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24.3. The sampled-data control system shown in Fig. P24.3 contains a first-order hold, for which the transfer function is G
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= (1 + Ts) 1 - f -Ts *
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(a) Determine G(z) for the closed-loop response. (b) Determine the value of K for which the closed-loop response is on the verge of instability by means of the root locus method. Sketch the root locus diagram. (c) If a zero-order hold were used in place of the first-order hold, what would be the value of K for the system to be on the verge of instability
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24.4. One can show that for the sampled-data system in Fig. P24.4
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G(z) = KI(Z + a> z(z - b)
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where CY = 0.517, b = 0.607, K1 is proportional to K. Draw accurately the root-locus diagram and from it determine the ultimate value of K, above which the system is unstable.
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I-~-Ts s
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CHAPTER
MODIFIED Z-TRANSFORMS
The inversion of the Z-transform C(z) gives information about c(t) only at sam-
pling instants. This, of course, is a result of introducing the sampling switch. The simplicity of the mathematics of Z-transforms must be paid for by the limited information about c(t). For some processes, knowing the response at sampling instants is quite sufficient. However, if one wants information between sampling instants (intersample information, as it is called), a procedure other than the use of Z-transforms is required. One method, which can be very tedious except for first-order processes, is to compute the continuous response c(t) by solving the differential equations describing the process. If one were to go through this much effort, there would be little reason to use Z-transforms in the first place. Another method for finding intersample information is to use the modified Z-transform. This method is nearly as easy to use as the ordinary Z-transform and gives the intersample information about the response at any desired time between sampling instants. Another reason for introducing the modified Z-transform is to have a method for obtaining the pulse transfer function of a system that includes a transport lag (eeTS) for which 7 is not equal to an integral number of sampling periods (7 f; nT). The development of such a pulse transfer function will be described in the next chapter. 384
MODIFIED &TRANSFORMS
DEVELOPMENT OF MODIFIED Z-TRANSFORM
Consider the process shown in Fig. 25.1 in which a fictitious delay e -hTs has been placed after the block G(s). The value of A is between 0 and 1. The use of ordinary Z-transforms will give c(nT), which is obtained by inverting C(z). From Eq. (23.6), C(z) can be found quite simply from the expression:
C(z) = G(z)F(z)
(25.1)
To obtain c(t) at times other than sampling instants, c(t) is delayed (or translated) by an amount AT before sampling. The choice of A gives the desired intersample value of c(t). Before developing the definition of the modified Z-transform, Fig. 25.2 provides a simple example that will clarify the timing of signals in Fig. 25.1. In Fig. 25.2 f(t) = u(t), a unit-step function, G(s) = l/(s + l), and A = 0.7. By studying Fig. 25.2, one can see the nature of the signal at each position in the diagram. Notice that the continuous signal c*(t) from the delay block is the response c(t) shifted by 0.7T to the right. The sampled response c:(t) consists of a train of impulses; the magnitudes of which equal the values of c(t) at 0.3T into each sampling period. As will be shown later c~(nT) gives the intersample information that is provided by the modified Z-transform. We may now turn to the general development of the modified Z-transform. From Fig. 25.1, we may write CA(S) = GA(~)F*@) where GA(S) = G(s)edhTS Taking the starred transform of this expression: (25.2)
C;(s) = G;(s)F*(s)
(25.3)
The basis for performing this last step has been discussed in detail in connection with Eq. (23.5). To develop the modified Z-transform, consider separately the processing of c(t) as shown in Fig. 25.3. To find the Z-transform C*(z) corresponding to C:(S),
f( ) F(s)
f*(t) F*(s) F(z)
G(s)
c(t) C(s)
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