On enlarging backward reachable sets via Zonotopic set membership

Abstract

Abstract-While a number of efficient methods have been proposed for approximating backward reachable sets, no synthesis method via backward reachable sets has been developed for estimating and enlarging the region of attraction (RA). This paper shows how to use backward reachable sets to enlarge the estimate of the RA of linear discrete-time systems, by using an optimal static feedback controller. Two controller design methods are provided: the first method enlarges the estimate of the RA via invariant sets, whose existence is ensured by zonotope containment; the second method provides the optimal control input by using Lyapunov stability and quadratic stabilization. The backward reachable set is represented by zonotopes which give a good compromise between accuracy and efficiency. The effectiveness of both methods is illustrated by a numerical example.

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

“…That restriction can be relaxed to any class of sets for which a reasonable number of constraints can con rm zonotope containment; for example, intersections of slabs, or even convex polygons with a modest number of faces. It may even be possible to allow full zonotopes using [14,Lemma 3], albeit at the cost of swapping a small number of constraints (such as (36)) for a full linear matrix inequality.…”

Set operations and order reductions for constrained zonotopes

“…With c, Φ, Γ 1 , Γ 2 , and β as decision variables in this optimization problem, (28) consists of only linear constraints and thus an LP or QP can be formulated based on the norm used to minimize the vector φ, where Φ = diag(φ). In the following example, an LP is formulated by minimizing φ ∞ subject to (28). Computing RPI set Z using Theorem 6 requires solving an LP with n 2 g + n g (n w + 2) + n decision variables.…”

“…Inner-approximations are particularly important when computing backward reachable sets that define a set of initial states for which a system will enter a specified target region after some allotted time [27]. While there are existing techniques for zonotopes [3,28], inner-approximation techniques for constrained zonotopes are lacking.…”

Scalable Zonotopic Under-approximation of Backward Reachable Sets for Uncertain Linear Systems

“…Particularly, the existence of the environment player leads to a Minkowski subtraction step in the sequential backward reachable set computation, but zonotopes are not closed under Minkowski subtraction [2]. Therefore, the idea of time-reversing [18] and zonotopic backward reachable sets [11] were explored only for deterministic systems, but using zonotopes for uncertain systems' backward reachability, to the best of our knowledge, is still missing.…”