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This article is concerned with the stability and <inline-formula><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula>-gain analysis of periodic piecewise positive systems with constant time delay. <inline-formula><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&#x2013;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&#x2013;Krasovskii functional to characterize the <inline-formula><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><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula>-gain, and an unweighted upper bound of the <inline-formula><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula>-gain of the system is given.

ieee transactions on automatic control

Stability and <inline-formula><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula>-Gain Analysis of Periodic Piecewise Positive Systems With Constant Time Delay

Stability and $L_{1}$-Gain Analysis of Periodic Piecewise Positive Systems With Constant Time Delay