Finite-Time Stabilization of Linear Systems with Time-varying Delays using New Integral Inequalities

Chantapish Zamart, Thaned Rojsiraphisal

Abstract

In this research we propose stability conditions for guaranteeing finite-time stabilization of linear systems with time-varying delay. Based on Lyapunov theory, improved finite-time stability and stabilization criteria of the linear systems are formulated in the form of linear matrix inequalities. To obtain the improved stability criteria, we first propose  two new  inequalities in the forms of free-matrix based inequality for bounding the integral of the form$\int_{t-d_2}^{t-d_1} e^{\alpha(t-s)} \dot{x}^T (s) R \dot{x}(s)ds$ which is occurred in the derivative of Lyapunov-Krasovskii functional. By desiring a proper state-feedback controller, the non-linear terms occuring in the formulation can be managed without defining new variables.  At the end, two  numerical examples are given to show that the new criteria are praticable and can be applied to the case of continuous but not differentiable delay function.