_ bZnfl

ANSI/AIM Code 128 Scanner In NoneUsing Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications.

Making Code 128 In NoneUsing Barcode creation for Software Control to generate, create Code 128C image in Software applications.

+ K(l - d)z + K(d - b) = 0

Decode Code 128 Code Set B In NoneUsing Barcode scanner for Software Control to read, scan read, scan image in Software applications.

Code 128 Creation In Visual C#.NETUsing Barcode generator for .NET Control to generate, create USS Code 128 image in Visual Studio .NET applications.

(26.13)

Code 128 Drawer In .NET FrameworkUsing Barcode maker for ASP.NET Control to generate, create Code128 image in ASP.NET applications.

Make Code-128 In Visual Studio .NETUsing Barcode creator for Visual Studio .NET Control to generate, create Code 128C image in .NET applications.

Note that the order of the characteristic equation increases if (1) the delay time a r is increased for a fixed sampling period T; (2) the sampling rate increases (lower 7 ) for a fixed delay time.

Printing ANSI/AIM Code 128 In Visual Basic .NETUsing Barcode drawer for .NET framework Control to generate, create Code 128 Code Set B image in Visual Studio .NET applications.

Barcode Drawer In NoneUsing Barcode printer for Software Control to generate, create barcode image in Software applications.

STABILITY

Making EAN13 In NoneUsing Barcode creator for Software Control to generate, create European Article Number 13 image in Software applications.

GTIN - 12 Generation In NoneUsing Barcode maker for Software Control to generate, create UCC - 12 image in Software applications.

The stability of the sampled-data system represented by Eq. (26.13) can be examined by applying the Routh test as discussed in Chap. 24. As the order of the characteristic equation increases, the effort involved in determining stability criteria greatly increases. A few cases corresponding to various values of n in Eq. (26.13) are presented here.

Create Code128 In NoneUsing Barcode generator for Software Control to generate, create Code-128 image in Software applications.

Barcode Encoder In NoneUsing Barcode generator for Software Control to generate, create bar code image in Software applications.

Case 1: n = 0

Making ABC Codabar In NoneUsing Barcode creation for Software Control to generate, create USS Codabar image in Software applications.

Generate Code39 In Objective-CUsing Barcode encoder for iPhone Control to generate, create Code-39 image in iPhone applications.

For this case, the delay is less than one sampling period, i.e., O<aTI T The characteristic equation given by IQ. (26.13) becomes z2 + [K(l - d) - b]z - K(b - d) = 0 For convenience, this may be written z2+yz-a=0 where y = [K(l - d) - b] and (Y = K(b - d). (26.15) (26.14)

Barcode Generation In JavaUsing Barcode creator for BIRT reports Control to generate, create barcode image in Eclipse BIRT applications.

Paint Linear Barcode In VB.NETUsing Barcode generation for .NET framework Control to generate, create 1D image in .NET framework applications.

SAMPLED-DATA

Bar Code Generation In NoneUsing Barcode creation for Font Control to generate, create barcode image in Font applications.

Recognizing Code 39 Extended In Visual Studio .NETUsing Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.

CONTROL

Drawing Bar Code In Objective-CUsing Barcode encoder for iPad Control to generate, create barcode image in iPad applications.

European Article Number 13 Generation In JavaUsing Barcode generator for BIRT reports Control to generate, create EAN-13 image in BIRT applications.

SYSTEMS

Applying the bilinear transformation given by Eq. (24.4) to Eq.(26.15) (w + 1)2 w + 1 wy-a (w - 1)2 + After algebraic rearrangement, one obtains w2(1 + y - a) + w[2(1 + (I)] + 1 - y - (Y = 0 Replacing y by K(l - d) - b and (Y by K(b - d) gives w*[l-b+K(l-b)]+2w[l+K(b-d)]+l-[K(l-d)-b+K(b-d)] or = o

gives (26.16)

w*(l - b)(l + K) + 2w[l + K(b - d)] + 1 - [K(l - d) - b + K(b - d)] = 0 The coefficient of w* is positive since b = edT is always positive and less than one. For stability, the Routh test requires that all coefficients be positive; therefore 1 + K(b - d) > 0 and 1 + b - K(l + b - 2d) > 0 These inequalities may be rewritten

KCd - b

(26.17) (26.18)

Both of these inequalities must be satisfied simultaneously. The best way to understand the result is to plot the stability boundaries as shown in Fig. 26.2 where the ultimate gain K, is plotted as a function of a. Recall that d is a function of u as shown under Eq. (26.11). For stability, K must fall under the boundary as indicated in the figure. One can see that K, reaches a maximum at a max in Fig. 26.2. The value of amax is determined by the intersection of the two constraints, which leads to

exp (hd =

(1 + b)* b(3 + b)

For the value of T/T of 0.8 used in Fig. 26.2, one can compute that a ,,,= = 0.304. Figure 26.2 shows that up to a certain point, adding delay time to a system with T/r = 0.8 increases the ultimate gain. One suggested design criterion is to set K/K,, = 0.5. The assumption is made that the relative stability of the loop is constant, for constant K/K,. This would imply for this particular example that adding delay time to the loop up to a = 0.304 will be beneficial, since increasing K gives less steady-state offset and also faster system response. However, the relative stability of the sampled-data system does not remain the same for constant K/K,. Figure 26.3 illustrates this point. The response of the system to a unit-

SAMPLED-DATA CONTROL OF A FIRST-ORDER PROCESS WITH TRANSPORT LAG

FIGURE 26-2 Stability boundary of a sampleddata system as a function of delay time.

step change in set point is shown for various amounts of delay time in the loop. Although K/K,, is maintained at 0.5, increasing the delay time toward amax has definitely destabilized the system. Hence, use of constant K/K,, is not a good design rule for sampled-data systems and the conclusion that control can be improved by adding delay time is false.

Case 2: n = 1

For this case, T C UT I: 2T One can see from Eq. (26.13) that the order of the characteristic equation is three. Using the same stability analysis as for the case for n = 0, one can show that stability requires that the following inequalities hold simultaneously (2d + 1 - 3b)K < 3 - b (2d - 1 - b)K < 1 + b (d-b)2K2 + (1 - b)(l f b - d)K - 1 < 0

2.07 TIT- 0.6 KIK,, = 0.5

(26.19) (26.20) (26.21)

FIGURE%-3