Hybrid system

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 or control graph). 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 modelling dynamic phenomena.

In general, the state of a hybrid system is defined by the values of the continuous variables and a discrete control mode. The state changes either continuously, according to a flow condition, or discretely according to a control graph. Continuous flow is permitted as long as so-called invariants hold, while discrete transitions can occur as soon as given jump conditions are satisfied. Discrete transition may be associated with events.

Examples

Hybrid systems have been used to model several systems, including physical systems with impact, logic-dynamic controllers, and even Internet congestion.

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 acceleration 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.

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, a hybrid program 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.

Tools

• HyTech: A Model Checker for Hybrid Systems
• HSolver: Verification of Hybrid Systems
• PHAVer: Polyhedral Hybrid Automaton Verifyer
• KeYmaera: A Hybrid Theorem Prover for Hybrid Systems

See also

• Sliding mode control
• Variable structure system
• Variable structure control
• Joint spectral radius

References

• Henzinger, Thomas A. (1996), "The Theory of Hybrid Automata", 11th Annual Symposium on Logic in Computer Science (LICS), IEEE Computer Society Press, pp. 278–292, http://www.eecs.berkeley.edu/~tah/Publications/the_theory_of_hybrid_automata.html 
• Alur, Rajeev; Courcoubetis, Costas; Halbwachs, Nicolas; Henzinger, Thomas A.; Ho, Pei-Hsin; Nicollin, Xavier; Olivero, Alfredo; Sifakis, Joseph et al. (1995), "The algorithmic analysis of hybrid systems.", Theoretical Computer Science 138: 3-34, http://www.eecs.berkeley.edu/~tah/Publications/the_algorithmic_analysis_of_hybrid_systems.html 
• Goebel, Rafal; Sanfelice, Ricardo G.; Teel, Andrew R. (2009), "Hybrid dynamical systems", IEEE Control Systems Magazine 29 (2): 28–93, doi:10.1109/MCS.2008.931718 

External links

• IEEE CSS Committee on Hybrid Systems