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Lyapunov-stable Neural Control for State and Output Feedback: A Novel Formulation
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Proposes a neural Lyapunov framework that jointly learns NN controllers, observers and Lyapunov functions for state and output feedback, using PGD counterexamples during training and post-hoc alpha,beta-CROWN verification to certify a larger ROA.
Problem
Obtaining formal Lyapunov stability guarantees and region-of-attraction certificates for neural network controllers on nonlinear systems, especially with output feedback, is hard because existing methods rely on expensive SOS, MIP or SMT solvers and restrict Lyapunov derivative constraints to an exp
Approach
Reformulates the Lyapunov derivative condition so it is enforced only inside the implicitly defined invariant sublevel set (intersected with a box), trains with cheap PGD counterexamples instead of complete verifiers, and certifies post-training with alpha,beta-CROWN branch-and-bound on GPU, plus a learned NN observer for output feedback.
Key results
- On the pendulum and path tracking, the certified ROA from the new formulation is a strict superset of all baseline ROAs (DITL, NLC, UNL).
- Verification runtime for models from DITL: cartpole 129s vs 448s and PVTOL 217s vs 1935s for DITL's MIP-based verifier; path tracking 11.7s vs 13.3s.
- First certified neural Lyapunov controllers with output feedback for the pendulum (angle observation) and 2D quadrotor (lidar), with verification taking 94s and 8.9hrs respectively.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace DITL's MIP verifier with α,β-CROWN | Verification of DITL-trained models · Pendulum state | verification runtime: 7.8s -> 11.3s | Hurt |
| Replace DITL's MIP verifier with α,β-CROWN | Verification of DITL-trained models · Path tracking | verification runtime: 13.3s -> 11.7s | Helped |
| Replace DITL's MIP verifier with α,β-CROWN | Verification of DITL-trained models · Cartpole | verification runtime: 448s -> 129s | Helped |
| Replace DITL's MIP verifier with α,β-CROWN | Verification of DITL-trained models · PVTOL | verification runtime: 1935s -> 217s | Helped |
| Restrict Lyapunov derivative constraints to the certifiable ROA instead of enforcing them across the entire region of interest | Proposed formulation · Inverted pendulum and path tracking | certified ROA | Helped |
| Use the previous formulation that enforces the Lyapunov derivative condition over the entire region (ablation) | Quadrotor state feedback · 2D quadrotor | verification success | Hurt |
| Use the previous formulation that enforces the Lyapunov derivative condition over the entire region (ablation) | Quadrotor output-feedback control · 2D quadrotor with lidar | verification success | Hurt |
Surprising findings
- Prior verification code for PVTOL (DITL/NLC) had a bug that made the region-of-interest much smaller than intended; the corrected version failed to train the model.
Limitations
- Scalability to higher-dimensional systems remains limited; output feedback only tested up to 8 dimensions.
- PVTOL corrected region could not be trained successfully by the new version of the prior code, so comparison relies on the original region.
- Quadrotor output-feedback verification takes 8.9 hours.
Experimental setup
Datasets: simulated dynamics of inverted pendulum, path tracking, 2D quadrotor, cartpole, PVTOL (from prior works)
Benchmarks: inverted pendulum ROA path tracking ROA cartpole verification runtime PVTOL verification runtime 2D quadrotor state and output feedback ROA
Metrics: certified ROA size verification runtime
Baselines: DITL NLC UNL LQR quadratic Lyapunov function
Builds on: alpha,beta-CROWN neural Lyapunov control CEGIS Luenberger observer
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