Stability and $L_{1}$-Gain Analysis of Periodic Piecewise Positive Systems With Constant Time Delay
Bohao Zhu, James Lam, Xiaochen Xie, Xiaoqi Song, Ka‐Wai Kwok
Abstract
This article is concerned with the stability and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> -gain analysis of periodic piecewise positive systems with constant time delay. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\lambda$</tex-math></inline-formula> -exponential stability, which is applied to characterize the decay rates of the considered systems, is investigated first. A copositive Lyapunov–Krasovskii functional is used to obtain a sufficient stability condition. The stability condition characterizes the convergent speed of the state by the system matrices and the size of the time delay. One can also apply the Lyapunov–Krasovskii functional to characterize the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> -gain of the systems. By taking advantage of the periodic property of the system, linear inequalities are employed to characterize the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> -gain, and an unweighted upper bound of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula> -gain of the system is given.