Generalized homogeneous methodologies and new solutions to control problems of nonlinear systems

In the field of nonlinear control, homogeneous systems theory is a powerful tool for stability analysis and controller design of nonlinear systems, mainly owing to the nice properties established for homogeneous systems. However, the applicability of homogeneous systems theory is largely limited since the extant definition of homogeneity is very restrained. In practice, more and more nonlinear models are being developed to capture the complexity of real phenomena in control systems, which are not necessarily homogeneous and therefore cannot be handled by the traditional homogeneity-based approaches. This dissertation focuses on the development of generalized homogeneous methodologies and their applications to general nonlinear systems by relaxing the restriction imposed on the uniform homogeneous degree. In this way, the methodologies obtained in this dissertation will provide new strategies to simplify the stabilizer design process and offer new solutions to analyze and control a wider class of nonlinear systems which cannot be handled by the existing approaches.

This dissertation presents a series of methodologies and solutions to control problems of nonlinear systems. Firstly, a polynomial Lyapunov function and a continuously differentiable (C1) global stabilizer with detailed design schemes are recursively constructed based on a modified definition of homogeneity, namely homogeneity with monotone degrees (HWMD). Secondly, for a class of 3-dimensional nonlinear systems, existence condition and explicit design method of smooth (C infinity) controllers are proposed. Several special cases are investigated to show the advantages of HWMD over the existing approaches based on the traditional homogeneity in designing C1 and C infinity controllers. Thirdly, by adopting and modifying HWMD, a new constructive Lyapunov function and a globally stabilizing small controller are recursively designed in a bottom-to-top manner. This new method not only encompasses the existing results but also achieves more general results for global stabilization of upper-triangular systems. Finally, extensions and applications including finite-time stabilizer design and global stabilization of time-delayed systems are discussed.

The significance of this research lies in the development of generalized homogeneous methodologies and more importantly, providing new solutions to control problems of nonlinear systems, particularly in the following directions: (i) continuously differentiable and smooth controllers are constructed by utilizing the flexibility from using HWMD; (ii) a wider class of nonlinear systems, which cannot be handled by existing results, now can be stabilized by using HWMD; (iii) finite-time stability is achieved for a class of nonsmooth systems; and (iv) time-delayed systems can be stabilized by using HWMD-based controllers.