State estimation over packet-dropping channels

Centralized approximate optimal estimation for cyber‐physical systems under joint cyber‐attacks

“…Begin from P B[0] k ≤ P * [0] k , after q step iterations, we can get P B[q] k ≤ P * [q] k by using ( 19), (37), and (38). For the q + 1 channel, one has…”

“…By (39), there is $$$ {\mathbf{E}}_{Y_k,{\boldsymbol{\Gamma}}_k^a}\left[{P}_k\right]\le {\mathbf{E}}_{{\boldsymbol{\Gamma}}_k^a}\left[{P}_k^A\right] $$$. It follows from Reference 38 Lemma 4.4.2 that under Assumption 1, $$$ {\mathbf{E}}_{{\boldsymbol{\Gamma}}_k^a}\left[{P}_k^A\right] $$$ is bounded when $$$ {\gamma}^{\left[i\right]}>1-\Lambda {(A)}^{-1} $$$ and set its upper bound as $$$ {T}^A $$$. Since $$$ {\theta}^{\left[i\right]}=1-{\gamma}^{\left[i\right]} $$$, for $$$ {\theta}^{\left[i\right]}<\Lambda {(A)}^{-1} $$$, the following inequality hold: …”