State Tracking and Fault Diagnosis for Dynamic Systems Using Labeled Uncertainty Graph

. 2015 Nov 5;15(11):28031–28051. doi: 10.3390/s151128031

Algorithm 1: Roll forward process

1: Input: Initial belief state  $B_{0}^{1}, ..., B_{0}^{n}$ ; Number of the particles N
2: Output: LUG with the most likely belief state  $B^{0}, ... B^{i}$  at each time step
3: Sample N particles using the prior probability distribution  $P (B_{0})$ 
4: Add the initial belief state  $B_{0}^{1}, ..., B_{0}^{n}$  to proposition layer  $P_{0}$ 
5: For each time-step t >0 do
6:    For each belief state  $B_{t}^{i}$  in  $P_{t}$  do
7:      If all the particles can be assigned according to a set of obtained a posteriori transitions
             probability  ${P_{R}^{1}, P_{R}^{2}, ...}$  Then break
8:      Execute possible transitions  $k : = B_{t}^{i} \to B_{t + 1}^{j}$  and store the corresponding effect  $φ_{k}$  into  $ε_{t + Δ}$ 
9:      If the successor belief state  $B_{t + 1}^{j}$  is consistent with observation  $y_{t}$ 
10:          Save the belief state  $B_{t + 1}^{j}$  into proposition layer  $P_{t + 1}$ 
11:          Calculate the a posteriori transitions probability  $P_{R}^{k}$ 
12:          Insert  $P_{R}^{k}$  into a set of obtained a posteriori transitions probability  ${P_{R}^{1}, P_{R}^{2}, ...}$ 
13:      Else
14:          Recalculate the normalization term  $α$ 
15:          Update the set of obtained a posteriori transitions probability  ${P_{R}^{1}, P_{R}^{2}, ...}$ 
16:      End If
17:    End For
18:  Assign the particles for the belief state  $B_{t + 1}$  in  $P_{t + 1}$  according to a set of obtained a posteriori transitions probability  ${P_{R}^{1}, P_{R}^{2}, ...}$ 
19: End For