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Learning Neural Network Safe Tracking Controllers from Backward Reachable Sets

Yuezhu Xu, Mohamed Serry, Jun Liu, S. Sivaranjani

TL;DR

The paper tackles safe tracking for nonlinear discrete-time systems under disturbances with reach-avoid constraints. It introduces a framework that integrates zonotopic backward reachable sets along a nominal trajectory to guide training of a neural-network tracking controller, producing a closed-form online policy. Safety is reinforced with conformal prediction to provide distribution-free, finite-sample guarantees on meeting the reach-avoid spec. Empirical results on a discrete-time Dubin's car show the approach achieves consistent safety performance and can generalize beyond computed BRS boundaries, outperforming a purely optimization-based baseline in some cases.

Abstract

The design of tracking controllers that closely follow a reference trajectory while ensuring safety and robustness against disturbances is a challenging problem in the control of autonomous systems. In this work, we propose a neural network-based safe tracking control framework for nonlinear discrete-time systems with reach-avoid specifications in the presence of disturbances. Our approach begins with generation of a nominal trajectory using standard trajectory synthesis approaches, followed by construction of safe zonotopic backward reachable sets along the nominal trajectory. The states lying within the backward reachable sets are guaranteed to satisfy safe reachability specifications. Then, our key insight is to leverage the computed backward reachable sets to inform the architecture and training of a neural network-based tracking controller such that the neural network drives the system's states through these backward reachable sets, thereby improving the likelihood of safe reachability. We perform formal verification with conformal prediction to achieve statistical safety guarantees on the performance of the learned neural controller. The performance of our approach is illustrated through a numerical example on the discrete-time Dubin's car model.

Learning Neural Network Safe Tracking Controllers from Backward Reachable Sets

TL;DR

The paper tackles safe tracking for nonlinear discrete-time systems under disturbances with reach-avoid constraints. It introduces a framework that integrates zonotopic backward reachable sets along a nominal trajectory to guide training of a neural-network tracking controller, producing a closed-form online policy. Safety is reinforced with conformal prediction to provide distribution-free, finite-sample guarantees on meeting the reach-avoid spec. Empirical results on a discrete-time Dubin's car show the approach achieves consistent safety performance and can generalize beyond computed BRS boundaries, outperforming a purely optimization-based baseline in some cases.

Abstract

The design of tracking controllers that closely follow a reference trajectory while ensuring safety and robustness against disturbances is a challenging problem in the control of autonomous systems. In this work, we propose a neural network-based safe tracking control framework for nonlinear discrete-time systems with reach-avoid specifications in the presence of disturbances. Our approach begins with generation of a nominal trajectory using standard trajectory synthesis approaches, followed by construction of safe zonotopic backward reachable sets along the nominal trajectory. The states lying within the backward reachable sets are guaranteed to satisfy safe reachability specifications. Then, our key insight is to leverage the computed backward reachable sets to inform the architecture and training of a neural network-based tracking controller such that the neural network drives the system's states through these backward reachable sets, thereby improving the likelihood of safe reachability. We perform formal verification with conformal prediction to achieve statistical safety guarantees on the performance of the learned neural controller. The performance of our approach is illustrated through a numerical example on the discrete-time Dubin's car model.

Paper Structure

This paper contains 14 sections, 4 theorems, 42 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Problem Formulation
  4. Neural-Network Tracking Control using Backward Reachable Sets
  5. Backward Reachable Sets
  6. Architecture Design and Training of Neural Tracking Controllers
  7. Neural Network Architecture Design
  8. Sampling Methods for Training Data
  9. Optimization for Neural Network Training
  10. Formal Verification with Conformal Prediction
  11. Experiments
  12. Consistent and Reliable Performance of Online Neural Network Tracking Controllers
  13. Extending beyond Backward Reachable Sets
  14. Discussion and Future Work

Key Result

Lemma 1

For all $(x,u)\in {T}_{x,k}\times {T}_{u,k}$,

Figures (3)

  • Figure 1: $O_1,O_2,O_3$ are obstacles to avoid and the green set is the target set. The nominal starting points $\tilde{x}_0$ are: $[4,0.5,-\pi/4]^\top$, $[0.5,4.5,-\pi/4]^\top$, and $[1,1,0]^\top$ respectively. For visual clarity, only 100 trajectories are shown out of 1000 simulated.
  • Figure 2: Trajectories generated with the initial states uniformly sampled from the enlarged initial zonotopes $\Lambda_0^{\tau}$. For visual clarity, only 100 trajectories are shown out of 1000 simulated for each case. The original initial zonotope and the sampled initial states are highlighted in the inset in each plot.
  • Figure 3: $O_1,O_2,O_3$ are obstacles to avoid and the green set is the target set. The nominal starting points is $[1,1,0]^\top$ and the inital point we pick outside $\Lambda_0$ is $[1,0.8,0]^T$.

Theorems & Definitions (4)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Proposition 1: Reach-Avoid Guarantee lindemann2024formal