Combining zonotopes and support functions for efficient reachability analysis of linear systems

Abstract

Abstract-Reachability analysis is an important technique for formally verifying continuous systems, as well as for guaranteed state estimation, stability analysis, and controller synthesis. We present a detailed assessment of the computational efficiency for the reachability analysis of linear systems with respect to the two most scalable set representations: zonotopes and support functions. As a result, we propose representing reachable sets as a combination of support functions and zonotopes. This mix of representations can be converted to polyhedra of desired (directional) precision, at a higher precision compared to exclusively using support functions or zonotopes. The benefits are shown by an in-depth analysis of computational complexity and by numerical experiments.

Invariant, viability and discriminating kernel under-approximation via zonotope scaling

“…A key shortcoming of the current formulation is its restriction to discrete time. Unfortunately, the typical approaches to soundly mapping continuous time reachability into discrete time reachability (for example, see [3]) cannot be used in our approach, so we are exploring alternatives.…”

“…Ellipsoid and support vector parametric representations of viability constructs were explored in [15,16]. Zonotopic representations for reachability were introduced in [9] and have since been extensively explored; for example [3,4,10]. Our work was inspired, however, by the papers [18,20] and in particular [19], which utilizes a convex optimization to select zonotope generator weights to construct a control scheme that will drive a set of initial states into the smallest possible set of nal states.…”

Reach Set Approximation through Decomposition with Low-dimensional Sets and High-dimensional Matrices

“…In the context of control engineering, the sequence of sets {V(k)} k usually represents contributions from nondeterministic inputs or noise, ⊕ denotes the Minkowski sum between n-dimensional sets, Φ is a given real n ×n matrix, and the set X(0) accounts for uncertain initial states. Numerous works present strategies for solving equation (1) in the form of ellipsoids [35,36], template polyhedra such as zonotopes [4,23] or support functions [12,[19][20][21]38], or a combination [3]. The problem also generalizes to hybrid systems with piecewise affine dynamics [8,28].…”

“…• We provide a new decomposition approach to solve equation (1) and analyze the approximation error. • We address both the dense time and the discrete time instances of the reachability problem for general LTI systems of the form (2)- (3). 1…”

Direct Verification of Linear Systems with over 10000 Dimensions

“…Reachable state set computation tools typically track states using data structures such as polyhedra [15], zonotopes [17], support functions [18], Taylor models [11], or combinations of these [2]. Recently, a data structure and associated reachability method was proposed which stores states in a data structure called a generalized star set [14].…”