Stability and Finite-time Stability of Dynamical Systems with Time-delay

Abstract

In this thesis, we study the problem of finite-time stability and finite-time boundedness of dynamical systems with time-delay. Firstly, we address problem formulation for a class of linear switched singular systems with time-delay and exogenous disturbance for both continuous-time and discrete-time cases. By using the state-space singular value decomposition and monomial coordinate transformation methods, we propose necessary and sufficient conditions for the positivity of the considered systems. Then, we design a class of quasi-alternative switching signals to analyze the switching behaviors of the considered systems consisting of stable (bounded) and unstable (unbounded) subsystems. Based on a mode-dependent average dwell time (MDADT) switching, composing of a slow mode-dependent average dwell time (SMDADT) switching and a fast mode-dependent average dwell time (FMDADT) switching, the underlying systems whose subsystems are stable (bounded) and unstable (unbounded) can be stabilized in a finite-time interval. By establishing novel copositive Lyapunov-Krasovskii functionals and adopting the MDADT switching strategy, we formulate new delay-dependent sufficient conditions guaranteeing finite-time stability of the considered systems in terms of linear vector inequalities and linear matrix inequalities. Finally, we consider finite-time boundedness for a class of linear switched singular systems with time-delay and exogenous disturbance for both continuous-time and discretetime cases. By utilizing the same technique mentioned above, we also derive new delaydependent criteria ensuring finite-time boundedness of the considered systems in terms of linear vector inequalities and linear matrix inequalities.