Solvability of the direct Lyapunov first matching condition in terms of the generalized coordinates

Abstract

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.