. Despite these advances in the stability analysis of fuzzy neural networks, most research results were restricted to fuzzy neural networks without external disturbance. With the existing results, it is difficult to guarantee the stability of fuzzy neural networks with external disturbance. Thus, it is desirable to investigate some stability conditions of fuzzy neural networks with external disturbance. This situation motivates our investigation.On the other hand, model uncertainties always exist, along with a lack of statistical information on the signals in real physical systems. This has led in recent years to an interest in anℋ∞ approach [9]. Analysis and synthesis in an ℋ∞ framework have advantages such as effective disturbance attenuation, less sensitivity to uncertainties, and many practical applications. This paper provides an answer to the question of whether an ℋ∞ stability condition can be obtained for T-S fuzzy neural networks. To the best of our knowledge, the ℋ∞ analysis of T-S fuzzy neural networks has not yet been reported in the literature.In this paper, we present newℋ∞ stability conditions for T-S fuzzy Hopfield neural networks. The conditions proposed in this paper are a new contribution to the stability analysis of fuzzy neural networks. In contrast to the existing stability conditions for fuzzy neural networks, an advantage of the proposed conditions is to attenuate the effect of external disturbance to a prescribed level. This paper is organized as follows. In Section 2, new ℋ∞ stability conditions are derived. In Section 3, a numerical example is given, and finally, conclusions are presented in Section 4.


## 2. Newℋ∞ Stability Conditions

Given a prescribed level of noise attenuationγ>0, the purpose of this paper is to find conditions such that the fuzzy neural network (2) with J(t)=0 is asymptotically stable (limt→∞x(t)=0) and
(3)∫0∞xT(t)x(t)dt<γ2∫0∞JT(t)J(t)dt,
under zero-initial conditions for all nonzero J(t)∈L2[0,∞), where L2[0,∞) is the space of square integrable vector functions over [0,∞).Now we derive anℋ∞ stability condition of the T-S fuzzy Hopfiel neural network (2) in the following theorem.Theorem 1.
For a given levelγ>0, the T-S fuzzy Hopfiel neural network (2) is ℋ∞ stable if
(4)‖Wi‖<1Lϕki-1-(1+1γ2)‖P‖2,‖P‖<ki-11+1/γ2,ki>1,P=PT>0,
where P satisfies the Lyapunov inequality AiTP+PAi<-kiI for i=1,…,r.Proof.
We consider the functionV(t)=xT(t)Px(t). Its time derivative along the trajectory of (2) satisfies
(5)V̇(t)<∑i=1rhi(ω){-kixT(t)x(t)+2xT(t)PWiϕ(x(t))+2xT(t)PJ(t)}.
By Young’s inequality [10], we have
(6)2xT(t)PWiϕ(x(t))≤xT(t)PPTx(t)+ϕT(x(t))WiTWiϕ(x(t))≤‖P‖2‖x(t)‖2+‖Wi‖2‖ϕ(x(t))‖2≤‖P‖2‖x(t)‖2+Lϕ2‖Wi‖2‖x(t)‖2,2xT(t)PJ(t)≤1γ2xT(t)PPTx(t)+γ2JT(t)J(t)≤1γ2‖P‖2‖x(t)‖2+γ2‖J(t)‖2.
Substituting (6) into (5), we finally obtain
(7)V̇(t)<∑i=1rhi(ω)×{-(ki-(1+1γ2)‖P‖2-Lϕ2‖Wi‖2)×‖x(t)‖2+γ2‖J(t)‖2(1+1γ2)}=-∑i=1rhi(ω)(ki-1-(1+1γ2)‖P‖2-Lϕ2‖Wi‖2)×‖x(t)‖2+∑i=1rhi(ω){-‖x(t)‖2+γ2‖J(t)‖2}.
If the following condition is satisfied
(8)(ki-1-(1+1γ2)‖P‖2-Lϕ2‖Wi‖2)>0,
we have
(9)V̇(t)<-‖x(t)‖2+γ2‖J(t)‖2.
Integrating both sides of (9) from 0 to ∞ gives
(10)V(∞)-V(0)<-∫0∞xT(t)x(t)dt+γ2∫0∞JT(t)J(t)dt.
Since V(∞)≥0 and V(0)=0, we have the relation (3). The condition (8) implies
(11)‖Wi‖2<1Lϕ2(ki-1-(1+1γ2)‖P‖2),‖P‖2<ki-11+1/γ2,
for i=1,…,r. This completes the proof.Corollary 2.
WhenJ(t)=0, the condition (4) ensures that the fuzzy neural network (2) is asymptotically stable.Proof.
WhenJ(t)=0, from (9), we have
(12)V̇(t)<-‖x(t)‖2<0,∀x(t)≠0.
This relation ensures that the fuzzy neural network (2) is asymptotically stable from Lyapunov stability theory. This completes the proof.Next, we propose a new LMI-based condition for theℋ∞ stability of the fuzzy neural network (2). This LMI condition can be facilitated readily via standard numerical algorithms [11, 12].Theorem 3.
For a given levelγ>0, the T-S fuzzy Hopfiel neural network (2) is ℋ∞ stable if there exist a positive symmetric matrix P and a positive scalar ϵ such that
(13)[AiTP+PAi+(ϵLϕ2+1)IPWiPWiTP-ϵI0P0-γ2I]<0,
for i=1,…,r.Proof.
Consider the functionV(t)=xT(t)Px(t). By Young’s inequality [10], it is clear that the following relation is satisfied:
(14)ϵ[Lϕ2xT(t)x(t)-ϕT(x(t))ϕ(x(t))]≥0.
By using (14), the time derivative of V(t) along the trajectory of (2) is
(15)V̇(t)=∑i=1rhi(ω){xT(t)[AiTP+PAi]x(t)+2xT(t)PWiϕ(x(t))+2xT(t)PJ(t)}≤∑i=1rhi(ω){xT(t)[AiTP+PAi]x(t)+2xT(t)PWiϕ(x(t))+2xT(t)PJ(t)+ϵ[Lϕ2xT(t)x(t)-ϕT(x(t))ϕ(x(t))]}=∑i=1rhi(ω)[x(t)ϕ(x(t))J(t)]T×[AiTP+PAi+(ϵLϕ2+1)IPWiPWiTP-ϵI0P0-γ2I][x(t)ϕ(x(t))J(t)]+∑i=1rhi(ω){-xT(t)x(t)+γ2JT(t)J(t)}.
If the LMI (13) is satisfied, we have
(16)V̇(t)<-xT(t)x(t)+γ2JT(t)J(t).
Integrating both sides of (16) from 0 to ∞ gives
(17)V(∞)-V(0)<-∫0∞xT(t)x(t)dt+γ2∫0∞JT(t)J(t)dt.
Since V(∞)≥0 and V(0)=0, we have the relation (3). This completes the proof.Corollary 4.
WhenJ(t)=0, the LMI condition (13) ensures that the fuzzy neural network (2) is asymptotically stable.Proof.
WhenJ(t)=0, from (16), we have
(18)V̇(t)<-xT(t)x(t)<0,∀x(t)≠0.
This relation ensures that the fuzzy neural network (2) is asymptotically stable from Lyapunov stability theory. This completes the proof.Remark 5.
The proposed scheme can be applied to several real-world problems. For example, in image-based visual servo control of an unknown aerial robot vehicle [13, 14], a possible application of the proposed scheme is to use fuzzy neural networks to build a mathematical model from experimental data and then check the ℋ∞ stability of this model. If the ℋ∞ stability of this model is guaranteed by Theorem 1 or Theorem 3, we can use this model in the implementation of the nonlinear controller. Learning algorithms already exist, which ensures error convergence for fuzzy neural networks [15]. Even in presence of model mismatching, the identification error remains bounded. Therefore, from the point of view of control, the ℋ∞ stability analysis for fuzzy neural networks is a prerequisite for successful applications of the networks.Remark 6.
The condition (4) in Theorem 1 is a matrix norm-based criterion for checking the ℋ∞ stability of fuzzy neural networks. But the condition (13) in Theorem 3 is an LMI based criterion. This condition can be facilitated readily via several numerical algorithms [11, 12]. Hence, this condition is computationally attractive.


## 3. Numerical Example

Consider the following T-S fuzzy Hopfield neural network:(19)FuzzyRule1:IFω1isμ11and…ωsisμ1sTHENẋ(t)=A1x(t)+W1ϕ(x(t))+J(t),(20)FuzzyRule2:IFω1isμ21and…ωsisμ2sTHENẋ(t)=A2x(t)+W2ϕ(x(t))+J(t),
where
(21)x(t)=[x1(t)x2(t)],J(t)=[J1(t)J2(t)],ϕ(x(t))=[tanh(x1(t))tanh(x2(t))],A1=[-2.800-4.1],A2=[-3.900-2.8],W1=[-10.40-0.1],W2=[0.2-0.80.40.5].
The fuzzy membership functions are taken as h1(ω)=sin2(x1(t)) and h2(ω)=cos2(x1(t)). By applying Theorem 3 via the Matlab LMI Control Toolbox [12], we have the following feasible solution:
(22)P=[0.69250.01010.01010.9734],ϵ=0.8563,
with the ℋ∞ performance index γ=0.6. When the initial condition is given by x(0)=[2.90.8]T and the external disturbance Ji(t)(i=1,2) is given by a Gaussian noise with mean 0 and variance 1, Figure 1 shows the trajectories of state vector x(t). This simulation result confirms that the proposed condition guarantees to reduce the effect of the external disturbance J(t) on the state vector x(t).Figure 1
Responses of the state vectorx(t).


## 4. Conclusion

In this paper, as our main contribution, we establish new conditions for the weights of the interconnection matrix of fuzzy neural networks, in order to ensureℋ∞ stability. These conditions also guarantee asymptotic stability for external input identically equal to zero.



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