Trajectory-Learned Actuator-Aware Funnel Network

1. Integration point: place this module at the policy head of a controller for a differentiable simulator, neural ODE, world model, or model-based RL system. The network receives the current state, target, time, and optionally the normalized error $z$; it outputs an unconstrained vector $v_\theta$, which is converted to actuator-valid actions by the tanh parameterization. Do not train on demonstrated actions. 2. Pseudocode: interpolate demonstrations at training times; compute the empirical error quantile and fit a smooth positive $\rho(t)$; sample states near and inside the funnel; evaluate $u=\bar u\odot\tanh(v_\theta)$; roll out the plant or learned dynamics; minimize task loss plus $\beta\max(0,\max_k|e_k|/\rho_k-1)^2+\gamma\max(0,\dot V+\lambda V)^2$. Estimate $\dot V$ with automatic differentiation through the dynamics or a finite difference over two rollout steps. Optionally use an active-set loop: identify samples with funnel or Lyapunov violations, increase their weights, and freeze already feasible samples. 3. Computed from the paper's mechanism: the funnel is inferred from state trajectories, the policy is state-feedback rather than action imitation, and actuator feasibility is enforced during synthesis. Estimated empirically: $A_e$, $B$, disturbance terms, $\dot V$, and the local density radius around demonstrations. 4. First cheap experiment: use a 1D or 2D saturated integrator and compare behavioral cloning from actions, an unconstrained MLP policy, and this funnel controller on held-out initial states. Sweep actuator limit $\bar u$ and funnel radius $\rho$. The quantitative prediction is a sharp feasibility boundary near $k_{\max}=\bar u/\rho$ in the scalar system: above this requested gain, funnel violations or saturation should rise rapidly; below it, normalized error should decay approximately as $|z(t)|\sim e^{-\lambda t/2}$ and the violation rate should remain below $\alpha$ on trajectories close to the demonstration set. A second prediction is that shrinking demonstration spacing should reduce local funnel violations monotonically, while conservative performance degrades gracefully outside the demonstrated region.