Source-linked AI summary
Learning for Safety-Critical Control with Control Barrier Functions
Andrew Taylor, Andrew Singletary, Yisong Yue, Aaron Ames
TL;DR
Model uncertainty, including parametric error and unmodeled dynamics, challenges safe controller synthesis. The paper combines Control Barrier Functions with episodic machine learning to learn uncertainty’s effect on safety and update a controller. It demonstrates improved Segway safety in simulation and experimentally on hardware.
Problem
Parametric error and unmodeled dynamics can make model-based safety guarantees unreliable, creating a need to account for uncertainty in safe control.
Method
The method episodically learns how model uncertainty affects Control Barrier Function dynamics and incorporates the learned information into an optimization-based safety controller.
Results
The method improves the safety of a Segway system in simulation and experimentally on hardware.
Takeaways & Limitations
Integrating episodic learning with Control Barrier Functions can incorporate learned unmodeled dynamics and parametric error into safety-critical control.
Takeaways & Limitations
The approach assumes the true system retains the model’s actuation capability and the same relative degree for the safety function.
Abstract
from arXiv · showhide
Modern nonlinear control theory seeks to endow systems with properties of stability and safety, and have been deployed successfully in multiple domains. Despite this success, model uncertainty remains a significant challenge in synthesizing safe controllers, leading to degradation in the properties provided by the controllers. This paper develops a machine learning framework utilizing Control Barrier Functions (CBFs) to reduce model uncertainty as it impact the safe behavior of a system. This approach iteratively collects data and updates a controller, ultimately achieving safe behavior. We validate this method in simulation and experimentally on a Segway platform.
1. Introduction
Model uncertainty can undermine safe control in complex applications, motivating learning-based methods that improve safety while accounting for imperfect dynamics. The paper proposes episodic learning integrated with CBF-based controller synthesis and demonstrates it on a Segway in simulation and hardware.
- Model uncertainty from parametric error and unmodeled dynamics can cause unsafe behavior in autonomous driving, robotics, and aerospace control.
- CBF safety guarantees require an accurate system model, while robust formulations may restrict behavior and adaptive methods cover only restricted uncertainty classes.
- Exhaustive or upfront data collection scales poorly and may miss states entered by a learning-augmented controller.
- The proposed approach learns model uncertainty as it affects CBF evolution, then synthesizes a controller that enforces safety.
- The method uses episodic learning to update the controller and is demonstrated on a Segway in simulation and experimentally on hardware.
2. Control Barrier Functions
Control Barrier Functions encode safe sets and provide conditions under which controllers render those sets forward invariant. The resulting safety-critical optimization filters a nominal controller, but its guarantees depend strongly on model accuracy.
- Safety is formalized by a set S that must remain forward invariant, with S represented as the 0-superlevel set of a continuously differentiable function h.
- A Control Barrier Function is a function h for which an extended class K∞ function α satisfies the CBF condition throughout the safe set.
- If h is a CBF, any Lipschitz controller whose actions satisfy the CBF constraint renders the system safe with respect to S.
- The safety-critical controller filters a potentially unsafe hand-designed controller by selecting a nearest safe action through optimization.
- Because the dynamics model appears in the safety constraint, the controller’s safety guarantees require strong assumptions about model accuracy.
3. Uncertainty Models & Learning
The paper models uncertainty from parametric error and unmodeled dynamics, then learns how that uncertainty affects CBF dynamics to improve safety. Its estimator framework uses supervised regression to capture these effects.
- Parametric error and unmodeled dynamics make the true system functions differ from the nominal model used for control synthesis.
- Model uncertainty appears in the CBF time derivative, making it impossible to verify whether a control input satisfies the safety inequality.
- The method assumes any CBF valid for the nominal model remains valid for the uncertain system.
- This assumption corresponds to matching relative degrees and retaining the model’s actuation capability, while motivating data-driven learning instead of potentially conservative robust CBFs.
- The estimator is learned through supervised regression over functions representing uncertainty effects in the system and CBF dynamics.
4. Episodic Learning Framework
The episodic framework alternates between collecting data with an intermediate controller and synthesizing an updated controller. It addresses sequential-data limitations while moderating changes through trust-weighted blending.
- Conventional supervised learning is challenged because sequential state-input data are dependent rather than independently and identically distributed.
- Each episode runs an intermediate controller, aggregates newly collected data with prior data, and solves a new empirical risk minimization problem.
- The updated estimator is incorporated into a CBF-QP controller that finds the nearest safe input to the hand-designed controller.
- Controller safety degrades when the learned estimator contains error, so newly synthesized controllers are used to improve the estimator.
- The learned controller is blended with the nominal controller using trust weights w_j ∈[0, 1] to limit how quickly system behavior changes.
- DaCBarF takes an initial CBF derivative estimate, model classes, loss function, initial conditions, nominal controller, experiments, and trust coefficients, and outputs an aggregated dataset, updated estimate, and controller.
5. Simulation & Experimental Results
Simulation and hardware experiments on a Segway compare model-based and learning-augmented CBF controllers under uncertainty. Learning kept the system within the safe set and maintained nonnegative barrier values.
- Segway setup: The Segway state comprises horizontal position and velocity, pitch angle, and pitch-angle rate, with motor voltage percentage producing wheel torque.
- Segway setup: The Segway safety barrier limits pitch angle and pitch-angle rate relative to an equilibrium angle, with coefficient c weighting pitch-rate effects.
- Simulation: 15% parameter perturbations caused the model-based controller to leave the safe set and drive h below zero in simulation.
- Simulation: After 10 experiments, the learning-augmented controller kept the simulated state within the 0-superlevel set and maintained h above 0 despite model uncertainty.
- Hardware experiment: On the physical Segway, the desired velocity profile violated the barrier function when safety was enforced using model information alone.
- Hardware experiment: Three hardware episodes trained neural-network estimators, which were incorporated with w_j = 1 and produced improved safety in Figure 2.
6. Related Work
The paper situates its approach among learning methods for safe control, including forward invariance, region-of-attraction, Lyapunov, reachable-set, and performance-based safety approaches.
- The paper combines CBF-based model-driven safety with supervised learning to learn model uncertainty as it affects safe behavior.
- Forward Invariance with Synthesis: Forward-invariance methods learn safe regions or control policies using reinforcement learning, demonstrations, Gaussian processes, or adaptive CBF updates.
- Region-of-Attraction Estimation: Region-of-attraction estimation learns states from which convergence to an equilibrium is guaranteed, imposing convergence beyond forward invariance.
- Learning Lyapunov Functions: Lyapunov-function methods learn certificates or policies for stability-based safety guarantees.
- Reachable Set Learning: Reachable-set learning estimates reachable states or unmodeled dynamics to limit unsafe states, including through HJI reachability and Gaussian processes.
- Performance-Based Safety: Performance-based safety maintains a minimum performance threshold while exploring policies or tuning controller parameters.
7. Conclusion
The episodic framework integrates Control Barrier Functions with learning to account for uncertainty in safety control, improving Segway safety in simulation and hardware experiments.
- The method learns unmodeled dynamics and parametric error as they affect system safety, then incorporates that information into an optimization-based controller.
- The approach improves the safety of a Segway system in simulation and experimentally on hardware.