Parameter identification based on lag synchronization via hybrid feedback control in uncertain drive-response dynamical networks
- Hongming Liu^{1},
- Weigang Sun^{1}Email author and
- Ghada Al-mahbashi^{2}
https://doi.org/10.1186/s13662-017-1181-7
© The Author(s) 2017
Received: 5 December 2016
Accepted: 12 April 2017
Published: 26 April 2017
Abstract
In this paper, we use a hybrid feedback control method to study lag synchronization in uncertain drive-response dynamical networks with a feature that the unknown system parameter exists in the node dynamics. We then design two hybrid feedback control methods to achieve the lag synchronization including the linear and adaptive feedback control. With the designed controllers and update laws for the system parameter in the node dynamics, we obtain two theorems on the lag synchronization based on the LaSalle invariance principle. When the lag synchronization is achieved, we identify the unknown system parameter. Finally, we provide two numerical examples to verify the efficiency of the proposed control schemes.
Keywords
1 Introduction
Complex networks [1, 2] have attracted considerable attention as a fundamental tool in understanding the dynamical behaviors of real systems, such as internet, World Wide Web, food webs, electrical power grids, metabolic networks, scientific citation web and fractal networks [3]. The dynamics of complex networks has been an interesting issue with focus on the interplay between the local node dynamics and the overall topological structures. As a typical collective behavior, synchronization of complex networks has been widely investigated because of many applications in engineering [4]. Among many types of synchronization of complex networks, inner synchronization inside a network and outer synchronization between two coupled networks are striking. Generally the method of synchronization is to transform the networked systems into some low dimensional systems and obtain the criteria of synchronization by the master stability function or linear matrix inequality [5, 6]. When the inner synchronization may not happen inside a network with unappropriate topological connections and node dynamics, some controlling (e.g., the adaptive, feedback, pinning and impulsive control) methods are employed for realizing the synchronization (see [7–11] and many references cited therein).
To study the outer synchronization between two coupled networks, Li et al. applied the open-plus-closed-loop control to achieve outer synchronization [12]. This original work theoretically and numerically demonstrated the feasibility of this type of synchronization. In reality, when the trains arrive at the platform in subway systems, the inner and outer doors simultaneously open or close, showing that both inner synchronization and outer synchronization happen. Afterwards, the expanded works on the outer synchronization, such as introducing the adaptive control, effect of noise and fractional-order node dynamics, can be found in the literature [13–17].
It should be noted that all of the above-mentioned works on the synchronization focus on the known node dynamics and topological connections. This assumption cannot be used in many real networks with evolving or adaptive couplings, e.g., flocks of robots [18]. Recently the synchronization of dynamical networks with unknown information has received increasing attention. Zheng used the impulsive control to study the synchronization of uncertain complex-variable chaotic delayed systems [19]. Wu and Lu studied the outer synchronization and parameter identification between two networks with time-varying connections by designing the adaptive controllers [20]. By designing the nonlinear controllers, we achieved the generalized outer synchronization between two uncertain networks where the couplings of each network are unknown nonlinear functions [21].
During the studies on the synchronization of complex networks, most of the existing works studied the complete synchronization. Apart from this type of synchronization, lag synchronization is an interesting phenomenon, which is referred to as the coincidence of the states between two coupled systems, where one system is delayed by a finite time. Presently lag synchronization has been observed in lasers, neural models and electronic circuits [22] and applied in secure communication [23]. Recently the lag synchronization of coupled networks has become a new issue in the research of complex networks. Li et al. studied the successive lag synchronization on nonlinear dynamical networks by the linear feedback control [24]. Zhao et al. considered the lag synchronization between two different networks based on the state observer and designed the corresponding adaptive controllers [25]. Projective lag synchronization in drive-response dynamical networks was studied by proposing a hybrid feedback control method [26, 27]; however they did not consider the parameter identification. To reduce the number of control nodes, the lag synchronization by pinning control was studied in [28, 29].
Inspired by the above discussions, we study lag synchronization in drive-response dynamical networks by the method proposed in [26, 27] and identify the unknown parameter in the node dynamics. By designing the hybrid feedback controllers, we achieve the lag synchronization. With the proposed controllers and update laws for the system parameter, we obtain two theorems on the lag synchronization and identify the unknown system parameter when the lag synchronization happens. In addition, this control method is effective for the drive system with and without mismatched terms. Our findings may help a deeper understanding of the consensus or agreement of the connected agents.
The rest of this paper is organized as follows. In Sections 2 and 3, preliminaries and network models are given. Section 4 studies the lag synchronization by linear and adaptive feedback control. Numerical examples are shown to verify the efficiency of the proposed adaptive schemes in Section 5. Finally, conclusions are drawn in Section 6.
Notations: Throughout this paper, some necessary notations are first introduced. The norm of a vector x is \(\|x\|=\sqrt{x^{T}x}\). The norm of a matrix A is \(\|A\|=\sqrt{\lambda_{\max}(A^{T}A)}\), where \(\lambda_{\max}(A^{T}A)\) denotes the maximal eigenvalue of matrix \(A^{T}A\). ⊗ is the Kronecker product. \(I_{n}\) is an identity matrix of size n.
2 Preliminaries
Assumption 1
Lemma 1
(LaSalle invariance principle) [30]
Let \(\Omega\subset D\) be a compact set that is positively invariant with respect to \(\dot{x}=f(x)\). Let \(V: D\rightarrow R\) be a continuously differentiable function such that \(\dot{V}(x)\leq0\) in Ω. Let E be the set of all points in Ω where \(\dot{V}(x)=0\). Let M be the largest invariant set in E. Then every solution starting in Ω approaches M as \(t\rightarrow\infty\).
3 Model presentation
4 Lag synchronization analysis
4.1 Linear feedback control
In this subsection, we study the lag synchronization in drive-response networks (2) and (3) with the designed linear feedback controllers. The main results are summarized in the following theorem.
Theorem 1
Proof
Remark 1
In this theorem, we design a hybrid feedback control method to realize lag synchronization, including \(u_{i1}(t)\) being the nonlinear feedback control and \(u_{i2}(t)\) being the linear feedback control.
Remark 2
For this linear feedback control, the expected feedback gains \(d_{i}\) need satisfy \(d_{i}\geq L+\lambda_{\max}((P+P^{T})/2)+1\). As many studies on the linear feedback control show [24, 28], the theoretical values of \(d_{i}\) are much bigger than those needed in practice. To deal with the shortcoming of linear feedback control, we will use the adaptive technique to achieve the lag synchronization.
4.2 Adaptive feedback control
In this subsection, we apply the adaptive controllers to achieve the lag synchronization and identify the unknown parameter in the node dynamics. The corresponding adaptive controllers and update laws are then designed.
Theorem 2
Proof
Remark 3
It is noted that the mismatched term \(\Delta(t)\) has no effect on the derivative of Lyapunov function \(V(t)\), which shows that this method is robust for some control systems with parameters perturbation and noise disturbance.
5 Numerical analysis
We introduce the quantity \(E(t)=\max_{i=1,\ldots,N}\| y_{i}^{r}(t)-x^{d}(t-\tau)\|\) to measure the lag synchronization process and take the inner coupling matrix \(\Gamma=I_{3}\).
5.1 Lag synchronization by linear feedback control
5.2 Lag synchronization by adaptive feedback control
6 Conclusions
In the current study, we have studied lag synchronization in drive-response dynamical networks with an uncertain parameter vector in the node dynamics by the hybrid feedback control method. By employing the linear and adaptive feedback controllers, we have designed two types of control schemes and updated laws for the system parameter and derived two criteria on the lag synchronization. Simultaneously, we have identified the unknown system parameter when the lag synchronization is achieved. In the numerical simulations, we have provided two examples to show the validity of the proposed control schemes. Our hybrid control method is effective for the driving system with parameter perturbation and noise disturbance and it also holds that the disturbance is in the response networks. In the future, how to derive the domain of the lag in the synchronization of drive-response networks is underway.
Declarations
Acknowledgements
This work was supported by the Zhejiang Provincial Natural Science Foundation of China (No. LY16A010014) and the National Natural Science Foundation of China (No. 61673144).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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