Proportional + Derivative Controller

8/4/2019 Proportional + Derivative Controller

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Pneumatic Proportional + DerivativeController


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Pneumatic Proportional +

Derivative Controller In partial Fullfillment of Diploma in Instrumentation & Control.

(Gujarat Technological University)Sem V Electronic and Pneumatic

Intrumentation.

ByJojo John 096170317032

Patel Mihir - 096170317030


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Learning Objective Explain the importance of Pneumatic

Proportional + Derivative ControllerCircuits in the process industries.

The PD algorithm is a feedback controller used within the processindustries. It has been successfully usedfor over 50 years. It is a robust easilyunderstood algorithm that can provideexcellent control performance desPDtethe varied dynamic characteristics of process plant.


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Pneumatic Proportional & DerivativeControllers

Pneumatic Proportional + Derivative (PD) controllerswere developed because of the desirable property thatsystems with open loop transfer functions of type 1 orabove have zero steady state error with respect to a stepinput.

The Pneumatic Proportional-Derivative (PD) algorithmcomputes and transmits a controller output (CO) signalevery sample time, T, to the final control element (e.g.,valve, variable speed pump). The computed CO fromthe PD algorithm is influenced by the controller tuningparameters and the controller error, e(t).


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Pneumatic Proportional & Derivative Controllers

The schematic arrangement of the pneumaticproportional controller is shown in Fig.1. Theproportional plus derivative (PD) action can also begenerated in a pneumatic controller by introducing arestrictor in the line towards the feedback bellows inFig. 1. This particular arrangement is shown separatelyin Fig. 4, all other parts remaining same as in Fig.1. Thearea of opening of the restrictor is small, so that thetime constant associated with changing the pressureinside the feedback bellows is appreciable. In order toexplain the generation of PD action, we need to studyin detail the performance of the section shown in Fig.4. Since air inside the feedback bellows is confined, thecompressibility of air needs to be considered.


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Let Po

and P f

are the pressures before and after the restrictor respectively. The mass flow rate of the fluid sGthrough the restrictor is proportional to the square root of the pressure differencebetween P o and P f . In general, we can write:

Gs= f ( Po , P f ) The above nonlinear expression can be linearised by considering

the incremental changes , sogpand fp of the values of variables sG,Po and P f at the operating point as:

Gs = C 1 Po + C 2 P f Where,

C1 = G s,o and C 2 = G s,0P o, 0 Pf, 0

are the slopes of the curves Gs vs. P o and G s vs. P f at the operating point. These values can be obtained either experimentally, or fromtheoretical considerations.


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In fact, C1 = -C2; this can be ascertained from thefact that if Po and P f both change from theoperating point by the same amount (so that P o

P f = 0) there is no change in the pressure drop,and so there will also be no change in mass flow rate and gs = 0. From (8). We obtain C 1 = C 2 and

(8) can be rewritten as:gs = C 1 ( Po P f )

Further, the pressure P f inside the feedback

bellows can also be obtained from the expression:


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Pf = MRT f Vf

where M = Mass of the gas inside the bellows

V f

= Volume of the gas inside the bellowsT f = Temperature of the gas (constant) Let m and v f be the changes in mass of the gas

and volume of the gas from the operating point corresponding change in pressure by f p from theoperating point.

From (10), one can also obtain the linearisedexpression around the operating point as:


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Function of the Pneumatic Proportional TermAs with the P-Only controller, the Pneumatic Proportionalterm of the PD controller, Kce(t), adds or subtracts fromCO bias based on the size of controller error e(t) at each timet.

As e(t) grows or shrinks, the amount added to CO bias growsor shrinks immediately and proportionately. The past historyand current trajectory of the controller error have noinfluence on the Pneumatic Proportional term computation.

The plot below (click for a large view) illustrates this idea

for a set point response. The error used in the PneumaticProportional calculation is shown on the plot: At time t = 25 min, e(25) = 60 56 = 4 At time t = 40 min, e(40) = 60 62 = 2


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Where and are constants. The negative signassociated with is due to the fact that increasein volume causes decrease in pressure.

Now the change in volume inside the bellows isdue to the displacement of the free end, and

V f = A f Again the force balance condition at the

feedback bellows gives:

K f Z = P f A f


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It is clear that the introduction of the restrictor in thefeedback bellows introduces a time constant in thefeedback path. Further, by varying the restrictor area,C 1 can be changed (refer (8)), thus changing the time

constant d. Other parts in the block diagram for P -controller shown in Fig. 3 remains the same. For the sake of simplicity, let us assume the link lengths

=. In that case we can develop the simplified blockdiagram for the system shown in Fig. 1 with themodified feedback bellows configuration shown in Fig.4. The simplified block diagram can be expressed asshown in Fig.5, wherefrom we obtain,

Version


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Thus by simple introduction of a restrictor inthe line connecting the feedback bellows cantransform the pneumatic P-controller in Fig. 1to a P-D controller.

Note that with a lagged feedback signal (asseen from the transfer function of thefeedback block) the closed loop transferfunction provides a net lead, with the sametime constant as the lag time constant of the

feedback path.


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Bibliography

Bibliography means that we have refer whiledoing project like, which web sites we gone

through and which books we have referred orwhich magazine we have seen to implementthis project. There are many books and manyweb sites that can help us in different ways toimplement the project and give us properguidance to implement our system in the rightdirection.


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Bibliography

Human Resource Management ( ATUL ) D.R.Patel

Y.R.Joshi

Website Referred : www.google.com www.msdn.microsoft.com www.templatewise.com www.controlguru.com
http://www.google.com/http://www.msdn.microsoft.com/http://www.templatewise.com/http://www.controlguru.com/http://www.controlguru.com/http://www.controlguru.com/http://www.templatewise.com/http://www.msdn.microsoft.com/http://www.google.com/


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Project Guided by:

Mrs. Jyoti A Mishra.

(Lect. In EPDGovt. Polytechnic A' Bad.)


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Documents

Proportional + Derivative Controller