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Neural Network-based Co-design of Output-Feedback Control Barrier Function and Observer

Vaishnavi Jagabathula, Ahan Basu, Pushpak Jagtap

TL;DR

The work addresses safe control under partial observability by co-designing a neural network-based observer, controller, and control barrier function using an augmented state formulation. By formulating CBF conditions on the augmented state $\tilde{x}$ and converting the problem into a scenario-optimized, Lipschitz-certified training loop, the approach delivers safety guarantees without requiring exact observer convergence or handcrafted barrier templates. The method is validated on nonlinear, partially observed systems including a DC motor, a pendulum, and a three-tank system, demonstrating safety under input constraints and robustness to imperfect state estimates. This co-design framework enables practical deployment of safe controllers in real-world partially observed dynamical systems, while highlighting scalability challenges in high dimensions.

Abstract

Control Barrier Functions (CBFs) provide a powerful framework for ensuring safety in dynamical systems. However, their application typically relies on full state information, which is often violated in real-world scenarios due to the availability of partial state information. In this work, we propose a neural network-based framework for the co-design of a safety controller, observer, and CBF for partially observed continuous-time systems. By formulating barrier conditions over an augmented state space, our approach ensures safety without requiring bounded estimation errors or handcrafted barrier functions. All components are jointly trained by formulating appropriate loss functions, and we introduce a validity condition to provide formal safety guarantees beyond the training data. Finally, we demonstrate the effectiveness of the proposed approach through several case studies.

Neural Network-based Co-design of Output-Feedback Control Barrier Function and Observer

TL;DR

The work addresses safe control under partial observability by co-designing a neural network-based observer, controller, and control barrier function using an augmented state formulation. By formulating CBF conditions on the augmented state and converting the problem into a scenario-optimized, Lipschitz-certified training loop, the approach delivers safety guarantees without requiring exact observer convergence or handcrafted barrier templates. The method is validated on nonlinear, partially observed systems including a DC motor, a pendulum, and a three-tank system, demonstrating safety under input constraints and robustness to imperfect state estimates. This co-design framework enables practical deployment of safe controllers in real-world partially observed dynamical systems, while highlighting scalability challenges in high dimensions.

Abstract

Control Barrier Functions (CBFs) provide a powerful framework for ensuring safety in dynamical systems. However, their application typically relies on full state information, which is often violated in real-world scenarios due to the availability of partial state information. In this work, we propose a neural network-based framework for the co-design of a safety controller, observer, and CBF for partially observed continuous-time systems. By formulating barrier conditions over an augmented state space, our approach ensures safety without requiring bounded estimation errors or handcrafted barrier functions. All components are jointly trained by formulating appropriate loss functions, and we introduce a validity condition to provide formal safety guarantees beyond the training data. Finally, we demonstrate the effectiveness of the proposed approach through several case studies.

Paper Structure

This paper contains 14 sections, 4 theorems, 14 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and problem statement
  3. System Description and Problem Formulation
  4. Control Barrier Function (CBF) for Augmented System
  5. Proof:
  6. Neural Network based Observer and CBF
  7. Proof:
  8. Methodology
  9. Neural Network Architecture
  10. Formulation of Loss Functions
  11. Training of Neural Networks with Formal Guarantees
  12. Proof:
  13. Simulation Results
  14. Conclusion

Key Result

Theorem 1

Consider the partially observed system $S$ as in dyn, its state observer $\hat{S}$ as in dyn_obs, and the augmented system $\tilde{S}$. The initial and unsafe sets are ${X_0,{\mathsf{X}}_u \subset D}$. Suppose there exists a continuously differentiable function $B$ under some controller $g$ that sat

Figures (5)

  • Figure 1: Simultaneous training of neural networks
  • Figure 2: Implementation of trained state-observer based controller neural network for safety
  • Figure 3: DC Motor (a) Actual and estimated trajectories starting from different initial conditions (b) Top: $B(x,\hat{x}) > 0$ for all $t \geq 0$ and Bottom: The satisfaction of condition \ref{['cbf_3']} for different initial conditions, (c) The CBF loss \ref{['cbf_loss_fn_nn']} goes to zero, while the observer loss is minimized, (d) Controller in cbf_obs_3 ensures safety only if observer and actual state converges.
  • Figure 4: Pendulum (a) Actual and estimated trajectories starting from different initial conditions, (b) The CBF $B(x,\hat{x})$ value is greater than zero for all time $t\geq 0$ for different initial conditions, (c) The condition \ref{['cbf_3']} is satisfied for different initial conditions. (d) Comparision of pendulum trajectories with cbf_obs_3.
  • Figure 5: Three Tank System: The trained controller ensures the states $x_1, x_2, x_3$ remain in the safe region.

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2
  • Remark 3
  • Remark 4