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Pinning synchronization for directed networks with node balance via adaptive intermittent control

Abstract

In this paper, the problem of synchronization control for directed networks with node balance is investigated. First, a dynamical model of directed network is proposed. Additionally, a new adaptive intermittent scheme is introduced to realize pinning synchronization and some novel criteria are derived by constructing a piecewise auxiliary function and utilizing piecewise analysis method and the theory of series. Based on those criteria, a feasible bound of the rate of control time is given. Finally, some examples with numerical simulations are given to demonstrate the effectiveness of the results derived.

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Acknowledgments

This work was supported by National Natural Science Foundation of Peoples Republic of China (Grants Nos. 61164004, 61473244, 11402223), Natural Science Foundation of Xinjiang University (Grant No. BS120101), Project funded by China Postdoctoral Science Foundation (Grant No. 2013M540782 and No. 2014T70953), Natural Science Foundation of Xinjiang (Grant No. 2013211B06), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20136501120001).

Author information

Correspondence to Cheng Hu.

Appendices

Appendix 1

Proof of Theorem 1

Proof

Construct a piecewise function described by

$$\begin{aligned} W(t)=\left\{ \begin{array}{lllll} \displaystyle \frac{1}{2}e^{-\mu (t-k\hat{T})}\sum _{i=1}^l \frac{1}{\varepsilon _i}\Big (d_i-d_i(t)\Big )^2,\\ \displaystyle \qquad \qquad k\hat{T}\le t\le (k+\delta )\hat{T},\\ \displaystyle \frac{1}{2}e^{2\beta (t-k\hat{T}\!-\!\delta \hat{T})\!-\!\mu \delta \hat{T}}\sum _{i=1}^l \frac{1}{\varepsilon _i}\Big (d_i\!-\!d_i(k\hat{T}\!+\!\delta \hat{T})\Big )^2,\\ \displaystyle \qquad \qquad (k+\delta )\hat{T}<t<(k+1)\hat{T} \end{array} \right. \end{aligned}$$

for \(k\in Z^+\), in which \(d_i\) is a positive constant to be determined later. It follows from (10) that \(W(t)\) is continuous except for \(t=(k+1)\hat{T}\) with \(k\in Z^+\) and

$$\begin{aligned} W((k+1)\hat{T})=W_+((k+1)\hat{T})=e^{\alpha \hat{T}}W_-((k+1)\hat{T}),\nonumber \\ \end{aligned}$$
(24)

where \(\alpha =\mu \delta -2\beta (1-\delta )\) and \(\alpha >0\) from condition (2) in Theorem 1, \(W_+((k+1)\hat{T})\) and \(W_-((k+1)\hat{T})\) denote the right limit and the left limit of \(W(t)\) at time \((k+1)\hat{T}\), respectively.

Introduce the following Lyapunov function

$$\begin{aligned} V(t)= U(t)+W(t), \end{aligned}$$
(25)

where

$$\begin{aligned} U(t)=\frac{1}{2}\sum _{i=1}^N\mathbf {e}_{\mathbf{i}}(t)^T{\mathbf {e}}_{\mathbf{i}}(t). \end{aligned}$$

Evidently, \(U(t)\) is continuous for all \(t\ge 0\), and \(V(t)\) is continuous except for \(t=(k+1)\hat{T}\) with \(k\in Z^+\) and it is right continuous at \(t=(k+1)\hat{T}\).

For \(k\hat{T}\le t\le (k+\delta )\hat{T}\), the upper right derivative of \(V(t)\) with respect to time \(t\) along the error system (12) can be calculated as follows:

$$\begin{aligned}&D^+{V}(t)\nonumber \\&\quad =\displaystyle \sum _{i=1}^N{\mathbf {e}}_{\mathbf{i}}(t)^T\Big [\mathbf {f}({\mathbf {e}}_{\mathbf{i}}(t) +\mathbf {s}(t))-\mathbf {f}(\mathbf {s}(t))\Big ]\nonumber \\&\qquad +\,c\sum _{i=1}^N\sum _{j=1}^N a_{ij}{\mathbf {e}}_{\mathbf{i}}(t)^T{\varvec{\Gamma }} \mathbf{e}_{\mathbf{j}}(t) -\sum _{i=1}^l d_i(t){\mathbf {e}}_{\mathbf{i}}(t)^T\mathbf {e}_{\mathbf{i}}(t)\nonumber \\&\qquad -\,\frac{\mu }{2}\exp {\Big [-\mu \big (t-k\hat{T}\big )\Big ]}\sum _{i=1}^l \frac{1}{\varepsilon _i}\Big (d_i-d_i(t)\Big )^2\nonumber \\&\qquad -\,\exp {\Big [-\mu \big (t-k\hat{T}\big )\Big ]}\sum _{i=1}^l \big (d_i-d_i(t)\big ){\mathbf {e}}_{\mathbf{i}}(t)^T\mathbf {e}_{\mathbf{i}}(t)\nonumber \\&\quad \le \displaystyle \sum _{i=1}^N\theta {\mathbf {e}}_{\mathbf{i}}(t)^T {\mathbf {e}}_{\mathbf{i}}(t)-\exp {(-\mu \delta \hat{T})}\sum _{i=1}^l d_i{\mathbf {e}}_{\mathbf{i}}(t)^T {\mathbf {e}}_{\mathbf{i}}(t)\nonumber \\&\qquad -\,\frac{\mu }{2}\exp {\Big [-\mu \big (t-k\hat{T}\big )\Big ]}\sum _{i=1}^l \frac{1}{\varepsilon _i}\Big (d_i-d_i(t)\Big )^2\nonumber \\&\qquad +\,c\sum _{i=1}^N\sum _{j=1}^N a_{ij}{\mathbf {e}}_{\mathbf{i}}(t)^T{\varvec{\Gamma }} \mathbf{e}_{\mathbf{j}}(t)\nonumber \\&\quad = \displaystyle \sum _{i=1}^N\sum _{k=1}^ne_i^k(t)\theta e_i^k(t) -e^{-\mu \delta \hat{T}}\sum _{i=1}^l\sum _{k=1}^ne_i^k(t)d_ie_i^k(t)\nonumber \\&\qquad +\,c\sum _{i=1}^N\sum _{j=1}^N\sum _{k=1}^n \gamma _k e_i^k(t)\frac{a_{ji}+a_{ij}}{2} e_j^k(t)\nonumber \\&\qquad -\,\frac{\mu }{2}\exp {\Big [-\mu \big (t-k\hat{T}\big )\Big ]}\sum _{i=1}^l \frac{1}{\varepsilon _i}\Big (d_i-d_i(t)\Big )^2\nonumber \\&\quad = \displaystyle \sum _{k=1}^n ({\tilde{\mathbf {e}}^\mathbf{k}}(t))^T\Big [\theta {\mathbf {I}}_{\mathbf{N}}+c\gamma _k\frac{\mathbf {A}^T+\mathbf {A}}{2} - {\hat{\mathbf {D}}} \big ) \Big ]{\tilde{\mathbf {e}}^\mathbf{k}}(t)\nonumber \\&\qquad -\,\frac{\mu }{2}\exp {\Big [-\mu \big (t-k\hat{T}\big )\Big ]}\sum _{i=1}^l \frac{1}{\varepsilon _i}\Big (d_i-d_i(t)\Big )^2\nonumber \\&\quad = \displaystyle -\frac{\mu }{2}\sum _{k=1}^n \left( {\tilde{\mathbf {e}}^\mathbf{k}}(t)\right) ^T{\tilde{\mathbf {e}}^\mathbf{k}}(t)\!+\! \sum _{k=1}^n ({\tilde{\mathbf {e}}^\mathbf{k}}(t))^T(\mathbf {G}\!-\!{\hat{\mathbf {D}}}){\tilde{\mathbf {e}}^\mathbf{k}}(t)\nonumber \\&\qquad -\,\frac{\mu }{2}\exp {\Big [-\mu \big (t-k\hat{T}\big )\Big ]}\sum _{i=1}^l \frac{1}{\varepsilon _i}\Big (d_i-d_i(t)\Big )^2,\nonumber \\ \end{aligned}$$
(26)

where \({\tilde{\mathbf {e}}^\mathbf{k}}=(e_1^k,e_2^k,\ldots ,e_N^k)^T\) for \(k=1,2,\ldots ,n\),

$$\begin{aligned} {\hat{\mathbf {D}}}&= \text{ diag }(e^{-\mu \delta \hat{T}}d_1,\ldots ,e^{-\mu \delta \hat{T}}d_l,\underbrace{0,\ldots ,0}_{N-l}),\\ \mathbf {G}&= \left( \theta +\frac{\mu }{2}\right) {\mathbf {I}}_{\mathbf{N}} +c\gamma _k\frac{\mathbf {A}^T+\mathbf {A}}{2}. \end{aligned}$$

Obviously, \(\mathbf {G}\) is a real symmetric matrix, by Lemma 1, \(\mathbf {G}-{\hat{\mathbf {D}}}<0\) is equivalent to

$$\begin{aligned} \mathbf {G}_l=\left( \left( \theta +\frac{\mu }{2}\right) {\mathbf {I}}_{\mathbf{N}}+c\gamma _k\frac{\mathbf {A}^T+\mathbf {A}}{2}\right) _l<0 \end{aligned}$$
(27)

when \(d_i\) (\(i=1,2,\ldots ,l\)) are sufficiently large.

In addition, according to Lemma 2 and condition (13),

$$\begin{aligned}&\lambda _{\max }\Big (\left( \theta +\frac{\mu }{2}\right) {\mathbf {I}}_{\mathbf{N}} +c\gamma _k\frac{\mathbf {A}^T+\mathbf {A}}{2}\Big )_l\le \theta \\&\quad +\frac{\mu }{2}+c\gamma _k\lambda _{\max }\Big (\frac{\mathbf {A}^T +\mathbf {A}}{2}\Big )_l\le 0, \end{aligned}$$

which implies that \(\mathbf {G}_l<0\) and then \(\mathbf {G}-{\hat{\mathbf {D}}}<0\), this together with (26), we have

$$\begin{aligned} D^+V(t)\le -\mu V(t) \end{aligned}$$
(28)

for \(k\hat{T}\le t\le (k+\delta )\hat{T}\).

Similarly, for \((k+\delta )\hat{T}<t<(k+1)\hat{T}\),

$$\begin{aligned} D^+{V}(t)&\le \displaystyle \sum _{k=1}^n \left( {\tilde{\mathbf {e}}^\mathbf{k}}(t)\right) ^T\Big [\theta {\mathbf {I}}_{\mathbf{N}}+c\gamma _k\frac{\mathbf {A}^T+\mathbf {A}}{2} \Big ]{\tilde{\mathbf {e}}^\mathbf{k}}(t)\\&+\,2\beta W(t). \end{aligned}$$

Since the graph of (3) is strongly connected, then it follows from Ref. [40] that the coupling matrix \(\mathbf {A}=(a_{ij})_{N\times N}\) and \(\mathbf {A}^T\) are irreducible. According to Lemma 3, the eigenvalues of \(\frac{\mathbf {A}+\mathbf {A}^T}{2}\) can be denoted by \(0=\hat{\lambda }_1>\hat{\lambda }_2\ge \hat{ \lambda }_3\ge \ldots \ge \hat{ \lambda }_N\). Besides, it is easy to see that \(\frac{\mathbf {A}^T+\mathbf {A}}{2}\) is a real symmetric matrix, then there exists a unitary matrix \(\mathbf {P}=(P_1,\ldots , P_N)\) satisfying \(\mathbf {P}^T\mathbf {P}={\mathbf {I}}_{\mathbf{N}}\) such that

$$\begin{aligned} \frac{\mathbf {A}^T+\mathbf {A}}{2}=\mathbf {P}^T\mathbf {QP}, \end{aligned}$$

where \(\mathbf {Q}=\text{ diag }(\hat{\lambda }_1, \hat{\lambda }_2,\ldots ,\hat{\lambda }_N)\). Therefore,

$$\begin{aligned} D^+{V}(t)&\le \displaystyle \sum _{k\!=\!1}^n \left( {\mathbf {P}\tilde{\mathbf{e}}^\mathbf{k}}(t)\right) ^T\Big [\theta {\mathbf {I}}_{\mathbf{N}}\!+\!c\gamma _k\mathbf {Q}\!-\!\beta {\mathbf {I}}_{\mathbf{N}} \Big ]\left( {\mathbf {P}\tilde{\mathbf{e}}^\mathbf{k}}(t)\right) \\&\displaystyle +\,\beta \sum _{k=1}^n \left( {\tilde{\mathbf {e}}^\mathbf{k}}(t)\right) ^T{\tilde{\mathbf {e}}^\mathbf{k}}(t)+2\beta W(t) \\&= \displaystyle \sum _{k=1}^n \sum _{i=1}^N \Big (\theta +c\gamma _k\hat{\lambda }_i-\beta \Big )\big (y_i^k(t)\big )^2\\&\displaystyle +\,\beta \sum _{k=1}^n \left( {\tilde{\mathbf {e}}^\mathbf{k}}(t)\right) ^T{\tilde{\mathbf {e}}^\mathbf{k}}(t)+2\beta W(t)\\&\le \displaystyle 2\beta V(t), \end{aligned}$$

where \({\mathbf {P}}\tilde{\mathbf{e}}^\mathbf{k}(t)={\mathbf {y}}^\mathbf{k}(t)=\left( y_1^k(t),\ldots ,y_N^k(t)\right) ^T\) for \(k=1,2,\ldots ,n\).

Hence, for \((k+\delta )\hat{T}< t<(k+1)\hat{T}\),

$$\begin{aligned} D^+V(t)\le 2\beta V(t). \end{aligned}$$
(29)

In the following, we will prove that

$$\begin{aligned} \lim _{t\rightarrow +\infty }U(t)=0. \end{aligned}$$

By virtue of (24), (28) and (29), we can derive that

$$\begin{aligned} V_-((k+1)\hat{T})&\le \displaystyle e^{2\beta (1-\delta )\hat{T}}V(k\hat{T}+\delta \hat{T})\\&\le \displaystyle e^{-\alpha \hat{T}}V_+(k\hat{T})\\&= \displaystyle e^{-\alpha \hat{T}}U(k\hat{T})+W_-(k\hat{T})\\&= \displaystyle e^{-\alpha \hat{T}}V_-(k\hat{T})+(1-e^{-\alpha \hat{T}})W_-(k\hat{T}), \end{aligned}$$

which implies that

$$\begin{aligned}&V_-((k+1)\hat{T})-V_-(k\hat{T})\nonumber \\&\quad \le \displaystyle (e^{-\alpha \hat{T}}-1)V_-(k\hat{T})+(1-e^{-\alpha \hat{T}})W_-(k\hat{T})\nonumber \\&\quad =\displaystyle (e^{-\alpha \hat{T}}-1)U(k\hat{T}), \end{aligned}$$
(30)

and then

$$\begin{aligned} V_-((k+1)\hat{T})-V(0)\le (e^{-\alpha \hat{T}}-1)\sum _{i=0}^k U(i\hat{T}), \end{aligned}$$

it shows that

$$\begin{aligned} \sum _{i=0}^{\infty } U(i\hat{T})\le \frac{V(0)}{1-e^{-\alpha \hat{ T}}}, \end{aligned}$$
(31)

therefore, by the theory of series,

$$\begin{aligned} \lim _{i\rightarrow +\infty }U(i\hat{T})=0. \end{aligned}$$
(32)

In addition, for \(k\hat{T}\le t< (k+1)\hat{T}\), in view of the nonnegativity of \(d_i(t)\), it is easy to estimate that

$$\begin{aligned} D^+{U}(t)&\le \displaystyle \sum _{i=1}^N{\mathbf {e}}_{\mathbf{i}}(t)^T\Big [\mathbf {f}({\mathbf {e}}_{\mathbf{i}}(t) +\mathbf {s}(t))-\mathbf {f}(\mathbf {s}(t))\Big ]\\&\displaystyle +\,c\sum _{i=1}^N\sum _{j=1}^N a_{ji} {\mathbf {e}}_{\mathbf{i}}(t)^T{\varvec{\Gamma }} \mathbf{e}_{\mathbf{j}}(t)\\&\le \displaystyle \sum _{k=1}^n ({\tilde{\mathbf {e}}^\mathbf{k}}(t))^T\Big [\theta {\mathbf {I}}_{\mathbf{N}} +c\gamma _k\frac{\mathbf {A}^T+\mathbf {A}}{2} \Big ]{\tilde{\mathbf {e}}^\mathbf{k}}(t)\\&\le \displaystyle 2\hat{\lambda }U(t), \end{aligned}$$

where \(\hat{\lambda }=\max \,\{\hat{\lambda }_i,\ i=1,2,\ldots ,n\}\), \(\hat{\lambda }_i\) is the largest eigenvalue of the matrix \(\theta {\mathbf {I}}_{\mathbf{N}}+c\gamma _i\frac{\mathbf {A}^T+\mathbf {A}}{2}\). In view of this, we have

$$\begin{aligned} U(t)\le U(k\hat{T})e^{2\hat{\lambda }(t-k\hat{T})},\quad k\hat{T}\le t< (k+1)\hat{T}. \end{aligned}$$
(33)

Evidently, \(k\rightarrow \infty \) when \(t\rightarrow \infty \), this combines with (32) and (33), we obtain

$$\begin{aligned} \lim _{t\rightarrow +\infty }U(t)=0. \end{aligned}$$

Therefore, the asymptotical synchronization of the controlled network (6) is realized, and the proof of Theorem 1 is completed.\(\square \)

Appendix 2

Proof of Corollary 1

Proof

From inequality (14),

$$\begin{aligned} \theta +c\gamma \lambda _{l+1}<0. \end{aligned}$$

Denote \(\mu =-2(\theta +c\gamma \lambda _{l+1})\) and \(\beta =\theta \), then condition (13) holds and

$$\begin{aligned} \mu \delta -2(1-\delta )\beta =-2(\theta +c\gamma \delta \lambda _{l+1})>0. \end{aligned}$$

It follows from Theorem 1 that the network (6) is globally asymptotically synchronized.\(\square \)

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Hu, C., Jiang, H. Pinning synchronization for directed networks with node balance via adaptive intermittent control. Nonlinear Dyn 80, 295–307 (2015). https://doi.org/10.1007/s11071-014-1869-0

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Keywords

  • Adaptive intermittent control
  • Directed network
  • Node balance
  • Pinning strategy
  • Synchronization