Stability and Stabilization of Systems with Time Delay. Limitations and Opportunities

Control systems often operate in the presence of delays, primarily due to the time it takes to acquire the information needed for decision-making, to create control decisions, and to execute these decisions, as shown in Figure 1. Systems with delays arise in engineering, biology, physics, operations research, and economics. In traffic-flow models, the drivers’ delayed reactions, which combine sensing, perception, response, selection, and programming delays, must be considered [1–3]. These delays are critical in accounting for human behavior, analyzing traffic-flow stability, and designing collision-free traffic flow using adaptive cruise controllers [4]. Material distribution and supply-chain systems are composed of interconnected supply-demand points that share products and information in order to regulate inventories and respond to customer demands [5]. Sources of delay in supply chains include decision-making, transportation-line delivery, and manufacturing facilities that work with lead times [6]. These delays, which influence every stage of the supply-demand chain, deteriorate inventory regulation, thereby causing financial losses, inefficiencies, and reduced quality-of-service [7]. In process control, delay terms arise from mass-transport phenomena in stirred-tank reactors and flow-temperature-composition control [8, 9]. In milling processes, the flexibility of the cutting tool prevents a tooth from precisely machining the desired chip thickness, causing the following tooth to encounter the uncut portion of the chip in the form of an additional force [10, 11]. In this setting, the delay arises since the forces affecting the dynamics are associated with past events. In the milling process, the delay is the tooth-passing period, which is related to the spindle speed. If the