Detection and Isolation of Simultaneous Additive and Parametric Faults in Nonlinear Stochastic Dynamical Systems

Table 4

Statistics of the magnitudes 𝑋 and 𝑌 under the 𝐻^sim₁ hypotheses after the filtering process.

Hypothesis	𝐻^sim1
𝐸[𝑋]	𝑘𝜑𝐿𝑃^2𝑇−⟨𝜑𝐿𝑃,𝜑𝐿𝑃⟩
Var[𝑋]	Var[𝑋∣𝐻0]
𝐸[𝑌]	𝑅𝜖+𝑘^2𝜑𝐿𝑃^2𝑇+⟨𝜑𝐿𝑃,𝜑𝐿𝑃⟩−2𝑘⟨𝜑𝐿𝑃,𝜑𝐿𝑃⟩
Var[𝑌]	𝑅^2𝜖𝜆^2𝑇^2(2𝜆𝑇−1)+4𝑘^2Var[𝑋∣𝐻0]+4𝑉𝜑^2𝜖−8𝑘𝑉𝜑𝜑𝜖
Cor[𝑋,𝑌]	2𝑘Var[𝑋∣𝐻0]−2𝑉𝜑𝜑𝜖