Hybrid system are of
interest to a diverge group of researchers. Research topics include the
mathematics and specification part, the verification and simulation implementation
of these formal notions, and the physical semantics leg which tries to
put the mathematical and implementation activity in a sound physical framework.

A mathematical paradigm for hybrid
systems relies on the dynamic system notion of flows in phase space. An
excellent book on dynamic systems is *Differential
Equations, Dynamical Systems, and Linear Algebra* by Hirsh and
Smale. The hybrid dynamic system paradigm is well phrased by Guckenheimer
and Johnson in their "Planar Hybrid Systems"
article in the Hybrid Systems III issue. Discrete transitions occur in
between continuous modes of operation when state variables reach threshold
values. These transitions may discontinuously change the field gradient
that determines flow and may cause discontinuous jumps in the state vector
values. Several implementations
of this hybrid dynamic systems modeling paradigm have been developed, either
from the discrete systems perspective or from the continuous systems point
of view.

Discrete system behavior is well
captured by finite state machines and these can be extended to include
continuous behavior. Initially, this resulted in timed automata, which
contain clocks that can be reset to 0 to constitute the continuous dimension.
The resulting continuous behavior, therefore, is monotonously increasing
with a constant rate and can be reset at discrete events. As a next step,
hybrid automata were introduced that do not enforce the monotonicity constraint
and allow clocks with different rates. These formalisms are especially
useful in analyzing synchronization and communication behavior. For more
information, refer to Alur, Courcoubetis, Henzinger, Halbwachs, Ho, Nicollin,
Olivero, Sifakis, and Yovine, "The Algorithmic Analysis
of Hybrid Systems" in *Theoretical Computer Science.*

In the continuous systems realm,
there is a common mathematical framework in the form of differential equations,
possibly extended by algebraic constraints (so called DAEs). State of the
art modeling packages apply this formalism to handle large continuous models.
To cope with the modeling complexity, hierarchy and object-oriented notions
such as inheritance are applied. Presently, the international Modelica
initiative aims to standardize the object-oriented model specification
language. Models are specified as noncausal interactions between objects,
and a compilation stage flattens the model structure and compiles a causal
DAE. Noncausal modeling is first developed by the bond graph formalism
and a good book on the subject is *System Dynamics:
A unified approach* by Karnopp, Margolis, and Rosenberg. In the
DAE framework, discrete mechanisms are introduced by conditionally (de)activating
algebraic constraints, and, in some cases, some of the differential equations.
This hybrid DAE formalism is powerful in modeling complex dynamic system
behavior.

By nature, hybrid automata are well
suited for verification purposes. To deal with complex continuous behavior,
the continuous phase space is discretized into piecewise linear behaviors.
In this context, linear refers to the constant rate of change constraint
as opposed to the classical linear systems notion where dynamic behavior
typically is exponential (i.e., linearity refers to the solutions not the
underlying equations). An excellent book on systems theory is *Introduction
to Dynamic Systems: Theory, Models, and Applications* by Luenberger.
The piecewise linearization of a continuous phase space is studied in detail
by Stursberg, Kowalewski, Hoffmann, and Preußig in their *Hybrid
Systems IV* article "Comparing Timed and Hybrid
Automata as Approximations of Continuous Systems".

Hybrid DAEs are a typical simulation
formalism. Along with a continuous numerical approximation, root-finding
to locate the discrete event time, event iteration to handle sequences
of discrete events, and reinitialization needs to be handled. Cellier describes
this in his thesis Combined Continuous/Discrete System
Simulation by Use of Digital Computers: Techniques and Tools*.
*Discontinuities that result from model abstraction (e.g., bouncing
ball) can be conveniently modeled by hybrid DAEs as small, localized, conditional
constraints. The elaborate discrete control mechanisms that are part of
embedded control systems can be modeled by Petri nets, which requires a
deterministic implementation.

The simulation and verification
research takes a control perspective and relies on 'good models'. Such
models are obtained by including physical constraints as developed in the
field of thermodynamics. An excellent book (though in German) on the subject
is *Energie und Entropie: Eine Einfuhrung in die
Thermodynamik* by Falk and Ruppel. The resulting physical semantics
for hybrid dynamic systems enables one to analyze mathematical singularities
such as described by Utkin in *Sliding Modes in
Control and Optimization*. Bond graphs apply thermodynamic concepts
to provide a systematic modeling approach from physical concepts all the
way to a mathematical representation that inherently enforces conservation
of energy and continuity of power. Much work on including discontinuous
phenomena in this theoretical modeling framework is done by Mosterman and
Biswas, which resulted in Mosterman's dissertation *Hybrid
Dynamic Systems: A hybrid bond graph modeling paradigm and its application
in diagnosis.*

Finally, a lot of research is conducted
to develop formal models for powerful hybrid system specification formalisms.
Branicky presents a thorough study of this in his dissertation *Studies
in Hybrid Systems*. This interesting strand of research has close
ties with the formal methods community, which is an emerging factor in
system specification for verification and, ultimately, automated code generation.
This then addresses the eventual synthesis goal where an embedded control
system is formulated as a system of equations with unknown control variables
that are solved algorithmically.

In my opinion this describes the
texture of the research performed by the hybrid dynamic systems community.
A good indication of where we stand is the annual Hybrid System workshop
of the IEEE Technical Committee on Hybrid Dynamical Systems. At present, we have
moved beyond solving small, illustrative, problems and hybrid system methodologies
are deployed in such projects as real-time hardware in the loop simulation
and consumer electronics control verification. A suite of simulation and
verification tools is readily available in the public domain.

Future research will apply hybrid
dynamic system concepts and notions to increasingly complex systems for
analysis and synthesis purposes based on sound theoretical principles.
I look forward to new developments, enjoy being part of it, and hope this
site contributes to advancing the state of the art!

Pieter J. Mosterman