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The invariance conditions (6) are satisfied at the three boundary points shown labeled with the directions of the dynamics F and the normal directions N S .
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Context 1
... the boundary of S, which we denote as ∂S, the normal cone N S is the set of outward-pointing normal directions. Fig. 1 illustrates three boundary points with the corresponding normal cone at those points. On the left of Fig. 1 is a point where ∂S is locally smooth, and the normal cone is equivalent to the traditional normal vector, on the right is a non-smooth point where S is locally convex and the normal cone is non-trivial, and in the center is a ...
Context 2
... the boundary of S, which we denote as ∂S, the normal cone N S is the set of outward-pointing normal directions. Fig. 1 illustrates three boundary points with the corresponding normal cone at those points. On the left of Fig. 1 is a point where ∂S is locally smooth, and the normal cone is equivalent to the traditional normal vector, on the right is a non-smooth point where S is locally convex and the normal cone is non-trivial, and in the center is a non-smooth point where S is locally concave and the normal cone is equal to the zero vector N S (x) = {0}. ...
Context 3
... invariance conditions in (6) relate the angle between the outward-pointing normal cone N S and the direction of the dynamics F , so that the set S is invariant if the elements of F point "into" the set, as illustrated in Fig. 1. In the rest of this paper, we will use the invariance conditions (6) to construct sets that are the boundaries of ...
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Citations
... This is evident from some recent works such as [57]- [59], where closed polytopes are constructed using path planning algorithms on radial graphs for arbitrary nonlinear systems in continuous time, but limited to systems in R 2 . Another illustration of this can be found in [60], [61], where simplex-based approaches are used for arbitrary nonlinear systems. Besides these limited works, few computation-oriented results are available for generic nonlinear systems [36]. ...
... Proof. This follows from Proposition 3 in [60]. ...
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... The set is the region of interest. Functions f(·), (·) are assumed Lipschitz, because this is required for uniqueness of solutions, which is needed for the existence of an invariant set [2,8,34]. ...
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... Previous research has studied computational methods that find asymptotic approximations to largest or smallest invariant sets or tolerable disturbance sets. Given a disturbance set and a set of starting points for a continuous-time nonlinear system, our previous work [6] considers the problem of finding the smallest invariant set that contains the given starting points. Given a disturbance set and a state constraint set of a discrete-time linear system, the work in [7] studies the largest invariant set that is contained within the given state constraint. ...
... This is relevant to this paper because we will use state sets S with piecewise linear boundaries, which are non-smooth at the intersections between the boundary segments. When evaluating invariance in this special case, the non-smooth intersections can essentially be ignored, as shown in [6]. ...
... (2) associated with the simplexes ∂ i ∆ and ∆. Proof: See [6] for a proof. Polytopes are a broader class of sets with piecewise-linear boundaries that includes simplexes. ...
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