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Decoding Code 128 In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Creating GS1 128 In ObjectiveC Using Barcode printer for iPhone Control to generate, create EAN / UCC  14 image in iPhone applications. 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 firstorder hold. Code39 Drawer In None Using Barcode maker for Online Control to generate, create Code 39 Extended image in Online applications. Barcode Printer In .NET Framework Using Barcode encoder for Reporting Service Control to generate, create bar code image in Reporting Service applications. FIGURE P233 EAN13 Supplement 5 Drawer In Java Using Barcode drawer for Android Control to generate, create GS1  13 image in Android applications. Bar Code Recognizer In Java Using Barcode Control SDK for BIRT Control to generate, create, read, scan barcode image in BIRT reports applications. 23.4. For the sampleddata 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 sampleddata control system shown in Fig. P23.5, determine c(nT) K = 0.2. Sketch the continuous response c(t). FIGURE P235 CHAPTER
STABILITY
We have seen in Example 23.3 of the previous chapter on the proportional, sampleddata control of a firstorder 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 higherorder processes can be quite complicated. Fortunately, one may develop general rules of stability that resemble those for continuous systems. Consider the response of a sampleddata 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 zl, 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: 21 .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 splane. When the characteristic equation expressed in the sdomain is transformed to the zdomain 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 splane in Fig. 24.1. By applying the transformation z = eTS , we may write z = eTs = eTaejoT
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 zplane, this result states that stability for a sampleddata 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

