Inventory Control of Switched Production Systems: LMI Approach

Abstract

This paper focuses on the problem of inventory control of production systems. The main contribution of the paper is that, for the first time, production systems are modeled as switched linear systems and the production problem is formulated as a switched $H\infty$ control problem with a piecewise-affine control law. The switching variable for the production systems modeled in this paper is the stock level. When the stock level is positive, some of the produced parts are being stored. The stocked parts may deteriorate with time at a given rate. When the stock level is negative it leads to backorders, which means that orders for production of parts are coming in and there is no stocked parts to immediately meet the demand. A switched linear model is used and it is shown that the inventory control problem can be solved using switched control theory. More specifically, a state feedback controller that forces the stock level to be kept close to zero, even when there are fluctuations in the demand, will be designed in this paper using $H\infty$, control theory. The synthesis of the gains of the state feedback controller that quadratically stabilizes the production dynamics and at the same time rejects the external demand fluctuation (treated as a disturbance) are determined by solving a given set of linear matrix inequalities (LMIs). A numerical example is provided to show the effectiveness of the developed method.