Doctor of Philosophy / Department of Mechanical and Nuclear Engineering / Warren N. White / There are a number of different types of mechanical systems which can be termed as
underactuated. The degrees of freedom (DOF) of a system are defined by the system’s number of
independent movements. Underactuated mechanical systems have fewer actuators than DOF.
Some examples such as satellites, air craft, overhead crane loads, and missiles have at least one
unactuated DOF.
The work presented here develops a nonlinear control law for the asymptotic stabilization
of underactuated systems. This is accomplished by finding the solution of matching conditions
that arise from Lyapunov’s second method, analogous to the dissipation of energy. The direct
Lyapunov approach (DLA) offers a wide range of applications for underactuated systems due to
the fact that the algebraic equations, ordinary differential equations, and partial differential
equations stemming from the matching conditions are more tractable than those appearing in
other approaches.
Two lemmas of White et al. (2007) are applied for the positive definiteness and
symmetry condition of the KD matrix which is used to define an analogous kinetic energy for the
system. The defined KD matrix and the Lyapunov candidate function are developed to ensure
stability. The KD matrix is analogous to the mass matrix of the dynamic system. The candidate
Lyapunov function, involving the analogous kinetic energy and an undefined potential of the
generalized position coordinates, is presented. By computing the time derivative of the
Lyapunov candidate function, three equations called matching conditions emerge and parts of
their solution provide the nonlinear control law that stabilizes the system.
This dissertation presents the derivation of the DLA, provides a new method to solve the
first matching condition (FMC), and shows the tools for the control law design. The stability is
achieved from the proper shape of the potential, the positive definiteness of the KD matrix, and
the non-positive rate of change of the Lyapunov function. The ball and beam, the inverted
pendulum cart, and, a more complicated system, the ball and arc are presented to demonstrate the
importance of the results because the methods to solve the matching equations, emerging from
the system examples, are simple and easier. The presented controller design formulation satisfies
the FMC exactly without introducing control law terms that are quadratic in the velocities or
approximations. This methodology allows the development of the first nonlinear stabilizing
control law for the ball and arc system, a simple and effective formulation to find a control law
for the inverted pendulum cart, and a stabilizing control of the ball and beam apparatus without
the necessity of approximations to solve the FMC. To illustrate the formulation, the derivation is
performed using the symbolic manipulation program Maple and it is simulated in the
Matlab/Simulink environment.
The dissertation on the solvability of the first matching condition for stabilization is
organized into six different chapters. The introduction of the problem and the previous
approaches are presented in Chapter 1. Techniques for solving of the first matching condition, as
well as the limitations, are provided in Chapter 2. The application of this general strategy to the
ball and beam system appears in Chapter 3. Chapter 4 and 5 present the application of the
method to the ball and arc apparatus and to the inverted pendulum cart, respectively. The
difficulties for each application are also presented. Particularly, Chapter 5 shows the application
of the produced material to obtain an easier formulation for the inverted pendulum cart compared
to previous published controller examples. Finally, some conclusions and recommendations for
future work are presented.
| Identifer | oai:union.ndltd.org:KSU/oai:krex.k-state.edu:2097/13736 |
| Date | January 1900 |
| Creators | Garcia Batista, Deyka Irina |
| Publisher | Kansas State University |
| Source Sets | K-State Research Exchange |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
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