Intelligent Modeling for Batch Polymerization Reactors with Unknown Inputs

. 2023 Jun 29;23(13):6021. doi: 10.3390/s23136021

Algorithm 1 Recursive Expectation-Maxization Algorithm

Data:

{\hat{x}}_{0}

{\hat{P}}_{0}

a_{0}

y_{N}

u_{N}

Q_{N}

R_{N}

γ_{N}

Result:

{\hat{x}}_{N}

a_{N}

1: for

N = 1, 2, \dots

do
2:

{\overset{ˇ}{x}}_{N} = A {\hat{x}}_{N - 1} + M a_{N - 1} + B u_{N - 1}

{\overset{ˇ}{P}}_{N} = A {\hat{P}}_{N - 1} A^{T} + Q_{N - 1}

K_{N} = {\overset{ˇ}{P}}_{N} H^{T} {[H {\overset{ˇ}{P}}_{N} H^{T} + R_{N}]}^{- 1}

{\hat{x}}_{N} = {\overset{ˇ}{x}}_{N} + K_{N} (y_{N} - H {\overset{ˇ}{x}}_{N})

{\hat{P}}_{N} = (I - K_{N} H) {\overset{ˇ}{P}}_{N}

a_{N} = (1 - γ_{N}) a_{N - 1} + γ_{N} (inv (M) ({\hat{x}}_{N} - A {\hat{x}}_{N - 1} - B u_{N - 1}))

8: end for
9: † Remarks: If matrice

M

is not of full column-rank, it should be computed by using any pseudo-inverse.