New Stability Analysis Results on Switched Systems and Fuzzy Systems

Abstract

The switched systems and fuzzy systems are common models to describe the non-linear dynamics in practical engineering. And the problem of stability analysis of those systems is of paramount importance. In this thesis we intend to analyze the relationship between some existing methods of stability analysis on switched systems and fuzzy systems, and investigate new approaches to improve these results in term of simplicity and conservativeness. The main content of this thesis can be divided into two branches: stability analysis of second-order switched systems and stability analysis of Takagi-Sugeno (T-S) fuzzy systems.
On the topic of stability analysis of switched systems, we will start from the Lyapunov theory and then propose the concept of phase function. By exploring the properties of phase function, the phase-based stability analysis will be made. And a unified approach for the stability analysis of second-order switched system will be obtained. Compared with existing works, the stability condition obtained here shows advantages in terms of theoretical analysis and numerical computation.
In addition, we will use the phase-based method to see whether the existence of common quadratic Lyapunov functions (CQLFs) for every pair of subsystems can ensure the overall system stability. With additional properties of phase function, the previous stability condition can be further extended. By analysing the equivalent algebraic and linear matrix inequality (LMI) expression of such a new condition, it can be verified that the existence of CQLF for every pair of subsystems will be sufficient for the system stability.
Regarding the stability analysis of T-S fuzzy systems, we will describe the distri-bution of membership functions in a unified membership space. In this way, a graphical approach is provided to analyse the conservativeness of membership-dependent stability conditions. Following this idea, we will use membership function extrema to construct a simple and tighter convex polyhedron, which encloses the membership trajectory and generates less conservative LMI condition.
As an application of the above membership-dependent analysis method, we will investigate the stability problem of a reconfigurable metamorphic palm control system based on its T-S fuzzy model. Firstly the obtained dynamic model of the metamorphic palm will be transformed to a T-S fuzzy model. Then the palm dy-namic boundaries will be used as extrema to reduce the conservativeness in palm controller design and ensure a wider range of palm re-configuration operations.
Overall, the research in this thesis provides a basic theoretical review of existing stability analysis approaches on specific switched systems and fuzzy systems, and novel approaches have been proposed as an improvement. The results will be of potential use for researchers in the area of systems engineering.

Date of Award	2019
Original language	English
Awarding Institution	King's College London
Supervisor	Hak-Keung Lam (Supervisor) & Kaspar Althoefer (Supervisor)