Control of Piecewise Smooth Systems: Generalized Absolute Stability and Applications to Supercavitating Vehicles
Control of Piecewise Smooth Systems: Generalized Absolute Stability and Applications to Supercavitating Vehicles
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Date
2008-04-28
Authors
Lin, Guojian
Advisor
Abed, Eyad H.
Balachandran, Balakumar
Balachandran, Balakumar
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DRUM DOI
Abstract
Many systems in engineering applications are modeled as piecewise
smooth systems. The piecewise smoothness presents great challenges
for stability analysis and control synthesis for these systems.
Over the years, the theory of absolute stability has been one of
the few tools developed by control theory researchers to meet
these challenges. For systems in which the nonlinearity is known
to be bounded within certain sectors, many stability and control
problems can be addressed using results from absolute stability
theory.
During the last few decades, many important advances have been
made in the study of the absolute stability. In these studies, it
is commonly assumed that the sector bound for the system
nonlinearity is \textit{symmetric} with respect to the origin in
state space. However, in many practical engineering systems, the
nonlinearity does not satisfy such a symmetry assumption. To study
stability and control problems for these systems, in this work the
author studies generalized absolute stability problems involving
\textit{asymmetric} sector bounds. Nonlinear systems with Lure'
structure are considered. For second-order systems, conditions
that are both necessary and sufficient for generalized absolute
stability are obtained. These conditions can be easily tested in
engineering applications. For general finite-order systems,
sufficient conditions are provided for generalized absolute
stability. The derived conditions may be easily tested by using
numerical tools for linear programming. With the generalizations
in this work, absolute stability theory becomes a more powerful
tool in the sense that it applies to an extended class of
piecewise smooth systems in which the nonlinearities can be
asymmetric with respect to the state variables.
This work includes general theoretical questions as well as
detailed investigations of an application to models of
supercavitating vehicles. For these high-speed underwater
vehicles, the dive-plane motion is naturally modeled as a
piecewise smooth system with a dead zone. The strong nonlinear
planing force plays an important role in determining the
dive-plane dynamics. To design control laws that stabilize the
dive-plane motion, the necessary and sufficient condition for
generalized absolute stability of second-order systems is applied
to a reduced-order model obtained through the backstepping control
approach. The obtained sufficient conditions for generalized
absolute stability of finite-order systems can also be
successfully applied for stabilizing the dive-plane motion. In
comparison with alternative control approaches, control designs
with the aid of theoretical findings in generalized absolute
stability lead to stability that is robust to the modeling errors
in the nonlinearity such as the magnitude, local slope and the
dead zone location.
The dissertation also includes basic results on bifurcation and
bifurcation control of supercavitating vehicles. The presence of
bifurcations in the dive-plane dynamics is demonstrated, and
control techniques for modifying the bifurcation behavior to
improve the vehicle dynamic performance are developed. These
results complement the absolute stability results to give a more
complete picture of the dynamics and control of supercavitating
vehicles.