We propose a method to outer bound forward reachable sets on finite horizons for uncertain nonlinear systems with polynomial dynamics. This method makes use of time-dependent polynomial storage functions that satisfy appropriate dissipation inequalities that account for time-varying uncertain parameters, L2 disturbances, and perturbations characterized by integral quadratic constraints (IQCs) with both hard and soft factorizations. In fact, to our knowledge, this is the first result introducing IQCs to reachability analysis, thus allowing for various types of uncertainty, including unmodeled dynamics. The generalized S-procedure and Sum-of-Squares techniques are used to derive algorithms with the goal of finding the tightest outer bound with a desired shape. Both pedagogical and practically motivated examples are presented, including a 7-state F-18 aircraft model.
A method is proposed to compute robust inner-approximations to the backward reachable set for uncertain nonlinear systems. It also produces a robust control law that drives trajectories starting in these sets to the target set. The method merges dissipation inequalities and integral quadratic constraints (IQCs) with both hard and soft IQC factorizations. Computational algorithms are presented using the generalized S-procedure and sum-of-squares techniques. The use of IQCs in backward reachability analysis allows for a variety of perturbations including parametric uncertainty, unmodeled dynamic…
In this work, we perform safety analysis of linear dynamical systems with uncertainties. Instead of computing a conservative overapproximation of the reachable set, our approach involves computing a statistical approximate reachable set. As a result, the guarantees provided by our method are probabilistic in nature. In this paper, we provide two different techniques to compute statistical approximate reachable set. We have implemented our algorithms in a python based prototype and demonstrate the applicability of our approaches on various case studies. We also provide an empirical comparison b…
The literature on reachability analysis methods for neural networks currently only focuses on uncertainties on the network's inputs. In this paper, we introduce two new approaches for the reachability analysis of neural networks with additional uncertainties on their internal parameters (weight matrices and bias vectors of each layer), which may open the field of formal methods on neural networks to new topics, such as safe training or network repair. The first and main method that we propose relies on existing reachability analysis approach based on mixed monotonicity (initially introduced fo…
A method is presented to obtain an inner-approximation of the backward reachable set (BRS) of a given target tube, along with an admissible controller that maintains trajectories inside this tube. The proposed optimization algorithms are formulated as nonlinear optimization problems, which are decoupled into tractable subproblems and solved by an iterative algorithm using the polynomial S-procedure and sum-of-squares techniques. This framework is also extended to uncertain nonlinear systems with L_2 disturbances and L_{\infty} parametric uncertainties. The effectiveness of the method is demons…
One often wishes for the ability to formally analyze large-scale systems---typically, however, one can either formally analyze a rather small system or informally analyze a large-scale system. This work tries to further close this performance gap for reachability analysis of linear systems. Reachability analysis can capture the whole set of possible solutions of a dynamic system and is thus used to prove that unsafe states are never reached; this requires full consideration of arbitrarily varying uncertain inputs, since sensor noise or disturbances usually do not follow any patterns. We use Kr…
Efficiently handling time-triggered and possibly nondeterministic switches for hybrid systems reachability is a challenging task. In this paper we present an approach based on conservative set-based enclosure of the dynamics that can handle systems with uncertain parameters and inputs, where the uncertainties are bound to given intervals. The method is evaluated on the plant model of an experimental electro-mechanical braking system with periodic controller. In this model, the fast-switching controller dynamics requires simulation time scales of the order of nanoseconds. Accurate set-based com…
A neural ordinary differential equations network (ODE-Net)-enabled reachability method (Neuro-Reachability) is devised for the dynamic verification of networked microgrids (NMs) with unidentified subsystems and heterogeneous uncertainties. Three new contributions are presented: 1) An ODENet-enabled dynamic model discovery approach is devised to construct the data-driven state-space model which preserves the nonlinear and differential structure of the NMs system; 2) A physics-data-integrated (PDI) NMs model is established, which empowers various NM analytics; and 3) A conformance-empowered reac…
Tsinghua University（清华大学）；Harvard University（哈佛大学）；University of Science and Technology Beijing（北京科技大学）
Safe reinforcement learning (RL) that solves constraint-satisfactory policies provides a promising way to the broader safety-critical applications of RL in real-world problems such as robotics. Among all safe RL approaches, model-based methods reduce training time violations further due to their high sample efficiency. However, lacking safety robustness against the model uncertainties remains an issue in safe model-based RL, especially in training time safety. In this paper, we propose a distributional reachability certificate (DRC) and its Bellman equation to address model uncertainties and c…
Guaranteeing safety for robotic and autonomous systems in real-world environments is a challenging task that requires the mitigation of stochastic uncertainties. Control barrier functions have, in recent years, been widely used for enforcing safety related set-theoretic properties, such as forward invariance and reachability, of nonlinear dynamical systems. In this paper, we extend this rich framework to nonlinear discrete-time systems subject to stochastic uncertainty and propose a framework for assuring risk-sensitive safety in terms of coherent risk measures. To this end, we introduce risk …
Reachability analysis is at the core of many applications, from neural network verification, to safe trajectory planning of uncertain systems. However, this problem is notoriously challenging, and current approaches tend to be either too restrictive, too slow, too conservative, or approximate and therefore lack guarantees. In this paper, we propose a simple yet effective sampling-based approach to perform reachability analysis for arbitrary dynamical systems. Our key novel idea consists of using random set theory to give a rigorous interpretation of our method, and prove that it returns sets w…
We propose algorithms for performing model checking and control synthesis for discrete-time uncertain systems under linear temporal logic (LTL) specifications. We construct temporal logic trees (TLT) from LTL formulae via reachability analysis. In contrast to automaton-based methods, the construction of the TLT is abstraction-free for infinite systems, that is, we do not construct discrete abstractions of the infinite systems. Moreover, for a given transition system and an LTL formula, we prove that there exist both a universal TLT and an existential TLT via minimal and maximal reachability an…
Backward reachability analysis is essential to synthesizing controllers that ensure the correctness of closed-loop systems. This paper is concerned with developing scalable algorithms that under-approximate the backward reachable sets, for discrete-time uncertain linear and nonlinear systems. Our algorithm sequentially linearizes the dynamics, and uses constrained zonotopes for set representation and computation. The main technical ingredient of our algorithm is an efficient way to under-approximate the Minkowski difference between a constrained zonotopic minuend and a zonotopic subtrahend, wh…
Neural Networks (NNs) can provide major empirical performance improvements for robotic systems, but they also introduce challenges in formally analyzing those systems' safety properties. In particular, this work focuses on estimating the forward reachable set of closed-loop systems with NN controllers. Recent work provides bounds on these reachable sets, yet the computationally efficient approaches provide overly conservative bounds (thus cannot be used to verify useful properties), whereas tighter methods are too intensive for online computation. This work bridges the gap by formulating a con…
Neural Networks (NNs) can provide major empirical performance improvements for closed-loop systems, but they also introduce challenges in formally analyzing those systems' safety properties. In particular, this work focuses on estimating the forward reachable set of \textit{neural feedback loops} (closed-loop systems with NN controllers). Recent work provides bounds on these reachable sets, but the computationally tractable approaches yield overly conservative bounds (thus cannot be used to verify useful properties), and the methods that yield tighter bounds are too intensive for online comput…
NeuroHJR: Hamilton-Jacobi Reachability-based Obstacle Avoidance in Complex Environments with Physics-Informed Neural Networks
Department of Physical Science, Indian Institute of Science Education and Research（物理科学系，印度科学教育与研究学院）；Department of Aerospace Engineering, Indian Institute of Science（航空航天工程系，印度科学研究院）；Department of Computer Science and Engineering, Shiv Nadar University（计算机科学与工程系，施瓦斯纳大学）
Recent advances in machine learning technologies and sensing have paved the way for the belief that safe, accessible, and convenient autonomous vehicles may be realized in the near future. Despite tremendous advances within this context, fundamental challenges around safety and reliability are limiting their arrival and comprehensive adoption. Autonomous vehicles are often tasked with operating in dynamic and uncertain environments. As a result, they often make use of highly complex components, such as machine learning approaches, to handle the nuances of sensing, actuation, and control. While…
Safe, Real-Time Active Model Discrimination and Fault Diagnosis for Nonlinear Systems via Differentiable Reachability
Institute of Robotics and Intelligent Machines (IRIM), Georgia Institute of Technology（佐治亚理工学院机器人与智能机器研究所）；Department of Aerospace Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校航空航天工程系）
Finite-Sample-Based Reachability for Safe Control with Gaussian Process Dynamics
ETH AI Center, ETH Zurich（苏黎世联邦理工学院人工智能中心）；Institute for Dynamic Systems and Control, ETH Zurich（苏黎世联邦理工学院动态系统与控制研究所）；Department of Computer Science, ETH Zurich（苏黎世联邦理工学院计算机科学系）
Verification and synthesis of Cyber-Physical Systems (CPS) are challenging and still raise numerous issues so far. In this paper, based on a new concept of mixed sets defined as function images of symbol type domains, a compositional approach combining eager and lazy evaluations is proposed. Syntax and semantics are explicitly distinguished. Both continuous (interval) and discrete (signed, boolean) symbol types are used to model dependencies through linear and polynomial functions, so leading to mixed zonotopic and polynotopic sets. Polynotopes extend sparse polynomial zonotopes with typed sym…
RTD-RAX: Fast, Safe Trajectory Planning for Systems under Unknown Disturbances
Georgia Institute of Technology（佐治亚理工学院）；School of Electrical and Computer Engineering（电子与计算机工程学院）；School of Mechanical Engineering（机械工程学院）；School of Civil and Environmental Engineering（土木与环境工程学院）
