Schur-Constrained Neural Derivative Feedback

Integrate this at the action head of a neural feedback policy in an RL or model-based control system. Let the policy receive the current observation \(y_k\), maintain the previous observation \(y_{k-1}\) in its recurrent controller state, and output a base action \(\pi_\theta(y_k)\). Add a trainable derivative gain matrix \(K_D\) and form the action as \(u_k=\pi_\theta(y_k)-K_D(y_k-y_{k-1})/h\); optionally clip the final action to the actuator range. If a differentiable world model is available, estimate local plant Jacobians \(B_k=\partial u_{k+1}/\partial u_k\) and \(C_k=\partial y_{k+1}/\partial x_k\), or use a local linear system-identification model. At every training step compute \(M_k=C_kB_kK_D\). Estimate \(\rho(M_k)\) by power iteration: initialize a random vector \(v\), repeatedly set \(v\leftarrow M_kv/\|M_kv\|_2\), and use \(\hat\rho=\|M_kv\|_2\) after 5-10 iterations; for non-normal matrices, use several random initial vectors and take the maximum estimate. Add \(\lambda_{stab}[\max(0,\hat\rho-(1-\varepsilon))]^2\) to the policy loss, with \(\varepsilon\in[0.05,0.2]\). The paper supplies the fast-mode mechanism and Schur constraint; Jacobian estimation, the margin, and the differentiable penalty are engineering adaptations. First test on a linear or lightly nonlinear cart-pole or quadrotor benchmark with a two-layer MLP policy, comparing a finite-difference derivative branch with and without the spectral penalty across sampling intervals \(h\). Measure episode-return variance, action oscillation, instability rate, and performance after abruptly reducing \(h\). Success is fewer divergent rollouts and lower high-frequency action energy at equal return, while the unregularized controller should show failures correlated with \(\hat\rho(CBK_D)\ge 1\).