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23.1. (a) For the open-loop system shown in Fig. P23.1, determine c(nT) for R = s(t), u(t), tu(t) when T = 1 and T = 0.5. Sketch the continuous response c(t) for each disturbance. (b) Repeat part a for the case in which the zero-order hold is removed. Note: The complete solution to this problem requires the solution of 12 openloop problems. c
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23.2. For the sampled-data process in Fig. P23.2, determine (4 C(z). (b) c(nT) for several values of n. (c) Plot the continuous response, c(t).
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FIGURE P 23-2 23.3. Consider the transfer function
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If the function X(t) shown in Fig. P23.3 is fed to an impulse sampler, which is followed by Gh 1 , determine the output Y(t) . Present your results graphically. The term Ghi is called a first-order hold.
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23.4. For the sampled-data control system shown in Fig. P23.4, determine c(nT) for K = 1 and K = 2. Sketch the continuous response c(t). Determine the ultimate controller gain.
23.5. For the sampled-data control system shown in Fig. P23.5, determine c(nT) K = 0.2. Sketch the continuous response c(t).
FIGURE P23-5
CHAPTER
STABILITY
We have seen in Example 23.3 of the previous chapter on the proportional, sampled-data control of a first-order system that the question of stability arises. By solving the response c(H) for this relatively easy problem [Eq. (23.36)] we were able to derive the conditions for stability [Eq. (23.37)]. Using this same approach for finding conditions for stability for higher-order processes can be quite complicated. Fortunately, one may develop general rules of stability that resemble those for continuous systems. Consider the response of a sampled-data system to be of the form:
Fl(z)
( ) = 1 + G(z) = (z - zl)(z -
F2(z)
z2) ... (z - zd
(24.1)
To obtain the response c(S), we may expand the right side by the method of partial fractions to obtain (24.2) In anticipation of an entry in the table of transform pairs, each term within the parentheses of Eq. (24.2) is written as z/(z - zJ. The term z-l, which has been placed outside the parentheses to balance the z placed inside, will simply shift the time response by T units and in no way affect the conclusion on stability to be given in the following discussion. For the moment, consider the roots z 1, ~2, . . . to be real. We have seen from previous examples that the inverse of a typical term z/(z - z,i) is:
2-1 .-L = z; i Z -Zi I
(24.3)
STABILITY
For this term to contribute a bounded response to c(nT) requires that (zil < 1. We shall now extend this special case of real roots, which has been presented to introduce the subject, to the general case of roots being complex. GENERAL CONDITIONS FOR STABILITY The general conditions for stability of a continuous system are that the roots of the characteristic equation fall in the left half of the complex plane. Before the sampled signal C*(S) is changed to the form C(z) by introducing the transformation z = eTS, the characteristic equation of the sampled system is 1 + G*(s) = 0 We may apply the general stability criterion and require that all roots of the characteristic equation be in the left half of the s-plane. When the characteristic equation expressed in the s-domain is transformed to the z-domain through the transformation z = eTs, we get 1 + G(z) = 0 Consider a typical stable root of the characteristic equation to have the value s = --a+ jw where u > 0 This root is shown in the complex s-plane in Fig. 24.1. By applying the transformation z = eTS , we may write
z = eTs = e-TaejoT
This expression for z,, a complex number, is of the form z = IZI 42 where M or IzI is the magnitude of the complex number and 8 or &z is the angle associated with the complex number. Since cr > 0, emTa < 1 therefore IzI < 1 In terms of the complex z-plane, this result states that stability for a sampled-data control system requires that the roots of the characteristic equation 1 + G(z) = 0 fall within the unit circle as shown in Fig. 24.2.
z = Mej'
Se---
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