In the Laboratory for Information and Decision Systems (LIDS), we study fundamental questions related to modeling, analysis, and control of hybrid systems.
So far, this work has been carried out by Dr. Michael Branicky, working with Prof. Sanjoy Mitter. We have also collaborated with Prof. Vivek Borkar, who visits LIDS from time to time. Others in the lab who have an interest in hybrid systems include Prof. Munther Dahleh and graduate student Jorge Goncalves.
Hybrid systems involve both continuous-valued and discrete variables. Their evolution is given by equations of motion that generally depend on all variables. In turn these equations contain mixtures of logic, discrete-valued or digital dynamics, and continuous-variable or analog dynamics. The continuous dynamics of such systems may be continuous-time, discrete-time, or mixed (sampled-data), but is generally given by differential equations. The discrete-variable dynamics of hybrid systems is generally governed by a digital automaton, or input-output transition system with a countable number of states. The continuous and discrete dynamics interact at ``event'' or ``trigger'' times when the continuous state hits certain prescribed sets in the continuous state space.
Hybrid control systems are control systems that involve both continuous and discrete dynamics and continuous and discrete controls. The continuous dynamics of such a system is usually modeled by a controlled vector field or difference equation. Its hybrid nature is expressed by a dependence on some discrete phenomena, corresponding to discrete states, dynamics, and controls.
More Technical:
[Source:
Michael S. Branicky,
Studies in Hybrid Systems, Sc.D. thesis, MIT, June 1995]
The notion of dynamical system has a long history as an important conceptual
tool in science and engineering.
It is the foundation of our formulation of hybrid dynamical systems.
Briefly, a dynamical system is a system
Examples of dynamical systems abound, including autonomous ODEs, autonomous difference equations, finite automata, pushdown automata, Turing machines, Petri nets, etc. As seen from these examples, both digital and analog systems can be viewed in this formalism. The utility of this has been noted since the earliest days of control theory.
Briefly, a hybrid dynamical system is an indexed collection of dynamical systems along with some map for ``jumping'' among them (switching dynamical system and/or resetting the state). This jumping occurs whenever the state satisfies certain conditions, given by its membership in a specified subset of the state space. Hence, the entire system can be thought of as a sequential patching together of dynamical systems with initial and final states, the jumps performing a reset to a (generally different) initial state of a (generally different) dynamical system whenever a final state is reached.
A controlled hybrid dynamical system adds maps that allow us to make decisions about whether and where to jump whenever the state satisfies certain other conditions.
Some prominent researchers in the control theory area include
(but are not limited to!):
Panos Antsaklis,
Michael Branicky,
Roger Brockett,
Peter Caines,
Michael Heymann,
Wolf Kohn,
George Meyer,
Sanjoy Mitter,
Anil Nerode,
Peter Ramadge,
Shankar Sastry,
and
Pravin Varaiya.
Some prominent researchers in the computer science are include
(but are not limited to!):
Rajeev Alur,
Tom Henzinger,
Nancy Lynch,
Oded Maler,
Zohar Manna, and
Amir Pnueli.
(b) Check out some of Michael Branicky's Hybrid Systems publications.
(c) Check out the pages of the IEEE Working Group on Hybrid Dynamical Systems.