Adaptive Deep Neural Network-Based Control Barrier Functions

TL;DR

This work addresses safety for uncertain nonlinear systems by integrating adaptive deep neural networks with control barrier functions (aDCBFs). It introduces a real-time, pre-training-free DNN adaptation law based on a least-squares identification error, coupled with a state-derivative observer, to obtain a bounded parameter error and a data-driven CBF constraint. A Lyapunov-based analysis provides uniform ultimate boundedness of the estimation errors under a persistence of excitation assumption, and an optimization-based controller enforces forward invariance of the safe set $\mathcal{S}$. The method extends to intermittent state feedback by using open-loop DNN predictions and a modified CBF with a maximum dwell-time condition, and simulations on adaptive cruise control and non-polynomial dynamics demonstrate improved safety performance and reduced conservativeness compared to baselines.

Abstract

Safety constraints of nonlinear control systems are commonly enforced through the use of control barrier functions (CBFs). Uncertainties in the dynamic model can disrupt forward invariance guarantees or cause the state to be restricted to an overly conservative subset of the safe set. In this paper, adaptive deep neural networks (DNNs) are combined with CBFs to produce a family of controllers that ensure safety while learning the system's dynamics in real-time without the requirement for pre-training. By basing the least squares adaptation law on a state derivative estimator-based identification error, the DNN parameter estimation error is shown to be uniformly ultimately bounded. The convergent bound on the parameter estimation error is then used to formulate CBF-constraints in an optimization-based controller to guarantee safety despite model uncertainty. Furthermore, the developed method is extended for use under intermittent loss of state-feedback. A switched systems analysis for CBFs is provided with a maximum dwell-time condition during which the feedback can be unavailable. Comparative simulation results demonstrate the ability of the developed method to ensure safety in an adaptive cruise control problem and when feedback is lost, unlike baseline methods. Results show improved performance compared to baseline methods and demonstrate the ability of the developed method to ensure safety in feedback-denied environments.

Figures (5)

• Figure 1: An illustration of the sets $\Omega$ and ${\cal S}$ in $\mathbb{R}^{2}$. On the blue set $\Omega$ the universal function approximation property holds. Flows generated by the CBF are constrained to the red set ${\cal S}$, where $\left\Vert \dot{f}\left(x\right)\right\Vert \le\bar{\dot{f}}$.
• Figure 2: The value of the barrier functions over time for the ACC problem. A negative value of $B$ indicates the follower vehicle remains in the safe set.
• Figure 3: The state trajectory of the closed-loop system in Section \ref{['sec:Example']} using the developed aDCBF approach (black line) compared to the same control scheme without the ResNet approximation of the dynamics (green line).
• Figure 4: Comparison plots of the control inputs for the developed and baseline methods. The gray regions represent the time periods where state feedback is made unavailable.
• Figure 5: The top plot shows the desired versus actual value of each position state for the baseline and developed methods over the first 14 seconds of the simulation. The bottom plot shows the position tracking error for the two methods over the first 14 seconds of the simulation. The gray region represents the time period where state feedback is made unavailable.

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Remark 1
• Lemma 1
• Theorem 1
• Definition 3
• Lemma 2
• Theorem 2