State estimation via prediction-based scheme for linear time-varying uncertain networks with communication transmission delays and stochastic coupling

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Table

NPBSE Algorithm:
Step 1: Initialize the related values.
Step 2: Calculate the estimator parameter matrix $G_{i,s}$
by the expression (17(17) $\begin{array}{rl}{G}_{i,s}& =(1+{\sigma }_1+{\sigma }_2+{\sigma }_3+{\sigma }_4)(A_{i,s}+c{\beta }_{ii}\mathrm{\Gamma })\\ & \times {\mathcal{Z}}_{i,s}{C}_{i,s}^T{\mathrm{\Xi }}_{i,s}^{-1},\end{array}$(17) ).
Step 3: Update the prediction estimation ${\hat{x}}_{j,s\mid s-dij}$
by using (5(5) $\begin{array}{rl}{\hat{x}}_{j,s-d_{ij}+1|s-d_{ij}}& =A_{j,s-d_{ij}}{\hat{x}}_{j,s-d_{ij}},\\ {\hat{x}}_{j,s-d_{ij}+2|s-d_{ij}}& =A_{j,s-d_{ij}+1}{\hat{x}}_{j,s-d_{ij}+1|s-d_{ij}},\\ & ⋮\\ {\hat{x}}_{j,s-1|s-d_{ij}}& =A_{j,s-2}{\hat{x}}_{j,s-2|s-d_{ij}},\\ {\hat{x}}_{j,s|s-d_{ij}}& =A_{j,s-1}{\hat{x}}_{j,s-1|s-d_{ij}}.\end{array}$(5) ).
Step 4: Obtain the state estimation ${\hat{x}}_{j,s+1}$ according
to (6(6) $\begin{array}{rl}{\hat{x}}_{i,s+1}& =A_{i,s}{\hat{x}}_{i,s}+c{\beta }_{ii}\mathrm{\Gamma }{\hat{x}}_{i,s}+c\sum _{j=1,j\ne i}^N{\beta }_{ij}\mathrm{\Gamma }{\hat{x}}_{j,s\mid s-d_{ij}}\\ & +G_{i,s}(y_{i,s}-C_{i,s}{\hat{x}}_{i,s}),\end{array}$(6) ).
Step 5: Design the upper bound of estimation error
covariance ${\mathcal{Z}}_{i,s+1}$ by utilizing (19(19) $\begin{array}{rl}{\mathcal{Z}}_{i,s+1}& =(1+{\sigma }_1+{\sigma }_2+{\sigma }_3+{\sigma }_4)(A_{i,s}+c{\beta }_{ii}\mathrm{\Gamma }){\mathcal{Z}}_{i,s}\\ & \times (A_{i,s}+c{\beta }_{ii}\mathrm{\Gamma }{)}^T+2(N-1)c^2\\ & \times (1+{\sigma }_1^{-1}+{\sigma }_5+{\sigma }_6+{\sigma }_7)\sum _{j=1,j\ne i}^N{\beta }_{ij}^2\mathrm{\Gamma }\\ & \times ({\mathcal{X}}_{j,s}+{\hat{x}}_{j,s\mid s-d_{ij}}{\hat{x}}_{j,s\mid s-d_{ij}}^T){\mathrm{\Gamma }}^T\\ & +c^2\sum _{j=1}^N({\beta }_{ij}+\epsilon )(1-{\beta }_{ij}+\epsilon )\mathrm{\Gamma }{\mathcal{X}}_{j,s}{\mathrm{\Gamma }}^T\\ & +N(1+{\sigma }_2^{-1}+{\sigma }_5^{-1}+{\sigma }_8+{\sigma }_9)c^2{\epsilon }^2\sum _{j=1}^N\mathrm{\Gamma }{\mathcal{X}}_{j,s}{\mathrm{\Gamma }}^T\\ & +(1+{\sigma }_4^{-1}+{\sigma }_7^{-1}+{\sigma }_9^{-1}+{\sigma }_{10}^{-1})\mathrm{tr}\left(H_{i,s}{\mathcal{X}}_{i,s}{H}_{i,s}^T\right)\\ & \times M_{i,s}{M}_{i,s}^T-(1+{\sigma }_1+{\sigma }_2+{\sigma }_3+{\sigma }_4{)}^2\\ & \times (A_{i,s}+c{\beta }_{ii}\mathrm{\Gamma }){\mathcal{Z}}_{i,s}{C}_{i,s}^T{\mathrm{\Xi }}_{i,s}^{-1}{C}_{i,s}{\mathcal{Z}}_{i,s}\\ & \times (A_{i,s}+c{\beta }_{ii}\mathrm{\Gamma }{)}^T+W_{i,s}.\end{array}$(19) ).
Step 6: Let s = s + 1. Go back to Step 2.

Figure 1. The curves of $x_{1,s}$ and ${\hat{x}}_{1,s}$.

Figure 2. The curves of $x_{2,s}$ and ${\hat{x}}_{2,s}$.

Figure 3. The curves of $x_{3,s}$ and ${\hat{x}}_{3,s}$.

Figure 4. log(MSE) and upper bounds.

Figure 5. log(MSE) with/without compensation (500 iterations).

Table 1. log($\mathrm{tr}\{{\mathcal{Z}}_{1,s+1}\})$ and ${\beta }_{ij}$ (Node 1).

time(s)	···	65	66	67	68	69	···
Case I	···	30.389	30.82	31.195	31.842	32.363	···
Case II	···	32.587	33.047	33.457	34.133	34.68	···
Case III	···	33.666	34.138	34.565	35.255	35.815	···

Table 2. log($\mathrm{tr}\{{\mathcal{Z}}_{2,s+1}\})$ and ${\beta }_{ij}$ (Node 2).

time(s)	···	65	66	67	68	69	···
Case I	···	31.378	31.747	32.188	32.883	33.262	···
Case II	···	33.54	33.94	34.412	35.136	35.549	···
Case III	···	34.601	35.017	35.504	36.243	36.672	···

Table 3. log($\mathrm{tr}\{{\mathcal{Z}}_{3,s+1}\})$ and ${\beta }_{ij}$ (Node 3).

time(s)	···	65	66	67	68	69	···
Case I	···	31.487	32.135	32.481	32.987	33.585	···
Case II	···	33.528	34.262	34.55	35.101	35.803	···
Case III	···	34.54	35.314	35.576	36.149	36.9	···