We present a risk-aware safety certification method for autonomous, learning enabled control systems. Focusing on two realistic risks, state/input delays and interval matrix uncertainty, we model the neural network (NN) controller with local sector bounds and exploit positivity structure to derive linear, delay-independent certificates that guarantee local exponential stability across admissible uncertainties. To benchmark performance, we adopt and implement a state-of-the-art IQC NN verification pipeline. On representative cases, our positivity-based tests run orders of magnitude faster than SDP-based IQC while certifying regimes the latter cannot-providing scalable safety guarantees that complement risk-aware control.

Learning-based neural network (NN) control policies have shown impressive empirical performance in a wide range of tasks in robotics and control. However, formal (Lyapunov) stability guarantees over the region-of-attraction (ROA) for NN controllers with nonlinear dynamical systems are challenging to obtain, and most existing approaches rely on expensive solvers such as sums-of-squares (SOS), mixed-integer programming (MIP), or satisfiability modulo theories (SMT). In this paper, we demonstrate a new framework for learning NN controllers together with Lyapunov certificates using fast empirical falsification and strategic regularizations. We propose a novel formulation that defines a larger verifiable region-of-attraction (ROA) than shown in the literature, and refines the conventional restrictive constraints on Lyapunov derivatives to focus only on certifiable ROAs. The Lyapunov condition is rigorously verified post-hoc using branch-and-bound with scalable linear bound propagation-based NN verification techniques. The approach is efficient and flexible, and the full training and verification procedure is accelerated on GPUs without relying on expensive solvers for SOS, MIP, nor SMT. The flexibility and efficiency of our framework allow us to demonstrate Lyapunov-stable output feedback control with synthesized NN-based controllers and NN-based observers with formal stability guarantees, for the first time in literature. Source code at https://github.com/Verified-Intelligence/Lyapunov_Stable_NN_Controllers

We present a novel method for imitation learning for control requirements expressed using Signal Temporal Logic (STL). More concretely we focus on the problem of training a neural network to imitate a complex controller. The learning process is guided by efficient data aggregation based on counter-examples and a coverage measure. Moreover, we introduce a method to evaluate the performance of the learned controller via parameterization and parameter estimation of the STL requirements. We demonstrate our approach with a flying robot case study.

This paper introduces a model-based approach for training feedback controllers for an autonomous agent operating in a highly nonlinear environment. We desire the trained policy to ensure that the agent satisfies specific task objectives, expressed in discrete-time Signal Temporal Logic (DT-STL). One advantage for reformulation of a task via formal frameworks, like DT-STL, is that it permits quantitative satisfaction semantics. In other words, given a trajectory and a DT-STL formula, we can compute the robustness, which can be interpreted as an approximate signed distance between the trajectory and the set of trajectories satisfying the formula. We utilize feedback controllers, and we assume a feed forward neural network for learning these feedback controllers. We show how this learning problem is similar to training recurrent neural networks (RNNs), where the number of recurrent units is proportional to the temporal horizon of the agent's task objectives. This poses a challenge: RNNs are susceptible to vanishing and exploding gradients, and na\"{i}ve gradient descent-based strategies to solve long-horizon task objectives thus suffer from the same problems. To tackle this challenge, we introduce a novel gradient approximation algorithm based on the idea of dropout or gradient sampling. We show that, the existing smooth semantics for robustness are inefficient regarding gradient computation when the specification becomes complex. To address this challenge, we propose a new smooth semantics for DT-STL that under-approximates the robustness value and scales well for backpropagation over a complex specification. We show that our control synthesis methodology, can be quite helpful for stochastic gradient descent to converge with less numerical issues, enabling scalable backpropagation over long time horizons and trajectories over high dimensional state spaces.

The integration of neural networks into safety-critical systems has shown great potential in recent years. However, the challenge of effectively verifying the safety of Neural Network Controlled Systems (NNCS) persists. This paper introduces a novel approach to NNCS safety verification, leveraging the inductive invariant method. Verifying the inductiveness of a candidate inductive invariant in the context of NNCS is hard because of the scale and nonlinearity of neural networks. Our compositional method makes this verification process manageable by decomposing the inductiveness proof obligation into smaller, more tractable subproblems. Alongside the high-level method, we present an algorithm capable of automatically verifying the inductiveness of given candidates by automatically inferring the necessary decomposition predicates. The algorithm significantly outperforms the baseline method and shows remarkable reductions in execution time in our case studies, shortening the verification time from hours (or timeout) to seconds.

Several methods have been proposed recently to learn neural network (NN) controllers for autonomous agents, with unknown and stochastic dynamics, tasked with complex missions captured by Linear Temporal Logic (LTL). Due to the sample-inefficiency of the majority of these works, compositional learning methods have been proposed decomposing the LTL specification into smaller sub-tasks. Then, separate controllers are learned and composed to satisfy the original task. A key challenge within these approaches is that they often lack safety guarantees or the provided guarantees are impractical. This paper aims to address this challenge. Particularly, we consider autonomous systems with unknown and stochastic dynamics and LTL-encoded tasks. We assume that the system is equipped with a finite set of base skills modeled by trained NN feedback controllers. Our goal is to check if there exists a temporal composition of the trained NN controllers - and if so, to compute it - that will yield a composite system behavior that satisfies the assigned LTL task with probability one. We propose a new approach that relies on a novel integration of automata theory and data-driven reachability analysis tools for NN-controlled stochastic systems. The resulting neuro-symbolic controller allows the agent to generate safe behaviors for unseen complex temporal logic tasks in a zero-shot fashion by leveraging its base skills. We show correctness of the proposed method and we provide conditions under which it is complete. To the best of our knowledge, this is the first work that designs verified temporal compositions of NN controllers for unknown and stochastic systems. Finally, we provide extensive numerical simulations and hardware experiments on robot navigation tasks to demonstrate the proposed method.

Differentiable physics enables efficient gradient-based optimizations of neural network (NN) controllers. However, existing work typically only delivers NN controllers with limited capability and generalizability. We present a practical learning framework that outputs unified NN controllers capable of tasks with significantly improved complexity and diversity. To systematically improve training robustness and efficiency, we investigated a suite of improvements over the baseline approach, including periodic activation functions, and tailored loss functions. In addition, we find our adoption of batching and an Adam optimizer effective in training complex locomotion tasks. We evaluate our framework on differentiable mass-spring and material point method (MPM) simulations, with challenging locomotion tasks and multiple robot designs. Experiments show that our learning framework, based on differentiable physics, delivers better results than reinforcement learning and converges much faster. We demonstrate that users can interactively control soft robot locomotion and switch among multiple goals with specified velocity, height, and direction instructions using a unified NN controller trained in our system. Code is available at https://github.com/erizmr/Complex-locomotion-skill-learning-via-differentiable-physics.

We report on a study that employs an in-house developed simulation infrastructure to accomplish zero shot policy transferability for a control policy associated with a scale autonomous vehicle. We focus on implementing policies that require no real world data to be trained (Zero-Shot Transfer), and are developed in-house as opposed to being validated by previous works. We do this by implementing a Neural Network (NN) controller that is trained only on a family of circular reference trajectories. The sensors used are RTK-GPS and IMU, the latter for providing heading. The NN controller is trained using either a human driver (via human in the loop simulation), or a Model Predictive Control (MPC) strategy. We demonstrate these two approaches in conjunction with two operation scenarios: the vehicle follows a waypoint-defined trajectory at constant speed; and the vehicle follows a speed profile that changes along the vehicle's waypoint-defined trajectory. The primary contribution of this work is the demonstration of Zero-Shot Transfer in conjunction with a novel feed-forward NN controller trained using a general purpose, in-house developed simulation platform.

Over the past decade, neural network (NN)-based controllers have demonstrated remarkable efficacy in a variety of decision-making tasks. However, their black-box nature and the risk of unexpected behaviors and surprising results pose a challenge to their deployment in real-world systems with strong guarantees of correctness and safety. We address these limitations by investigating the transformation of NN-based controllers into equivalent soft decision tree (SDT)-based controllers and its impact on verifiability. Differently from previous approaches, we focus on discrete-output NN controllers including rectified linear unit (ReLU) activation functions as well as argmax operations. We then devise an exact but cost-effective transformation algorithm, in that it can automatically prune redundant branches. We evaluate our approach using two benchmarks from the OpenAI Gym environment. Our results indicate that the SDT transformation can benefit formal verification, showing runtime improvements of up to 21x and 2x for MountainCar-v0 and CartPole-v0, respectively.

Neural networks have shown great success in many machine learning related tasks, due to their ability to act as general function approximators. Recent work has demonstrated the effectiveness of neural networks in control systems (known as neural feedback loops), most notably by using a neural network as a controller. However, one of the big challenges of this approach is that neural networks have been shown to be sensitive to adversarial attacks. This means that, unless they are designed properly, they are not an ideal candidate for controllers due to issues with robustness and uncertainty, which are pivotal aspects of control systems. There has been initial work on robustness to both analyse and design dynamical systems with neural network controllers. However, one prominent issue with these methods is that they use existing neural network architectures tailored for traditional machine learning tasks. These structures may not be appropriate for neural network controllers and it is important to consider alternative architectures. This paper considers rational neural networks and presents novel rational activation functions, which can be used effectively in robustness problems for neural feedback loops. Rational activation functions are replaced by a general rational neural network structure, which is convex in the neural network's parameters. A method is proposed to recover a stabilising controller from a Sum of Squares feasibility test. This approach is then applied to a refined rational neural network which is more compatible with Sum of Squares programming. Numerical examples show that this method can successfully recover stabilising rational neural network controllers for neural feedback loops with non-linear plants with noise and parametric uncertainty.