barcode reading in asp.net THE Z-TRANSFORM in Software

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THE Z-TRANSFORM
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We have now reached the point where the Z-transform can be introduced. The Z-transform is simply the Laplace transform of the impulse-modulated function, f*(t), in which z = e Ts . Keeping this point in mind, we may write Eq. (22.6) in the form F(z) = ZCf(t)} = L@*(t)} = ~f(nT)e- TsI,=,rs n=O or F(z) = ZCf(t)} = -j-f(nT)z- (22.8) n=O In the definition of the Z-transform given by either of these equations, we have expressed the Z-transform by F(z) or ZCf(t)}. lko examples of the use of this definition of the Z-transform will be given.
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Example 22.1. The unit step.
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(22.7)
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f(t) = u(t) = 1
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therefore, f(G) = 1
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From Eq. (22.8) we write for n 2 0
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Z{u(t>} = -g-n
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This infinite series has the sum z/(z - 1); therefore the result is
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SAMPLING AND ZTRANSFORMS
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Example
exponential
function.
f(t) = e-%(t)
below
This infinite series has the sum as shown
iqe- u(t)} =
- ; T T I
Table of lkansform Pairs
Tables have been prepared that give the Z-transforms of various functions. A short table of transform pairs is given in Table 22.1. This table was adapted from an extensive table in Tou(1959). Table 22.1 includes for each.function of t listed, the Laplace transform, F(s), the Z-transform, F(z), and the modified Z-transform, F(z ,m). The modified Z-transform will be discussed later. An example of a Ztransform pair from this table is
2 e -at .. z _ e-aT
Note that this is the same as Example 22.2 with a = l/r. Tables of Ztransforms are very useful in obtaining the transients for sampled-data systems and they are used in much the same way as tables of ordinary Laplace transforms are used for continuous systems.
SUMMARY
One reason for studying sampled-data control is to be able to describe mathematically a process in which the flow of signals is interrupted periodically. An example of such a system is one that contains a chemical analyzer (e.g., a chromatograph) that produces a measured value of composition after a fixed processing time. Another reason for studying sampled-data control is to be able to describe the operation of a microprocessor-based controller. The form of sampling used in practical applications is clamping, a process of sampling that holds a signal constant between sampling instants. It was shown that clamping is produced by sending an impulse modulated signal through a zero-order hold. lko forms of the Laplace transform of the impulse-modulated function f *(t ) were presented. One of these forms was used to define the Z-transform in which the Laplace variable s is replaced by z through use of the transformation z = e Ts. A short table of Z-transforms was provided. The Z-transform will be used in the next chapter to compute the response of sampled-data systems at sampling instants.
10 11 12 13 14 15 16
T2E-oT
(s : a)3
1 (s + .)k+l a s(s + a)
a s2(s + a) , -E-al
2(z -
l T)2 (1 - l -oT)Z
TZE-amT
2 m+ (2m + l)~- ~ + 2p-2aT z - E-aT (z - E-=T)2 (z - cPT)3
t l-z-= sinoor cos otJt 1 - COSWOt
e-of _ ,-bt
&z - ( - l - T) oT)
a(z - l)(z - E z sinooT z2 - 22 coswoT + 1
z(z - coswoT)
(z - l)(z - E-aT)
l z _ E-aT e
-amT
mT - I/a
-nmT
z - l
a(z - E-T)
z sin mooT + sin (1 - m)woT z2 - 22 coswoT + 1
z cosmooT -
z2 - 22 cosooT + 1 +i- z2 - 22 coswoT + 1 Z-Z
z _ e-=T z _ ,-bT
cos(1 - m)woT z2-2zcosooT+1
z(z - coswoT)
1 - - z cosmooT - cos(l - m)woT z - l z2 - 22 cosooT + 1
-al lT L z _ E-oT
l7 (s + ,,P+ b)
18 (b - a)@ + c) (s + a)(s + b)
(c - a)E-0' + (b - C)E-b'
(b - c)z iEZZ$+___ z _ ,-bT Z-E
(c - u)E- =mT + (b - C)E-bmT z me-aT z - E-bT Z
ab s(s + a)(s + b)
1+-L -o* _ &e-b * + u - b u - b
(a - b)(z - E-=T) - (a - b);: - E-bT)
z i 1 + (a - &; :-aT) - (a _
;;cib::-bTj
SAMPLED-DATA CONTROL SYSTEMS
APPENDIX 22A An Alternate Form of F*(s)
Another form for F*(s) which is useful in proofs and derivations can be obtained by the application of a Fourier series expansion. (See Tou, 1959 for more detail). An outline of the derivation is given below. A Fourier series representation of a periodic function g(t) may be written
g(t) = 2 ckejkost k=-m
wherej = fi and the coefficients c k are obtained from the following equation in which the integration is done over one period T. For this application, it is convenient to choose the period from -Tl2 to Tl2.
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