Consensus-Based Estimation for Type-2 Fuzzy Time-Delay Systems Under Deception Attacks and Partial Information Exchange

Abstract

This paper addresses the problem of consensus-based state estimation for interval type-2 Takagi–Sugeno (T–S) fuzzy time-delay systems over a wireless sensor network under deception attacks, partial information exchange and switching sensor-network topologies. The switching-topology-dependent estimators are constructed using interval type-2 fuzzy models considering the cases where the premise variables are available and unavailable. By means of the mode-dependent dwell time approach, sufficient conditions are established such that the estimation error system is exponentially and ultimately bounded. In addition, a fuzzy Lyapunov functional is utilized to derive the existence conditions of estimators for the case that the premise variable are unavailable. Two illustrative examples are provided to verify the effectiveness of the proposed method.

Appendixes

Appendix A: The proof of the instability on the system in Example 1

Let $V (k) = x_1(k) x_2(k)$ be a function defined on ${\mathcal {D}} = \{(x_1(k), x_2(k)) : x_1(k)>0, x_2(k)>0 \}$. It is clear that $V (k)>0$ on ${\mathcal {D}}$ and $V (k)=0$ on the boundary of ${\mathcal {D}}$.

The variation of V(k) is given by

$$\begin{aligned} \triangle V (k)= \, & {} V (k+1) -V (k)\nonumber \\= \, & {} x_1(k+1) x_2(k+1)-x_1(k) x_2(k)\nonumber \\= \, & {} (0.17\omega _1(x_1(k)) \nonumber \\&+1.95\omega _2(x_1(k)))x_1^2(k)-0.3x_1(k) x_2(k) \nonumber \\&+(0.01\omega _1(x_1(k))\nonumber \\&+0.05\omega _2(x_1(k)))x_1(k)x_1(k-d(k))\nonumber \\&+0.02x_1(k)x_2(k-d(k)). \end{aligned}$$

(54)

If $|x_1(k-d(k))|\le |x_1(k)|$ and $|x_2(k-d(k))|\le |x_2(k)|$, then

$$\begin{aligned} \triangle V (k)\ge & {} (0.16\omega _1(x_1(k)) +1.9\omega _2(x_1(k)))x_1^2(k)\nonumber \\&-0.32x_1(k) x_2(k)\nonumber \\= \, & {} \omega _1(x_1(k))(0.16x_1^2(k)-0.32x_1(k) x_2(k))\nonumber \\&+\omega _2(x_1(k))(1.9x_1^2(k)-0.32x_1(k) x_2(k)) \end{aligned}$$

(55)

where the fact $\omega _1(x_1(k))+\omega _2(x_1(k))=1$ is applied. Since $\omega _1(x_1(k))>0$ and $\omega _2(x_1(k))>0$, $\triangle V (k) >0$ for all $(x_1(k),x_2(k))\in {\mathcal {D}} \cap {\mathcal {B}}$, where ${\mathcal {B}}= \{(x_1(k), x_2(k)) : x_2(k)<0.5x_1(k), x_1^2(k)+x_2^2(k) ~\text { small} \}$. By Chetaev’s theorem [54], the system in Example 1 is unstable.

Appendix B: The gain matrices of the concerned observers in Example 1

$$\begin{aligned}&{\varvec{L}}_{111}= \left[ \begin{array}{c} -1.1155\\ -0.9924 \end{array} \right] , \ {\varvec{L}}_{211}=\left[ \begin{array}{c} -0.8998\\ -2.5877\end{array} \right] , \\&{\varvec{K}}_{1121}=\left[ \begin{array}{cc} 0.0262 &{} -0.2857\\ 0.5458 &{} -2.8686\end{array} \right] , \ {\varvec{K}}_{2121}=\left[ \begin{array}{cc} 0 &{} 0.3953\\ 0 &{} 0.5976 \end{array} \right] , \\&{\varvec{L}}_{121}=\left[ \begin{array}{c} -0.1385\\ -0.5709\end{array} \right] , \ {\varvec{L}}_{221}=\left[ \begin{array}{c} -0.4586\\ -1.5231\end{array} \right] , \\&{\varvec{K}}_{1211}=\left[ \begin{array}{c} 0.5411\\ -0.0117 \end{array} \right] , \ {\varvec{K}}_{2211}=\left[ \begin{array}{c} 0.\\ 0.1929\end{array} \right] , \\&{\varvec{K}}_{1231}=\left[ \begin{array}{c} 0.4335\\ 0.3825\end{array} \right] , \ {\varvec{K}}_{2231}=\left[ \begin{array}{c} 1.5058\\ 1.6308\end{array} \right] , \\&{\varvec{L}}_{131}=\left[ \begin{array}{c} -0.8630\\ -1.2539\end{array} \right] , \ {\varvec{L}}_{231}=\left[ \begin{array}{c} -0.5349\\ -2.7045\end{array} \right] , \\&{\varvec{K}}_{1321}=\left[ \begin{array}{cc} 0.0328 &{} 1.0634\\ 0.4605 &{} -0.5131 \end{array} \right] , \ {\varvec{K}}_{2321}=\left[ \begin{array}{cc} 0 &{} 1.5057\\ 0 &{} 2.0479\end{array} \right] , \\&{\varvec{L}}_{112}=\left[ \begin{array}{c} -0.9615\\ -0.8433\end{array} \right] , \ {\varvec{L}}_{212}=\left[ \begin{array}{c} -0.8556\\ -2.5370\end{array} \right] , \\&{\varvec{K}}_{1122}=\left[ \begin{array}{cc} -0.0385 &{} -0.0041\\ -0.6816 &{} 0.5078\end{array} \right] , \ {\varvec{K}}_{2122}=\left[ \begin{array}{cc} 0.0867 &{} 0.0254\\ -0.0644 &{} 0.5640\end{array} \right] , \\&{\varvec{K}}_{1132}=\left[ \begin{array}{c} 0.0093\\ -0.3521\end{array} \right] , \ {\varvec{K}}_{2132}=\left[ \begin{array}{c} 0.1365\\ -0.0506\end{array} \right] , \\&{\varvec{L}}_{122}=\left[ \begin{array}{c} -0.0974\\ -0.4552\end{array} \right] , \ {\varvec{L}}_{222}=\left[ \begin{array}{c} -0.3083\\ -1.3159\end{array} \right] , \\&{\varvec{K}}_{1212}=\left[ \begin{array}{c} 1.5365\\ 0.2145\end{array} \right] , \ {\varvec{K}}_{2212}=\left[ \begin{array}{c} 1.3554\\ 2.8348\end{array} \right] , \\&{\varvec{L}}_{132}=\left[ \begin{array}{c} -0.7028\\ -0.5624\end{array} \right] , \ {\varvec{L}}_{232}=\left[ \begin{array}{c} -0.8756\\ -2.6001\end{array} \right] , \\&{\varvec{K}}_{1312}=\left[ \begin{array}{cc} 1.1272 &{} 0.0059\\ -0.6880 &{} 0.3596 \end{array} \right] , \ {\varvec{K}}_{2312}=\left[ \begin{array}{cc} 1.0195 &{} 0.0091\\ 0.7218 &{} 0.3564\end{array} \right] . \end{aligned}$$