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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

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  • ReceivedMay 31, 2019
  • AcceptedSep 16, 2019
  • PublishedMar 30, 2020

Abstract

This paper addresses the problem of event-triggered attack-tolerant tracking control for a networked nonlinear system under spasmodic denial-of-service (DoS) attacks. Compared with some existing results, theexact duration of DoS attacks is not required while only assuming attainable bounds of attack frequency and duration. First, a new event-triggered attack-tolerant fuzzy tracking controller is proposed, to reduce the amount of sensor data transmissions over the sensor-to-controller (S-C) channel while counteracting unknown DoS attacks. Second, for the purpose of performance analysis and synthesis, a unified event-triggered Takagi-Sugeno (T-S) fuzzy switched model is established, which accounts for a suitable attack-resilient event-triggered communication scheme and unknown DoS jamming signals.Third, using piecewise Lyapunov-Krasovskii functional (PLKF) analysis technique, a newcriterion is derived to ensure exponential stability of the resulting switched tracking error system while achieving a weighted $H_{\infty}$ performance level. Additionally, the relationship among the parameters of a DoS attack signal, thetriggering parameters, the fuzzy controller gains, the sampling period, andthe decay rate can be quantitatively characterized. Moreover, the triggering matrix parameter and fuzzy controller gains are obtained by finding a feasible solution to a set of linear matrix inequalities (LMIs).Finally, numerical verification is performed to demonstrate the effectiveness of the proposed control design method.


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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