By Ильин В.П.

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E. satisfies We refer to the smallest such integer N as the index of PWO. PWO was moreover shown to be decidable. More precisely, the index of pathwise observability was shown to be bounded above by numbers (actually, in [1]) depending only on the number of modes (or pairs) of the system, and Even though the numbers were not given explicitly, they were shown to be smaller than numbers (again, in [1]) that, moreover, satisfy (we let denote the row range space of a matrix M): Theorem 1 If is a path of length for some ) and a path then there exists a prefix of arbitrary length such that of and thus Even though this theorem was not explicitly stated in [1], its proof can easily be derived from that of [1, Theorem 1].

Is the bound on the This theorem is proved in Appendix. The proof uses the assumption that is nonempty and takes closed convex values. 3 Abstraction with Timing Information So far we have used only the sign of the derivative of to determine the switching conditions of the hybrid system. However, the time for the system to move from one box to its adjacent boxes is omitted, in other words, in the approximating system this time can be anything between 0 and To obtain a more precise abstraction, we can include information about the time the system can stay in a location, which we call staying time.

3]). Moreover, there is a closed form solution for Therefore, understanding the effect of applying an element of to an affine set is equivalent to understanding the effect of applying a Givens rotation. 2. Let X be an affine set of dimension in and assume that If with then there exists an affine set S such that X and Y are both faces of S. Moreover, there is a closed form solution for the affine constraints that determine S. Affine Hybrid Systems 23 Fig. 2. 1. Right: The set S with X and Y as faces.

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