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Proportional-Only Offset

A fundamental limitation of proportional control has to do with its response to changes in setpoint and changes in process load. A “load” in a controlled process is any variable subject to change which has an impact on the variable being controlled (the process variable), but is not subject to correction by the controller. In other words, a “load” is any variable in the process we cannot or do not control, yet affects the process variable we are trying to control.

In our hypothetical heat exchanger system, the temperature of the incoming process fluid is an example of a load:

 

 

If the incoming fluid temperature were to suddenly decrease, the immediate effect this would have on the process would be to decrease the outlet temperature (which is the temperature we are trying to maintain at a steady value). It should make intuitive sense that a colder incoming fluid will require more heat input to raise it to the same outlet temperature as before. If the heat input remains the same (at least in the immediate future), this colder incoming flow must make the outlet flow colder than it was before. Thus, incoming feed temperature has an impact on the outlet temperature whether we like it or not, and the control system has no way to regulate how warm or cold the process fluid is before it enters the heat exchanger. This is precisely the definition of a “load.”

Of course, it is the job of the controller to counteract any tendency for the outlet temperature to stray from setpoint, but as we shall soon see this cannot be perfectly achieved with proportional control alone.

Let us carefully analyze the scenario of sudden inlet fluid temperature decrease to see how a proportional controller would respond. Imagine that previous to this sudden drop in feed temperature, the controller was controlling outlet temperature exactly at setpoint (PV = SP) and everything was stable. Recall that the equation for a proportional controller is as follows:

 

m = Kpe + b

Where,

  m = Controller output

  e = Error (difference between PV and SP)

  Kp = Proportional gain

  b = Bias

We know that a decrease in feed temperature will result in a decrease of outlet temperature with all other factors remaining the same. From the equation we can see that a decrease in process variable (PV) will cause the Output value in the proportional controller equation to increase. This means a wider-open steam valve, admitting more heating steam into the heat exchanger. All this is good, as we would expect the controller to call for more steam as the outlet temperature drops. But will this action be enough to bring the outlet temperature back up to setpoint where it was prior to the load change? Unfortunately it will not, although the reason for this may not be evident upon first inspection.

In order to prove that the PV will never go back to SP as long as the incoming feed temperature has dropped, let us imagine for a moment that somehow it did. According to the proportional controller equation, this would mean that the steam valve would resume its old pre-load-change position, only letting through the original flow rate of steam to heat the process fluid. Obviously, if the incoming process fluid is colder than before, and the flow rate is unchanged, the same amount of heat input (from steam) will be inadequate to maintain the outlet temperature at setpoint. If it were adequate, the outlet temperature never would have decreased and the controller never would have had to adjust the steam valve position at all. In other words, if the steam valve goes back to its old position, the outlet temperature will fall just as it did when the incoming flow suddenly became colder. This tells us the controller cannot bring the outlet temperature exactly to setpoint by proportional action alone.

What will happen is that the controller’s output will increase with falling outlet temperature, until there is enough steam flow admitted to the heat exchanger to prevent the temperature from falling any further. But in order to maintain this greater flow rate of steam (for greater heating effect), an error must develop between PV and SP. In other words, the process variable (temperature must deviate from setpoint in order for the controller to call for more steam, in order that the process variable does not fall any further than this.

This necessary error between PV and SP is called proportional-only offset, sometimes less formally known as droop. The amount of droop depends on how severe the load change is, and how aggressive the controller responds (i.e. how much gain it has). The term “droop” is very misleading, as it is possible for the error to develop the other way (i.e. the PV might rise above SP due to a load change!). Imagine the opposite load-change scenario, where the incoming feed temperature suddenly rises instead of falls. If the controller was controlling exactly at setpoint before this upset, the final result will be an outlet temperature that settles at some point above setpoint, enough so the controller is able to pinch the steam valve far enough closed to stop any further rise in temperature.

Proportional-only offset also occurs as a result of setpoint changes. We could easily imagine the same sort of effect following an operator’s increase of setpoint for the temperature controller on the heat exchanger. After increasing the setpoint, the controller immediately increases the output signal, sending more steam to the heat exchanger. As temperature rises, though, the proportional algorithm causes the output signal to decrease. When the rate of heat energy input by the steam equals the rate of heat energy carried away from the heat exchanger by the heated fluid (a condition of energy balance, the temperature stops rising. This new equilibrium temperature will not be at setpoint, assuming the temperature was holding at setpoint prior to the human operator’s setpoint increase. The new equilibrium temperature indeed cannot ever achieve any setpoint value higher than the one it did in the past, for if the error ever returned to zero (PV = SP), the steam valve would return to its old position, which we know would be insufficient to raise the temperature of the heated fluid.

An example of proportional-only control in the context of electronic power supply circuits is the following op-amp voltage regulator, used to stabilize voltage to a load with power supplied by an unregulated voltage source:


 

In this circuit, a zener diode establishes a “reference” voltage (which may be thought of as a “setpoint” for the controlling op-amp to follow). The operational amplifier acts as the proportionalonly controller, sensing voltage at the load (PV), and sending a driving output voltage to the base of the power transistor to keep load voltage constant despite changes in the supply voltage or changes in load current (both “loads” in the process-control sense of the word, since they tend to influence voltage at the load circuit without being under the control of the op-amp).

If everything functions properly in this voltage regulator circuit, the load’s voltage will be stable over a wide range of supply voltages and load currents. However, the load voltage cannot ever precisely equal the reference voltage established by the zener diode, even if the operational amplifier (the “controller”) is without defect. The reason for this incapacity to perfectly maintain “setpoint” is the simple fact that in order for the op-amp to generate any output signal at all, there absolutely must be a differential voltage between the two input terminals for the amplifier to amplify. Operational amplifiers (ideally) generate an output voltage equal to the enormously high gain value (AV ) multiplied by the difference in input voltages (in this case, Vref Vload). If Vload (the “process variable”) were to ever achieve equality with Vref (the “setpoint”), the operational amplifier would experience absolutely no differential input voltage to amplify, and its output signal driving the power transistor would fall to zero. Therefore, there must always exist some offset between Vload and Vref (between process variable and setpoint) in order to give the amplifier some input voltage to amplify.

The amount of offset is ridiculously small in such a circuit, owing to the enormous gain of the operational amplifier. If we take the op-amp’s transfer function to be Vout = AV (V(+) V()), then we may set up an equation predicting the load voltage as a function of reference voltage (assuming a constant 0.7 volt drop between the base and emitter terminals of the transistor):

 

If, for example, our zener diode produced a reference voltage of 5.00000 volts and the operational amplifier had an open-loop voltage gain of 250,000, the load voltage would settle at a theoretical value of 4.9999772 volts: just barely below the reference voltage value. If the op-amp’s open-loop voltage gain were much less – say only 100 – the load voltage would only be 4.94356 volts. This still is quite close to the reference voltage, but definitely not as close as it would be with a greater op-amp gain!

Clearly, then, we can minimize proportional-only offset by increasing the gain of the process controller gain (i.e. decreasing its proportional band). This makes the controller more “aggressive” so it will move the control valve further for any given change in PV or SP. Thus, not as much error needs to develop between PV and SP to move the valve to any new position it needs to go. However, too much controller gain and the control system will become unstable: at best it will exhibit residual oscillations after setpoint and load changes, and at worst it will oscillate out of control altogether. Extremely high gains work well to minimize offset in operational amplifier circuits, only because time delays are negligible between output and input. In applications where large physical processes are being controlled (e.g. furnace temperatures, tank levels, gas pressures, etc.) rather than voltages across small electronic loads, such high controller gains would be met with debilitating oscillations.

If we are limited in how much gain we can program in to the controller, how do we minimize this offset? One way is for a human operator to periodically place the controller in manual mode and move the control valve just a little bit more so the PV once again reaches SP, then place the controller back into automatic mode. In essence this technique adjusts the “Bias” term of the controller equation. The disadvantage of this technique is rather obvious: it requires frequent human intervention. What’s the point of having an automation system that needs periodic human intervention to maintain setpoint?

A more sophisticated method for eliminating proportional-only offset is to add a different control action to the controller: one that takes action based on the amount of error between PV and SP and the amount of time that error has existed. We call this control mode integral, or reset. This will be the subject of the next section.

Go Back to Lessons in Instrumentation Table of Contents


Comments (1)Add Comment
0
offset in proportional control
written by gwenael, April 06, 2016
I fear the explanation of offset above when "If the incoming fluid temperature were to suddenly decrease," is not correct. The limiting thing with proportional control is that the valve opening (in %) is proportional to difference between set point (desired value) and measurement (actual value) and this opening value depends on a gain that is initially preset. And this gain will be set just so that it can compensate the given max "perturbation" foreseen (the max drop in incoming temp of water AT GIVEN CONDITONS, flowrate, pressure, density....) smoothly, means restore the desired set outlet temperature after a reasonable time. Lets say te gain is set so as to compensate a 5 degree drop in outlet temperature due to similar decrease in inlet temperature by a 20 % additional opening of the steam valve (BUT for a given flow unchanged). In this case the controler WILL RESTORE the set point. This is the purpose of the controler and there would be no reason to have a controler if he cannot do any control (we read everywhere that a P only controler will alway leave an offset ! i beleive this is wrong, its the case only under certain unforeseen circonstances). So the P only controler can effectivelly control a change, wether permanent or transient IN THE CONTROLED VARIABLE all other parameters staying the same.
But what he cannot deal with is a changes in the other variables : for example if the temperature of incoming water stay the same but flow rate is increased, witch happens when you have an increased demand of hot water for example (change of set point). Then the increased flow of water will again lower the outlet water temperature as the opening of the valve cannot reheat at previous temperature all this increased amount of water. Lets assume that the additional flow of incoming water is such that it make the outlet temperature drop again of 5 degree C. Then the controler action will be the same as in the first case, it will open 20 % additionally the valve. BUT now these 20%, corresponding to the same heat power than previously must heat up an increased volume of water. The outlet temperature (depending on the load change but lets assume its a relativelly small change) will effectivelly also increase but let say only of 4 C now not of 5 C, in the same period as previous scenario (the target (set point) calibrated previously for a given flow rate, will not be achieved in this case, the heat is not enough.). So now we have indeed still an error because temperature is still below set point ( 4 C ) but the valve has closed back anyway because the error has been lowered from what it was at the begining. The proportional controler anyway still will react as there is an error and the valve will open (less tan previously).This cycle theorically will repeat indefinitivelly but with lower and lower opening, hence will practically stabilise (valve dead band). But it will never reach the set point, it will reach another equilibrium point depending on the initial load change and the total new capacity.

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