|
The use of approximation techniques
to discretise continuous time systems has its roots in numerical integration, and the two
commonly used approaches are:
- backward difference method
- bilinear transformation
Backward
Difference Method
Starting in the Laplace domain, the
Laplace operator ‘s’ is replaced by a first order difference operator,
i.e.
where  , and Ts is the sampling
interval.
't' denotes current time, and the
term  is regarded as a time
shift operator. The index indicates how many integer multiples of the sampling interval is
involved in the time shift, with the sign of the index denoting whether it is a forward (plus
sign) or backward (minus sign) shift. For example,
 or 
To simplify notation, this can be written as
 or 
that is, the sampling interval is treated
as a 'unit' of time.
Since the Laplace operator ‘s’
is equivalent to the differential operator  in the time domain, it should be obvious that the backward
difference method is the well known Euler's approximation to a
differential. As such, the use of this approximation suffers from the same accuracy
limitations. The sampling interval must be sufficiently small for accurate conversion from
the continuous time domain to the discrete time domain. A more forgiving conversion,
especially with high order systems, is the Bilinear Transform.
Bilinear
Transform
The Bilinear Transform is also known as Tustin's
Rule as well as the more familiar Trapezoidal Rule used in
numerical integration. Here, the Laplace operator ‘s’ is replaced by:
where the term has the same meaning as before. Using this
transformation, the sampling interval can be much larger, although the ensuing algebraic
manipulations become more tedious. The benefit of using this transformation is only
obvious with high order systems that have poles of quite different magnitudes.
Example:
As an example, consider a first-order
ODE, relating some output y(t) to an input u(t),
This system has the following Laplace
transfer function:
Applying the backward difference method,
i.e. 
Rearrangement gives the recursive difference
equation
Now, if we use the Bilinear Transform, we
obtain
i.e. 
Therefore, given the Laplace transfer
function of a continuous system, the application of either approximation method will
quickly yield discretised models. Both methods are useful in that any continuous time
controller design may be quickly discretised using either conversion techniques.
|
