ABSTRACT
Notice that there is a very simple rule that describes how one may perform this translation: regard the variable z as a 'forward-shift' operator, which has the effect of advancing the time index of any variable on which it operates. For example, consider 'ZUk' to be the same as 'Uk + 1'. Similarly, regard Z-l as a 'backward shift' operator.* Then simply rewrite an equation such as (16) as
where A(z) = l-alZ-1-... -anz-n , B(z) = bo + b 1z-1 + ... + bnz-n ,
and
(30)
(31)
The following simple example will be useful in Chapter XIII. Suppose a single variable y is measured, but that it takes one whole period to perform the measurement. What is the transfer function corresponding to this delay? Let 1/ represent the measurements of y that are obtained. Then, assuming the measurement is exact, the appropriate difference equation is
Using z as the shift operator this becomes
1/(z) _ 1 y(z) - z
(32)
(33)
(34)
As a second example (which will also be useful later on) suppose that we wish to design an economic control policy in the form of equation (15), but which has the feature that the policy instrument Uk will only be set to zero if not only the error ek is zero, but also the cumulative error
~~o ei is zero. A control policy with this feature is said by control engineers to incorporate 'integral action'. Such 'integral action' is often used, because it ensures that the error ek is eventually returned to zero, even in the presence of sustained exogenous disturbances - such as a onceand-for·all rise in the price of oil, for example.
