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Barrier-Riccati Synthesis for Nonlinear Safe Control with Expanded Region of Attraction

Hassan Almubarak, Maitham F. AL-Sunni, Justin T. Dubbin, Nader Sadegh, John M. Dolan, Evangelos A. Theodorou

TL;DR

The paper addresses safe control for nonlinear systems by embedding barrier-state dynamics into a Riccati-based framework. By factorizing the barrier-state dynamics and applying SDRE to the augmented system, the authors derive BaS-SDRE, which computes online, state-dependent feedback that enforces safety while achieving near-optimal performance. A key contribution is a matrix-inequality condition that guarantees semi-global asymptotic safe stability and yields an enlarged region of attraction. Numerical results on an unstable constrained linear system and a cluttered planar quadrotor demonstrate improved constraint handling, scalability, and robustness near safety boundaries, highlighting the practical potential of principled nonlinear safe control in safety-critical environments.

Abstract

We present a Riccati-based framework for safety-critical nonlinear control that integrates the barrier states (BaS) methodology with the State-Dependent Riccati Equation (SDRE) approach. The BaS formulation embeds safety constraints into the system dynamics via auxiliary states, enabling safety to be treated as a control objective. To overcome the limited region of attraction in linear BaS controllers, we extend the framework to nonlinear systems using SDRE synthesis applied to the barrier-augmented dynamics and derive a matrix inequality condition that certifies forward invariance of a large region of attraction and guarantees asymptotic safe stabilization. The resulting controller is computed online via pointwise Riccati solutions. We validate the method on an unstable constrained system and cluttered quadrotor navigation tasks, demonstrating improved constraint handling, scalability, and robustness near safety boundaries. This framework offers a principled and computationally tractable solution for synthesizing nonlinear safe feedback in safety-critical environments.

Barrier-Riccati Synthesis for Nonlinear Safe Control with Expanded Region of Attraction

TL;DR

The paper addresses safe control for nonlinear systems by embedding barrier-state dynamics into a Riccati-based framework. By factorizing the barrier-state dynamics and applying SDRE to the augmented system, the authors derive BaS-SDRE, which computes online, state-dependent feedback that enforces safety while achieving near-optimal performance. A key contribution is a matrix-inequality condition that guarantees semi-global asymptotic safe stability and yields an enlarged region of attraction. Numerical results on an unstable constrained linear system and a cluttered planar quadrotor demonstrate improved constraint handling, scalability, and robustness near safety boundaries, highlighting the practical potential of principled nonlinear safe control in safety-critical environments.

Abstract

We present a Riccati-based framework for safety-critical nonlinear control that integrates the barrier states (BaS) methodology with the State-Dependent Riccati Equation (SDRE) approach. The BaS formulation embeds safety constraints into the system dynamics via auxiliary states, enabling safety to be treated as a control objective. To overcome the limited region of attraction in linear BaS controllers, we extend the framework to nonlinear systems using SDRE synthesis applied to the barrier-augmented dynamics and derive a matrix inequality condition that certifies forward invariance of a large region of attraction and guarantees asymptotic safe stabilization. The resulting controller is computed online via pointwise Riccati solutions. We validate the method on an unstable constrained system and cluttered quadrotor navigation tasks, demonstrating improved constraint handling, scalability, and robustness near safety boundaries. This framework offers a principled and computationally tractable solution for synthesizing nonlinear safe feedback in safety-critical environments.

Paper Structure

This paper contains 13 sections, 2 theorems, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Contributions and Organization
  4. Safety-Critical Optimal Control Problem Formulation
  5. Safety-Critical Optimal Control Problem
  6. Safety Embedded State Dependent Riccati Equation Control
  7. Factorized Barrier States and Extended Safety Embedded Systems
  8. Safety Embedded SDRE Control
  9. Local Safe Stability and ROA Expansion Condition
  10. Numerical Simulations
  11. Unstable Constrained Linear System
  12. Planar quadrotor
  13. Conclusions

Key Result

Corollary 1

Let $u=K(\bar{x})$ be a continuous feedback controller such that the origin of the safety embedded closed-loop system, $\dot{\bar{x}}= \bar{A}(\bar{x}) \bar{x} + \bar{g}(\bar{x}) K(\bar{x})$, is asymptotically stable. Then, there exists an open set $\mathcal{A}_{\text{safe}} \subseteq \mathcal{S}$ s

Figures (3)

  • Figure 1: Simulations of the closed-loop system under the BaS-LQR controller (dotted) and under the proposed BaS-SDRE controller (solid) starting from different initial conditions (black circles) with the unsafe region shown as a dark red circle.
  • Figure 2: Simulation results show the system trajectory, barrier state evolution, and feedback gains under the BaS-SDRE controller. As the barrier state grows, the BaS-SDRE adapts its gains to capture the system’s nonlinear behavior, differing from the fixed LQR gains.
  • Figure 3: A planar quadrotor navigating in two different obstacle courses. Our method successfully stabilizes the quadrotor at the target position, while vanilla SDRE and BaS-LQR can not, having their trajectories crossing unsafe regions.

Theorems & Definitions (6)

  • Definition 1: blanchini1999set
  • Definition 2
  • Corollary 1: almubarak2023BaSTheorey
  • Theorem 1
  • proof
  • Remark 1