Safety Verification of Unknown Dynamical Systems via Gaussian Process Regression

Abstract

Leveraging autonomous systems in safety-critical scenarios requires verifying their behaviors in the presence of uncertainties and black-box components that influence the system dynamics. In this article, we develop a framework for verifying partiallyobservable, discrete-time dynamical systems with unmodelled dynamics against temporal logic specifications from a given inputoutput dataset. The verification framework employs Gaussian process (GP) regression to learn the unknown dynamics from the dataset and abstract the continuous-space system as a finite-state, uncertain Markov decision process (MDP). This abstraction relies on space discretization and transition probability intervals that capture the uncertainty due to the error in GP regression by using reproducible kernel Hilbert space analysis as well as the uncertainty induced by discretization. The framework utilizes existing model checking tools for verification of the uncertain MDP abstraction against a given temporal logic specification. We establish the correctness of extending the verification results on the abstraction to the underlying partially-observable system. We show that the computational complexity of the framework is polynomial in the size of the dataset and discrete abstraction. The complexity analysis illustrates a trade-off between the quality of the verification results and the computational burden to handle larger datasets and finer abstractions. Finally, we demonstrate the efficacy of our learning and verification framework on several case studies with linear, nonlinear, and switched dynamical systems.

Strategy synthesis for partially-known switched stochastic systems

“…Assumption 1 is a standard assumption [18,31], which is intimately related to the continuity of д u , as discussed in Section 4. For instance, assuming that κ is the widely used squared exponential kernel, we obtain that H κ (X ) is a space of functions that is dense with respect to the set of continuous functions on a compact set X ⊂ R n , i.e., members of H κ (X ) can approximate any continuous function on X arbitrarily well [32].…”

“…For a compact set 𝑋 ⊂ R 𝑛 , let 𝜅 : R 𝑛 ×R 𝑛 → R >0 be a given kernel and H 𝜅 (𝑋 ) the reproducing kernel Hilbert space (RKHS) of functions over 𝑋 corresponding to 𝜅 with norm ∥ • ∥ 𝜅 [31]. Then, for each 𝑢 ∈ 𝑈 and 𝑖 ∈ {1, ..., 𝑛}, 𝑔 Assumption 1 is a standard assumption [18,31], which is intimately related to the continuity of 𝑔 𝑢 , as discussed in Section 4. For instance, assuming that 𝜅 is the widely used squared exponential kernel, we obtain that H 𝜅 (𝑋 ) is a space of functions that is dense with respect to the set of continuous functions on a compact set 𝑋 ⊂ R 𝑛 , i.e., members of H 𝜅 (𝑋 ) can approximate any continuous function on 𝑋 arbitrarily well [32].…”

“…The advantage of GP regression is in its ability to quantify the bound on the uncertainty in the learning process as derived in [9,13,24,31]. This has led to an increased use of GPs in safe learning frameworks, e.g., [3,5,18,28,33]. In most work, the main objective is to learn safe policies via reinforcement learning with the exception of work [18], which considers a safety verification problem of unknown systems with noisy measurements.…”

“…This has led to an increased use of GPs in safe learning frameworks, e.g., [3,5,18,28,33]. In most work, the main objective is to learn safe policies via reinforcement learning with the exception of work [18], which considers a safety verification problem of unknown systems with noisy measurements. The proposed framework uses GPs to construct a Markov abstraction for an invariant set (safety) analysis from a noisy dataset.…”

“…Conf. on Hybrid Systems: Computation and Control (HSCC 2021) [18], which considers a safety verification problem of unknown systems with noisy measurements. The proposed framework uses GPs to construct a Markov abstraction for an invariant set (safety) analysis from a noisy dataset.…”