Differentiable dynamical systems.

This dissertation concerns the dynamics and control of an autonomous underwater
vehicle (AUV) which uses internal actuators to stabilize its horizontalplane
motion. The demand for high-performance AUVs are growing in the field of
ocean engineering due to increasing activities in ocean exploration and research.
New generations of AUVs are expected to operate in harsh and complex ocean environments.
We propose a hybrid design of an underwater vehicle which uses internal
actuators instead of control surfaces to steer. When operating at low speeds or in
relatively strong ocean currents, the performances of control surfaces will degrade.
Internal actuators work independent of the relative
ows, thus improving the maneuvering
performance of the vehicle.
We develop the mathematical model which describes the motion of an underwater
vehicle in ocean currents from first principles. The equations of motion of a
body-fluid dynamical system in an ideal fluid are derived using both Newton-Euler
and Lagrangian formulations. The viscous effects of a real fluid are considered separately.
We use a REMUS 100 AUV as the research model, and conduct CFD simulations to compute the viscous hydrodynamic coe cients with ANSYS Fluent. The
simulation results show that the horizontal-plane motion of the vehicle is inherently
unstable. The yaw moment exerted by the relative flow is destabilizing.
The open-loop stabilities of the horizontal-plane motion of the vehicle in
both ideal and real fluid are analyzed. In particular, the effects of a roll torque and
a moving mass on the horizontal-plane motion are studied. The results illustrate
that both the position and number of equilibrium points of the dynamical system
are prone to the magnitude of the roll torque and the lateral position of the moving
mass.
We propose the design of using an internal moving mass to stabilize the
horizontal-plane motion of the REMUS 100 AUV. A linear quadratic regulator
(LQR) is designed to take advantage of both the linear momentum and lateral position
of the internal moving mass to stabilize the heading angle of the vehicle. Alternatively,
we introduce a tunnel thruster to the design, and use backstepping
and Lyapunov redesign techniques to derive a nonlinear feedback control law to
achieve autopilot. The coupling e ects between the closed-loop horizontal-plane
and vertical-plane motions are also analyzed.

Ban and Kalies [3] proposed an algorithmic approach to compute attractor-
repeller pairs and weak Lyapunov functions based on a combinatorial multivalued
mapping derived from an underlying dynamical system generated by a continuous
map. We propose a more e cient way of computing a Lyapunov function for a Morse
decomposition. This combined work with other authors, including Shaun Harker,
Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes
the process of nding a global Lyapunov function for Morse decomposition very e -
cient. One of the them is to utilize highly memory-e cient data structures: succinct
grid data structure and pointer grid data structures. Another technique is to utilize
Dijkstra algorithm and Manhattan distance to calculate a distance potential, which is
an essential step to compute a Lyapunov function. Finally, another major technique
in achieving a signi cant improvement in e ciency is the utilization of the lattice
structures of the attractors and attracting neighborhoods, as explained in [32]. The
lattice structures have made it possible to let us incorporate only the join-irreducible
attractor-repeller pairs in computing a Lyapunov function, rather than having to use
all possible attractor-repeller pairs as was originally done in [3]. The distributive lattice structures of attractors and repellers in a dynamical
system allow for general algebraic treatment of global gradient-like dynamics. The
separation of these algebraic structures from underlying topological structure is the
basis for the development of algorithms to manipulate those structures, [32, 31].
There has been much recent work on developing and implementing general compu-
tational algorithms for global dynamics which are capable of computing attracting
neighborhoods e ciently. We describe the lifting of sublattices of attractors, which
are computationally less accessible, to lattices of forward invariant sets and attract-
ing neighborhoods, which are computationally accessible. We provide necessary and
su cient conditions for such a lift to exist, in a general setting. We also provide
the algorithms to check whether such conditions are met or not and to construct the
lift when they met. We illustrate the algorithms with some examples. For this, we
have checked and veri ed these algorithms by implementing on some non-invertible
dynamical systems including a nonlinear Leslie model.