Safety and stability are common requirements for robotic control systems; however, designing safe, stable controllers remains difficult for nonlinear and uncertain models. We develop a model-based learning approach to synthesize robust feedback controllers with safety and stability guarantees. We take inspiration from robust convex optimization and Lyapunov theory to define robust control Lyapunov barrier functions that generalize despite model uncertainty. We demonstrate our approach in simulation on problems including car trajectory tracking, nonlinear control with obstacle avoidance, satellite rendezvous with safety constraints, and flight control with a learned ground effect model. Simulation results show that our approach yields controllers that match or exceed the capabilities of robust MPC while reducing computational costs by an order of magnitude.
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Direction finding and positioning systems based on RF signals are significantly impacted by multipath propagation, particularly in indoor environments. Existing algorithms (e.g MUSIC) perform poorly in resolving Angle of Arrival (AoA) in the presence of multipath or when operating in a weak signal regime. We note that digitally sampled RF frontends allow for the easy analysis of signals, and their delayed components. Low-cost Software-Defined Radio (SDR) modules enable Channel State Information (CSI) extraction across a wide spectrum, motivating the design of an enhanced Angle-of-Arrival (AoA) solution. We propose a Deep Learning approach to deriving AoA from a single snapshot of the SDR multichannel data. We compare and contrast deep-learning based angle classification and regression models, to estimate up to two AoAs accurately. We have implemented the inference engines on different platforms to extract AoAs in real-time, demonstrating the computational tractability of our approach. To demonstrate the utility of our approach we have collected IQ (In-phase and Quadrature components) samples from a four-element Universal Linear Array (ULA) in various Light-of-Sight (LOS) and Non-Line-of-Sight (NLOS) environments, and published the dataset. Our proposed method demonstrates excellent reliability in determining number of impinging signals and realized mean absolute AoA errors less than $2^{\circ}$.
Gradient flows are a powerful tool for optimizing functionals in general metric spaces, including the space of probabilities endowed with the Wasserstein metric. A typical approach to solving this optimization problem relies on its connection to the dynamic formulation of optimal transport and the celebrated Jordan-Kinderlehrer-Otto (JKO) scheme. However, this formulation involves optimization over convex functions, which is challenging, especially in high dimensions. In this work, we propose an approach that relies on the recently introduced input-convex neural networks (ICNN) to parametrize the space of convex functions in order to approximate the JKO scheme, as well as in designing functionals over measures that enjoy convergence guarantees. We derive a computationally efficient implementation of this JKO-ICNN framework and experimentally demonstrate its feasibility and validity in approximating solutions of low-dimensional partial differential equations with known solutions. We also demonstrate its viability in high-dimensional applications through an experiment in controlled generation for molecular discovery.
With the advancement of affordable self-driving vehicles using complicated nonlinear optimization but limited computation resources, computation time becomes a matter of concern. Other factors such as actuator dynamics and actuator command processing cost also unavoidably cause delays. In high-speed scenarios, these delays are critical to the safety of a vehicle. Recent works consider these delays individually, but none unifies them all in the context of autonomous driving. Moreover, recent works inappropriately consider computation time as a constant or a large upper bound, which makes the control either less responsive or over-conservative. To deal with all these delays, we present a unified framework by 1) modeling actuation dynamics, 2) using robust tube model predictive control, 3) using a novel adaptive Kalman filter without assuminga known process model and noise covariance, which makes the controller safe while minimizing conservativeness. On onehand, our approach can serve as a standalone controller; on theother hand, our approach provides a safety guard for a high-level controller, which assumes no delay. This can be used for compensating the sim-to-real gap when deploying a black-box learning-enabled controller trained in a simplistic environment without considering delays for practical vehicle systems.
We present $\mathcal{CL}_1$-$\mathcal{GP}$, a control framework that enables safe simultaneous learning and control for systems subject to uncertainties. The two main constituents are contraction theory-based $\mathcal{L}_1$ ($\mathcal{CL}_1$) control and Bayesian learning in the form of Gaussian process (GP) regression. The $\mathcal{CL}_1$ controller ensures that control objectives are met while providing safety certificates. Furthermore, $\mathcal{CL}_1$-$\mathcal{GP}$ incorporates any available data into a GP model of uncertainties, which improves performance and enables the motion planner to achieve optimality safely. This way, the safe operation of the system is always guaranteed, even during the learning transients. We provide a few illustrative examples for the safe learning and control of planar quadrotor systems in a variety of environments.
We present the deep neural network multigrid solver (DNN-MG) that we develop for the instationary Navier-Stokes equations. DNN-MG improves computational efficiency using a judicious combination of a geometric multigrid solver and a recurrent neural network with memory. DNN-MG uses the multi-grid method to classically solve on coarse levels while the neural network corrects interpolated solutions on fine ones, thus avoiding the increasingly expensive computations that would have to be performed there. This results in a reduction in computation time through DNN-MG's highly compact neural network. The compactness results from its design for local patches and the available coarse multigrid solutions that provides a "guide" for the corrections. A compact neural network with a small number of parameters also reduces training time and data. Furthermore, the network's locality facilitates generalizability and allows one to use DNN-MG trained on one mesh domain also on different ones. We demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around an obstacle. For these, our method significantly improves the solutions as well as lift and drag functionals while requiring only about half the computation time of a full multigrid solution. We also show that DNN-MG trained for the configuration with one obstacle can be generalized to other time dependent problems that can be solved efficiently using a geometric multigrid method.
Two types of second-order in time partial differential equations (PDEs), namely semilinear wave equations and semilinear beam equations are considered. To solve these equations with exponential integrators, we present an approach to compute efficiently the action of the matrix exponential as well as those of related matrix functions. Various numerical simulations are presented that illustrate this approach.
This paper addresses the consensus of a class of uncertain nonlinear fractional-order multi-agent systems (FOMAS). First a fractional non-fragile dynamic output feedback controller is put forward via the output measurements of neighboring agents, then appropriate state transformation reduced the consensus problem to a stability one. A sufficient condition based on direct Lyapunov approach, for the robust asymptotic stability of the transformed system and subsequently for the consensus of the main system is presented. Additionally, utilizing S-procedure and Schur complement, the systematic stabilization design algorithm is proposed for fractional-order system with and without nonlinear term. The results are formulated as an optimization problem with linear matrix inequality constraints. Simulation results are given to verify the effectiveness of the theoretical results.
Solving the Schr\"odinger equation is key to many quantum mechanical properties. However, an analytical solution is only tractable for single-electron systems. Recently, neural networks succeeded at modeling wave functions of many-electron systems. Together with the variational Monte-Carlo (VMC) framework, this led to solutions on par with the best known classical methods. Still, these neural methods require tremendous amounts of computational resources as one has to train a separate model for each molecular geometry. In this work, we combine a Graph Neural Network (GNN) with a neural wave function to simultaneously solve the Schr\"odinger equation for multiple geometries via VMC. This enables us to model continuous subsets of the potential energy surface with a single training pass. Compared to existing state-of-the-art networks, our Potential Energy Surface Network PESNet speeds up training for multiple geometries by up to 40 times while matching or surpassing their accuracy. This may open the path to accurate and orders of magnitude cheaper quantum mechanical calculations.
Accurate models of robot dynamics are critical for safe and stable control and generalization to novel operational conditions. Hand-designed models, however, may be insufficiently accurate, even after careful parameter tuning. This motivates the use of machine learning techniques to approximate the robot dynamics over a training set of state-control trajectories. The dynamics of many robots, including ground, aerial, and underwater vehicles, are described in terms of their SE(3) pose and generalized velocity, and satisfy conservation of energy principles. This paper proposes a Hamiltonian formulation over the SE(3) manifold of the structure of a neural ordinary differential equation (ODE) network to approximate the dynamics of a rigid body. In contrast to a black-box ODE network, our formulation guarantees total energy conservation by construction. We develop energy shaping and damping injection control for the learned, potentially under-actuated SE(3) Hamiltonian dynamics to enable a unified approach for stabilization and trajectory tracking with various platforms, including pendulum, rigid-body, and quadrotor systems.
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.
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