A Physical Model-Based Observer Framework for Nonlinear Constrained State Estimation Applied to Battery State Estimation

. 2019 Oct 11;19(20):4402. doi: 10.3390/s19204402

Algorithm 1 . EKF Algorithm with FMU evaluations

Set k = 1

(k \in ℕ^{+})

and {\hat{x}}_{0}^{+} = E (x_{0})

Set P_{0}^{+} = E ((x_{0} - {\hat{x}}_{0}^{+}) {(x_{0} - {\hat{x}}_{0}^{+})}^{T})

while (break==false) do
Predict estimation by means of the FMU:

{\hat{x}}_{k}^{-} =

f_{k | k - 1} ({\hat{x}}_{k - 1}^{+}, u_{k - 1})

F_{k - 1} = \exp (\underset{J_{k - 1}}{\underset{⏟}{{\partial f / \partial x |}_{{\hat{x}}_{k - 1}^{+}}}} \cdot T_{s})

P_{k}^{-} = F_{k - 1} P_{k - 1}^{+} F_{k - 1}^{T} + Q

Correct estimation with current measurements:

H_{k} = {\partial h / \partial x |}_{{\hat{x}}_{k}^{-}}

K_{k} = P_{k}^{-} H_{k}^{T} \cdot {(H_{k} P_{k}^{-} H_{k}^{T} + R)}^{- 1}

{\hat{x}}_{k}^{+} = {\hat{x}}_{k}^{-} + K_{k} \cdot (y_{k}^{m} - h ({\hat{x}}_{k}^{-}))

P_{k}^{+} = (I - K_{K} \cdot H_{k}) \cdot P_{k}^{-}

k = k + 1

end while