OPTIMAL ATTRACTORS AND SWITCHED KANBAN CONTROL IN DISCRETE STRANGE BILLIARDS

Abstract

Switched buffer systems are discretized switched flow systems. They are discrete, but can be handled with the same mathematical framework as hybrid systems. In particular, we show that switched buffer systems form discrete strange billiards. For any initialization, discrete strange billiards lead to a finite periodic attractor. The periods and the buffer switching rates are the most important performance measures for the practical use of switched buffer systems. Both measures depend on the sum of objects in the buffers. For each object sum multiple suboptimal attractors may exist. Onto which attractor the system converges depends on the initial buffer levels. We propose a switched Kanban control algorithm that forces the system onto the attractor with minimum buffer switching rate independently of the initialization of the buffer levels. This algorithm needs to perform only one single additional buffer switching operation. Also Heijunka control of switched buffer systems is considered in the framework of strange billiards. We show that Heijunka forms equilateral triangles in the state space. Heijunka is able to yield low buffer switching rates, but only if the buffer levels are optimally initialized. For suboptimal buffer level initializations very high (bad) buffer switching rates can be achieved. In an illustrative example, our proposed switched Kanban control algorithm outperforms conventional Kanban control by about 29% and Heijunka control by about 5%.