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Transient response of a sampled-data system.
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response, c(nT), gives only values of c(t) at sampling instants. The continuous response c(t) is obtained from basic knowledge of the first-order system. Offset. From Eq. (23.36), one can see that for a stable system, c(m) = K/(l + K). This is the same result that would be obtained for a continuous system under proportional control. In terms of offset, we have Offset = r(w) - c(m) = 1 - Kl(1 + K) = l/(-l + K) This result should not be surprising, for once the transient terms have disappeared, which is always the condition under which offset is determined, the sampled-data system is equivalent to the continuous system.
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The methods used to obtain the response of open-loop and closed-loop sampleddata systems are similar to those used for continuous systems. The block diagram for a sampled-data system contains one or more sampling switches. For an openloop system, the response at sampling instants is obtained by expressing the Ztransform of the output C(z) as the product of the pulse transfer function G(z) and the Z-transform of the forcing function F(z) : C(z) = G(z)F(z). This expression is analogous to the one used for continuous systems: C(S) = G(s)F(s). The inversion of C(z) can be obtained by (1) partial fraction expansion and use of a table of Z-transforms or (2) by long division. The method using partial fraction expansion gives an analytical result that can be used to find the response at any sampling instant. The process of long division must be continued until the particular output term of interest is reached; for this reason long division is better suited for obtaining the response during the first few sampling periods. Because of the sampling switches present in a sampled-data system, obtaining the expression for the closed-loop response C(z) requires considerable effort. To assist in this effort, a table relating Z-transform outputs to a variety of closed-loop sampled-data configurations was presented. The response of a sampled-data system containing a transport lag can be obtained easily as an analytical expression; this is in contrast to the difficulty one has for continuous systems that contain transport lag.
OPEN-LDOP
CLOSEDLOOP
RESPONSE
APPENDIX 23A AI. Derivation of Eq. (23.3)
The basic definition of F*(s) given by Eq. (22.9) is
F (s) = ; 5 F(s + jko,) k=-co
Replacing s by s + jnw,, where n is an integer, gives F*(s + jnw,) = $ 2 F(s + j(n + k)o,) kc-m Let p = n + k, then the above equation becomes
m F*(s +
jmo,)
= k 2 F(s + jcLwd p=-cc
Since the limits on k am UJ and -03, the limits on p am the same. By the definition of Eq. (22.9), the term on the right is simply F*( S) and we may write F*(s + jno,) = F*(s) which is Eq. (23.3). We describe this relation by stating that F*(s) is periodic in s with frequency w , .
AIL Taking the Starred tiansform of GcGpG&)C*(s) in Eq. (23.18)
In obtaining the closed-loop transfer function for the system in Fig. 23.4, we took the starred transform of G,G,G&)C*(s) in Eq. (23.18) to get G,G,Gh *(s)C*(s). An explanation of this step is as follows. For convenience, let GcGpGh(s) = Gi(s), Consider the block diagram in Fig. 23.6 in which Gi(s) operates on the sampled value of C, which is C*(s). From this diagram, we write Y(s) = Gl(s)C*(s)
CM -+~~~T* yN
G&l =G& $,(s) -- -- Yys) T FIGURE 23-6 Taking the stamd transform of G,G,Gh(s)C*(s) in Q. (23.18).
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Taking the starred transform of both sides of this equation and using the definition of a starred function given by Eq. (22.9) give
Y*(S) = $ f: Y(s + jnw,) = f 5 Gl(s + jnw,)C*(s n=-co n=-03
From Eq. (23.3), we can write C*(s + jnw,) = C*(s) therefore
+ jaws)
f 5 Gl(s + jno,) n=-cc 1
The term in brackets, according to the definition of Eq. (22.9), is G:(s). We can now write Y*(s) = C*(s)G;(s) Converting Gi to the original variables gives
Y*(s) = C*(s)G,G,G,, *(s)
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