# Trajectory-Learned Actuator-Aware Funnel Network

- ID: 1339
- Canonical URL: https://synthcore.org/idea/1339/trajectory-learned-actuator-aware-funnel-network
- API JSON: https://synthcore.org/api/idea/1339.json
- API Markdown: https://synthcore.org/api/idea/1339.md
- Verification status: beats_baseline
- Source: [arXiv:2607.23876](https://arxiv.org/abs/2607.23876)
- Category: dynamics
- Solves: stability, accuracy, generalization
- ML areas: optimizer, training-dynamics, rl, world-model
- Math tags: control-theory, dynamical-systems, optimization, statistics
- Ratings: usefulness 8/10; difficulty 6/10; novelty 7/10

## Idea description

Construct a prescribed-performance funnel directly from state-only demonstrations, then train a state-feedback neural network whose output is bounded and whose gain is optimized to keep the tracking error inside that funnel. The controller should not imitate actions; it should reproduce the demonstrated transient and steady-state error geometry while explicitly reducing feedback authority whenever actuator saturation would make the funnel infeasible.

## Mathematical statement

Let $x\in\mathbb{R}^{n}$ be the plant state, $r(t)$ the desired trajectory, $e=x-r$, and $u\in\mathbb{R}^{m}$ the control input with componentwise limits $|u_i|\leq\bar u_i$. From demonstrations $x^{(j)}(t)$, estimate a positive funnel radius $\rho_k(t)$ for each state component, for example $\rho_k(t)=\epsilon_k+q_{1-\alpha}(|x_k^{(j)}(t)-r_k(t)|)$ followed by positive smoothing, where $q_{1-\alpha}$ is the empirical $(1-\alpha)$ quantile and $\epsilon_k>0$. Use the normalized error $z_k=e_k/\rho_k(t)$ and train a gain network $K_\theta(x,t)$ in a bounded controller $u=\operatorname{diag}(\bar u)\tanh(v_\theta(x,t))$. For a locally linear error model $\dot e\approx A_e(x,t)e+B(x)u+d(x,t)$, a sufficient contraction-style condition is $\dot V\leq-\lambda V$ for $V=\frac12 z^\top Pz$, $P\succ0$, with $\dot V=z^\top P[\dot e/\rho-e\dot\rho/\rho^2]$ componentwise. The actuator-authority condition is explicit in the scalar case: if $\dot e=-k e$ and $|e|\leq\rho$, then $k\rho\leq\bar u$, hence $k\leq\bar u/\rho$. The neural analogue penalizes or constrains the worst-case required control over sampled funnel states.

## Key formulas

- $$\rho_k(t)=\epsilon_k+q_{1-\alpha}\left(\left\{|x_k^{(j)}(t)-r_k(t)|\right\}_{j=1}^{N}\right),\qquad z_k(t)=\frac{x_k(t)-r_k(t)}{\rho_k(t)}.$$
- $$u_\theta(x,t)=\operatorname{diag}(\bar u)\tanh\!\left(v_\theta(x,t)\right),\qquad |u_{\theta,i}|<\bar u_i.$$
- $$V(z)=\frac12z^\top Pz,\qquad \dot V=z^\top P\left[\operatorname{diag}(\rho)^{-1}\dot e-\operatorname{diag}\!\left(\frac{\dot\rho}{\rho^2}\right)e\right]\leq-\lambda V,$$
- $$k\rho\leq\bar u\quad\Longrightarrow\quad k\leq k_{\max}:=\frac{\bar u}{\rho}.$$

## Implementation notes

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.

## Verification

- Status: beats_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: helps
- Verdict: Built a state-only quantile funnel estimator and actuator-aware scalar saturated-integrator controller MVP in funnel_mvp.py. The mechanism checks succeeded: observed gain clipped exactly at the predicted k_max=ubar/rho, authority scaling was linear, and the normalized-error derivative identity had 0.2% relative error. However, the funnel-aware controller did not beat the fixed-gain saturated baseline on this setup, so the mechanism was verified but no performance win was observed.

### Mechanism check

- Verdict: Built a state-only quantile funnel estimator and actuator-aware scalar saturated-integrator controller MVP in funnel_mvp.py. The mechanism checks succeeded: observed gain clipped exactly at the predicted k_max=ubar/rho, authority scaling was linear, and the normalized-error derivative identity had 0.2% relative error. However, the funnel-aware controller did not beat the fixed-gain saturated baseline on this setup, so the mechanism was verified but no performance win was observed.
- Confidence: 8/10
- Limitations: This is a scalar toy MVP rather than a trained neural network, and it uses a known integrator instead of learned or differentiable dynamics. The inferred funnel shrinks over time, held-out initial states include points outside its initial radius, and no multidimensional, noisy-dynamics, action-imitation, or training-compute comparison was tested.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.0081
- Paired wins: 8/8
- Benchmark verdict: idea better (significant)

## Artifacts

- [bench_funnel.py](https://synthcore.org/code/357/bench_funnel.py)
- [bench_report.json](https://synthcore.org/code/357/bench_report.json)
- [funnel_mvp.py](https://synthcore.org/code/357/funnel_mvp.py)
- [report.md](https://synthcore.org/code/357/report.md)
- [report_bench_2026-08-31T091800.md](https://synthcore.org/code/357/report_bench_2026-08-31T091800.md)
- [results.json](https://synthcore.org/code/357/results.json)
- [Download all files as ZIP](https://synthcore.org/download/357)

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.