Discrete-Time Systems

Discrete-time models provide mathematical relations between the system variables at these time instants. This chapter develops the mathematical properties of discrete-time models. It also provides a concise review of material covered in basic courses on control and system theory. Following this, it explains why difference equations result from digital control of analog systems. Because the purpose of z -transformation is often to simplify the solution of time domain problems, it is essential to inverse-transform z -domain functions. In this context, the chapter demonstrates the procedure to obtain the z -transform of a given time sequence and the time sequence corresponding to a function of z , which is used to solve linear time-invariant (LTI) difference equations using the z -transform. Thereafter, it describes the procedure of obtaining the z -transfer function of an LTI system, and obtaining the time response of an LTI system using its transfer function or impulse response sequence. In most engineering applications, it is necessary to control a physical system or plantso that it behaves according to given design specifications. Typically, the plant is analog, the control is piecewise constant, and the control action is updated periodically. This arrangement results in an overall system that is conveniently described by a discrete-time model.