Mathematical modelling of systems with state dependent delays. Applications for conveyer belt plants

The paper refers to a special class of delay systems namely those systems which delays incurred to the input variable dependent on the value of the system state. Such systems can be found in various industries with conveyor belt (food, chemical, metallurgy) where certain characteristics of the conveyed material, at the time when it leaves the conveyor must meet certain conditions. These characteristics depend on the residence time on the conveyor belt inside that plant. In our proposed dynamical model, the main idea is to consider that time intervals as components of a state vector. In this way the mathematical model of the plant with conveyor belt can be expressed as a serial connection between two abstract objects: AOSP=Abstract Object Speed-Period and AOPCV=Abstract Object Period-Controlled Variable. The general form of this model is described by a State-Dependent Delay Differential Equation (SDDDE). In the case of a plant with a single specific zone, the AOSP is described by an integral equation. Using this integral description of the AOSP, the finite memory property is proved and other properties are also analysed. Time responses to different specific input signals are derived by analytical calculation and compared with results obtained by numerical simulation.