Stabilization of Time Scale-Type Nonlinear Systems With Stochastic Impulses

Different from continuous-time or discrete-time (CoD-T) systems, time scale-type systems (TTSs) can operate in a continuous-discrete mode, bridging the gap between continuous and discrete states. As a result, conventional methods such as the average impulsive interval (AII) and the impulsive density function (IDF) are not suitable for characterizing the frequency of impulses (FoI) required for stability analysis of TTSs with impulses. This is due to the possibility that all impulses may fall into the complementary set of time scales. In light of this, the number of impulses satisfying AII or IDF may not be sufficient to ensure stability. In response to this challenge, we introduce a novel definition in this paper, termed time scale-type impulsive density (TTID). The key feature of TTID lies in the consideration of the graininess function, which is determined by time scales, allowing for a more accurate characterization of the number of impulses required for the stability of the TTSs. Employing the proposed TTID, the criteria of asymptotical stability are presented for TTSs with stochastic and Markov-type impulses, respectively. The results presented in this paper encompass and generalize the discoveries of some existing CoD-T impulsive systems, which can be regarded as special cases of our results. We shed new light on the advantages of the TTID and the effectiveness of the theoretical results by an application of voltage control for micro-grids.