Non-fragile memory filtering of T-S fuzzy delayed neural networks based on switched fuzzy sampled-data control☆
Introduction
Over the last there decades, neural networks (NNs) have garnered wide-scale attention since they have been comprehensively investigated and developed in a variety of science and engineering fields, such as signal processing, pattern recognition, smart antenna arrays, image processing, associative memory, industrial automation, combinatorial optimization problems, and so on [1], [2], [3], [4]. Meanwhile, it is a well known phenomenon that time delay necessarily exists in natural and man-made systems and cannot be neglected, for example, time delay effects can not be omitted in the communication systems mainly due to the limited switching speed of the hardware [5]. However, the existence of time delay often results in undesired dynamics, including divergence, oscillation, or even instability of NNs [6], [7], [8], [9], [10]. Hence, it is all-important to study the stability of delayed neural networks (DNNs). This issue has recently attracted increasing attention, a great number of literatures about the dynamic behaviors of DNNs have been published and have left many important results [11], [12], [13], [14], [15], [16].
In the mathematical modeling of DNNs, parameter variations are also ubiquitous due to the system structure variation, environmental disturbance, parameter fluctuations and estimation errors, which would result in complex dynamical behaviors [17], [18], [19]. Therefore, we must take full account of the parameter uncertainties, when considering stability of DNNs, in which case, one must deal with the robust stability of DNNs. In recent years, many robust stability algorithms and criteria have been derived in [20], [21], [22], [23].
Meanwhile, fuzzy systems in the form of the Takagi-Sugeno fuzzy model (TSFM) [24] have provoked rapidly widespread interests from mathematics and engineering fields in recent years [25], [26]. Because T-S fuzzy systems (TSFSs) have excellent ability in analyzing, synthesizing, and approximating complex dynamical behaviors by a set of IF-THEN rules [27], [28]. Consequently, finite-time stability analysis, fault-tolerant control, reliable dissipative filtering, and integral sliding mode control, memory feedback control problems for TSFSs have gained increasing attention, and many effective methods and important results have been reported in [29], [30], [31], [32], [33], [34]. Based on the above discussion, the concept of incorporating fuzzy logic has been extended to describe DNNs via TSFM and some stability criteria have been derived [35], [36], [37], [38], [39], [40], [41]. Whereas it is worth noting that all the methods mentioned above on the stability analysis of TSFDNNs are independent of fuzzy membership functions (FMFs). Besides, some related processing techniques are scarce, such as the Wirtinger's inequality and time-dependent Lyapunov-like functional [35], [36], [37], [38], [39], [40], [41]. To this end, a good issue is whether we can propose an FMFs dependent approach by taking these neglected factors into account to develop less conservative conditions for TSFDNNs? This is the first motivation for this paper.
On the other hand, with the development of communication and digital technology, sampled-data control (SDC) theory has attracted much attention [42], [43], [44], [45], [46]. Compared with some continuous control [35], [36], [37], [38], [39], [40], [41], the SDC shows more advantages, such as efficiency, high reliability, and easy installation. Accordingly, the investigation of fuzzy SDC of TSFDNNs is very important. Very recently, many interesting stability conditions have been obtained on the filtering issue of TSFDNNs via fuzzy SDC [47], [48], [49]. By using LKF together with the zero function, the authors considered the sampled-data filtering of uncertain TSFNNs with interval time-varying delays [47]. Based on employing the Newton-Leibniz formulation and SDC way, the authors obtained a good design of a state estimator for TSFNNs with Markovian jumping parameters [48]. It should be noted that the relationships of , and were not taken fully considered [47], [48]. In order to reduce the conservatism of obtained results [47], [48], the authors investigated the sampled-data filtering problem of TSFDNNs by using the relaxed reciprocally convex inequality and some novel integral inequalities [49]. Nevertheless, these approaches [47], [48], [49] suffer from the following drawbacks: (1) the chosen LKFs in [47], [48], [49] needs to satisfy the positive condition inside the sampling intervals; (2) the information on the actual sampling pattern has not been fully utilized in [47], [48], [49], which may result in some conservatism; (3) the fuzzy SDC with a constant signal transmission delay in [50] have not yet considered. Thus, how to construct the appropriate LKF for the sampled-data filtering problem of TSFNNs to get a larger sampling period is the another motivation of this paper.
Motivated by the above consideration, in this paper, our attention focuses mainly on investigating the switched fuzzy non-fragile SDC issue for TSFDNNs with randomly occurring time-varying parameters uncertainties (ROTVPUs) and variable sampling rates. A new switched fuzzy memory SDC scheme that is more general than the conventional SDC one is used to deal with this problem. By constructing a augmented loose-looped FMFs dependent LKF that involves some novel terms and relaxed matrices, several optimized control algorithms are presented such that the resultant TSFDNNs is stable. The primary competitive advantage of the modified loose-looped FMFs dependent LKF is that it enables us to take full advantage of the variable characteristics related to the actual sampling pattern and the information of the time derivative of FMFs. The main contributions of this paper are as follows: (1) A switched fuzzy memory SDC scheme is firstly developed for the resultant TSFDNNs; (2) A new loose-looped FMFs dependent LKF with relaxed conditions is constructed; (3) Several optimized control algorithms with both a larger sampling period and upper bound of time-varying delays are obtained for the resultant TSFDNNs.
Notation: Notations used in this paper are fairly standard: denotes the n-dimensional Euclidean space, is the set of all dimensional matrices; I denotes the identity matrix of appropriate dimensions, T stands for matrix transposition, the natation (respectively ), for means that the matrix is real symmetric positive definite (respectively, positive semi-definite); denotes block diagonal matrix with diagonal elements , the symbol ⁎ represents the elements below the main diagonal of a symmetric matrix, is defined as .
Section snippets
Preliminaries
Consider the following TSFDNNs with ROTVPUs: where is the neuron state vector of the network, is the measurement output, is the neuron activation function, is the in initial state vector. , is a constant input vector, is the
Main results
In this section, we will give the stability analysis of uncertain TSFDNNs filter. We first give a better estimation algorithm design for the close-loop TSFDNNs filter to be asymptotically stable. When , , , , , and , the resultant TSFDNNs (22) can be rewritten as For the sake of simplicity of matrix representation,
Numerical examples
In this section, one numerical example is presented to demonstrate the potential benefits and effectiveness of our main results for TSFDNNs.
Example 1 Consider the following fuzzy non-fragile memory SDC filtering of TSFDNNs with switched topologies : IF is and ⋯ is THEN Fuzzy Rule 1: IF is and ⋯ is THEN Fuzzy Rule 2: IF is and ⋯ is THEN
Conclusions
In this paper, we have investigated the non-fragile memory filtering problem of TSFDNNs with ROTVPUs and variable sampling rates. Under the consideration of communication delay and state switched signals, we proposed a new fuzzy switched memory sampled-data control scheme with variable sampling. Together with FMFs dependent LKF and some novel integral inequality, we designed improved control algorithms. The relaxed conditions derived in this paper make full use of the variable characteristics
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This work was supported by the National Natural Science Foundation of China under Grant (No. 61703060, 61802036, 61701048, 11601474 and 11461082, 61873305), and the Fundamental Research Funds for the Central Universities, Southwest Minzu University (2019NQN07), the Opening Fund of Geomathematics Key Laboratory of Sichuan Province (scsxdz2018zd02 and scsxdz2018zd04), Guangxi Natural Science Foundation Project (No. 2017GXNSFBA198179) and Basic Ability Promotion Project for Young and Middle-aged Teachers in Universities of Guangxi (No. 2018KY0214).