[Paper Review] Robust Safe Control Synthesis with Disturbance Observer-Based Control Barrier Functions

Ersin Daş, Richard M. Murray|arXiv (Cornell University)|Jan 15, 2022

Advanced Control Systems Optimization9 citations

TL;DR

This paper proposes a disturbance observer-based control barrier function (DO-CBF) framework that enables robust, safety-critical control for nonlinear systems under time-varying disturbances. By estimating the unmodeled dynamics of the CBF using a high-gain input observer with a single tunable parameter, the method formulates a robust CLF-CBF quadratic program that guarantees safety and stability even under uncertainty, validated on adaptive cruise control and Segway platforms with relative degree two CBFs.

ABSTRACT

In a complex real-time operating environment, external disturbances and uncertainties adversely affect the safety, stability, and performance of dynamical systems. This paper presents a robust stabilizing safety-critical controller synthesis framework with control Lyapunov functions (CLFs) and control barrier functions (CBFs) in the presence of disturbance. A high-gain input observer method is adapted to estimate the time-varying unmodelled dynamics of the CBF with an error bound using the first-order time derivative of the CBF. This approach leads to an easily tunable low order disturbance estimator structure with a design parameter as it utilizes only the CBF constraint. The estimated unknown input and associated error bound are used to ensure robust safety and exponential stability by formulating a CLF-CBF quadratic program. The proposed method is applicable to both relative degree one and higher relative degree CBF constraints. The efficacy of the proposed approach is demonstrated using a numerical simulations of an adaptive cruise control system and a Segway platform with an external disturbance.

Motivation & Objective

• To address the challenge of maintaining safety and stability in real-time control systems under time-varying disturbances and unmodeled dynamics.
• To extend existing CBF and CLF-based control frameworks to handle disturbances without requiring full system models or worst-case bounds.
• To develop a low-complexity, tunable disturbance estimation method that leverages only the first-order CBF constraint.
• To ensure robust safety for systems with relative degree greater than one, which are often excluded in standard CBF approaches.
• To demonstrate the method's efficacy on practical systems such as adaptive cruise control and Segway platforms with unmatched disturbances.

Proposed method

• A high-gain input observer is used to estimate the time-varying disturbance affecting the control barrier function (CBF), based solely on the first-order derivative of the CBF.
• The disturbance estimation error is bounded exponentially, and the bound is used to construct a robust safety constraint in the CLF-CBF quadratic program.
• The method formulates a robust CLF-CBF-QP that incorporates both the estimated disturbance and its error bound, ensuring forward invariance of the safe set.
• The approach is applicable to both relative degree one and higher CBF constraints, overcoming limitations of prior methods restricted to relative degree one.
• For unmatched disturbances, such as in the Segway example, the method adapts by estimating the effect of the disturbance on the CBF derivative directly.
• The design requires tuning only one high-gain parameter, enabling practical implementation with minimal complexity.

Experimental results

Research questions

• RQ1 Can a low-complexity disturbance observer be designed using only the first-order CBF constraint to estimate unmodeled dynamics?
• RQ2 How can the estimated disturbance and its error bound be integrated into a CLF-CBF-QP to ensure robust safety under uncertainty?
• RQ3 Can the proposed method guarantee safety for systems with relative degree greater than one, such as pitch angle constraints in a Segway?
• RQ4 How does the performance of the DO-CBF framework compare to standard CBF-QP under external disturbances?
• RQ5 What is the impact of disturbance estimation accuracy on the stability and safety of the closed-loop system?

Key findings

• The proposed disturbance observer accurately estimates the effect of unmodeled dynamics on the CBF derivative, with the estimation error bounded exponentially.
• The Segway simulation shows that the DO-CBF approach maintains safety on a 20° inclined surface, while the nominal CBF-QP leads to unsafe behavior due to unmodeled disturbances.
• The method successfully handles unmatched disturbances, as demonstrated in the Segway example where the disturbance affects the system dynamics non-matching the control input.
• The control barrier function $ h(x) = \pi/10 - \theta^2 $ remains positive throughout the simulation under the DO-CBF controller, confirming safe operation.
• The disturbance estimation error bound $ M_{b_d} $ is used effectively in the robust CBF constraint, ensuring that the safety condition is satisfied despite uncertainty.
• The framework achieves robust safety with only one tunable design parameter, simplifying implementation compared to prior methods requiring worst-case disturbance bounds.