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barcode reader project in c#.net STABILITY in Software
STABILITY Code 128B Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Drawing Code 128C In None Using Barcode generator for Software Control to generate, create Code128 image in Software applications. CRITERION
USS Code 128 Reader In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Code 128 Code Set B Printer In Visual C#.NET Using Barcode printer for .NET Control to generate, create Code 128C image in VS .NET applications. The purpose of this section is to translate the stability definition into a more simple criterion, one that can be used to ascertain the stability of control systems of the form shown in Fig. 14.3. Code128 Generation In VS .NET Using Barcode generation for ASP.NET Control to generate, create Code 128 Code Set B image in ASP.NET applications. Paint ANSI/AIM Code 128 In .NET Framework Using Barcode maker for Visual Studio .NET Control to generate, create ANSI/AIM Code 128 image in Visual Studio .NET applications. FIGURE 143 Basic singleloop control system.
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Paint GS1  12 In None Using Barcode drawer for Software Control to generate, create UPC A image in Software applications. Bar Code Generation In None Using Barcode encoder for Software Control to generate, create barcode image in Software applications. CHARACTERISTIC EQUATION. From the block diagram of the control system
European Article Number 13 Printer In None Using Barcode creation for Software Control to generate, create EAN13 image in Software applications. GTIN  128 Printer In None Using Barcode drawer for Software Control to generate, create GS1 128 image in Software applications. (Fig. 14.3), we obtain by the methods of Chap. 12 C =
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Code 128A Drawer In Java Using Barcode creation for BIRT reports Control to generate, create USS Code 128 image in Eclipse BIRT applications. Barcode Printer In None Using Barcode generation for Font Control to generate, create bar code image in Font applications. (14.4) Encode GS1 128 In Visual Basic .NET Using Barcode maker for Visual Studio .NET Control to generate, create UCC  12 image in VS .NET applications. USS Code 128 Maker In Visual Basic .NET Using Barcode creation for Visual Studio .NET Control to generate, create Code128 image in .NET framework applications. In order to simplify the nomenclature, let G = GiG2H. We call G the openloop transferfunction because it relates the measured variable B to the set point R if the feedback loop of Fig. 14.3 is disconnected from the comparator (i.e., if the loop is opened). In terms of the openloop transfer function G, Eq. (14.4) becomes Data Matrix Generator In Visual Studio .NET Using Barcode generation for VS .NET Control to generate, create Data Matrix 2d barcode image in Visual Studio .NET applications. Encoding USS Code 128 In ObjectiveC Using Barcode creation for iPad Control to generate, create Code 128 Code Set B image in iPad applications. SR+$J
(14.5) In principle, for given forcing functions R(s) and U( S), Eq. (14.5) may be inverted to give the control system response. To determine under what conditions the system represented by Eq. (14.5) is stable, it is necessary to test the response to a bounded input. Suppose a unitstep change in set point is applied. Then GW (s) GIG:! C(s) = 1 = l+Gs s(s  rl)(s  12) . . . (s  r,) where rl,r2, . . . . rn are the n roots of the equation 1 + G(s) = 0 (14.7) (14.6) and F(s) is a function that arises in the rearrangement to the righthand form of Eq. (14.6). Equation (14.7) is called the characteristic equation for the control system of Fig. 14.3. For example, for the control system of Fig. 14.1 the step response is C(s) = GG2 s(1 + G) = (71s + l)(QS + 1) which may be rearranged to
C(s) = s /I 1+ , (71s + l)(QS + l)(T3S + 1) Kc(T3s + 1) s[T1T273s3 I (7172 + 7173 + ~2~3)s~ I (71 I 72 I 73)s + (1 i Kc)] This is equivalent to
C(s) = Kc(73s 4s  rd(s + l)/T,T273  r2M  r3) where r 1, r2, and r3 are the roots of the characteristic equation
717273s~
(7172 T2T3)S2
+ 72 + 73)s + (1 + K,) = 0 (14.8) LINEAR CLOSEDLOOP SYSTEMS
Evidently, for this case the function F(s) in Eq. (14.6) is F(s) = (71s + l)(QS +
7152 73 1)(73s
+ 1) In Chap. 3, the qualitative nature of the inverse transforms of equations such as Eq. (14.6) was discussed. It was shown that (see Fig. 3.1 and Table 3.1), if there are any of the roots ~1, r2, . . . , rn in the right half of the complex plane, the response C(t) will contain a term that grows exponentially in time and the system is unstable. If there are one or more roots of the characteristic equation at the origin, there is an s in the denominator of Eq. (14.6) (where m 2 2) and the response is again unbounded, growing as a polynomial in time. This condition specifies m as greater than or equal to 2, not 1, because one of the s terms in the denominator is accounted for by the fact that the input is a unitstep (l/s) in Eq. (14.6). If there is a pair of conjugate roots on the imaginary axis, the contribution to the overall step response is a pure sinusoid, which is bounded. However, if the bounded input is taken as sin w t, where o is the imaginary part of the conjugate roots, the contribution to the overall response is a sinusoid with an amplitude that increases as a polynomial in time. It is evident from Eq. (14.5) that precisely the same considerations apply to a change in U. Therefore, the definition of stability for linear systems may be translated to the following criterion: a linear control system is unstable if any roots of its characteristic equation are on, or to the right of, the imaginary axis. Otherwise the system is stable. It is important to note that the characteristic equation of a control system, which determines its stability, is the same for setpoint or load changes. It depends only on G(s), the openloop transfer function. Furthermore, although the rules derived above were based on a step input, they are applicable to any input. This is true, first, by the definition of stability and, second, because if there is a root of the characteristic equation in the right half plane, it contributes an unbounded term in the response to any input. This follows from Eq. (14.5) after it is rearranged to the form of Eq. (14.6) for the particular input. Therefore, the stability of a control system of the type shown in Fig. 14.3 is determined solely by its openloop transfer function through the roots of the characteristic equation. Example 14.1. In terms of Fig. 14.3, a control system has the transfer functions 0.5s + 1 G1 3 los (PI (stirred controller) tank) G2 = 1

