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 4. Simplified Newton descent search algorithm for constrained estimation projection.

Set μ_{o} = 0

and k = - 1

and i t e r = 1

and m a x_i t e r = 30

Set α = {(\nabla c {({\hat{x}}_{k}^{+})}^{T} \cdot \nabla c ({\hat{x}}_{k}^{+}))}^{- 1}

while

| α \cdot c ({\hat{x}}_{k}^{+} + μ_{k + 1} \cdot \nabla c ({\hat{x}}_{k}^{+})) | \geq ϵ_{Newton}

and

iter < \max_iter

k = k + 1

Δ μ_{k + 1} = - α \cdot c ({\hat{x}}_{k}^{+} + μ_{k} \cdot \nabla c ({\hat{x}}_{k}^{+}))

μ_{k + 1} = μ_{k} + Δ μ_{k + 1}

i t e r = i t e r + 1

end while
Solution:

{\hat{x}}_{k}^{P} = {\hat{x}}_{k}^{+} + μ_{k + 1} \cdot \nabla c ({\hat{x}}_{k}^{+})