Modified z transformation

When the deadtime in a process is an integer multiple of the sampling period, the function in the Laplace domain converts easily into z in the z domain, where dead time D = kT. When the dead time is not an integer multiple of the sampling period, we can use modified z transforms to handle the situation. 

First we must define modified z transformation. This is a dialect of our... 

Dutch. Then we will give an example which illustrates how modified z transforms can be used to handle noninteger values of deadtime. 

We define the right-hand side of Eq. (18.126) as the modified z transformation of the original function X(,) . The defining equation for any function/( is... 

The most important use of modified z transforms is to handle deadtimes that are not integer multiples of the sampling period. From the development above, it should be clear that the regular z transformation of a function with a deadtime that is a fraction of the sampling period is just the modified z-transformation of the function with no deadtime. 

We need to derive the modified z transformation of a few simple functions. 

Using the definition of the modified z transformation given in Eq. (18.127) gives... 

Note that this modified z transform does not depend on the variable m since the function is constant in between the sample periods. 

Notice that this modified z transformation is a function of both m and z. 

Pulse transfer functions for modified z transforms are defined in the same way as for regular z transforms. For a system with input m, and output x, ), the pulse transfer function is... 

Therefore, the Kalman designed algorithm is specified by obtaining the modified Z-transform of the process pulse transfer function ot get P(z) and Q(z) and then substituting these values into equation (19). 

Suppose that a function is translated by a noninteger multiple of the sampling period T (i.e., td = kT + AT, where 0 < A < 1). Such systems can be handled by an extension of the z-transforms known as the modified z-transform. The interested reader can consult Appendix 28A. 

By inverting the modified z-transform of a function we can find the value of the function between sampling instants (see Appendix 28A). 

From the defining eq. (28A.1) we notice that the modified z-transform of a function is characterized by two variables, z and m. The first is a complex variable having the same meaning as in the normal z-transform, while m is a parameter denoting the fraction of a sampling period present in the delay. Table 28A.1 shows the modified z-transforms of several common functions. 

We can invert modified z-transforms through long division, taking care to divide separately terms involving m and those that do not include m. This allows us to find the value of a sampled-value function between sampling instants. For example, suppose that we have a signal with... 

Modified z-transform, 606-7 Modulus of complex numbers, 319 MOS memory, 555 Multicapacity process, 187, 193-94, 212-14... 

VII.19 Compute the inverse of the following modified z-transforms and find the behavior of y(t) between sampling instants. 

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