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1T 2T 3T 4T t FIGURE 25-2 Example to illustrate the development of the modified Z-transform (A = 0.7 or m = 0.3).
we may use the definition of Eq. (22.8) to write (25.4) n=O Since we work only with functions of t for which the function is zero for t < 0, we have for n = 0 c(0 - AT) = c( -AT) = 0
CA(Z) = 2 c(nT - AT)z-
MODIFED
ZTRANSFORMS
Equation (25.4) may now be written CA(Z) = Tc(nT - hT)z-
(25.5)
If we let m = 1 - A, we may write for the argument of c in Eq. (25.5) nT - AT = nT - (1 - m)T = (n - l)T + mT Equation (25.5) may now be written CA(Z) = 2 c[(n - l)T + mT]z-
(25.6)
(25.7)
If we let n = n - 1, Eq. (25.7) may be written CA(Z) = 5 c[(n + ~)T]z- z-~ n =O or CA(Z) = z-l 2 c[(n + m)T]z- n =O (25.9) (25.8)
This last expression is the definition of the modified Z-transform. Replacing the index n with n, to avoid an awkward symbol in the definition, we have the expression for the modified Z-transform: CA(Z) = c(z,m) = z-l 2c[(n + m)T]z- n=O (25.10)
The symbol C(z,m) has replaced CA(Z) and m = 1 - A. Tables of transform pairs have been developed that relate a function f(t) to its modified Z-transform. Table 22.1 provides the modified Z-transforms for the functions of t listed in the table.
PULSE TRANSFER FUNCTION FOR MODIFIED Z-TRANSFORM
Returning to Eq. (25.3), we may write c;(s) = G;W *W Writing this in terms of Z-transforms, we have
CA(z)
(25.11)
GA(Z)F(Z)
(25.12)
The convention has been established to replace the Z-transform of the delayed function, such as C,+(Z) in Eq. (25.12), with the symbol C(z,m) where m is
SAMPLED-DATA CONTROL SYSTEMS
related to A by the relation m = 1 - A. Changing the subscripted symbols in Eq. (25.12) according to this convention gives the equivalent expression
C(z,m) = .G(z,m)F(z)
(25.13)
where G(z ,m) is the Z-transform corresponding to GA(S). Remember that GA(S) is simply the transfer function for the process G(s) multiplied by the transfer function for the fictitious delay; thus: GA(s) = G(s)~-* ~ (25.14)
To find G(z. ,m) one may refer to a table of transforms and find the entry G(z ,m) corresponding to the desired G(s). To find the output of the block diagram in Fig. 25.4 at times other than sampling instants, one uses the modified Z-transform and writes
SUMMARY OF USE OF THE MODIFIED Z-TRANSFORM.
C(z,m) = G(z,m)F(z)
(25.15)
It should be realized that C (z , m) is simply a Z-transform and that it can be inverted by the same procedures used for inverting ordinary Z-transforms. Furthermore, the inversion gives information about the response only at sampling instants. The result from the inversion of the modified Z-transform gives the values of c(t) between sampling instants. By choosing m between 0 and 1, one can obtain the values of c(t) at any desired time within the sampling intervals. For convenience, one may apply the following rule to determine the effect of the size of m on the time into the sampling intervals.
Rule. Inversion of C (z , m) gives a response that is equivalent to stepping back one sampling period in c(t) and advancing mT units of time. Thus, for nz = 1, there is no delay and the result is the same as that obtained from the ordinary Z-transform. This can be seen from the fact that A = 1 - m = 1 - 1 = 0. On the other extreme, for m = 0, one has a delay that approaches a full sampling period.
The system presented earlier in Fig. 25.2 for the purpose of explaining the timing of the various signals will now be worked as an example using the method
just discussed. Example 25.1. Obtain by means of ordinary and modified Z-transforms the response for the system shown in Fig. 25.5. Determine the response c(t) at sampling instants and at times positioned 0.3T from the beginning of sampling instants.
c*(r) I --A , ----T c*(s)
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