Discrete systems can be described by their difference equations. They are, therefore, represented in the state-space form as:

x_d is called the state of the system, and contains n elements for an nth-order system. In matrix notation, x_d is a column matrix of dimension nx1. u and y have the same meaning as for continuous systems.

The fundamental difference between continuous and discrete systems is that the discrete or digital system operates on samples of the sensed plant data, rather than on the continuous signal. The dynamics of the controller are represented by recursive algebraic equations, known as difference equations, that have the form shown in Equations 8.

The sampling of any signal occurs repetitively at instants in time that are T seconds apart. T is called the sampling period of the controller. In complex systems, the sampling period is not a constant, but is, instead, a function of time and the instantaneous state of the controller. The signal being sampled is usually maintained at the sampled value in between sampling instances. This is called zero-order-hold (ZOH). Determining an appropriate sampling period is a crucial design decision for discrete and sampled systems.

One major problem to avoid with sampling is aliasing. This is a phenomenon where a signal at a frequency₀ produces a component at a different frequency ₁ simply because the sampling is occurring too infrequently. The general rule of thumb for such situations is as follows:

If you want to avoid aliasing in a signal with a maximum frequency of , the sampling frequency_s is calculated from . This is a lower limit for _s. If you want to obtain a reasonably smooth time response, then .

The sampling rate for sampling the states of a discrete system must follow the above criterion to avoid aliasing.