Stability of delayed systems: delay-compensatory impulsive control

This paper investigates the stability characteristics of systems exhibiting delayed and unstable dynamics, along with destabilizing impulse gains, using a method called delay-compensatory impulsive control. A new control strategy, termed delay-compensatory impulsive control, is introduced, which incorporates and utilizes impulse delays to counteract the instability induced by unstable impulse gains. Initially, a compensation criterion is formulated, taking into account both unstable impulse gains and impulse delays concurrently. Subsequently, a modified Halanay impulsive delay inequality with real impulse gain is proposed. Next, sufficient conditions for achieving exponential stability in delayed impulse systems are established, employing the aforementioned impulse delay compensation criterion and the relaxed Halanay inequality. Finally, the comparative simulation not only illustrates the advantages of the proposed control algorithm but also proves that the impulse delay can be used to mitigate the adverse effects of unstable impulse gain on the controlled dynamic system.