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barcode reader integration with asp.net FIGURE 328 Characteristics of true relay with dead zone. in Software
FIGURE 328 Characteristics of true relay with dead zone. Read Code 128C In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128C Generator In None Using Barcode generator for Software Control to generate, create ANSI/AIM Code 128 image in Software applications. METHODS
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Creating EAN13 Supplement 5 In None Using Barcode drawer for Software Control to generate, create EAN13 image in Software applications. Code 3 Of 9 Creator In None Using Barcode generator for Software Control to generate, create Code39 image in Software applications. over a small segment of the trajectory as it crosses the E axis. The result of this graphical calculation is o = 9.2 radknin. The frequency thus computed for the error signal is, for obvious physical reasons, the same as the frequency of the controlled signal, C . However, the amplitude of C , which is of more direct interest, is not the same as the amplitude of E . It may be found in this case by noting from Eqs. (32.10) and (32.11) that c = ii  E Generating Bar Code In None Using Barcode drawer for Software Control to generate, create bar code image in Software applications. Generate EAN / UCC  14 In None Using Barcode generation for Software Control to generate, create EAN / UCC  14 image in Software applications. It is therefore clear from Fig. 32.9 that C attains a maximum value near the switching points where C = kO.17 Reverting to the original variables, it follows that the water temperature will oscillate with an amplitude of (0.17)(25) = 4.25OF The effect of a small dead zone, 2~ = 2(.01)(25) = OS F, is thus quite significant. In practice, the width of this dead zone is usually an adjustable design parameter. This width is always chosen as a compromise. The wider it is made, the Code 11 Creation In None Using Barcode generator for Software Control to generate, create Code11 image in Software applications. Print UCC.EAN  128 In Visual Studio .NET Using Barcode maker for Reporting Service Control to generate, create UCC  12 image in Reporting Service applications. 500 NONLINEAR CONTROL
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ECC200 Creator In None Using Barcode generator for Word Control to generate, create Data Matrix 2d barcode image in Microsoft Word applications. Barcode Maker In .NET Framework Using Barcode generator for ASP.NET Control to generate, create barcode image in ASP.NET applications. We now wish to consider the phaseplane behavior of the chemical reactor of Chap. 31. This study is based on the paper by Aris and Amundson (1958). For convenience, the dynamic equations are reproduced here: dX/t = ;(X&  XA)  keEtRT;A
dt dT  = $To  T) + dt Defining the dimensionless variables Ft 7=V these equations y=XA XAo become *= CpT x&W eo = CPTO
k(AH)e E RT
Q(T) xA  PVC, (31.17) Q,@H>
dy = 1  y  r(y, (3) dr  = e.  8 + r(y, e)  q(e) dr
kb where r(y, 0) = ye [ECP/Rx,+JAH)t ] (32.19) 0) = Q(T) FPQ,&W
As a control heatremoval function q(O), Aris and Amundson chose the form 0) = we  em + K,(e  edi (32.20) where 8, is the dimensionless mean temperature of water in the cooling coil. This indicates that the heat removal is always proportional to the difference between the reactor temperature and mean coolingwater temperature. In addition, the term in brackets indicates that proportional control on the coolingwater flow rate is present. The flow rate is increased by an amount proportional to the difference between the actual reactor temperature 8 and the desired steadystate temperature 8,. This increase in coolingwater flow rate is assumed for convenience to cause an approximately proportional increase in heat removal. The constant U is a dimensionless analog of Us A, the overall heattransfer rate. htl IXODS
OF PHASEPLANE ANALYSIS
As a specific numerical example, Ark and Amundson selected the following values for constants: kV =e 25
EC, = R+,,(AW
8, = 2 80 = 8, = 1.75 U=l Under these conditions, Eqs. (32.19) become
dr  = 1  y _ ye5w2l/e) d9  = 1.75  e + ye50( n1 e)  (e  1.75)[1 + K&J  2)] d7
(32.21) It can be seen that there is a critical point of Eqs. (32.21) at y=4=Ys
e=2=e, and this is the location at which control is desired. This point has the correct steadystate temperature and a 50 percent conversion of reactant. In addition, there may be two more critical points of Eq. (32.21) depending on the proportional control constant K,, as will be discussed below. Since we are primarily interested in control about 8,) we make use of Liapunov s theorem on local stability, presented earlier. Linearizing Eq. (32.21) in deviation variables 8  8, and y  ys by using Taylor s series yields d(y dr d7
= 2(y  ys)  6.25(8  es) (32.22) w  0,) where ys = i. As we have seen before, the solution to this linear system is y  y, = cleslt + c2eszt 8  8, = c3es1 + cqeszt where, in this case, sr and s2 are the roots of [see Eq. (31.6) and the steps following this equation] (32.23) 4 4 According to the Routh criteria, all coefficients in this characteristic equation must be positive in order that the real parts of the roots sr and s:! be negative. Hence, we can see immediately from Eq. (32.23) that, in order to achieve a stable node or focus, it is necessary that K, > 9.

