Explicit solutions for a class of inherently nonlinear systems and applications to controller design
In practice, more and more nonlinear models are being developed to capture the complexity of real phenomena in control systems. It is well-preconized that nonlinear systems are vastly more difficult to analyze than linear ones. In the nonlinear regime, many of the most basic questions remain unanswered. Among them, the problem of finding the explicit formula of a solution for a nonlinear system is very difficult to solve because the linear superposition is no longer available for linear systems. On the other hand, explicit solutions to the nonlinear equations are of fundamental importance for behavior analysis. In this thesis, we develop new theorems for property analysis and tools for controller design for some inherently nonlinear systems.
In this thesis, we first develop a method based on homogeneous systems theory to systematically solve explicit solution for some 2D nonlinear systems. The sufficient the conditions for the the existence of explicit solutions are given for systems with zero and positive homogeneous degrees. By comparing the explicit solution with the numerical solution, we show that explicit solutions will eliminate the approximation error which is accumulated by the numerical method and becomes larger and larger as the time elapses. Second, we apply the use of explicit solutions to design controllers which will yield explicit solutions for the nonlinear systems. The availability of the explicit solutions will have the advantages for verifying the performance of the controller. Finally, we extend our theories to construct explicit solutions for higher-dimensional dynamic systems whose solutions are asymptotically stable.
The systematic approaches for constructing the explicit solutions of some nonlinear systems developed in this thesis provides us new means for analyzing system properties, which were inadequately addressed earlier due to the lack of useful linearizations. With further research in this direction, more powerful tools for behavior analysis and controller design can be achieved.