Hybrid system
Encyclopedia
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.
s with impact, logic-dynamic controllers
, and even Internet
congestion.
. A mathematical description of the bouncing ball follows. Let
be the height of the ball and 
be the velocity of the ball. A hybrid system describing the ball is as follows:
When
, flow is governed by
,
where
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
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 
. 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.
It is noteworthy that the dynamical model is complete if and only if one adds the contact force between the ground and the ball. Indeed, without forces, one cannot properly define the bouncing ball and the model is, from a mechanical point of view, meaningless. The simplest contact model that represents the interactions between the ball and the ground, is the complementarity relation between the force and the distance (the gap) between the ball and the ground. This is written as
Such a contact model does not incorporate magnetic forces, nor gluing effects (guess why!). When the complementarity relations are in, one can continue to integrate the system after the impacts have accumulated and vanished: the equilibrium of the system is well-defined as the static equilibrium of the ball on the ground, under the action of gravity compensated by the contact force
. One also notices from basic convex analysis that the complementarity relation can equivalently be rewritten as the inclusion into a normal cone, so that the bouncing ball dynamics is a differential inclusion into a normal cone to a convex set. See Chapters 1, 2 and 3 in Acary-Brogliato's book cited below (Springer LNACM 35, 2008). See also the other references on non-smooth mechanics.
, 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
.
As an unified simulation approach for hybrid system analysis, there is a method based on DEVS
formalism in which integrators for differential equations are quantized into atomic DEVS
models. This methods generates traces of system behaviors in discrete event system manner which is different from discrete time systems. Detailed of this approach can be found in references [Kofman2004] [CF2006] [Nutaro2010] and the software tool PowerDEVS
.
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
) and jump (described by a difference equation or control graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
). Often, the term "hybrid dynamic system" is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
, 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
Flow conditioning
Flow conditioning ensures that the “real world” environment closely resembles the “laboratory” environment for proper performance of inferential flowmeters like orifice, turbine, coriolis, ultrasonic etc.- Types of Flow :...
, 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 systemPhysical system
In physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
s with impact, logic-dynamic controllers
Controller (control theory)
In control theory, a controller is a device which monitors and affects the operational conditions of a given dynamical system. The operational conditions are typically referred to as output variables of the system which can be affected by adjusting certain input variables...
, and even Internet
Internet
The Internet is a global system of interconnected computer networks that use the standard Internet protocol suite to serve billions of users worldwide...
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 collisionInelastic collision
An inelastic collision, in contrast to an elastic collision, is a collision in which kinetic energy is not conserved.In collisions of macroscopic bodies, some kinetic energy is turned into vibrational energy of the atoms, causing a heating effect, and the bodies are deformed.The molecules of a gas...
. A mathematical description of the bouncing ball follows. Let
be the height of the ball and be the velocity of the ball. A hybrid system describing the ball is as follows:When
, flow is governed by,where
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
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 . Effectively, this describes the nature of the inelastic collision.The bouncing ball is an especially interesting hybrid system, as it exhibits Zeno
Zeno of Elea
Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...
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.
It is noteworthy that the dynamical model is complete if and only if one adds the contact force between the ground and the ball. Indeed, without forces, one cannot properly define the bouncing ball and the model is, from a mechanical point of view, meaningless. The simplest contact model that represents the interactions between the ball and the ground, is the complementarity relation between the force and the distance (the gap) between the ball and the ground. This is written as
Such a contact model does not incorporate magnetic forces, nor gluing effects (guess why!). When the complementarity relations are in, one can continue to integrate the system after the impacts have accumulated and vanished: the equilibrium of the system is well-defined as the static equilibrium of the ball on the ground, under the action of gravity compensated by the contact force
. One also notices from basic convex analysis that the complementarity relation can equivalently be rewritten as the inclusion into a normal cone, so that the bouncing ball dynamics is a differential inclusion into a normal cone to a convex set. See Chapters 1, 2 and 3 in Acary-Brogliato's book cited below (Springer LNACM 35, 2008). See also the other references on non-smooth mechanics.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 automatonHybrid automaton
In automata theory, a hybrid automaton is a mathematical model for precisely describing systems in which digital computational processes interact with analog physical processes. A hybrid automaton is a finite state machine with a finite set of continuous variables whose values are described by a...
, a hybrid program or a hybrid Petri net
Petri net
A Petri net is one of several mathematical modeling languages for the description of distributed systems. A Petri net is a directed bipartite graph, in which the nodes represent transitions and places...
. 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
Hybrid bond graph
A hybrid bond graph is a graphical description of a physical dynamic system with discontinuities . Similar toa regular bond graph, it is an energy-based technique...
.
As an unified simulation approach for hybrid system analysis, there is a method based on DEVS
DEVS
DEVS abbreviating Discrete Event System Specification is a modular and hierarchical formalism for modeling and analyzing general systems that can be discrete event systems which might be described by state transition tables, and continuous state systems which might be described by differential...
formalism in which integrators for differential equations are quantized into atomic DEVS
DEVS
DEVS abbreviating Discrete Event System Specification is a modular and hierarchical formalism for modeling and analyzing general systems that can be discrete event systems which might be described by state transition tables, and continuous state systems which might be described by differential...
models. This methods generates traces of system behaviors in discrete event system manner which is different from discrete time systems. Detailed of this approach can be found in references [Kofman2004] [CF2006] [Nutaro2010] and the software tool PowerDEVS
PowerDEVS
PowerDEVS is a general purpose software tool for DEVS [Zeigler76] modeling and simulation oriented to the simulation of hybrid systems. The environment allows defining atomic DEVS models in C++ language that can be then graphically coupled in hierarchical block diagrams to create more complex...
.
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
- PowerDEVSPowerDEVSPowerDEVS is a general purpose software tool for DEVS [Zeigler76] modeling and simulation oriented to the simulation of hybrid systems. The environment allows defining atomic DEVS models in C++ language that can be then graphically coupled in hierarchical block diagrams to create more complex...
A general purpose software tool for DEVS modeling and simulation oriented to the simulation of hybrid systems
See also
- Sliding mode controlSliding mode controlIn control theory, sliding mode control, or SMC, is a nonlinear control method that alters the dynamics of a nonlinear system by application of a discontinuous control signal that forces the system to "slide" along a cross-section of the system's normal behavior. The state-feedback control law is...
- Variable structure system
- Variable structure controlVariable structure controlVariable structure control, or VSC, is a form of discontinuous nonlinear control. The method alters the dynamics of a nonlinear system by application of a high-frequency switching control. The state-feedback control law is not a continuous function of time; it switches from one smooth condition to...
- Joint spectral radiusJoint spectral radiusIn mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years this notion has found applications in a large number of engineering fields and is still a topic of active research.-General description:The...