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1 + G,G,G/, *(s)
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W*(s)
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1 + G&G/, *(s)
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(23.20)
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The starred functions of s appearing in Eq. (23.20) can be converted to functions of z formally by letting z = eTs as was done in the previous chapter to obtain the definition of a Z-transform. The result is
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G,G,R(z) + G,U(z)
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( ) = 1 + G(z) where G(z) = GcGpGh(z) 1 + G(z)
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(23.21)
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The overbar above a group of terms in Eq. (23.21), such as G&J, is useful in reminding one that the functions of s corresponding to each term in the group are multiplied together to form one function of s before the starred transform or the equivalent Z-transform of the group of terms is taken. For example, one cannot obtain G&J(z) by looking up the Z-transform of G,,(S) and then looking up the Z-transform of U(s) in tables and multiplying these two functions of z together, i.e., it is wrong to write Wrong UG,(z) = U(z)G,(z) The correct procedure is to obtain one function of s, written as G&l(s), and to use this combined function of s to look up its Z-transform. Although the overbar is useful to remind one that the terms must be multiplied together before taking the Z-transform, this convention is not always used in the literature or in this book, for it is often difficult and inconvenient to place a bar above a group of terms. For this reason, the overbar will not be used in the equations following this section. If a group of terms are multiplied together followed by the argument z, the overbar above the terms will be understood. The examples to follow should clear up this rather subtle, mysterious taking of a Z-transform of a group of terms. The two expressions on the right side of Eq. (23.21) may be said to contain the transfer functions relating C to R and C to U. However, this is not strictly true, for R cannot be separated from G,G, as in the case of a continuous system.
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SAMPLED-DATA
CONTROL
SYSTEMS
To clarify this important point, the expression for C(z) for a change in set point only (i.e., U = 0) is GcG,Rtz 1 ( ) = 1 + G(z) It is wrong to write (23.22)
C(z) R(z)
G&k)
1 + G(z)
Wrong
In other words, the term G,G$(z) must be worked out for each R(s) to be studied. A similar comment applies to the term G,V(z). These subtle points in the correct use of Eq. (23.21) can be made much clearer by the example shown later.
Table Relating Z-lhnsform to Sampled-Data Systems
Outputs
Obtaining the expression C(z) in Eq. (23.22) for the sampled:data block diagram of Fig. 23.4 requires considerable effort. As we shall see in later chapters, other sampled-data block diagrams occur for which an expression for C(z) is needed. In the literature (Tou, 1959), one can find tables of various types of sampled-data block diagrams with the corresponding expressions for C(z). A short table, which will be useful later, is shown in Table 23.1. This table also lists the modified Ztransform C(z ,m), which will be discussed in Chap. 25. (Notice that the overbar is not used in this table.) It is important to know how to interpret the entries in the table. For the diagram in Fig. 23.4, we see that item 2 in Table 23.1 applies. For this case (23.23) The expression GR(z) in Eq. (23.23) is equal to G,G$(z) in Fig. 23.4 and GH(z) is equal to GcGpGh(z). Using these equivalent expressions for GR(z) and GH(z), we write directly from Table 23.1
G,G,Rtz 1
( ) = 1 + GcGpGh(z) This agrees with Eq. (23.22).
Example 23.3. Closed-loop response. For the diagram shown in Fig. 23.4, let
Gp = l/(~s + 1)
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