Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Safe Controller Synthesis Using Lyapunov-based Barriers for Linear Hybrid Systems with Simplex Architecture

Sunandan Adhikary, Soumyajit Dey

TL;DR

The paper tackles safety guarantees for hybrid cyber-physical systems under real-time constraints by designing backup safe controllers (BSCs) within a simplex architecture. It combines Lyapunov-based decay (via quadratic Lyapunov functions) with barrier-function safety to synthesize multiple BSCs at different sampling rates, each ensuring recovery to the primary operating region within a specified deadline while remaining inside the safe region. A state-dependent switching policy (SCAP) and a resource-aware runtime mechanism (GLBF) enable safe, adaptive switching between BSCs and the primary controller under bandwidth limits. The proposed SDP-based synthesis achieves larger safe forward-invariant regions and timely convergence with favorable computation times, as demonstrated on aircraft longitudinal dynamics and inverted pendulum benchmarks. This approach provides a practical, schedulable pathway to ensure safety with minimal resource overhead in real-time embedded control systems.

Abstract

Modern cyber-physical systems often have a two-layered design, where the primary controller is AI-enabled or an analytical controller optimising some specific cost function. If the resulting control action is perceived as unsafe, a secondary safety-focused backup controller is activated. The existing backup controller design schemes do not consider a real-time deadline for the course correction of a potentially unsafe system trajectory or constrain maximisation of the safe operating region as a synthesis criterion. This essentially implies an eventual safety guarantee over a small operating region. This paper proposes a novel design method for backup safe controllers (BSCs) that ensure invariance across the largest possible region in the safe state space, along with a guarantee for timely recovery when the system states deviate from their usual behaviour. This is the first work to synthesise safe controllers that ensure maximal safety and timely recovery while aiming at minimal resource usage by switching between BSCs with different execution rates. An online safe controller activation policy is also proposed to switch between BSCs (and the primary optimal controller) to optimise processing bandwidth for control computation. To establish the efficacy of the proposed method, we evaluate the safety and recovery time of the proposed safe controllers, as well as the activation policy, in closed loops with linear hybrid dynamical systems under budgeted bandwidth.

Safe Controller Synthesis Using Lyapunov-based Barriers for Linear Hybrid Systems with Simplex Architecture

TL;DR

The paper tackles safety guarantees for hybrid cyber-physical systems under real-time constraints by designing backup safe controllers (BSCs) within a simplex architecture. It combines Lyapunov-based decay (via quadratic Lyapunov functions) with barrier-function safety to synthesize multiple BSCs at different sampling rates, each ensuring recovery to the primary operating region within a specified deadline while remaining inside the safe region. A state-dependent switching policy (SCAP) and a resource-aware runtime mechanism (GLBF) enable safe, adaptive switching between BSCs and the primary controller under bandwidth limits. The proposed SDP-based synthesis achieves larger safe forward-invariant regions and timely convergence with favorable computation times, as demonstrated on aircraft longitudinal dynamics and inverted pendulum benchmarks. This approach provides a practical, schedulable pathway to ensure safety with minimal resource overhead in real-time embedded control systems.

Abstract

Modern cyber-physical systems often have a two-layered design, where the primary controller is AI-enabled or an analytical controller optimising some specific cost function. If the resulting control action is perceived as unsafe, a secondary safety-focused backup controller is activated. The existing backup controller design schemes do not consider a real-time deadline for the course correction of a potentially unsafe system trajectory or constrain maximisation of the safe operating region as a synthesis criterion. This essentially implies an eventual safety guarantee over a small operating region. This paper proposes a novel design method for backup safe controllers (BSCs) that ensure invariance across the largest possible region in the safe state space, along with a guarantee for timely recovery when the system states deviate from their usual behaviour. This is the first work to synthesise safe controllers that ensure maximal safety and timely recovery while aiming at minimal resource usage by switching between BSCs with different execution rates. An online safe controller activation policy is also proposed to switch between BSCs (and the primary optimal controller) to optimise processing bandwidth for control computation. To establish the efficacy of the proposed method, we evaluate the safety and recovery time of the proposed safe controllers, as well as the activation policy, in closed loops with linear hybrid dynamical systems under budgeted bandwidth.
Paper Structure (12 sections, 4 theorems, 9 equations, 4 figures, 1 algorithm)

This paper contains 12 sections, 4 theorems, 9 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prelimineries
  3. Convergence in Lyapunov Sense
  4. Safety Invariance using Barrier Function
  5. Hybrid Systems Stably Operating with Multiple Sampling Periods
  6. Problem Formulation & Solution Framework
  7. Step 1: Minimum Exponential Decay Rate Derivation from Recovery Deadline
  8. Step 2: Constraint Synthesis for Desired Recovery Rate and Safety Guarantee
  9. Step 3: Synthesis of Multi-Rate BSCs and Safe & Stable Switching Logic
  10. Step 4: Incorporating Resource-awareness in Algorithmic Framework for SCAP
  11. Experiments
  12. Concluding Remarks

Key Result

Proposition 1

If there exists a quadratic Lyapunov function with decay rate $\alpha_i\in (\alpha_{ref,i},1)$ such that a trajectory, starting from anywhere in $\mathcal{R}_{S}\setminus \mathcal{R}_{P}$ recovers back to $\mathcal{R}_{P}$($\subset \mathcal{R}_{S}$) within $\delta k$ sampling iterations; then the mi

Figures (4)

  • Figure 1: Simplex Setup with POC and BSC
  • Figure 2: Overview of the proposed framework
  • Figure 3: Adaptive Scheduling of BSCs:$\quad$ (a) Aircraft Longitudinal Dynamics (on left),$\quad$ (b) Inverted Pendulum Dynamics (on right)
  • Figure 4: State Recovery Under BSCs:$\quad$ (a) Aircraft Longitudinal Dynamics (on left),$\quad$ (b) Inverted Pendulum Dynamics (on right)

Theorems & Definitions (7)

  • Definition 1: Globally Uniformly Exponentially Stable
  • Definition 2
  • Proposition 1
  • Lemma 1: timely recovery guarantee
  • Definition 3
  • Lemma 2: safety guarantee
  • Lemma 3: Safe & Stable Switchability Guarantee