Asynchronous dissipative filtering for nonhomogeneous Markov switching neural networks with variable packet dropouts
Introduction
In general, nonlinearities exist extensively in a variety of physical systems, and linear systems’ technophobia always cannot be applied to nonlinear systems directly. Neural networks (NNs) have gained much attention for its capacity in dealing with nonlinearity. Up to now, NNs have been applied in various fields, for instance, image processing, signal processing, automatic control, (Ali and Balasubramaniam, 2009, Ali et al., 2019, Cheng et al., 2020c, Liu et al., 2020, Shen, Wang, and Qiao, 2017, Wu et al., 2013). On the other hand, the major work is to squeeze out finite-state performances when investigating the stability of NNs. NNs consisting of finite modes, which switch among modes over different time interval. Due to this reason, considerable attention has been shifted into Markov switching NNs (MSNNs). Note that MSNNs cover the nonlinearities and stochastic variations as special cases, and many valuable results in the area of the theory and applications of MSNNs have been presented in Ali et al., 2017, Ali et al., 2020, Lu et al., 2018, Saravanakumar et al., 2016, Tao et al., 2019, Wang et al., 2020, Zhang et al., 2017, Zhang et al., 2018, Zhang et al., 2015 and Zhu and Chao (2011). Specially, in Zhang et al. (2017), a systematic Markovian system approach has been proposed to deal with the non-uniform sampling, measurement size reduction and active topology switching, which is helpful to save energy.
As is well known, transition probability rates (TPRs) play a critical role on the performance of MSNNs. In aforementioned MSNNs (Wang et al., 2020, Zhang et al., 2017, Zhang et al., 2018), TPRs are presupposed to be time invariant. However, such assumption limits the application of MSNNs due to the inaccurateness in acquiring the information of TPR. To overcome the aforementioned shortage, the uncertain TPRs are introduced, namely, partially known TPRs (Kao, Xie, & Wang, 2014) and time-varying TPRs (Aberkane, 2011, Cheng et al., 2020b, Hua et al., 2020, Kao et al., 2014, Shen et al., 2020, Yin et al., 2015). In fact, TPRs of Markov switching systems (MSSs) may alter over time, and nonhomogeneous MSSs have been explored (Aberkane, 2011, Cheng et al., 2020b, Hua et al., 2020, Shen et al., 2020, Yin et al., 2015), where time-varying TPRs are described as by convex polytope sets. However, to our knowledge, except for Xu, Wu, and Pan (2020), the researches on MSNNs subject to time-varying TPRs are quite few, to explore the dynamic behavior of nonhomogeneous MSNNs (NMSNNs) motivates our curiosity in this work.
The issue of filtering remains a hot topic in the networked control systems, because of the different locations of targeting plant and filter, the signals are exchanged by communication networks. To save the network sources, many efficient methods are developed, such as, event-triggering techniques (Wu et al., 2018, Xu and Wu, 2020), quantized control (Cheng et al., 2020a, Cheng et al., 2020d, Shen, Tan, et al., 2017). For the unreliable of sensors and network caused by network-induced delay, packet dropout, quantizer, and so on, which may lead to asynchronization between plant and filter. In the reported work of MSSs (Aberkane, 2011, Hua et al., 2020, Kao et al., 2014, Yin et al., 2015), the filter remains mode-independent or mode-synchronous, in which the useful information of target plant is omitted. Stem from above discussion, the asynchronous filtering technique for MSSs has been developed (Shen et al., 2019, Song et al., 2017, Wu et al., 2017), where the hidden Markov model (HMM) has been proved to be an efficient method to depict the relationship between plant and filter. As stated in Shen et al., 2019, Song et al., 2017 and Wu et al. (2017) that, HMM-based filter can be decreased to special cases, such as synchronous, mode-independent, cluster observation. Added by HMM strategy, the asynchronous control for MSS has been discussed in Wu et al. (2017). Nevertheless, the problem for NMSNNs by means of HMM strategy has not been explored yet, which motivates the present work.
Summarize the above discussion, a new attempt to consider the asynchronous filter for NMSNNs with VPDs is studied in this work. The major contributions are outlined as blow: (1) A more general plant is constructed, where the TPRs of nonhomogeneous Markovian chain are characterized by polytope approach. (2) The practical phenomena of mode-dependent VPDs is constructed, which is depicted by the bounded uncertainty. (3) By adopting the HMM strategy, the asynchronization relationship between nonhomogeneous plant modes and filter modes is revealed.
Notations The notations utilized in this work are same as that in Cheng et al. (2020b).
Section snippets
Problem formulations
Consider the discrete-time NMSNNs in the following form: where symbolizes the neuron state, means the measurement output, stands for the neuron signal to be decided, denotes the external disturbance and belonging to . denotes the neuron activation function (NAF).
Main results
In this section, the SMSS condition for FENMSNN (8) will be formulated in Theorem 1, and designed nonstationary filter gains will be established in Theorem 2.
Theorem 1 The FENMSNN (8) is called SMSS with a strictly -dissipative , if there exist matrices , , and diagonal matrices , , such that where
Computational experiments
Consider the NMSNNs (1) subject to two neurons and three operation modes:
Similar to Zhang, Yu, Wang, and Ong (2012), the nonlinear NAFs are given as
Conclusions
In this work, an issue of asynchronous filtering for NMSNNs with VPDs is studied. The discrete-time nonhomogeneous Markov process is adopted to depict the modes switching of target plant, where time-varying transition probabilities are revealed by utilizing a polytope technology. By means of the Bernoulli distributed sequence, VPDs and the HMM scheme, the more general resilient filters are developed. Finally, the effectiveness of the developed filter scheme is verified by a simulation example.
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.
Acknowledgments
This work was supported by Guangxi Science and Technology Base and Specialized Talents (No. Guike AD18281026), Guangxi Natural Science Foundation Project (No. 2017GXNSFBA198179, No. 2019JJA110041, No. 2019AC20190), the Training Program for 1,000 Young and Middle-aged Cadre Teachers in Universities of Guangxi Province, the Science and Technology Joint Foundation of Guizhou Province (No. LKM[2013]21), and the National Natural Science Foundation of China (No. 61703150).
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2021, Applied Mathematics and ComputationCitation Excerpt :Specifically, benefit from the HMM, the filter/controller information can be detected and the original plant information is hidden. Recently, the HMM has been applied in many complexity systems and many interesting results have appeared [15–18]. Meanwhile, the transition probability (TP) plays a crucial role since it affects the MJSs performance.