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

Table 2

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

Hypothesis	𝐻^sim1
𝐸[𝑋]	𝑘𝜙𝑎𝜑𝐿𝑃+𝑘𝜑𝐿𝑃^2𝑇
Var[𝑋]	Var[𝑋∣𝐻0]
𝐸[𝑌]	𝑅𝜖+𝑘^2𝜙𝑎+𝑘^2𝜑𝐿𝑃^2𝑇+2𝑘𝑘𝜙𝑎𝜑𝐿𝑃
Var[𝑌]	𝑅^2𝜖𝜆^2𝑇^2(2𝜆𝑇−1)+4𝑘^2Var[𝑋∣𝐻0]+8𝑘𝑘𝜙𝑎𝑉𝜑𝜖+8𝑘^2𝜙𝑎𝑅𝜖𝑇^2𝜆^2−𝑇^2𝜆^24+𝜆𝑇−1
Cor[𝑋,𝑌]	2𝑘Var[𝑋∣𝐻0]+2𝑘𝜙𝑎𝑉𝜑𝜖