barcode reader integration with asp.net FIGURE 32-8 Characteristics of true relay with dead zone. in Software

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FIGURE 32-8 Characteristics of true relay with dead zone.
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ANALYSIS
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FIGURE 32-9 Phase plane for system of Fig. 32.4 using relay with characteristics of Fig. 32.8.
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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
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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
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lower will be the limit-cycle frequency, thus saving excessive switching or chatter. However, the limit-cycle amplitude increases with dead-zone width, decreasing the quality of control.
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The Exothermic Chemical Reactor
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We now wish to consider the phase-plane 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) - ke-EtRT;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 heat-removal 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 cooling-water temperature. In addition, the term in brackets indicates that proportional control on the cooling-water 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 steady-state temperature 8,. This increase in cooling-water 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 heat-transfer rate.
htl IXODS
OF PHASE-PLANE 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 _ ye5w2-l/e)
d9 - = 1.75 - e + ye50( n-1 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 steady-state 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.
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