Aliasing

Stable linear systems have property that the steady state response to sinusoidal excitations is sinusoidal with same frequency as that of the input. But digital control systems behave in a much more complicated way because sampling will create signals with new frequencies.

Aliasing is an effect of the sampling that causes different signals to become indistinguishable. Due to aliasing, the signal reconstructed from samples may become different than the original continuous signal. This can drastically deteriorate the performance if proper care is not taken.

2 Inherently Sampled Systems

Sampled data systems are natural descriptions for many phenomena. In some cases sampling occurs naturally due to the nature of measurement system whereas in some cases it occurs because information is transmitted in pulsed form. The theory of sampled data systems thus has many applications.

1. Radar: When a radar antenna rotates, information about range and direction is naturally obtained once per revolution of the antenna.
2. Economic Systems: Accounting procedures in economic systems are generally tied to the calendar. Information about important variables is accumulated only at certain times, e.g., daily, weekly, monthly, quarterly or yearly even if the transactions occur at any point of time.
3. Biological Systems: Since the signal transmission in the nervous system occurs in pulsed form, biological systems are inherently sampled.

All these discussions indicate the need for a separate theory for sampled data control systems or digital control systems.

3 How Was Theory Developed ?

1. Sampling Theorem: Since all computer controlled systems operate at discrete times only, it is important to know the condition under which a signal can be retrieved from its values at discrete points. Nyquist explored the key issue and Shannon gave the complete solution which is known as Shannon's sampling theorem. We will discuss Shannon's sampling theorem in proceeding lectures.

2. Difference Equations and Numerical Analysis: The theory of sampled-data system is closely related to numerical analysis. Difference equations replaced the differential equations in continuous time theory. Derivatives and integrals are evaluated numerically by approximating them with differences and sums.

3. Transform Methods: Z-transform replaced the role of Laplace transform in continuous domain.

4. State Space Theory: In late 1950's, a very important theory in control system was developed which is known as state space theory. The discrete time representation of state models are obtained by considering the systems only at sampling points.