On estimating neuronal states of delayed neural networks based on canonical Bessel-Legendre inequalities
Introduction
Since John Hopfield established a Hopfield net and convinced its ability of finding acceptable solutions for the Travelling Salesman problem in the 1980s, neural networks have been a hot research issue and have found many applications in a wide range of disciplines, to mention a few, quantum chemistry, artificial intelligence, data mining, and medical diagnosis [1], [2], [3]. Many applications of neural networks, such as image processing, pattern recognition and geoscience, depend on neuronal state information. In relatively large-scale neural networks, usually, the neuronal state information is not fully available in the network outputs. In order to employ the neural networks to complete certain missions, it is significant to estimate the neuronal state information through available measurements. As a consequence, a plenty of results have been reported on designing different types of estimators for neural networks, see [4], [5], [6]. Among these estimators, a Luenberger estimator, which gives an estimation error dependent only on the initial neuronal states, is applied to a range of practical estimations, e.g. to neural networks with quantization and data dropouts [7], and to neural networks with stochastically damaged samplings under Round-Robin scheduling [8].
The theory of dissipativity, which was first proposed in [9], plays a significant role in neural networks, and gives a forceful instrument in depicting essential system behaviors including stability and passivity. Dissipativity theory has many applications in circuit systems, internal combustion engine, robotics, and draws considerable attention from relevant researchers (see [10], [11], [12] and references therein). On the other hand, due to amplifiers’ limited switching speed, time delay is an inherent feature for both artificial and biological neural networks, and becomes one of the main factors affecting the dynamic behavior of neural networks or even causing instability [13], [14], [15], [16]. Therefore, the state estimation problem of delayed neural networks subject to a dissipativity performance index has received increasing attention in the last decade (see [17], [18], [19] for some recent publications). Estimating of delayed neural networks is first studied in [20], where a delay-independent criterion is established on the existence of Luenberger estimators. Since the obtained criterion is irrespective of the delay’s bounds, it inevitably brings somewhat conservatism especially for neural networks with very small delays. To reduce the conservatism, a number of efforts have been devoted to setting up delay-dependent criteria for DNNs, where Lyapunov-Krasovskii functional method combined with some integral inequalities have played a key role, see [21], [22], [23], [24], [25], [26], [27], [28], [29].
Lyapunov-Krasovskii functional (LKF) method has been well recognized as a powerful tool to deal with stability analysis of DNNs. Regarding the LKF method, one critical issue is how to estimate the cross product term appeared in the time-derivative of the LKF. For this issue, several approaches, such as model transformation, free-weighting matrices and integral inequalities, are commonly used in the literature [30], [31]. Compared with the first two methods, Jesen integral inequality(JII) appears to be simpler since it directly gives an estimation for the quadratic integral term without employing model transformations or introducing any free matrices, and thus is widely used in traditional methods. To reduce the conservatism of JII, some new integral inequality methods have been developed, such as Wirtinger-based inequality [32] and the auxiliary-based integral inequality [33]. Recently, the Bessel-Legendre inequality, which generalizes the above mentioned integral inequalities and gives smaller upper bounds on certain integral items, has gained increasing attention and resulted in a number of results [34], [35], [36], [37]. On the other hand, a tight bound on the quadratic integral term does not necessarily lead to a less conservative criterion. The effectiveness of a tight inequality will be greatly reduced if using a LKF less related to the inequality [38]. Therefore, how to construct a proper augmented LKF which takes the Legendre-polynomials into account and how to estimate the time derivative of the corresponding LKF are not easy tasks. To the author’s best knowledge, there have been few results on using the N-order Bessel-Legendre inequality to set up delay-dependent criteria for the state estimation issue of DNNs, which inspires our current research.
This paper deals with the dissipativity state estimation problem for DNNs. Applying a new augmented LKF and the N-order Bessel-Legendre inequality, a hierarchical delay-dependent criterion is established on existence of desired state estimators. The main contributions are as follows: First, a proper augmented LKF, which incorporate the vectors of the N-order Bessel-Legendre inequality, is tailored for performance analysis of DNNs. Second, a novel bounded real lemma (BRL) is derived such that the estimation error system is globally asymptotically stable and satisfies a dissipativity performance. In particular, compared with some existing results, smaller upper bounds are obtained for certain integral terms by using the N-order Bessel-Legendre inequality and the quadratic delay coefficients are absorbed through introducing some new vectors, which makes our results less conservative. Based on this, the state estimators can be designed via the linear matrix inequality (LMI) approach. Last, the BRL is proven to form a hierarchy, which means that the conservatism of the BRL reduces with the increase of the order N of the Bessel-Legendre inequality.
The arrangement of the paper is as follows. Problem formulation and some preliminary knowledge are presented in Section 2. In Section 3, we introduce the main results which comprise a criterion on dissipativity performance analysis and the explicit expressions of state estimators. In section 4, we prove that the criterion forms a hierarchy. Three example are provided in Section 5 and the paper is concluded in Section 6.
Notations: The notations are standard. Throughout the paper, P > 0 (P ≥ 0) means that matrix P is positive (semi)definite. I denotes an identity matrix of appropriate dimensions. and are the n-dimensional Euclidean space and the set of all n × m real matrices, respectively. The space of square-integrable vector functions over [0, ∞) is denoted by . diag{⋅⋅⋅} and col{⋅⋅⋅} stand for a block-diagonal matrix and a block-column matrix (vector), respectively. In symmetric block matrices, ‘⁎’ denotes the term that is induced by symmetry. He{X} represents .
Section snippets
Problem formulation and preliminary knowledge
Consider the following delayed neural network described bywhere is the neural state vector, is the network measurement output, is the vector to be estimated, is the neuron activation function, is the noise disturbance belonging to φ(ν) is the initial condition function,
Dissipativity state estimation for DNNs
We first give a BRL under which the estimation error system (5) is globally asymptotically stable with a prescribed dissipativity performance. Before proceeding, we make the following denotations. Proposition 1 Let N > 0 be a positive integer, λ > 0 be a prescribed scalar, h, μ1, μ2 be given constants satisfying (2), and Ψ1 ≤ 0, Ψ2, Ψ3 > λI be real matrices of appropriate dimensions. For a given estimator parameter K, the estimation
Hierarchy of LMI Criteria
In this section, we will prove that Proposition 1 forms a hierarchy. In other words, the conservatism of Proposition 1 reduces with the increase of the order N of the Bessel-Legendre inequality. Proposition 3 Consider the estimation error system (5) with given allowable sets for the time-varying delay δ(t) ∈ [0, h] and its derivative . Let N be a positive integer, λ > 0 be a given scalar, Ψ1 ≤ 0, Ψ2, Ψ3 > λI be real matrices of appropriate dimensions, and K be the given estimator parameter. If
Numerical Examples
In this section, three examples are given to show the validity of our results.
Example 1. Consider the DNN (1) with the following parameters:and the neuron activation function f(x(t)) satisfies (3) with
This example is given to make a comparison with some existing results ([43, Th 3], [44, Th 2]). For we compute the allowable maximum h under which the DNN is stable. Applying Proposition 1 (set N
Conclusion
In this paper, we address the dissipativity state estimation problem for neural networks with time-varying delays. An augmented LKF, which incorporates the vector information of the N-order Bessel-Legendre inequality, has been constructed. Based on this, we present a novel BRL under which the estimation error system is globally asymptotically stable and strictly -dissipative. Moreover, the estimator parameter can be determined via the linear matrix inequality approach. We also show
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work was supported by the Natural Science Foundation of China (Grant Nos. 61973201, 61803244 and 61803243).
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