By G. S. Virk (auth.)
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38) 26 Digital Computer Control Systems where Tis the sampling interval. 6 Discrete Approximations Whenever we introduce sampling into continuous operations, the continuous transfer functions need to be transformed into z-domain transfer functions. 40) to give the "exact" z-transform. However this substitution is awkward and leads to complicated terms which are difficult to handle. It turns out that far simpler "approximate" transformations can be used to give adequate description for most cases.
0347, giving G()- 0 0347 (z+ 1)2 z - . 61) Starting with a continuous system, we have presented several methods for obtaining discrete equivalents. These included the numerical integration methods where the continuous transfer function, G (s), is represented as a differential equation and is used to derive a difference equation by approximating the integration. When G (s) is not in factored form the integration method is straightforward to apply, whereas the pole-zero mapping technique is suited to factorised transfer functions.
18) Another alternative approach is obtained by using the following inverse transform formula, which is derived from the definitions of the Laplace and z-transforms (see Kuo [70]) y (kT) = 2 1 . 19) where r is a closed path (usually a circle) in the z-plane which encloses all the singularities of Y (z) zk- 1 . 21) We will inverse z-transform this using the three methods mentioned above. 760z- 4 + ... 23) represents a unit impulse at t = kT. 7k for k = 0, 1, 2, 3, ... 25) and so y ( kT) is an exponentially increasing signal with a steady state value of unity.
