Stability and hybrid L2×ℓ2 gain analysis of a class of impulsive switched systems

This work investigates the stability and the hybrid L₂×ℓ₂ gain properties of impulsive switched systems. Our study is motivated by a numerical example, which shows that a single switch can cause a large change in the L₂ gain of a switched system compared with the L₂ gains of its subsystems. Next, our main theorem presented and proven in this work finds a large class of switching signals, uniformly over which an impulsive switched system is globally exponentially stable with no input. Meanwhile, upper bounds on its L₂×ℓ₂ gain from the continuous and discrete inputs to the mixed output can also be found by our theorem, which are related to the L₂ gains of its continuous-time subsystems and ℓ₂ gains of its discrete-time subsystems. Finally, we apply our theoretical result on the H_∞ control of linear systems subjected to Denial-of-Service attack and persistent resetting. Both the theory and simulations show that when the control is not intermittently lost “too often”, or when the system is frequently resetting, the stability of the closed-loop system is preserved.