Bayesian neural networks (BNNs) retain NN structures with a probability distribution placed over their weights. With the introduced uncertainties and redundancies, BNNs are proper choices of robust controllers for safety-critical control systems. This paper considers the problem of verifying the safety of nonlinear closed-loop systems with BNN controllers over unbounded-time horizon. In essence, we compute a safe weight set such that as long as the BNN controller is always applied with weights sampled from the safe weight set, the controlled system is guaranteed to be safe. We propose a novel two-phase method for the safe weight set computation. First, we construct a reference safe control set that constraints the control inputs, through polynomial approximation to the BNN controller followed by polynomial-optimization-based barrier certificate generation. Then, the computation of safe weight set is reduced to a range inclusion problem of the BNN on the system domain w.r.t. the safe control set, which can be solved incrementally and the set of safe weights can be extracted. Compared with the existing method based on invariant learning and mixed-integer linear programming, we could compute safe weight sets with larger radii on a series of linear benchmarks. Moreover, experiments on a series of widely used nonlinear control tasks show that our method can synthesize large safe weight sets with probability measure as high as 95% even for a large-scale system of dimension 7.

Abstract There is a pressing need for learning controllers to endow systems with properties of safety and goal-reaching, which are crucial for many safety-critical systems. Reinforcement learning (RL) has been deployed successfully to synthesize controllers from user-defined reward functions encoding desired system requirements. However, it remains a significant challenge in synthesizing provably correct controllers with safety and goal-reaching requirements. To address this issue, we try to design a special hybrid polynomial-DNN controller which is easy to verify without losing its expressiveness and flexibility. This paper proposes a novel method to synthesize such a hybrid controller based on RL, low-degree polynomial fitting and knowledge distillation. It also gives a computational approach, by building and solving a constrained optimization problem coming from verification conditions to produce barrier certificates and Lyapunov-like functions, which can guarantee every trajectory from the initial set of the system with the resulted controller satisfies the given safety and goal-reaching requirements. We evaluate the proposed hybrid controller synthesis method on a set of benchmark examples, including several high-dimensional systems. The results validate the effectiveness and applicability of our approach.

Abstract In this paper, we propose a safe reinforcement learning approach to synthesize deep neural network (DNN) controllers for nonlinear systems subject to safety constraints. The proposed approach employs an iterative scheme where alearnerand averifierinteract to synthesize safe DNN controllers. Thelearnertrains a DNN controller via deep reinforcement learning, and theverifiercertifies the learned controller through computing a maximal safe initial region and its corresponding barrier certificate, based on polynomial abstraction and bilinear matrix inequalities solving. Compared with the existing verification-in-the-loop synthesis methods, our iterative framework is a sequential synthesis scheme of controllers and barrier certificates, which can learn safe controllers with adaptive barrier certificates rather than user-defined ones. We implement the tool SRLBC and evaluate its performance over a set of benchmark examples. The experimental results demonstrate that our approach efficiently synthesizes safe DNN controllers even for a nonlinear system with dimension up to 12.