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Piecewise-linear ${H}_{\infty }$ controller synthesis with applications to inventory control of switched production systems. (English) Zbl 1097.90006
Summary: This paper focuses on the problem of inventory control of production systems. The main contribution of the paper is that production systems are modeled as constrained switched linear systems and the inventory control problem is formulated as a constrained switched ${H}_{\infty }$ problem with a piecewise-affine (PWA) 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 perishable produced parts are being stored and will 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 are no stocked parts to immediately meet the demand. A state feedback controller that forces the stock level to be kept close to zero (sometimes called a just-in-time policy), even when there are fluctuations in the demand, will be designed in this paper using ${H}_{\infty }$ control theory. The synthesis 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) is cast as a set of linear matrix inequalities (LMIs). Two numerical examples are provided to show the effectiveness of the proposed method.
##### MSC:
 90B05 Inventory, storage, reservoirs 93B36 ${H}^{\infty }$-control 90B30 Production models