Improved result on state estimation for complex dynamical networks with time varying delays and stochastic sampling via sampled-data control
Introduction
A complex dynamical networks (CDNs) is a large set of interconnected nodes, where each nodes corresponds to its respective dynamical networks based on the network topology, some of these nodes are usually coupled, which give rise to a variety of collective complexities in the overall dynamical properties of CDNs (Strogatz, 2001). In real world, a large number of practical systems can be represented as CDNs. As science and technology have been developed rapidly, CDNs can be seen everywhere, and have been regarded as a fundamental tool to understand dynamical behavior and the response of the practical systems. Complex dynamical network model, and the edges represent complicate connections between those individual systems. Moreover, CDNs are classified into different categories such as small-world networks (Watts & Strogatz, 1998), random networks and scale-free networks (Wang & Chen, 2003). It was possible to explore the deeper behavior in the complex network (Hooper, DeDeo, Caldwell Hooper, Gurven, & Kaplan, 2013). Complex dynamical network may be visible everywhere, and have been regarded as a fundamental tool to apprehend dynamical behavior and the reaction of the realistic systems, such as, internet, food webs, smart phone cell graphs, transportation networks, which includes biological structures, the World Wide Web (WWW), electricity distribution networks, chemical systems, image processing. Thus, much attention has been paid to the research on their characters including topological structure, dynamical evolution, node diversity, meta complication and so on.
Time delay can be found in many physical processes due to the finite switching speed of amplifiers, finite signal propagation time in biological networks, memory effects, nuclear reactors, population dynamic models, aircraft stabilization, chemical engineering systems, and so on. On the other hand, time delay exhibits as a typical characteristic of the signal transmission between neurons and, therefore, becomes one of the main sources to cause instability and poor performances of complex dynamical networks. In recent decades, considerable interest has been devoted to the time delay structures due to their extensive applications in practical systems which include circuit principle, chemical processing, bio engineering, complicated dynamical networks (Pan, Cao, & Hu, 2015), and automatic control and many others (Park, Kwon, Park, et al., 2012).
State estimation has become one of the popular topics, some profound results are established (Shen, Wang, & Liu, 2012). In Liu, Wang, Liang, and Liu (2008), synchronization and state estimation are investigated for discrete-time complex networks with distributed delays, the addressed problem of synchronization and state estimation methods can be mainly divided into two categories: centralized and distributed algorithms in the centralized state estimation all measured data are communicated to the fusion or control center for processing. It can provide the optimal state estimate for the entire network, due to the smallest information loss. But the computational burden at the fusion or control center increases exponentially when the system becomes large (Li, Wang, Song and Fei, 2011). At present, there have been much research activity focused on the distributed methods to static state estimation for many practical domains. Furthermore, the node dynamics are coupled with the states and the node inner coupling matrices of are identical (Li, Wang et al., 2011, Liang et al., 2009, Liu et al., 2008).
On the other hand, many control approaches including impulsive distributed control (Guan, Liu, Feng, & Wang, 2010), adaptive pinning control (Su et al., 2013), hybrid and impulsive control (Li & Cao, 2017), intermittent control, sampling data control (Lee, Wu, & Park, 2012), randomly occurring control (Tang & Wong, 2013), have been proposed to deal with synchronization problem for CDNs (Fridman, 2010, Shi et al., 2017, Wu et al., 2013). Sampled-data systems contain mainly two approaches, the first one is based on the lifting technique in which the problem is transformed to an equivalent finite-dimensional discrete control problem. Compared with some continuous control approaches, sampled-data control method has many advantages, along with easy installation, small size, maintenance with low cost, and efficiency in realizing the synchronization of CDNs (Li, Zhang, Hu and Nie, 2011). Thus, the investigation of sampled-data exponential synchronization for CDNs with the coupling time-varying delay is important in each concept and application.
In sampled-data control systems choosing proper sampling interval is very important for designing suitable controllers. As is known to all, a longer sampling interval will lead to lower communication channel occupation, by converting the sampling techniques are assumed to be implemented in a deterministic way, few actuation of the controller, and less signal transmission. In Shen et al. (2012), the sampled-data is significant to investigate the synchronization control design problem under a bigger sampling period and system overall performance in detail. Furthermore, the input delay approach is one of the most popular approaches to analysis and synthesis of sampled-data systems (Jeeva Sathya Theesar, Banerjee, & Balasubramaniam, 2012). In other words, there is a vital need to investigate the sampled-data synchronization control problem for a class of dynamical networks. Unfortunately, although sampled-data control technologies have been developed relatively well in control theory, the stochastic sampled data state estimation problem for complex dynamical networks with sampling period has so far received very little attention (Wu, Park, Su, Song, & Chu, 2012).
Motivated by this, we study state estimation for complex dynamical networks with time-varying and delay stochastic sampling via sampled-data control. The multiple sampling period considered here is assumed to be time-varying that switches between two different values in a random way with given probability. By applying an input-delay approach, the probabilistic sampling state estimator is transformed into a continuous time-delay system with stochastic parameters in the system matrices, where the purpose is to design a state estimator to estimate the network states through available output measurements. The solvability of derived conditions depends on not only the size of the delay and the sampling period, but also the probability of taking values of the sampling period. Finally, a numerical example is provided to demonstrate the effectiveness of the results.
The main contributions of this paper are summarized as follows:
1. The state estimation is investigated for CDN, where the sampling period considered here is assumed to be Bernoulli distribution which can be further extended to the case with multiple stochastic sampling periods.
2. New Lyapunov–Krasovskii functional with triple and fourth integral terms is constructed.
3. Additionally the Wirtinger-based single and double integral inequality technique is taken into account with the other inequalities Lemma 2.5, Lemma 2.6 when estimating the time-derivative of the LKFs.
4. The conditions in our main results are given as linear matrix inequalities easily, which can be solved by using Matlab LMI toolbox.
The rest of this paper is organized as follows: In Section 2 problem formulation and preliminaries are briefly outlined. In Section 3, we established some synchronization criteria for the proposed models with or without a virtual synchronization target. We also showed the effectiveness of the theoretical results with a numerical example in Section 4. Some conclusions are given in Section 5.
Notations: The notations used in this paper are quite and fairly standard. Throughout this paper, and denote, the n-dimensional Euclidean space and the set of all real matrices. means the occurrence probability of the event . means the occurrence probability of conditional on . refers to the Euclidean vector norm. represents the transpose of matrix . I is the identity matrix with compatible dimension. means that and are symmetric matrices, and that is positive definite. The symbol represents the elements below the main diagonal of a symmetric matrix. The Kronecker product of matrices and is a matrix in and denoted as . In this paper, if not explicitly stated, matrices are assumed to have compatible dimensions.
Section snippets
Problem formulation and preliminaries
We consider the following delayed complex dynamical networks consisting of N coupled nodes of the form where is the state vector of the ith node, is the output of the ith node, and are some constant matrices, are two unknown but sector-bounded non-linear vector functions. is the outer-coupling matrix of the networks representing the coupling strength and
Main results
In this section, we derive state estimator for complex networks with stochastic sampling and time varying delays such that the concerned system is asymptotically stable. Functional, novel delay-dependent criterion for the closed-loop error system (18) with differentiable time varying delay is derived in terms of LMIs.
Theorem 3.1 Under Assumption 1, for given sampled-data controller gain matrix K, and positive constants, the error system (18) is asymptotically stable, if there exist
Numerical example
In this section, the case studies are carried out by a numerical example. The allowable maximum state estimation complex dynamical rates of each sampled-data scheme are calculated based on Theorem 3.1 by using the LMI toolbox.
Example 1 We consider complex dynamical networks with three nodes and state vector of each node being two dimensional (i.e) (, ).
Conclusion
This paper has discussed the issue of state estimation problem of complex dynamical networks (CDNs) for a class of sampled-data with probabilistic sampling. An improved Wirtinger-based single double integral inequality technique and input-delay approach are utilized to derive the results. The probabilistic sampling state estimator is transformed into a continuous time-delay system with stochastic parameter in the system matrices. A convex procedure for designing stabilization controllers has
Acknowledgment
This work was supported by NBHM grant. 2-48(5)/2017/NBHMR.P/-R-D II/14088.
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