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Because of the nature of digital
devices, signals from plant have to be converted into a suitable form before it can be
transferred for processing by a computer. Similarly, signals generated by a computer must
be presented in a form suitable for receipt by the plant. The important pieces of hardware
that achieve these tasks are the:
The Sampler
The sampler is essentially a switch,
operating usually at fixed intervals of time. When the 'switch' closes, it grabs or
samples the output of the transmitting device. It then transfers the sampled signal to a
receiver. The sampler can operate on both continuous or discrete signals.
Thus if the source signal is continuous,
the output of the sampler is a series of pulses, and the magnitude of each pulse is equal
to the magnitude of the continuous signal at the instant of sampling as shown in the
figure below.
ADCs and DACs
ADCs converts sampled voltage or current signals to
their binary equivalent while DACs converts binary signals to continuous signals such as
voltages or currents. These converters provide the interface between a computer and the
external environment.
Signal Hold Devices
The output of a sampler is a train of pulses,
regardless of whether the source is continuous or discrete. Thus the output of a computer
after digital-to-analog conversion is also a train of pulses. If this is a control signal,
then unless the device receiving this signal, say a pump or valve, has integration
capabilities, then the process will be driven by pulses. This is obviously not acceptable.
So, in process control applications, the signal from the DAC is always 'held' using
hardware known as signal hold devices. The most common is the Zero-Order-Hold, where each
pulse is held until the next pulse comes along, that is:
Result of Zero-Order-Hold
Mathematics
of Sampled Data Systems
Before proceeding into a discussion on
digital controller design, the mathematical tools required will first be covered. The
design and analysis of continuous time control systems are usually done using Laplace
Transforms, simplifying tremendously, the solution of equations involving
time-differentials. Laplace Transforms are therefore continuous transforms.
There are two routes to the design and
analysis of Digital Controllers. One is to approximate the Laplace operator ‘s’
to derive difference equations while the other is to use a variant of the Laplace
Transforms that is applicable to sampled data systems. These are respectively known as:
- Approximation of differentials
- z-Transforms
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