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SCIENCE CHINA Information Sciences, Volume 63 , Issue 5 : 150207(2020) https://doi.org/10.1007/s11432-019-2691-4

Event-triggered attack-tolerant tracking control design for networked nonlinearcontrol systems under DoS jamming attacks

Xiaoli CHEN , Youguo WANG *
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  • ReceivedMay 31, 2019
  • AcceptedSep 16, 2019
  • PublishedMar 30, 2020
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Abstract


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 61771256).


Supplement

Appendix

Proof of Lemma 1

Because system (16) is a hybrid fuzzy switched system, we will estimate the upper bound of $V_{i}\left(~t\right)~$ from the two cases mentioned below.

Case 1. When DoS jamming attack signals are sleeping, i.e., $t\in I_{k,n}\cap~\mathcal{G}_{1,n-1}$, for any given $k\in~\varphi~\left( n\right)~$, taking the time derivation of $V_{1}\left(~t\right)~$ along the trajectory of the fuzzy switched delay system (16), one has \begin{eqnarray}\dot{V}_{1}\left( t\right) &\leq &-2\alpha _{1}V_{1}\left( t\right) +2\alpha _{1}X^{\rm T}\left( t\right) \mathcal{P}_{1}X\left( t\right) +2X^{\rm T}\left( t\right) \mathcal{P}_{1}\dot{X}\left( t\right) +X^{\rm T}\left( t\right) \mathcal{Q}_{1}X\left( t\right) \\ & &-X^{\rm T}\left( t-\tau _{M}\right) {\rm e}^{-2\alpha _{1}\tau _{M}}\mathcal{Q} _{1}X\left( t-\tau _{M}\right) +\tau _{M}\dot{X}^{\rm T}\left( t\right) \left( \mathcal{R}_{1}+\mathcal{Z}_{1}\right) \dot{X}\left( t\right) \\ & &-{\rm e}^{-2\alpha _{1}\tau _{M}}\sum_{r=1}^{3}I_{r}+2\sum_{r=4}^{6}I_{r}, \tag{49} \end{eqnarray} where $\nu~_{1}=X\left(~t\right)~-X\left(~t-\tau~_{M}\right)~-\int_{t-\tau _{M}}^{\rm~T}\dot{X}\left(~s\right)~{\rm~d}s$, $\nu~_{2}=X\left(~t\right)~-X\left( t-\rho_{k,n}\left(~t\right)~\right)~-\int_{t-\rho_{k,n}\left(~t\right) }^{\rm~T}\dot{X}\left(~s\right)~{\rm~d}s$, $\nu~_{3}=X\left(~t-\rho_{k,n}\left( t\right)~\right)~-X\left(~t-\tau~_{M}\right)~-\int_{t-\tau~_{M}}^{t-\rho _{k,n}\left(~t\right)~}\dot{X}\left(~s\right)~{\rm~d}s$, and \begin{eqnarray*}I_{1} &=&\int_{t-\tau _{M}}^{\rm T}\dot{X}^{\rm T}\left( s\right) \mathcal{Z}_{1} \dot{X}\left( s\right) {\rm d}s, I_{2}=\int_{t-\rho _{k,n}\left( t\right) }^{\rm T}\dot{ X}^{\rm T}\left( s\right) \mathcal{R}_{1}\dot{X}\left( s\right) {\rm d}s, \\ I_{3} &=&\int_{t-\tau _{M}}^{t-\rho_{k,n}\left( t\right) }\dot{X}^{\rm T}\left( s\right) \mathcal{R}_{1}\dot{X}\left( s\right) {\rm d}s, I_{4}=\underset{l=1}{ \overset{r}{\sum }}\underset{s=1}{\overset{r}{\sum }}\xi _{l}\xi _{s}^{k,n}\zeta ^{\rm T}\left( t\right) M_{1ls}\nu _{1}, \\ I_{5} &=&\underset{l=1}{\overset{r}{\sum }}\underset{s=1}{\overset{r}{\sum }} \xi _{l}\xi _{s}^{k,n}\zeta ^{\rm T}\left( t\right) N_{1ls}\nu _{2}, I_{6}= \underset{l=1}{\overset{r}{\sum }}\underset{s=1}{\overset{r}{\sum }}\xi _{l}\xi _{s}^{k,n}\zeta ^{\rm T}\left( t\right) S_{1ls}\nu _{3}. \end{eqnarray*} Using the element inequality to deal with the integral terms in (49), we obtain \begin{eqnarray}\dot{V}_{1}\left( t\right) &\leq &-2\alpha _{1}V_{1}\left( t\right) + \underset{l=1}{\overset{r}{\sum }}\underset{s=1}{\overset{r}{\sum }}\xi _{l}\xi _{s}^{k,n}\zeta ^{\rm T}\left( t\right) [\Pi _{11ls}^{1}+\tau _{M}N_{1ls}{\rm e}^{2\alpha _{1}\tau _{M}}\mathcal{R}_{1}^{-1}N_{1ls}^{\rm T} \\ & &+\tau _{M}M_{1ls}{\rm e}^{2\alpha _{1}\tau _{M}}\mathcal{Z}_{1}^{-1}M_{1ls}^{\rm T}+ \tau _{M}S_{1ls}{\rm e}^{2\alpha _{1}\tau _{M}}\mathcal{R}_{1}^{-1}S_{1ls}^{\rm T}+ \tau _{M}\Gamma_{1ls}^{\rm T}\left( \mathcal{R}_{1}+\mathcal{Z}_{1}\right) \Gamma_{1ls}]\zeta \left( t\right) . \tag{50} \end{eqnarray} Based on Assumption 3, it follows that \begin{equation}\dot{V}_{1}\left( t\right) +2\alpha _{1}V_{1}\left( t\right) \leq \underset{ l=1}{\overset{r}{\sum }}\gamma _{l}\xi _{l}^{2}\zeta ^{\rm T}\left( t\right) \Pi _{1ll}\zeta \left( t\right) +\underset{l=1}{\overset{r-1}{\sum }}\underset{ s>l}{\overset{r}{\sum }}\xi _{l}\xi _{s}^{k,n}\zeta ^{\rm T}\left( t\right) \left[ \gamma _{s}\Pi _{1ls}+\gamma _{l}\Pi _{1sl}\right] \zeta \left( t\right), \tag{51}\end{equation} where $\Pi~_{1ls}=\Pi~_{11ls}^{1}+\tau~_{M}N_{1ls}{\rm~e}^{2\alpha~_{1}\tau~_{M}} \mathcal{R}_{1}^{-1}N_{1ls}^{\rm~T}+\tau~_{M}M_{1ls}{\rm~e}^{2\alpha~_{1}\tau~_{M}} \mathcal{Z}_{1}^{-1}M_{1ls}^{\rm~T}+\tau~_{M}S_{1ls}{\rm~e}^{2\alpha~_{1}\tau~_{M}} \mathcal{R}_{1}^{-1}S_{1ls}^{\rm~T}$ $+\tau~_{M}\Gamma_{1ls}^{\rm~T}\left(~\mathcal{R}_{1}+\mathcal{Z}_{1}\right) \Gamma_{1ls}$, $\Pi~_{11ls}^{1}$, $N_{1ls}$, $M_{1ls}$, $S_{1ls}$ and $ \Gamma_{1ls}$ are defined in (19)–(eq-Lemma condition~2 ).

Define $\beta~_{ls}=\frac{\gamma~_{l}}{\gamma~_{s}}~\left(~l,s=1,2,\ldots~,r\right)~$. Notice that $\gamma~_{l}$ and $\gamma~_{s}\in~\left[~\gamma _{\min~},\gamma~_{\max~}\right]~$, and then $\beta~_{ls}\in~\left[~\beta~_{\min },\beta~_{\max~}\right]~$. Therefore, from (51), it yields that \begin{equation}\gamma _{s}\Pi _{1ls}+\gamma _{l}\Pi _{1sl}<0\Leftrightarrow \Pi _{1ls}+\beta _{ls}\Pi _{1sl}<0. \tag{52}\end{equation} Applying the matrix convex property, it follows from (52) that \begin{equation}\Pi _{1ls}+\beta _{ls}\Pi _{1sl}<0\Leftrightarrow \left\{ \begin{array}{c} \Pi _{1ls}+\beta _{\min }\Pi _{1sl}<0, \\ \Pi _{1ls}+\beta _{\max }\Pi _{1sl}<0. \end{array} \right. \tag{53}\end{equation} Thus, by combining (19)–(21), for $t\in~I_{k,n}\cap~\mathcal{G}_{1,n-1}$, we have \begin{equation*}\dot{V}_{1}\left( t\right) +2\alpha _{1}V_{1}\left( t\right) \leq 0.\end{equation*} In view of the arbitrariness of $k$, for any $t\in~\mathcal{G }_{1,n-1}$, one has \begin{equation}V_{1}\left( t\right) \leq {\rm e}^{-2\alpha _{1}\left( t-t_{1,n-1}\right) }V_{1}\left( t_{1,n-1}\right) . \tag{54}\end{equation}

Case 2. When DoS jamming attack signals are active, i.e., $t\in \mathcal{G}_{2,n-1}$, following Case 1, we obtain \begin{equation}\dot{V}_{2}\left( t\right) -2\alpha _{1}V_{2}\left( t\right) \leq \underset{ l=1}{\overset{r}{\sum }}\gamma _{l}\xi _{l}^{2}\zeta ^{\rm T}\left( t\right) \Pi _{2ll}\zeta \left( t\right) +\underset{l=1}{\overset{r-1}{\sum }}\underset{ s>l}{\overset{r}{\sum }}\xi _{l}\xi _{s}\zeta ^{\rm T}\left( t\right) \left[ \gamma _{s}\Pi _{2ls}+\gamma _{l}\Pi _{2sl}\right] \zeta \left( t\right), \tag{55}\end{equation} where $\Pi~_{2ls}=\Pi~_{11ls}^{2}+\tau~_{M}N_{2ls}\mathcal{R} _{2}^{-1}N_{2ls}^{\rm~T}+\tau~_{M}M_{2ls}\mathcal{Z}_{2}^{-1}M_{2ls}^{\rm~T}+\tau _{M}S_{2ls}\mathcal{R}_{2}^{-1}S_{2ls}^{\rm~T}+\tau~_{M}\Gamma_{2ls}^{\rm~T}\left( \mathcal{R}_{2}+\mathcal{Z}_{2}\right)~\Gamma~_{2ls}$. The remaining proof is similar to Case 1, and we can obtain that for $t\in~\mathcal{G}_{2,n-1}$, $\dot{V}_{2}\left(~t\right) \leq~2\alpha~_{2}V_{2}\left(~t\right)~$, which implies \begin{equation}V_{2}\left( t\right) \leq {\rm e}^{2\alpha _{2}\left( t-t_{2,n-1}\right) }V_{2}\left( t_{2,n-1}\right) . \tag{56}\end{equation} According to (54)–(56), the estimation of $ V_{i}\left(~t\right)~$ in (22) is obtained.

Proof of Theorem 2

Let $-\gamma~^{2}\bar{w}^{\rm~T}\left(~t\right)~\bar{w}\left(~t\right) +{\rm~e}^{\rm~T}\left(~t\right)~e\left(~t\right)~\triangleq~J\left(~t\right)~$. Similar to Lemma 1, for $\bar{w}\left(~t\right)~\neq~0$, from (33)–(35), we have \begin{equation}\frac{{\rm d}V_{1}\left( t\right) }{{\rm d}t}+2\alpha _{1}V_{1}\left( t\right) +J\left( t\right) \leq 0, t\in \mathcal{G}_{1,n}, \tag{57}\end{equation} and \begin{equation}\frac{{\rm d}V_{2}\left( t\right) }{{\rm d}t}-2\alpha _{2}V_{2}\left( t\right) +J\left( t\right) \leq 0, t\in \mathcal{G}_{2,n}. \tag{58}\end{equation} Now, for $t\in~\left[~g_{k},g_{k}+s_{k}\right)~$, multiplying $\frac{1}{\mu _{2}}{\rm~e}^{-2\alpha~_{1}\left(~g_{k}-t\right)~}$ on both sides of (eq-TH2 guanxi ) yields \begin{equation}\frac{1}{\mu _{2}}{\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }\left[ \frac{ {\rm d}V_{1}\left( t\right) }{{\rm d}t}+2\alpha _{1}V_{1}\left( t\right) \right] \leq \frac{1}{\mu _{2}}{\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }\left[ -J\left( t\right) \right] . \tag{59}\end{equation} Integrating both sides of (59) from $t=g_{k}$ to $ g_{k}+s_{k}$ yields \begin{equation}\int_{g_{k}}^{g_{k}+s_{k}}\frac{1}{\mu _{2}}{\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }\left[ \frac{{\rm d}V_{1}\left( t\right) }{{\rm d}t}+2\alpha _{1}V_{1}\left( t\right) \right] {\rm d}t\leq \int_{g_{k}}^{g_{k}+s_{k}}\frac{1}{ \mu _{2}}{\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }\left[ -J\left( t\right) \right] {\rm d}t. \tag{60}\end{equation} Summing both sides of (60), one has \begin{equation}\sum\limits_{k=0}^{n}\int_{g_{k}}^{g_{k}+s_{k}}\frac{1}{\mu _{2}}{\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }\left[ \frac{{\rm d}V_{1}\left( t\right) }{{\rm d}t}+2\alpha _{1}V_{1}\left( t\right) \right] {\rm d}t\leq \sum\limits_{k=0}^{n}\int_{g_{k}}^{g_{k}+s_{k}}\frac{1}{\mu _{2}}{\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }\left[ -J\left( t\right) \right] {\rm d}t. \tag{61}\end{equation} Similarly, for $t\in~\left[~g_{k}+s_{k},g_{k+1}\right)~$, it follows from ( 58) that \begin{equation}\sum\limits_{k=0}^{n}\int_{g_{k}+s_{k}}^{g_{k+1}}{\rm e}^{2\alpha _{2}\left( g_{k}+s_{k}-t\right) }\left[ \frac{{\rm d}V_{2}\left( t\right) }{{\rm d}t}-2\alpha _{2}V_{2}\left( t\right) \right] {\rm d}t\leq \sum\limits_{k=0}^{n}\int_{g_{k}+s_{k}}^{g_{k+1}}{\rm e}^{2\alpha _{2}\left( g_{k}+s_{k}-t\right) }\left[ -J\left( t\right) \right] {\rm d}t. \tag{62}\end{equation} Now, $\forall~t\in~\lbrack~0,g_{n+1})$, adding both sides of (eq-TH2 guanxi4 ) and (62), and using Theorem 1, we have \begin{eqnarray}& &\sum\limits_{k=0}^{n}\int_{g_{k}}^{g_{k}+s_{k}}\frac{{\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }}{\mu _{2}}\left[ -J\left( t\right) \right] {\rm d}t+\sum\limits_{k=0}^{n}\int_{g_{k}+s_{k}}^{g_{k+1}}{\rm e}^{2\alpha _{2}\left( g_{k+1}-t\right) }\left[ -J\left( t\right) \right] {\rm d}t \\ & & \geq \sum\limits_{k=0}^{n}\int_{g_{k}}^{g_{k}+s_{k}}\frac{{\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }}{\mu _{2}}\left[ \frac{{\rm d}V_{1}\left( t\right) }{{\rm d}t }+2\alpha _{1}V_{1}\left( t\right) \right] {\rm d}t+\sum\limits_{k=0}^{n} \int_{g_{k}+s_{k}}^{g_{k+1}}{\rm e}^{2\alpha _{2}\left( g_{k}+s_{k}-t\right) }\left[ \frac{{\rm d}V_{2}\left( t\right) }{{\rm d}t}-2\alpha _{2}V_{2}\left( t\right) \right]{\rm d}t \\ & & \geq \frac{V_{1}\left( g_{n+1}\right) -V_{1}\left( 0\right) }{\mu _{2}} +\sum\limits_{k=0}^{n}V_{1}(g_{k}+s_{k})\left( \frac{{\rm e}^{2\alpha _{1}s_{\min }}}{\mu _{2}}-\mu _{1}{\rm e}^{2\alpha _{2}d_{\max }+2\left( \alpha _{1}+\alpha _{2}\right) \tau _{M}}\right). \tag{63} \end{eqnarray} Noting $V_{1}\left(~g_{n+1}\right)~\geq~0$, $V_{1}(g_{k}+s_{k})\geq~0$, $ V_{1}\left(~0\right)~=0$, and (23), it follows from (63) that \begin{equation}\sum\limits_{k=0}^{n}\left( \int_{g_{k}}^{g_{k}+s_{k}}\frac{{\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }}{\mu _{2}}\left[ -J\left( t\right) \right] {\rm d}t+\int_{g_{k}+s_{k}}^{g_{k+1}}{\rm e}^{2\alpha _{2}\left( g_{k+1}-t\right) }\left[ -J\left( t\right) \right] {\rm d}t\right) \geq 0. \tag{64}\end{equation} Using Assumption 2, if $t\in~\left[~g_{k},g_{k}+s_{k}\right)~$, $k\in \left\{~0,1,2,\ldots~,n\right\}~$, $n\in~\mathbb{N}$, then \begin{equation}1\leq {\rm e}^{-2\alpha _{1}\left( g_{k}-t\right) }\leq {\rm e}^{2\alpha _{1}s_{k}}\leq {\rm e}^{2\alpha _{1}s_{\max }}. \tag{65}\end{equation} On the other hand, if $t\in~\left[~g_{k}+s_{k},g_{k+1}\right)~$, then \begin{equation}1\leq {\rm e}^{2\alpha _{2}\left( g_{k+1}-t\right) }\leq {\rm e}^{2\alpha _{2}\left( g_{k+1}-g_{k}-s_{k}\right) }\leq {\rm e}^{2\alpha _{2}d_{\max }}. \tag{66}\end{equation} Combining (64)–(66), for $\bar{w}\left(~t\right)\in~L_{2}\left[~0,+\infty~\right)~$, we obtain $\left\Vert~e\left(~t\right)~\right\Vert~_{2}\leq~\tilde{\gamma}~^{\ast~} \left\Vert~\bar{w}\left(~t\right)~\right\Vert~_{2}, $ where $\tilde{\gamma}~^{\ast~}=\sqrt{\frac{\rho~_{\max~}}{\rho~_{\min~}}}\gamma~$, $\rho~_{\min~}$ and $\rho~_{\max~}$ have been defined in Theorem 2.

When $\bar{w}\left(~t\right)~=0$, according to (57) and ( 58), we get $\frac{{\rm~d}V_{1}\left(~t\right)~}{{\rm~d}t}+2\alpha _{1}V_{1}\left(~t\right)~\leq~0$ for $t\in~\mathcal{G}_{1,n}$ and $\frac{ {\rm~d}V_{2}\left(~t\right)~}{{\rm~d}t}-2\alpha~_{2}V_{2}\left(~t\right)~\leq~0$ for $ t\in~\mathcal{G}_{2,n}$. Then, by applying Theorem 1, it follows that the fuzzy switched system (16) is GES. Therefore, based on Definition 3, the proof is complete.


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