Robust Successor and Precursor Sets of Hybrid Systems Using Hybrid Zonotopes

Abstract

This paper proposes methods for reachability analysis of nonlinear systems in both open loop and closed loop with advanced controllers. The methods combine hybrid zonotopes, a construct called a state-update set, functional decomposition, and special ordered set approximations to enable linear growth in both reachable set memory complexity and computational complexity with time. Facilitating this combination are new identities for constructing nonconvex sets that contain nonlinear functions and for efficiently converting a collection of polytopes from vertex representation to hybrid zonotope representation. Numerical examples demonstrate reachability analysis of a continuous-time nonlinear system in closed loop with a neural network controller trained using nonlinear model predictive control and a high-dimensional logical system.

Backward Reachability Analysis of Neural Feedback Systems Using Hybrid Zonotopes

“…Given the target set T = [[0.95, 1.05]] × [[0.95, 1.05]], Algorithm 1 is employed to over-approximate the BRSs. Note that we use 10 breakpoints to over-approximate the nonlinear term x 3 1 via SOS and OVERT and then H f (X p t , U p t ) is constructed by following a similar procedure as the example in [20]. Figure 2 shows the simulation results of the over-approximated BRSs (cyan), the one-epoch-refined overapproximated BRSs (dark green), and the samples located inside the true BRSs (red) for two steps.…”

“…The over-approximation method based on SOS was originally developed for solving nonlinear and nonconvex optimization problems [25]. In recent work, SOS approximations were utilized to compute the forward reachable set of nonlinear dynamical systems [20]. The SOS approximation S of a scalar-valued function was defined in [20, Definition 3], while the identity that converts S into an HZ was provided in [20,Theoem 4].…”

“…In recent work, SOS approximations were utilized to compute the forward reachable set of nonlinear dynamical systems [20]. The SOS approximation S of a scalar-valued function was defined in [20, Definition 3], while the identity that converts S into an HZ was provided in [20,Theoem 4]. The maximum SOS approximation error δ for some common nonlinear functions was provided in [24].…”

Backward Reachability Analysis of Neural Feedback Systems Using Hybrid Zonotopes

“…This letter aims to compute the exact BRS of a neural feedback system where the controller is a Feedforward Neural Network (FNN) with Rectified Linear Unit (ReLU) activation functions. The main mathematical tool used is Hybrid Zonotope (HZ), which can compactly represent a finite union of polytopic sets [18], [19], [20], [21]. This work builds on our previous work [21], which shows that an FNN with ReLU activation functions can be exactly represented by an HZ and provides algorithms to compute the exact and approximated forward reachable sets of neural feedback systems.…”