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Input-to-State Stabilizing Neural Controllers for Unknown Switched Nonlinear Systems within Compact Sets

Bhabani Shankar Dey, Ahan Basu, Pushpak Jagtap

TL;DR

The paper tackles safety and ISS for unknown switched nonlinear systems operating in a compact region. It develops a dwell-time–based framework that couples multiple Lyapunov functions with neural networks to learn mode-dependent ISS control Lyapunov functions and controllers from data, while enforcing forward invariance via control barrier functions. Formal guarantees are achieved by embedding Lyapunov decrease, forward-invariance, and Lipschitz validity conditions into the training process, enabling provable correctness despite unknown dynamics. A key result is that, when a common ISS-CLF exists, the framework recovers ISS and safety under arbitrary switching; otherwise, ISS is ensured under a dwell-time constraint dictated by inter-mode Lyapunov comparisons. Numerical experiments on a switched Lotka–Volterra system illustrate that the learned certificates keep the state within the safe set and achieve ISS under the prescribed switching, underscoring the practical impact of data-driven, certified stability for unknown switched systems.

Abstract

This paper develops a neural network based control framework that ensures system safety and input-to-state stability (ISS) for general nonlinear switched systems with unknown dynamics. Leveraging the concept of dwell time, we derive Lyapunov based sufficient conditions under which both safety and ISS of the closed-loop switched system are guaranteed. The feedback controllers and the associated Lyapunov functions are parameterized using neural networks and trained from data collected over a compact state space via deterministic sampling. To provide formal stability guarantees under the learned controllers, we introduce a validity condition based on Lipschitz continuity assumptions, which is embedded directly into the training framework. This ensures that the resulting neural network controllers satisfy provable correctness and stability guarantees beyond the sampled data. As a special case, the proposed framework recovers ISS and safety under arbitrary switching when a common Lyapunov function exists. Simulation results on a representative switched nonlinear system demonstrate the effectiveness of the proposed approach.

Input-to-State Stabilizing Neural Controllers for Unknown Switched Nonlinear Systems within Compact Sets

TL;DR

The paper tackles safety and ISS for unknown switched nonlinear systems operating in a compact region. It develops a dwell-time–based framework that couples multiple Lyapunov functions with neural networks to learn mode-dependent ISS control Lyapunov functions and controllers from data, while enforcing forward invariance via control barrier functions. Formal guarantees are achieved by embedding Lyapunov decrease, forward-invariance, and Lipschitz validity conditions into the training process, enabling provable correctness despite unknown dynamics. A key result is that, when a common ISS-CLF exists, the framework recovers ISS and safety under arbitrary switching; otherwise, ISS is ensured under a dwell-time constraint dictated by inter-mode Lyapunov comparisons. Numerical experiments on a switched Lotka–Volterra system illustrate that the learned certificates keep the state within the safe set and achieve ISS under the prescribed switching, underscoring the practical impact of data-driven, certified stability for unknown switched systems.

Abstract

This paper develops a neural network based control framework that ensures system safety and input-to-state stability (ISS) for general nonlinear switched systems with unknown dynamics. Leveraging the concept of dwell time, we derive Lyapunov based sufficient conditions under which both safety and ISS of the closed-loop switched system are guaranteed. The feedback controllers and the associated Lyapunov functions are parameterized using neural networks and trained from data collected over a compact state space via deterministic sampling. To provide formal stability guarantees under the learned controllers, we introduce a validity condition based on Lipschitz continuity assumptions, which is embedded directly into the training framework. This ensures that the resulting neural network controllers satisfy provable correctness and stability guarantees beyond the sampled data. As a special case, the proposed framework recovers ISS and safety under arbitrary switching when a common Lyapunov function exists. Simulation results on a representative switched nonlinear system demonstrate the effectiveness of the proposed approach.
Paper Structure (26 sections, 11 theorems, 37 equations, 1 figure, 1 algorithm)

This paper contains 26 sections, 11 theorems, 37 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Background
  3. Notations
  4. System Description
  5. Proof:
  6. Proof:
  7. Control Barrier Function
  8. Proof:
  9. Problem Formulation and Control Objective
  10. State and Input Constraints
  11. Unknown System
  12. Switching Signals and Dwell Time
  13. Neural ISS Control Lyapunov Function
  14. Proof:
  15. Formal verification
  16. ...and 11 more sections

Key Result

Theorem 2.6

The subsystem in eq:system is said to be ISS within the state space ${\mathbf{X}}$ with respect to the external input ${\mathsf{w}}$, if there exists an ISS-CLF $V_p$ under the forward invariant controller $g_p$ as defined in Definition def:ISS-Lf.

Figures (1)

  • Figure 1: (a) Evolution of Predator and Prey Population with respect to time corresponding to switched system \ref{['eq:predator_prey']}, (b) Switching Sequence.

Theorems & Definitions (21)

  • Definition 2.1: ISS vu2007input
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Corollary 2.9
  • Definition 2.10
  • ...and 11 more