2 - Systems and signals

Abstract

A system can be seen as an assembly of components that interact with each other. Examples are a car suspension or an RC circuit. A system in this definition has one or more measurable properties referred to as the output signals. These properties (e.g., speed, temperature, or voltage) are typically functions of time. Moreover, a system provides the means to influence the output signals, for example, by changing a motor drive voltage or heating power. An input to the system that has the purpose of influencing the output signals is referred to as an input signal. Input signals are generally also functions of time. To design a feedback control system, each of its components must have a known and defined response (output signal) to a defined input signal. A mathematical description that predicts the output signal for a given input signal is called a model. Frequently, the system follows a set of rules that allows it to be called linear. For linear systems, the mathematical model is generally an ordinary differential equation with constant coefficients. For a step input signal (i.e., a defined change of the input signal at an arbitrary point t=0 to a new value that remains constant at t>0), solutions of a differential equation are exponential functions or exponentially decaying oscillations. For sinusoidal input signals, the output signal is sinusoidal with the same frequency but altered amplitude and phase. In this chapter, we present two example systems (the RC circuit and the spring-mass-damper system) and explain their mathematical models.