Algorithm 1 Bi-level Optimization for Deep Robust MHE

Input: Predicted state ${\overline{x}}_t$; Initialization: Initial tuning parameter ${\omega }_0$; for $i=1$ to $N_{\mathrm{epochs}}$ do     for $t=0$ to T do        Outer Optimization (Forward Pass):       Solve the MHE problem (10) by substituting $\mathcal{G}\left(\omega \right)$ to obtain $\widehat{\mathbf{x}}$;        Compute batch loss $L_{track}$ using (36);        Outer Optimization (Backward Pass):        Compute gradient components ${\displaystyle \frac{\partial L_{track}\left(\widehat{\mathbf{x}}\right)}{\partial \widehat{\mathbf{x}}}}$, ${\displaystyle \frac{\partial \widehat{\mathbf{x}}\left(\mathcal{G}\right)}{\partial \mathcal{G}}}$;        Inner Optimization (neural network):        Compute ${\displaystyle \frac{\partial L_{neuro}\left(\omega \right)}{\partial \omega }}$ using machine learning tool;        Update $\omega$ using gradient-based optimization;        Update adaptive weightings $\mathcal{G}\left(\omega \right)$ using (34);     end for end for return $\omega$.