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 work, we develop a framework for verifying discrete-time dynamical systems with unmodelled dynamics and noisy measurements against temporal logic specifications from an input-output dataset. The verification framework employs Gaussian process (GP) regression to learn the unknown dynamics from the dataset and abstracts 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 created from noisy measurements to the underlying 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.

@en

Verifying the performance of safety-critical, stochastic systems with complex noise distributions is difficult. We introduce a general procedure for the finite abstraction of nonlinear stochastic systems with nonstandard (e.g., non-affine, non-symmetric, non-unimodal) noise distributions for verification purposes. The method uses a finite partitioning of the noise domain to construct an interval Markov chain (IMC) abstraction of the system via transition probability intervals. Noise partitioning allows for a general class of distributions and structures, including multiplicative and mixture models, and admits both known and data-driven systems. The partitions required for optimal transition bounds are specified for systems that are monotonic with respect to the noise, and explicit partitions are provided for affine and multiplicative structures. By the soundness of the abstraction procedure, verification on the IMC provides guarantees on the stochastic system against a temporal logic specification. In addition, we present a novel refinement-free algorithm that improves the verification results. Case studies on linear and nonlinear systems with non-Gaussian noise, including a data-driven example, demonstrate the generality and effectiveness of the method without introducing excessive conservatism.

@en

We present a data-driven framework for strategy synthesis for partially-known switched stochastic systems. The properties of the system are specified using linear temporal logic (LTL) over finite traces (LTLf), which is as expressive as LTL and enables interpretations over finite behaviors. The framework first learns the unknown dynamics via Gaussian process regression. Then, it builds a formal abstraction of the switched system in terms of an uncertain Markov model, namely an Interval Markov Decision Process (IMDP), by accounting for both the stochastic behavior of the system and the uncertainty in the learning step. Then, we synthesize a strategy on the resulting IMDP that maximizes the satisfaction probability of the LTLf specification and is robust against all the uncertainties in the abstraction. This strategy is then refined into a switching strategy for the original stochastic system. We show that this strategy is near-optimal and provide a bound on its distance (error) to the optimal strategy. We experimentally validate our framework on various case studies, including both linear and non-linear switched stochastic systems.

@en

Autonomous systems often have complex and possibly unknown dynamics due to, e.g., black-box components. This leads to unpredictable behaviors and makes control design with performance guarantees a major challenge. This paper presents a data-driven control synthesis framework for such systems subject to linear temporal logic on finite traces (LTLf) specifications. The framework combines a baseline (offline) controller with a novel online controller and refinement procedure that improves the baseline guarantees as new data is collected. The baseline controller is computed offline on an uncertain abstraction constructed using Gaussian process (GP) regression on a given dataset. The offline controller provides a lower bound on the probability of satisfying the LTLf specification, which may be far from optimal due to both discretization and regression errors. The synergy arises from the online controller using the offline abstraction along with the current state and new data to choose the next best action. The online controller may improve the baseline guarantees since it avoids the discretization error and reduces regression error as new data is collected. The new data are also used to refine the abstraction and offline controller using local GP regression, which significantly reduces the computation overhead. Evaluations show the efficacy of the proposed offline-online framework, especially when compared against the offline controller.@en