Nearly Optimal Nonlinear Safe Control with BaS-SDRE

2025-04-21

AI Summary

This paper addresses the challenge of simultaneously ensuring safety and optimality for nonlinear systems operating near unsafe regions. We propose the Barrier-state-based State-Dependent Riccati Equation (BaS-SDRE) safe optimal control method. Our approach uniquely embeds a barrier state (BaS) into the extended linearization framework to construct a safety-augmented dynamical model, and solves its state-dependent Riccati equation (SDRE) to approximate the solution of the Hamilton–Jacobi–Bellman (HJB) equation—thereby unifying safety constraints and near-optimal control synthesis online. Unlike conventional linearized safety designs, BaS-SDRE enables accurate modeling of strongly nonlinear safety dynamics. Experimental validation on an unstable constrained linear system, an angular-rate-limited flight control system, and a quadrotor platform operating in dense environments demonstrates that BaS-SDRE significantly improves the trade-off between safety margin near boundaries and overall control performance.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The State-Dependent Riccati Equation (SDRE) approach has emerged as a systematic and effective means of designing nearly optimal nonlinear controllers. The Barrier States (BaS) embedding methodology was developed recently for safe multi-objective controls in which the safety condition is manifested as a state to be controlled along with other states of the system. The overall system, termed the safety embedded system, is highly nonlinear even if the original system is linear. This paper develops a nonlinear nearly optimal safe feedback control technique by combining the two strategies effectively. First, the BaS is derived in an extended linearization formulation to be subsequently used to form an extended safety embedded system. A new optimal control problem is formed thereafter, which is used to construct a safety embedded State-Dependent Riccati Equation, termed BaS-SDRE, whose solution approximates the solution of the optimal control problem's associated Hamilton-Jacobi-Bellman (HJB) equation. The BaS-SDRE is then solved online to synthesize the nearly optimal safe control. The proposed technique's efficacy is demonstrated on an unstable, constrained linear system that shows how the synthesized control reacts to nonlinearities near the unsafe region, a nonlinear flight control system with limited path angular velocity that exists due to structural and dynamic concerns, and a planar quadrotor system that navigates safely in a crowded environment.

Problem

• Develops nearly optimal safe control for nonlinear systems
• Combines SDRE and BaS for safety embedded control
• Solves BaS-SDRE online for optimal safe feedback

Innovation

• Combines BaS and SDRE for safe control
• Forms safety embedded system via BaS
• Solves BaS-SDRE online for optimality