Wireless Ad Hoc and Sensor Networks

2BackgroundIn this chapter, we provide a brief background on dynamical systems,mainly covering the topics that will be important in a discussion ofprotocol and algorithm development for networking such as congestioncontrol, admission control, scheduling, and neural network (NN) applicationsin closed-loop control of networks. It is quite common for noncontrolengineers working in wireless networking systems and controlapplications to have little understanding of feedback control and dynamicalsystems. Many of the phenomena they observe are not attributableto properties of NN but to the properties of feedback control systems.Control applications of dynamical systems are a complex area with severalfacets. An incomplete understanding of any one of these can lead toincorrect conclusions being drawn, with inaccurate attributions ofcauses — many are convinced that often the exploratory, regulatory, andbehavioral phenomena observed in NN control systems are completelybecause of the NN; in fact, most are due to the rather remarkable natureof feedback itself. Included in this chapter are discrete-time systems,computer simulation, norms, and stability.2.1 Dynamical SystemsMany systems in nature, including neurobiological systems, are dynamicalin nature, in the sense that they are acted upon by external inputs,have internal memory, and behave in certain ways that are captured bythe notion of the development of activities through time. According tothe notion of systems defined by Whitehead (1953), it is an entity distinctfrom its environment, whose interactions with the environment can becharacterized through input and output signals. An intuitive feel fordynamic systems is provided by Luenberger (1979), which includesmany examples.49