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barcode reading in asp.net G(z) = KS in Software
G(z) = KS Code 128C Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Draw Code 128 Code Set C In None Using Barcode printer for Software Control to generate, create Code128 image in Software applications. where b = emT . Using the transformation given by Eq. (24.4), we obtain for 1 + G(z) = 0 1+ K(l b) = 0 w+l b w  l or Code 128 Reader In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. Code 128C Generation In C#.NET Using Barcode creation for VS .NET Control to generate, create Code 128 image in .NET framework applications. 1 + KU b)(w 1) = o
ANSI/AIM Code 128 Creation In .NET Framework Using Barcode encoder for ASP.NET Control to generate, create Code 128B image in ASP.NET applications. Generate Code 128 In .NET Using Barcode generation for .NET framework Control to generate, create Code 128 Code Set B image in VS .NET applications. (24.12) Encoding Code 128C In Visual Basic .NET Using Barcode creation for Visual Studio .NET Control to generate, create Code 128C image in VS .NET applications. GS1128 Maker In None Using Barcode creator for Software Control to generate, create GS1128 image in Software applications. w + 1  b(w  1) Paint Bar Code In None Using Barcode drawer for Software Control to generate, create barcode image in Software applications. Code 39 Full ASCII Encoder In None Using Barcode drawer for Software Control to generate, create Code39 image in Software applications. (24.13) Bar Code Printer In None Using Barcode printer for Software Control to generate, create barcode image in Software applications. EAN13 Supplement 5 Creation In None Using Barcode creation for Software Control to generate, create EAN / UCC  13 image in Software applications. Rearranging this result in polynomial form for applying the Routh test gives
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Making Code 128 Code Set B In None Using Barcode generation for Online Control to generate, create ANSI/AIM Code 128 image in Online applications. Encoding EAN / UCC  13 In Java Using Barcode printer for Android Control to generate, create European Article Number 13 image in Android applications. The Routh array for this expression becomes
R o w 1t e r m
(K + l)(l  b) (1 + b)  K(l  b) Since b = emT/r is always positive and less than one and K is positive, the first element in the array is positive. For stability, the Routh test requires that all elements of the first column be positive. Therefore, (1 + b)  K(l  b) > 0 1+ b > K(l b) or l+b KCl  b
for stability
This is the same result given by Rq. (23.37), which was obtained from an expression for c(nT>. We shall now use the root locus method on the same example. For this simple problem, there is only one pole of the openloop transfer function, G(z), which is located at b as shown in Fig. 24.5. The root locus consists of one branch that moves from the pole at b along the real axis to the left. The intersection of this branch with the unit circle at z =  1 gives us the stability boundary. Using the magnitude criterion of root locus construction gives KUb) = 1 1%  Pll We can obtain the value of K at the stability boundary by solving for K, thus K ~ hiPll _ l+b l  b l  b zplane
FIGURE
245 Root locus plot for Example 24.1.
SAMPLEDDATAcoNTRoLSYSTEMS
Since the root locus branch moves to the left with increase of K, we see that l+b Kclb for stability For this simple example, the root locus method is easier, for one does not need to use a transformation and the root locus diagram is very simple. However, for higherorder systems, the apparent advantage of the root locus method over the Routh test is lost. To appreciate the details of applying the stability criterion to sampleddata systems, the reader is encouraged to work a few of the more complex problems at the end of this chapter. Other methods for determining stability of sampleddata systems include the SchurCohn criterion and the Jury criterion (see Jury, 1964 and Tou, 1959). The Jury criterion is a simplification of the SchurCohn criterion. These methods, which can be applied directly to the characteristic equation written in the zdomain, can detect roots outside the unit circle of the zplane. Since these methods require the evaluation of highorder determinants, they are limited to simple systems. SUMMARY
The presence of sampling in a control system contributes to instability. The criterion for stability of a sampleddata system requires that the roots of the characteristic equation, 1 + G(z) = 0, fall within the unit circle of the complex zplane. Based on this criterion, two methods were developed to determine stability: (a) the modified Routh test and (b) the root locus method. To use the Routh test, one
must first apply the bilinear transformation, which maps the inside of the unit circle into the left half of the wplane. The usual rules of the Routh test am then applied to the transformed characteristic equation. Using the root lqcus method is simpler, for one simply constructs the root locus diagram from the poles and zeroes of the openloop transfer function G(z). When a branch of the root locus diagram crosses the unit circle, the system becomes unstable. It is of interest to note that systems having transport lag can be analyzed easily for stability in sampleddata systems by either the modified Routh test or the root locus method; this was not the case for continuous systems having transport lag. PROBLEMS
24.1. For the system shown in Fig. P23.5, determine the ultimate gain by use of the Routh test and by use of the root locus method. 24.2. For the control system shown in Fig. P24.2, determine (a) an expression for C(z) when a unitstep change occurs in U, R remaining 0. (b) the stability criteria for the control system. (c) plot the continuous response c(t) for at least a period of time equal to 2T. Obtain this information from basic knowledge of firstorder systems. Note: Clamping is not present in this system, for them is no zeroorder hold in the block diagram.

