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Transfer of Safety Controllers Through Learning Deep Inverse Dynamics Model

Alireza Nadali, Ashutosh Trivedi, Majid Zamani

TL;DR

This work proposes integratingverse dynamics -- a neural network that suggests required action given a desired successor state of the target system with the barrier certificate of the source system to provide formal proof of safety, and proposes a validity condition that, when met, guarantees correctness of the controller.

Abstract

Control barrier certificates have proven effective in formally guaranteeing the safety of the control systems. However, designing a control barrier certificate is a time-consuming and computationally expensive endeavor that requires expert input in the form of domain knowledge and mathematical maturity. Additionally, when a system undergoes slight changes, the new controller and its correctness certificate need to be recomputed, incurring similar computational challenges as those faced during the design of the original controller. Prior approaches have utilized transfer learning to transfer safety guarantees in the form of a barrier certificate while maintaining the control invariant. Unfortunately, in practical settings, the source and the target environments often deviate substantially in their control inputs, rendering the aforementioned approach impractical. To address this challenge, we propose integrating \emph{inverse dynamics} -- a neural network that suggests required action given a desired successor state -- of the target system with the barrier certificate of the source system to provide formal proof of safety. In addition, we propose a validity condition that, when met, guarantees correctness of the controller. We demonstrate the effectiveness of our approach through three case studies.

Transfer of Safety Controllers Through Learning Deep Inverse Dynamics Model

TL;DR

This work proposes integratingverse dynamics -- a neural network that suggests required action given a desired successor state of the target system with the barrier certificate of the source system to provide formal proof of safety, and proposes a validity condition that, when met, guarantees correctness of the controller.

Abstract

Control barrier certificates have proven effective in formally guaranteeing the safety of the control systems. However, designing a control barrier certificate is a time-consuming and computationally expensive endeavor that requires expert input in the form of domain knowledge and mathematical maturity. Additionally, when a system undergoes slight changes, the new controller and its correctness certificate need to be recomputed, incurring similar computational challenges as those faced during the design of the original controller. Prior approaches have utilized transfer learning to transfer safety guarantees in the form of a barrier certificate while maintaining the control invariant. Unfortunately, in practical settings, the source and the target environments often deviate substantially in their control inputs, rendering the aforementioned approach impractical. To address this challenge, we propose integrating \emph{inverse dynamics} -- a neural network that suggests required action given a desired successor state -- of the target system with the barrier certificate of the source system to provide formal proof of safety. In addition, we propose a validity condition that, when met, guarantees correctness of the controller. We demonstrate the effectiveness of our approach through three case studies.
Paper Structure (13 sections, 1 theorem, 15 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 1 theorem, 15 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Discrete-time Control Systems
  4. Safety and Control Barrier Certificate
  5. Neural Networks
  6. Problem Definition
  7. Controller Synthesis using Inverse Dynamics
  8. Validity Condition
  9. Experiments
  10. Inverted Pendulum
  11. DC Motor
  12. Quadrotor Drone
  13. Conclusion and Future work

Key Result

Theorem 4

Consider a source system $\mathfrak{S}$ = ($\mathcal{X}, U,f)$ with a control barrier certificate $B: \mathcal{X} \rightarrow \mathbb{R}$ with the corresponding parameter $\eta$ in (eq_barr_1)-(eq_barr_3), corresponding controller $k: \mathcal{X} \rightarrow U$, and a target system $\hat{\mathfrak{S

Figures (5)

  • Figure 1: This figure depicts the dynamics for a series of inverted pendula (angular position on $x$ axis and angular velocity on $y$) differing in their weights and length. The leftmost figure is for the source system for which a CBC has already been computed. We show the gradual failure of the source CBC with changes in the dynamics of an inverted pendulum (details in Section \ref{['sec:expts']}). The blue enclosed region and black square region indicate the zero level set of the CBC and the initial set, respectively. Moreover, the yellow regions show the violation in a key condition (i.e., decrement in barrier value along transitions) of CBC , while the purple regions show its satisfaction.
  • Figure 2: Transfer of Safety Controllers Through Learning Deep Inverse Dynamics Model.
  • Figure 3: Some state sequences showing the evolution of the target inverted pendulum (Figure \ref{['fig_traj']}) and the corresponding input (Figure \ref{['fig_inp']}). The areas marked with red indicate the unsafe set. We denote the initial set by the dotted black square.
  • Figure 4: Barrier certificate for the target system. Figure on the left indicates the value of the CBC when using the control policy of the source system, whereas Figure on the right depicts the same CBC but with the learned inverse dynamics controller.
  • Figure 5: Some state sequences showing the evolution of the target drone (Figure \ref{['fig_traj']}) and the corresponding input (Figure \ref{['fig_inp']}). We denote the initial set by the dotted black square.

Theorems & Definitions (3)

  • Definition 1
  • Definition 3
  • Theorem 4