On the stability of Markov feedback loops of a microsystem

In the field of control engineering, there has been a growing emphasis on systems with feedback loops that involve stochastic factors. Simultaneously, with the advancements in probability theory, feedback loops with stochastic processes have gained practicality due to their customizability and the potential for practical applications. Essentially, the effect of adding additive random signals in linear feedback systems can be viewed as the introduction of filtered stochastic noise. However, when random signals enter the feedback loop multiplicatively, a richer set of state behaviors emerges. This study utilizes multiplicative feedback loops to construct Markov chains and provides a probabilistic modeling approach to investigate the stability and asymptotic behavior of Markov feedback loops, offering insights for microsystems. This paper derives properties of such stochastic processes and provides corresponding theoretical proofs. The theoretical foundation of this probabilistic modeling can be applied in economics, biological variation processes, epidemic control predictions, and linear perturbation models, offering a theoretical perspective for feedback systems with uncertain triggers.