Prescribed-Performance Hidden-State Observer

Integrate the method into a GRU, small diagonal state-space model, or recurrent world model that produces prediction $\hat y_k$ and hidden state $h_k$. Define $e_{1,k}=y_k-\hat y_k$ and maintain observer variables $\hat e_{1:n,k}$ outside the main network. Optionally correct the recurrent state with $h_{k+1}=F_\theta(h_k,x_k)+K_\theta(\hat e_{1,k}-e_{1,k})$. At each step initialize or update $r_k=(r_0-r_\infty)e^{-l_ok}+r_\infty$, calculate $\xi_k=(e_{1,k}-\hat e_{1,k})/r_k$, clamp it to $[-1+\epsilon,1-\epsilon]$, compute $T_k=\log((1+\xi_k)/(1-\xi_k))$, and apply the discrete observer equations above. Add the envelope barrier to the task loss; if derivative labels are unavailable, omit the derivative matching term and retain the barrier and one-step output consistency. The formula, transformed innovation, and gain scaling come from the paper; $n$, gains, envelope parameters, and correction gain are hyperparameters estimated by validation. First test on Mackey--Glass prediction and noisy sinusoid tracking with a one-layer GRU against a vanilla GRU and fixed-gain observer. Sweep $l_o$ and $\gamma_i$. Measure normalized error, long-horizon RMSE, transformed-innovation magnitude, and first envelope violation. The prediction is a sharp degradation boundary as $l_o$ or gains increase: stable runs keep $|e_1-\hat e_1|/r_k<1$, while unstable runs show rapidly increasing transformed innovation. Reducing $r_\infty$ should improve accuracy only until this boundary.