We address the problem of verifying closed-loop contraction in nonlinear control systems whose controller and contraction metric are both parameterized by neural networks. By leveraging interval analysis and interval bound propagation, we derive a tractable and scalable sufficient condition for closed-loop contractivity that reduces to checking that the dominant eigenvalue of a symmetric Metzler matrix is nonpositive. We combine this sufficient condition with a domain partitioning strategy to integrate this sufficient condition into training. The proposed approach is validated on an inverted pendulum system, demonstrating the ability to learn neural network controllers and contraction metrics that provably satisfy the contraction condition.

This paper presents a robust neural control design for a three-drone slung payload transportation system to track a reference path under external disturbances. The control contraction metric (CCM) is used to generate a neural exponentially converging baseline controller while complying with control input saturation constraints. We also incorporate the uncertainty and disturbance estimator (UDE) technique to dynamically compensate for persistent disturbances. The proposed framework yields a modularized design, allowing the controller and estimator to perform their individual tasks and achieve a zero trajectory tracking error if the disturbances meet certain assumptions. The stability and robustness of the complete system, incorporating both the CCM controller and the UDE compensator, are presented. Simulations are conducted to demonstrate the capability of the proposed control design to follow complicated trajectories under external disturbances.

We present a novel robust control framework for continuous-time, perturbed nonlinear dynamical systems with uncertainty that depends nonlinearly on both the state and control inputs. Unlike conventional approaches that impose structural assumptions on the uncertainty, our framework enhances contraction-based robust control with data-driven uncertainty prediction, remaining agnostic to the models of the uncertainty and predictor. We statistically quantify how reliably the contraction conditions are satisfied under dynamics with uncertainty via conformal prediction, thereby obtaining a distribution-free and finite-time probabilistic guarantee for exponential boundedness of the trajectory tracking error. We further propose the probabilistically robust control invariant (PRCI) tube for distributionally robust motion planning, within which the perturbed system trajectories are guaranteed to stay with a finite probability, without explicit knowledge of the uncertainty model. Numerical simulations validate the effectiveness of the proposed robust control framework and the performance of the PRCI tube.

Control contraction metrics (CCMs) provide a framework to co-synthesize a controller and a corresponding contraction metric -- a positive-definite Riemannian metric under which a closed-loop system is guaranteed to be incrementally exponentially stable. However, the synthesized controller only ensures that all the trajectories of the system converge to one single trajectory and, as such, does not impose any notion of optimality across an entire trajectory. Furthermore, constructing CCMs requires a known dynamics model and non-trivial effort in solving an infinite-dimensional convex feasibility problem, which limits its scalability to complex systems featuring high dimensionality with uncertainty. To address these issues, we propose to integrate CCMs into reinforcement learning (RL), where CCMs provide dynamics-informed feedback for learning control policies that minimize cumulative tracking error under unknown dynamics. We show that our algorithm, called contraction actor-critic (CAC), formally enhances the capability of CCMs to provide a set of contracting policies with the long-term optimality of RL in a fully automated setting. Given a pre-trained dynamics model, CAC simultaneously learns a contraction metric generator (CMG) -- which generates a contraction metric -- and uses an actor-critic algorithm to learn an optimal tracking policy guided by that metric. We demonstrate the effectiveness of our algorithm relative to established baselines through extensive empirical studies, including simulated and real-world robot experiments, and provide a theoretical rationale for incorporating contraction theory into RL.

Contraction metrics are crucial in control theory because they provide a powerful framework for analyzing stability, robustness, and convergence of various dynamical systems. However, identifying these metrics for complex nonlinear systems remains an open challenge due to the lack of scalable and effective tools. This paper explores the approach of learning verifiable contraction metrics parametrized as neural networks (NNs) for discrete-time nonlinear dynamical systems. While prior works on formal verification of contraction metrics for general nonlinear systems have focused on convex optimization methods (e.g. linear matrix inequalities, etc) under the assumption of continuously differentiable dynamics, the growing prevalence of NN-based controllers, often utilizing ReLU activations, introduces challenges due to the non-smooth nature of the resulting closed-loop dynamics. To bridge this gap, we establish a new sufficient condition for establishing formal neural contraction metrics for general discrete-time nonlinear systems assuming only the continuity of the dynamics. We show that from a computational perspective, our sufficient condition can be efficiently verified using the state-of-the-art neural network verifier $α,\!β$-CROWN, which scales up non-convex neural network verification via novel integration of symbolic linear bound propagation and branch-and-bound. Built upon our analysis tool, we further develop a learning method for synthesizing neural contraction metrics from sampled data. Finally, our approach is validated through the successful synthesis and verification of NN contraction metrics for various nonlinear examples.

Contraction theory is an analytical tool to study differential dynamics of a non-autonomous (i.e., time-varying) nonlinear system under a contraction metric defined with a uniformly positive definite matrix, the existence of which results in a necessary and sufficient characterization of incremental exponential stability of multiple solution trajectories with respect to each other. By using a squared differential length as a Lyapunov-like function, its nonlinear stability analysis boils down to finding a suitable contraction metric that satisfies a stability condition expressed as a linear matrix inequality, indicating that many parallels can be drawn between well-known linear systems theory and contraction theory for nonlinear systems. Furthermore, contraction theory takes advantage of a superior robustness property of exponential stability used in conjunction with the comparison lemma. This yields much-needed safety and stability guarantees for neural network-based control and estimation schemes, without resorting to a more involved method of using uniform asymptotic stability for input-to-state stability. Such distinctive features permit systematic construction of a contraction metric via convex optimization, thereby obtaining an explicit exponential bound on the distance between a time-varying target trajectory and solution trajectories perturbed externally due to disturbances and learning errors. The objective of this paper is therefore to present a tutorial overview of contraction theory and its advantages in nonlinear stability analysis of deterministic and stochastic systems, with an emphasis on deriving formal robustness and stability guarantees for various learning-based and data-driven automatic control methods. In particular, we provide a detailed review of techniques for finding contraction metrics and associated control and estimation laws using deep neural networks.

Learning-enabled control systems have demonstrated impressive empirical performance on challenging control problems in robotics, but this performance comes at the cost of reduced transparency and lack of guarantees on the safety or stability of the learned controllers. In recent years, new techniques have emerged to provide these guarantees by learning certificates alongside control policies -- these certificates provide concise, data-driven proofs that guarantee the safety and stability of the learned control system. These methods not only allow the user to verify the safety of a learned controller but also provide supervision during training, allowing safety and stability requirements to influence the training process itself. In this paper, we provide a comprehensive survey of this rapidly developing field of certificate learning. We hope that this paper will serve as an accessible introduction to the theory and practice of certificate learning, both to those who wish to apply these tools to practical robotics problems and to those who wish to dive more deeply into the theory of learning for control.

We present a method for providing statistical guarantees on runtime safety and goal reachability for integrated planning and control of a class of systems with unknown nonlinear stochastic underactuated dynamics. Specifically, given a dynamics dataset, our method jointly learns a mean dynamics model, a spatially-varying disturbance bound that captures the effect of noise and model mismatch, and a feedback controller based on contraction theory that stabilizes the learned dynamics. We propose a sampling-based planner that uses the mean dynamics model and simultaneously bounds the closed-loop tracking error via a learned disturbance bound. We employ techniques from Extreme Value Theory (EVT) to estimate, to a specified level of confidence, several constants which characterize the learned components and govern the size of the tracking error bound. This ensures plans are guaranteed to be safely tracked at runtime. We validate that our guarantees translate to empirical safety in simulation on a 10D quadrotor, and in the real world on a physical CrazyFlie quadrotor and Clearpath Jackal robot, whereas baselines that ignore the model error and stochasticity are unsafe.

This paper presents an approach to trajectory-centric learning control based on contraction metrics and disturbance estimation for nonlinear systems subject to matched uncertainties. The approach uses deep neural networks to learn uncertain dynamics while still providing guarantees of transient tracking performance throughout the learning phase. Within the proposed approach, a disturbance estimation law is adopted to estimate the pointwise value of the uncertainty, with pre-computable estimation error bounds (EEBs). The learned dynamics, the estimated disturbances, and the EEBs are then incorporated in a robust Riemann energy condition to compute the control law that guarantees exponential convergence of actual trajectories to desired ones throughout the learning phase, even when the learned model is poor. On the other hand, with improved accuracy, the learned model can help improve the robustness of the tracking controller, e.g., against input delays, and can be incorporated to plan better trajectories with improved performance, e.g., lower energy consumption and shorter travel time.The proposed framework is validated on a planar quadrotor example.

We present a motion planning algorithm for a class of uncertain control-affine nonlinear systems which guarantees runtime safety and goal reachability when using high-dimensional sensor measurements (e.g., RGB-D images) and a learned perception module in the feedback control loop. First, given a dataset of states and observations, we train a perception system that seeks to invert a subset of the state from an observation, and estimate an upper bound on the perception error which is valid with high probability in a trusted domain near the data. Next, we use contraction theory to design a stabilizing state feedback controller and a convergent dynamic state observer which uses the learned perception system to update its state estimate. We derive a bound on the trajectory tracking error when this controller is subjected to errors in the dynamics and incorrect state estimates. Finally, we integrate this bound into a sampling-based motion planner, guiding it to return trajectories that can be safely tracked at runtime using sensor data. We demonstrate our approach in simulation on a 4D car, a 6D planar quadrotor, and a 17D manipulation task with RGB(-D) sensor measurements, demonstrating that our method safely and reliably steers the system to the goal, while baselines that fail to consider the trusted domain or state estimation errors can be unsafe.