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Additional info for [Magazine] PC World. Special Bonus Collection. Vol. 5: Power Guide to Home Electronics

Example text

Similarly, definitions of predecessors, R-invariance and (local) contractivity for modes extend naturally to those for patterns. – Let π be a pattern. Then the mappings n n n n P ostπ : 2R → 2R and P reπ : 2R → 2R are affine transformations. The image of a convex set X by P ostπ (respectively, P reπ ) is therefore a convex set. Given a convex set R ⊂ Rn and a pattern π, in order to show that a convex subset X is R-invariant via π, it suffices to show that every vertex of X is mapped via P ostπ onto a point of R.

At this level of abstraction, the behavior of a switch is idealized as having two discrete states: an open circuit and a short circuit. In a circuit with K switches, there are 2K possible modes. In practice, however, not all these modes are admissible. Some of them are not feasible because of the physical characteristics of the switches, while others are banned by the designer because of safety considerations. Because of the restricted choice of circuit elements, the resulting systems have the desirable property that the continuous dynamics of each mode are linear or affine, which justifies (A1).

We have simplified their formalism here by removing the notion of output and observation sets. What has been called sampled switched system here, denoted by Σ, corresponds actually in [TAB 09, Chapter 11] to the notion of a symbolic system associated with a switched affine system Σ, denoted by Sτ (Σ). The idea of approximating the state trajectories of dynamic systems using zonotopes comes from [KÜH 98]. Using zonotopes for enclosing the uncertainty, we can avoid the wrapping effect, which leads to exponential fast-growing enclosures when using an iteratively naive approximation (such as the bounding box).

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