<inline-formula> <tex-math notation="LaTeX">$H_{\infty }$ </tex-math> </inline-formula> Control of Two-Time-Scale Markovian Switching Production-Inventory Systems

Abstract

Production-inventory systems are complex due to their multitime-scale, switching, and random fluctuation nature. How to reduce its bullwhip effects has been a main challenge and can be recast as an $H_{\infty }$ control problem. Hence, we model a transfer production line consisting of two facilities with different running time scales as a switched time-varying delay system. We introduce a Markovian chain to orchestrate the switching among the activation of different subsystems that is triggered by fluctuations of customers’ demand. We therefore study the $H_{\infty }$ control of two-time-scale production-inventory systems with Markovian jumping parameters and time-varying delays. To handle the different sampling rates of two types of facilities, lifting technique is introduced to eliminate the drastic ripples for the facility with fast-rate sampling. Based on the Lyapunov-Krasovskii theory, sufficient conditions are given such that the two-time-scale production-inventory system is exponentially stabilizable with an efficient attenuation of the bullwhip effects. A potassium carbonate production-inventory system is used to illustrate the effectiveness of the proposed methods.