Source-linked AI summary
Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes
Pushpak Jagtap, George J. Pappas, Majid Zamani
TL;DR
The paper addresses safety-controller synthesis for unknown nonlinear systems, where prior GP control work mainly focused on stability or tracking. It learns the dynamics with a GP accuracy bound and computes CBF-based controllers that account for model uncertainty. In a jet-engine example, the resulting controller has a reported safety-probability lower bound of 0.9839.
Problem
The paper addresses how to synthesize safety controllers for unknown dynamics when prior GP-based control work mainly focused on stability or tracking.
Source: §I · p. 1, Notations · p. 2Method
It learns unknown control-affine dynamics with a Gaussian process and computes control barrier functions and controllers that account for model confidence.
Source: §I · p. 1Results
0.9839 is the reported lower bound on the probability that trajectories under the jet-engine controller satisfy the safety specification.
Source: §V · p. 5Takeaways & Limitations
Under the associated control policy, trajectories starting in the initial set remain outside the unsafe set.
Source: §IV · p. 3
Abstract
from arXiv · showhide
This paper focuses on the controller synthesis for unknown, nonlinear systems while ensuring safety constraints. Our approach consists of two steps, a learning step that uses Gaussian processes and a controller synthesis step that is based on control barrier functions. In the learning step, we use a data-driven approach utilizing Gaussian processes to learn the unknown control affine nonlinear dynamics together with a statistical bound on the accuracy of the learned model. In the second controller synthesis steps, we develop a systematic approach to compute control barrier functions that explicitly take into consideration the uncertainty of the learned model. The control barrier function not only results in a safe controller by construction but also provides a rigorous lower bound on the probability of satisfaction of the safety specification. Finally, we illustrate the effectiveness of the proposed results by synthesizing a safety controller for a jet engine example.
I. INTRODUCTION
The paper addresses safety-controller synthesis when system dynamics are unknown, a setting where prior GP control work mainly focused on stability or tracking. It combines GP learning with control barrier functions (CBFs) to synthesize safety controllers for unknown nonlinear control-affine systems.
I. INTRODUCTION
- Conventional safety-controller synthesis requires a precise system model, which may be unavailable analytically in some control applications.The introduction notes that sensor or simulation data can help identify unmodeled dynamics.Source: §I · p. 1
- Prior GP-based control work mainly addressed stability or tracking rather than safety or complex logic specifications.The paper also notes that existing CBF results for unknown dynamics lacked formal guarantees and did not suit bounded control input sets.Source: §I · p. 1
- The paper combines Gaussian process models and CBFs to synthesize safety controllers for unknown nonlinear control-affine systems.It learns the GP model with a probabilistic accuracy guarantee, then computes CBFs and associated controllers accounting for model confidence.Source: §I · p. 1
- A jet-engine example illustrates the proposed results.Source: §I · p. 1
II. PRELIMINARIES AND PROBLEM DEFINITION
The paper considers continuous-time nonlinear control-affine systems with unknown drift dynamics and known input dynamics, and assumes noisy measurements and bounded RKHS complexity. Its objective is a controller providing a data-dependent lower bound on the probability that trajectories from the initial set avoid the unsafe set.
II. PRELIMINARIES AND PROBLEM DEFINITION
- The system class is nonlinear, continuous-time, and control-affine, with state in a compact set and control input in U.Source: Notations · p. 2
- The unknown drift f has low complexity under an RKHS norm and a known kernel, while the input map g is known.Measurements are modeled as y = f(x) + w with additive Gaussian noise.Source: Notations · p. 2
- The target is a controller with a data-dependent lower bound on the probability that trajectories from any initial state avoid the unsafe set.The problem is posed for initial set X0 and unsafe set X1.Source: Notations · p. 2
- The method is sound when it returns a solution, but a reported failure does not establish that no solution exists.Source: Notations · p. 2
III. GAUSSIAN PROCESS MODEL
The paper models each component of the unknown vector field with a Gaussian process and uses posterior means and uncertainty to bound model error. The resulting accuracy guarantee depends on finite-sample count and information gain.
III. GAUSSIAN PROCESS MODEL
- Each component of the unknown vector field is approximated with a Gaussian process, with a mean function and covariance kernel.The kernel measures similarity between states.Source: §III · p. 2
- Given noisy training measurements, the posterior for each component at a state has a normal distribution with mean μ_j(x) and covariance ρ_j(x).Source: §III · p. 2
- Proposition 3.1 bounds model error with high probability using the GP mean and standard deviation.The stated bound involves the number of finite data samples N and information gain γ_j; γ_j quantifies maximum mutual information between samples and the actual function.Source: §III · p. 3
IV. CONTROL BARRIER FUNCTIONS
The section uses control barrier functions to characterize safety-preserving controllers, first for known dynamics and then for systems modeled with Gaussian processes. Under the stated conditions, the learned-model result gives a probability bound on avoiding the unsafe set and can be extended to temporal specifications.
IV. CONTROL BARRIER FUNCTIONS
- A control barrier function and its associated policy keep trajectories from any x0 in X0 outside X1 for all t ≥ 0 under the theorem's conditions.The barrier conditions prevent B from increasing along the trajectory from its initially safe value.Source: §IV · p. 3
- For the learned Gaussian-process model, satisfying the stated barrier conditions guarantees avoidance of X1 for all t ≥ 0 with probability at least (1 − ε)^n.The theorem uses the GP mean and covariance together with the result bounding model accuracy.Source: §IV · p. 3
- The safety conditions associate a state-feedback controller with the barrier, while the unknown-system policy must work for arbitrary d ∈ D.For known dynamics, the policy applies the selected input u at the current state along the trajectory.Source: §IV · pp. 3–4
- The approach can extend to deterministic co-Büchi automaton specifications by decomposing them into safety properties and combining their probability bounds and controllers.The resulting policy is a hybrid control policy for the temporal specification.Source: §IV · p. 4
A. Computation of Control Barrier Functions
For finite input sets, the section proposes counterexample-guided inductive synthesis to search for parametric control barrier functions. SMT solvers find candidate coefficients, and counterexamples iteratively refine the data samples until the candidate is verified or the search fails to find a solution.
A. Computation of Control Barrier Functions
- With a finite input set, CEGIS searches for a parametric control barrier function using feasibility solvers such as SMT solvers.The function uses user-defined basis functions with unknown coefficients.Source: A. Computation of Control Barrier… · p. 4
- A barrier found by the lemma yields a corresponding controller that selects an input satisfying the derivative condition for any arbitrary d ∈ D.The controller selects from the finite input set.Source: A. Computation of Control Barrier… · p. 4
- SMT solvers find candidate coefficients on finite data samples, then the algorithm searches for counterexamples violating the feasibility condition.A counterexample is added to the samples before the coefficients are recomputed.Source: A. Computation of Control Barrier… · p. 4
- If no counterexample is feasible, the candidate is a control barrier function; if a counterexample exists, CEGIS adds it and repeats.The algorithm may run forever, terminate with a barrier function, or terminate with a counterexample showing no solution for the considered structure.Source: A. Computation of Control Barrier… · p. 4
V. CASE STUDY
The case study applies Gaussian-process learning and control-barrier synthesis to an unknown nonlinear Moore–Greitzer jet-engine model. The resulting controller has a reported safety-probability lower bound of 0.9839.
V. CASE STUDY
- The case study uses the nonlinear Moore–Greitzer jet-engine model in no-stall mode, with X0 as the initial set and X1 as the unsafe set.The state variables are defined from mass flow and pressure rise, and the state space is X = [−1, 3] × [−4, 4].Source: §V · pp. 4–5
- The study learns the unknown jet-engine dynamics from 35 samples and reports a maximum standard deviation ρmax of 0.0360.The samples are collected by simulation; the GP uses a squared-exponential kernel.Source: §V · p. 5
- Figure 2 tracks the probability lower bound and maximum standard deviation ρmax as the number of data samples N increases.The text reports that the probability lower bound tightens as N increases.Source: Figure 2 · p. 5, §V · p. 5
- Using 106 realizations, the probability interval is [0.9839, 0.9905] at confidence 1 − 10−10; thus, the lower bound (1−ϵ)2 can be chosen as 0.9839.For a fixed error bound, increasing the number N of data samples used to learn GPs tightens the lower bound on the probability in (8).Source: §V · p. 5
- The authors compute an order-2 polynomial control barrier function using the counterexample-guided inductive synthesis approach.Source: §V · p. 5
- 0.9839 is the lower bound on the probability that trajectories from any initial state in X0 satisfy the safety specification under the synthesized controller.The closed-loop trajectories from several initial states in X0 are also shown.Source: §V · p. 5
VI. CONCLUSIONS
The paper proposes control-policy synthesis for unknown control-affine systems that enforces safety specifications. It computes control barrier functions using Gaussian-process models learned with high confidence.
VI. CONCLUSIONS
- The proposed scheme synthesizes control policies that enforce safety specifications for unknown control-affine systems.Source: §VI · p. 5
- Figure 3 depicts closed-loop trajectories from X0, the zero level set of B(x), and the regions of interest.Source: Figure 3 · p. 6
- The method computes control barrier functions using models learned through Gaussian processes with high confidence.Source: §VI · p. 6