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Variable Structure Systems

The standard theory of dynamical systems is mostly concerned with systems that evolve in time according to a set of rules depending smoothly on the current state of the system.

However, in electrical engineering as well as in other fields, for instance in impact and friction phenomena in mechanical engineering, one is often confronted with systems that are most easily modeled as going through a succession of periods of smooth evolution separated by instantaneous events that mark transitions of one set of laws of evolution to another. We will refer to these systems as Variable Structure Systems (VSS), and they are part of a wider class known as hybrid systems in the literature. To come up with a precise formulation of systems with events is a nontrivial matter, in particular because one has, in general, to allow for the possibility that a state jump is associated with events and so it would be too restrictive to require solutions to be continuous, let alone differentiable. A special case of VSS can be described in the framework of Linear Complementarity Systems (LCS), for which precise mathematical results can be formulated. The theory behind LCS generalizes the observation that, for an ideal diode, both current and voltage can have only a sign, and their product must always be zero.