Stability analysis of dynamic nonlinear systems by means of Lyapunov matrix-valued functions

Abstract

The method of Lyapunov matrix-valued functions is critically examined for its capability, applicability and overall functionality in the adequate construction and development of an appropriate Lyapunov function for the stability analysis of dynamic nonlinear systems. This method provides an analytical methodology of Lyapunov function construction by effectively exploiting indispensible information relating to the internal dynamics of the nonlinear system, gained by means of hierarchical nonlinear system decomposition. While relatively computationally intensive in its application, when compared to traditional scalar Lyapunov function construction techniques, as well as to vector Lyapunov function approaches, in terms of practical applicability and the successful acquirement of a suitable Lyapunov function, it is found that the method of Lyapunov matrix-valued functions outperforms its predecessors for both linear and nonlinear dynamic systems. Furthermore, in order to present a comprehensive investigation and analysis of the researched methodology, a linear system simplification is proposed, and two variations on the Lyapunov matrix-valued function method are also put forward. A critical analysis of the investigated technique ensues, whereby both its virtues and weaknesses in terms of practical applicability and relative improvement on pre-existing techniques are highlighted. Finally, the stability of a practical, real-world case study is analysed, namely, the Buckling Beam system, where it is found that the combination of the Lyapunov matrix-valued function theory development, as well as its extension to practical, real-world applications