Nonlinear Impulsive and Hybrid Dynamical Systems
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Author(s)
Nersesov, Sergey G
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Abstract
Modern complex dynamical systems typically possess a
multiechelon hierarchical hybrid structure characterized by
continuous-time dynamics at the lower-level units and logical
decision-making units at the higher-level of hierarchy. Hybrid
dynamical systems involve an interacting countable collection of
dynamical systems defined on subregions of the partitioned state
space. Thus, in addition to traditional control systems, hybrid
control systems involve supervising controllers which serve to
coordinate the (sometimes competing) actions of the lower-level
controllers. A subclass of hybrid dynamical systems are impulsive
dynamical systems which consist of three elements, namely, a
continuous-time differential equation, a difference equation, and
a criterion for determining when the states of the system are to
be reset. One of the main topics of this dissertation is the
development of stability analysis and control design for impulsive
dynamical systems. Specifically, we generalize Poincare's
theorem to dynamical systems possessing left-continuous flows to
address the stability of limit cycles and periodic orbits of
left-continuous, hybrid, and impulsive dynamical systems. For
nonlinear impulsive dynamical systems, we present partial
stability results, that is, stability with respect to part of the
system's state. Furthermore, we develop adaptive control framework
for general class of impulsive systems as well as energy-based
control framework for hybrid port-controlled Hamiltonian systems.
Extensions of stability theory for impulsive dynamical systems
with respect to the nonnegative orthant of the state space are
also addressed in this dissertation. Furthermore, we design
optimal output feedback controllers for set-point regulation of
linear nonnegative dynamical systems. Another main topic that has
been addressed in this research is the stability analysis of
large-scale dynamical systems. Specifically, we extend the theory
of vector Lyapunov functions by constructing a generalized
comparison system whose vector field can be a function of the
comparison system states as well as the nonlinear dynamical system
states. Furthermore, we present a generalized convergence result
which, in the case of a scalar comparison system, specializes to
the classical Krasovskii-LaSalle invariant set theorem. Moreover,
we develop vector dissipativity theory for large-scale dynamical
systems based on vector storage functions and vector supply rates.
Finally, using a large-scale dynamical systems perspective, we
develop a system-theoretic foundation for thermodynamics.
Specifically, using compartmental dynamical system energy flow
models, we place the universal energy conservation, energy
equipartition, temperature equipartition, and entropy
nonconservation laws of thermodynamics on a system-theoretic
basis.
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Date
2005-06-23
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Dissertation