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Integral (Reset) Control of Dead Time in Visual Studio .NET
Integral (Reset) Control of Dead Time Denso QR Bar Code Recognizer In Visual Studio .NET Using Barcode Control SDK for .NET framework Control to generate, create, read, scan barcode image in VS .NET applications. Generate QR Code JIS X 0510 In VS .NET Using Barcode creation for .NET Control to generate, create Quick Response Code image in VS .NET applications. Proportional control is obviously rejected for most applications demanding a band wider than a few percent. So another control mode is needed. An integral controller is a device whose output is the time integral of the deviation: QR Code Recognizer In VS .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Barcode Printer In .NET Using Barcode drawer for VS .NET Control to generate, create barcode image in .NET framework applications. 1 = e dt R / Bar Code Recognizer In .NET Framework Using Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications. QR Drawer In Visual C# Using Barcode encoder for VS .NET Control to generate, create QR Code image in Visual Studio .NET applications. (1.6) Painting QR Code JIS X 0510 In VS .NET Using Barcode printer for ASP.NET Control to generate, create QR Code 2d barcode image in ASP.NET applications. QR Code Printer In VB.NET Using Barcode generation for .NET Control to generate, create QR Code ISO/IEC18004 image in .NET applications. where R is the time const.ant of the controller, known as integral or reset time. As long as a deviation exists, this controller will change Data Matrix 2d Barcode Creation In Visual Studio .NET Using Barcode drawer for .NET Control to generate, create DataMatrix image in VS .NET applications. GTIN  13 Creation In .NET Using Barcode generator for Visual Studio .NET Control to generate, create GTIN  13 image in .NET applications. FIG 1.7. The response to a load change illustrates how the proportional band affects both damping and offset. Making Bar Code In .NET Framework Using Barcode drawer for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. Leitcode Maker In .NET Framework Using Barcode creator for VS .NET Control to generate, create Leitcode image in VS .NET applications. Time
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(1.7) Response to a step input is shown in Fig. 1.8. Before using an integral controller in a closed loop, its gain and phase characteristics must be defined. Again we are primarily interested in these properties at the natural period of the loop, TV. Introducing a sinusoidal input to the controller, e = A sin 2a t To The controller output mill be the time integral of the input: 1 112 = edt=i R / R A sin 2a 4 dt 70 > Extraction of the appropriate item from a table of definite integrals enables us to solve the above equation: m = g+os2*;) +wlo where 71~0 is the output at time zero. In order to evaluate phase and gain properties, the output must be reduced to the same form as the input, using the trigonometric identity cosz=sin(5+X) We can convert 112 into a sine function: Understanding
Feedback
Control
The phase shift of the integrator is the angle of the output minus the angle of the input: nz
90 (1.8) An integrator exhibits a phase lag of 90 regardless of the period of the input. The gain of an integrator is the amplitude of the output over the amplitude of the input: G = A70/2aR A =2:R
(1.9) e FIG 1.9. Adjusting reset time affects the damping.
Dynamic Elements in the Control Loop
FIG 1.10. Increasing reset time
trades recolrery for damping, although 7O is unaffected.
Time
In closing the loop, the sum of the phase shift of the dead time and the integral controller must equal 7r at the natural period TV: lr 2lrTd lr= ~ 2 70
180 = 90  360 z
Solving for 70, 7 o  4Td (1.10) Notice that the period is twice that for proportional control, because only 90 of phase shift was allowed to take place in the deadtime element. To sust ain oscillations, the loop gain must be 1.0. Since the deadtime gain is already 1.0, the integrator gain for this condition must also be 1.0. Solving for reset time, GR = pR = 1 . 0 lr R=+ T To summarize, a dead time of 1 min would cycle with a period of 4 min, sustained by a reset time of 2/r, or about 0.63 min. Quarternmplitude damping can be achieved by halving the gain, which means doubling the reset time. Figure 1.9 shows the entire situation. Again, the controller has but one adjustment, which only affects damping. The period of oscillation and the integral time for f/lamplitude damping have been established by the process. Use of the integral controller has avoided the previously encountered proportional offset, but at the cost of reduction in speed of response. The response of a deadtime process under integral control to a gradual load change is pictured in Fig. 1.10. The rate of recovery is slow xhen the reset time is too long. With a proper amount of reset, the measurement will cross the set point during the first cycle, exhibiting $iamplitude damping. Proportionalplusreset Control
(1.11) This controller combines the best features of the proportional and integral modes in that proportional offset is eliminated with little loss 2w +PR
Reset
Gspasc \ +p=o
GR~100~,/2,,RP +R =90" E        G,,,~w J~+OZ
.#&on%,,/2nR
Proportionaltreset
FIG 1.11. The resultant gain is the square root of the sum of the squares of the components.
of response speed.
The controller is represented as follows: m=T(e+$/edt) Having already found the performance characteristics of each of the modes individually on a deadtime process, intuition dictates that the performance of the combination will be somewhere in between, e.g., depending on the particular combination of sett ings of proportional and reset. An infinite combination of settings can be found t,o provide constant damping. We have already seen that for s/4amplitude damping, 100=05 P * or &= 0.5 depending on the control mode used. For the twomode controller, then, the sum of the gains must equal 0.5. The proportional and integral components of gain are out of phase with each other, however. So their resultant gain must be the vector sum of the two components. Figure 1.11 shows the relationship between the vectors. 200 P B 0 _ 100 P /_ _N / / / / 0 0.5 1.0 ro/2rR / A/ //Reset Proportional _ FZG 1.12. A plot of gain vs. 7O for the proportionalplusreset controller shows the contributions of the components.

