ℋ_∞ Stability Conditions for Fuzzy Neural Networks

Abstract

This paper presents a novel approach to assess the stability of fuzzy neural networks. First, we propose a new condition for the ℋ_∞ 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.

1. Introduction

In this paper, we consider the following Takagi-Sugeno (T-S) fuzzy Hopfield neural network: 𝐅𝐮𝐳𝐳𝐲𝐑𝐮𝐥𝐞𝑖:𝐈𝐅𝜔₁is𝜇_𝑖1and…𝜔_𝑠is𝜇_𝑖𝑠(𝑡)𝐓𝐇𝐄𝐍𝑥̇=𝐴_𝑖𝑥(𝑡)+𝑊_𝑖𝜙(𝑥(𝑡))+𝐽(𝑡), (1) where 𝑥(𝑡)=[𝑥₁(𝑡)⋯𝑥_𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the state vector, 𝐴_𝑖=diag{−𝑎_(𝑖,1),…,−𝑎_(𝑖,𝑛)}∈𝑅^𝑛×𝑛(𝑎_(𝑖,𝑘)>0,𝑘=1,…,𝑛) is the self-feedback matrix, 𝑊_𝑖∈𝑅^𝑛×𝑛 is the connection weight matrix, 𝜙(𝑥(𝑡))=[𝜙₁(𝑥(𝑡))⋯𝜙_𝑛(𝑥(𝑡))]^𝑇∶𝑅^𝑛→𝑅^𝑛 is the nonlinear function vector satisfying the global Lipschitz condition with Lipschitz constant 𝐿_𝜙>0, 𝐽(𝑡)∈𝑅^𝑛 is an external input vector, 𝜔_𝑗(𝑗=1,…,𝑠) is the premise variable, 𝜇_𝑖𝑗(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: 𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜔)𝐴_𝑖𝑥(𝑡)+𝑊_𝑖𝜙(𝑥(𝑡))+𝐽(𝑡), (2) where 𝜔=[𝜔₁,…,𝜔_𝑠], ℎ_𝑖(𝜔)=𝑤_𝑖(𝜔)/∑^𝑟_𝑗=1𝑤_𝑗(𝜔), and 𝑤_𝑖∶𝑅^𝑠→[0,1](𝑖=1,…,𝑟) is the membership function of the system with respect to the fuzzy rule 𝑖. ℎ_𝑖 can be regarded as the normalized weight of each IF-THEN rule, and it satisfies ℎ_𝑖(𝜔)≥0,∑^𝑟_𝑖=1ℎ_𝑖(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 an ℋ_∞ 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 new ℋ_∞ 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. New ℋ_∞ Stability Conditions

Given a prescribed level of noise attenuation 𝛾>0, the purpose of this paper is to find conditions such that the fuzzy neural network (2) with 𝐽(𝑡)=0 is asymptotically stable (lim_𝑡→∞𝑥(𝑡)=0) and ^∞₀𝑥^𝑇(𝑡)𝑥(𝑡)𝑑𝑡<𝛾^2∞₀𝐽^𝑇(𝑡)𝐽(𝑡)𝑑𝑡, (3) 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 an ℋ_∞ stability condition of the T-S fuzzy Hopfiel neural network (2) in the following theorem.

Theorem 1. For a given level 𝛾>0, the T-S fuzzy Hopfiel neural network (2) is ℋ_∞ stable if 𝑊_𝑖<1𝐿_𝜙𝑘_𝑖1−1−1+𝛾²‖𝑃‖²,‖𝑃‖<𝑘_𝑖−11+1/𝛾²,𝑘_𝑖>1,𝑃=𝑃^𝑇>0, (4) where 𝑃 satisfies the Lyapunov inequality 𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖<−𝑘_𝑖𝐼 for 𝑖=1,…,𝑟.

Proof. We consider the function 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡). Its time derivative along the trajectory of (2) satisfies 𝑉̇(𝑡)<^𝑟_𝑖=1ℎ_𝑖(𝜔)−𝑘_𝑖𝑥^𝑇(𝑡)𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝑊_𝑖𝜙(𝑥(𝑡))+2𝑥^𝑇(𝑡)𝑃𝐽(𝑡). (5) By Young’s inequality [10], we have 2𝑥^𝑇(𝑡)𝑃𝑊_𝑖𝜙(𝑥(𝑡))≤𝑥^𝑇(𝑡)𝑃𝑃^𝑇𝑥(𝑡)+𝜙^𝑇(𝑥(𝑡))𝑊^𝑇_𝑖𝑊_𝑖𝜙(𝑥(𝑡≤‖𝑃‖))₂‖𝑥(𝑡)‖²+𝑊_𝑖²𝜙(𝑥(𝑡))²≤‖𝑃‖²‖𝑥(𝑡)‖²+𝐿²_𝜙𝑊_𝑖²‖𝑥(𝑡)‖²,2𝑥^𝑇(𝑡)(𝑡)≤1𝑃𝐽𝛾²𝑥^𝑇(𝑡)𝑃𝑃^𝑇𝑥(𝑡)+𝛾²𝐽^𝑇(𝑡)𝐽(𝑡)≤1𝛾²‖𝑃‖²‖𝑥(𝑡)‖²+𝛾²‖𝐽(𝑡)‖². (6) Substituting (6) into (5), we finally obtain 𝑉̇(𝑡)<^𝑟_𝑖=1ℎ_𝑖(𝜔)×−𝑘_𝑖−11+𝛾²‖𝑃‖²−𝐿²_𝜙𝑊_𝑖²×‖𝑥(𝑡)‖²+𝛾²‖𝐽(𝑡)‖²=−^𝑟_𝑖=1ℎ_𝑖(𝜔)𝑘_𝑖1−1−1+𝛾²‖𝑃‖²−𝐿²_𝜙𝑊_𝑖²×‖𝑥(𝑡)‖²+^𝑟_𝑖=1ℎ_𝑖(𝜔)−‖𝑥(𝑡)‖²+𝛾²‖𝐽(𝑡)‖². (7) If the following condition is satisfied 𝑘_𝑖1−1−1+𝛾²‖𝑃‖²−𝐿²_𝜙𝑊_𝑖²>0, (8) we have 𝑉̇(𝑡)<−‖𝑥(𝑡)‖²+𝛾²‖𝐽(𝑡)‖². (9) Integrating both sides of (9) from 0 to ∞ gives 𝑉(∞)−𝑉(0)<−^∞₀𝑥^𝑇(𝑡)𝑥(𝑡)𝑑𝑡+𝛾^2∞₀𝐽^𝑇(𝑡)𝐽(𝑡)𝑑𝑡. (10) Since 𝑉(∞)≥0 and 𝑉(0)=0, we have the relation (3). The condition (8) implies 𝑊_𝑖²<1𝐿²_𝜙𝑘_𝑖1−1−1+𝛾²‖𝑃‖²,‖𝑃‖²<𝑘_𝑖−11+1/𝛾², (11) for 𝑖=1,…,𝑟. This completes the proof.

Corollary 2. When 𝐽(𝑡)=0, the condition (4) ensures that the fuzzy neural network (2) is asymptotically stable.

Proof. When 𝐽(𝑡)=0, from (9), we have 𝑉̇(𝑡)<−‖𝑥(𝑡)‖²<0,∀𝑥(𝑡)≠0. (12) 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 the ℋ_∞ 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 level 𝛾>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 𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖+𝜖𝐿²_𝜙+1𝐼𝑃𝑊_𝑖𝑃𝑊^𝑇_𝑖𝑃−𝜖𝐼0𝑃0−𝛾²𝐼<0, (13) for 𝑖=1,…,𝑟.

Proof. Consider the function 𝑉(𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡). By Young’s inequality [10], it is clear that the following relation is satisfied: 𝜖𝐿²_𝜙𝑥^𝑇(𝑡)𝑥(𝑡)−𝜙^𝑇(𝑥(𝑡))𝜙(𝑥(𝑡))≥0. (14) By using (14), the time derivative of 𝑉(𝑡) along the trajectory of (2) is 𝑉̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝜔)𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝑊_𝑖𝜙(𝑥(𝑡))+2𝑥^𝑇(𝑡)(𝑡≤𝑃𝐽)_𝑟𝑖=1ℎ_𝑖(𝜔)𝑥^𝑇(𝑡)𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝑊_𝑖𝜙(𝑥(𝑡))+2𝑥^𝑇(𝑡)(𝑡)𝐿𝑃𝐽+𝜖²_𝜙𝑥^𝑇(𝑡)𝑥(𝑡)−𝜙^𝑇(𝑥(𝑡𝜙(𝑥(𝑡=))))_𝑟𝑖=1ℎ_𝑖(𝜔)𝑥(𝑡)𝜙(𝑥(𝑡𝐽(𝑡)))^𝑇×𝐴^𝑇_𝑖𝑃+𝑃𝐴_𝑖+𝜖𝐿²_𝜙+1𝐼𝑃𝑊_𝑖𝑃𝑊^𝑇_𝑖𝑃−𝜖𝐼0𝑃0−𝛾²𝐼𝑥(𝑡)𝜙(𝑥(𝑡𝐽(𝑡)+))_𝑟𝑖=1ℎ_𝑖(𝜔)−𝑥^𝑇(𝑡)𝑥(𝑡)+𝛾²𝐽^𝑇(𝑡)𝐽(𝑡). (15) If the LMI (13) is satisfied, we have 𝑉̇(𝑡)<−𝑥^𝑇(𝑡)𝑥(𝑡)+𝛾²𝐽^𝑇(𝑡)𝐽(𝑡). (16) Integrating both sides of (16) from 0 to ∞ gives 𝑉(∞)−𝑉(0)<−^∞₀𝑥^𝑇(𝑡)𝑥(𝑡)𝑑𝑡+𝛾^2∞₀𝐽^𝑇(𝑡)𝐽(𝑡)𝑑𝑡. (17) Since 𝑉(∞)≥0 and 𝑉(0)=0, we have the relation (3). This completes the proof.

Corollary 4. When 𝐽(𝑡)=0, the LMI condition (13) ensures that the fuzzy neural network (2) is asymptotically stable.

Proof. When 𝐽(𝑡)=0, from (16), we have 𝑉̇(𝑡)<−𝑥^𝑇(𝑡)𝑥(𝑡)<0,∀𝑥(𝑡)≠0. (18) 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: 𝐅𝐮𝐳𝐳𝐲𝐑𝐮𝐥𝐞1:𝐈𝐅𝜔₁is𝜇₁₁and…𝜔_𝑠is𝜇_1𝑠(𝑡)𝐓𝐇𝐄𝐍𝑥̇=𝐴₁𝑥(𝑡)+𝑊₁𝜙(𝑥(𝑡))+𝐽(𝑡),𝐅𝐮𝐳𝐳𝐲𝐑𝐮𝐥𝐞2:𝐈𝐅𝜔₁is𝜇₂₁and…𝜔_𝑠is𝜇_2𝑠(𝑡)𝐓𝐇𝐄𝐍𝑥̇=𝐴₂𝑥(𝑡)+𝑊₂𝜙(𝑥(𝑡))+𝐽(𝑡), (19)(20) where 𝑥(𝑡)=𝑥₁(𝑡)𝑥₂(𝑡)(𝑡)=𝐽,𝐽¹(𝑡)𝐽₂(𝑡),𝜙(𝑥(𝑡=𝑥))tanh₁(𝑡)tanh𝑥₂(𝑡),𝐴₁=−2.800−4.1,𝐴₂=,𝑊−3.900−2.8₁=−10.40−0.1,𝑊₂=.0.2−0.80.40.5 (21) 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: 𝑃=0.69250.01010.01010.9734,𝜖=0.8563, (22) 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 𝐽_𝑖(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 vector 𝑥(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 ensure ℋ_∞ stability. These conditions also guarantee asymptotic stability for external input identically equal to zero.