skip to main content

Title: Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming
There has been an increasing interest in using neural networks in closed-loop control systems to improve performance and reduce computational costs for on-line implementation. However, providing safety and stability guarantees for these systems is challenging due to the nonlinear and compositional structure of neural networks. In this paper, we propose a novel forward reachability analysis method for the safety verification of linear time-varying systems with neural networks in feedback interconnection. Our technical approach relies on abstracting the nonlinear activation functions by quadratic constraints, which leads to an outer-approximation of forward reachable sets of the closed-loop system. We show that we can compute these approximate reachable sets using semidefinite programming. We illustrate our method in a quadrotor example, in which we first approximate a nonlinear model predictive controller via a deep neural network and then apply our analysis tool to certify finite-time reachability and constraint satisfaction of the closed-loop system.  more » « less
Award ID(s):
Author(s) / Creator(s):
; ; ;
Date Published:
Journal Name:
IEEE Conference on Decision and Control
Page Range / eLocation ID:
5929 to 5934
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. null (Ed.)
    We present a data-driven method for computing approximate forward reachable sets using separating kernels in a reproducing kernel Hilbert space. We frame the problem as a support estimation problem, and learn a classifier of the support as an element in a reproducing kernel Hilbert space using a data-driven approach. Kernel methods provide a computationally efficient representation for the classifier that is the solution to a regularized least squares problem. The solution converges almost surely as the sample size increases, and admits known finite sample bounds. This approach is applicable to stochastic systems with arbitrary disturbances and neural network verification problems by treating the network as a dynamical system, or by considering neural network controllers as part of a closed-loop system. We present our technique on several examples, including a spacecraft rendezvous and docking problem, and two nonlinear system benchmarks with neural network controllers. 
    more » « less
  2. 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, which consists of all possible values of the uncertainties and the linearization error. This Minkowski difference needs to be represented as a constrained zonotope to enable subsequent computation, but, as we show, it is impossible to find a polynomial-size representation for it in polynomial time. Our algorithm finds a polynomial-size under-approximation in polynomial time. We further analyze the conservatism of this under-approximation technique, and show that it is exact under some conditions. Based on the developed Minkowski difference technique, we detail two backward reachable set computation algorithms to control the linearization error and incorporate nonconvex state constraints. Several examples illustrate the effectiveness of our algorithms. 
    more » « less
  3. This paper proposes a method to generate feasible trajectories for robotic systems with predefined sequences of switched contacts. The proposed trajectory generation method relies on sampling-based methods, optimal control, and reach-ability analysis. In particular, the proposed method is able to quickly test whether a simplified model-based planner, such as the Time-to-Velocity-Reversal planner, provides a reachable contact location based on reachability analysis of the multi-body robot system. When the contact location is reachable, we generate a feasible trajectory to change the contact mode of the robotic system smoothly. To perform reachability analysis efficiently, we devise a method to compute forward and backward reachable sets based on element-wise optimization over a finite time horizon. Then, we compute robot trajectories by employing optimal control. The main contributions of this study are the following. Firstly, we guarantee whether planned contact locations via simplified models are feasible by the robot system. Secondly, we generate optimal trajectories subject to various constraints given a feasible contact sequence. Lastly, we improve the efficiency of computing reachable sets for a class of constrained nonlinear systems by incorporating bi-directional propagation (forward and backward). To validate our methods we perform numerical simulations applied to a humanoid robot walking. 
    more » « less
  4. This paper aims to enhance the computational efficiency of safety verification of neural network control systems by developing a guaranteed neural network model reduction method. First, a concept of model reduction precision is proposed to describe the guaranteed distance between the outputs of a neural network and its reduced-size version. A reachability-based algorithm is proposed to accurately compute the model reduction precision. Then, by substituting a reduced-size neural network controller into the closed-loop system, an algorithm to compute the reachable set of the original system is developed, which is able to support much more computationally efficient safety verification processes. Finally, the developed methods are applied to a case study of the Adaptive Cruise Control system with a neural network controller, which is shown to significantly reduce the computational time of safety verification and thus validate the effectiveness of the method. 
    more » « less
  5. Robust motion planning entails computing a global motion plan that is safe under all possible uncertainty realizations, be it in the system dynamics, the robot’s initial position, or with respect to external disturbances. Current approaches for robust motion planning either lack theoretical guarantees, or make restrictive assumptions on the system dynamics and uncertainty distributions. In this paper, we address these limitations by proposing the robust rapidly-exploring random-tree (Robust-RRT) algorithm, which integrates forward reachability analysis directly into sampling-based control trajectory synthesis. We prove that Robust-RRT is probabilistically complete (PC) for nonlinear Lipschitz continuous dynamical systems with bounded uncertainty. In other words, Robust-RRT eventually finds a robust motion plan that is feasible under all possible uncertainty realizations assuming such a plan exists. Our analysis applies even to unstable systems that admit only short-horizon feasible plans; this is because we explicitly consider the time evolution of reachable sets along control trajectories. Thanks to the explicit consideration of time dependency in our analysis, PC applies to unstabilizable systems. To the best of our knowledge, this is the most general PC proof for robust sampling-based motion planning, in terms of the types of uncertainties and dynamical systems it can handle. Considering that an exact computation of reachable sets can be computationally expensive for some dynamical systems, we incorporate sampling-based reachability analysis into Robust-RRT and demonstrate our robust planner on nonlinear, underactuated, and hybrid systems. 
    more » « less