Hybrid systems exhibit dynamics which evolve both continuously and
in discrete jumps. Such behavior arises in many contexts, both in
man-made systems and in nature. Continuous systems which have a phased
operation,
such as insect motion and biological cell growth and division, are
well-suited to be
modeled as hybrid systems, as are
continuous systems which are controlled by a discrete logic, such as
a chemical plant controlled with valves and pumps, or the
autopilot modes for controlling an aircraft. Hybrid systems are
also natural models for systems comprised of many interacting
subsystems or processes, such as air or ground transportation systems.
In these examples, the continuous dynamics model system motion, biochemical
or chemical reactions, while the discrete dynamics model the sequence of
contacts or collisions in the gait cycle,
cell divisions, valves and pumps switching, and coordination
protocols. In all of these examples, the system dynamics are complex enough
that traditional analysis and control methods are not computationally
feasible.
To understand the behavior of hybrid systems, to simulate, and
to control these systems, theoretical advances and analytical tools are
needed.
In this talk, I will present a method that we have designed
for analyzing and controlling hybrid systems, and will focus on the
numerical methods that we are now developing to achieve efficient
computation
of the control law. These methods will be presented in the context of
two applications: flight management system design, and the design
of collision avoidance maneuvers for automated air traffic control.
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