Intelligence Sampling Control Algorithm for T-S Fuzzy Networked Control Systems via Cloud Server Storage Method Under DoS Attack

Abstract

This work studies the $H_{\infty }$ performance of T-S fuzzy networked control systems under denial-of-service (DoS) attacks. First, the weight coefficient is introduced to measure the damage to the control signal caused by DoS attacks. Then, novel looped Lyapunov-Krasovskii functions are constructed based on the fuzzy membership function, and the nonlinear problem of the system is considered under the premise of reducing the initial condition constraints. Next, the issue of finding the suitable sampling period is transformed into an optimization problem of finding the optimal sampling period to reduce sampling times. An intelligent sampling controller is designed to ensure the asymptotic stability of the system. Finally, the effectiveness of the proposed method is verified with a truck trailer model.

Appendices

Appendix 1

$$\begin{aligned} \theta _{1}= & {} \int ^{t}_{t_{k}}\frac{x^T}{t-t_{k}}ds,\ \theta _{4}=\int ^{t_{k+1}}_{t}\int ^{t_{k+1}}_{u}\frac{x^T}{(t_{k+1}-t)^2}duds,\\ \theta _{2}= & {} \int ^{t}_{t_{k}}\int ^{t_{k}}_{u}\frac{x^T}{(t-t_{k})^2}duds,\ \theta _{3}=\int ^{t_{k+1}}_{t}\frac{x^T}{t_{k+1}-t}ds,\\ \alpha= & {} \, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,col\{x,\underline{h}_{k}\theta _{1},\underline{h}^2_{k}\theta _{2}, \underline{h}_{k}\theta _{3},\overline{h}^2_{k}\theta _{4}\},\\ \varepsilon= & {} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,col\{x,x_{d},x_{\bar{d}}\},\overline{h}_{k}=t_{k+1}-t,\\ \underline{h}_{k}= &\,\,\,\,\,\,\, {} \,\,\,t-t_{k},\theta (t,t_{k})=col\{\theta _{1},\theta _{2},\theta _{3},\theta _{4}\},\\ \xi= & {} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,col\{x,x_{d},x_{\bar{d}},\dot{x},\dot{x}_{d},\dot{x}_{\bar{d}},\theta _{k},x_{k},x_{k+1},\omega \}, \end{aligned}$$

Appendix 2

$$\begin{aligned} \varGamma= & {} \varXi _{1}+\varXi _{2}+\varXi _{3}+\text {sym}\{\eta \varphi \}+z^Tz-\gamma ^2\omega ^T\omega ,\\ \varXi _{1}= & {} \sum \limits _{i = 1}^r\dot{\psi }_{i}(\delta )\varPhi ^T_{1}P_{i}\varPhi _{1}+\text {sym}\{\sum \limits _{i = 1}^r\psi _{i}(\delta )\varPhi ^T_{1}P_{i}\varPhi _{1}\},\\ \varXi _{2}= & {} \sum \limits _{i = 1}^r\dot{\psi }_{i}(\delta )\varPhi ^T_{5}S_{i}\varPhi _{6}+\sum \limits _{i = 1}^r\psi _{i}(\delta )e^T_{4}S_{i}\varPhi _{6}\\{} & {} -\sum \limits _{i = 1}^r\psi _{i}(\delta )\varPhi ^T_{5}S_{i}e_{4}+\sum \limits _{i = 1}^r\dot{\psi }_{i}(\delta )\overline{h}_{k}\underline{h}_{k}\varPhi ^T_{3}R_{i}\varPhi _{3}\\{} & {} +\sum \limits _{i = 1}^r\psi _{i}(\delta )\overline{h}_{k}\varPhi ^T_{3}R_{i}\varPhi _{3}-\sum \limits _{i = 1}^r\psi _{i}(\delta )\underline{h}_{k}\varPhi ^T_{3}R_{i}\varPhi _{3}\\{} & {} +\text {sym}\{\sum \limits _{i = 1}^r\psi _{i}(\delta )\overline{h}_{k}\underline{h}_{k}\varPhi ^T_{3}R_{i}\varPhi _{4}\},\\ \varXi _{3}= & {} \sum \limits _{i = 1}^r\psi _{i}(\delta )\overline{h}_{k}e^T_{4}Q_{i}e_{4}+\sum \limits _{i = 1}^r\psi _{i}(\delta )\underline{h}_{k}e^T_{4}U_{i}e_{4}\\{} & {} +\textrm{sym}\{W^T_{1}L_{1}\varPhi _{5}+W^T_{1}L_{2}\varPhi _{7}+W^T_{1}L_{3}\varPhi _{8}\}\\{} & {} +\textrm{sym}\{W^T_{2}L_{5}\varPhi _{9}-W^T_{2}L_{4}\varPhi _{5}+W^T_{2}L_{6}\varPhi _{10}\},\\ \varPhi _{1}= & {} col\{e_{1},\underline{h}_{k}e_{7},\underline{h}^2_{k}e_{8},\overline{h}_{k}e_{9},\overline{h}^2_{k}e_{10}\},\\ \varPhi _{2}= & {} col\{e_{4},e_{1},\underline{h}_{k}(e_{1}-e_{7}),-e_{1},\overline{h}_{k}(e_{1}-e_{9})\},\\ \varPhi _{3}= & {} \{e_{1},e_{2},e_{3}\},\varPhi _{4}=\{e_{4},(1-\dot{d})e_{5},e_{6}\},\\ \varPhi _{5}= & {} col\{e_{1}-e_{11}\},\varPhi _{6}=col\{e_{12}-e_{1}\},\\ \varPhi _{7}= & {} col\{e_{1}+e_{11}-2e_{7}\},\varPhi _{9}=col\{e_{11}+e_{1}-2e_{9}\},\\ \varPhi _{8}= & {} col\{e_{1}-e_{11}+6e_{7}-12e_{8}\},\\ \varPhi _{10}= & {} col\{e_{11}-e_{1}+6e_{9}-12e_{10}\},\\ W_{1}= & {} col\{e_{1},e_{11},e_{7},e_{8}\},W_{2}=col\{e_{12},e_{1},e_{9},e_{10}\},\\ \overline{Q}_{i}= & {} \sum \limits _{i = 1}^r\psi _{i}(\delta )Q_{i}-\ell _{i}\overline{h}_{k}Q_{i},\\ \overline{U}_{i}= & {} \ell _{i}\underline{h}_{k}U_{i}+\sum \limits _{i = 1}^r\psi _{i}(\delta )U_{i},\\ \eta= & {} e^T_{1}G_{1}+e^T_{11}G_{2}+e^T_{4}G_{3},\\ \varphi= & {} A_{1i}e_{1}+C_{1i}e_{5}+B_{1i}K_{j1}e_{11}\\{} & {} +(1+\lambda )B_{1i}K_{j2}e_{5}+D_{1i}e_{13}-e_{4},\\ z= & {} A_{2i}e_{1}+C_{2i}e_{5}+B_{2i}K_{j1}e_{11}\\{} & {} +(1+\lambda )B_{2i}K_{j2}e_{5}+D_{2i}e_{13}-e_{4},\\ e_{i}= & {} [0_{n\times (i-1)n}\quad I_{n\times n}\quad 0_{n\times (13-i)}],\quad i=1,2,\cdot \cdot \cdot ,13. \end{aligned}$$

Appendix 3

$$\begin{aligned} \bar{\varGamma }= & {} \bar{\varXi }_{1}+\bar{\varXi }_{2}+\bar{\varXi }_{3}+\text {sym}\{\bar{\eta }\bar{\varphi }\}+\bar{z}^T\bar{z}-\gamma ^2\omega ^T\omega ,\\ \bar{\varXi }_{1}= & {} \sum \limits _{i = 1}^r\dot{\psi }_{i}(\delta )\varPhi ^T_{1}\bar{P}_{i}\varPhi _{1}+\text {sym}\{\sum \limits _{i = 1}^r\psi _{i}(\delta )\varPhi ^T_{1}\bar{P}_{i}\varPhi _{1}\},\\ \bar{\varXi }_{2}= & {} \sum \limits _{i = 1}^r\dot{\psi }_{i}(\delta )\varPhi ^T_{5}\bar{S}_{i}\varPhi _{6}+\sum \limits _{i = 1}^r\psi _{i}(\delta )e^T_{4}\bar{S}_{i}\varPhi _{6}\\{} & {} -\sum \limits _{i = 1}^r\psi _{i}(\delta )\varPhi ^T_{5}\bar{S}_{i}e_{4}+\sum \limits _{i = 1}^r\dot{\psi }_{i}(\delta )\overline{h}_{k}\underline{h}_{k}\varPhi ^T_{3}\bar{R}_{i}\varPhi _{3}\\{} & {} +\sum \limits _{i = 1}^r\psi _{i}(\delta )\overline{h}_{k}\varPhi ^T_{3}\bar{R}_{i}\varPhi _{3}-\sum \limits _{i = 1}^r\psi _{i}(\delta )\underline{h}_{k}\varPhi ^T_{3}\bar{R}_{i}\varPhi _{3}\\{} & {} +\text {sym}\{\sum \limits _{i = 1}^r\psi _{i}(\delta )\overline{h}_{k}\underline{h}_{k}\varPhi ^T_{3}\bar{R}_{i}\varPhi _{4}\},\\ \bar{\varXi }_{3}= & {} \sum \limits _{i = 1}^r\psi _{i}(\delta )\overline{h}_{k}e^T_{4}\bar{Q}_{i}e_{4}+\sum \limits _{i = 1}^r\psi _{i}(\delta )\underline{h}_{k}e^T_{4}\bar{U}_{i}e_{4}\\{} & {} +\textrm{sym}\{W^T_{1}L_{1}\varPhi _{5}+W^T_{1}L_{2}\varPhi _{7}+W^T_{1}L_{3}\varPhi _{8}\}\\{} & {} +\textrm{sym}\{W^T_{2}L_{5}\varPhi _{9}-W^T_{2}L_{4}\varPhi _{5}+W^T_{2}L_{6}\varPhi _{10}\},\\ \bar{\overline{Q}}_{i}= & {} \sum \limits _{i = 1}^r\psi _{i}(\delta )\bar{Q}_{i}-\ell _{i}\overline{h}_{k}\bar{Q}_{i},\\ \bar{\overline{U}}_{i}= & {} \ell _{i}\underline{h}_{k}\bar{U}_{i}+\sum \limits _{i = 1}^r\psi _{i}(\delta )\bar{U}_{i},\\ \bar{\eta }= & {} e^T_{1}+\kappa _{1}e^T_{11}+\kappa _{2}e^T_{4},\\ \bar{\varphi }= & {} A_{1i}Xe_{1}+C_{1i}Xe_{5}+B_{1i}Y_{j1}e_{11}\\{} & {} +(1+\lambda )B_{1i}Y_{j2}e_{5}+D_{1i}Xe_{13}-Xe_{4},\\ z= & {} A_{2i}Xe_{1}+C_{2i}Xe_{5}+B_{2i}Y_{1j}e_{11}\\{} & {} +(1+\lambda )B_{2i}Y_{2j}e_{5}+D_{2i}Xe_{13}-Xe_{4}. \end{aligned}$$