Hybrid system
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A hybrid system is a dynamic system that exhibits both continuous and discrete dynamic behavior — a system that can both flow (described by a differential equation) and jump (described by a difference equation). Often, the term "hybrid dynamic system" is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic, or electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modeling dynamic phenomena.
In general, a hybrid system can be described by a few pieces of information. The state of the system consists of vector signals, which can change according to dynamic laws in the system data. The data includes a flow equation, f(x), which describes the continuous dynamics, a flow set, C, in which flow is permitted, a jump equation, g(x), which describes the discrete dynamics, and a jump set, D, in which discrete state evolution is permitted.
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[edit] Examples
Hybrid systems have been used to model several systems, including physical systems with impact, logic-dynamic controllers, and even Internet congestion.
[edit] Bouncing ball
A canonical example of a hybrid system is the bouncing ball, a physical system with impact. Here, the ball (thought of as a point-mass) is dropped from an initial height and bounces off the ground, dissipating its energy with each bounce. The ball exhibits continuous dynamics between each bounce; however, as the ball impacts the ground, its velocity undergoes a discrete change modeled after an inelastic collision. A mathematical description of the bouncing ball follows. Let x1 be the height of the ball and x2 be the velocity of the ball. A hybrid system describing the ball is as follows:
When 
, flow is governed by 
, where g is the force due to gravity. These equations state that when the ball is above ground, it is being drawn to the ground by gravity.
When 
, jumps are governed by 
, where 0 < γ < 1 is a dissipation factor. This is saying that when the height of the ball is zero (it has impacted the ground), its velocity is reversed and decreased by a factor of γ. Effectively, this describes the nature of the inelastic collision.
The bouncing ball is an especially interesting hybrid system, as it exhibits Zeno behavior. Zeno behavior has a strict mathematical definition, but can be described informally as the system making an infinite number of jumps in a finite amount of time. In this example, each time the ball bounces it loses energy, making the subsequent jumps (impacts with the ground) closer and closer together in time.
[edit] Other modeling approaches
Two basic hybrid system modeling approaches can be classified, an implicit and an explicit one. The explicit approach is often represented by a hybrid automaton or a hybrid Petri net. The implicit approach is often represented by guarded equations to result in systems of differential algebraic equations (DAEs) where the active equations may change, for example by means of a hybrid bond graph.
