A method is presented to learn neural network (NN) controllers with stability and safety guarantees through imitation learning (IL). Convex stability and safety conditions are derived for linear time-invariant plant dynamics with NN controllers by merging Lyapunov theory with local quadratic constraints to bound the nonlinear activation functions in the NN. These conditions are incorporated in the IL process, which minimizes the IL loss, and maximizes the volume of the region of attraction associated with the NN controller simultaneously. An alternating direction method of multipliers based algorithm is proposed to solve the IL problem. The method is illustrated on an inverted pendulum system, aircraft longitudinal dynamics, and vehicle lateral dynamics.
Imitation learning (IL) is a learning paradigm which can be used to synthesize controllers for complex systems that mimic behavior demonstrated by an expert (user or control algorithm). Despite their popularity, IL methods generally lack guarantees of safety, which limits their utility for complex safety-critical systems. In this work we consider safety, formulated as set-invariance, and the associated formal guarantees endowed by Control Barrier Functions (CBFs). We develop conditions under which robustly-safe expert controllers, utilizing CBFs, can be used to learn end-to-end controllers (wh…
Safety-Guaranteed Imitation Learning from Nonlinear Model Predictive Control for Spacecraft Close Proximity Operations
Space Applications Group, e:fs TechHub GmbH（空间应用组，e:fs科技枢纽 GmbH）；Faculty of Electrical Engineering and Information Technology, Ingolstadt University of Applied Sciences（电气工程与信息科技学院，因戈尔施塔特应用技术大学）
Safe Human-to-Humanoid Motion Imitation Using Control Barrier Functions
New York University Abu Dhabi (NYUAD)（纽约大学阿布扎比分校）；New York University（纽约大学）；NYUAD Center for Artificial Intelligence and Robotics (CAIR)（纽约大学阿布扎比人工智能与机器人中心）
Control policies from imitation learning can often fail to generalize to novel environments due to imperfect demonstrations or the inability of imitation learning algorithms to accurately infer the expert's policies. In this paper, we present rigorous generalization guarantees for imitation learning by leveraging the Probably Approximately Correct (PAC)-Bayes framework to provide upper bounds on the expected cost of policies in novel environments. We propose a two-stage training method where a latent policy distribution is first embedded with multi-modal expert behavior using a conditional var…
Imitation Learning of MPC with Neural Networks: Error Guarantees and Sparsification
Control and Cyber-Physical Systems Laboratory, Technical University of Darmstadt（达姆施塔特工业大学控制与信息物理系统实验室）；Dept of Chemical & Biomolecular Engineering, University of California, Berkeley（加州大学伯克利分校化学与生物分子工程系）
Deploying safety-critical controllers in practice necessitates the ability to modulate uncertainties in control systems. In this context, robust control barrier functions -- in a variety of forms -- have been used to obtain safety guarantees for uncertain systems. Yet the differing types of uncertainty experienced in practice have resulted in a fractured landscape of robustification -- with a variety of instantiations depending on the structure of the uncertainty. This paper proposes a framework for generalizing these variations into a single form: parameterized barrier functions (PBFs), which…
Deep-MPC: A DAGGER-Driven Imitation Learning Strategy for Optimal Constrained Battery Charging
School of Aerospace and Mechanical Engineering, The University of Oklahoma（俄克拉荷马大学航空航天与机械工程学院）；Institute for Resilient Environmental & Energy Systems, The University of Oklahoma（俄克拉荷马大学弹性环境与能源系统研究所）；Faculty of Mathematics, Physical and Natural Science, Catholic University of Sacred Heart（圣心天主教大学数学、物理与自然科学学院）
This paper addresses the trajectory planning for multiple autonomous underwater vehicles (AUVs) in strong waves that can disturb the AUVs' trajectory tracking ability and cause obstacle and inter-vehicle collisions. A novel approach based on Hamilton-Jacobi differential game formulation and model predictive control is proposed to generate safety-guaranteed trajectories for AUV systems considering wave disturbance that is either time-varying state-independent or time-varying state-dependent. This approach plans safety-guaranteed trajectories for multiple AUVs and considers the shapes of AUVs, w…
This paper presents a safe robust policy iteration (SR-PI) algorithm to design controllers with satisficing (good enough) performance and safety guarantee. This is in contrast to standard PI-based control design methods with no safety certification. It also moves away from existing safe control design approaches that perform pointwise optimization and are thus myopic. Safety assurance requires satisfying a control barrier function (CBF), which might be in conflict with the performance-driven Lyapunov solution to the Bellman equation arising in each iteration of the PI. Therefore, a new develop…
Coordinated Science Laboratory (CSL)（协调科学实验室）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
When applying imitation learning techniques to fit a policy from expert demonstrations, one can take advantage of prior stability/robustness assumptions on the expert's policy and incorporate such control-theoretic prior knowledge explicitly into the learning process. In this paper, we formulate the imitation learning of linear policies as a constrained optimization problem, and present efficient methods which can be used to enforce stability and robustness constraints during the learning processes. Specifically, we show that one can guarantee the closed-loop stability and robustness by posing…
Approaches to keeping a dynamical system within state constraints typically rely on a model-based safety condition to limit the control signals. In the face of significant modeling uncertainty, the system can suffer from important performance penalties due to the safety condition becoming overly conservative. Machine learning can be employed to reduce the uncertainty around the system dynamics, and allow for higher performance. In this article, we propose the safe uncertainty learning principle, and argue that the learning must be properly structured to preserve safety guarantees. For instance…
Refined Barrier Conditions for Finite-Time Safety and Reach-Avoid Guarantees in Stochastic Systems
KLSS, ISCAS（中国科学院自动化研究所）；University of Chinese Academy of Sciences（中国科学院大学）；College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院）；Software Engineering Institute, East China Normal University（华东师范大学软件学院）
Cyber-physical systems (CPS) are required to operate safely under fault and malicious attacks. The simplex architecture and the recently proposed cyber resilient architectures, e.g., Byzantine fault tolerant++ (BFT++), provide safety for CPS under faults and malicious cyber attacks, respectively. However, these existing architectures make use of different timing parameters and implementations to provide safety, and are seemingly unrelated. In this paper, we propose an analytical framework to represent the simplex, BFT++ and other practical cyber resilient architectures (CRAs). We construct a h…
Automated vehicles require efficient and safe planning to maneuver in uncertain environments. Largely this uncertainty is caused by other traffic participants, e.g., surrounding vehicles. Future motion of surrounding vehicles is often difficult to predict. Whereas robust control approaches achieve safe, yet conservative motion planning for automated vehicles, Stochastic Model Predictive Control (SMPC) provides efficient planning in the presence of uncertainty. Probabilistic constraints are applied to ensure that the maximal risk remains below a predefined level. However, safety cannot be ensur…
We present a safe-by-design approach to path planning and control for nonlinear systems. The planner uses a low fidelity model of the plant to compute reference trajectories by solving an MPC problem, while the plant being controlled utilizes a feedback control law that tracks those trajectories with an upper-bound on the tracking error. Our main goal is to allow for maximum permissiveness (that is, room for constraint feasibility) of the planner, while maintaining safety after accounting for the tracking error bound. We achieve this by parametrizing the state and input constraints imposed on …
Control schemes for autonomous systems are often designed in a way that anticipates the worst case in any situation. At runtime, however, there could exist opportunities to leverage the characteristics of specific environment and operation context for more efficient control. In this work, we develop an online intermittent-control framework that combines formal verification with model-based optimization and deep reinforcement learning to opportunistically skip certain control computation and actuation to save actuation energy and computational resources without compromising system safety. Exper…
In this paper we seek to quantify the ability of learning to improve safety guarantees endowed by Control Barrier Functions (CBFs). In particular, we investigate how model uncertainty in the time derivative of a CBF can be reduced via learning, and how this leads to stronger statements on the safe behavior of a system. To this end, we build upon the idea of Input-to-State Safety (ISSf) to define Projection-to-State Safety (PSSf), which characterizes degradation in safety in terms of a projected disturbance. This enables the direct quantification of both how learning can improve safety guarante…
Technical University of Munich（慕尼黑工业大学）；Wayne State University（韦恩州立大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Deep reinforcement learning (DRL) has achieved tremendous success in many complex decision-making tasks of autonomous systems with high-dimensional state and/or action spaces. However, the safety and stability still remain major concerns that hinder the applications of DRL to safety-critical autonomous systems. To address the concerns, we proposed the Phy-DRL: a physical deep reinforcement learning framework. The Phy-DRL is novel in two architectural designs: i) Lyapunov-like reward, and ii) residual control (i.e., integration of physics-model-based control and data-driven control). The concur…
This paper details the theory and implementation behind practically ensuring safety of remotely piloted racing drones. We demonstrate robust and practical safety guarantees on a 7" racing drone at speeds exceeding 100 km/h, utilizing only online computations on a 10 gram micro-controller. To achieve this goal, we utilize the framework of control barrier functions (CBFs) which give guaranteed safety encoded as forward set invariance. To make this methodology practically applicable, we present an implicitly defined CBF which leverages backup controllers to enable gradient-free evaluations that e…
This paper introduces differentiable higher-order control barrier functions (CBF) that are end-to-end trainable together with learning systems. CBFs are usually overly conservative, while guaranteeing safety. Here, we address their conservativeness by softening their definitions using environmental dependencies without loosing safety guarantees, and embed them into differentiable quadratic programs. These novel safety layers, termed a BarrierNet, can be used in conjunction with any neural network-based controller, and can be trained by gradient descent. BarrierNet allows the safety constraints…
Leveraging recent developments in black-box risk-aware verification, we provide three algorithms that generate probabilistic guarantees on (1) optimality of solutions, (2) recursive feasibility, and (3) maximum controller runtimes for general nonlinear safety-critical finite-time optimal controllers. These methods forego the usual (perhaps) restrictive assumptions required for typical theoretical guarantees, e.g. terminal set calculation for recursive feasibility in Nonlinear Model Predictive Control, or convexification of optimal controllers to ensure optimality. Furthermore, we show that the…
Pacific Northwest National Laboratory（太平洋西北国家实验室）；Battelle Memorial Institute（巴特尔纪念研究所）
We develop a novel form of differentiable predictive control (DPC) with safety and robustness guarantees based on control barrier functions. DPC is an unsupervised learning-based method for obtaining approximate solutions to explicit model predictive control (MPC) problems. In DPC, the predictive control policy parametrized by a neural network is optimized offline via direct policy gradients obtained by automatic differentiation of the MPC problem. The proposed approach exploits a new form of sampled-data barrier function to enforce offline and online safety requirements in DPC settings while …
When autonomous robots interact with humans, such as during autonomous driving, explicit safety guarantees are crucial in order to avoid potentially life-threatening accidents. Many data-driven methods have explored learning probabilistic bounds over human agents' trajectories (i.e. confidence tubes that contain trajectories with probability $δ$), which can then be used to guarantee safety with probability $1-δ$. However, almost all existing works consider $δ\geq 0.001$. The purpose of this paper is to argue that (1) in safety-critical applications, it is necessary to provide safety guarantees…
This paper aims at the safety-critical control synthesis of network systems such that the satisfaction of the safety constraints can be guaranteed. To handle the large state dimension of such systems, an assume-guarantee contract is used to break the large synthesis problem into smaller subproblems. Parameterized signal temporal logic (pSTL) is used to formally describe the behaviors of the subsystems, which we use as the template for the contract. We show that robust control invariant sets (RCIs) for the subsystems can be composed to form a robust control invariant set for the whole network s…
Autonomous systems like aircraft and assistive robots often operate in scenarios where guaranteeing safety is critical. Methods like Hamilton-Jacobi reachability can provide guaranteed safe sets and controllers for such systems. However, often these same scenarios have unknown or uncertain environments, system dynamics, or predictions of other agents. As the system is operating, it may learn new knowledge about these uncertainties and should therefore update its safety analysis accordingly. However, work to learn and update safety analysis is limited to small systems of about two dimensions du…
University of California, Berkeley（加州大学伯克利分校）；California Institute of Technology（加州理工学院）
Modern nonlinear control theory seeks to develop feedback controllers that endow systems with properties such as safety and stability. The guarantees ensured by these controllers often rely on accurate estimates of the system state for determining control actions. In practice, measurement model uncertainty can lead to error in state estimates that degrades these guarantees. In this paper, we seek to unify techniques from control theory and machine learning to synthesize controllers that achieve safety in the presence of measurement model uncertainty. We define the notion of a Measurement-Robus…
We study the asymptotic behavior of deterministic, continuous-time imitation dynamics for population games over networks. The basic assumption of this learning mechanism -- encompassing the replicator dynamics -- is that players belonging to a single population exchange information through pairwise interactions, whereby they get aware of the actions played by the other players and the corresponding rewards. Using this information, they can revise their current action, imitating the one of the players they interact with. The pattern of interactions regulating the learning process is determined …
