Event-triggered consensus control for DC microgrids based on MKELM and state observer against false data injection attacks

Abstract

This paper proposes an event-triggered distributed control method based on state observer and multi-kernel extreme learning machine (MKELM) for direct current microgrid, which can suppress the false data injection attacks (FDIA) in system. Firstly, considering the problem that most of the methods have poor suppression capability for discrete attacks in sensor signals, an MKELM-based suppression method for sensor attack is proposed. MKELM is used to model the microgrid system, the output current estimated by MKELM is employed in the controller instead of the current value measured by sensors, thus avoiding the FDIA present in sensors. Secondly, an event-triggered state observer-based distributed secondary control method with H_∞ consensus performance is designed for suppressing the FDIA in communication link. In the controller, a state variable is constructed by deriving the microgrid system expression as a first-order multi-agent system, and then a state observer is designed to observe the transmitted information for isolating and suppressing the attack signals. And it is proved theoretically that the proposed event-triggered controller can achieve H_∞ consensus convergence. Finally, simulation analyses verify the effectiveness of the method for suppressing FDIA in DC microgrid system.

Appendices

Appendix

Appendix 1: The proof of Theorem 1.

Let $e_{{\overline{d}_{i} }} = \overline{d}_{i} - \hat{\overline{d}}_{i}$. According to Eq. (22), the derivative of $e_{{\overline{d}_{i} }}$ is

$$\dot{e}_{{\overline{d}_{i} }} = \eta_{i} \left( {\upsilon_{i} \hat{\overline{d}}_{i} - \left| {e_{Xi} } \right|} \right)$$

(36)

Consider the following Lyapunov function.

$$V_{1} = \frac{1}{2}{\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} + \frac{1}{2}\sum\limits_{i = 1}^{n} {\frac{1}{{\eta_{i} }}e_{{\overline{d}_{i} }}^{2} } ,$$

(37)

where $e_{{\overline{d}_{i} }} = \overline{d}_{i} - \hat{\overline{d}}_{i}$.

According to Definition 1, let the mathematical expectation of the infinitesimal operator for V₁ be

$$\begin{gathered} {\rm E}\left\{ {\Im V_{1} } \right\} = {\rm E}\left\{ {{\mathbf{e}}_{X}^{{\text{T}}} {\dot{\mathbf{e}}}_{X} + \sum\limits_{i = 1}^{n} {\frac{1}{{\eta_{i} }}e_{{\overline{d}_{i} }} \dot{e}_{{\overline{d}_{i} }} } } \right\} \\ = {\rm E}\left\{ { - \lambda {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{L}}^{*} {\mathbf{e}}_{X} + {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{d}} - {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{f}} + \lambda {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{\tilde{\gamma }v}} + \sum\limits_{i = 1}^{n} {\frac{1}{{\eta_{i} }}e_{{\overline{d}_{i} }} \dot{e}_{{\overline{d}_{i} }} } } \right\} \\ \end{gathered}$$

(38)

According to Eq. (29), ${\rm E}\left\{ {\lambda {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{\tilde{\gamma }v}}} \right\} = {\rm E}\left\{ {\lambda {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{\gamma v}}} \right\}$. Based on Lemma 1, let ${\mathbf{M}}_{1} = {{\varvec{\upgamma}}},{\mathbf{M}}_{2}^{ - 1} = {\mathbf{\gamma \gamma }}$, then

$$\begin{gathered} {\rm E}\left\{ {\lambda {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{\gamma v}}} \right\} \le {\rm E}\left\{ {\frac{1}{2}\lambda \beta {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} + \frac{1}{2}\lambda \beta^{ - 1} {\mathbf{v}}^{{\text{T}}} {\mathbf{\gamma \gamma v}}} \right\} \\ \le {\rm E}\left\{ {\frac{1}{2}\lambda \beta {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} } \right\} + \frac{1}{2}\lambda \beta^{ - 1} \overline{v}^{{2}} \lambda_{\max } \left( {{\mathbf{\gamma \gamma }}} \right) \\ \end{gathered}$$

(39)

Furthermore, according to Eqs. (22) and (36),

$$\begin{gathered} {\rm E}\left\{ {{\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{d}} - {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{f}} + \sum\limits_{i = 1}^{n} {\frac{1}{{\eta_{i} }}e_{{\overline{d}_{i} }} \dot{e}_{{\overline{d}_{i} }} } } \right\} \hfill \\ \le {\rm E}\left\{ {\sum\limits_{i = 1}^{n} {\left| {e_{Xi} } \right|\overline{d}_{i} } - \sum\limits_{i = 1}^{n} {e_{Xi} f_{i} } + \sum\limits_{i = 1}^{n} {\frac{1}{{\eta_{i} }}e_{{\overline{d}_{i} }} \dot{e}_{{\overline{d}_{i} }} } } \right\} \hfill \\ \le {\rm E}\left\{ {\sum\limits_{i = 1}^{n} {\left| {e_{Xi} } \right|\left( {e_{{\overline{d}_{i} }} - \hat{\overline{d}}_{i} } \right)} - \sum\limits_{i = 1}^{n} {e_{Xi} \frac{{e_{Xi} \hat{\overline{d}}_{i}^{2} }}{{\left| {e_{Xi} } \right|\left| {\hat{\overline{d}}_{i} } \right| + \upsilon_{i} }}} + \sum\limits_{i = 1}^{n} {e_{{\overline{d}_{i} }} \left( {\upsilon_{i} \hat{\overline{d}}_{i} - \left| {e_{Xi} } \right|} \right)} } \right\} \hfill \\ \le {\rm E}\left\{ { - \sum\limits_{i = 1}^{n} {\left| {e_{Xi} } \right|\hat{\overline{d}}_{i} } - \sum\limits_{i = 1}^{n} {\frac{{\left| {e_{Xi} } \right|^{2} \hat{\overline{d}}_{i}^{2} }}{{\left| {e_{Xi} } \right|\left| {\hat{\overline{d}}_{i} } \right| + \upsilon_{i} }}} + \sum\limits_{i = 1}^{n} {\upsilon_{i} e_{{\overline{d}_{i} }} \left( {\overline{d}_{i} - e_{{\overline{d}_{i} }} } \right)} } \right\}. \hfill \\ \end{gathered}$$

(40)

Since

$$e_{{\overline{d}_{i} }} \overline{d}_{i} \le \frac{1}{2}\overline{d}_{i}^{2} { + }\frac{1}{2}e_{{\overline{d}_{i} }}^{2} ,$$

(41)

then Eq. (40) can be written as

$$\begin{gathered} {\rm E}\left\{ {{\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{d}} - {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{f}} + \sum\limits_{i = 1}^{n} {\frac{1}{{\eta_{i} }}e_{{\overline{d}_{i} }} \dot{e}_{{\overline{d}_{i} }} } } \right\} \hfill \\ \le {\rm E}\left\{ {\sum\limits_{i = 1}^{n} {\left| {e_{Xi} } \right|\hat{\overline{d}}_{i} } - \sum\limits_{i = 1}^{n} {\frac{{\left| {e_{Xi} } \right|^{2} \hat{\overline{d}}_{i}^{2} }}{{\left| {e_{Xi} } \right|\left| {\hat{\overline{d}}_{i} } \right| + \upsilon_{i} }}} + \sum\limits_{i = 1}^{n} {\upsilon_{i} \left( {\frac{1}{2}\overline{d}_{i}^{2} { + }\frac{1}{2}e_{{\overline{d}_{i} }}^{2} - e_{{\overline{d}_{i} }}^{2} } \right)} } \right\} \hfill \\ \le {\rm E}\left\{ {\sum\limits_{i = 1}^{n} {\frac{{\upsilon_{i} \left| {e_{Xi} } \right|\hat{\overline{d}}_{i} }}{{\left| {e_{Xi} } \right|\left| {\hat{\overline{d}}_{i} } \right| + \upsilon_{i} }}} - \frac{1}{2}\sum\limits_{i = 1}^{n} {\upsilon_{i} e_{{\overline{d}_{i} }}^{2} } + \frac{1}{2}\sum\limits_{i = 1}^{n} {\theta_{i} \upsilon_{i} \overline{d}_{i}^{2} } } \right\} \hfill \\ \le {\rm E}\left\{ {\sum\limits_{i = 1}^{n} {\upsilon_{i} \left( {1 + \frac{1}{2}\overline{d}_{i}^{2} } \right)} - \frac{1}{2}\sum\limits_{i = 1}^{n} {\upsilon_{i} e_{{\overline{d}_{i} }}^{2} } } \right\} \hfill \\ \end{gathered}$$

(42)

Substituting Eq. (39) and Eq. (42) into Eq. (38) yields

$$\begin{gathered} {\rm E}\left\{ {\Im V_{1} } \right\} \le {\rm E}\left\{ { - \lambda {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{L}}^{*} {\mathbf{e}}_{X} + \frac{1}{2}\lambda \beta {\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} - \frac{1}{2}\sum\limits_{i = 1}^{n} {\upsilon_{i} e_{{\overline{d}_{i} }}^{2} } + \sum\limits_{i = 1}^{n} {\upsilon_{i} \left( {1 + \frac{1}{2}\overline{d}_{i}^{2} } \right)} } \right\} + \frac{1}{2}\lambda \beta^{ - 1} \overline{v}^{{2}} \lambda_{\max } \left( {{\mathbf{\gamma \gamma }}} \right) \hfill \\ \le {\rm E}\left\{ { - \lambda \left( {\lambda_{0} - \frac{1}{2}\beta } \right){\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} } \right\} + \rho \hfill \\ \le - \lambda \left( {\lambda_{0} - \frac{1}{2}\beta } \right){\rm E}\left\{ {\left\| {{\mathbf{e}}_{X} } \right\|^{2} } \right\} + \rho \hfill \\ \end{gathered}$$

(43)

where $\rho = \sum\limits_{i = 1}^{n} {\upsilon_{i} \left( {1 + \frac{1}{2}\overline{d}_{i}^{2} } \right)} + \frac{1}{2}\lambda \beta^{ - 1} \overline{v}^{{2}} \lambda_{\max } \left( {{\mathbf{\gamma \gamma }}} \right)$. Since $\lambda_{0} > \frac{1}{2}\beta$,

$${\rm E}\left\{ {\left\| {{\mathbf{e}}_{X} } \right\|} \right\} \le \overline{\rho }.$$

(44)

where $\overline{\rho } = \sqrt {\frac{\rho }{{\lambda \left( {\lambda_{0} - \frac{1}{2}\beta } \right)}}}$ is the upper bound of the estimation error. Therefore, the state estimation error of the observer is finally consensus bounded and the proof is finished.

Remark 5: As shown in Eq. (44), the $\overline{\rho }$ is affected by $\overline{d}_{i}$, ${\mathbf{\gamma v}}$ and the parameters $\lambda ,\beta$. Larger disturbances $d_{i}$ and frequent FDIA attacks may lead to larger estimation errors, but $\overline{\rho }$ can be kept arbitrarily small by choosing appropriate parameters.

Appendix 2: The proof of Theorem 2.

Let the Lyapunov function be

$$V_{2} \left( t \right) = \frac{1}{2}{{\varvec{\upvarepsilon}}}^{{\text{T}}} {{\varvec{\upvarepsilon}}} + nl{\text{e}}^{ - t}$$

(45)

According to Definition 1, let the mathematical expectation of the infinitesimal operator for V₂ be

$$\begin{gathered} {\rm E}\left\{ {\Im V_{2} } \right\} = {\rm E}\left\{ {{{\varvec{\upvarepsilon}}}^{{\text{T}}} {\dot{\mathbf{\varepsilon }}} - nl{\text{e}}^{ - t} } \right\} \\ = {\rm E}\left\{ { - k{{\varvec{\upvarepsilon}}}^{{\text{T}}} {\mathbf{L}}^{*} {{\varvec{\upvarepsilon}}} + k{{\varvec{\upvarepsilon}}}^{{\text{T}}} {\mathbf{L}}^{*} {\mathbf{e}}_{X} + k{{\varvec{\upvarepsilon}}}^{{\text{T}}} {\mathbf{e}} + {{\varvec{\upvarepsilon}}}^{{\text{T}}} {\mathbf{d}} - nl{\text{e}}^{ - t} } \right\} \\ \end{gathered}$$

(46)

Since

$${\rm E}\left\{ {k{{\varvec{\upvarepsilon}}}^{{\text{T}}} {\mathbf{L}}^{*} {\mathbf{e}}_{X} } \right\} \le {\rm E}\left\{ {\frac{k}{2}\lambda_{\max } \left( {{\mathbf{L}}^{*} {\mathbf{L}}^{{*{\text{T}}}} } \right){{\varvec{\upvarepsilon}}}^{{\text{T}}} {{\varvec{\upvarepsilon}}} + \frac{k}{2}{\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} } \right\},$$

(47)

$${\rm E}\left\{ {k{{\varvec{\upvarepsilon}}}^{{\text{T}}} {\mathbf{e}}} \right\} \le {\rm E}\left\{ {\frac{k}{2}{{\varvec{\upvarepsilon}}}^{{\text{T}}} {{\varvec{\upvarepsilon}}} + \frac{k}{2}{\mathbf{e}}^{{\text{T}}} {\mathbf{e}}} \right\},$$

(48)

$${\rm E}\left\{ {{{\varvec{\upvarepsilon}}}^{{\text{T}}} {\mathbf{d}}} \right\} \le {\rm E}\left\{ {\frac{1}{{2\sigma^{2} }}{{\varvec{\upvarepsilon}}}^{{\text{T}}} {{\varvec{\upvarepsilon}}} + \frac{{\sigma^{2} }}{2}{\mathbf{d}}^{{\text{T}}} {\mathbf{d}}} \right\},$$

(49)

where $\sigma$ is a positive number. Then Eq. (46) can be expressed as

$${\rm E}\left\{ {\Im V_{2} } \right\} \le {\rm E}\left\{ {{{\varvec{\upvarepsilon}}}^{{\text{T}}} \left[ { - k{\mathbf{L}}^{*} + \left( {\frac{k}{2}\lambda_{\max } \left( {{\mathbf{L}}^{*} {\mathbf{L}}^{{*{\text{T}}}} } \right) + \frac{k}{2} + \frac{1}{{2\sigma^{2} }}} \right)I_{n} } \right]{{\varvec{\upvarepsilon}}} + \frac{k}{2}{\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} + \frac{k}{2}{\mathbf{e}}^{{\text{T}}} {\mathbf{e}} + \frac{{\sigma^{2} }}{2}{\mathbf{d}}^{{\text{T}}} {\mathbf{d}} - nl{\text{e}}^{ - t} } \right\}.$$

(50)

Since $\frac{k}{2}{\mathbf{e}}^{{\text{T}}} {\mathbf{e}} = \frac{k}{2}\sum\limits_{i = 1}^{n} {\left| {e_{i} } \right|^{2} }$, according to the trigger condition of Eq. (24), we can get

$$\begin{gathered} \frac{k}{2}\sum\limits_{i = 1}^{n} {\left| {e_{i} \left( t \right)} \right|^{2} } \le \frac{k}{2}\sum\limits_{i = 1}^{n} {\left( {\left| {\xi_{i} \left( t \right)} \right|^{2} + l{\text{e}}^{ - t} } \right)} \\ \le \frac{k}{2}\sum\limits_{i = 1}^{n} {\left| {\xi_{i} \left( t \right)} \right|^{2} } + \frac{kln}{2}{\text{e}}^{ - t} . \\ \end{gathered}$$

(51)

Let $\xi_{i}^{a} = \sum\limits_{{j \in N_{i} }} {a_{ij} \left( {\hat{X}_{j} - \hat{X}_{i} } \right)}$, $\xi_{i}^{b} = X_{{{\text{ref}}}} - \hat{X}_{i}$. Then $u_{i} = \xi_{i}^{a} + \xi_{i}^{b}$, and Eq. (51) can be written as

$$\frac{k}{2}\sum\limits_{i = 1}^{n} {\left| {e_{i} \left( t \right)} \right|^{2} } \le k\sum\limits_{i = 1}^{n} {\left( {\left| {\xi_{i}^{a} \left( t \right)} \right|^{2} + \left| {\xi_{i}^{b} \left( t \right)} \right|^{2} } \right)} + \frac{kln}{2}{\text{e}}^{ - t} .$$

(52)

Since

$$\begin{gathered} \left| {\xi_{i}^{a} } \right| = \left| {\sum\limits_{{j \in N_{i} }} {a_{ij} \left( {\hat{X}_{j} - X_{{{\text{ref}}}} + X_{{{\text{ref}}}} - \hat{X}_{i} } \right)} } \right| \\ = \left| {\sum\limits_{{j \in N_{i} }} {a_{ij} \left( {\xi_{i}^{b} - \xi_{j}^{b} } \right)} } \right| \\ \le \sum\limits_{{j \in N_{i} }} {a_{ij} \left( {\left| {\xi_{i}^{b} } \right| + \left| {\xi_{j}^{b} } \right|} \right)} \\ \le \tilde{N}\left| {\xi_{i}^{b} } \right| + \sqrt {\tilde{N}} \left\| {\xi^{b} } \right\|, \\ \end{gathered}$$

(53)

$$\left\| {\xi^{b} } \right\|^{2} \le 2\left\| {{\varvec{\upvarepsilon}}} \right\|^{2} + 2\left\| {{\mathbf{e}}_{X} } \right\|^{2} ,$$

(54)

then

$$\begin{gathered} \sum\limits_{i = 1}^{n} {\left| {\xi_{i}^{a} \left( t \right)} \right|^{2} } \le \left( {\tilde{N} + \sqrt {n\tilde{N}} } \right)^{2} \left\| {\xi^{b} } \right\|^{2} \\ \le 2\left( {\tilde{N} + \sqrt {n\tilde{N}} } \right)^{2} \left( {\left\| {{\varvec{\upvarepsilon}}} \right\|^{2} + \left\| {{\mathbf{e}}_{X} } \right\|^{2} } \right) \\ \end{gathered}$$

(55)

So far, combining Eqs. (51)-(55), we can get

$$\begin{gathered} {\rm E}\left\{ {\Im V_{2} } \right\} \le {\rm E}\left\{ {{{\varvec{\upvarepsilon}}}^{{\text{T}}} \left[ { - k{\mathbf{L}}^{*} + \left( {\frac{k}{2}\lambda_{\max } \left( {{\mathbf{L}}^{*} {\mathbf{L}}^{{*{\text{T}}}} } \right) + \frac{k}{2} + \frac{1}{{2\sigma^{2} }} + k\left( {\tilde{N} + \sqrt {n\tilde{N}} } \right)^{2} } \right)I_{n} } \right]{{\varvec{\upvarepsilon}}}} \right. \hfill \\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \left( {\frac{k}{2} + k\left( {\tilde{N} + \sqrt {n\tilde{N}} } \right)^{2} } \right){\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} + \frac{{\sigma^{2} }}{2}{\mathbf{d}}^{{\text{T}}} {\mathbf{d}}} \right\} \hfill \\ \end{gathered}$$

(56)

To achieve the dsired H_∞ consensus performance, Eq. (57) is introduced.

$$J\left( t \right) = {\rm E}\left\{ {\Im V_{2} + \frac{1}{2}{{\varvec{\upvarepsilon}}}^{{\text{T}}} {{\varvec{\upvarepsilon}}} - \frac{{\sigma^{2} }}{2}{\mathbf{\varsigma }}^{{\text{T}}} {\mathbf{\varsigma }}} \right\}$$

(57)

where ${\mathbf{\varsigma }} = \left[ {{\mathbf{e}}_{X} ,{\mathbf{d}}} \right]^{{\text{T}}}$.

Substituting Eq. (56) into Eq. (57) yields

$$\begin{gathered} J\left( t \right) \le {\rm E}\left\{ {{{\varvec{\upvarepsilon}}}^{{\text{T}}} \left[ { - k{\mathbf{L}}^{*} + \left( {\frac{k}{2}\lambda_{\max } \left( {{\mathbf{L}}^{*} {\mathbf{L}}^{{*{\text{T}}}} } \right) + \frac{k}{2} + \frac{1}{{2\sigma^{2} }} + k\left( {\tilde{N} + \sqrt {n\tilde{N}} } \right)^{2} } \right)I_{n} } \right]{{\varvec{\upvarepsilon}}}} \right. \\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \left( {\frac{k}{2} + k\left( {\tilde{N} + \sqrt {n\tilde{N}} } \right)^{2} } \right){\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} + \frac{1}{2}{{\varvec{\upvarepsilon}}}^{{\text{T}}} {{\varvec{\upvarepsilon}}} - \frac{{\sigma^{2} }}{2}{\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} } \right\} \\ \le {\rm E}\left\{ {\left( { - k\lambda_{0} + \frac{k}{2}\lambda_{\max } \left( {{\mathbf{L}}^{*} {\mathbf{L}}^{{*{\text{T}}}} } \right) + \frac{k}{2} + \frac{1}{{2\sigma^{2} }} + k\left( {\tilde{N} + \sqrt {n\tilde{N}} } \right)^{2} + \frac{1}{2}} \right){{\varvec{\upvarepsilon}}}^{{\text{T}}} {{\varvec{\upvarepsilon}}}} \right. \\ \left. {\;\;\;\;\;\;\;\;\;\;\;\; + \left( {\frac{k}{2} + k\left( {\tilde{N} + \sqrt {n\tilde{N}} } \right)^{2} - \frac{{\sigma^{2} }}{2}} \right){\mathbf{e}}_{X}^{{\text{T}}} {\mathbf{e}}_{X} } \right\}. \\ \end{gathered}$$

(58)

According to Eqs. (34) and (35), it can be guaranteed that $J(t) < 0$. Then Eq. (59) can be obtained by integrating $J(t) < 0$.

$${\rm E}\left\{ {V_{2} \left( {t_{{\text{f}}} } \right) - V_{2} \left( 0 \right) + \int_{0}^{{t_{{\text{f}}} }} {\left( {\frac{1}{2}{{\varvec{\upvarepsilon}}}^{{\text{T}}} {{\varvec{\upvarepsilon}}} - \frac{{\sigma^{2} }}{2}{\mathbf{\varsigma }}^{{\text{T}}} {\mathbf{\varsigma }}} \right)} dt} \right\} < 0$$

(59)

Therefore, the proposed event-triggered controller can achieve the desired H_∞ consensus convergence performance for the system.

Next, it will be shown that the time interval between the two events is positive, i.e., the trigger condition can avoid Zeno behavior.

During $t \in \left[ {t_{k}^{i} ,t_{k + 1}^{i} } \right)$, $\frac{{d\left| {e_{i} \left( t \right)} \right|}}{dt} \le \left| {\dot{e}_{i} } \right|$ holds.

Since $e_{i} = u_{i} \left( {t_{k}^{i} } \right) - u_{i} \left( t \right)$,

$$\begin{gathered} \left| {\dot{e}_{i} \left( t \right)} \right| = \left| { - \dot{u}_{i} \left( t \right)} \right| \\ = \left| {\sum\limits_{{j \in N_{i} }} {a_{ij} \left( {\dot{\hat{X}}_{i} \left( t \right) - \dot{\hat{X}}_{j} \left( t \right)} \right) + g_{i} \dot{\hat{X}}_{i} \left( t \right)} } \right|. \\ \end{gathered}$$

(60)

According to the observer in Eq. (21), let $\theta_{i} = \sum\limits_{{j \in N_{i} }} {a_{ij} \left( {e_{Xi} - e_{Xj}^{*} } \right)} + g_{i} e_{Xi}$, then Eq. (60) can be written as

$$\begin{gathered} \left| {\dot{e}_{i} \left( t \right)} \right| = \left| {\sum\limits_{{j \in N_{i} }} {a_{ij} \left( {\left( {ku_{i} \left( {t_{k}^{i} } \right) + f_{i} + \lambda \theta_{i} } \right) - \left( {ku_{j} \left( {t_{k}^{i} } \right) + f_{j} + \lambda \theta_{j} } \right)} \right)} + g_{i} \left( {ku_{i} \left( {t_{k}^{i} } \right) + f_{i} + \lambda \theta_{i} } \right)} \right| \\ = \left| {k\left( {\sum\limits_{{j \in N_{i} }} {a_{ij} \left( {u_{i} \left( {t_{k}^{i} } \right) - u_{j} \left( {t_{k}^{i} } \right)} \right)} + gu_{i} \left( {t_{k}^{i} } \right)} \right)} \right. + \left( {\sum\limits_{{j \in N_{i} }} {a_{ij} \left( {f_{i} - f_{j} } \right) + } g_{i} f_{i} } \right) \\ \;\;\;\;\;\left. { + \lambda \left( {\sum\limits_{{j \in N_{i} }} {a_{ij} \left( {\theta_{i} - \theta_{j} } \right) + } g_{i} \theta_{i} } \right)} \right| \\ \le \tau_{k}^{i} \le \left| {e_{i} } \right| + \tau_{k}^{i} , \\ \end{gathered}$$

(61)

where

$$\begin{gathered} \tau_{k}^{i} = \mathop {\max }\limits_{{t \in \left[ {t_{k}^{i} ,t_{k + 1}^{i} } \right)}} \left\{ {\left| {k\left( {\sum\limits_{{j \in N_{i} }} {a_{ij} \left( {u_{i} \left( {t_{k}^{i} } \right) - u_{j} \left( {t_{k}^{i} } \right)} \right)} + gu_{i} \left( {t_{k}^{i} } \right)} \right) + \left( {\sum\limits_{{j \in N_{i} }} {a_{ij} \left( {f_{i} - f_{j} } \right) + } g_{i} f_{i} } \right)} \right|} \right. \hfill \\ \;\;\;\;\;\;\;\left. {\; + \lambda \left( {4\tilde{N}^{2} + 2g_{i} \tilde{N} + 2\tilde{N} + g_{i} } \right)\overline{\rho } + \lambda \left( {2\tilde{N}^{2} + g_{i} \tilde{N}} \right)\overline{v}} \right\}. \hfill \\ \end{gathered}$$

(62)

From Eq. (61), it can be further derived that

$$\left| {e_{i} \left( t \right)} \right| \le \tau_{k}^{i} \left( {{\text{e}}^{{t - t_{k}^{i} }} - 1} \right)$$

(63)

According to the trigger condition in Eq. (24),

$$\begin{gathered} \sqrt {\frac{l}{k}{\text{e}}^{ - t} } < \sqrt {\left| {e_{i} \left( t \right)} \right|^{2} - \frac{h}{k}\left| {u_{i} \left( t \right)} \right|^{2} } \\ < \left| {e_{i} \left( t \right)} \right| \\ < \tau_{k}^{i} \left( {{\text{e}}^{{t - t_{k}^{i} }} - 1} \right). \\ \end{gathered}$$

(64)

Then Eq. (65) can be further derived from Eq. (64).

$$t_{k + 1}^{i} - t_{k}^{i} > \ln \left( {\frac{1}{{\tau_{k}^{i} }}\sqrt {\frac{l}{k}{\text{e}}^{ - t} } + 1} \right) > 0$$

(65)

Therefore, this event trigger condition can avoid Zeno behavior.