ABSTRACT
JAGAN: “dk3189_c002” — 2006/3/13 — 11:46 — page 75 — #1
In this chapter, we provide a brief background on dynamical systems, mainly covering the topics that will be important in a discussion of standard discretetime adaptive control and neural network (NN) applications in closed-loop control of dynamical systems. It is quite common for noncontrol engineers working in NN system and control applications to have little understanding of feedback control and dynamical systems. Many of the phenomena they observe are not due to properties of NN but to properties of feedback control systems. NN applications in dynamical systems are a complex area with several facets. An incomplete understanding of any one of these can lead to incorrect conclusions being drawn, with inaccurate attributions of causes — many are convinced that often the exploratory, regulatory, and behavioral phenomena observed in NN control systems are completely due to the NN, while in fact most are due to the rather remarkable nature of feedback itself. Included in this chapter are discretetime systems, computer simulation, norms, stability and passivity definitions, and discrete-time adaptive control (referred to as self-tuning regulators [STRs]).
Many systems in nature, including neurobiological systems, are dynamical in nature, in the sense that they are acted upon by external inputs, have internal memory, and behave in certain ways that are captured by the notion of the development of activities through time. According to the notion of systems defined by Alfred North Whitehead (1953), it is an entity distinct from its environment, whose interactions with the environment can be characterized through input and output signals. An intuitive feel for dynamic systems is provided by Luenberger (1979), which includes many examples.
