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Stability Analysis of Switched Linear Systems with Neural Lyapunov Functions
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Introduces ReLU neural Lyapunov functions for estimating the joint spectral radius of switched linear systems, giving approximation guarantees via convex geometry and benchmarking against ellipsoidal and SOS methods.
Problem
Stability analysis of discrete-time switched linear systems via the joint spectral radius (JSR), which is NP-hard to approximate and undecidable in general; open questions concern accuracy guarantees, sample complexity, and validity of neural Lyapunov functions as norms.
Approach
Represent the norm/Lyapunov function as a bias-free ReLU network (positively homogeneous CPWL function) trained on sampled unit-sphere points to minimize the max ratio V(A_i x)/V(x); combine Barvinok's polytope approximation and McMullen's face bound with Hertrich et al.'s ReLU expressivity result to derive depth and width bounds for a given JSR precision.
Key results
- On the 2D system (JSR 8.6881), best 10-neuron 1-layer ReLU network gives 8.6910 vs ellipsoidal 9.5868 and SOS degree-4 8.7203.
- On the 8D system (JSR 1), ReLU network with 10-30 neurons runs in roughly the time of SOS degree-2, but the network overfits and can underestimate the JSR; ellipsoidal gives 2.4286 and SOS degree-4 gives 1.0006 (~258 s).
- L1 regularization and incremental learning are proposed to mitigate overfitting in the 8D case.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| increase network width from 5 to 10 neurons (ablation) | ReLU neural Lyapunov approximation · Σ2 | JSR approximation precision: 9.0251 -> 8.6969 | Helped |
| increase network width from 5 to 10 neurons (ablation) | ReLU neural Lyapunov approximation · Σ2 | seed variability (standard deviation): 0.8800 -> 0.0056 | Helped |
| add L1 regularization (ablation) | ReLU network, 1 hidden layer, 15 neurons · Σ8 | JSR approximation | Helped |
| add sample points incrementally during training, combined with regularization (ablation) | ReLU neural Lyapunov approximation · Σ8 | JSR approximation validity | Helped |
| post-process the learned network into a norm using the sample-set convex hull (ablation) | ReLU neural Lyapunov approximation · Σ2 | JSR approximation: 8.5788 -> 9.4157 | Mixed |
Surprising findings
- Neural approximations can fall below the true JSR because training samples do not guarantee a valid norm, so the learned bound is not a formal upper bound.
- Neural approach is slower than ellipsoidal and SOS in the low-dimensional example despite better precision.
Limitations
- Sample-based training gives no formal guarantee that the learned function is a valid norm or upper bound on the JSR.
- Overfitting in higher dimensions requires more samples, increasing computation time.
- Experiments limited to two example systems and a single architecture family.
Experimental setup
Datasets: synthetic switched linear systems (2D example and 8D example)
Benchmarks: 2D switched system (Ahmadi et al. 2014 style example) 8D switched system with 8 modes
Metrics: JSR upper bound approximation computation time mean and std over 20 seeds
Baselines: ellipsoidal approximation (SDP) SOS approximation of degree 4
Builds on: Hertrich et al. 2021 ReLU expressivity Barvinok 2013 polytope approximation Blondel-Nesterov-Theys ellipsoidal JSR bound Parrilo-Jadbabaie SOS JSR approximation
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