


  Lyapunov functions 

  A solution to threedimensional findpath problem Author: Sharma, Bibhya Nand. Institution: University of the South Pacific. Award: M.Sc. Subject: Lyapunov functions , Stability Date: 1998. Call No.: Pac QA 871 .S47 1998 BRN: 731241 Copyright:Under 10% of this thesis may be copied without the authors written permission Abstract: Alexander Mikhailovich Liapunov (18571918) was a Russian mathematician and mechan ical engineer, who in 1892, published The General Problem of the Stability of Motion. In his classical memoir, he outlined two approaches for stability of dynamical systems. These are popularly known as Liapunov's first method and Liapunov's second method. The first method depends on finding approximate solutions of the differential equations. Whereas the second method requires no such knowledge of the solutions and it provides valuable qualitative information regarding stability of the equilibrium state of the system. Hence, it is an ideal approach to study stability of realistic models that usually involve large amplitudes of motion and contains several equilibria. In the general form, these systems are only rarely tractable and finding exact solutions of such models are literally impossi ble. Consequently one must either rely on approximations, recourse to partial/complete linearization or generate solutions numerically. However, in many practical situations, finding approximated solutions can be as difficult as finding the exact ones. Lineariza tion may not predict realistically, the true behaviour of the original system, although complete linearization has been very common with the mathematicians in formulating physical laws. Also while we can witness present events by generating numerical solutions via computer for example, we really do not have any knowledge of the future events. Liapunov's direct method provides an alternative way of analysing realistic models and also enables us to examine the qualitative, rather than the quantitative behaviour of the solution. Consequently, linearization of nonlinear dynamical systems is not required. The method provides us with a special function called the Liapunov function with which we can explore the whole class of solutions; for example those that precisely converge to the equilibrium points (asymptotic stability) and ones that diverge and leave the defined boundary of stability (instability). We would then be able to confirm our predictions by using a computer to process the available information. In this thesis, the problem of generating collisionfree trajectories of point objects moving to their targets in a closed "busy" setting is considered. This problem is known as the findpath problem. Research on twopoint systems in a horizontal plane using Liapunov's direct method has been considered by Stonier (1990) and Vanualailai et. al. (1994, 1995, 1998). One of the major aims of this research is to further extend the capabilities of the Liapunov function constructed by Vanualailai and Ha (1998) for collisionfree paths of three point objects. Next, we would like to generalise the Liapunov function so that it works efficiently for an npoint system. Hence, ideally we can have a workspace clut tered with moving objects that reach their targets, while avoiding many other fixed and moving obstacles in their paths. The second major aim of this research is to consider threedimensional systems rather than the conventional twodimensional systems. This generates a realistic picture of the solutions of the system and it will be easier to under stand the related problems for future research with minimal restrictive measures. The outline of this thesis is as follows: Chapter 1 deals primarily with a brief discussion of the stability of dynamical systems, introduction to Liapunov's methods, a detailed review of the history of "findpath problem" and progress made in the area through Liapunov's second method, and finally definitions of stability via the Liapunov function. Chapter 2 considers the dynamics of two point objects or masses in threedimensional space. A Liapunov function is constructed that guarantees the stability of the system. Controllers are extracted from the Liapunov function that control the objects and maneu ver them, amongst the fixed and moving antitargets, to their designated target centers. Simulations are taken that show the applicability of the method. Chapter 3 considers the dynamics of three point objects in threedimensional space. The approach is similar to Chapter 2, nevertheless, many of the important results appear. In the second part of the chapter, two extra fixed antitargets are incorporated into the sys tem to show the additional capabilities of the Liapunov function. Many more simulations are considered to show the applicability of the Liapunov method. In Chapter 4, the Liapunov function from Chapter 3 is generalised to work for an n point system. Generalised controllers are constructed for controlling the trajectory of the ith object. Chapter 5 discusses the effects of two essential parameters associated with the findpath problem, namely, control and the convergence parameters. Various changes to the values of each parameter is seen and analysed. It also looks briefly at RK4 method which was used to generate data for the collisionfree paths. Finally, Chapter 6 deals with the discussion of the research findings and how it may be useful for research in the future. It also gives an account of the difficulties faced and suggestions on how we can improve to obtain better results.


