Algorithm 2. Nonlinear gradient search algorithm for MHE

Set $j=0$ $\left(j\in {\mathbb{N}}^{+}\right)$ and ${\mathit{\xi }}_k^0={\left({\mathit{x}}_{\mathrm{int},k-M}^T,\dots ,{\mathit{x}}_{\mathrm{int},k}^T\right)}^T$

while$\left|g\left({\mathit{\xi }}_k^j\right)-g\left({\mathit{\xi }}_k^{j+1}\right)\right|>10\cdot ϵ$do

if unconstrained then

${\mathit{r}}_j=-\mathbf{\nabla }g\left({\mathit{\xi }}_k^j\right)$

elseif constrained then

${\mathit{r}}_{\mathrm{pro},j}=-{\mathit{P}}^{†}\left(\mathbf{A}\right)\cdot \mathbf{\nabla }g\left({\mathit{\xi }}_k^j\right)$

${\mathit{r}}_{\mathrm{f},j}={\mathit{r}}_{\mathrm{pro},j}-\left({\mathit{G}}_{\mathrm{pa},i}{\mathit{r}}_{\mathrm{pro},j}\right){\mathit{G}}_{\mathrm{pa},i}^T$

end if

Determine step size via line search: ${\eta }_j=\underset{0\le \eta }{\mathrm{argmin}} g\left({\mathit{\xi }}_k^j+\eta \cdot {\mathit{r}}_j\right)$

Optimization step: ${\mathit{\xi }}_k^{j+1}={\mathit{\xi }}_k^j+{\eta }_j{\mathit{r}}_j$

$j=j+1$

end while