Minimizing Conservatism in Safety-Critical Control for Input-Delayed Systems via Adaptive Delay Estimation

TL;DR

This work addresses safety guarantees for unknown input-delayed systems by integrating DaCBFs with an online adaptive delay estimation framework. It introduces a disturbance-observer-based approach to bound delay-induced disturbances and employs two nonlinear programs to iteratively shrink the feasible delay-uncertainty set, thereby tightening the state-prediction error bound used in DaCBFs. The main contributions are (i) a DOB-based treatment of delay effects, (ii) online tightening of the delay bound set $\Xi_{t_j}$ and the corresponding $e_{t_j,\max}$, and (iii) theoretical analysis showing monotonically nonincreasing prediction errors and reduced conservatism, validated on automated connected-vehicle simulations. The results demonstrate substantial safety guarantees with substantially less conservatism, enabling more responsive safety-critical control in delay-prone cyber-physical systems.

Abstract

Input delays affect systems such as teleoperation and wirelessly autonomous connected vehicles, and may lead to safety violations. One promising way to ensure safety in the presence of delay is to employ control barrier functions (CBFs), and extensions thereof that account for uncertainty: delay adaptive CBFs (DaCBFs). This paper proposes an online adaptive safety control framework for reducing the conservatism of DaCBFs. The main idea is to reduce the maximum delay estimation error bound so that the state prediction error bound is monotonically non-increasing. To this end, we first leverage the estimation error bound of a disturbance observer to bound the state prediction error. Second, we design two nonlinear programs to update the maximum delay estimation error bound satisfying the prediction error bound, and subsequently update the maximum state prediction error bound used in DaCBFs. The proposed method ensures the maximum state prediction error bound is monotonically non-increasing, yielding less conservatism in DaCBFs. We verify the proposed method in an automated connected truck application, showing that the proposed method reduces the conservatism of DaCBFs.

Figures (2)

• Figure 1: A block diagram of the proposed method for ensuring safety in unknown input-delayed systems. The proposed method consists of three components: delay and disturbance estimations, online adaptive update of input delay bound, and DaCBFs with minimum conservatism. The estimated input delay and disturbance are leveraged to update the bound set of input delay and the maximum state prediction error bound, $e_{t_j, \textnormal{max}}$. Given the updated maximum delay estimation error bound, $\tilde{D}_{\textnormal{max}}$ from the online adaptive algorithm \ref{['pm:nlp1']} and \ref{['pm:nlp2']}, $e_{t_j, \textnormal{max}}$ is updated and then used in the robust condition in DaCBFs, monotonically non-increasing over time if $\bm{u}(t)$ is not constant. The proposed method ensures less conservative than the previous work, DaCBFs YK2024ECC.
• Figure 2: The performance of the proposed method and DaCBFs YK2024ECC in an automated connected vehicle application under the condition, $D=0.5$. (a) shows the safety regulations of each method. (b) presents barrier function values of each method, indicating the performance of conservatism. (c) and (d) show the results of the two nonlinear programs from \ref{['pm:nlp1']} and \ref{['pm:nlp2']}, and the plots show that the set, $\Xi_{t_j}$ is gradually reduced, including the true delay $D$. Note that the update algorithm initiates after $2$ seconds because we assume that the initial $\Tilde{D}_{\textnormal{max}}$ is $2$.

Theorems & Definitions (11)

• Lemma 1: BertinoDelay2022
• Theorem 1: YK2024ECC
• Corollary 1: YK2024ECC
• Theorem 2: YK2024ECC
• Lemma 2
• Proposition 1
• proof
• Corollary 2
• proof
• Lemma 3