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

This article is concerned with the stability and
$L_{1}$
-gain analysis of periodic piecewise positive systems with constant time delay.
$\lambda$
-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
$L_{1}$
-gain of the systems. By taking advantage of the periodic property of the system, linear inequalities are employed to characterize the
$L_{1}$
-gain, and an unweighted upper bound of the
$L_{1}$
-gain of the system is given.